The Challenge of d and f Electrons. Theory and Computation 9780841216280, 9780841212459, 0-8412-1628-2

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ACS SYMPOSIUM SERIES 394

The Challenge of d and f Electrons Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.fw001

Theory and Computation Dennis R. Salahub, EDITOR Universitéde Montréal

Michael C. Zerner, EDITOR University of Florida

Developed from a symposium sponsored by the Divisions of Inorganic Chemistry and of Physical Chemistry of the American Chemical Society and the Division of Physical and Theoretical Chemistry of the Canadian Society for Chemistry at the Third Chemical Congress of North America (195th National Meeting of the American Chemical Society), Toronto, Ontario, Canada, June 5-11, 1988

American Chemical Society, Washington, DC 1989

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.fw001

Library of Congress Cataloging-in-Publication Data The Challenge of d and f electrons. (ACS symposium series, ISSN 0065-6156; 394) "Developedfroma symposium sponsored by the Divisions of Inorganic Chemistry and of Physical Chemistry of the American Chemical Society and the Division of Physical and Theoretical Chemistry of the Canadian Society for Chemistry at the Third Chemical Congress of North America (195th National Meeting of the American Chemical Society), Toronto, Ontario, Canada, June 5-11, 1988." Includes bibliographies and indexes. 1. Molecular orbitals—Congresses. 2. Quantum chemistry—Congresses. I. Salahub, Dennis R., 1946- . II. Zerner, Michael C. III. American Chemical Society. Division of Inorganic Chemistry. IV. American Chemical Society. Division of Physical Chemistry. V. Canadian Society for Chemistry. Division of Physical and Theoretical Chemistry. VI. Chemical Congress of North America (3rd: 1988: Toronto, Ont.) VII. American Chemical Society. Meeting (195th: 1988: Toronto, Ont.) VIII. Series. GD461.C37 1989 541.2'2 89-6926 ISBN 0-8412-1628-2 CIP

Copyright ©1989 American Chemical Society All Rights Reserved. The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner's consent that reprographic copies of the chapter may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, MA 01970, for copying beyond mat permitted by Sections 107 or 108 or the U.S. Copyright Law. This consent does not extend to copying or transmission by any means-graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating a new collective work, for resale, or for information storage and retrieval systems. The copying fee for each chapter is indicated in the code at the bottom of thefirstpage of the chapter. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of anyrightor permission to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. Registered names, trademarks, etc., used in this publication, even without specific indication thereof, are not to be considered unprotectedbylaw. PRINTED IN THE UNITED STATES OF AMERICA

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

ACS Symposium Series M. Joan Comstock, Series Editor

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.fw001

1989 ACS Books Advisory Board Paul S. Anderson

Mary A. Kaiser

Merck Sharp & Dohme Research Laboratories

E. I. du Pont de Nemours and Company

Alexis T. Bell

Michael R. Ladisch

University of California—Berkeley

Harvey W. Blanch University of California—Berkeley

Malcolm H. Chisholm Indiana University

Purdue University

John L. Massingill Dow Chemical Company

Daniel M. Quinn University of Iowa

James C. Randall Alan Elzerman

Exxon Chemical Company

Clemson University

Elsa Reichmanis John W. Finley Nabisco Brands, Inc.

AT&T Bell Laboratories

C. M. Roland

Natalie Foster

U.S. Naval Research Laboratory

Lehigh University

Stephen A. Szabo

Marye Anne Fox The University of Texas—Austin

Conoco Inc.

Wendy A. Warr Imperial Chemical Industries

G. Wayne Ivie U.S. Department of Agriculture, Agricultural Research Service

Robert A. Weiss University of Connecticut

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.fw001

Foreword The ACS S Y M P O S I U M S E R I E S was founded in 1974 to provide a medium for publishing symposia quickly in book form. The format of the Series parallels that of the continuing A D V A N C E S IN C H E M I S T R Y S E R I E S except that, in order to save time, the papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form. Papers are reviewed under the supervision of the Editors with the assistance of the Series Advisory Board and are selected to maintain the integrity of the symposia; however, verbatim reproductions of previously pub­ lished papers are not accepted. Both reviews and reports of research are acceptable, because symposia may embrace both types of presentation.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.pr001

Preface THE G O A L O F T H E O R E T I C A L C H E M I S T R Y is to predict chemistry, either with simple mathematical expressions or with the more complicated expressions that researchers must solve on a computer, rather than by working with test tube and beaker in the lab. Theorists have long dreamed of synthesizing and characterizing the properties of conceptualized materials and their stabilities and reactivities without early recourse to bench experiments. Computer experiments are already of considerable aid to the experimentalist in prescreening compounds of desired character. The day is fast approaching when organic chemists will routinely use computational procedures to examine targeted molecules for specific properties before attempting synthesis and characterization of these systems. Quantum chemical experiments already aid our understanding of chemical processes, and such computations are often included with experimental results in publications. The impact of such work is continually growing. Entire industries are being transformed at the most fundamental level, perhaps most strikingly so i n the pharmaceutical industry. Progress has been somewhat slower in the development of theoretical tools for transition metal systems. The localized nature of d and f electrons, for example, often not only makes molecular orbital calculations difficult but also makes the utility of such calculations uncertain. In addition, the chemistry of these systems requires consideration of large molecules, clusters, surfaces, and bulk systems. Phenomena as diverse as medicine, catalysis, and high-temperature superconductivity are complex, and they require the most modern techniques for their accurate study. Methods that are of proven value for the chemistry of hydrogen and the main group elements, methods that predict molecular structure with an accuracy rivaling experiment, are proving inadequate, even qualitatively, when applied to the more challenging metal-containing systems. The symposium on which this book is based presented recent advances in the theory and computation of systems containing d and f electrons and applications to a number of the complex systems that have recently been examined using these techniques. Leading experimentalists presented work that would greatly benefit from advanced theory. The latest developments in molecular orbital theories, in correlation ix In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.pr001

calculations, and in local and nonlocal spin density theories were discussed, as well as the effects that calculations using these techniques have had on our way of thinking about these systems. The foundations of future scientific, technological, and industrial revolutions are currently being laid in the laboratories and computer rooms dedicated to understanding d and f electrons. This book contains contributions from most of the leading scientists in this area. It presents a snapshot of the state of the art as it existed in June 1988. We take this opportunity to thank the sponsors of this symposium again, whose support helped make this meeting the dynamic forum it was. Financial support came from Eastman Kodak Company of Rochester, NY; IBM Corporation of Yorktown Heights, NY; Multiflow Computer Company of Branford, CT; and the Petroleum Research Fund of the American Chemical Society. We also take this opportunity to thank Tom Ellis in Montreal and Susanne Gaddy in Gainesville for their cheerful editorial assistance. D E N N I S R.

SALAHUB

Universitéde Montréal Montréal, Québec H3C 3J7, Canada MICHAEL

C.

ZERNER

University of Florida Gainesville, F L 32611 December 1, 1988

x In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 1 Quantum

Chemistry

Throughout

the

Periodic

Table 1

2

Dennis R. Salahub and Michael C. Zerner Département de Chimie, Universitéde Montréal, C.P. 6128, Succursale A, Montréal, Québec H3C 3J7, Canada Quantum Theory Project, Williamson Hall, University of Florida, Gainesville, FL 32611

1

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch001

2

An overview is presented of the state-of-the-art for quantum chemical calculations for d- and f- electron systems. The present role and the potential of ab­ initio, density functional and semi-empirical methods are discussed with reference to contemporary developments in related experimental disciplines. Progress towards a true computational chemistry including the transition metals, lanthanides, and actinides is outlined with emphasis both on achievements and on the remaining barriers. Imagine a chemist, a modern-day Rip Van Winkle, awake and refreshed following a twenty-year sleep and finding himself at the instrument exhibit at an ACS national meeting. Poor fellow! His search for rotating evaporators and simple bench-top IR's will not be an easy one. If there are any, they are lost in a sea of television screens; a dazzling display of rotating, vibrating, pulsating, dancing, reacting, technicolor molecules. A few well placed questions and Rip learns that, while he was sleeping the sleep of the just, there was quite a revolution going on, that now chemists and, it seems, in large numbers are using computer programs to simulate the molecules they will perhaps eventually make react. A bit more courage, and a few more questions and he learns that byand-large the molecules dancing on the screen are organic - carbon, hydrogen, nitrogen, oxygen, with an odd halogen, or sulfur or phosphorus. Finding the colorful transition metals that he liked so much before he went to sleep turns out to be another difficult task; and when he finally finds a demonstrator to explain how they are handled, after much hemming and hawing, he learns that the methodological basis for the computations is not the same as it is for the lighter, s-p, elements. The organic computational chemistry that so dazzled our allegorical eyes-wide-open chemist is based on the combined efforts of a large number of theoretical and experimental chemists over the last few decades. This combined experience has resulted, only 0097-6156/89/0394-0001$06.00A) c 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch001

2

THE CHALLENGE

OF d AND f ELECTRONS

r e c e n t l y , i n a s e t of t o o l s t h a t can p r o v i d e the e n t i r e c h e m i s t r y community, s p e c i a l i s t and n o n - s p e c i a l i s t a l i k e , w i t h a manual and a s e t of "specs", i n much the same way as any o t h e r i n s t r u m e n t . The u s e r , once he has read, or has been p r o p e r l y i n f o r m e d , t h a t the i n s t r u m e n t i s a p p r o p r i a t e f o r h i s problem (an o r g a n i c system w i t h N atoms and a r e q u i r e d accuracy of x k c a l / m o l e ) can go ahead and use t h e s e enormously complex programs w i t h the g r e a t e s t of ease to h e l p him i n h i s work. That the same i s not t r u e i n g e n e r a l f o r problems i n v o l v i n g t r a n s i t i o n metals or l a n t h a n i d e s or a c t i n i d e s can be a t t r i b u t e d above a l l e l s e t o the d i f f i c u l t y of h a n d l i n g e l e c t r o n c o r r e l a t i o n f o r these systems. The d and f e l e c t r o n s are c o n f i n e d t o a s m a l l volume and t h e r e are many of them, p a r t i c u l a r l y i f m e t a l - m e t a l bonds are i n c l u d e d . T h i s c o r r e l a t i o n problem w i l l be a t the core of t h i s paper and indeed of the whole volume. But t h i s i s not the o n l y problem when i t comes to comparing d and f o r b i t a l systems w i t h systems t h a t do not c o n t a i n these t r o u b l e makers. Transition metal systems are most o f t e n o p e n - s h e l l systems, unlike their organic cousins. The t r e a t m e n t of o p e n - s h e l l systems i s more d i f f i c u l t both a t the H a r t r e e - F o c k l e v e l , and a t the p o s t H a r t r e e Fock l e v e l . Of importance i n t r a n s i t i o n metal complexes i s a comparison of the r e l a t i v e e n e r g i e s of complexes of d i f f e r i n g s p i n m u l t i p l i c i t i e s , and t h i s comparison i s made very d i f f i c u l t . The Hartree-Fock procedure greatly favors states of higher m u l t i p l i c i t y , and t h i s b i a s i s , a g a i n , o n l y c o r r e c t e d by a h i g h l e v e l c o r r e l a t e d t h e o r y (see DAVIDSON, f o r e x a m p l e ) . In a d d i t i o n , the g r e a t wealth of s t a t e s t h a t l i e near i n energy f o r t r a n s i t i o n metal systems o f t e n make the SCF s t e p r e q u i r e d i n most t r e a t m e n t s much more troublesome. F o r o r g a n i c systems, SCF i s u s u a l l y (but not a l w a y s ! ) not a p r o b l e m . There are i n g e n e r a l t h r e e common ways used today t o go about examining the e l e c t r o n i c s t r u c t u r e of t r a n s i t i o n metal systems. A l l of these are concerned i n one way o r another i n s o l v i n g (or getting around) the c o r r e l a t i o n problem and we w i l l g i v e an o v e r v i e w of a l l t h r e e , u n d e r l i n i n g t h e i r r e s p e c t i v e s t r e n g t h s and weaknesses: "standard" ab i n i t i o t e c h n i q u e s (the s u r e s t and o f t e n the b e s t way to go f o r cases where the c o m p u t a t i o n a l e f f o r t does not render them i m p r a c t i c a l or i m p o s s i b l e ) , D e n s i t y F u n c t i o n a l methods where the c o r r e l a t i o n i s handled through an e l e c t r o n gas model ( s t i l l an "ab i n i t i o " framework but p r a c t i c a b l e f o r much l a r g e r systems) and s e m i - e m p i r i c a l where " n a t u r e ' s c o r r e l a t i o n " , namely e x p e r i m e n t a l d a t a , are used t o a d j u s t p a r a m e t e r s , r e s u l t i n g i n very r a p i d methods t h a t , when used j u d i c i o u s l y , can p r o v i d e a c c u r a t e r e s u l t s f o r very l a r g e systems. I n t h i s b r i e f overview we can only touch on very few a s p e c t s of a v a s t and growing f i e l d . Our o n l y g o a l here i s t o p r o v i d e some s t r u c t u r e so t h a t newcomers, i n p a r t i c u l a r , may more r a p i d l y put the v a r i o u s q u e s t i o n s and the v a r i o u s techniques into proper p e r s p e c t i v e , and o b t a i n a f i r s t i m p r e s s i o n of the p r e s e n t s t a t e of the f i e l d and o f where i t i s g o i n g , as we see i t . Taken a l o n g w i t h the o t h e r chapters of the book ( r e f e r r e d to by the name of an a u t h o r i n c a p i t a l l e t t e r s ) and the o t h e r r e f e r e n c e s c i t e d , we b e l i e v e t h a t t h i s i m p r e s s i o n s h o u l d be r e a s o n a b l y u n b i a s e d .

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch001

1.

SALAHUB & ZERNER

Quantum Chemistry Throughout the Periodic Table

Before s t a r t i n g the m e t h o d o l o g i c a l o v e r v i e w , i t i s w o r t h w h i l e t o remind o u r s e l v e s b r i e f l y of the e x p e r i m e n t a l c o n t e x t . The presence of d and f e l e c t r o n s , whose wave f u n c t i o n s can be e i t h e r c o r e - l i k e , e s s e n t i a l l y a t o m i c , o r more d i f f u s e , and thus t r u e valence like, or a n y t h i n g i n between, does c o n f e r special, sometimes u n i q u e , p r o p e r t i e s . The f a s c i n a t i o n and the u t i l i t y of the t a r g e t systems f o r many of the c h a p t e r s of t h i s book are owed, i n the f i n a l a n a l y s i s , t o the range of b e h a v i o r t h a t e l e c t r o n s i n these wave f u n c t i o n s demonstrate. The f o u r " e x p e r i m e n t a l " c h a p t e r s of the book treat subjects ranging from ESR and other s p e c t r o s c o p i e s of c l u s t e r s i n m a t r i c e s (WELTNER and VAN Z E E ) , t o the thermodynamics and phase t r a n s i t i o n s of heavy fermions (OTT and F I S K ) , t o e l e c t r o n d e n s i t y maps (COPPENS), t o energy c o n v e r s i o n i n p h o t o s y n t h e s i s (SMITH and GRAY). These r e p r e s e n t o n l y a very s m a l l s a m p l i n g of the many c h a l l e n g e s b e i n g o f f e r e d to quantum c h e m i s t r y by r e c e n t e x p e r i m e n t a l advances. Other c h a p t e r s w i l l mention aspects of inorganic synthesis, of reaction mechanisms, of homogeneous, heterogeneous, and enzymatic c a t a l y s i s , of c l u s t e r beams, of s u r f a c e s c i e n c e , of s u p e r c o n d u c t i v i t y and magnetism, of NMR and Mossbaur s p e c t r o s c o p y , of b i o m o l e c u l e s and much, much more. In f a c t t h i s f i e l d p r o v i d e s a v e r i t a b l e smorgasbord of e x p e r i m e n t a l data waiting for interpretation and guidance from quantum chemistry. A good smorgasbord is stocked with individual d e l i c a c i e s but a l s o has a theme - ours has the d and f e l e c t r o n s t o p r o v i d e c o n t i n u i t y , coherence and, i f you w i l l , the f l a v o r t h a t each of the authors of t h i s volume so w e l l e n j o y . Ab-initio Calculations The term " a b - i n i t i o " i s o f t e n taken to i n c l u d e those methods which, g i v e n an i n i t i a l c h o i c e f o r the g e n e r a l form of the N - e l e c t r o n wave function, ( e . g . one or many d e t e r m i n a n t s ) a t t e m p t to s o l v e the Schrodinger Equation H¥ = E¥ H = K i n e t i c energy o p e r a t o r + P o t e n t i a l energy

operator

E= m o l e c u l a r energy ¥=Y(r1,r2,....rN) w i t h o u t i n t r o d u c i n g any e m p i r i c a l p a r a m e t e r s . T h i s does not mean t h a t no a p p r o x i m a t i o n s are made (indeed, the whole a r t i s i n the a p p r o x i m a t i o n s ) but only t h a t r e c o u r s e i s not t a k e n to experiment o t h e r than, p e r h a p s , a - p o s t e r i o r i , t h a t there r e s u l t s agreement w i t h experiment. Table I g i v e s a b r i e f summary of some of the main c l a s s e s of ab i n i t i o methods w i t h t h e i r c h a r a c t e r i z i n g f e a t u r e s . A good e n t r y i n t o the d e t a i l s of these methods i s the book by Szabo and O s t l u n d (J_). Recent i s s u e s of the J o u r n a l of C h e m i c a l P h y s i c s , C h e m i c a l P h y s i c s L e t t e r s , and the I n t e r n a t i o n a l J o u r n a l of Quantum C h e m i s t r y c o u l d a l s o be c o n s u l t e d . Examples of a p p l i c a t i o n s w i l l be found i n the chapters of WILLIAMSON and HALL (GVB,CASSCF), DEDIEU and BRANCHADELL (CASSCF,CI), DAVIDSON ( H F , M P P T , C I ) , NOVARRO

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

3

THE CHALLENGE OF d AND f ELECTRONS

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch001

Table I .

Some C l a s s e s of Ab I n i t i o Methods 4

Hartree-Fock HF (or SCF)

one ( s p i n adapted) determinant ¥ =U are e v a l u a t e d as the i n t e g r a l s h(i,j)

=

g = = - = / / d (1 ) d ( 2 ) ^ ( 1 ) ^ ( 2 ) T

T

[1-P(12)]

^(1)^(2)

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

1.

SALAHUB & ZERNER

Quantum Chemistry Throughout the Periodic Table

t h e n the method i s g e n e r a l l y c l a s s i f i e d as " a b - i n i t i o " . It is important to stress here that a l l successful semi-empirical e l e c t r o n i c s t r u c t u r e H a m i l t o n i a n s have t h i s same form, and abi n i t i o and s e m i - e m p i r i c a l THEORIES are the same. The Table i n the a b - i n i t i o section that describes theories, i s also appropriate here: SCF, C I , MCSCF, CPMET, e t c . Semi-empirical theories generally parameterize h and g. U s u a l l y e n t i r e c l a s s e s of i n t e g r a l s i n g are n e g l e c t e d , and those t h a t remain, a l o n g w i t h h , a r e p a r a m e t e r i z e d t o compensate f o r t h i s omission. The p a r a m e t e r i z a t i o n may be based on model a b - i n i t i o c a l c u l a t i o n s , o r d i r e c t l y on e x p e r i m e n t , and o f t e n the proponents of these two d i f f e r e n t p o i n t s of view d o n ' t t a l k t o one a n o t h e r ! A common g o a l , though, i s t h a t of r e d u c i n g the N g i n t e g r a l s , where N i s the s i z e of the b a s i s s e t , t o one of N o r N • F o r t r a n s i t i o n m e t a l systems, i t i s p r o b a b l y f a i r t o l i m i t our a t t e n t i o n t o t h r e e methods, o r r a t h e r t h r e e c l a s s e s of methods s i n c e a number of v a r i a n t s e x i s t f o r e a c h . These are summarized i n Table I V . 4

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch001

3

Table I V .

Some S e m i - E m p i r i c a l Methods Used f o r d and f E l e c t r o n Systems

Extended H u c k e l EH REX(Re1. EH)

h ( i , i ) r e l a t e d to i o n i z a t i o n potentials, h ( i , j ) empirical param. s e t p r o p o r t i o n a l t o overlap (S). S included i n secular Eq. (H-E(i)S)c(i)=0

p i and sigma elec. large systems not self-consist, relativistic vers.

I t e r a t i v e EH IEH, SCCEH ITEREX

As above, i t e r a t i o n t o charge c o n s i s t e n c y u s u a l l y based on e x t r a p . between I . P . ' s o f atomic i o n s .

As above•

Fenske-Hall

As above: h ( i , j ) w i t h terms p r o p o r t i o n a l t o S as w e l l as k i n e t i c energy t e r m s .

Highly e f f e c t i v e f o r complexes; iterative.

N e g l e c t of Differential Overlap. CNDO, INDO NDDO

one c e n t e r p a r t of h ( i , i ) from I P ' s . N u c l e a r a t t r a c t , i n c l u d e d . CNDO c o n t a i n s a l l < i , j / i , j > ; INDO i n add. a l l one-center < i , j / k , l > ; NDDO i n add. a l l r e m a i n i n g t w o - c e n t e r Coulomb t y p e . h ( i , j ) usually proportional to S. Two-elec. i n t s often parameterized.

Large complexes C I , MBBT, MCSCF versions. Useful f o r geometry e s t i m a t e and for e l e c t r o n i c spectroscopy.

Relativistic version.

The Extended Huckel method, as i t i s known today, was f i r s t implemented by Lohr and Lipscomb i n s t u d i e s of boron h y d r i d e s (23)

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

13

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch001

14

THE CHALLENGE OF d AND f ELECTRONS

and l a t e r extended t o o r g a n i c systems by Hoffmann ( 2 4 ) . I t has been enormously f r u i t f u l , p r o v i d i n g a framework f o r the a n a l y s i s o f o r b i t a l i n t e r a c t i o n s which pervades modern o r g a n i c c h e m i s t r y . The a p p l i c a t i o n of the p r i n c i p l e s b e h i n d the Extended H u c k e l method, however, actually stem from a p p l i c a t i o n on t r a n s i t i o n metal complexes, initially by W o l f s b e r g and Helmholz, and then by B a l l h a u s e n , and B a l l h a u s e n and Gray ( 2 5 ) . I t e r a t i v e schemes f o r t r a n s i t i o n - m e t a l s compounds were d e v i s e d by B a l l h a u s e n and Gray and by Z e r n e r and Gouterman ( 2 6 ) . More r e c e n t l y , p r i m a r i l y through the work of the s c h o o l o f Hoffmann, i t has been adapted t o s o l i d s and surfaces and a g a i n i s p r o v i d i n g i n s i g h t t h a t o f t e n cannot be o b t a i n e d n e a r l y so r e a d i l y w i t h o t h e r more e l a b o r a t e techniques (27). Indeed, i t c o u l d reasonably be argued t h a t i f one had t o choose one t e c h n i q u e t o "do" c h e m i s t r y ( l u c k i l y one d o e s n ' t have t o make t h i s c h o i c e ! ) then EH, o r i t s i t e r a t i v e v e r s i o n s , w i t h i t s easy t o i n t e r p r e t r e s u l t s , would p r o b a b l y be the b e s t c h o i c e f o r a g e n e r a l overview of a wide v a r i e t y of systems. I t can p r o v i d e i n v a l u a b l e guidance f o r q u e s t i o n s t h a t depend on o r b i t a l symmetry, o r b i t a l e n e r g i e s and o r b i t a l o v e r l a p . A good d e a l of u n d e r s t a n d i n g can be based o n l y on these s i m p l e c o n c e p t s ! The F e n s k e - H a l l method, which i s i n the s p i r i t of the Extended H u c k e l Methods (28) but i n c l u d e s k i n e t i c energy terms as w e l l as much of the electrostatics, p u r p o r t s to g r e a t e r a c c u r a c y , but q u a l i t a t i v e l y y i e l d s results s i m i l a r to simple IEH. F o r more q u a n t i t a t i v e , and more d e t a i l e d treatments o f , f o r example, m u l t i p l e t s and o t h e r a s p e c t s of s p e c t r o s c o p y , one must i n c l u d e most of the proper e l e c t r o s t a t i c s . The methods d e s c r i b e d g e n e r i c a l l y as Z e r o - D i f f e r e n t i a l Ovelap (ZDO) types do t h i s i n a h i e r a r c h y of t e c h n i q u e s ; CNDO, Complete N e g l e c t of D i f f e r e n t i a l O v e r l a p (29), INDO, I n t e r m e d i a t e N e g l e c t of D i f f e r e n t i a l O v e r l a p ( 3 0 ) , and NDDO, N e g l e c t o f D i f f e r e n t i a l D i a t o m i c O v e r l a p (_31_), a l l of which are g e n e r a l i z a t i o n s of the e a r l i e r P a r i s e r - P a r r - P o p l e (PPP) method d e v i s e d f o r p i e l e c t r o n s o n l y ( 3 2 ) . A t present there is o n l y one g e n e r a l l y a v a i l a b l e program of t h i s nature t h a t includes t r a n s i t i o n metals, dubbed f o r un-acknowledged reasons ZINDO ( 3 3 ) , although other interesting v a r i a t i o n s are being developed ( 3 4 J . An a p p l i c a t i o n of the INDO method i s found i n the c h a p t e r by LOEW, i n which s p i n s t a t e s of l a r g e p o r p h i n a t o F e ( I I I ) complexes are examined, a task very d i f f i c u l t by a b - i n i t i o t h e o r i e s , as d i s c u s s e d i n the c h a p t e r by DAVIDSON, and one s t i l l a w a i t i n g a t h e o r y of m u l t i p l e t s t r u c t u r e i n DF methods. V a r i o u s CNDO and INDO schemes have a l s o been proposed f o r t r a n s i t i o n - m e t a l c l u s t e r s and f o r c h e m i s o r p t i o n on them. Although t h e r e have been some s u c c e s s e s , i t remains t r u e t h a t both the l e v e l of theory b e i n g used and the p a r a m e t e r i z a t i o n are both very much experimental. The s e m i - e m p i r i c a l methods c o n t i n u e t o y i e l d guidance to e x p e r i m e n t a l i s t s and t h e o r i s t a l i k e , i n a wide v a r i e t y of f i e l d s . But i n a sense the a p p l i c a t i o n s of these methods have a moving target. As a b - i n i t i o and d e n s i t y f u n c t i o n a l methods improve, becoming more r a p i d and a p p l i c a b l e t o a w i d e r range of q u e s t i o n s , those a p p r o p r i a t e f o r s e m i - e m p i r i c a l s t u d i e s become more and more complex. But t h e r e i s a l s o a fundamental q u e s t i o n h e r e . Any c a l c u l a t i o n t h a t i s performed on any system can be, and w i l l be,

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

1. salahub & zerner

Quantum Chemistry Throughout the Periodic Table

performed at greater accuracy. At what level of theory and computation may the problem at hand be considered to be solved? Extended Huckel, the simplest of these theories, is apparently enough to describe a great deal of the actual chemistry of a complex; however, we s t i l l do not have ANY theory capable of reliably reproducing, for example, the ESR spectroscopy of these systems. Clearly the computational method of choice will depend on the problem at hand, and this will continue to be so.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch001

Concluding Remarks It has been impossible, in these few pages,to capture even a small fraction of the advances and of the excitement surrounding the treatment of d- and f- electron systems in contemporary quantum chemistry. The term quantum chemistry has been used advisedly, as opposed to either theoretical chemistry, with its connotations of dynamics and "things other than electronic structure calculations", or computational chemistry with its connotations of black-box programs and user-friendly graphics software. Very little has been done in either of these directions for the transition metals and even less for the lanthanides and actinides. Indeed, the excitement in the field is largely due to the fact that there is so much left to discover, and to the fact that the tools to allow pioneering discoveries are now just becoming available. At the Toronto meeting we saw the first signs of progress towards both theoretical and computational chemistry for the d- and f- electron systems. Dynamics are definitely on the agenda for the next few years as are the development of convenient computer interfaces and the extraction of force fields and the like that will allow more and more experimentalists in inorganic chemistry to spend some of their time fruitfully at the computer keyboard. Our friend Rip had better not go to sleep for another twenty years! Literature Cited 1. 2. 3.

4. 5. 6. 7. 8. 9.

Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory MacMillan, New York, 1982. Salahub, D. R. Adv. Chem. Phys. 1987, 69, 447. e.g. Huzinaga, S.; Andzelm,J.;Klobukowski,M.;RadzioAndzelm, E.; Sakai, Y.; Tatewaki, H. Gaussian Basis Sets for Molecular Calculations; Elsevier, Amsterdam, 1984 and references therein. e.g. Huzinaga, S.; Klobukowski, M.; Sakai, Y. J. Phys. Chem. 1984, 88, 4880 and references therein. Pyykko, P. Chem. Rev. 1988, 88, 1. Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864. Kohn, W.; Sham, L. J . Phys. Rev. 1965, 140, A1133. Lundqvist, S.; March, N. H., Eds. Theory of the Inhomogeneous Electron Gas; Plenum, New York, 1983. Dahl, J. P.; Avery, J., Eds. Local Density Approximations in Quantum Chemistry and Solid State Physics; Plenum, New York, 1984.

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10. Dreizler, R. M.; da Providencia, J., Eds. Density Functional Methods in Physics; Plenum, New York, 1984. 11. Slater, J. C. Adv. Quantum Chem. 1972, 6, 1; The SelfConsistent Field for Molecules and Solids; Vol. 4, McGraw-Hill, New York, 1974. 12. Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. 13. Perdew, J. P.; Zunger, A. Phys. Rev. B 1981, 23, 5048. 14. Ceperley, D. M.; Alder, B. J . Phys. Rev. Lett. 1980, 45, 566. 15. Langreth, D. C.; Mehl, M. J. Phys. Rev. B 1983, 28, 1809; erratum 1984, 29, 2310 16. Becke, A. D. J. Chem. Phys. 1986, 84, 4524. 17. DePristo, A. E.; Kress, J . D. J. Chem. Phys. 1987, 86, 1425. 18. Perdew, J. P.; Yue, W. Phys. Rev. B 1986, 33, 8800; Perdew, J. P. Phys. Rev. B 1986, 33, 8822. 19. Johnson, K. H. Adv. Quantum Chem. 1973, 7, 143. 20. Muller, J . E.; Jones, R. O.; Harris, J . J. Chem. Phys. 1983, 79, 1874 and references therein. 21. Delley, B.; Ellis, D. E.; Freeman, A. J.; Baerends, E. J.; Post, D. Phys. Rev. 1983, 27, 2132 and references therein. 22. Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. J. Chem. Phys. 1979, 71, 3386, 4993. 23. Lohr, L. L.; Lipscomb, W. N. J. Chem. Phys. 1963, 38, 1604. 24. Hoffmann, R. J . Chem. Phys. 1964, 39, 1397: ibid. 40, 2047; 40, 2474; 40, 2480; 40, 2745. 25. Ballhausen, C. J; Gray, H. B. Inorg. Chem. 1962, 1, 111: ibid. Molecular Orbital Theory, Benjamin Press, New York, 1964. 26. Zerner, M.; Gouterman, M. Theoret. Chim. Acta. 1966, 4, 44. 27. Hoffmann, R. Rev. Mod. Phys. 1988, 60, 601. 28. Fenske, R.; Hall, M. B. Inorg. Chem. 1972, 11, 768. 29. Pople, J. A.; Santry, D. P.; Segal, G. A. J. Chem. Phys. 1965, 43, S129; Pople, J. A.; Segal, G. A. J. Chem. Phys. 1965, 43, S136; ibid. 1966, 44, 3289. 30. Pople, J. A; Beveridge, D. L.; Dobosh J. Chem. Phys. 1967, 47, 2026. 31. Pople, J. A.; Beveridge, D. L. Approximate Molecular Orbital Theory, McGraw Hill, New York, 1970. 32. Parr, R. G. Quantum Theory of Molecular Electronic Structure, Benjamin Press, New York, 1963. 33. Ridley, J. E.; Zerner, M. C. Theoret. Chim. Acta, 1973, 32, 111; Bacon, A.; Zerner, M. C. Theoret. Chim. Acta. 1979, 53, 21; Zerner, M. C.; Loew, G. H.; Kirchner, R. F.; MuellerWesterhoff, U. T. J. Am. Chem. Soc. 1980, 102, 589. 34. Lipinski, J. Intern. J. Quantum Chem. 1988, 34, 423. RECEIVED

March 2, 1989

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter Optimizations

of

the

Ti(IV)

2

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of

Tetrahedral

Complexes

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

A Basis Set and Correlation Study of Tetrachlorotitanium and Trichloromethyltitanium Rodney L. Williamson and Michael B. Hall Department of Chemistry, Texas A&M University, College Station, TX 77843 Optimization of the geometry of TiCl and TiCl CH at the SCF level results in Ti-Cl bond lengths longer than the experimental values, even when d- and f-type polarization functions are added to the basis set. The bond lengths remain too long even as the Hartree-Fock limit is approached because the SCF level of theory over-estimates the noble-gas-like Cl···Cl repulsions, which hinder close Ti-Cl approach. The Ti-C-H angle of T i C l C H is calculated to be close to tetrahedral geometry with little flattening of the hydrogen atoms, which apparently was observed in the electron diffraction. These same calculations do predict the anomalously low methyl-rocking frequency for thetitaniumcomplex in agreement with the experimental IR. This low methyl rocking frequency is due to stabilization of the Ti-C bond during the rocking motion by low lying empty d-orbitals on Ti. The large positive geminal hydrogen coupling constant observed in the NMR experiment is due primarily to the σ-donor and π-acceptor character of the TiCl moiety and not to any flattening of the methyl group. 4

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In the past few years, geometry optimizations of transition metal complexes have seen increased attention and improvement. Two minimal basis set studies (1-2) optimized metal-ligand bond lengths with errors of 0.05-0.25 A. The errors decreased for most ligands in studies with moderate basis sets (3-8) with the exception of metalcyclopentadienyl (Cp) bond lengths which showed errors ranging from 0.16-0.24 A. Further work (9-12) has shown that the error in calculating the metal-Cp bond lengths with very large basis sets is primarily correlation error. Two recent studies with 3-21G type basis sets have predicted the equilibrium geometries of some transition metal complexes with reasonable accuracy. In one of these studies (13), metal-carbonyl and metal-Cp distances were predicted to be 0.03 A and 0.15 A longer, respectively, than experimental values. It was shown that these long metal-Cp distances could be significantly reduced by including electron correlation. In the other study Q4), the optimized metal-carbonyl distances averaged 0097-6156/89/0394-0017$06.00/0 c 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

18

THE CHALLENGE OF d AND f ELECTRONS

0.11 A longer than the experimental values. The optimized geometries for a large number of tetrahedral metal-halide complexes gave M - F bond lengths shorter than experimental values and M - C l bond lengths longer than experimental values. In order to show how different basis sets and electron correlation affect bond lengths and angles we report here the results of self-consistent-field (SCF) and generalized-valence-bond (GVB) geometry optimizations of two tetrahedral titanium(IV) chloride complexes, T i C l and TiCl3CH . For TiCl3CH we also did complete-active-space-self-consistent-field (CASSCF) geometry optimizations. We chose TiCLt because of its high symmetry, which greatly simplifies the calculation and interpretation, and we chose TiCl3CH3 as a second Ti(IV) complex because of its "unique" geometry. This geometry, which was reported by Berry et al. (15) from electron diffraction (ED), appeared to have a flattened methyl group due to three agostic hydrogens. A n agostic hydrogen is defined as a hydrogen atom covalently bonded simultaneously to both a carbon atom and to a transition metal atom (16). Agostic hydrogens have been reported for several titanium alkyl complexes (16-19) such as [TiCl3(Me2PCH2CH PMe )R] (R=Et,Me). In these complexes, the geometry of the alkyl ligand containing the agostic hydrogen is distorted from the geometry it would have if it were bonded only to an organic substrate. In the methyl case this distortion is observed as a rocking of the methyl group such that one hydrogen atom moves toward the metal atom while the other two move away from the metal. The X-ray crystal structure of the above methyl complex (17) shows T i - C - H angles of 70(2)°,105(4)°, and 117(3)°. A later neutron diffraction study (12) on the same complex showed Ti-C-H angles of 93.5(2)°, 118.4(2)° and 112.9(2)°. If the complex did not have an agostic hydrogen, one would expect the Ti-C-H angles to be close to 109.5°. An ab initio molecular orbital study (2Q) of the agostic hydrogen interaction in Ti(CH3)(PH3)Cl3 reported direct interaction between the C-H a-bond and an unoccupied T i d-orbital. This type of interaction involving a single hydrogen bent towards the metal center is typical of most complexes with an agostic hydrogen. However, Berry et al. (15) reported all three methyl hydrogens "flattened" toward the titanium atom, the first such complex with three agostic hydrogens. In addition to the ED results, they reported a large difference between the methyl rocking vibrational frequencies of TiCl3CH3 and GeCl3CH3 and explained this difference as resulting from the flattening of the methyl hydrogens. They also did C and H N M R studies and found the H,H coupling constant to have a large positive value, which they suggested was further evidence of hydrogen flattening. We optimized the geometry of TiCl3CH3 to determine if the calculations would predict the symmetrical flattening of the methyl hydrogens. The model complexes TiH3CH3 and GeH3CH3 were used to calculate vibrational frequencies to determine if the calculation would predict the large difference between the rocking frequencies of the Ge and Ti complexes. Some of these results have been reported in a preliminary communication (21). And finally, we estimated the H,H coupling constant for the Is orbitals of two methyl hydrogens on TiCbCH3, CH4, and CH3CI using the method of Pople and Santry (22). Although this method is known to have serious problems in predicting the sign and absolute magnitude of the H,H coupling constant (22), it does correctly predict the direction and size of the change of the coupling constant resulting from changes in the nature of the methyl substituent or changes in geometry of the compound.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

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Geometric Optimizations ofTetrahedral Complexes 19

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

Computational Details We optimized the geometries of the Ti complexes using three different titanium basis sets. The first titanium basis set (I) is a (432-421-31) used in previous calculations (12), the second titanium basis set (H) is a Huzinaga (24) (5333-53-5) modified to a (533211-5211-3111) by splitting off the most diffuse s, p, and d functions and adding s, p, and d functions with exponents 1/3 the values of the split-off functions, and the third titanium basis set (III) is the titanium basis set II with an additional ftype polarization function (£=0.55) (25). For the main group atoms, basis set I is Cl(3321-321), C(321-21), H(21), basis set II is Cl(5321-521), C(421-31), H(31), basis set HI is Cl(531111-4211), C(721-41), H(31), basis set IV is Cl(5321-521-1), C(421-31-l), H(31-l), basis V is Cl(533-53), basis V I is Cl(53111-5111), basis VII is Cl(533-53-l), basis VIII is Cl(533-53-ll), and basis IX is Cl(5321-521-ll). Basis sets I, II, and VI-IX are Huzinaga (24) basis sets which were all modified (expect for basis set V) by singly or doubly splitting the most diffuse s and p functions. Basis sets IV, and VII-IX were further modified by adding polarization functions. Basis set IH is an unmodified Dunning-Hay-Huzinaga (26) basis set. The basis sets used for the geometry optimizations and force constant calculations of TiH3CH3 and GeH3CH3 are 3-21G type basis sets used in previous geometry optimization (13). The H,H coupling constant calculations were done using fully contracted titanium and chlorine basis sets I and carbon and hydrogen basis sets EL The geometry of TiCl3CH3 and TiCU were optimized in staggered C3 and Td symmetry, respectively. The G V B calculations (27) involve perfect-pairing for all seven sigma bonds for TiC^CTb and for the four Ti-Cl sigma bonds for TiCU. The first of two CAS SCF calculations on TiCbCH3 contains eight electrons in the eight orbitals (8/8) made up of four a-bonding and four a-antibonding T i - C and C - H orbitals. The second contains eight electrons in eleven orbitals (8/11) made up of the eight orbitals in the 8/8 calculation plus two T i - C rc-bonding orbitals and one additional Ti-C a-bonding orbital. A l l of the calculations were done with G A M E S S (Generalized Atomic and Molecular Electronic Structure Systems) except for the TiCU calculations with f-type polarization functions which were done with Q U E S T (QUantum Electronic STructure). These programs were run on a C R A Y X - M P at C R A Y Research in Mendota Heights, an FPS-264 at the Cornell National Supercomputer Facility, an I B M 3090/200 at Texas A & M University, and the Department of Chemistry's V A X 11/780 andFPS-164. V

Results and Discussion TiCl^ Geometry Optimization. The optimized Ti-Cl bond lengths (see Table I) for TiCU with a variety of basis sets are all longer than the experimental value and differ from that by 0.016 to 0.051 A. Splitting the Cl(533-53) basis set, which allows the orbitals freedom to expand or contract, only decreased the Ti-Cl bond length an average 0.005 A. However, when we add d-type polarization functions to the chlorine basis set the Ti-Cl bond distance decreases an average 0.026 A. The effect of adding polarization functions is also seen in deformation density maps of TiCU- The deformation density of TiCU without d-functions on CI (Figure la) shows a buildup of density in the Ti-Cl bonding region and the CI lone-pair regions. When a polarization function is added to CI (Figure lb) density in the bonding region increases and density in the chlorine lone-pair region decreases. The

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

ELECTRONS

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THE CHALLENGE OF d AND f

Figure 1. Electron deformation density plots of TiCU in Cl-Ti-Cl plane: a) deformation density without d-functions b) deformation density with d-functions. Contours are geometric beginning at ±0.001 e~au- and incremented by doubling the previous contour value. 3

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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Geometric Optimizations of Tetrahedral Complexes 21

Table I. Ti-Cl Bond Distances for TiCU

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

Tjtagiurn Pasis Sets Method I Chlorine Basis 1. SCF Cl(3321-321) (I) 2.197 2. SCF Cl(533-53) (V) 2.203 3. SCF 2.201 Cl(5321-521) (II) 4. SCF C1(531U-5111)(VI) 2.200 5. SCF 0(531111-4211) (HI) 2.202 6. SCF Cl(533-53-l) (VH) 7. SCF Cl(5321-521-1) (IV) 8. SCF Cl(533-53-ll)(Vni) 9. SCF C1(5321-521-11)(IX) GVB 10. Cl(5321-521) (II) Experimental 2.170(2)* Reference 28, the error is reported at a confidence level of 2.5

II 2.209 2.221 2.213 2.214 2.219 2.186 2.187 2.189 2.191 2.234

in

2.196

2.181

a.

origin of the shift in deformation density from the lone-pair region on CI to the Ti-Cl bonding region is also seen in orbital plots (Figures 2a-d) of both the Ti-Cl a-bonding andrc-bondingmolecular orbitals (MO). The value in the Ti-Cl bonding region of the ai M O with d-functions (Figure 2b) is greater than the value in the bonding region of the ai M O without d-functions (Figure 2a). The CI t ic-orbitals also show a similar increase in the Ti-Cl bonding region (Figures 2e,f). Molecular orbital plots of the other orbitals (Figures 2c-d,g-j) do not show any difference between the plots with dfunctions and the plots without d-functions. As we improve the TiCU wavefunction by adding f-type polarization functions to the titanium basis set the Ti-Cl bond distance shortens further. When a chlorine basis set without d-functions is used, the addition of an f-function on titanium shortens the Ti-Cl bond 0.017 A. However, when a chlorine basis set with d-functions is used, the Ti-Cl bond only shortens 0.006 A to 2.181 A. Although this Ti-Cl distance is the shortest of all the optimized geometry calculations, it is still 0.011 A longer than the experiment As the wavefunction approaches the Hartree-Fock limit one would expect the TiC l bond distance to be shorter than the experiment because of the lack of bond-pair correlation. The bond-pair correlation added by the G V B wavefunction lengthened the Ti-Cl bond 0.021 A, because the G V B wavefunction adds only limited left-right correlation and none of the dynamical correlation. For most A - B bonds, the calculated bond lengths at the SCF level are too short, and the correlation added by a G V B calculation accounts for a major portion of the non-dynamical correlation error in the SCF wavefunction. But for Ti-Cl bonds, both the SCF and G V B calculations predict too long a bond distance because they do not include necessary dynamical atomic correlation of the CI atoms. In TiCU, each CI satisfies the octet rule by sharing a pair of electrons with Ti. Thus, to each other, the CI atoms appear as noble gas atoms. As is well known (29i 30). the Hartree-Fock wavefunction does not adequately describe the attraction of two noble-gas-type atoms. SCF level calculations (29) of the H e , N e , and A r potentials show the atoms in these dimers to be too strongly repulsive at close 2

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THE CHALLENGE OF d AND f ELECTRONS

Figure 2. Wavefunction plots of T i C l molecular orbitals in the C l - T i plane: a, ^ without d functions; b, ^ with d functions. Contours are geometric beginning at ±0.001 e~au~ and incremented by doubling the previous contour value. Continued on next page. 4

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WILLIAMSON & HALL

Geometric Optimizations of Tetrahedral Complexes 23

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

2.

Figure 2. Continued. Wavefunction plots of T i C l molecular orbitals in the C l - T i plane: c, t a without d functions; d, ^ a with d functions. Contours are geometric beginning at ±0.001 e"au~ and incremented by doubling the previous contour value. Continued on next page. 4

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THE CHALLENGE OF d AND f ELECTRONS

Figure 2. Continued Wavefunction plots of T i C l molecular orbitals in the C l - T i plane: e, t 7T without d functions; f, t * with d functions. Contours are geometric beginning at ±0.001 e~au~ and incremented by doubling the previous contour value. Continued on next page. 4

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In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Geometric Optimizations of Tetrahedral Complexes 25

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WILLIAMSON & HALL

Figure 2. Continued Wavefunction plots of T i C l molecular orbitals in the C l - T i plane: g, e without d functions; h, e with d functions. Contours are geometric beginning at ±0.001 e"au~ and incremented by doubling the previous contour value. Continued on next page. 4

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Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

THE CHALLENGE OF d AND f ELECTRONS

Figure 2. Continued. Wavefunction plots of T i C l molecular orbitals in the C l - T i plane: i , tj without d functions; j , i with d functions. Contours are geometric beginning at ±0.001 e"au~ and incremented by doubling the previous contour value. 4

x

3

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

2. WILLIAMSON & HALL

Geometric Optimizations of Tetrahedral Complexes 27

distances and lacking a van der Waals minimum at long distances. Because the CI atoms are noble-gas-like, the C l - C l interactions in T i C U sterically hinder the CI atoms from bonding close to the T i atom. The accurate determination of the Ti-Cl bond distance will require a large configuration-interaction calculation. T i C h C H * Geometry Optimizations. The results of complete SCF geometry optimizations on T i C l C H using basis sets I-IV are shown in Table II. As the basis sets are improved, the Ti-C and C-H bond distances show the smallest changes of 3

3

Table II. SCF Geometries for C l T i C H

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

3

Basis Set M Ligand 1. I I 2. n II n 3. HI 4. n IV ED Experiment* Reference 15

3

Bond Distances (A) C-H Ti-C Ti-Cl 1.087 2.229 2.011 1.092 2.251 2.013 2.012 1.095 2.258 1.091 2.020 2.219 2.042 1.158 2.185

Ti-C-H 107.9 108.3 108.2 108.2 101.0

Angles O Cl-Ti-C 102.4 103.4 103.2 103.7 105.2

a

0.009 A and 0.008 A , respectively, and the Ti-Cl bond distance shows the largest change of 0.039 A. The T i - C - H and Cl-Ti-C angles vary only 0.4° and 1.3°, respectively. The geometry optimizations with T i and ligand basis sets II give Ti-C and C-H bond distances 0.029 A and 0.066 A shorter and a Ti-Cl bond length 0.066 A longer than the ED results. When polarization functions are added to the CI, C and H functions (basis set IV), the differences between the E D and the calculated Ti-Cl and T i - C bond lengths decrease to 0.034 A and 0.022 A, respectively, and the analogous difference for the C-H bond distance increases to 0.067 A. For basis set II, the Cl-Ti-C and Ti-C-H angles are 1.8° smaller and 7.2° larger, respectively, than the E D angle. When polarization functions are added to all atoms but T i , the differences decrease to 1.5° and 7.1°, respectively. Although most of these optimized geometric parameters are in good agreement (±0.03 A, ±1.5°) with the ED results, the optimized Ti-C-H angle and C-H distance do not agree with the ED result. In the G V B geometry (see Table III), the already long Ti-Cl distance increases 0.035 A, the Ti-C bond lengthens 0.168 A, and the Ti-C-H and Cl-Ti-C angles both decrease 4.4° and 3.9°, respectively. Previous work by Ditchfield and Seidman (21)

Table HI. G V B Geometries for C l T i C H 3

Basis Set M Ligand 1. IH in 2. in in in 3. in ED Experiment* Reference 15

Bond Distances (A) C-H Ti-Cl Ti-C 1.108 2.296 2.180 ED 1.111 ED ED 1.109 2.295 2.042 1.158 2.185

3

Angle? (°) Ti-C-H Cl-Ti-C 99.3 103.8 ED 106.5 100.5 105.5 105.2 101.0

a

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

28

THE CHALLENGE OF d AND f ELECTRONS

has shown that the addition of electron correlation to the SCF wavefunction in A H (A=C,N,0,F) molecules usually has a small effect on bond lengths and angles. Although an increase in the bond length is not surprising for a G V B calculation, the magnitude of the increase in the Ti-C bond length is surprising. Because the G V B wavefunction overemphasizes dissociation, the C ^ T i and CH3 moieties have too much radical-like C^Ti- and CH3 character. As the fragments become more radical­ like they flatten toward their equilibrium planar geometry. Even with the long Ti-C bond distance, the Ti-C-H angle is still 2.8° larger than the ED angle. If there were only steric repulsions between the hydrogens and titanium, the bond angles would increase as the fragments are forced back together. On the other hand, if there were attractive agostic hydrogen interactions, the hydrogens would bend towards the titanium atom as the fragments are brought closer, which would decrease the angle closer to the ED value. When we shorten the Ti-C bond distance to the E D bond length and optimize the remaining geometric parameters at the G V B level, the Ti-C-H and Cl-Ti-C angles increase 1.7° and 1.3°, respectively. When, in addition, we shorten the Ti-Cl bond length and the increase the Cl-Ti-C angle to the ED values, the Ti-C-H bond angle increases an additional 1.0°, to a value 5.5° larger than the reported ED angle. Thus, the bending of the hydrogen atoms away from the titanium atom as the Ti-C distance is shortened shows that the titanium-hydrogen interactions are repulsive and not attractive. We improved the calculation further by using a CASSCF wavefunction with an active space of 8 electrons in 8 orbitals (8/8). The results of the geometry optimization (see Table IV) give a Ti-C bond length of 2.119 A and a Ti-C-H angle of 105.4° (the Ti-Cl bond distance and Cl-Ti-C angle were frozen at the ED values). In

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

n

Table IV. CASSCF Geometries for C l T i C H 3

Basis Set M Ligand Method II II SCF II II 8/8 CAS II II 8/11 CAS ED Experiment* Reference 15

Bond Distances (A) C-H Ti-Cl Ti-C 1.092 2.251 2.013 ED 2.119 1.112 ED 2.106 1.112 2.042 1.158 2.185

3

Angles (°) Ti-C-H Cl-Ti-C 103.4 108.3 105.4 ED ED 106.2 105.2 101.0

a

contrast to the SCF geometries, but like the G V B geometry, the CASSCF Ti-C bond distance is longer than the E D value and the Ti-C-H angle shows a slightly larger decrease from the tetrahedral angle. However, when we increase the size of the active space to 11 orbitals (8/11), with additional Ti-C c and n bonding orbitals, the Ti-C bond shortens 0.013 A and the Ti-C-H angle increases 0.8°. This CASSCF includes those virtual T i drc-orbitals which are involved in the qualitative description of the agostic hydrogen interaction. The increase one observes in the Ti-C-H angle as the Ti-C bond distance shortens, which one also observed in the G V B geometry optimizations, shows that the Ti-C-H angle is very sensitive to changes in the Ti-C bond distance. If the wavefunction were improved by including even more correlation, one would expect the Ti-C bond distance to shorten further toward the ED value and as the Ti-C bond shortens, one would also expect the Ti-C-H angle to increase towards a tetrahedral angle.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

2. WILLIAMSON & HALL

Geometric Optimizations of Tetrahedral Complexes 29

Vibrational Frequencies Calculations. Berry et al. ( I D reported that the methyl rocking frequency of TiCbCH3 is much lower than the analogous frequency of GeCbCHs. They presumed this anomalously low frequency of TiCbCHs to result from the "flattening" of the methyl hydrogens. We calculated the vibration frequencies of the hypothetical model complexes TiH3CH3 and GeH3CH3 after first optimizing their geometries. The results of the geometry optimizations (see Table V) Table V . SCF Geometries for MH3CH3 (M = Ti, Ge) Angles (°) M-C-H H-M-C 109.9 108.3 110.7 110.3

Bond Distances (A)

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1. 2.

Metal Ti Ge

Basis I I

M-H 1.710 1.533

M-C 2.035 1.959

C-H 1.092 1.086

show the expected differences in geometry between the T i and Ge complexes. The vibration frequencies were calculated by taking finite differences of energy gradients beginning at these optimized geometries. The calculated frequencies, when compared with the experimental frequencies (see Table VI), show errors expected of calculations at the SCF level. However, Table VI. MX3CH3 Vibrational Frequencies in cirr

Mode e C H rock i CH3 deform. e CH3 deform. a CH3 stretch C H stretch e Reference 15 3

a

3

Ge 978 1468 1611 3190 3270

Calcul. (X-Ti 577 1360 1564 3122 3208

Diff. 401 108 47 68 62

a

Ge 825 1246 1403 2940 3019

Exper. (X=C\) Ti Diff 580 245 1052 194 1375 28 2894 46 2980 39 a

when one compares the SCF differences between the T i and Ge model complexes with the experimental differences between the T i and Ge chloride complexes, the differences are in good agreement. The large difference in the methyl rocking frequency observed in the experiment is predicted by the SCF calculation, the same SCF calculation that predicts no flattening of the methyl hydrogen atoms. If the large difference between the T i and Ge methyl rocking frequency is not the result of hydrogen flattening, what then is the cause of the observed difference in the methyl rocking frequencies? To answer the above question we did point by point SCF calculations of the model complexes as the methyl group rocks. A plot of the resulting energies for each point calculated is shown in Figure 3 and, as expected, the titanium complex potential curve is flatter than the germanium complex potential curve. When we rock the methyl group on the model complexes +45°, the titanium complex is destabilized by

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

THE CHALLENGE OF d AND f ELECTRONS

• Ge • Ti

Rock Angle (degrees)

Figure 3. Relative energy of

TUH3CH3

and GeH3CH3 for the methyl rock

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

2. WILLIAMSON & HALL

Geometric Optimizations of Tetrahedral Complexes 31

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

-1

1

12.9 kcal m o l and the germanium complex is destabilized by 34.9 kcal m o l . Eisenstein and Jean (22) using extended Huckel calculations on the titanium model complex also found the rocking motion to be weakly destabilizing. At the equilibrium geometry the deformation density plot (Figure 4a) shows a large buildup of density aligned symmetrically along the Ti-C bond axis. As the methyl group rocks (Figures 4b and 4c) the density buildup is no longer symmetric about the Ti-C bond axis, but rather shifted to one side of the axis. Although the methyl group in Figures 4b and 4c has been tilted 45° from equilibrium the maximum density between the Ti and C atoms has only tilted an average 22°. As was seen in the titanium complex, the germanium complex at equilibrium geometry (Figure 5a) shows a similar symmetric buildup of density along the Ge-C bond axis. However, as the methyl group rocks 45° (Figures 5b and 5c) the density shifts to one side of die axis with an average tilt of 35°, 13° larger than the tilt in the T i complex. It is also interesting to note that the density in the C-H bond region decreases slightly for both the Ge and T i complexes as the hydrogen is tilted toward the metal. The density between the metal and the other hydrogen in the plot remains unchanged for the germanium complex but increases slightly for the titanium complex as the methyl group is rocked awayfromequilibrium. Goddard, Hoffmann, and Jemmis (33) in a study of alkyl tantalum complexes have suggested that rehybridization of the metal-carbon bond with empty d-orbitals on the metal stabilizes the rocked methyl group. Titanium, like tantalum, has low-lying empty d-orbitals that allow facile rehybridization of the metal-carbon bond. Germanium, on the other hand, has used its s- and p-orbitals and the empty d-orbitals are at very high energy, therefore no empty orbitals are available for rehybridization. The difference in the ease of rehybridization explains the much lower methyl rocking frequency for the titanium complex than for the germanium complex. The deformation density plots do not contradict this explanation. Furthermore, this explanation is supported by the total metal-carbon overlap populations which actually increase for the Ti-methyl rock but decrease for the Ge-methyl rock. A n increase in the titanium d^-carbon and p -carbon more than compensates for the decrease in the d 2-carbon overlap populations as the methyl group is rocked. The germanium complex however, shows no change in any germanium d-carbon overlap population and the very slight increase in the p -carbon overlap population does not compensate for the loss in the p -carbon overlap population as the methyl group is rocked. x

z

x

z

Geminal H.H Coupling Constant. Using the method of Pople and Santry (22) we calculated the difference between the H,H coupling constant of CICH3 and HCH3 to be 1.6 Hz, which is in good agreement with the experimentally determined value (34). The calculated H,H coupling constant for TiCbCH3 at the optimized geometry is 8.0 H z larger than the coupling constant for HCH3. Although this change is smaller than the 23.7 Hz change observed experimentally (15.34). this simple model does predict a change which is both large and positive. When the optimized geometry of TiCl3CH3 is flattened to the E D geometry the H,H coupling constant increases only 3.6 Hz. By comparing the increase in the coupling constant of X C H 3 as X changes from H to T i C l 3 with the increase in the coupling constant as the hydrogens in T i C l 3 C H 3 are flattened, we see that the flattening of the methyl hydrogens has a relatively small effect on the coupling constant when compared with the large effect of the T i C ^ substituent. However, Green and Payne (35) using extended Huckel calculations observed the effect of the substituent on the coupling constant to be only half of the

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

THE CHALLENGE OF d AND f ELECTRONS

Figure 4. Deformation density of TiH3CH in the H-C-Ti-H plane: a) equilibrium geometry b) methyl rocked 45° counterclockwise c) methyl rocked45° clockwise. Contours are geometric beginning at±0.001 e au and incremented by doubling the previous contour value. 3

_

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

3

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

WILLIAMSON & HALL

Geometric Optimizations of Tetrahedral Complexes

a

b

c Figure 5. Deformation density of GeHsCHsin the H-C-Ge-H plane: a) equilibrium geometry b) methyl rocked 45° counterclockwise c) methyl rocked 45° clockwise. Contours are geometric beginning at ±0.001 e~au and incremented by doubling the previous contour value. -3

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

34

THE CHALLENGE OF d AND f ELECTRONS

effect of the geometry change on the coupling constant, which is opposite to what we predict using ab initio calculations. We believe that ab initio calculations describe the true nature of this titanium complex more accurately than the semi-empirical extended Hiickel calculations. It is the strong a-donor and weak ic-acceptor character of the TiCl3 substituent which results in the observed positive shift in the coupling constant. Thus, one can explain the coupling constant without postulating any geometric distortions.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

Conclusions Optimization of the geometry of TiCL* at the SCF level results in a Ti-Cl bond length which is longer than the experiment, even when d- and f-type polarization functions are added to the basis set. For covalently bonded systems one expects a wavefunction at the Hartree-Fock limit to give bond lengths shorter than the experiment i f they are not sterically crowded. Because the Hartree-Fock wavefunction overestimates the CI—CI repulsions, the Ti-Cl bond distances remain long, even in large basis sets. Geometry optimizations of TiCl3CH3 also show long Ti-Cl bond distances. The T i - C - H angle is close to tetrahedral geometry with little, if any, flattening of the hydrogen atoms. Because of the known problem that electron diffraction has with determining the positions of hydrogen atoms, the large difference between the optimized value and the electron diffraction value for the T i - C - H angle is not surprising. Our calculations correctly predict the anomalously low methyl-rocking frequency for the titanium complex without hydrogen flattening. This low methylrocking frequency is due to stabilization of the rocking motion by low-lying empty dorbitals on Ti. The large positive geminal hydrogen coupling constant is primarily due to the a-donor and xc-acceptor character of the TiCb moiety and not the flattening of the methyl hydrogens. We do not believe agostic interactions are strong enough to symmetrically flatten a methyl group. The key to an agostic interaction is the low-energy rocking distortion. Thus, we would expect all agostic methyls to be rocked with the H-C-H angles close to those in similar organic compounds. Acknowledgments The authors would like to thank the National Science Foundation (Grant C H E 8619420) and C R A Y Research for support of the work. This research was conducted on an IBM-3090 and FPS-264 at the Cornell National Supercomputer Facility, a resource for the Center for Theory and Simulation in Science and Engineering at Cornell University, which is funded in part by the National Science Foundation, New York State, and the I B M Corporation, on a C R A Y X - M P at C R A Y Research, Mendota Heights, Minnesota, and on a V A X 11/780 and FPS-164 at the Chemistry Department of Texas A & M University. We would also like to thank Dr. Martyn F. Guest at SERC, Daresbury Laboratory, Warrington, U K . for making his version of GAMESS available and Drs. Thomas Dunning and Ron Shepard at Argonne National Laboratory for making QUEST available. Literature Cited 1. Pietro, W.J.; Hehre, W.J. J. Comp. Chem. 1983, 4, 241.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

2. WILLIAMSON & HALL

Geometric Optimizations of Tetrahedral Complexes 35

2. Seijo, L.; Barandiaran, Z.; Klobukowski, M.; Huzinaga, S. Chem. Phys. Lett. 1985, 117, 151. 3. Faegri Jr., K.; Almolf, J. Chem. Phys. Lett. 1984, 107, 121; 4. Luthi, H.P.; Ammeter, J.H.; Almolf, J.; Faegri Jr., K . J. Chem. Phys. 1982, 77, 2002. 5. Pitzer, R.M.; Goddard, J.D.; Schaefer, H.F. J. A m . Chem. Soc. 1981, 103, 5681. 6. Yates, J.H.; Pitzer, R . M . J. Chem. Phys. 1979, 70, 4049. 7. Kataura, K.; Sakaki, S.; Morokuma, K . Inorg. Chem. 1981, 20, 2292. 8. Spangler, D.; Wendoloski, J.J.; Dupuis, M.; Chen, M . M . L . ; Schaefer, H.F. J. Am. Chem. Soc. 1981, 103, 3985. 9. Guest, M.F.; Hillier, I.H.; Vincent, M.; Rosi, M. J. Chem. Soc. Chem.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

Commun. 1986, 438. 10. Luthi, H.P.; Siegbahn, P.E.M.; Almolf, J. J. Phys. Chem. 1985, 89, 2156. 11. Luthi, H.P.; Siegbahn, P.E.M.; Almolf, J.; Faegri Jr., K.; Heiberg, A . Chem. Phys. Lett. 1984, 111, 1. 12. Almolf, J.; Faegri Jr., K.; Schilling, B.E.R.; Luthi, J.P. Chem. Phys. Lett. 1984, 106, 266. 13. Williamson, R.L.; Hall, M.B. Int. J. Quantum Chem., Quantum Chem. Symp. 21, 1987, 503. 14. Dobbs, K.D.; Hehre, W.J. J. Comput. Chem. 1987, 8, 861. 15. Berry, A.; Dawoodi, Z.; Derome, A . E.; Dickinson, J. M.; Downs, A . J.; Green, J. C.; Green, M . L . H.; Hare, P. M . ; Payne, M . P.; Rankin, W. H . ; Robertson, H . E . J. J. Chem. Soc. Chem. Commun. 1986, 519. 16. Brookhart, M.; Green, M . L . H. J. Organomet. Chem. 1983, 250, 395. 17. Dawoodi, Z.; Green, M . L . H.; Mtetwa, V . S. B.; Prout, K . J. J. Chem. Soc. Chem. Commun. 1982, 1410. 18. Dawoodi, Z.; Green, M . L . H.; Mtetwa, V . S. B.; Prout, K . J. J. Chem. Soc. Chem. Commun. 1982, 802. 19. Dawoodi, Z.; Green, M . L . H.; Mtetwa, V . S. B.; Prout, K.; Schultz, A . J.; Williams, J. M.; Koetzle, T. F. J. Chem. Soc. Dalton Trans. 1986, 1629. 20. Obara, S.; Koga, N . Morokuma, K . J. Organomet. Chem. 1984, 270, C33. 21. Williamson, R.L.; Hall, M . B . J. Am. Chem. Soc. 1988, 110, 4428. 22. Pople, J.A.; Santry, D.P. Mol. Phys. 1963, 8, 1. 23. Maciel, G.E.; McIver Jr., J . W . ; Ostlund, N.S.; Pople, J.A. J. A m . Chem. Soc. 1970, 92, 4151. 24. Gaussian Basis Sets for Molecular Calculations: Huzinaga, S., Ed.; Amsterdam: Elsevier, 1984 25. Bauschlicher Jr., C.W.; Seigbahn, P.E.M. Chem. Phys. Lett. 1984, 104, 331. 26. Dunning Jr., T.H.; Hay, P.J. In Methods of Electronic Structure Theory; Schaefer, H.F., Ed.; Plenum Press: New York, 1977; Vol. 4 Chapter 1. 27. Bobrowicz, F.W.; Goddard, W.A. In Methods of Electronic Structure Theory: Schaefer, H.F., Ed.; Plenum Press: New York, 1977; Vol. 4 Chapter 4. 28. Morino, Y.; Uehara, H . J. Chem. Phys. 1966, 45, 4543. 29. Gilbert, T.L.; Wahl, A . C . J. Chem. Phys. 1967, 47, 3425. 30. Schaefer, H.F.; McLaughlin, D.R.; Harris, F.E.; Alder, B.J. Phys. Rev. Lett. 1970, 25, 988. 31. Ditchfield, R.; Seidman, K . Chem. Phys. Lett. 1978, 54, 57. 32. Eisenstein, O.; Jean, Y . J. Am. Chem. Soc. 1985, 107, 1177.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

36

33. Goddard, R.J.; Hoffmann, R.; Jemmis, E.D. J. Am. Chem. Soc. 1980, 102, 7667. 34. Pople, J.A.; Bothner-By, A . A . J. Chem. Phys. 1965, 42, 1339. 35. Green, J. C.; Payne, M.P. Magn. Res. Chem. 1987, 25, 544.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch002

R E C E I V E D December 9, 1988

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 3

LCGTO—Xα

Study on the Agostic Interaction in Cl TiCH 3

3

N . Rösch and P. Knappe Lehrstuhl für Theoretische Chemie, Technische Universität München, 8046 Garching, Federal Republic of Germany The geometry and selected vibrational frequencies of Cl TiCH have been calculated using the LCGTO-Xα method. The methyl group is found essentially undistorted, thus providing no indication for an agostic interaction.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch003

3

3

We would like to comment on the findings of Williamson and Hall on tetrahedral T i f l V ) complexes (1,2) by reporting some results from our L C G T O - X a calculations on the geometry of C l T i C H and of related titanium complexes (Knappe, P.; Rosch, N . J . Organometal. Chem.. in press). Our study was prompted by the experimental finding (3) in C l T i C H of an unusual agostic interaction at an a carbon atom and by results of an Extended Hiickel analysis (4) of the agostic interaction in tetrahedral and octahedral titanium complexes. For T i C U (in TSL

L

H

W C I A

l

U„

2

^

c

1

H

I , *H %

A

l

C = C H

c-c c=c

Η Rh'

^Rrt

=

""Rh Cl^ >L

Rh Cl^ I >L S

-Ho

1

fast / - x c

Η

rate determing step

c

Η H

[RhCI(H)L (Ç-ÇH)]

'Rh

3

cr I s

L

n

ι H

dominant catalytic cycle Scheme 1

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

6.

KOGA & MOROKUMA

Potential Energy Surface of Olefin Hydrogénation

t i o n o f a l k a n e . The r a t e - d e t e r m i n i n g s t e p has been b e l i e v e d t o be the o l e f i n i n s e r t i o n s t e p . We u s e d PH^ i n s t e a d o f PR~ and C^H^ as a model o f o l e f i n . The model c a t a l y t i c c y c l e we s t u d i e d i s shown i n Scheme 2. A l l the e q u i l i b r i u m and t r a n s i t i o n s t a t e s t r u c t u r e s were o p t i m i z e d by the RHF energy g r a d i e n t method. W h i l e two p h o s p h i n e s a r e always t r a n s t o each o t h e r i n a l l the i n t e r m e d i a t e s i n the H a l p e m mechanism, Brown e t a l . have v e r y r e c e n t l y p r o p o s e d a d i f f e r e n t mechanism, i n which the c i s b i p h o s p h i n e i n t e r m e d i a t e s p l a y an e s s e n t i a l r o l e ( 4 ) . T h e i r m o l e c u l a r m o d e l i n g c a l c u l a t i o n s where the van der Waals energy i s c a l c u l a t e d between s u b s t i t u t e d o l e f i n s and the Rh fragment w i t h b u l k y t r a n s p h o s p h i n e s have s u g g e s t e d t h a t when the s u b s t i t u e n t s on the o l e f i n a r e b u l k y , the s t e r i c r e p u l s i o n i s too l a r g e f o r the o l e f i n t o c o o r d i n a t e . T h e i r NMR e x p e r i m e n t s have shown the e x i s t e n c e o f the f o l l o w i n g e q u i l i b r i u m ( E q u a t i o n 1) i n w h i c h an i n t e r m e d i a t e w i t h a p a i r o f c i s phosphines can be formed. I n f a c t , ^ R h C l C P R ^ ) ^ has been d e t e c t e d i n the c a t a l y t i c system.

H

H

Phi

Rh-

•pPh, b

PPh

I

.ΛΝ

ΡΡΠ

'

a

(1)

a

- Rh ' .

3

+ PPh

j ^ pph CI

3

b

Cl

3

Based on these r e s u l t s , t h e y have s u g g e s t e d t h a t the i n t e r m e d i a t e s o f the c a t a l y t i c system have two c i s p h o s p h i n e s . I n Scheme 3 i s shown the new mechanism. I n t h i s Scheme, the key s t e p i s i s o m e r i z a t i o n (Equation 2), t r a n s - H R h C l ( P R ) t o c i s - H R h C l ( P R > presum­ a b l y through p s e u d o r o t a t i o n . 2

PPh

Rri j PPh

2

2

I

Η

.

3

3

2

H

3

I .,Λ CI

3

Η —

,Λ ™3 Ρ

Rh. j ^PPh CI

(2) 3

Then, the o l e f i n i n s e r t i o n and the r e d u c t i v e e l i m i n a t i o n take p l a c e from the r e s u l t a n t c i s b i p h o s p h i n e complex. I n t h i s a r t i c l e , we w i l l compare the e n e r g e t i c s o f the 'conven­ t i o n a l ' H a l p e m mechanism w i t h t h a t o f the Brown mechanism. The b a s i s f u n c t i o n s u s e d a r e the 3-21G f o r e t h y l e n e and h y d r i d e s , the ST0-2G f o r ' s p e c t a t o r ' l i g a n d s , PH^ and CI, and v a l e n c e double z e t a b a s i s f u n c t i o n s f o r Rh w i t h e f f e c t i v e c o r e p o t e n t i a l r e p l a c i n g the c o r e e l e c t r o n s (up t o 4p) (5a.b.6). I n a d d i t i o n , we c a r r i e d out the MP2 c a l c u l a t i o n s a t s e l e c t e d , R H F - o p t i m i z e d s t r u c t u r e s w i t h a l a r g e r b a s i s s e t , which c o n s i s t s o f u n c o n t r a c t e d (3s,3p,4d) f u n c t i o n s from the above v a l e n c e DZ s e t f o r Rh, 4-31G f o r the e t h y l group, (10s,7p)/[3s2p] f o r Ρ and CI, and ( 4 s ) / [ 3 s ] f o r the h y d r i d e s (5ce ) . The b a s i s s e t u s e d i s l i m i t e d and the e l e c t r o n c o r r e l a t i o n t a k e n i n t o account f o r a few c r i t i c a l s t e p s i s m i n i m a l . T h e r e f o r e ,

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

79

THE CHALLENGE OF d AND f ELECTRONS

C H 2

6

A*-* Reductive e l i m i n a t i o n ^ / ^

J. Rh—Cl ^

H

2

I Oxidati

CH -Rh-Cl 2

H-CH

2

L

t

H 5b

l^-CH^CHz

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

Isomerization

Olefin coordination

g

L u Olefin insertion

Scheme 2

Scheme 3

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

6.

KOGA & MOROKUMA

Potential Energy Surjhce of Olefin Hydrogénation

the r e s u l t s p r e s e n t e d quantitative .

h e r e s h o u l d be

considered

t o be

semi-

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

H a l p e m mechanism S i n c e our c a l c u l a t i o n s on the H a l p e m mechanism have been p u b l i s h e d (2)» we w i l l g i v e a b r i e f summary f o r comparison i n a s u c c e e d i n g s e c t i o n . The p o t e n t i a l energy p r o f i l e shown i n F i g u r e 1 i s cons t r u c t e d from the e n e r g e t i c s o f the e l e m e n t a r y r e a c t i o n s i n v o l v e d i n the H a l p e m mechanism. The o p t i m i z e d s t r u c t u r e s a r e shown i n F i g u r e 2. The f i r s t s t e p o f the H^ o x i d a t i v e a d d i t i o n i s e x o t h e r m i c and l e a d s t o the d i h y d r i d e complex 3 . D u r i n g t h i s s t e p , t h e r e may be an H^ complex 2 from which o x i d a t i v e a d d i t i o n t a k e s p l a c e w i t h a l m o s t no a c t i v a t i o n b a r r i e r . The ethylene c o o r d i n a t i o n that follows r e q u i r e s no a c t i v a t i o n energy. The r e s u l t a n t e t h y l e n e d i h y d r i d e complex 4 i s i n the v a l l e y o f the p o t e n t i a l energy s u r f a c e o f the c a t a l y t i c c y c l e . E t h y l e n e i n s e r t i o n r e q u i r e s a much h i g h e r a c t i v a t i o n energy o f 18 k c a l / m o l and i s endothermic by 16 k c a l / m o l a t the RHF l e v e l . The t r a n s e t h y l h y d r i d e complex, the d i r e c t p r o d u c t o f e t h y l e n e i n s e r t i o n , i s u n s t a b l e due t o the c i s e f f e c t o f CI t o be mentioned below and the t r a n s e f f e c t o f H and o 5 ' T h e r e f o r e i s o m e r i z a t i o n takes p l a c e t o g i v e more s t a b l e e t h y l h y d r i d e comp l e x e s , w h i c h have e t h y l and h y d r i d e c i s t o each o t h e r and a r e the s t a r t i n g p o i n t o f the f i n a l reductive e l i m i n a t i o n step. This i s o m e r i z a t i o n p r o c e e d s t h r o u g h h y d r i d e and c h l o r i d e m i g r a t i o n . The f i n a l r e d u c t i v e e l i m i n a t i o n step r e q u i r e s a s u b s t a n t i a l energy b a r r i e r o f 15 k c a l / m o l . The p o t e n t i a l energy p r o f i l e i s smooth w i t h o u t e x c e s s i v e b a r r i e r s and too s t a b l e i n t e r m e d i a t e s which would b r e a k the sequence o f s t e p s . The r a t e - d e t e r m i n i n g s t e p i s found t o be o l e f i n i n s e r t i o n f o l l o w e d by i s o m e r i z a t i o n , s u p p o r t i n g the H a l p e m mechanism. I s o m e r i z a t i o n o f the e t h y l h y d r i d e complex i s an i m p o r t a n t p a r t o f the r a t e - d e t e r m i n i n g s t e p . These two r e a c t i o n s , e x o t h e r m i c o v e r a l l , has an o v e r a l l b a r r i e r h e i g h t o f about 20 k c a l / m o l . The t r a n s e t h y l h y d r i d e complex, the p r o d u c t o f e t h y l e n e i n s e r t i o n , may n o t be a l o c a l minimum (per MP2 c a l c u l a t i o n ) and t h e s e two s t e p s may w e l l be a combined s i n g l e s t e p . The a c t i v a t i o n b a r r i e r o f r e d u c t i v e e l i m i n a t i o n , though s u b s t a n t i a l , i s s m a l l e r t h a n t h a t o f the r e v e r s e o f the r a t e d e t e r m i n i n g s t e p ( i s o m e r i z a t i o n and 0-hydrogen e l i m i n a t i o n ) . T h i s i s a v e r y i m p o r t a n t r e q u i r e m e n t o f a good o l e f i n hydrogénation catalyst. I f t h i s r e v e r s e r e a c t i o n i s easy, i t would l e a d t o u n d e s i r a b l e o l e f i n i s o m e r i z a t i o n . F o r i n s t a n c e , i n the same hydrogén a t i o n c y c l e c a t a l y z e d by a Pt system, we f o u n d t h a t the r a t e determining step i s r e d u c t i v e e l i m i n a t i o n r a t h e r than o l e f i n i n s e r t i o n . T h i s p o t e n t i a l energy p r o f i l e i s e x p e c t e d t o g i v e o l e f i n i s o m e r i z a t i o n through s u c c e s s i v e o l e f i n i n s e r t i o n / ^ - h y d r o g e n e l i m i n a t i o n . Thus the Pt complex i s n o t a good c a t a l y s t f o r o l e f i n hydrogénation. On the o t h e r hand, the p o t e n t i a l p r o f i l e o f the W i l k i n s o n c a t a l y s t i n d i c a t e s an e f f i c i e n t hydrogénation w i t h o u t isomerization. c

H

I t i s i m p o r t a n t , as mentioned above, t h a t the o l e f i n i n s e r t i o n i s r a t e - d e t e r m i n i n g . T h e r e f o r e , we have compared the r e a c t i o n e n e r g e t i c s between H R h C l ( P H ) ( C H ) and H R h H ( P H ) ( C H ) , and 2

3

2

2

4

2

3

2

2

4

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

81

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

-40

-30

-20 -

-10

0

kcal/mole|

Olefin insertion

Isomerization

Reductive elimination

F i g u r e 1. P o t e n t i a l energy p r o f i l e o f the e n t i r e c a t a l y t i c c y c l e i n the H a l p e m mechanism f o r o l e f i n h y d r o g é n a t i o n , i n k c a l / m o l a t the RHF l e v e l , r e l a t i v e t o l+C^H^+H^. Numbers i n p a r e n t h e s e s a r e the MP2 energy a t the RHF o p t i m i z e d g e o m e t r i e s , r e l a t i v e t o 4 .

coordination

Oxidative addition Olefin

TS(5b+1)

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

ο

00

KOGA & MOROKUMA

Potential Energy Surface of Olefin Hydrogénation

H-? _. 2-280 I P s t-tuw 0-8631

H ;

A6

1

164-8

1.514 \ 2.302 ^Rhs—CI \-i ^— 145.5 M

H

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

Π

^ C = C ^ H

H ,

2.600 χ 1 2-575

n

2-282

5 7 2

^/.:R[>—Cl 160-8 TS(2-3)

W 2 C

/

« * ;

^ f L ^ R f t — CI 83.ΐΊ^925 2·3Α2 Η,1.504

/

^ t e o

Π

H1-587

TS(4-5)

H 1.081

1 1 8 1

Ha Hi

'l547

| 79-9\ 2.137

H

91>/W1-621

2 7 2

N

H

1026* 2.O6I 17^17^

' · / 115.3 H 751

H

H

2-247

V 2.095^-:

2

87-3 Α - Ο 0 2 Ό u^i-Rh^—a n

1.558

1.553/Η^λ®:- α VH

HU72 5

b

2.313

5a

1 H

Η

Λ

"7.2^\ .OOS Îoœ

TS(5+5a)

5

H

Γ1539

C 2-185 _165.1

4 ? ^ ^ —

CI

1-524 TS(5b+1)

F i g u r e 2. O p t i m i z e d s t r u c t u r e s ( i n À and deg) o f some i m p o r t a n t s p e c i e s . TS(2->3), f o r i n s t a n c e , denotes t h e t r a n s i t i o n s t a t e c o n n e c t i n g 2 and 3. Though p r a c t i c a l l y a l l t h e g e o m e t r i c a l p a r a m e t e r s were o p t i m i z e d , o n l y e s s e n t i a l v a l u e s a r e shown. Two PH3's, one above and one below the p l a n e o f paper, a r e o m i t t e d for c l a r i t y .

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

84

f o u n d t h a t , i n the former complex, the weak t r a n s i n f l u e n c e o f CI makes the Rh-H bond t o be b r o k e n s t r o n g e r . I n a d d i t i o n the s t r o n g CI c i s e f f e c t makes the Rh-C bond t o be formed weaker, resulting in e n d o t h e r m i c e t h y l e n e i n s e r t i o n . These two e f f e c t s o f CI combined appears t o be e s s e n t i a l t o make e t h y l e n e i n s e r t i o n the r a t e determining step. firown, mechanism The f i r s t p o i n t o f d i f f e r e n c e o f the Brown mechanism from the H a l p e r n mechanism i s i s o m e r i z a t i o n o f H R h C l ( P H ~ ) . T h e r e f o r e , we have i n v e s t i g a t e d the s t a b i l i t y o f isomers o f R R h C l ( P H ) , 3. The o p t i m i z e d s t r u c t u r e s o f H R h C l ( P H ~ ) a r e shown i n F i g u r e 3. The most s t a b l e isomer i s found t o be 3, the t r a n s i n t e r m e d i a t e o f the H a l p e r n mechanism. A l l the o p t i m i z e d s t r u c t u r e s b u t 3 a r e n e a r l y square p y r a m i d a l (though o p t i m i z a t i o n was done w i t h o u t symmetry restriction). The most s t a b l e square p y r a m i d a l isomer i s 3a w i t h a p i c a l H and b a s a l c i s p h o s p h i n e s . 3b w i t h a p i c a l p h o s p h i n e and b a s a l c i s h y d r i d e s i s n e x t . The r e m a i n i n g t h r e e i s o m e r s , 3c, 3d, and 3e a r e much more u n s t a b l e ; the e n e r g i e s r e l a t i v e t o 3 a r e 33, 37 and 39 k c a l / m o l , r e s p e c t i v e l y . Two h y d r i d e s w i t h s t r o n g t r a n s i n f l u e n c e a r e l o c a t e d t r a n s t o each o t h e r i n 3c. T h i s makes 3c 12 k c a l / m o l l e s s s t a b l e t h a n 3b i n which two h y d r i d e s a r e c i s . The l e a s t s t a b l e i s o m e r s , 3d and 3e, have a p i c a l CI. Comparison o f the s t a b i l i t y among the isomers o f 3 l e a d s t o the o r d e r o f a p i c a l p r e f e r e n c e : H>PH^>C1. H w i t h the s t r o n g e s t t r a n s i n f l u e n c e p r e f e r s the a p i c a l p o s i t i o n t h a t i s t r a n s t o the v a c a n t s i t e , and the most weakly t r a n s - i n f l u e n c i n g CI a t the a p i c a l position g i v e s the most u n s t a b l e isomers o f 3d and 3e. 3b and 3c a r e inbetween, i n a c c o r d w i t h the s t r e n g t h o f PH~ t r a n s i n f l u e n c e . S i n c e 3a and 3b a r e low i n energy, we have i n v e s t i g a t e d i s o m e r i z a t i o n from 3 t o 3a and 3b. Brown e t a l . have p r o p o s e d t h a t the c i s i n t e r m e d i a t e o f the c a t a l y t i c c y c l e i s 3b i n w h i c h one o f the b u l k y p h o s p h i n e s i s t r a n s t o o l e f i n and thus the v a c a n t c o o r d i n a t i o n s i t e i s l e s s crowded. There a r e two p o s s i b l e pathways f o r i s o m e r i z a t i o n o f 3 as shown i n Scheme 4. The i n t e r m e d i a t e s o f the second pathway a r e u n s t a b l e 3d and 3c, and i t i s u n l i k e l y t h a t i s o m e r i z a t i o n t a k e s p l a c e t h r o u g h them. The e a s i e r i s o m e r i z a t i o n pathway i s through 3b t o 3a. The t r a n s i t i o n s t a t e f o r PH^ m i g r a t i o n c o n n e c t i n g 3 and 3b have been l o c a t e d , as shown i n F i g u r e 3, w i t h the a c t i v a t i o n b a r r i e r f o r i s o m e r i z a t i o n from 3 t o 3b o f 27 k c a l / m o l (See Scheme 4 ) . T h e r e f o r e , one can c o n c l u d e i s o m e r i z a t i o n from 3 t o 3b o r 3a i s r a t h e r d i f f i c u l t ( c f . 18 k c a l / m o l , the a c t i v a t i o n energy o f the r a t e - d e t e r m i n i n g s t e p i n the H a l p e r n mechanism a t the same l e v e l o f c a l c u l a t i o n ) . S e t t i n g a s i d e t h i s h i g h a c t i v a t i o n b a r r i e r f o r a moment, the r e m a i n i n g s t e p s o f the c a t a l y t i c c y c l e i n the Brown mechanism w i l l be d i s c u s s e d . As shown i n the energy p r o f i l e o f the H a l p e r n mechanism, the e l e m e n t a r y r e a c t i o n s i n v o l v e d h e r e a r e e x p e c t e d t o be v e r y easy, and thus we have j u s t d e t e r m i n e d the s t r u c t u r e s and energies of intermediates but not of t r a n s i t i o n s t a t e s (Figure 5). The r e l a t i v e e n e r g i e s o f the i n t e r m e d i a t e s , shown i n F i g u r e 4, would be a good i n d i c a t o r o f the b a r r i e r o f each e l e m e n t a r y r e a c t i o n c o n n e c t i n g them; an endothermic r e a c t i o n r e q u i r e s a l a r g e a c t i v a t i o n energy and the a c t i v a t i o n b a r r i e r o f an e x o t h e r m i c s t e p i s low. 2

2

2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

2

3

2

2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Potential Energy Surface of Olefin Hydrogénation

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

KOGA & MOROKUMA

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

THE CHALLENGE OF d AND f ELECTRONS

F i g u r e 3. O p t i m i z e d s t r u c t u r e s ( i n  and deg) o f isomers o f H ^ R h C l i P H ^ ) ^ and the 3-*3b t r a n s i t i o n s t a t e , and t h e i r e n e r g i e s ( i n k c a l / m o l ) r e l a t i v e t o 3.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

H

2

9

0

complex isomerization

Olefin coordination

Dihydride

olefin insertion

Isomerization

complex

Ethyl hydride

Reductive elimination

F i g u r e 4. P o t e n t i a l energy p r o f i l e o f t h e c a t a l y t i c c y c l e i n t h e Brown mechanism f o r o l e f i n hydrogénation, i n k c a l / m o l a t t h e RHF l e v e l , r e l a t i v e t o 1+C H.+H .

Oxidative addition

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

THE CHALLENGE OF d AND f ELECTRONS

F i g u r e 5. O p t i m i z e d s t r u c t u r e s ( i n  and deg) o f some i m p o r t a n t s p e c i e s i n the Brown mechanism.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

6. KOGA & MOROKUMA

Potential Energy Surface of Olefin Hydrogénation

E t h y l e n e c o o r d i n a t i o n t o 3b, t h e Brown's i n t e r m e d i a t e , g i v e s 4a w h i c h i s h i g h e r i n energy t h a n 4 by 6 k c a l / m o l . Olefin insertion o f 4a c a n l e a d t o 5c o r 5d. S i n c e 5c i s much more s t a b l e t h a n 5d, o l e f i n i n s e r t i o n g i v i n g 5c would take p l a c e e x c l u s i v e l y . The i n s t a b i l i t y o f 5d w i t h an a p i c a l CI i s s i m i l a r t o t h a t o f 3d and 3e d i s c u s s e d above. The e t h y l group and t h e h y d r i d e i n 5c a r e c i s t o each o t h e r and thus r e d u c t i v e e l i m i n a t i o n might take p l a c e d i r e c t l y w i t h o u t i s o m e r i z a t i o n . However, r e d u c t i v e e l i m i n a t i o n from a d fivec o o r d i n a t e complex would f a v o r a t r a n s i t i o n s t a t e where t h r e e l i g a n d s b u t C H,. and H a r e i n the same p l a n e , as shown below.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

9

\

L

The r e a s o n f o r t h i s p r e f e r e n c e i s t h a t t h e d o n a t i o n and back d o n a t i o n between a deformed a l k a n e and a m e t a l fragment shown below i s e x p e c t e d t o f a c i l i t a t e easy bond exchange.

C (

^DËO ® ®

c l

donation

back-donation

Therefore, p r i o r to reductive e l i m i n a t i o n , i s o m e r i z a t i o n should t a k e p l a c e from 5c t o e t h y l h y d r i d e complexes w h i c h have H o r C^H, as an a p i c a l group, as shown below.

Et

Rh-

H

Rh-

y Et E t h y l m i g r a t i o n from 5c l e a d s t o 5c i t s e l f , and h y d r i d e m i g r a t i o n g i v e s 5d, an u n s t a b l e i n t e r m e d i a t e ; n e i t h e r o f t h e s e g i v e s a p i c a l H o r C^H^. The two r e m a i n i n g m i g r a t i o n s g i v e a more s t a b l e e t h y l h y d r i d e complexes and they have e i t h e r an a p i c a l H o r C^H^; CI m i g r a t i o n l e a d s t o s t a b l e 5e and PH~ m i g r a t i o n r e s u l t s i n 5a w i t h t r a n s p h o s p h i n e s , the i n t e r m e d i a t e or t h e H a l p e r n mechanism. The t r a n s i t i o n s t a t e f o r r e d u c t i v e e l i m i n a t i o n o f 5e t o g i v e l a has been determined, as shown i n F i g u r e 5, and i t has an a c t i v a t i o n energy o f 14.8 k c a l / m o l . 5a i s more s t a b l e t h a n 5e by 5 k c a l / m o l and the a c t i v a t i o n energy f o r r e d u c t i v e e l i m i n a t i o n from 5a t o r e g e n e r a t e 1 i s c a l c u l a t e d t o be about 15 k c a l / m o l , comparable w i t h t h e 5e-+la b a r r i e r . T h e r e f o r e , the system i s e x p e c t e d t o r e t u r n t o t h e

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

89

90

THE CHALLENGE OF d AND f ELECTRONS

H a l p e r n mechanism, i f i t i s n o t p r e v e n t e d f o r some r e a s o n steric).

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

Comparison between two

(eg.

mechanisms

I n the Brown mechanism, s e t t i n g a s i d e the h i g h energy r e q u i r e d f o r i s o m e r i z a t i o n from 3 t o 3b, the f i n a l s t e p o f the r e d u c t i v e e l i m i n a t i o n would r e q u i r e the h i g h e s t a c t i v a t i o n energy. O l e f i n i n s e r t i o n , the r a t e - d e t e r m i n i n g s t e p i n the H a l p e r n mechanism, i s an easy p r o c e s s i n t h i s mechanism. The k i n e t i c s o f the Brown mechanism i s thus e x p e c t e d t o be c o m p l e t e l y d i f f e r e n t from t h a t o f t h e H a l p e r n mechanism. T h e r e f o r e , i n the c a s e s i n which o l e f i n i n s e r t i o n has been f o u n d t o be r a t e - d e t e r m i n i n g , the H a l p e r n mechanism i s c l e a r l y more c o n s i s t e n t and a c c e p t a b l e . The above f e a t u r e o f the Brown mechanism t h a t r e d u c t i v e e l i m i n a t i o n i s more d i f f i c u l t t h a n o l e f i n i n s e r t i o n may be r e l a t e d t o the n a t u r e o f c a t a l y s t h a v i n g a c h e l a t i n g b i d e n t a t e l i g a n d such as DIPHOS. H a l p e r n have a l s o i n v e s t i g a t e d the hydrogénation ( E q u a t i o n 3) ( 2 ) , where i s o m e r i z a t i o n from the t r a n s - t o c i s b i p h o s p h i n e complex i s n o t n e c e s s a r y .

( s )

Rh' *

S ( S )

ï + COOCH, > PnCH,CH

(3)

NHCOCHj

They have f o u n d t h a t a t the low temperature, r e d u c t i v e e l i m i n a t i o n i s r a t e - l i m i t i n g (ΔΗ -17.0 k c a l / m o l - 4 0 ° C ) . The p r e s e n t c a l c u l a t i o n s u s i n g PH^ as the phosphine and e t h y l e n e as the o l e f i n , however, does n o t e x c l u d e the p o s s i b i l i t y o f the c i s mechanism c o m p l e t e l y , s i n c e the s t e r i c f a c t o r has n o t been t a k e n i n t o a c c o u n t . The c i s mechanism might be a c c e s s i b l e i n the c a s e where o l e f i n i s too b u l k y t o c o o r d i n a t e t o the t r a n s phosphine complex. U s i n g the above c a l c u l a t i o n s as a g u i d e , h e r e we c o n s i d e r q u a l i t a t i v e l y what i s e x p e c t e d t o take p l a c e when a v e r y b u l k y o l e f i n i s h y d r o g e n a t e d . L e t us assume t h a t i n such a c a s e the e q u i l i b r i u m ( E q u a t i o n 1) g e n e r a t e s the c i s i n t e r m e d i a t e . Then, the r e a c t i o n r o u t e w i l l pass t h r o u g h 3b-+4a-»5c, each o f w h i c h i s s t e r i c a l l y n o t too crowded. I s o m e r i z a t i o n o f 5c i s n o t a l l o w e d t o l e a d t o 5a, w h i c h has the b u l k y a l k y l group c i s t o two p h o s p h i n e s and i s overcrowded. Thus i s o m e r i z a t i o n o f 5c has t o l e a d t o 5e, w h i c h i s i n t r i n s i c a l l y (without s t e r i c e f f e c t ) only s l i g h t l y l e s s s t a b l e but s t e r i c a l l y l e s s crowded t h a n 5a. R e d u c t i v e e l i m i n a t i o n o f 5e w i l l r e q u i r e an a c t i v a t i o n energy comparable t o t h a t o f 5a->l and g e n e r a t e cis-RhCl(PH ) , l a . 3

2

There a r e two p o s s i b i l i t i e s i n the r e a c t i o n s o f l a . The f i r s t i s t h a t l a i s o m e r i z e s t o 1 due t o the s t e r i c r e p u l s i o n and t h a t the same r e a c t i o n p a t h l-*2-*3->3b i s f o l l o w e d . The second p o s s i b i l i t y i s t h a t H^ o x i d a t i v e a d d i t i o n t o l a t a k e s p l a c e t o g i v e d i r e c t l y 3a and thus i n the subsequent c a t a l y t i c c y c l e s i n t e r m e d i a t e s always have c i s p h o s p h i n e s . O l e f i n c o o r d i n a t i o n t o 3a i s p r o h i b i t e d b e c a u s e o f the s t e r i c r e p u l s i o n between the b u l k y o l e f i n and two b u l k y phos­ p h i n e s c i s t o the o l e f i n . Thus 3a-»3b i s o m e r i z a t i o n has t o take p l a c e b e f o r e the c a t a l y t i c c y c l e p r o c e e d s .

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

6. KOGA & MOROKUMA

Potential Energy Surface of Olefin Hydrogénation

91

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch006

One can consider that the Halpern mechanism and the cis mech­ anism are two extremes. The Halpern mechanism is most widely accepted and our ab initio MO calculations support this from the point of view of intrinsic electronic energy. However, there may be cases where the steric effect overshadows the electronic effect. There may also be cases where both effects are important. It may be, as Collman et al. said, that "this multistep reaction is very complicated. Like a chameleon, the dominant reaction mechanism changes when the nature of the catalyst, the ligands, or the substrate is altered" (8). Conclusions In this work, we have compared the potential energy profiles of the model catalytic cycle of olefin hydrogénation by the Wilkinson catalyst between the Halpern and the Brown mechanisms. The former is a well-accepted mechanism in which all the intermediates have trans phosphines, while in the latter, proposed very recently, phosphines are located cis to each other to reduce the steric repulsion between bulky olefin and phosphines. Our ab initio calculations on a sterically unhindered model catalytic cycle have shown that the profile for the Halpern mechanism is smooth without too stable intermediates and too high activation barrier. On the other hand, the key cis dihydride intermediate in the cis mechanism is electron­ ically unstable and normally the sequence of elementary reactions would be broken. Possible sequences of reactions can be proposed from our calculation, if one assumes that steric effects of bulky olefin substituents prohibits some intermediates or reactions to be realized. Literature Cited. 1. Koga, N.; Morokuma, K. Tod.Phys.Organomet.Chem., in press. 2. (a) Koga, N.; Daniel, C.; Han, J.; Fu, X.Y.; Morokuma, K. J.Am.Chem.Soc., 1987, 109, 3455. (b) Daniel, C.; Koga, N.; Han, J.; Fu, X.Y.; Morokuma, K. J.Am.Chem.Soc., 1988, 110, 3773. 3. (a) Halpern, J.; Wong, C.S. J.Chem.Soc.Chem.Commun., 1973, 629. (b) Halpern, J. In Organotransition Metal Chemistry: Ishii, Y.; Tsutsui, M., Eds.; Plenum: New York, 1975; p109. (c) Halpern, J.; Okamoto, T.; Zakhariev, A. J.Mol.Catal., 1976, 2, 65. 4. Brown, J.M.; Evans, P.L.; Lucy, A.R. J . Chem. Soc. Perkin Trans. II, 1987, 1589. 5. (a) Binkley, J.S.; Pople, J.A.; Hehre, W.J. J.Am.Chem.Soc., 1980, 102, 939. (b) Hehre, W.J.; Stewart, R.F.; Pople, J.A. J.Chem.Phys., 1969, 51, 2657. (c) Ditchfield, R.; Hehre, W.J.; Pople, J.A. J.Chem. Phys., 1971, 54, 724. (d) Huzinaga, S.; Andzelm, J.; Kłobukowski, M.; Radzio-Andzelm, E.; Sakai, Y.; Tatewaki, H. Gaussian Basis Sets for Molecular Calculations: Elsevier: Amsterdam, 1984. 6. Hay, P.J.; Wadt, W.R. J . Chem. Phys., 1985, 82, 270. 7. Chan, A.S.C.; Halpern, J . J.Am.Chem.Soc.. 1980, 102, 838. 8. Collman, J.P.; Hegedus, L.S.; Norton J.R.; Finke, R.G. Principles and Applications of Organotransition Metal Chemistry: University Science Books: Mill Valley, 1987; p.535. RECEIVED

January 18, 1989

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 7

Ab Initio Studies of Transition Metal Dihydrogen Chemistry Edward M . Kober and P. Jeffrey Hay

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

Los Alamos National Laboratory, Los Alamos, NM 87545

Examples of transition metal complexes containing dihydrogen ligands are investigated using ab initio electronic structure calculations employing effective core potentials. Calculated geometrical structures and relative energies of various forms of WL (H ) complexes (L = CO, PR ) are reported, and the influence of the ligand on the relative stabilities of the dihydrogen and dihydride forms is studied. The possible intramolecular mechanisms for H - D scrambling involving Cr(CO) (H ) are investigated by examining the structures and energies of various polyhydride species. 5

2

3

4

2 2

The recent discoveries of a new class of metal complexes involving molecular hydrogen has spawned numerous experimental(l,2) and theoretical (2—5) investigations to understand the bonding and reactivity of these systems. In this paper we discuss recent theoretical calculations using ab initio electronic structure techniques. In the first part of this article we review calculations on W dihydrogen species of d W complexes with emphasis on predictions of structures and energies of various chemical forms and on comparisons with available experimental information. In the second part a mechanistic problem involving scrambling of H / D mixtures to HD by Cr dihydrogen complexes is addressed. In this section we hope to illustrate the role of for theory in helping to distinguish between various reaction mechanisms in transition metal chemistry. Before proceeding to the specific examples of dihydroge ?hfmistry, it is worthwhile to summarize the particular challenges transition metal and actinide compounds present to this type of approach. There is a striking contrast to most compounds of first— and second—row main—group elements where reasonably accurate bond lengths and bond ancles of stable species can be predicted at the SCF Hartree-Fock level with small basis sets and thermochemical quantities can be computed with reasonable accuracy by perturbative techniques to electron correlation (fi). Accurate studies of molecular properties or transition states of chemical reactions are feasible using multi-configuration SCF ( M C - S C F ) and configuration interaction (CI) techniques. In contrast, for transition metal compounds one encounters a e

2

2

0097-6156/89/0394-O092$06.00/0 c 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

7. KOBER & HAY

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Transition Metal Dihydrogen Chemistry

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

situation where metal ligand distances are predicted to be 0.05-0.25 Â too long at the SCF level of calculation even with very accurate basis sets (Ζ-1Ω) and such approaches as Moller—Plesset perturbation theory can be unreliable for treating electron correlation effects (11,12). The relatively poor description of the ground state by a single configuration and the presence of numerous low-lying electronic states with very different electron correlation energies often requires sophisticated M C - S C F and CI treatments to obtain reliable molecular geometries and bond energies (13). In addition there are additional challenges arising from the sheer number of electrons in transition metal compounds and the relativistic effects which become increasingly important in second— and third—transition series compounds. These latter difficulties have been largely circumvented by the development of relativistic effective core potentials(li) to replace the chemically inert core electrons and to incorporate the relativistic effects on the valence electrons into the effective potential.

Tungsten Dihydrogen Complexes In the molecular dihydrogen species W(CO)3(PR3)2(H2) first characterized by Kubas et al. in both X - r a y and neutron diffraction studies (1,2), the dihydrogen is bonded in τ/ sideways fashion to the d metal center. Over 50 compounds involving many of the transition metals have since been synthesized by various workers. In addition, reanalysis(15) of existing hydrides such as FeH4(PR ) have now been found to be formulated as molecular hydrogen complexes, i.e., as Fe(H )(H)2(PR3)3. In this section we review briefly our previous theoretical calculations(5) on two prototypical dihydrogen complexes W(CO)3(PH ) (H ) and W f P H s M ^ ) . The calculations employed a relativistic effective core potential ( E C P - 1 ) to replace the inner [Xe] (4f ) core on W and a nonrelativistic E C P on Ρ with a flexible gaussian basis to describe the valence electrons of the system. Details of the calculation are given in Ref. 5. Structures 1—3 in Fig. 1 exemplify the modes of H bonding to a W ( C O ) ( P H ) fragment: two sideways bonded (rçS-coordinated) forms (1 and 2) and the end-on bonded (^Coordinated) form (3). Using a rigid W ( C O ) ( P H ^ fragment, the geometries of these three forms of H coordination nave been optimized using Hartree—Fock wave functions. The sideways bonded species (Table I) are found to be stable with respect to the fragments 4 by 17 kcal/mol and more stable than the end-on form, which is bound by only 10 kcal/mol. Little difference in energy is observed between the two-sideways bonded form with the H axis parallel either to the P - W - P axis or to the C—W-C axis. The former orientation is slightly favored, leading to a rotational barrier about the midpoint of the W - H bond of 0.3 kcal/mol. The calculated structure of the lower energy η form (1) shows a slight lengthening (from 0.74 to 0.796 À) of the H - H bond from uncomplexed H with a W—Η distance of 2.15 A . Recent low—temperature neutron diffraction studies show two equal W—Η bonds (1.89 ± 0.01 Â) and with the H lying exactly parallel to the P—W-P axis as predicted by the present calculations and having a Η—Η separation of 0.82 ± 0.01 Â. Although the calculations have correctly described the preferred mode of H binding, there remain some quantitative differences (Table I) between the theoretical and observed bond lengths. These differences are reduced considerably when one employs an effective core potential ( E C P - 2 ) which also treats the outermost 5s and 5p core orbitals of W as valence orbitals. The W - H and Η—Η distances are now calculated to be 1.93 and 0.81 Â, respectively, in much better agreement with the neutron diffraction results. 2

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94

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

THE CHALLENGE OF d AND f ELECTRONS

Fig. 1.

Structural forms of W(CO) (PR3)2(H ) species. 3

2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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95

Transition Metal Dihydrogen Chemistry

Table I. Calculated and Experimental W - H and H - H Bond Lengths and Rotational Barriers for W(CO)3(PR3)2(H ) Species

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

2

Method

R

R(W-H),À

R(H-H),A

Rotational barrier, kcal/mole

SCFcalc. ( E C P - 1 on W)

H

2.153

0.796

0.3

SCFcalc ( E C P - 2 on W)

H

1.932

0.806

1.3

Low-temp X - r a y diffraction

i-Pr

1.95*0.23

0.75*0.16

Low—temp i-Pr neutron diffraction

1.89*0.01

0.82*0.01

2.4

Examination of the Mulliken population analyses for the fragment and the η complex reveals an overall increase of 0.12 e on the W atom upon complexation and the total charge on each hydrogen has decreased slightly from 1.00 to 0.98 e. The σ-bonding orbitals of the W atom (6s, 6p , and 5 d | show a net increase of 0.13 e, while the τ—bonding orbit sus (6p and 5d ) undergo a net loss of 0.03 e. Although the other five ligands also influence the amount of charge on the metal, the above trends are consistent with a mechanism involving some σ-donation from the H2 ligand and a lesser degree of π—back—donation from the metal. Of the possible seven-coordinate dihydrides let us consider the least motion reaction in which the two W—L bonds originally parallel to the H—H axis bend back as two W - H bonds are formed. The energies of these species are compared with the r/S-dihydrogen forms and the fragments in Fig. 2. Both seven-coordinate dihydride species lie higher in energy than the dihydrogen from (17 and 11 kcal/mol, respectively) and are only slightly bound compared to W L + H . For the case of having all P H ligands in the W ( P H ) ( H ) complexes, a much different situation prevails concerning the oxidative addition reaction. In contrast to W ( C O W P H ) ( H ) , the seven-coordinate dihydride W ( P H ) ( H ) lies 3 kcal/mol below the η complex! Replacing the CO ligands by PR3 groups favors the oxidative addition reaction proceeding to completion rather than being arrested in the ^dihydrogen stage. This preference for ^Coordination in W(CO)3(PH ) (H) correlates with the overall stabilization of the 5d orbitals, and the 5 d orbital in particular, by the back—bonding CO ligands. When these ligands are replaced by the less stabilizing PH3 groups, the dihydride is the most favored form. This is consistent with experimental observations that the dihydrogen-dihydride equilibrium can be shifted depending on the basicity of the ligands. For example, in a series of complexes, M o ( C O ) ( R P C H P R ) H , the coordination mode changes from dihydrogen from R = F h to dihydride for the more basic R=Et(2). 2

2

z

y

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xz

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In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

z

yz

THE CHALLENGE OF d AND f ELECTRONS

V

I/'

o OC

XJ

I

°0. oc υ

H

oc

w

o

20

W + H 1

y

0°

L

ù

(3 (ô) FRAGMENTS L=PR

(0)

LU

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

(-7)

O υ LU CE

3

(-17)

7] -DIHYDROGEN COMPLEX 2

OCo o DIHYDRIDE COMPLEXES

1

I/

-W + H, υ

20

H

ce

o

(0)

LU Ζ LU

S-

20

(-19)

(-21)

(-33) L

FRAGMENTS L-PRo

η DIHYDROGEN COMPLEX 2

DIHYDRIDE COMPLEXES Fig. 2

Relative energies in kcal/mole of W ( C O ) ( P H ) 2 ( H 2 ) species (above) compared to W ( P H ) ( H ) species (below). 3

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In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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Transition Metal Dihydrogen Chemistry

H - D Exchange Involving CrfCO^Dihydrogen Species Background. The gas phase reaction H

+ D -ι 2HD

2

2

occurs only under severe conditions such as in shock tubes with an activation energy (~100 kcal/mole) comparable to the H—H bond energy. In fact, the kinetics have been interpreted in terms of a free radical mechanism involving H atoms rather than the bimolecular process indicated in the above equation. By contrast, several cases of facile H2—D exchange have been observed under thermal or low temperature conditions involving dihydrogen complexes 2

L M H + D -» L MHD + HD η ζ ζ η Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

0

0

Upmacis, Poliakoff and Turner(l£) observed HD exchange for mixtures of Cr(CO)4(H )2 and D but interestingly nol for mixtures ot Cr(CO)5(H ) and D . Kubas et al observe a similar phenomenon where HD is produced from reacting W(CO)3(PR )2(H ) with D either in solution or in the solid state. In addition other cases of H - D scrambling occur readily with metal hydride complexes(U) as in the case of Cp*ScH or Cp2ZrH where Cp* = 2

2

2

2

3

2

2

CsMe . Since the work of Upmacis et al. on Cr(CC>4)(H2)2 complexes is suggestive of an intramolecular mechanism, Burdett et α/.(1£) have examined various polyhydride structures as possible intermediates in this process using extended Huckel theory. These studies have led us to pursue ab initio electronic structure calculations of Cr(CO)4(H )2 species and possible mechanisms leading to H2—D exchange. Implicit in these studies is the assumption that there is rapid equilibrium between 5

2

2

Cr(CO) (H ) + D - Cr(CO) (H )(D ) + H 4

2

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2

2

2

which subsequently undergoes intramolecular exchange C r ( C O ) ( H ) ( D ) -, C r ( C O ) ( H D ) 4

2

2

4

2

although this is only inferred from the experimental studies. What is actually observed is Cr(CO) (HD) formation in a mixture of Cr(CO) (D ) and Cr(CO) (D )2 when reacted with H . 5

4

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2

Results of ab initio calculations. The calculations have been carried out on stable structures of Cr(CO)4(H )2 or its fragments at the Hartree—Fock level, where the structures have been optimized using gradient techniques with the modified GAUSSIAN82(I£) or the MESA(2Q) electronic structure codes. An effective core potential was used(H) to replace the [Ne] core of Cr with a [3s 3p 2d] contracted Gaussian basis to describe the 3s, 4s, 3p, 4p and 3d orbitals. A flexible [3s lp] bais was used for hydrogen and an S T O - 3 G basis was employed for C and 0 . (Some results are presented using an unpolarized (3s) hydrogen basis.) Of the possible forms for the parent molecule cis—Cr(CO)4(H2)2 the lowest geometrical structure is found to have the H2 molecules oriented in an 2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

98

THE CHALLENGE OF d AND f ELECTRONS

upright position relative to the equatorial plane (Fig. 3). The structure with both H lieands lying in the equatorial plane is 3.1 kcal/mole higher in energy, corresponding to a rotational barrier of 1.5 kcal/mole for the rotation of one H about the metal-H2 bond. Removal of one H 2 to form Cr(CO)4(H ) requires 15 kcal/mole and removal of the second H requires another 14 kcal/mole. At this level of calculation the dihydrogen-dihydride energies 2

2

2

2

Cr(CO) (H ) - Cr(CO) (H) 4

2

4

2

are comparable with the dihydride lower by 1 kcal/mole. The calculated C r - H and H - H bond lengths are 1.787 and 0.77 A respectively for the upright form. (Fig. 4). Some of the possible polyhydride forms—having either a square H 4 or a H 3 — H " species coordinated to Cr(CO)4—are found to be very high in energy (at least 50 kcal/mole) and hence are unlikely intermediates in the H 2 - D exchange reaction. Another pathway is shown schematically in Fig. 5, where the conversion of the bis—dihydrogen complex to a dihydrogen-dihydride complex. The process is symmetry allowed in the sense that the three relevant orbitals in the equatorial plane, the Cr d . and the two H σ orbitals of the bis dihydrogen species transform into the C r - H σ and H σ bonds of the dihydrogen dihydride species. The diagram is oversimplified in the sense that the orbital characters change qualitatively through the course of the reaction. The d - orbital is empty in the dihydrogen dihydride complex and the H bonding orbital actually descends from higher orbitals in the bis—dihydrogen complex. The dihydride form is calculated to lie 10 kcal/mole higher in energy with a C r - H and C r - H bond lengths of 1.729 and 1.923 Â, respectively. The transition state for the reaction 2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

4

2

2

x

2

y

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x

2

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Cr(CO) (H ) HCr(CO) 4

2

2

4

H (H) 2

2

was located at the SCF level assuming C symmetry and treating the H - C r - H bond angle between the two central Η atoms as the reaction coordinate. A n activation barrier of 24 kcal/mole (see Fig. 6) was found for this process—a relatively low barrier compared to some of the other polyhydride species. A non—C2V—pathway was also investigated where a similar barrier was also found. Because of the electron correlation effects can have a significant effect on calculated energies for chemical reactions, calculations on several of the above C r f C O ) ( H 2 ) species were carried out using configuration interaction (CI) techniques. These particular calculations consisted of single and double excitations (SDCI) with respect to the single Hartree—Fock configuration employed in the above studies. The optimized geometries from the SCF calculations were used for the respective species. No excitations were allowed from the inner 16 orbitals corresponding to the carbon Is, oxygen Is and 2s, and chromium 3s and 3p core orbitals. This resulted in 140,642 spin eigenfunctions in C 2 symmetry for the SDCI calculations. 2 v

4

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In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

KOBER & HAY

Transition Metal Dihydrogen Chemistry

Cr (CO) (H ) 4

2

2

STRUCTURES AND FRAGMENTS E(kcal/mole)

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

....

Λ

I

C rr" '

0.0

-..jA

+3.1

Cr

Cr

+H

+15.1

2

Η Cr^.

+H

+14.4

2

Η

Cr

Fig. 3.

+2H

2

+29.4

Relative energies in kcal/mole of Cr(CO)4(H )2 species. 2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

OPTIMIZED GEOMETRICAL

PARAMETERS

Bond Lengths (A) Cr - H

2

H - H

1.787

1.772

1.923

0.772

0.779

0.764

Cr - H,

1.729

Cr - C,

1.997

1.980

2.084

Cr - C

1.971

1.992

2.077

Ci - 0 ,

1.147

1.148

1.143

ci - o

1.150

1.146

1.142

2

2

2

Bond Anales idea) 46.9

44.9

42.1

2

90.8

93.6

94.2

«3

90.3

88.9

67.4

i

a

tt

. 4.

Calculated structural parameters for Cr(CO)4(H )2 species. 2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

7. KOBER & HAY

Transition Metal Dihydrogen Chemistry

CO

oc

H /

\

101

H

•

Cr Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

/ I V CO

Fig. 5.

OC

1

CO

Η

Correlation diagram for bis—dihydrogen to dihydrogen-dihydride forms of Cr(CO) (H2)2 species. 4

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

102

Fig. 6.

THE CHALLENGE OF d AND f ELECTRONS

Calculated energies from Hartree-Fock (SCF) and configuration interaction (CI) calculations for one possible path for H - D exchange involving Cr(CO)4(H )2. 2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

7. KOBER & HAY

103

Transition Metal Dihydrogen Chemistry Table II. Relative Energies (kcal/mole) for Cr(CO) (H )2 and Related Species 2

4

SCE CrfCOVH )2—upright Cr(CO) (H } —coplanar Cr(CO) (H )—coplanar Cr(CO) H (H ) Cr(CO) H + H 2

4

2

4

4

2

4

2

0.0 3.1b 22.5 29.7 43.3

0.0 3.1* 22.5 11.8 14.4

2

4

2

2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

*E = -532.823307 a.u. bE = -533.416149 a.u.

The CI results are contrasted with the SCF calculations in Fig. 6 and in Table II. The bis-dihydrogen complex C r ( C O ) ( H ) is 27 kcal/mole and 40 kcal/mole more stable, respectively, than the dihydrogen-dihydride species C r ( C O ) H ( H ) and the C r ( C O ) H + H fragment, respectively. These energies compare to 9 and 11 kcal/mole for the respective SCF calculations. The potential energy surface has also changed in that the C r ( C O ) ( H ) intermediate appeared to be a transition state at the SCF level, but actually lies below the energy of the C r ( C O ) ( H ) product at the CI level. Finally we compare the results ot the above ab initio calculations with the earlier extended Huckel theory (EHT) calculations of Burdett et al (IS). In this work, which helped to stimulate our own calculations, the authors emphasized that they were probing general trends and that reliable thermodynamic stabilities of M H species could not be obtained using this method. With these points in mind we compare the relative energies of tetrahydrogen species of C r ( C O ) in Table III. 4

4

2

2

4

2

2

2

2

4

4

2

2

n

4

Table III. Comparison of extended Huckel theory (EHT) results (lfi) with SCF ab initio calculations on C r ( C O ) tetranydrogen species 4

Rel. energy (kcal/mole) EHT SCF

C r ( C O ) fragment 4

ds-dihydrogen dihydrogen-ndihydride dihydride + H dihydrogen + H planar H linear H 3 triangular H 3 tetrahydride square H tetrahedral H

0

2

2

4

4

4

17 22 35 36 86 160

0 12 14 15 23 22 76 65 54

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

4

104

THE CHALLENGE OF d AND f ELECTRONS

While the earlier study did not investigate the H ( H ) species, the other intermediates involving "open " planar H3 and H4 moieties are actually placed at similar energies, while the intermediates involving closed species do not correspond with our findings. In particular, the (H )(H) species previously described as having two—electron triangular H3* and H ' ligands lies at considerably higher energy. There has, in fact, been considerable experimental activity to identify and isolate complexes containing H and H " ligands, but we find no evidence to support these forms for this particular class of complexes. 2

2

3

+

3

3

Discussion of Mechanisms. In summary, the above calculations have identified relatively low-energy (i.e., less than 30 kcal/mole) pathways for the conversion of H 2 - D 2 into HD as exemplified by the dihydrogen—dihydride species originally formed according to the process C r ( C O ) ( H ) ( D ) 1 Cr(CO) (H)(HD)(D) Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

4

2

2

4

as shown in Figs. 5 and 6. Once the incipient HD bond has begun to form, the reaction could proceed further by (a) rotation of HD about the C r - H D bond followed by collapse to Cr(CO) (HD) , (b) dissociation of HD to form Cr(CO)4(H)(D) followed by insertion of H or D , or (c) dissociation of HD followed by collapse to Cr(CO)4(HD), to mention some of the possibilities. Despite the above low-energy pathways discussed above, it is not clear whether they are actually involved in the liquid xenon experiments of Upmacis et al since relatively small barriers must be involved for any process at these temperatures. We would point out that the possibility of radical processes involving the presence of H atoms should also be examined thoroughly before these mechanisms are definitively understood. We observe that scrambling involving coordinated H and H atoms is much more facile than processes involving two H ligands. In cases such as 4

2

2

2

2

2

Cr(H )(H)(H) - C r ( H — H — H ) ( H ) -+ Cr(H)(H )(H) 2

2

and V ( C O ) ( H ) ( H ) -, V ( C O ) ( H — H — H ) -, V ( C O ) ( H ) ( H ) 5

2

5

5

2

the barriers involved in open H intermediates are only 10-13 kcal/mole. A second possibility to be considered is that the Cr(CO)4H dihydride species actually possesses a triplet ground state much lower in energy than the singlet species discussed in Fig. 5 and Table II. Such a species could react with H to form H—atom containing species. In summary, the nature of some likely reaction intermediates involved in HD formation from H and D complexes of Cr(CO)4 have been identified here. Unraveling the further details of the mechanisms involved in these fascinating complexes will require more extensive experimental and theoretical studies. 3

2

2

2

2

Acknowledgment This work was carried out under the auspices of the U.S. Department of Energy.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

7. K O B E R & I I A Y

Transition Metal Dihydrogen Chemistry

105

Literature Cited

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch007

1.

Kubas, G.J; Ryan, R.R.; Swanson, B.I.; Vergamini, P.J.; Wasserman, H. J. Am. Chem. Soc. 1984, 106, 451. (b) Kubas, G.J.; Ryan, R.R.; Wrobleski, D. ibid. 1986, 108, 1339. (c) Kubas, G.J.; Unkefer, G.J.; Swanson, B.I.; Fukishima, E. ibid. 1986, 108, 7000. 2. Kubas, G.J. Acc. Chem. Res. 1988, 21, 120 and references therein. 3. Saillard, J.-Y.; Hoffmann, R. J. A m . Chem. Soc. 1984, 106, 2006. 4. Jean, Y . ; Eisenstein, O., Volatron, F.; Maouche, B.; Sefta, F. J . Am. Chem. Soc. 1986, 108, 6587. 5. Hay, P . J . J . A m . Chem. Soc. 1987, 109, 705. 6. Hehre, W . J . ; Radom, L.; Schleyer, Paul v.R.; Pople, J . Α. Ab Initio Molecular Orbital Theory. Wiley: New York, 1986. 7. Spangler, D.; Wendoloski, J.L.; Dupuis, M . ; Chen M . M . L . ; Schaefer III, H.F. J. A m . Chem. Soc. 1981, 103, 3985. 8. Faegri, K . ; Almlof, J . Chem. Phys. Lett. 1984, 107 121. 9. Dobbs, K . D . ; Hehre, W . J . J . Computational Chem. 1987, 8, 861. 10. Williamson, R . L . ; Hall, M . B . Int. J. Quantum Chem. Symp. 1987, 21, 503. 11. Rohlfing, C . M . ; Martin, R.L. Chem. Phys. Lett. 1985, 115,104. 12. Rohfling, C . M . ; Hay, P . J . J . Chem. Phys. 1985, 83 4641. 13. Bauschlicher, Jr., C.W.; Walch, S.P.; Langhoff, S.R. Quantum Chemistry: The Challenge of Transition Metals and Coordination Chemistry; Veillard, Α., Ed.; Reidel: Dordrecht, Holland, 1985; p. 15. 14. (a) Hay, P.J.; Wadt, W.R. J. Chem. Phys. 1985, 82, 270. (b) Wadt, W. R.; Hay, P . J . ibid. 1985, 82, 274. (c) Hay P. J.; Wadt, W . R . ibid. 1985, 82, 299. 15. Crabtree, R.H.; Hamilton, D . G . J . A m . Chem. Soc. 1986, 108, 3124. 16. Upmacis, R . K . ; Poliakoff, M . ; Turner, J.J. J . A m . Chem. Soc. 1986, 108, 3645. 17. Thompson, M.E.; Baxter, S.M.; Bulls, A.R.; Burger, B . J . ; Nolan, M . C . ; Santarsiero, B.D.; Schaefer, W.P.; Bercaw, J.E. J . A m . Chem. Soc. 1987, 109, 203. 18. Burdett, J . K . ; Phillips, J.R.; Pourian, M . ; Upmacis, R. Inorg. Chem. 1987, 26, 3061. 19. Modified GAUSSIAN82 Program: J.S. Binkley and R.L. Martin. 20. M E S A program: P.W. Saxe and R.L. Martin. RECEIVED December 9, 1988

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 8

Activation of Small Molecules by Transition Metal Atoms Theoretical Interpretation of Low-Temperature Experiments with Cu, Pd, and Pt Atoms

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch008

O. A. Novaro Instituto de Física, Universidad Nacional Autónoma de México, 01000 México D.F., Mexico Theoretical-experimental results on transition metal atom-small mole­ cules systems are reported. The theoretical studies employ the PSHONDO-CIPSI sequence of programs that allow for variational and perturbational configuration-interaction calculations including up to 10 configurations for each ground and excited-states of the system. These theoretical results are contrasted with data from low-temperature ma­ trix isolation experiments on these same systems supported by infrared, visible-ultraviolet, epr and other spectroscopic techniques. Interesting correlations between theory and experiment are found, including the fol­ lowing: the photoactivation of H and methane by Cu atoms at low tem­ peratures are rationalized from a theoretical standpoint; the theoretical prediction of H activation by ground-state Palladium is verified exper­ imentally and the preference of insertion over abstraction reactions and the formation of Cu(N ) complexes serve to explain some extraordinary isotopic effects found in experiment. 6

2

2

2 n

Matrix isolation experimental techniques [1-10] stand out among many other mod­ ern chemical research methods with regard to their ability to provide direct comparisons with quantum mechanical calculations. The use of photoexcitation methods to induce reactions [7-9] as well as the applications of multiple spectroscopic techniques to study such photochemical reactions allows for close control of the reaction parameters. Most of the high temperature and entropy effects, otherwise very large in thermochemical re­ actions, are therefore not present here and thus some of the limitations associated with applications of precise quantum mechanical calculations to kinetic processes disappear. Specifically the low temperature studies which concern elementary interactions of small molecules and transition metal clusters or atoms isolated over "inert" solid matri­ ces [5-10] are of high interest, especially now that the Schrodinger equation representing such interactions can be solved to relatively high precision using ab-initio configurationinteraction methods. Among such methods we could mention the CASSCF and CCI, GVB-CI, Monstergauss, PSHONDO and CIPSI [11-15] among other methods and pro­ grams, many of them mentioned and described in this book. The fact is that theoretical physicists and chemists in the recent past have developed very accurate methods for the study of d- and /-electron systems. Therefore, while several low temperature exper­ iments concerning transition metal atoms or clusters and their interactions with small molecules have appeared in the literature, simultaneously many quantum mechanical calculations are appearing on the same type of systems. Rarely in the history of quan0097-6156/89/0394-O106$06.00/0 c 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch008

8. NOVARO

Activation of Small Molecules by Transition Metal Atoms 107

turn chemistry has a situation so tempting been found where theory can be compared with experiments concerning systems of great practical chemical interest and yet small enough and with such strict control of variables as to make direct comparisons feasible. However, actual theoretical experimental collaborations are not at all common. Some joint papers exist [16-18] and sometimes theoretical and a experimental papers on the same system are published back to back [19-20]. An example is that of Weltner and coworkers, who performed esr and endor studies of MnH supported over solid Argon at 4K [21] in order to compare with previous calculations on the same MnH molecules [22], among other similar efforts. The alternative situation of theoreticians directing their calculations of previous or current matrix isolation experiments is of course common also [22-31] but considering the potential benefits of following the experiments closely, we feel that in reality much more work should be done in this direction. In particular we believe that in order to be really relevant to the understanding of the kinetics, calcula­ tions should not only aim at obtaining the structure of a molecule but very importantly at determining reaction coordinates and activation barriers. This allows for predictions about feasibility and selectivity that may be contrasted with the experiments. In this chapter we shall review a few cases of such theoretical-experimental collaborations with­ out pretending to be exhaustive but rather as examples of the mutual influence of theory and experiment. Method The method used in our calculations is the ab-initio pseudopotential method of Durand et al. [32-34]. We apply it for the Cu, Pd and Pt metal atoms whose pseudopotentials are also given in the literature: that of Cu in [2£], that of Pd in [M], that of Pt in [3fi]. In every case all of the valence electrons as well as all the electrons of the outermost d-subshell are always treated explicitely and without restrictions. The basis sets used are always of double-zeta quality at least, those for Cu and H are given in [3Z], that of Pt in [3£], that of C by Pacchioni et al. [2£], that of Pd in [24], that of Ν by Daudey (Daudey, J.P. Preprint of the Laboratoire de Physique Quantique, Université Paul Sabatier, Toulouse, France, 1986). The convergence criterion of the SCF iteration energies was set at 10~ . Basis set superposition errors were systematically tested and corrected for by following the counterpoise correction of Kolos [22] when necessary. The basis sets were selected by thoroughly testing their accuracy in the reproduction of the energy splittings between the ground and lowest excited states (eg. the P- S and D- S splittings in Cu, the D- S splitting in Pt, etc.) but this is better seen by reading the original papers [36-37]. One limitation of these basis sets is the lack of /-polarization functions. In some instances it has been shown that their use introduces only marginal improvements in some calculations involving Cu [4Ω] or Pt [H]. The use of /-functions is however an open subject of current interest as is exemplified by several chapters of this book. We use the CIPSI algorithm [1£] which introduces configuration interaction by perturbation with multiconfigurational wave functions selected by an iterative process using the M0ller-Plesset barycentric values as proposed by Malrieu [42]. By this ap­ proach we then include many more configurations (typically of the order of a million or more) that interact effectively with the original reference states. These reference states correspond to more than one metal atom state, generally we take three or more states of each metal considered, say the ground and the lowest lying excited states. The im­ portance of this will be evident when discussing the comparison of the specific results with the experimental data. In every case a careful analysis of the configurations that are included in this large CI scheme is carried out trying to determine how their role during the process is to be understood. Such configurations represent very often a polarization of the d-subshell, which in many cases is closed or nearly closed, so that its relaxation substantially lowers the interaction energy between a transition metal atom and a small closed-shell molecule 6

2

3

2

l

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

2

2

108

THE CHALLENGE OF d AND f ELECTRONS

(eg. H2, C H or N2). As interesting as this and also many other trends shown by the configurations participating in the CI schemes are, it is not adequate to describe them within the overall fashion in which we shall discuss the results on several different systems in the present paper. The reader is therefore referred to the original articles to review this important aspect [36-37 40.43-44]. Comparisons with other theoretical methods are important. Our Cu calculations are based on the pseudopotential of Péllisier [35], who applied it to the C u system. A controversy arose when CASSCF-CCI calculations on the same system seemed to imply [4Q] that the CIPSI calculations matched the experimental energy too well for a non-relativistic method. Péllisier [45] replied by showing his pseudopotential included relativistic effects. On the other hand a recent CASSCF-CCI calculation on the Pt+H reaction was published [4£]. The author apparently was not aware of our previous work on the same system [25] and yet he obtained potential energy surfaces that are virtually identical to those obtained using CIPSI as is commented elsewhere [47]. He did make some comparisons with GVB-CI calculations [4£] on the same P t H systems concluding that both this method and his agreed well. From all this we must conclude that sometimes CIPSI, CASSCF-CCI and GVB calculations can lead to the same results and quite similar chemical pictures. Other methods also match well in their predictions. In one of our papers both the CIPSI and Monstergauss programs were used in the CuH-f H thermal reaction giving coinciding results in most aspects of the process [18]. 4

T

2

2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch008

2

Activation of Small Molecules by Closed-Shell Transition Metal Atom States The matrix isolation experiments using epr, ir, uv-visible and other spectroscopic tech­ niques on transition metal-olefin complexes [8,49] have naturally attracted the attention of theoretical chemists and calculations on the Ni-C H4 system were reported in one of the first theoretical-experimental papers mentioned in the introduction [15]. These results were later supplemented with a larger (double-zeta) basis set [5Q] and also [51] extended for a N i ( C H ) system. The main conclusions are that a net charge transfer of almost 1/5 of an electron from the metal to the ethylene is evident and that a dona­ tion and back donation mechanism consistent with a classical Dewar-Chatt-Duncanson model exists. The Ni-ethylene binding energy is 12.8 kcal/mol. Another system that has been studied theoretically is Cu-C H4 where, in con­ trast [54], it was suggested that a weak charge transfer from the olefin to the metal (0.164e) without the participation of the carbons and the unpaired electron remaining in the 45 - 4p hybridized orbital exists. This is indeed very far from the Dewar-ChattDuncanson model [52-53]. We shall now report our results on Pd-C H4 which are much more in coincidence with those of Ozin et al. [Ifi] and Siegbahn and coworkers [50-51]. 2

2

4

2

2

2

Palladium-Ethvlene Interaction The case of the interaction of Pd with C H is interesting because the Palladium ground state has a closed shell, d , configuration. This notwithstanding, the existence of a stable P d C H complex was established experimentally [&] and a net charge transfer from Palladium to the olefin carbons was reported. They also showed that the πbonded Pd-C H4 complex had very similar stretching modes to those observed for ethylene adsorption on Palladium surfaces [55], thus concluding that the complex is an acceptable model for this adsorbed species. This system is then interesting enough to justify a theoretical study and this was done by us using the methods described above [56-57]. The main conclusions of the matrix isolation experiments were confirmed by us as depicted in Fig. 1 where the Pd-fC H interaction energy curve as well as the geometrical and charge transfer properties of the complex are given. The binding energy of our complex, 47 Κ Joules/mole, was close to the value (54 Κ Joules/mole) of the desorption energy on Pd surfaces reported in [55], fulfilling the expectations of Huber, 2

4

10

2

4

2

2

4

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

8. NOVARO

Activation of Small Molecules by Transition Metal Atoms 109

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch008

f Energy (Kcal/mol) 9,0 \-

Figure 1. Potential energy curve and geometrical and charge transfer parameters of the Pd-ethylene complex.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

110

THE CHALLENGE OF d AND f ELECTRONS

Ozin and Power [g] that the P d - C 2 H complex is an acceptable model of the adsorbed olefin species. We also see from Fig. 1 that the C-C distance is lengthened and the C-H2 planes are rotated by 15° to the original plane of the ethylene, thus explaining the red shifts in the respective vibrational modes reported in [fi]. Also, their ultraviolet results show a charge transfer from the metal to the olefin, which is also evident in Fig. 1. The value of this donation is similar to that reported for the N 1 - C 2 H 4 complex [52] and we also find that a reasonable consistency of the Dewar-Chatt-Duncanson model exists for Palladium as for Nickel [52] but apparently not for C u C H [54] although the fact that the latter study included the cf-electron subshell in the pseudopotential may completely falsify this aspect of their results. At least our own studies of Cu reactions, to be described later on, systematically showed the need of having a flexible and explicit description of the d-subshell in order to obtain the very important avoided crossings and activation barriers and in general the multiple-well potential energy surfaces that will be discussed later on. 4

2

4

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch008

Ho Capture by Palladium Atoms Several calculations on the P d H 2 system exist [58-60] showing that a weakly bound A i ( C 2 v ) Palladium dihydride can be stable. However no attempts to depart from this C2V symmetry were carried out except for some analyses of the H2 positions around a Palladium dimer [fil]. We have studied both the side-on ( C 2 ) and the headon approaches of Pd to H2 showing that both present attractive curves without any activation barriers [12]. More recently we have studied a substantial part of the potential energy surface of the Pd(4 Cu + H

(2)

2

which produced high yields of ground state Copper atoms and hydrogen. Theoretical calculations using both the Monstergauss and CIPSI programs showed an energetically downhill reaction coordinate for the H+CuH addition reaction [IS]. The addition re­ action implies a linear approach of the H atom towards the Cu moiety of CuH. Also an abstraction reaction was studied, which implied a linear approach of H now towards the H moiety of HCu. The latter process does have an activation barrier of less than 7 kcal/mol [IS]. The CuH complex formed by either of these reactions spontaneously dissociates into the final products (H and ground state Copper atoms) thus explaining the experimental results of [64]. 2

2

Photoexcited Cu Activation of Ho In their study of the activation of H by photoexcitation of Cu atoms deposited in rare gas matrices, Ozing and coworkers [65-66] irradiated with 320nm photons to produce the 3d 4p ( P) «- 3d 4s ( S) transition. This was sufficient and necessary for the reaction of eq. (1) to take place. The efficiency of this photochemical process was originally attributed by them [££] to the radiationless transition of the Ρ state to the lower excited D state which hypothetical^ was the one that reacted with H . Our first calculations showed that the Ρ state itself is also capable [37] of capturing H effectively. Furthermore it was established that while the photoactivation by the 320nm was a sine qua non condition for the process to be triggered (without it Cu simply does not activate H ) right after it the process of eq. (1) proceeds regardless of whether Cu* suffers a radiationless transition to the lower excited state D or eventually to the ground state S of Cu. This was demonstrated [4£] from the relatively moderate (~ 20 kcal/mol) activation barriers that the potential energy curves of Cu^S), Cu( D) and Cu( P) present to H capture. These barriers are easily overcome by the great energy gain from the transition from the Ρ state (whose own potential curve is initially downhill in energy) to the lower D and S states. This process implies a very interesting mechanism involving Landau-Zener-Stuckelberg transitions and Herzberg-Teller couplings between the main A\ and P symmetry potential energy surfaces of the C Cu*+H system. These potential energy surfaces [44] are reproduced in Fig. 6 but the details of the reaction mechanisms envolving the three S, P and D states of Copper in activating H and eventually leading from the CuH energy minima of Fig. 6 towards the final products of eq. (1) cannot, for reasons of space, be reproduced here. For this the reader is referred to the original papers [37 43-44]. It is worth mentioning here however that the restriction to C symmetry in Fig. 6 is not a limitation for the actual chemical process of eq. (1) because, as was shown in [43], any deviation from C symmetry would only enhance the scission of H and the liberation of the CuH-|-H products because the activation barrier (~ 20 kcal/mol) would necessarily be lowered even more considering that both the Αχ and the B surfaces belong to a single representation A' of the C group when C symmetry is broken [21]. In conclusion: the activation of H by Cu is only feasible if the Ρ fc(abstraction). This of course would be disregarding any tunneling effects of an H moiety into the abstraction pathway barrier. However, such tunneling would not be present if D is used in the reaction. At this stage in conclusion we would necessarily infer from our theoretical results (M.E. Ruiz; G.A. Ozin and 0. Novaro, work in progress) that this reaction would indeed lead to no products of the type CuD or D, in complete agreement with the observed facts. 2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Activation of Small Molecules by Transition Metal Atoms 119

8. NOVARO

Nitrogen Matrix Hindrance of Ho Activation by Copper An amazing discovery (Gracie, C , M.Sc. Thesis, University of Toronto, 1985) was made when the photoexcitation reaction 2

Cu( P) + H + D ^ CuH + CuD + H + D 2

(6)

2

previously mentioned and which had a reaction rate ratio of about kjj/kj) ~ 1.5 [55] was carried out replacing the rare gas matrix by a N molecular matrix, with all other conditions (temperature, reactants, etc.) kept equal. The new results showed the reaction products 2

2

Cu( P) + H + D ^ CuH + H + D 2

2

(7)

2

with the CuD and D subproducts being much scarcer, by a factor of about one thousand (in effect, kjj/kj) « 10 ). The suggestion was made based on infrared, optical and epr spectroscopical observations (Ozin, G.A., Gracie, C. and Mattar, S.M., Toronto University, unpublished data, 1984) that a CuN complex existed of the type

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch008

3

2

Cu -

(8)

( Ν ΞΞ N ) „ 2

2

When a Copper atomic resonance line Ρ 1, i s a large z e r o - f i e l d s p l i t t i n g parameter (D) relative to X-band quanta; this usually occurs for molecules with large spin-orbit constants and/or through coupling to low-lying electronic states. (V , with a Σ ground state but D = 75 cm , i s therefore undetectable i n matrix ESR. ) These two conditions for not observing an ESR spectrum, even though there i s good reason to believe that a magnetic molecule has been matrix isolated (29), w i l l be used as circumstantial evidence supporting some of the ground states to be suggested here. q

2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

2

3

2

-1

Transition-Metal Diatomic Molecules Figure 1 shows the array of a l l possible diatomics derived from the first-row t r a n s i t i o n metals (except Zn). Ground states are not r e l i a b l y derived from Mossbauer data (30, 31 ) so that those given for the seven molecules containing Fe must be considered as suspect. To an e a r l i e r version of this Figure (32 ) we have added a designation of the number of valence 3d + 4s electrons i n the isoelectronic molecules occurring i n blocks perpendicular to the p r i n c i p a l diagonal of homonuclear molecules. Thus, C r , VMn, TiFe, and ScCo a l l have 12 valence electrons. C r i s known experimentally (as a l l the bold borders indicate) to have a Σg ground state. Can we then presume that the other three isoelectronic diatomics also have that ground state? (TiFe and ScCo in that series have been "inferred from experiment" to be Χ Σ; that w i l l be discussed below.) Similar groups of isoelectronic molecules are l i s t e d i n Table I, including those involving Zn. 2

2

1

1

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WELTNER & VAN ZEE

Diatomic and Monocarbonyl Molecules

215

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TRANSITION METAL DIATOMICS

11 GAS PHASE ES ESR/MATRIX 0

MOSSBAUER/MATRIX 1 θ '

H INFERRED FROM EXPT. THEORY

/ \

VALENCE ELECTRONS

Figure 1. The ground states of possible diatomics formed from the first-row transition metals, excluding Zn. Those i n bold borders are d e f i n i t e l y established, the others are derived as indicated. "Inferred from experiment" and "valence electrons" are explained i n the text.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

216

THE CHALLENGE OF d AND f ELECTRONS Table I.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

Valence Electrons 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

First-Row Transit!on-Metal Diatomics

Experi­ mentally Studied Molecules Sc

Experi­ mental Ground State

Isoelectronic Molecules

0

T i

2 TiV

1 4

*î Σ 9

g Cr TiCo, ScNi Mn VNi 0

2

z

g

4

E

g

2

ScTi ScV ScCr TiCr,, ScMn VCr, TiMn, ScFe VMn, TiFet, ScCot CrMn , VFe CrFe \ VCot, T i N i t , ScCu MnFe , CrCo, TiCu, ScZn 2 ' MnCo*, CrNi*, VCu, TiZn FeCo MnNi*, VZn * Co , FeNi , MnCu CoNi. * , FeCu*, MnZn . * CoCu , FeZn 2' CoZn NiZn c

F e

CrCu CrZn

Λ

2

N i

NiCu Cu CuZn Zn 2

1r+

0

J Ground state inferred from experiment - see text. Not detected i n ESR but believed to have been prepared.

Table I I . Mixed-Row Transition-Metal Diatomics

Valence Electrons 13 15 17 18 19 20 21 22 23

Experi­ mentally Studied Molecules YNi, VPd, CrAg, MnAg

Expéri­ menta 1 Ground State

Isovalent Molecules ScPt, YPt VPt, NbPd, NbPt MnAu FeAg* CoAg NiAg*

CrCd

CuAg, CuCd, AgZn, AuZn,

In Table II our d e f i n i t i o n of "isoelectronic" has been broadened to just requiring the same number of d + s electrons (regardless of

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

15.

WELTNER & VAN ZEE

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Diatomic and Monocarbonyl Molecules

p r i n c i p a l quantum numbers), so that mixed-row diatomic molecules can be included, and a more appropriate designation i s "isovalent". Table II l i s t s only experimentally-studied molecules and their "isovalent" counterparts, but there are, of course, a large number (465!) of possible transition-metal diatomics. [In these Tables an asterisk (*) indicates a molecule not observed i n the ESR but believed to have been prepared (29), while a dagger (t) indicates that the ground state of the molecule was "inferred from experi­ ment", as discussed below.] Four molecules i n column 4 of Table I, TiFe, ScCo with 12 valence electrons and VCo, TiNi with 14 valence electrons (see Figure 1) have been inferred to have Σ ground state (33 ). The reasoning i s that the addition or subtraction of one electron i n the dïr d6 da sa sa configuration of TiCo, ScNi would probably lead to closed s h e l l molecules. ScCu, although also i n that category, was considered more doubtful because of the reluctance of Cu to form multiple bonds (however, see below). As Figure 1 indicates, there are no experimental, or t h e o r e t i c a l , data on any of these putative Σ molecules, but the "inferences" are i n accord with the known ground states of C r and Mn . 1

4

4

2

2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

2

1

2

2

Pursuing the "Isovalent" P r i n c i p l e . The question, admittedly naive, i s whether the "isoelectronic" p r i n c i p l e of simple MO theory applies among these electronically-complex transition-metal molecules. It seems that exact adherrence to such a rule might be surprising i n these cases, but i f there i s a commonality among the lowest-lying states, i t i s worth pursuing. We have already considered two first-row metal series contain­ ing 12 and 14 valence electrons where there are hints of adherence to the p r i n c i p l e . However, there i s a more d e f i n i t e example of "isoelectronic" behavior where the ground states of four 13-electron molecules have been established to be Σ · These molecules are also i n the class of what have been referred to, after Gingerich, et a l . (34-36), as Brewer-Engel molecules (37); each involves two elements from opposite ends of the periodic table, i n this case Groups IIIB and VIII. Such molecules are expected to form strong multiple bonds and therefore to be of low spin. ScNi, S c P d , YNi, Y Pd not only a l l have S = 1/2 but the unpaired spin has similar charac­ t e r i s t i c s in the four molecules, as derived from hyperfine i n t e r ­ action with the indicated nuclei (38). The spin is largely (~70%) on the lighter atom and has about 30% s character throughout. Shim and Gingerich (36 ) have made an a l l - e l e c t r o n Hartree-Fock c a l c u l a ­ tion for YPd and find a Δ ground state, not i n agreement with the ESR result, but Σ and Π states l i e only about 0.2 eV higher i n energy. It i s l i k e l y that improvement i n the calculation could change the ordering of these levels. The resulting bonding i s unusual (and reminiscent of that i n metal carbonyls) in that the dr i c h Pd donates electrons to the d-poor Y and Y back donates s electrons. Their s i m i l a r i t i e s indicate that this type of bonding prevails in a l l four molecules. Extension of the experimental studies to La and Pt would be interesting, and a more extensive ab i n i t i o calculation, perhaps on ScNi, would be worthwhile. More recently i n our laboratory, Cheeseman has extended the e a r l i e r study of VNi (39) with 15 electrons to VPd, VPt, and NbNi to find that they also have Σ ground states (40). 2

45

4 5

1 0 5

89

8 9

2

2

+

2

4

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

1 0 5

218

THE CHALLENGE OF d AND f ELECTRONS

"Isoelectronic" S c (Χ Σ") and Y are bothersome, since only the former has an X-band ESR spectrum (41, 42) and theory finds the same ground state for both (43, 44). Walch and Bauschlicher (44) suggest that a more extensive calculation may lower the stronglybonded Σ + state below the Σ ~ i n the case of Y , but ESR cannot distinguish between that choice and the p o s s i b i l i t y of large zfs i n a Σ ~ ground state. Surprisingly, the binding of Cu has been puzzling i n the two diatomics CrCu and MnCu. CrCu does not appear to have a Σ ground state, as do CrAg and CrAu, and after considerable agonizing, the spectrum has been interpreted as Σ (45). Thus i t i s probably t r i p l y bonded with properties intermediate between C r and Cu * An exceptionally large e l e c t r i c f i e l d gradient at the Cu nucleus i n CrCu supports i t s anomalous ground state. The situation i n the 18 valence electron series MnCu, MnAg, MnAu i s similar but d i f f e r e n t . Whereas MnAg i s easily formed and characterized as a Σ molecule, MnCu (and MnAu) remains undetected i n the ESR (29). Other evidence for a preference for Σ ground states among this isoelectronic class i s that CrZn also has Χ Σ (29); however Co , even after many t r i a l s , was not observed v i a ESR. In summarizing this section, one can say that the present meager experimental data are i n encouraging support of the applica­ tion of the "isoelectronic" p r i n c i p l e to these diatomics. Both first-row and/or mixed-row diatomics containing 12, 13, 14, 15, and 18 valence electrons show indications of having Σ , Σ , Σ, Σ , and 2

1

2

5

2

5

4

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

2

2

7

7

7

2

1

2

1

4

7

Σ ground states, respectively. At present there are no definite discrepancies except i n the CrCu, CrAg, CrAu s e r i e s . It i s clear that theory has much to contribute here. Transition-Metal Monocarbonyls,

M-CO

Molecules

The experimental ground states of most of these molecules are un­ known. ESR has established VCO as Σ and CuCO as Σ , and recently also ScCO as Σ , but i t has f a i l e d to detect FeCO, which theory i n ­ dicates has a Σ ~ (or Σ~) ground state. CrCO(^) has also appar­ ently been detected via ESR. The present state of a f f a i r s i s sum­ marized simply i n Figure 2, where the o r i g i n of a given ground state i s indicated as experimental (E), theoretical (T), or suggested (?), usually from ESR evidence. The general scheme of bonding i s shown in Figure 3 for VCO (46). The c l a s s i c Dewar-Chatt-Duncanson model (47,48) involves donation by the CO 3σ o r b i t a l into the metal 3da + 4sa + 4ρσ o r b i t a l to form the 3σ VCO o r b i t a l and back donation by the metal an o r b i t a l s into the antibonding 2π o r b i t a l to form the 2π MO, weakening the CO bond. Experimentally, the effects of the metal-CO interaction are evidenced by lowering of the C-0 stretching frequency, v , below 2143 cm" , the value i n the free CO mole­ cule. The trend i n the CO stretching frequency varies as shown i n Figure 4 for first-row t r a n s i t i o n metals (49). An unknown frequency i n this Figure i s that of MnCO (and perhaps ScCO) and i t i s proposed to be high, i . e . , the bonding of Mn to CO to be weak. (This remains to be established by some other experiment.) The variation i n v , taken equivalent here to CO bond strength, may also be considered as inversely proportional to the M-C bond strength. Qualitatively, one can account for the bonding i n these MCO molecules by considering the 4 s 3 d ~ ·• 4 s 3 d ~ promotion energy of 6

2

4

5

3

1

c o

c o

2

n

2

1

n

1

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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Diatomic and Monocarbonyl Molecules

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

ScCO TiCO VCO CrCO MnCQFeCOCoCO NiCO CuCO ~4s

5σ

-ff:

4σ

-3d

16

2π

4

Σ(Τ,Ε)

'Φ(τ)

5

Σ(Ε)

7

Σ(Ε)

5

Σ(?)

5

Σ(τ)

»Δ(?)

•Σ(τ)

2

Σ(Ε,Τ)

Figure 2. Molecular o r b i t a l schemes and ground states of MCO molecules (M = first-row transition metal). Boxes here emphasize the only molecules with experimentally established ground states. Ground states are indicated as: theoretically calculated (T); experimentally determined (E); or suggested (?), based on absence of an ESR spectrum.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

220

THE CHALLENGE OF d AND f ELECTRONS Vatom

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

4—H

\

-H Figure (46).

3.

Molecular

orbital

scheme f o r the °Σ VCO molecule

ο ο 3

Se Ti V Cr Mn Fe Co Ni Cu Figure 4. Plot of the CO stretching frequencies, v , i n the first-row transition-metal monocarbonyl molecules MCO ( c i r c l e d points are tentative) (49). Ρ gives the variation of the atomic energy of promotion corresponding to 4s 3d ~ 4 s 3 d ~ , where η i s the number of valence electrons. Curve Β i s a crude indication of the expected trend i n CO bonding strength due to decreasing di\ backbonding from Sc to Cu. c o

2

1

n

1

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

n

2

15.

WELTNER & VAN ZEE

221

Diatomic and Monocarbonyl Molecules

the metal atom, the απ backbonding involved i n the DCD theory, and hybridization. The former yields the extreme variations (trace P) shown i n Figure 4 and provides a barrier to σ-bond formation. Pro­ motion and/or hybridization of the 4s electrons relieves the a n t i bonding between these metal electrons and the CO 3σ o r b i t a l , r e s u l t ­ ing i n a shorter M-C bond and stronger Mdïï back donation. Back donation i s expected to generally decrease as one proceeds from l e f t to right because the 3d o r b i t a l shrinks i n size (44, 50) · This e f f e c t tends to strengthen the CO bond as one moves across the tran­ s i t i o n series from l e f t to right, as depicted roughly (trace B) i n Figure 4. Then the sum of Β + Ρ should have the contours of v , which i s approximately true. What about the ground states? We w i l l consider each of the carbonyls as the 3d s h e l l i s f i l l e d up, beginning with ScCO. Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

c o

1 "λ ScCO. In the IR ScCO was presumably formed since a proper CO s h i f t was observed, but the bands were uncomfortably broad (49) · Then, although not firm, v = 1950 cm" , as given i n Figure 4, indicating a bound MCO molecule. Jeung and Kouteck^ (51) have made a MRD SDCI pseudopotential calculation and found a strongly bound Σ ~ lowest state corresponding to the configuration 2π 4σ shown i n Figure 2. A repulsive Π state i s next highest and then a strongly bound Π state (2Π 1θ 4σ ) about 0.2 eV higher. In these two bound states the Sc 4s electron i s polarized away from the CO i n the 4σ orbital. Also i n both states there i s a net electron transfer from Sc to CO. ESR experiments, after several t r i a l s , were successful i n detecting ScCO i n an argon matrix at 4 Κ (52). The ground state was found to be Σ , as predicted by theory, with the z e r o - f i e l d s p l i t ­ ting parameter | D | > 1 cm" . Hyperfine interaction with the S c nucleus was observed but substitution of C 0 produced no s p l i t ­ tings, indicating only a small spin density on the C atom of that ligand. The s character at the Sc atom i s estimated to be about 60% from the hyperfine parameters, and the very large g s h i f t i s reasonably accounted for by the low lying Π state founa by Jeung and Kouteck? (51). J

1

C Q

4

2

4

1

1

1

1

4

1

4 5

13

4

TiCO. GVB theory (perhaps the oldest theoretical treatment of an MCO molecule) finds the c l a s s i c a l σ donation of the CO nonbonding o r b i t a l to Ti da and d e r e a l i z a t i o n of T i απ to acquire CO π* character (53). Nonbonding T i electrons are polarized away from the CO i n 4s-Xdp o r b i t a l s . In our symbolism the ground state i s Φ (4σ 2π 1δ ) where 2π i s mainly 4ρπ rather than 3άπ. (It i s stated that this inducing of 4p character into the valence orbitals should be enhanced i n Sc and decrease rapidly i n proceeding past V. ) Ex­ perimentally, v has not been established; i n Figure 4 i t has been assumed to l i e along the line joining v for ScCO and VCO. An ESR spectrum for TiCO was not detected which i s i n accord with a Φ ground state. 3

2

1

1

c o

c o

3

5 1

1 3

VCO. V and C hyperfine structure (hfs) i n the ESR spectrum established the molecule as VCO and yielded a complete electronic and magnetic picture of a S = 5/2 ground state molecule including g tensor, hf parameters and z e r o - f i e l d - s p l i t t i n g parameter |D| (46). The d i s t r i b u t i o n of the five unpaired electrons i s i n approximate

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

222 Ί

accord with the (2π) (1δ) (4σ) configuration i n Figure 3. The σ electron has about 30% s character, the remainder being 3da and 4ρσ. The spin density on the CO i s quite small. There i s the interesting observation of two conformations of VCO; both were observed i n argon matrices, one i n neon, and the other i n krypton. It i s reasoned that one of these i s bent and that i t i s most l i k e l y the gas-phase form. E a r l i e r infrared studies by Hanlan, e t a l . (54), had suggested that VCO might be bent, and extended Htlckel theory (assuming low spin) indicated that there was a monotonie decrease i n energy of the molecule as the angle decreased. Thus, an ab i n i t i o calculation would be of real interest. A strong M-CO bond i s indicated by a decrease of 240 cm" (in s o l i d argon) i n v ( ) (see Figure 4). 1

5 4

c o

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

1

CrCO. The CO stretching frequency was observed at 1977 cnT (49), but there appear to be no other data on CrCO. Unpublished ESR spectra i n our laboratory showed uncharacteristic very broad lines when chromium metal was trapped i n an argon matrix containing CO i n various concentrations. A tentative analysis indicated a Σ mole­ cule, the reason for the extraordinary breadth of the lines was not understood. In Figure 2, the ground state i s tentatively given as Σ (2π 1δ 4σ 5σ ), i n agreement with the addition of one electron to the known VCO configuration. 7

7

2

2

1

1

MnCO. Two sets of authors agreed on the infrared spectrum of Mn + CO i n matrices (55,49), but the firm i d e n t i f i c a t i o n of a MnCO signal was i n doubt. I t was suggested by us that the molecule i s essen­ t i a l l y nonbonded and therefore that v = cm" . This was rationalized, as indicated i n Figure 4, by the high promotion energy of the Mn atom, implying that the 4s electrons provide a repulsive interaction with CO. There i s evidence of the existence of MnCO i n the gas phase, at least with a lifetime long enough to allow i o n i z a ­ tion to form MnCO , i f the suggested mechanism for i t s preparation i s correct. That ion and Mn , Mn were observed during photofrag­ mentation of Mn (CO) and believed to be produced by multiphoton ionization of the neutral MnCO (and Mn, Mn ) photoproducts (56). The signal shows f i r s t - o r d e r dependence upon the sample pressure. (Note that Mn i s proposed to be formed from Mn , which i s a van der Waals molecule as proposed for MnCO.) 2

1

4

0

1

c o

+

+

2

2

10

2

2

2

1

8

9

8

1

FeCO. Experimentally, IR i n matrices finds v cm" (57,58), indicating a relatively strong Fe-C bond, but EngeIking and Lineberger (59) estimate the bond energy as 1.0 ± 0.3 eV. Also the Mossbauer isomer s h i f t of -0.60 mm/s i s close to that of the free atom, -0.75 mm/s (59). Theory has not quite decided whether the ground state i s Σ ~ or Σ ~ and i s finding i t d i f f i c u l t to account for the isomer s h i f t (50,60-65). Thus, Guenzberger, et a l . (62), employing the discrete variational method with the Χα l o c a l approxi­ mation, place the 6 o r b i t a l below the σ and, on an aufbau basis, obtain a dir d6 4a , Σ ~ ground state. The calculated isomer s h i f t i s -0.12 mm/s, i n disagreement with experiment. A recent c a l c u l a ­ tion by Marathe, et a l . (65 ) (using SCF + MP4SDTQ) derived a spinquintet (d-?rdo 4σ 5σ , Σ " ) ground state rather than a spintriplet. Π states (such as π δ 4 σ 5 σ ) are calculated to l i e at least 1.4 eV higher i n energy. However, with the Σ " ground state c o

3

4

4

5

2

5

2

1

3

1

5

3

2

2

5

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15.

WELTNER & VAN ZEE

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Diatomic and Monocarbonyl Molecules

the calculated isomer s h i f t i s -0.11 mm/s, s t i l l i n poor agreement with experiment, although the calculated C-0 stretching force constant i s i n good agreement with that derived from the experi­ mental frequency. I t i s suggested that perhaps the matrix effects are large, which i s not consistent with the vibrational results. However, another recent calculation by Daoudi, et a l . (66) [using SCF with (CIPSI)] finds the Σ ~ state to be lower than the" !*" state by 0.4 eV, the Fe-C bond energy to be 1.34 eV and v ~ · I t i s disappointing that an ESR spectrum for FeCO was not observed since i t could resolve the m u l t i p l i c i t y problem i n the ground state, but i f Σ states are lowest, a large z e r o - f i e l d - s p l i t t i n g i s implied. 3

5

=

1

9

8

6 c m

c o

1

CoCO. v i n this molecule has been determined to be 1953 cm" i n s o l i d argon, but i t s ESR spectrum was not detected a t the time even though there was no question that Co atoms and CO were present i n the matrix (67 ). I t i s probable that the molecule contains at least one unpaired electron whether one reasons from FeCO or NiCO, so that i t s undetectability i n powder ESR spectra i s due either to o r b i t a l angular momentum or large zfs i n the ground state. The former seems more l i k e l y so that the Δ state suggested i n Figure 2 is reasonable.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

c o

4

NiCO. The dissociation energy has been determined but with rather large uncertainty, 29 ± 15 kcal/mole (68, see also 69), and the CO stretching frequency found to be 1996 cm"" (70,68). The electron a f f i n i t y has been measured (68). That i s e s s e n t i a l l y the extent of the experimental data except for observation of NiCO over the nickel surface under special conditions (71,72). Early theory predicted a Δ ground state, (73,74,75-81) but Rives and Fenske (82) were the f i r s t to show, using a many configuration wavefunction, that the Σ state i s s l i g h t l y lower (0.15 eV) than the Δ , but the bond distance i s much shorter i n the s i n g l e t (1.70 Â) and the binding energy (2.7 eV) and Ni-C v i b r a t i o n a l frequency (505 cm*" ) much larger. The most recent theoretical studies (83,84,50,85,86-96) have indicated that π bonding i s much more important than σ bonding, which i s i n fact repulsive. This i s "softened" by sda hybridization. The n i c k e l atom i s close to d rather than d , expected for zero valent Ni because the promotion to the S ( d ^ ) state i s energetic whereas the s d state i s almost degenerate with the ground state. 1

3

1

3

1

9

1 0

1

1

9

2

CuCO. Kasai and Jones (97) proved that CuCO has a Σ ground state and observed Cu and C hfs. The CO stretching frequency i s 2010 cm"* i n an argon matrix (98-100). Theory does not predict a bound complex (50,73,74,83-85,101-103 ) or at least only a "possible weak van der Waals interaction for the Σ state" (104). However, the ESR hf data indicate that the spin density d i s t r i b u t i o n i s p ( 4 S ) = +0.67, p ( 4 p a ) = +0.08, p(2sa) = +0.05 with the appreciable f r a c t i o n remaining probably i n the C(2pa) o r b i t a l . Thus, although weakly bound, the spin on Cu i s polarized, and more important, perhaps 20% of the spin i s on the CO ligand. 1 3

1

2

+

Cu

Cu

c

Acknowledgments The the

authors wish to thank their co-workers who have contributed to topics discussed here: S. B. H. Bach, C. A. Baumann, M.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

224

THE CHALLENGE OF d AND f ELECTRONS

Cheeseman, C. A. Taylor, R. L. DeKock, L. B. Knight, J r . , and M. T. Vala. This research was supported by National Science Foundation Grant CHE 8514585. Literature Cited 1. 2. 3. 4. 5. 6.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch015

7. 8. 9. 10. 11. 12.

13. 14.

15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25. 26.

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260-267.

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Saddei, D.; Freund, H. J.; Hohlmeicher, G. Chem. Phys. 1981, 55, 339-54. Howard, I. A.; Pratt, G. W.; Johnson, K. H.; Dresselhaus, G. J. Chem. Phys. 1981, 74, 3415-19. Bagus, P. S.; Roos, B. O. J . Chem. Phys. 1981, 75, 5961-62. Dunlap, Β. I.; Yu, H. L.; Antoniewicz, P. R. Phys. Rev. A 1982, 25, 7-13. You, X. Jiegou Huaxue 1983, 2, 183-188. Huzinaga, S.; Klobukowski, M.; Sakai, Y. J . Phys. Chem. 1984, 88, 4880-86. Kao, C. M.; Messmer, R. P. Phys. Rev. Β 1985, 31, 4835-47. Bauschlicher, C. W. Chem. Phys. Lett. 1985, 115, 387-391. Rohlfing, C. M.; Hay, P. J . J . Chem. Phys. 1985, 83, 464149. Madhavan, P. V.; Whitten, J . L. Chem. Phys. Lett. 1986, 127, 354-359. Carsky, P.; Dedieu, A. Chem. Phys. 1986, 103, 265-75. Kasai, P. H.; Jones, P. M. J . Am. Chem. Soc. 1985, 107, 81318. Huber, H.; Kuendig, E. P.; Moskovits, M.; Ozin, G. A. J . Am. Chem. Soc. 1975, 97, 2097-2106. Ozin, G. Α.; VanderVoet, A. Prog. Inorg. Chem. 1975, 19, 105-172. Moskovits, M.; Ozin, G. A. Vibrational Spectra and Structure (Durig, J., Ed. Elsevier, Amsterdam 1975). Bagus, P. S.; Hermann, K.; Seel, M. J . Vac. S c i . Technol. 1981, 18, 435-452. Bagus, P. S.; Nelin, C. J.; Bauschlicher, C. W., J r . J . Vac. S c i . Technol. 1984, A2, 905-909. Kuźminskii, M. Β.; Bagatuŕyants, Α. Α.; Kazanskii, V. Β. Izv. Akad. Nauk. SSSR, Ser. Khim. 1986 , 284-8. Merchán, M.; Nebot-Gil, I.; González-Luque, R.; O r t i , Ε. J. Chem. Phys. 1987, 87, 1690-1700.

R E C E I V E D December 9, 1988

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 16

Spin Density Functional Approach to the Chemistry of Transition Metal Clusters Gaussian-Type Orbital Implementation 1

1

2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

J. Andzelm , E. Wimmer , and Dennis R. Salahub 1

Cray Research, Inc., 1333 Northland Drive, Mendota Heights, M N 55120 Département de Chimie, Université de Montréal, Montréal, Québec H3C 3J7, Canada 2

In this contribution we review the accuracy and computational efficiency of the local spin density functional (LSDF) approach using linear combinations of Gaussian-type orbitals (LCGTO) for the calculation of the electronic structures, ground-state geometries, and vibrational properties of transition metal compounds and clusters. Specifically this is demonstrated for (1) bis(π-allyl) nickel where this approach gives an excellent qualitative and quantitative interpretation of the observed photoemission spectrum; (2) chemisorption of C atoms on a Ni(100) surface, where the present com­ putational approach determines the adsorption site of C as the four fold-hollow position above the surface with a calculated C - N i bond length of 1.79 Å(exp.: 1.75 ± 0.05 Å) and a vibrational frequency of 407 cm , (exp.: 410 cm ); (3) vibrational frequencies of C O on Pd: the calculations reveal that inclusion of surface/subsurface Pd-Pd motions couple significantly to the CO-Pd vibration leading to a reduction of the vibrational frequency by about 20% compared with a rigid substrate model. Inclusion of an external electrical field shows a stiffening of the C-O vibration with increased positive potential of the electrode; and (4) the electronic structure of Zn clusters where it is found that properties such as the first (s-electron) ionization potential converge rather slowly towards the value of the extended system requiring at least 20 transition metal atoms for an accurate description of the surface electronic structure. It is demonstrated that the computation of threeindex two-electron integrals can be achieved with a highly efficient vector/parallel algorithm based on recursive integral formulas recently published by Obara and Saika. Furthermore, we present the theoretical framework for L C G T O - L S D F gradient calculations. -1

-1

0097-6156/89/0394-0228$06.00/0 © 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

16. A N D Z E L M ET A L

Spin Density Functional Approach

229

There is growing evidence that local spin density functional (LSDF) theory (1) provides a unified theoretical framework for the study of electronic, geometric, and vibrational structures of solids, surfaces, interfaces, clusters and molecular systems (2,3) encompassing metallic as well as covalent bonding. Although widely used in solid state physics, including semiconductors, transition metals, lanthanides and actinides, density functional methods are still rather rarely applied to problems in chemistry. One of the reasons appears to be the lack of experience with this method i n addressing typical chemical questions such as molecular conformations, vibrations, and reactivity. Such investigations require the capability to calculate accurate analytical derivatives of the total energy with respect to displacements of the atomic nuclei. Evidently, the potential of L S D F gradient calculations has not yet been fully developed. However, i f such a goal could be achieved for molecules and large clusters including transition metals, one would not only have an additional theoretical/computational tool to investigate large molecules with first and second row atoms (which is the current domain of Hartree-Fock calculations) but one could also study organometallic compounds, investigate reactions on metallic surfaces, and simulate large and complex systems such as zeolites and enzyme catalysts at an unprecedented level of detail. In this paper, we present results obtained with a particular molecular orbital implementation of local spin density functional theory using a linear combination of Gaussian-type orbitals (LCGTO's). The results, derived from single-point total energy calculations, illustrate the applicability of the L C G T O - L S D F approach to questions of electronic structure, vibrational properties, and geometries of transition metal clusters and compounds and, in addition, shed light on the computational issues encountered in this kind of large-scale molecular simulation. Furthermore, we demonstrate that the L C G T O implementation allows a compact analytical formulation of energy gradients thus setting the stage for future exploitations of this chemically important feature. The paper is organized in the following way. First the key features of the L C G T O L S D F method are reviewed (3); for a more detailed description of L S D F calculations, the reader is referred to recent reviews (2,3). Four examples, discussed next, highlight the performance of the L C G T O - L S D F approach to predict (1) the electronic structure (photoelectron spectrum) of a transition metal complex, bis(TC-allyl) nickel; (2) chemisorption geometries of carbon atoms on the Ni(100) surface; (3) the vibrational properties o f C O on Pd including the influence of an external electric field; and (4) electronic properties of Z n clusters as a function of cluster size. The last example also illustrates the dependence of computational requirements on the size of the system.

The Linear Combination of Gaussian-Type Orbitals (LCGTO) Implementation Basis Sets. Since the suggestion of Boys (4), Gaussian-type basis functions have become the standard i n quantum chemical ab initio methods. In the Hartree-Fock theory the occurence of four-center two-electron integrals makes this choice a computational necessity. In local density functional theory, on the other hand, a conceptually simpler Hamiltonian gives greater freedom in selecting the variational basis set. For example, in many solid state calculations it has become common practice to use plane waves and augmented plane waves (5) with numerically generated radial functions i n a linearized form (6). A n elegant alternative consists in the use of numerically generated atomic orbitals as basis for molecular orbitals (7). Both approaches using numerical basis functions, the solid state linearized augmented plane wave (LAPW) method (6,8) and the molecular/cluster approach (7,9) have proven extremely useful in carrying out highprecision local spin density functional calculations for solids, surfaces, clusters, and

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

230

THE CHALLENGE OF d AND f ELECTRONS

molecules (2). On the other hand, the accurate evaluation of analytic energy derivatives within these implementations turns out to be a considerable challenge (10).

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

In the present studies, we use Gaussian-type basis functions for the molecular orbitals to solve the local density functional equations (11). This choice offers several advantages: (i) there is a wealth of experience from numerous ab initio Hartree-Fock calculations i n using G T O ' s for molecular calculations (12), (ii) computationally, this approach can be implemented in a highly efficient way, as will be discussed below, (iii) the analytic nature of the basis functions opens the possibility for accurate analytic calculations of energy gradients for geometry optimizations and density gradients for non-local corrections (13), and (iv) effective core potentials or model potentials can be readily incorporated (14). Besides the Gaussian basis set for the wavefunctions, there are two other sets of Gaussian expansions used in the present approach, one for the electron density and one for the exchange-correlation potential. The expansion of the electron density is used in the evaluation of Coulomb integrals. Hence the expansion coefficients of the electron density are chosen (lib) such as to minimize the error in the Coulomb energy arising from the difference between the "exact" electron density (i.e. the density originating directly from the wavefunctions) and the fitted electron density. A l l necessary steps to obtain the expansion coefficients of the electron density can be carried out analytically. On the other hand, the expansion coefficients for the exchange-correlation potential have to be obtained numerically by generating the values of the exchange-correlation potential on a grid, which are then used to fit a Gaussian expansioa After this numerical step, the matrix elements of the exchange-correlation potential operator are calculated analytically.

Integral Evaluation. In contrast to Hartree-Fock methods, the L C G T O - L S D F approach requires evaluation of only three-index integrals, thus representing an N algorithm (with Ν being the number of basis functions). A l l examples discussed below, except the Z n clusters, were calculated using the Hermite Gaussian basis originally implemented by Sambe and Felton (11a) and further developed by Dunlap, Connolly and Sabin (lib). In contrast, Cartesian, not Hermite, Gaussians are the most widely used choice i n ab initio quantum chemistry. The scheme of recursive computation of four-index cartesian Gaussian integrals, originally developed by Obara and Saika (15) for the Hartree Fock method, has now been reformulated for the three-index integrals needed i n the present method. A s shown below, a computationally highly efficient scheme results from this approach. 3

Two kinds of three-index integrals are needed, Coulomb integrals of the form (adopting the notation of Obara and Saika (15)) I = [a(l)b(l)llc(2)] c

(1)

where "II" refers to the l / r operator, and overlap-like integrals to calculate exchangecorrelation potentials and energies of the form 1 2

I

xc

= [a(l)b(l)c(l)]

(2)

Here a and b stand for orbital basis functions and c denotes Gaussian functions used in the fitting of the electron density or the exchange-correlation potential. Most of the time is spent in the computation of I . We can rewrite the original Obara and Saika formula (15) in a form suitable for computation of three-index integrals: c

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

16.

A N D Z E L M ET AL.

231

Spin Density Functional Approach

[àblICc+lj)]^ =

[abllc]

(m+1)

(m+1

+ Vi η N^c) {[ablKc-lj)]^ - (ρ/η) [abll(c-l.)] >} + ^ ( ζ + η ) ) {N.(a) [(a-lj)bllc]< > + N.(b) [a(b-ipilc]^ >) m

n+1

(3)

Here, l is a short-hand notation for a p , D or p function and function c+lj has an angular momentum one order higher than c. The recursive nature of Equation 3 allows to build, for example, integrals with d-functions from integrals with only s- and p- functions. The superscript (m) refers to the order of the incomplete Γ function. W. and Q are related to geometries and values of exponents of Gaussian functions; η , ζ and ρ depend on these exponents and N (c) is a generalized Kronecker delta. For details the reader is referred to the original paper by Obara and Saika (15), Equation 39. i

x

z

{

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

{

Expansion of the integral with a single function c, instead of a (or b) already cuts the number of arithmetic operations by as much as 30% for d-type integrals. Substantial savings in the time of integral evaluation occur i f we calculate entire groups of integrals with shared exponents in various symmetries. Particularly, we can design basis sets for the electron density and exchange-correlation potential with shared exponents without loosing accuracy in the fitting process (16,17). Hegarty and van der Velde (18) analyzed the number of arithmetic operations necessary to calculate 4-index integrals. The best algorithm (18) requires about 22 operations per integral with d functions. In contrast, the present method requires about 8 operations per 3-index integral. The new formulas for the integral calculations can be efficiently programmed on a vector computer and the algorithm is amenable to parallelization as will be shown below. As for Hartree-Fock calculations, storage of integrals becomes the computational bottleneck for systems with a large number of basis functions. In this case, a "direct S C F ' (19) scheme can be adopted (20). For the L C G T O - L S D F method we deal with a smaller number of integrals (of the order N rather than N as i n Hartree-Fock calculations) and therefore we may use the standard approach for up to about N=1000, as discussed below for the case of large Zn clusters. There is an additional advantage in the direct scheme as we have to calculate the full Hamiltonian matrix only once and then, in each iteration, add the changes to the matrix which correspond to modifications i n the density and the exchange-correlation potential, but only for those matrix elements where this change is greater then a threshhold. During the course of iterations towards selfconsistency, a smaller and smaller number of matrix elements needs to be updated. 3

4

Gradients. The calculation of energy gradients within the L S D F method using localized basis sets has been investigated by a number of researchers (10,21). However, there has been no published formulation for the case of the L C G T O - L S D F method. We will now give an outline of the formulas that have recently been implemented (22). Full details will be published i n due course. In this method, both Coulomb and exchange-correlation energies are calculated analytically once the fitting coefficients for the electron density, p with ρ = e , and exchange-correlation energy-density, e , and potential, μ , are obtained (11). The total energy ( E ^ p ) can then be expressed as r

s

E

LSDF =

Σ

P Ν

h

pq ί p ,

+

*r Pr ^

+

Σ

£

, , ^

>"*

Σ

8

U

« Pr P,

+ »

4

Here Ρ denotes the density matrix, h contains kinetic energy and electron-nuclear attraction operators; [pqllr] and [rllt] are Coulomb repulsion integrals with 3 and 2 indices, respectively, [pqs] denote one-electron 3-index integrals and U is the nuclear-nuclear repulsion term. The form of Equation 4 ensures (lib) that the Coulomb energies are accurate up to second order in the difference between the fitted density and the "exact" density obtained directly from the wavefunction. n

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

232

In order to obtain the energy gradient of E ^ p by differentiation of Equation 4 with respect to a nuclear coordinate, x, one can, in the first step, closely follow the procedure used in Hartree-Fock theory (23). In addition, derivatives of the density fitting coefficients occur in the present approach. These terms can be eliminated using the equation for the density fitting together with the normalization condition of the total density. At this point, the intermediate gradient formula is given by a E

LSDp/

=ς ρ - vt ς

a x

μ

α

a +

Z

ν

P r

Pq

{atyax + ς a[pqiir]/ax+ ς e aipqsi/ax j atriiti/ax + a u y a x - ς ^ w atpqi/ax γP r

8

P t

V

s

m

a x

{

Σ

(e

* s - V ) [pqs]} + Σ s

P

Ν

M

Σ,

Βεβχ

[pqs]

(5)

Here, W is an energy-weighted density matrix element as in the Hartree-Fock gradient formula {23). Equation 5 contains two "difficult" terms (the last two), the derivative of density matrix elements and the fitting coefficients e . It turns out that these terms can be eliminated by using the relationship

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

s

ap/ax^

-e

x c

x c

) = p ae /ax

(6)

xc

which can be obtained by differentiation of the fundamental L S D F formula (1) =

d

£

a

£

< xc Ρ > / P = xc

+

Ρ

a

E

xc /

d

7

Ρ

after multiplying both sides by p. In practice, μ and e are obtained through a fitting procedure. The fitted quantities do not correspond strictly to the original density and eq. (6) no longer holds exactly. However, with the fitting basis sets currently used, this approximation appears to be reasonable (22). Assuming therefore the validity of Equation 6, we obtain the following L S D F energy gradient formula which is valid also for the spin-polarized case and in the presence of non-local corrections to the Hamiltonian (20)) xc

a E

LSDF^

X

=

F

HFB

+

F

8

D

()

with F

S

P

+

HFB = pq pq {

Σ

γ Pr U

F is the Hellman-Feynman force plus the correction for the orbital basis set dependence on the nuclear coordinate, x. The term F is specific to the present L C G T O L S D F implementation. Equation 10 is equivalent to R F B

D

F = D

f

^p [a(r)/axll(p-p )] r

f

(11)

with ρ being the "exact" density and p the fitted density. Qearly, in the case of a perfect fit F vanishes. It is important to realize that within the L C G T O implementation, evaluation of L S D F gradients boils down to computations of 2 and 3 index integrals D

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

16.

ANDZELMETAL.

233

Spin Density Functional Approach

which can be accomplished by the same efficient technique of integral calculation as described above.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

Illustrative Examples Electronic Structure and Photoelectron Spectrum of bis(TC-allyl) nickel. The photoelectron (PE) spectrum of bis(TC-allyl) nickel has attracted considerable attention since it was first measured in 1972 by LLoyd and Lynaugh (24). Later experiments (25) established an accurate basis for comparisons with theory. The first theoretical investigation by Veillard et al. (26) using the Hartree Fock approach (cf. Table I) revealed that the interaction between d orbitals of N i and π* orbitals of the allyl group is responsible for most of the bonding. Koopmans* theorem was found to be invalid in this case and total energy A S C F calculations are required. However, even large-scale CI calculations (27) could not correctly identify the first band as ionization from the ligand π-type orbital (7a orbital assuming symmetry of the molecule). A semi-empirical Green function approach (28) (cf. Table I) provided an efficient calculation of all of the ionization potentials (IP's) and gave satisfactory assignment for most of the P E bands. Hancock et al. (29) performed scattered-wave (SW-Xot) calculations of IP's applying the transition state method. Compared with experiment, the calculated spectra are shifted considerably towards lower energies and especially ionizations out of π-type orbitals seem to be in error. u

The present calculations were performed with an all electron version of the L C G T O L S D F method. Triple zeta basis sets for nickel and carbon atoms including polarization functions were employed. Details of basis sets and the method of their optimization are given in Ref. 16. A recent neutron diffraction study (30) revealed a pronounced bending of the anti-hydrogen atoms (by 30°), as a result of their strong repulsion from the nickel atom. First we discuss results obtained assuming planar geometry of the allyl groups as this allows a direct comparisons with the other theoretical calculations of the P E spectra performed so far. The interaction diagram between the 3d and 4s orbitals of N i and the ( C H p fragment together with the resulting orbital levels of bis(TC-allyl) nickel are shown in Fig. 1. The basis set superposition error (BSSE) (17) is corrected for by using the same basis set in all calculations except for the isolated N i atom. This accounts for the splitting in the N i ( C ) case. 3

2

2h

Bonding is caused by the interaction between d-electrons of N i and the π* electrons of the ( C H ) fragment. By symmetry, bonding is allowed within the b and a manifolds. The donation of electrons from occupied 2b orbitals of the metal atom to the unoccupied 5b orbital of the ( C H ) fragment results in the bonding molecular orbital, 5b . Another bonding orbital, 10a is formed as a combination of 7a and 6a orbitals of Dis(TC-allyl) and nickel, respectively. This bonding effect is, however, largely cancelled by the antibonding partner, the 13a orbital. Close to the H O M O (6b ) there are levels of mainly 3

5

2

g

g

g

g

3

g

5

2

g

g

Mulliken charges reveals that the corresponding molecular orbitals (7a and l l b ) have some admixture of metal 4p states. The coupling with 4p states causes a transfer of the ligand*s charge to originally unoccupied 4p orbitals of the metal. This "back donation" mechanism was first discovered in SW-Χα calculations (29) and is confirmed by the present L C G T O - L S D F study. u

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

u

THE CHALLENGE OF d AND f ELECTRONS

234

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

Table I. Experimental and theoretical results for the photoelectron spectrum of bis(TC-allyl) nickel (in eV) The principal orbital character of the ionized electron is added in parenthesis

band exp. (a)

ASCF (b)

1 2

7.8 (π) 8.2 (d)

3

8.6 (d)

4

9.4 (d, π)

5 6 7

10.4 (π) 11.6 (π) 12.7

8

14.2

5.5 5.6 6.8

11.0

ASCF-CI (c)

GF (d)

SW-Xcc (e)

6.4 6.6 6.7

8.7 8.9 9.2 9.2 9.5 10.0

2.5 4.5

10.8

10.9 12.2 13.013.2 15.4 15.6

5.0 5.1 5.5 5.6 6.6 7.9-8.2 9.09.2

LCGTO-LSDF(f) planar bent

7.8 8.0 8.2 8.4

8.1 8.1 8.4 8.4

(π, (d, (d, (d,

9.2 9.5 10.7 11.9 12.112.5 13.6 13.8

9.4 (d, l i a ) 9.9 (d, π, 10.3 (π, lib*) 11.2 (π A 10a ) 11.8- (σ) 13.3 14.3 (σ) 14.6

(a) experimental data : Batich, Réf. 25 (b) Veillard; Rohmer et al., Réf. 26 (c) Moncrieft et al.; Hillier, Réf. 27 (d) semi-empirical Green Function calculation: Bohm and Gleiter, Réf. 28 (e) scattered-wave Χ α calculation: Hancock et al., Réf. 29 (f) present calculations: allyl groups are planar or hydrogen atoms are bent

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

7a ) 13aJ 12ap u

6bJ

Aj

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Figure 1. Orbital diagram of the L S D F one-particle energy levels of isolated nickel, nickeKC^), bis(TC-allyl) nickeKC^) and the ( C j H ^ C L ) fragment. • and Δ indicate spin up and spin down electrons. The three-dimensional structure of bis(7c-allyl)nickel is shown on the left-hand side.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

236

THE CHALLENGE O F d A N D f ELECTRONS

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch016

Using a planar geometry for the H atoms, the calculated L S D F A S C F values for the P E agree rather well (within 0.2 eV) with experiment (cf. Table I). Using a bent geometry (30), the total energy of the transition metal complex is lowered by 0.9 eV. The corresponding calculated photoelectron spectrum agrees less convincingly with experiment. The main bonding mechanism, i.e. donation from N i 3d to allyl π*, and back donation from allyl π to Ni-4p, remains, however unchanged. The d orbitals of N i are barely affected by this geometry change, and the important shifts are found in the ionizations from ligand π and σ orbitals. It should be noted that the experimental P E was taken in the gas phase while the "bent" structure was deduced from measurements on the crystalline solid (30). Clearly, a full geometry optimization of the isolated complex would settle the question of crystal packing effects and clarify the details of the photoelectron spectrum.

Chemisorption of Carbon Atoms on the Ni(100) Surface. The next example demonstrates the capability of the L C G T O - L S D F method to predict the geometrical structures and vibrational frequencies of carbon atoms chemisorbed on the Ni(100) surface. The C / N i system is of fundamental importance in catalytic petrochemical processes and thus has been the subject of many experimental and theoretical studies. Despite these efforts, many aspects such as the equilibrium position of the C atoms (above or below the surface N i atoms) remained unsettled. Recently, a joint experimental and theoretical study of the chemisorption of carbon on Ni(100) led to a clearer understanding of this system (31). It was found that in the ground state, C is adsorbed in four-fold hollow sites above the surface with a N i - C distance of 1.79 Â, in agreement with the experimental value of 1.75 ± 0.05 Â, obtained from surface extended energy loss fine structure (SEELFS) measurements. Furthermore, the calculated vibrational frequency of the perpendicular mode of the adsorbed C atom, 407 cm is in excellent agreement with the experimental value of 410 c m . _ 1

_ 1

Vibrational Frequency of CO Adsorbed on Pd Clusters. In the next example, we present results for the vibrational properties of a C O molecule on a Pd surface, modelled by a cluster of 14 transition metal atoms (32,33). Furthermore, the influence of an external electric field is investigated using a smaller cluster, P d C O (33,34). In both cases, C O is assumed to be bonded in the bridge position. The chemically inactive Pd core electrons are described by a relativistic model potential as described in detail in Ref. 14. 2

Two coupling modes are considered: for the P d C O cluster the first mode (denoted as h) represents vibration of the rigid C O molecule with respect to the transition metal surface. The second mode is either the Pd-Pd vibration within the plane of Pd surface atoms (r) or out-of-plane stretch of the surface/sub-surface Pd-Pd bond (z). The total energy surfaces (h,r) and (h,z) are calculated for discrete points and then fitted to a fourth order polynomial. Variational and Quantum Monte Carlo (QMC) methods were subsequently applied to calculate the ground and first excited vibrational states of each two-dimensional potential surfaces. The results of the vibrational frequences ω using both the variational and Q M C approach are displayed in Table II. 14

If one assumes a rigid substrate which, at first, seems reasonable because of the high mass of Pd compared with the C O molecule, a frequency of almost 500 cm" is obtained for the vibration of the entire (rigid) C O molecule perpendicular to the surface. This value is in significant disagreement with the experimental value of 340 cm" (35). As can be seen from Table II, a substantial lowering of this "beating" mode occurs due to anharmonic effects in the coupling of the ζ and h modes. In other words, the vibration of the surface/sub-surface Pd-Pd bond stretching couples to the C O beating mode and lowers 1

1

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

16.

ANDZELMETAL.

237

Spin Density Functional Approach 1

Table II. Vibrational frequencies (cm' ) of the two coupled modes (h,r) and (h,z) obtained from a harmonic (Har), variational (Var) and Quantum Monte Carlo ( Q M Q approach

mode

Har

Var

QMC

240 498

192 521

192 521

196 498

65 402

66 397

(h,r): ω

Γ , usually a determinant with η single f-states. The actual wave function for a p a r t i c u l a r experimental s i t u a t i o n h, for example an i n i t i a l state h = i or a f i n a l state h = f, i s

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch017

n

n

|h>=EC |f >

(8)

n h

t h i s being the so called configuration i n t e r a c t i o n (CI) or multiconfiguration (MC) schemes (see, for example (8-10), where the t o t a l number of f electrons n w i l l not, i n general, be an integer value. The configuration wave function |f > i s usually represented by a determinant for the " a c t i v e " space where only the ν valence e l e c ­ trons contribute to one column each: η columns for the f electrons, v-n-1 columns for the d electrons and one column for the (s-p) conduction electrons for each RE atom η - nf. In t h i s case the amount of atomic-like f-character: f

n

atomic n^ = ZnC, f hn η v

n 2

2

a

, < n η — f

/\ (9) 0

£

t o m i c

There should be some additional contributions to n ^ in Equation 9 , from the cross terms i n which can be important, mostly i f several C are of the same order of magnitude and at least two of these terms have η > 0. For a given h the parameter n | i s a suitable quantity to measure the amount of very l o c a l i z e d , atomic-like f-character, and i t s r e l a t i o n to the properties of the system. atomic i unique because i t depends on the choice of and ψ£» , but a consistent scheme w i l l lead to avoid t h i s , well known, shortcoming of population analysis. The configuration i n t e r a c t i o n ( i n the MC scheme) wave function |h> i s needed i n the condensed matter problem because the reduced symmetry of the c r y s t a l and, mainly, the e f f e c t of the scattering wave boundary conditions for energies above the i n t e r s t i t i a l po­ t e n t i a l or the existence of bonding states between the anions and the heavy metal ions, w i l l allow several types of coupling or cor­ r e l a t i o n between the f l e v e l and the valence or conduction e l e c ­ trons. This coupling, usually denoted by Δ , i s i n fact the sum of several contributions which are responsible for either MV, Kondo or other e f f e c t s . In general there w i l l be a dominant configuration n n

t o m : L C

n

s

n o t

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

252 η C

THE CHALLENGE OF d AND f ELECTRONS contribution

n+1

(°

r

C

(C Δ

n-P

n

~

1.0)

Ε

and

Ε

then a second with lower amplitude E

- ^ η + 1 "~ ίΐ ) » î}

b e i n

t h e

8

total

energy

computed

n

for a fixed f configuration. For example i n the d i f f e r e n t phases of the Ce problem the ground state i s usually described as i n a f configuration, but i f a core hole i s present or for the case of BIS spectroscopy the f configuration w i l l then take a larger weight. In the ground state of α-Ce the increased i n t e r a c t i o n between the f electrons and the conduction band w i l l bring i n the f° configuration because i n t h i s case the energy difference E ^ i c ^ a ^ g i - ^ f ^ ^ o s i s small; t h i s being one of the reasons for the complicated behaviour of Ce metal and Ce compounds. For these materials the CI method has been extensively used by Gunnarson and Schonhammer (13) (and re­ ferences therein). The c o e f f i c i e n t s Cjj w i l l i n general depend on the hybridization of the f electron wave functions which i s the o r i g i n of the forma­ tion of an f band. But, as already pointed out by Haldane (14), these materials tend to behave as "low density" or impurity l i k e systems at intermediate temperature. This consideration i s import­ ant because i n a material such as CeAl3 , at intermediate tempera­ tures, transport properties correspond to incoherent scattering by each RE atom at the maximum resonance scattering cross section (15), as expected from an impurity l i k e Kondo-resonance, and the coherence between the RE atoms, although fundamental for the under­ standing of the low temperature regime, can be introduced a poste­ riori. We can describe the process of pressure induced valence tran­ s i t i o n s (as i n F i g . 1 of Ref. (3)) as a three step phenomenon. The f i r s t , i n the pressure range Ρ 95

49

-

005

0

The s u b s c r i p t c i n d i c a t e s t h a t o n l y c o n n e c t e d diagrams have to be taken. The last equation follows after applying a linked cluster theorem. Furthermore the abbreviation < . . . · > = < Φ Ι . . . ιΦ > has been used. The energy i s evaluated a f t e r expanding 0

2 Σ i j l

Ε = -

+ Σ ijmn

Σ 11·

0

Dij

c

+ (

Dij . The c i r c l e s s y m b o l i z e atoms and t h e five segments the different d-orbitals. The d-electron occupancy p e r atom i s chosen t o be 2 . 5 . 0

Figure 2. Charge f l u c t u a t i o n s as f u n c t i o n o f d-band f i l l i n g n U . E l e c t r o n c o r r e l a t i o n s influence strongly the energy difference between nonmagnetic and magnetic states, l e a d i n g t o d r a s t i c changes o f the Stoner-Wohlfarth crit e r i o n for the onset of ferromagnetic order. The r e a s o n i s t h a t e l e c t r o n i c charge f l u c t u a t i o n s are smaller i n a ferromagnetically ordered than i n a nonmagnetic state. Therefore e l e c t r o n c o r r e l a t i o n s decrease the energy by a l a r g e r amount o f a s t a t e w h i c h i s n o n m a g n e t i c , t h a n o f a f e r r o m a g n e t i c s t a t e . F o r example, i n Fe t h e energy g a i n d u e t o f e r r o m a g n e t i c o r d e r i s 0 . 5 6 e V / a t o m when t h e HF a p p r o x i m a t i o n i s made a n d a r a t i o U / W = 0 . 4 4 i s assumed. When d e n s i t y and in addition spin correlations are i n c l u d e d , t h i s energy reduces t o 0.22 eV/atom and 0.15 e V / a t o m , r e s p e c t i v e l y (10) . I n t h e LDA t o t h e density functional theory, Hund's rule correlations are not taken into account, because t h e y are n o t p r e s e n t i n an unpolarized homogeneous electron gas from which the exchange-correlation potential i s taken. When t h e L S D approximation is applied instead, they are partially included. Spin correlations, however, modify the generalized Stoner parameter strongly ( 1 3 ) . The latter can be related to the exchange correlation energy E ( M ) f o r f i x e d m a g n e t i z a t i o n M by w r i t i n g

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch020

2

c

C

X C

Q

0

Mo E

X C

(M )

= E

0

x c

(0)

+

\

J dM I 2

x c

(M)

(13)

ο where I = I ( 0 ) i s the o r i g i n a l 4 d i s p l a y s the magnetic f i e l d parameter as obtained within mations for the case of Co. x c

Stoner parameter. Figure dependence o f the Stoner three different approxi­ I i i s the result of a

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

20.

STOLLHOFF & FULDE

285

Correlations in d and f Electron Systems

Ό

5

10

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch020

2

F i g u r e 3. Atomic spin correlations S as f u n c t i o n o f d-band filling n. U/W=0.5. (Adapted from r e f . 9.) 0

1.6

ι—ι—ι—ι—ι—ι—ι—r

F i g u r e 4. S t o n e r parameter I(M) and l o s s o f k i n e t i c energy D(M) (dashed line) for Co as functions of magnetization. 1^ - f u l l c o r r e l a t i o n c a l c u l a t i o n , I2 - neglecting spin correlations, I3 - n e g l e c t i n g a l l c o r r e l a t i o n s , r e s u l t i n g from J ^ j . (Adapted from r e f .

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

286

THE CHALLENGE OF d A N Df ELECTRONS

complete correlation calculation. I2 is the Stoner p a r a m e t e r when s p i n c o r r e l a t i o n s a r e n e g l e c t e d . Since spin correlations play no role for fully magnetic s t a t e s , b o t h c u r v e s become e q u a l a t M =1.6. W h i l e I2 does not d i s p l a y s i z e a b l e magnetic f i e l d dependencies and compares in this respect with results of LSD c o m p u t a t i o n s , 1^ i n c r e a s e s b y 20% f r o m M=0 t o M . 13 finally is the curve obtained when a l l exchange contributions (~Jij) to the interaction part of the H a m i l t o n i a n ( E q . 1) a r e t r e a t e d i n H F a p p r o x i m a t i o n . I2 a n d I3 m a y b e c o n s i d e r e d a s l o w e r a n d u p p l e r l i m i t o f t h e d e f i c i e n c i e s o f L S D . A l t h o u g h a c h a n g e o f I b y 20% may s e e m s m a l l , i t h a s t h e e f f e c t o f c h a n g i n g t h e C u r i e temperature T by approximately a f a c t o r o f two, because T ~ ( I N ( O ) - l ) / * a n d I N ( 0 ) * 1, However, even t h a t is not s u f f i c i e n t i n order to b r i n g the l a r g e r c a l c u l a t e d values for T i n agreement w i t h experiments. This is due t o the fact that Stoner theory does not contain fluctuations of the order parameter. For improved calculations of T see e . g . Ref. (14). Another point of considerable importance i s the n o n l o c a l character of the exchange. The l a t t e r always favours non-uniform distributions of electrons (or holes) a m o n g t h e d i f f e r e n t d o r b i t a l s , e . g . e g a n d t2g orbitals, when t h e system is cubic. Direct Coulomb interactions as well as correlations favour uniform occupations of the different atomic orbitals and therefore counteract the effect of nonlocality of the exchange. Despite this, the anisotropies caused by exchange are important, i n p a r t i c u l a r f o r b u l k N i (10) as w e l l as f o r i t s s u r f a c e (15). Finally it is of interest to compute spin c o r r e l a t i o n s between n e i g h b o r i n g s i t e s , i.e. m a x

m

a

x

c

1

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch020

c

c

c

2

S (o)

= , o

(14)

0

where δ denotes a n e a r e s t neighbor o f s i t e 0. IΨ> is c a l c u l a t e d by s t a r t i n g from the nonmagnetic S C F - s t a t e as b e f o r e , b u t b y i n c l u d i n g i n S ( s e e E q . 6) a l s o operators of the form O i j ( 1 , l + a ) = £ > i ( 1 ) S j (1+©) . The effect of these operators i s that additional ferromagnetic corre­ lations betweeen electrons on neighboring sites are b u i l t i n t o I V » > , e x c e p t f o r b a n d f i l l i n g s c l o s e t o ncf. The i n d e x 1 s t a n d s for l i g a n d o r b i t a l a n d we a s s u m e t h a t i t i s r a t h e r extended s o t h a t we m a y n e g l e c t C o u l o m b r e p u l s i o n s w i t h i n that orbital. The H a m i l t o n i a n t h e n r e a d s Η = ε,Σ σ

1σ1

χ

+ σ

c

f Σ φ σ

+ V Σ (φσ+ΐσ^σ) σ

σ

+

Un£nf * τ

(15) + with η = f f and U v e r y l a r g e . We w a n t t o d i s c u s s t h e s o l u t i o n s o f t h e e i g e n v a l u e problem f o r two e l e c t r o n s . F i r s t we s e t V = 0 . In that case, because ci>tf the ground s t a t e i s a quartet w i t h e n e r g y E = c i + C f , i . e . one e l e c t r o n i s i n t h e f o r b i t a l and the o t h e r i s i n the 1 o r b i t a l . The e x c i t e d s t a t e i s a s i n g l e t with E = 2 c i , i . e . both electrons are i n the 1 orbital. W h e n VfO i s t a k e n i n t o a c c o u n t , the grounds t a t e quartet s p l i t s i n t o a low l y i n g s i n g l e t f

σ

a

a

Q

s

72

•*o> =

(φΐ-φΐ)

U-(V/Ac)2)

I0>

- ^

φ|ΐΟ> (16)

with

energy

l*cl>

=

2

Ε =εχ+ε£-2ν /Δε 0

(1-(V/Ac)2)

and a

φ|ΐΟ>+

triplet

j2 ( φ ί - φ τ ) Ό > (17)

Φΐ

,0>

'*c2> = ? ε 3 > = ΦΪ»0> w i t h energy Et=ci+Cf. We h a v e s e t ε χ - ε £ = Δ ε 4 ) . The f o r m a t i o n o f t h e s i n g l e t ιΨ> w i t h triplet state of e x c i t a t i o n energy Ε =2ν characteristic feature of strongly correlated systems. When t h e f o r b i t a l i s e m b e d d e d i c o n d u c t i o n e l e c t r o n s , t h e energy g a i n due t o f o r m a t i o n becomes , ψ

0

β χ

ΔΕ= Dexp[

2N(Ô)V^

2

(see figure an e x c i t e d /Δε, is a f-electron n a sea of the singlet

]

2

(

1

8

)

i n s t e a d o f ΔΕ=2ν /Δε, as i n the case o f two o r b i t a l s . H e r e D i s t h e c o n d u c t i o n - e l e c t r o n band w i d t h and 2N(0) is their density of states. We h a v e set the Fermi energy equal to zero. The energy gain is usually identified with a characteristic temperature kgTi^E,

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch020

288

THE CHALLENGE OF d AND

f ELECTRONS

t h e Kondo t e m p e r a t u r e . The low l y i n g e x c i t a t i o n s l e a d t o h e a v y - f e r m i o n b e h a v i o u r when t h e i o n s w i t h f e l e c t r o n form a l a t t i c e . The above calculation suggests that a singlet f o r m a t i o n due to strong correlations with a triplet e x c i t e d s t a t e s h o u l d be f o u n d i n a p p r o p r i a t e m o l e c u l e s . The effect r e q u i r e s an e v e n t o t a l number o f v a l e n c e electrons. I n o r d e r t o d e t e c t i t one s h o u l d s e a r c h e.g. f o r m o l e c u l e s c o n t a i n i n g Ce, w h i c h a r e d i a m a g n e t i c , b u t w h i c h show a f - e l e c t r o n c o u n t c l o s e t o 1, when p h o t o emission experiments are performed. The f o r m a t i o n o f a s i n g l e t s t a t e due t o s t r o n g c o r r e l a t i o n s i m p l i e s a l s o a new k i n d o f e l e c t r o n - p h o n o n coupling. The e n e r g y g a i n Δ Ε due t o s i n g l e t f o r m a t i o n d e p e n d s on t h e h y b r i d i z a t i o n V, w h i c h i n t u r n d e p e n d s on p r e s s u r e Ρ o r volume Ω · In p a r t i c u l a r i n a s o l i d t h i s dependence ΔΕ(V) is very strong (see Eg. (18)), r e s u l t i n g i n a s t r o n g e l e c t r o n phonon c o u p l i n g . Its s t r e n g t h c a n be c h a r a c t e r i z e d by an e l e c t r o n i c G r u n e i s e n parameter

= η

-

din T din Ω

K ±y

< '

M e a s u r e d v a l u e s o f n a r e a s l a r g e a s 100-200 i n h e a v y f e r m i o n systems (17). One important problem, which i s presently under i n t e n s e i n v e s t i g a t i o n s i s t h a t of the Fermi s u r f a c e of s t r o n g l y c o r r e l a t e d f - e l e c t r o n systems. I t was a s u r ­ prise, at least to the present authors, that the m e a s u r e d F e r m i s u r f a c e o f t h e h e a v y - f e r m i o n s y s t e m UPt3 (18) i s v e r y much i n a c c o r d w i t h t h e one computed w i t h i n LDA ( 1 9 ) . T h e r e i s no a p r i o r i r e a s o n why t h e t o p o l o g y o f t h e F e r m i s u r f a c e s h o u l d come o u t c o r r e c t l y when e l e c t r o n c o r r e l a t i o n s a r e s t r o n g and a LDA i s made. But f o r UPt3 i t d o e s come o u t s u r p r i s i n g l y w e l l , a l t h o u g h t h e m e a s u r e d e f f e c t i v e masses a r e o f f b y a f a c t o r o f o r d e r 20 a s compared w i t h t h e c a l c u l a t e d o n e s . Detailed i n v e s t i g a t i o n s h a v e shown (20) t h a t t h e g o o d a g r e e m e n t in the case o f UPt3 i s due to a large spin-orbit s p l i t t i n g and a c r y s t a l - f i e l d (CEF) s p l i t t i n g , w h i c h i s much l e s s t h a n kgT^, i . e . t h e e n e r g y g a i n due t o s i n g l e t formation. In that case, the theory becomes a one-parameter ( w h i c h i s t h e e f f e c t i v e mass) t h e o r y , and t h e t o p o l o g y o f t h e F e r m i s u r f a c e due t o the heavy q u a s i p a r t i c l e s i s c o m p l e t e l y d e t e r m i n e d by t h e g e o m e t r y of the unit c e l l . I n c a s e s i n w h i c h t h e CEF s p l i t t i n g i s l a r g e r t h a n kgTR, one e x p e c t s d i f f e r e n c e s b e t w e e n t h e m e a s u r e d F e r m i s u r f a c e and t h e one w h i c h f o l l o w s from a p p l y i n g t h e LDA. I n o r d e r t o improve t h e computation o f t h e F e r m i s u r f a c e one c a n p r o c e e d a s f o l l o w s , at l e a s t f o r Ce compounds. One a p p l i e s t h e LDA t o t h e d e n s i t y f u n c t i o n a l theory f o r a l l e l e c t r o n s , except the f-electrons. The potential a c t i n g on t h e l a t t e r i s

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

20. STOLLHOFF & FULDE

289

Correlations in d andf Electron Systems

d e s c r i b e d by an energy dependent phase s h i f t $1=3(ε), for which a simple, phenomenological ansatz is made. Only these channels w i t h i n t h e 1=3 m a n i f o l d o b t a i n a phase s h i f t d i f f e r e n t from z e r o , w h i c h have t h e symmetry Γ o f the c r y s t a l - f i e l d ground state. The l a t t e r is usually known from inelastic neutron scattering experiments. T h e s l o p e 70.

suggest

the

of

to

by be

of

bodies

light)

as

two

known has

structure

it

been by

is

non-

does not

obey

assumes),

it

is

safe

to

(NRQM)

the

simplest

These for

simple

velocity

Bohr model

that

this

will

(NR)

(which

mechanics

conclude

to be

study

of

to

of

if

that

one-electron for

(which the

use

behaviour

appropriate

happen

understanding

that

the

comparable

considerations would

a proper

special

significantly

Newtonian

and m o l e c u l e s would not

Ζ predicts

a c c o r d i n g to

is

non-relativistic infinite')

systems moved a t

that

the

well

Schrôdinger equation

fast-moving

velocity

quantum m e c h a n i c s

even

charge

and m o l e c u l a r

Lorentz-invariant

predicted

atoms

these

not

also

Schrôdinger equation

atomic

However,

constitute is

relativity.

equation

of

light.

is

finite

of

non-relativistic

with

of

that

of

in

it

behaviour

(with

velocity

electrons

investigate

and c h e m i s t s .

relativistic; the

to

It

an

the of

atom atomic

therefore the

electronic

0097-6156/89A)394-0291$06.00/0 ο 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

292 structure (Z>75)

of

the

mechanics

atoms

which

relativity

is

is

in

involving

quantum m e c h a n i c s

conformity

mandatory.

with

Various

and

special

theory

Dirac

who

in

d i s c o v e r e d the

(4)

1928

(linear

in

only

Lorentz-invariant

of

is

momentum

electron

predictions Lamb and

shift forms

atomic that for

agree

the

(and H

basis

(6)

+

be

cannot

made

remain

theory

with

the

is

being

for of

the

for

decades because

not

for

to

(in

the

electrons

of

first

in

calculation

on the

relativity

and

electrons

of

shown b y

compared to

it

known

(7)

that

a t o m was

out

due

to

of

that an

came

the

the

electrons, total

angular

the

notation

i

(7)

and i

the

electrons

case

behaviour

who

of

i-1/2

Swirles

no

valence

that

of

valence

that

whereas

increased were

and

5d)

effect,

the (8).

-

which

d and

relativistic

and j

i+1/2,

A few

Hg

years

weakening

orbitals

s electrons, for

the

more

(i.e., for

Mayers

in

contracts thereby

important

Since

atomic

charge

indirect

mean

electron,

5d e l e c t r o n s

these

strongly

values.

5d e l e c t r o n s

relativity

the

the

less

larger

the

the

by

one-electron

the

unimportant

designate

introduced by

all

nuclear

This

-

was

of

relativistic

found

was

energy

viz.

b e i n g most

momentum j

dynamics

there

H E A V Y ATOMS

occupied by

in

the

primarily

their

the

orbitals

binding

shielding of

and

chemists

decreased s i g n i f i c a n t l y ;

effect,

although

sp>d>d.

level

the

systems of

EFFECTS

reliable

systems.

so

face

the

very

effect

and the

very

to

appreciable

correlation

calculations

(10-13),

(RIP),

is

electron

which happens

RELATIVISTIC

using

elements

due

very

and 5f

an i>0

are

is

these

i n v o l v i n g heavy

correlation

effect

affected

the

increase

calculation of

electrons

of

valence

near

order

6d)

destabilization

time

momentum o f

elements

c h e m i s t s must

the

effect.

electrons)

2

s and ρ e l e c t r o n s

(and

elements

also

of

especially valence

the

systems

quantum

the

splits

heavy

heavy

addition

amount

effects,

chemistry

(and

/

relativistic

in

These e f f e c t s

electrons

for

5d

interaction

±1/2.

in

the

1

relativistic

the

electrons

atom

indirect

and p

angular

relativistic

destabilized while

(s

therefore

direct

smaller

the

velocity

and are

increasing total

uranium

to

occupying penetrating

momenta

spend an a p p r e c i a b l e

that

for

the

due

electrons

angular

s t a b i l i z e d by they

293

Chemistry of Third-Row Transition Elements and Actinides

termed

to

one

are

the DFAO

small.

the

core,

accommodated i n

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

the

THE CHALLENGE OF d AND f ELECTRONS

294 valence

RMOs c o n s t i t u t i n g

general, The b a s i s four

thevalence

a r e composed o f s e v e r a l s e tused t o express

different (I)

types

DFAOs

wavefunction

DFAOs

which,

i n

o f theconstituent

thevalence

atoms.

RMOs c a n c o n s i s t o f u p t o

o f functions (14):

that

aren o tcompletely

filled

i n the isolated

atoms ; (II)

DFAOs be

completely

filled

significantly

i n the isolated

affected

atoms

by the formation

that

might

o f the

molecule ; (III)

E x c i t e d DFAOs u n o c c u p i e d i n t h e i s o l a t e d might

contribute

significantly

atoms

b u t which

to the formation

of the

molecule ; (IV)

Functions, needed

called

augmenting

to describe

valence

charge

small

functions

residual

distribution

(AF),

which a r e

distortions

onformation

of the

o f the

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

molecule. Although (I), the

s e t must

needed

same a t o m , basis,

A F s make features

called

selected

functions

atom

lost

o f type

thecoefficients

predicting

have

incorrect RCI

energy, The

techniques

theresults

well

importance

of

calculations significant

This

by using the

i s achieved by using the to construct

wavefunctions

properties

other

feature

as the computational

(14) and because

However,

o f space

we p r e s e n t

relativistic

a summary

D i r a c - F o c k SCF a s basis set

f o r AuH t o d e m o n s t r a t e t h e

as well

as electron

transition

correlation

elements. RELATIVISTIC

CALCULATIONS

binding

t h e 6s-6p

FOR_AuH

hybridization

o f thebonding so that set.

energy

and

distributions, etc.

t h e I s DFAO o f t h e h y d r o g e n a t o m .

( 1 4 ) show t h a t

more

can

than the

s e t f o r AuH c o n s i s t s o f t h e 5 d , 5 d a n d

the chemical basis

experimental

ionicities and

a r e remedied

D F S C F E B S AND

(RCI)

energy

INTERACTION

o u tu s i n g a n extended

functions

FULLY RELATIVISTIC

t h e g o l d atom p l u s

enter

carried

to the

o f thebinding

molecular

as well

fully

o f third-row

CONFIGURATION INTERACTION chemical basis

contribute

charge

here.

ab i n i t i o

of therelativistic

i n diatomics

INITIO

moments,

(III), f o r

i s deduced b y

a n RMO c a l c u l a t i o n

methodology

basis

importance o f

CONFIGURATION

(14).

molecular

i n bonding

i n the gold

set.

described elsewhere

o f our

The

t h e RMO a n d R C I

be repeated

o f 27 v a l e n c e

effects

The

from Both

as RCIc a l c u l a t i o n s

(EBS)

AB

cannot

they

o f overestimating

dipole

o f our

arefully

limitations

to the Hence t h e

played

5 d DFAOs

t o type

thefraction

calculations

e.g.,

details

(14).

d i s s o c i a t i o n products

wavefunctions.

role

i n t h e g o l d atom,

thebasis

been used to calculate

total

t o t h e RMOs.

_The

belonging

THE R E L A T I V I S T I C

u n o c c u p i e d RMOs r e s u l t i n g accurate

Since

t o DFAOs b e l o n g i n g

t h e 5d and

with which

from

o f RMO t h e o r y

t o perform

o f type

work.

consisting o f functions

(III).

elsewhere

orbitals

6p DFAOs

them

(CB),

and

o f which

i n detail

involving

by excluding

accurate

contributions

RMOs a n d b y c a l c u l a t i n g

defects

RIP

(II)

(II),

CORRELATION E F F E C T S : The

small

(I),

t h e 6p a n d

examining valence

only

types

arediscussed

example,

a l l the functions

o f thebonding can be understood b y u s i n g a

the chemical basis

from

hybridization

of

include

f o rquantitatively

AFs a r e constructed t o be orthogonal

essential

by

thebasis

AFs a r e only

However,

6 s DFAOs Our

6p a n d 6p DFAOs

since

DFSCF

i s n o ta do n o t

1.0 eV o u t _ o f t h e

o f 3 . 3 6 eV a r e l o s t

i f t h e 5 d and 5d

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

21.

MALLI

DFAOs two in

Chemistry of Third-Row Transition Elements and Actinides

and t h e i r

DFAOs which

energy

value.

The

5 d DFAOs

in

ten of

electrons 0.663

of

The

interaction

were

relativistic (the bond

shows

(R )

(a> ).

2102

for

R

the

non-relativistic

are

in

2.879

serious au and

1

improved wavefunctions 2.963

au and Our

2102

results

molecular

cm"

of

even

are

the

au and

the

R

features

chemical basis

of

1745

;

the

is

most

readily

from

the

a RIP

computation.

three

hybrid

orbital

in

interaction is,

which

is

with

however,

with α

the

responsible the

found

erroneously

interaction 5d-6s

whilst energy

the

direct

gap

of

polarity

AuH.

for

two

from p r e v i o u s

an approximate play

conclusion

is

incorrect

be

left

in

SCF c a l c u l a t i o n our

relativistic

the

5d-6s

a the

hybridization

hybridization calculation

solely

Is

greater

in

configuration

the

core

of

potential

relativity

role of

AuH. of

in

the

our Our

6s

to

interaction

(ECP)

that

eV

for

5d

5d 1 0

basis the

The

5d-6s in

the

greater

results Hay

et

al.

calculation

the

that

extended

1.682

the

e.g.,

chemical bond

result

DFAO. the

predicts

contrary

the

DFAOs

hybridization

calculations;

of

greater

because 5d

non-

degree

The

arises

the

The

the

significantly

5d-6s

from

DFAOs.

5d and

NR w a v e f u n c t i o n

rigorous

a value

the

stabilizes

reduces

the

i n view

core

formed

orbital 5d-6s

its

nonbonding o r b i t a l s .

effect

the

dominant

predicts

a π gold

important

hydrogen

destabilizes

treatment

to

to a

from

which

non-relativistic

and

an e f f e c t i v e

appear

the

conclusions are

less

of

bonding through

bond i s

effects to

two

reducing

This

revealed

understood

demonstrated

show t h a t

the

the

Moreover,

concluded from

cannot

of

the

the

relativity.

be

seriously underestimates

of

effect

above-mentioned

reported using

one

leading

case.

Our

also

relativistic

these

thus

6s

third

limit,

DFAO. in

that

gold

importance

relativistic

not

in

relativistic

combination

(11)

the

Is

absent

predicts

hybridization

indirect

be

calculation

relativistic

the

for

hydrogen

to

between

relativistic

s p i n and

of

RMOs o b t a i n e d

l o c a l i z e d AuH RMOs,

non-relativistic

are

bonding,

cannot

RMOs c a l c u l a t e d

orbitals

of the

β

localized valence

spin

latter

ω .

non-relativistic

σ

This

and

using

changed by

of

au

(18)

predictions

sets,

the

with

the

values

and

e

of

velocity

2.993

cm

using

with β

the

compared

using

become

theory.

of

contracts

calculations

for

basis

vibration

be

substantially

qualitative

The

large

calculations

The p r e d i c t i o n s

from RCI

both

A comparison

experimental

quantitative

AuH a r e

with

to

and

and

value

the

5d

vibration

predictions

respectively,

of

the

b y RMO c a l c u l a t i o n s

increases

3.431

with

resulting ,

the

AFs on

using a

significantly

respectively.

show t h a t

properties

Furthermore,

1

orbital

the

a

various

energies.

increasing

relativistic

values

cm" ,

with

at

only

experimental

several

and fundamental

respectively,

disagreement

2305

of

these

calculation

predicts

the

calculated

these

core,

RCI

wavefunction

addition

relativity

the e

core of

molecular

substantially

a n d u? ,

e

the 20%

was

from

simply by

Thus

e

in

AuH p r e d i c t e d

the

the

corresponding n o n - r e l a t i v i s t i c

that

and

e

frequency cm

predicted

performed

length

of

in

about

the

The bond l e n g t h

then

used)

is

methods

and e x a c t l y

latter

light

placed

energy

placed

Indeed our

an accurate

distances by both

including AFs.

frequency

are

are

set.

which

requires

valence

Au and H atoms.

set

eV,

the

internuclear

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

chemical basis

computation

configuration

of

the

these

binding

the

associated electrons

enter

295

D

1 0

core

does

in

AuH;

this

electrons set

e

of

calculation with

(EBS), AuH the

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

and

DF

296

THE CHALLENGE OF d AND f ELECTRONS

above-mentioned including 2.014




and of

|/?>).

the

We

the

express

|f m >

NRL the

molecular

components

with

terms

the au;

relativistic

total

through

order large

and

3.877

(58916eV!).

defined

|d m >

functions

component

two-component RMOs c a n b e

m >,

than

2165.19

Pyper

m

relativistic

bound at

relativistic

and |p

the

be

RMOs i n where

like

(the

au

3.5

calculated

shall

s

ThO

although

3.877

m

for

molecule

R -

|i m >

eV

molecule

the about

However,

3.5d)

R -

angstrom.

that

at

calculated

experimental

9.0

indicates

predict

NRL D

prediction.

The

0.529171

atoms.

c a n be

calculations

b)

1 au -

experimental

(negative)

au.

non-

separations

Relativistic

)

Relativistic

(ref.

the

18

doubly

(RMO) as

was

and

III.

various

b

ThO

appropriate

DF S C F a n d

various

for

Th atom

Thus,

the of

-0.437,

energies

(D )

dissociation

(au)

a

spinors as

calculated

-0.938,

set

the

orbitals

'cores'

9

Non-Relativistic

R

of

SCF method

taken

relativistic

dissociation

Table

the

atomic

the

The

-1.055, total

basis

thereby

molecular

ThO was

products

-1.246,

in

including

(numerical) for

valence 6d DFAOs

DF S C F w a v e f u n c t i o n .

elsewhere(14).

calculated

and

constructed via

wavefunction

antisymmetrized

the

7s

relativistic

were of

the

thus

6p,

valence

occupied valence spinors

and

6s,

299

equations the

of

the

functions number

momentum

m and correspond

Therefore,

|ρσα>,

have

|ρπβ>,

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

the |daa>,

THE CHALLENGE OF d AND f ELECTRONS

300 |άττ0>,

|faa>,

bonding in

|fc70> f u n c t i o n s ,

characteristics

question.

the

double

The v a l e n c e

group

representation has

the

It

-

of

|7el/2>

«

|6el/2>

-

|5el/2>

and Th

the

0

(using

the

with

calculated the

D

below

the

small

of

the

|2ρπα>

the

0

claimed

(lying

and

|6dπα>

below

the

8|e3/2>

value

of

D

due

e

au,

energy

about

molecular

corresponding -1.531,

-0.165

au

0.51 D

eV

to

the

2165

energies

at

neglect D

at

R -

-0.837, these

MOs;

NRL c a l c u l a t e d

orbital

energy

for

than

the

ThO a r e

-0.326,

between the

given

lowest

above.

relativistic

lying

valence

Thus

the

IP

of

low

by

-1.50

eV

compared to

the

experimental

u n d e r s t o o d as

the

calculated valence

N R L HOMO ( a t

pure

on the

au)

whereas hybrid with

consists

in of

the

contrast |7sa>

of

an

the

(-0.936)

coefficients

eV p r e d i c t e d

almost

calculated and

given

|βάσ> in

by

the

IP

RHOMO ( g i v e n

parenthesis.

HOMOs i s

NRL c a l c u l a t i o n of

|6d5> o r b i t a l

(-0.350)

the

-0.299

higher

eV.

4.49

for

much

the

The

whereas

3.877

corresponding difference

at

calculated

-0.315,

lie

energies

eV)

NRL

separations.

values

orbital

(16.33

for

eV

total

higher

yield

correlation

and 0.73 the

eV

(18). to

orbitals

easily

the

au

eV

expected

1.81

au

-0.948,-0.887,

0.6

9.00

electron

9 valence of

and

internuclear 3.877

in

whereas

of

eV!)

and

-6.00

au)

not

-1.140,

difference

of

is

for

the

is

IP

8.7

2ρπ 0

ionization eV

of

e

the the

6.14

However,

(58912

these

of

of

reported

R-3.877

between

predict au

involving

orbitals

calculated

(at

lies

e

correponding r e l a t i v i s t i c a

the

respectively.

DF S C F e n e r g y

NRL o r b i t a l

bonding

valence

βάσ

experimentally

NRL c a l c u l a t i o n s is

-0.379|6ρπ£1/2>

DF S C F w a v e f u n c t i o n

and 4.277

can be

and

valence

viz;

Koopman's theorem) is

e

The

3.877

the

just of

substantial

the

effects.

and

RMOs

Moreover,

R -

is

au

forms:

a n d 2s,

experimentally,

There

|8e3/2>

| βάσ>

6daal/2>-0.253|6paal/2>

The

be

Th atom w i t h

12s a n d

2saal/2>-0.123

(IP)

to

(using

irreducible

R =

-0.899 2ρσα1/2>+0.363

agreement

and

at

-0.215

excellent

valence

|9el/2>

-0.Ill|2sal/2>

-0.734|6paal/2>+0.629|2saal/2>

atoms,

the

the

RMO

following

2ρσ and

single

σ - h y b r i d on the

from

a bonding combination of

potential

A

diatomic)

-0.350|6daal/2>

calculated valence

the

valence

(18).

heteronuclear

as

additional

the

species

O.813|2p^l/2>+O.418|6d7r01/2>,

-

6d7r,

the

into

molecular

-0.815|2ρπα3/2>-0.417^πα3/2>+0.095|6ρπα3/2>. three

are

These

is

orbitals

next

RMO)

7s-6d

The v a l e n c e

|8e3/2> The

a

however

valence

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

of

for

insight

the

ThO d e s i g n a t e d

notation

the

can give

RMOs o f

form:

contribution

atoms.

RHOMO,

of

which

valence

RHOMO o f

-0.936|7scral/2>

consists

bonding Th

(AIR)

etc,

the

theoretical

following

|9el/2>

of

6.00

above)

orbitals Since

of

the

eV. Th

is

This

7s

Th

DFAO

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

R

=

atom,

consists the

1.50 too

of atom is

a

21.

MALLI

stabilized is

(due

differences the

in

easily

pure

the

(due the

|2s

the

Th

(due

to

in

the

it

can be

orbital

and

the

considerably

and

that

effects

due

to

atom

the

bonding

in

species

(>2000°C)

in

The

at

5f

electrons in

geometry

(linear)

of

the to

which

For 5f

the

i.e.,

6d a

DFAOs

total

its

core.

was

found

the

of

The at

to

a

the

Th0

is

the

the

the

U atom

U:

the

lowest

au;

diatomic. R -

3.55

the

2346

molecular

total

to

be

2s

the

at

(1

with 3

of

2

the

in

to

R -

3.55

UO p r e d i c t s

relativistic

calculation,

it

=

atom

a

D

Thus, e

of

should not

R -

for

at

of

R

»

6p, 0

7s,

atom;

atomic

[core]

contains

also

was

3.05,

kept

that

a

as

single for

the

calculation

only the

and

and

binding

limit

in

au

s t i l l 4.05

-0.11971

above

twice be

diatomic

-28145.039469

although

almost

This the

the

were

predict

eV)

2 +

2

valence

the

unbound by

27.211

au.

of

mentioned

energy

of

set

the

-0.14393,

fails

the

difference

and

following

where

at

in

the

performed

UO m o l e c u l e

s h o u l d be

UO m o l e c u l e

for

paramount

6d

basis

energy

while

(18). (U)

significant.

DFAOs

0

1

(U0 )

the

the

unbound by

hartree

for

the

were 2p

cm"

orbitals

structure

included

eV)

It

of

and

argon

5f

very

and

eV

in

isoelectronic

a valence

non-relativistic

hartrees energy

Th

of

corresponding non-relativistic

wavefunction the

the

7.8

0

2.5

T h O a n d UO b o n d s .

SCF w a v e f u n c t i o n

au p r e d i c t s

however, about

The

the

actinide

the

however,

be

is 5f

1 6

in

6

to

of

-

e

an

explanation

molecular

respectively.

spinors

6p 5f 6d7s

was

found

e

electronic

electrons

total

R — 3.55

DF

of

of U

roles

were

Γcore]

Is

a n d c*> x

role

and

of

the

wavefunction

4.55

au,

DFAO's

of

and

(-1.68

-0.17596

differ

temperatures

involving

calculations

electrons

viz

the

au u s i n g

au

determinant

6p

high

1

cm"

the

expected

-0.061739

it

energies

the

relativistic

spectrum

atomic

(bent)

2

terms

and U atoms

18

A n IR

825.0

valence

in

at

diatomic

unbound by au

and

of

the

bonding

ThO u s i n g

theories

6d

(-0.969)

DFAO

MO.

d i s s o c i a t i o n energy

eV.

-

β

exist a

Moreover,

molecular and

for

6s

explains

IPs,

significant the

|6sa>

effect)

is

the

the

|lel/2>

orbital

of

of

above

|lel/2>

whereas

predicted

very

investigate

and 4.55 of

configurations, electrons

ω

investigate

4.05

the

RMO

of

DFAO

the

diatomic.

5.6

of

UO m o l e c u l e ,

relativistic 80

to

to

found Th

ThO

of

relativity

3.55,

and

in

between

us

for

predicted

are

claimed

bond.

may b e

orbitals

3.05,

a hybrid

phase w i t h

led

order

prompted in

the

structure

actinide-oxygen

UO

be

the

6d

FOR UO

gas

15°K has

importance in

of

potential

electromic

with

to

relativistic

the

mentioned

valence

stabilization

NRL m o l e c u l a r

UO i s

the

ionization

matrix

atom;

participation

SCF CALCULATIONS The

an

the

orbitals

Th

that

there

the

the

bonding characteristics

relativistic

for

of

energies

stated

and

effect),

lying

The

direct

effect)

relativistic

lowest

calculated

the

Hence

DF

energies The

orbitals.

difference

the

relativistic

indirect

orbital

N R L MO i s

(-0.124)

atom

the

orbital

|6sa> v a l e n c e

and

and

direct

to

understood.

corresponding

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

to

destabilized

can be

301

Chemistry of Third-Row Transition Elements and Actinides

-0.973

eV;

UO m o l e c u l e the

the

DF S C F

is

total

NRL

much as

regarded

U0 at

as

the

better

than

DF SCF c a l c u l a t i o n . The

9|el/2>

relativistic consists

from

\5ΐπβ>

Malli

and

and Pyper

of

a

highest

occupied molecular

7sc7-6d m

functions.

s

(where

'e'

representation

ω -

1/2,3/2,5/2),

the

one-electron 9|el/2> -

T h e RHOMO 9 | e l / 2 >

denotes of

the

the

the

total

UO h a s

the

additional

RMO a n d 1 / 2 , 3 / 2 , 5 / 2

where ω i n d i c a t e s RMO),

of

two-dimensional

corresponds

angular

momentum

to

of

viz.

-0.873|7saal/2>

-0.3981 6 o V a l / 2 > + 0 . 1 8 3 | 5 ί π 0 1 / 2 >

- 0 . 1 7 6 | 5 f a a l / 2 > + 0 . 1 2 9 | 6d7r01/2> - 0 . 0 9 3 | 2 s a l / 2 > -0.069|2paal/2> It

is

an almost

contributions the

0

atom,

However,

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

are

where

the

following

from

n o n b o n d i n g RMO s i n c e

the the

|2saa>

coefficients

RMO 8 | e 3 / 2 >

form which

definitely

lying

clearly

involved

8|e3/2> ~

(-0.093)

in

are

just

shows

the

it

and

very

small

|2ρσα> (-0.069)

contains

DFAOs

given

below that

in

the

RHOMO h a s

|5fcS> a n d

bonding of

UO,

of

parentheses. the

16dn> D F A O s

of

U

viz,

0.727|2ρπα3/2> -0.465|5f*03/2> +0.344|6dπα3/2> -0.111|6ρπα3/2>

The v a l e n c e significant and, has

in

fact,

the

|5ίπα>

and

electrons contain

5f

in

the

DFAOs

the

We g i v e interaction

and

for

ground state

below

and

can o n l y be

destabilizes

these

DFAOs a r e

The

substantial

orbital

roles

the

expressions indicate

2ρπ DFAOs o f

0

clearly

their

for

the

energies 5f

to

in the

(electrons

p u s h e d up

the

associated (which

configuration)

u n d e r s t o o d due

which

spinors.

au)

\5£δβ>,

significantly

effect

very

atom

-0.502

actinides

electronic involved

U

the

results

so t h a t

clearly

the

in

contain the

of

for

(DFAOs)

bonding

6 d DFAOs a r e

energy

-0.38 These

spinors

UO a n d T h O d i a t o m i c s ,

also of

a n d 6 d DFAOs o f

respectively.

results

atomic

indicate

for

8 | e 3 / 2 > RMO a l s o

5f

+0.49

atomic

their

relativistic

UO w h i c h

atom,

-0.74,

5f

these

as v a l e n c e

the

the

significant

ThO,

Both

RMOs a l s o

of

very

6d and 5f

bonding

of

the

DFAO i n

that

indirect act

that

below

from

5 | e 3 / 2 > RMO ( w i t h o r b i t a l

|2ρπα> DFAOs, are

bonding. the)

the

coefficients

demonstrate

except

RMOs l y i n g

contributions

of

in

in

energy

the

and

valence

a n d 6 d DFAOs

in

respectively.

for

the

7|el/2>

and

π-bonding arising and the

due

6dπ a n d / o r

| 5 e 3 / 2 > RMOs to

the

5 ί π DFAOs

of

U

viz, 7|el/2> «

Ο . 7 8 8 | 2 ρ π 0 1 / 2 > +0.34616άπβ1/2>

-0.225 |

5ίπβ1/2>

-0.098|6ρπ01/2>, 5|e3/2> Although,

the

-0.74|5f*j33/2> +0.4915f*a3/2> 6 ρ π DFAO w a s n o t

these

valence

RMOs ( e x c e p t

noted

above);

the

|5f D F A O s

6 ρ σ DFAO o f

contribute

to

found

a very the

to

minor

be

involved

contribution

U contributes 61 e l / 2 )

-0.38 | 2ρπα3/2>. significantly to

a s much a s

RMO w h i c h h a s

the

8|el/2> the

|6da>

following

form: 6|el/2> «

-0.87|2ρσα1/2> +0.30|6daal/2> -0.29|6ρσα1/2> -0.29|5faal/2>

-0.2312s

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

in

as and

21. M A L L I The

calculated

relativistic

orbital

of the

7|el/2>,

8 | e 3 / 2 > RMOs a n d t h e RHOMO 9 | e l / 2 >

-0.5017,

-0.4551,

-0.4444,

-0.4281 and -0.2266

and u s i n g Koopman's theorem,

the calculated

e V f o r UO i s i n e x c e l l e n t

experimental molecular

value

calculated

agreement

o f 6 ± 0 . 5 eV ( 1 8 ) .

energies,

UO a r e c o l l e c t e d

e t c . , at various

i n Table

IV.

It

IV.

Calculated total

Non-Relativistic

(E

N R

(E

D F

+28144)

total

t h e UO m o l e c u l e

( E ^ ) and

i n au ( l a u -

( i n e V ) f o r U0 a t (R) i n a u ( l a u -

27.211 eV) various

0.529171

angstrom)

NON-RELATIVISTIC

D (eV)

(E

a

e

M R

+25379)

D (eV) e

-0.95728

-3.9165

-0.02107

-5.0213

3.447

-1.0394

-1.6800

-0.14147

-1.7451

3.877

-0.98150

-3.2573

-0.12528

-2.1855

4.277

-0.92535

-4.7852

-0.07869

-3.5142

-0.06620

-3.7930

-

a)

-

The experimental

(negative) respect

value

D

i s reported

e

indicates

t o b e 7.8 eV ( 1 8 ) .

the molecule

to be bound

A positive

(unbound)

Since

the lowest

total

molecular

e n e r g y was c a l c u l a t e d

at R —

a u , t h e c o r r e s p o n d i n g NRL c a l c u l a t i o n was a l s o p e r f o r m e d

3.55

au i n order

3.55

arising

to gain

insight

into

due t o r e l a t i v i t y .

It

eV w h i c h

i s about

energy

i s about

energy

at R -

ionization

2406 h a r t r e e s

3.55 a u .

potential

Moreover,

has

with

the following 9|el/2>NR -

than

that

the t o t a l

at R a D

value.

of

molecular

1.6 eV lower

o f 6 . 1 4 eV w h i c h ,

the experimental

e

NRL m o l e c u l a r

t h e NRHOMO 9 | e l / 2 > p r e d i c t s i s about

at R =

in

predicted by the

t h e DF SCFt o t a l

o f 4.42 eV, which

c o r r e s p o n d i n g DF SCF v a l u e

excellently

was f o u n d t h a t

however,

above

differences

f o r t h e UO m o l e c u l e

0 . 7 0 eV g r e a t e r

c o r r e s p o n d i n g DF SCF c a l c u l a t i o n ;

the

the major

a u , t h e NRL c a l c u l a t i o n p r e d i c t s

-0.97

with

t o t h e two a t o m s .

3.55

bonding

however,

the than

agrees

T h e 9 | e l / 2 > NRHOMO o f UO

expression, v i z . -0.539|5f*01/2>

-0.483|5faal/2>

-0.449|6daal/2>

-0.348|7saal/2>

-0.288|6άπβ1/2>

-0.213|2ρσα1/2>

+0.144|6paal/2> T h e NRHOMO h a s a m u c h l a r g e r 6dπ DFAOs to

contribution

from

the 5 ί π , 5fa,

o f t h e U a t o m a n d t h e 2 ρ σ DFAO o f t h e 0 a t o m

t h e RHOMO; h o w e v e r ,

atom

is

separations.

3.05

4.677

of

separations f o r

internuclear

RELATIVISTIC R (au)

potential

the corresponding

can be seen that

Relativistic

seperations

au, respectively;

The c a l c u l a t e d

) energies

and d i s s o c i a t i o n e n e r g i e s internuclear

with

5|e3/2>,

o f UO a r e

ionization

internuclear

t o be unbound a t a l l these

Table

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

eigenvalues

6|el/2>,

6.17

303

Chemistry of Third-Row Transition Elements and Actinides

the contribution

t o t h e NRHOMO i s m u c h s m a l l e r .

o f t h e 7 sa

Moreover,

6d DFAO the

consists mostly

(with

RMO 8 | e l / 2 >

|5f5>

(-0.465)

the

7|el/2>

the

NR 7 | e l / 2 >

(0.381),

has

6d

and

6|el/2>

a coefficient

from

contributions

(0.344)

from

o f t h e U atom.

contributions

from

t h e 2ρπ

DFAOs

above

(0.788),

|5f53/2>

(0.20)

t h e |2ρπ>

Similarly,

o f U; while,

(0.346)

e.g.,

(0.435), 6ρσ

6fa

the

i s a π - t y p e MO w i t h 6dπ

DFAO),

(0.727),

substantially;

o f 2pa (0.746),

( a n d \5fnfil/2>)

RMO a s d i s c u s s e d

and

major

5ίπ

(-0.225)

o f t h e U atom. The

7|el/2>

orbital

about

eigenvalues

a r e -0.4281

respectively, for

DFAOs

a n d |2ρπ>

RMOs a n d NRMOs d i f f e r

consists mostly

(-0.288)

o f 0.90) o f

(0.338)

\5£π>

corresponding DFAOs

and

the 8|e3/2>

( t h e NRMOs)

a n d-0.4444

i s clear

than

and

results

o f t h e RMO

(-0.2642)

i t

4 . 4 eV lower Our

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

(with

contributions

that

t h e NRL o r b i t a l

7|el/2>

therefore

clearly

demonstrate

predictions

o f t h e NRL a n d D F S C F c a l c u l a t i o n s

etc.,

as quantitative

energies,

f o r the diatomics

energies

MOs.

as well

total

eigenvalues are

t h e c o r r e s p o n d i n g DF SCF o r b i t a l

qualitative bonding,

8|e3/2> and

(-0.2989) a . u .

orbital

energies,

involving

that

differences

there

f o r the nature

dissociation

actinides

a r e marked

between t h e

duet o v e r y

relativistic

effects

i n such

RELATIVISTIC

EFFECTS

FOR D I P O L E MOMENTS O F D I A T O M I C S

of

energies significant

systems. O F HEAVY

ELEMENTS The

RIPhas

for

the diatomic

using

been

ab i n i t i o

chemical

basis

internuclear value

basis and

a u was

dipole

functions,

Moreover,

dipole

relativistic Pyper are

(14).

predicted

dipole

experimental

and

PbTe,

about

values

respectively,

turns

40%

out that

au differ

and

0.019

au,predicted with

TiH,

+

a n d NRL

and

27 ( E B 2 7 ) (STO)

f o r AuH ( 2 1 ) .

EB27) a n d

reported

and B i H ;

a u and

by Malli and dipole

moments

however, t h e

very

1.0623

dipole

well

with

au f o r T i l

moment

f o r

the

+

moment

calculated

wavefunction s e tused

the predicted

(at R

e

=

i n the calculation

dipole

s e twavefunctions) from

thevalues

b y t h e NRL w a v e f u n c t i o n s , of TiH, (CB

f o r AuH

i s smaller (by

b y t h e c o r r e s p o n d i n g NRL

thebasis

In the case +

a

In

asA B " .

predicted

considerably

v i z Ti H";

f o r which

f o r AuH u s i n g t h e and

au agree

a positive

the relativistic

polarity

+

orbital

reported

experimental

PbH

1.2655

chemical basis

respectively.

expected

where

PbH

distance.

20 (EB20)

s e t (EB20

(NRL)

t h e CB s e t r e l a t i v i s t i c

However,

0.371

been

BiH,

limit

a t the experimental

except

calculated

(18) o f 1.8137

that

and

calculated

(with a u and

depending upon

relativistic

BiH,

+

the dipole

the

have

moments and

the relativistic with

at present, HgH ,

PbTe

b y a 6p S l a t e r - t y p e

were

the relativistic

the wavefunction.

from sets

basis

i t s polarity

t o 50%) t h a n

wavefunction, of

moments

au) using

species

(CI) wavefunctions

f o r AuH,

AB i n d i c a t e s

It 2.8794

curves

extended

o f 1.9078

the

basis

b y a 6p DFAO,

Unfortunately,

wavefunctions)

species

calculated

interaction

notavailable

PbH ,

(WF) c a l c u l a t e d

e

extended

dipole +

T i l ,

as n o n - r e l a t i v i s t i c

( R ) o f each

moment

CB s e t ,

configuration

as well

CB s e t a u g m e n t e d

CB s e t a u g m e n t e d

TiH,

used f o r the internuclear

moments

using

(21) t o evaluate +

HgH ,

s e twavefunctions

separation

obtained

b y Ramos

s p e c i e s AuH, relativistic

o f 3.5884

addition, WFs,

adapted

although

moments

o f 1.372, f o r AuH,

of dipole

0.323 -0.120

T i H and

the dipole

set)wavefunction

the value

(using

o f 0.976,

moment

moment

predicts the (-0.12 au)

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

21.

MALLI

obtained

from

polarity

for

moments AuH, (/i

),

>

A*CB

EB27

Ε

>

^CB

is

at

au,

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

(μ

Ε Β

R

=

e

2.8794

by

the

moments

(21)

electron

Table

R

C

I

n

c

the *

internuclear W

^ R C I ^

that

at μ

ε

RCI

that

and

the

diatomic in

study

systems

Tables

effects

than of

Ε

Β

2

it

Ε

in β

2

fairly

set

A*

C B

and

lesser μ

polarity

the >

7

0.967

order: μ

Ζ Β

and

a

that

predicted

lesser

polarity both

by

chemical basis effect

can be

that

>

7

for CB

order:

that

the

dipole

the

of

i n v o l v i n g heavy

and

V-VI

are

au,

relativistic

first

μ

a

are

R C I

thereby

the

in

0.967

that

3.3794

wavefunctions

au,

are

R C I

the

relativistic

^ F S decrease

and / *

β

opposite

separations

decrease

thereby R «

ε

from

3.3794

(MRCI) μ

s

and / z

at

the

F

R =

β

the

A comparison of

predicted

MS c a l c u l a t i o n s ,

presented such

of

indicating

is

correlation of

*

limit

of

DF S C F LCAS

moments

(

au,

RCI

This

set

indicating a

respectively,

dipole

results

N

au,

27)

non-relativistic

using

A

^ R C I> except

wavefunctions. on

2.8794

set

predicted

the

=

e

>

^EB27

0.846

(MEB2 7^

a

Ti"H .

values

μ

respectively,

EB27

whereas

for

the

+

viz

^RCI » except

R

au,

) ,

Β

at

that

set >

^EB27

0.846

c o r r e s p o n d i n g N R L WF i n d i c a t e s molecule

calculated

(μ),

whereas ( μ

the this

indicates

C B

305

Chemistry of Third-Row Transition Elements and Actinides

set

relativity

atoms

or

ions

concluded from

relativistic

significant

our

and

for

dipole

systems.

V.

Dipole

Calculated

Moments

for

(μ)

LiH,

Til

by U s i n g Chemical Basis

b

and

PbTe

Wavefunctions

8

d

c

A-B

it ( a u )

LiH

2.575

Til

1.908

1.814

PbTe

1.266

1.062

a

Reproduced with

+

A B"

c

polarity.

reference

(18).

l e

EXP (2.367)

p e r m i s s i o n from au -

2.542

d

D.

U s i n g extended

Ref.

2.314

e

b

21.

A l l

Experimental

basis

values

values

function

of

indicate

from

Malli

and

Pyper

(14).

CONCLUDING We h a v e LCAS

REMARKS

c o n c l u s i v e l y shown f r o m

MS

calculations

quantitative diatomics

features

that of

properly

understood using

based

on

the

that,

for

Schrôdinger

the

chemistry

the

core

the

6p

to are

the

in

and

sixth of

gold

heavier 6p

row

DFAOs for

and v e r y the

not

In the

relativistic

well

as

(Z

>

Moreover, in

of

90)

in

cannot

non-relativistic and

it

it

is

theory

heavier

has

5d DFAOs

gold

safe

DF S C F

the

and bonding

atoms

compounds, whereas

involved

chemistry

as

addition, 5d

non-relativistic

the

fully

structure heavy

equation.

and mercury

are

initio

traditional

elements,

elements.

semi-empirical

significant

ab

qualitative

electronic

i n v o l v i n g heavy

the

the

be

theory

been

shown

participate they

to

chemistry

in

belong

state (in

predictions),

to

that contrast but

elements.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

they

306

THE CHALLENGE OF d AND f ELECTRONS

Table

VI.

AuH R e l a t i v i s t i c

Dipole

Non-Relativistic Values

at

Moment

Dipole

R ,

c

in

e

Curves,

(μ)

Moment

au

8

and

b

d

Wave f u n e t i o n s CB R(au)

+ CB

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

2. 6294 2.. 8 7 9 4

+

6DSL

f

6pDF

-

0.968 g

CB e

-

RCI

EB20

EB27

0.905

0.829

0.738 0.802

0.976

1.065

1.271

0.984

0.901

(1.373)

(1.443)

(1.925)

(1.434)

(1.363)

3. . 1 2 9 4

0.967

-

-

1.046

0.956

0.846

3. . 3 7 9 4

0.968

1.132

1.045

0.909

8

-

C a l c u l a t e d by u s i n g the

Chemical

Basis

functions,

(CB),

EB20

functions),

-

wavefunctions

extended b a s i s

(same

as

EB27 b u t

and r e l a t i v i s t i c b

wavefunction.

Non-relativistic

without values

polarization

6 p DFAO a s

(see

reference

We a l s o 5d,

structure

DFAOs

involved

relativistic

significant

in

hoped that

would fully

our

(and to

in

the

of

a u -

(RCI)

parentheses.

2.542

D.

centered

effects) the

the

prediction

of

for

the

the

knotty

gold

in

e.g.,

the

electronic

third-row the

formidable

for

dipole of

a

bottlenecks

systems. be

moment,

fairly and

properties

relativistic of

DF S C F c a l c u l a t i o n s h a v e b e e n b r o k e n ,

dual

of

such

shown t o

non-energetic

quality

that

are

calculation

effects

CB atom

18).

heavy

have been

f

on g o l d .

for

present

accurate

effects

All

Slater-type

DF SCF c a l c u l a t i o n s

and they

the

properties,

Thus,

l

in

associated electrons)

correlation

criterion

future.

d

6p'

(reference

e

initio

and a c t i n i d e s ,

accurate

R

diatomics

relativistic

relativistic

ab

of

given

2.75

relativistic

and e l e c t r o n

the

-

polarization

interaction

21.

ζ

(14);

polarization

MCDF c a l c u l a t i o n

their

the

non-energetic

supplement

wavefunction

the

quantum c h e m i s t s

Furthermore, is

(due

elements to

exponent

from

conclude from

5 f

are

CB plus

Experimental

and bonding o f

transition challenge

9

14).

e

polarity.

obtained

6d and 5f

definitely

the

Au H"

function with

plus

the

+

indicate

reference

including

configuration

R e p r o d u c e d w i t h p e r m i s s i o n from Ref. values

from

EB27

ab and

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

initio it

is

it

21.

MALLI

Chemistry of Third-Row Transition Elements and Actinides

gratifying for

that

performing

calculations

the computational reliable

for diatomics

It

i s hoped that

ab

initio

with

machinery fully

fully

c o n t a i n i n g heavy

is currently

relativistic

c o n t a i n i n g heavy

the a v a i l a b i l i t y

(all-electron)

polyatomics near

ab i n i t i o

and very

of faster

relativistic

atoms w i l l

hand

DF SCF heavy

atoms.

supercomputers

calculations

become

at

307

feasible

for i n the

future.

ACKNOWLEDGMENTS I

sincerely

inviting to

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

Yu, My for

Professors Dennis symposium.

Salahub

T h i s work

and Mike

Zerner

for

h a s b e e n made p o s s i b l e d u e

t h e c o o p e r a t i o n a n d t h e e n t h u s i a s m o f my c o l l e a g u e s a n d c o w o r k e r s

over to

thank

me t o t h i s

many y e a r s ;

in particular,

Dr. N.C. Pyper, for their thanks their

Research financial

also

D r . R.

contributions go t o

cordial

would

cooperation. through

like

t o a c k n o w l e d g e my d e b t

M e s s r s A . F . Ramos a n d D.

to the research reported

the operations

C o u n c i l o f Canada support

I

Arratia-Perez,

no.

is

in this

paper.

o f our Computing S e r v i c e s

The N a t u r a l

(NSERC) grant

staff

Sciences and Engineering

thanked

for their

continuous

A3598.

LITERATURE CITED 1. Schrödinger, Ε. Ann. Physik. 1926, 81, 109. 2. Klein, O. Z. Physik. 1926, 37, 895. 3. Gordon, W. Z. Physik. 1926, 40, 117. 4. Dirac, P. A. M. Proc. Roy. Soc. Lond. 1928, A117. 610. 5. Burrau, O. Kgl. Danske. Videnskab. Mat. Fys. 1927, 7, 14. 6. Dirac, P. A. M. Proc. Roy. Soc. Lond. 1928, A123. 714-33. 7. Mayers, D. F. Proc. Roy. Soc. Lond. 1957, A241. 93. 8. Swirles, B. Proc. Roy. Soc. Lond. 1935, A152. 625-49. 9. Boyd, R. G.; Larson, A. C.; Waber, J. T. Phys. Rev. 1963, 129, 1629-30. 10. Mingos, D. M. P. Phil. Trans. Roy. Soc. Lond. 1982, A308, 7583. 11. Hay, P. J.; Wadt, W. R.; Kahn, L. R.; Bobrowicz, F. W. J. Chem. Phys. 1978, 69, 984. 12. Ziegler, T.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys. 1981, 74, 1271. 13. Jiang, Y.; Alarez, S.; Hoffmann, R. Inorg. Chem. 1985, 24, 749-57. 14. Malli, G. L.; Pyper, N. C. Proc. Roy. Soc. Lond. 1986, A407. 377404. 15. Grant, I. P.; Mckenzie, B. J.; Norrington, P. H.; Mayers, D. F.; Pyper, N. C. Comput. Phys. Commun. 1980, 21, 207. 16. Brown, G. E.; Ravenhall, D. G. Proc. Roy. Soc. Lond. 1951, A208. 552-9. 17. Sucher, J. Phys. Rev. 1980, A22, 348-62. 18. Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules: Van Nostrand Reinhold, New York, 1979. 19. Pyykkö, P. Chem. Rev. 1988, 88, 563. 20. Lee, Y. S.; McLean, A. D. J. Chem. Phys. 1982, 76, 735.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE C H A L L E N G E OF d AND f ELECTRONS

308

21. 22. 23. 24. 25.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch021

26. 27. 28. 29. 30.

Ramos, A. F.; Pyper, N. C.; Malli, G. L. Phys. Rev. 1988, A 38 2729-2739. Katz, J. J.; Seaborg, G. T.; Morss, L. R. The Chemistry of the Actinide Elements. Chapman and Hall: London, 1986. Oetting, F. L.; Rand, M. H.; Ackermann, R. J. The Chemical Thermodynamics of Actinide Elements and Compounds Part 1: International Atomic Energy Agency: Vienna, 1976. Oetting, F. L.; Fuger, J. The Chemical Thermodynamics of Actinide Elements and Compounds Part 2: International Atomic Energy Agency: Vienna, 1976. Erdos, P.; Robinson, J. M. The Physics of Actinide Compounds: Plenum Press: New York, 1983. Handbook on the Physics and Chemistry of the Actinides Vols. 15; Freeman, A. J.; Lander, G. H. Eds.; North Holland: Amsterdam, 1987. Grant, I. P.; McKenzie, B. J.; Norrington, P. H.; Mayers, D. F.; Pyper, N. C. Computer Phys. Commun. 1980, 21, 218. Ackermann, R. J.; Rauh, E. G. Higher Temp. Sci. 1973, 5, 463; J. Chem. Phys. 1974, 60, 2266. Hildenbrand, D. L.; Murad, E. J. Chem. Phys. 1974, 61, 1232. Malli, G. L.; Oreg, J. J. Chem. Phys. 1975, 63, 830-841.

RECEIVED March 21, 1989

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 22

Relativistic Effective Potentials in Quantum Monte Carlo Studies Phillip A. Christiansen

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

Department of Chemistry, Clarkson University, Potsdam, NY 13676

An overview of quantum Monte Carlo electronic structure studies i n the context of recent effective potential implementations i s given. New results for three electron systems are presented. As long as care i s taken i n the selection of t r i a l wavefunctions, and appropriate frozen core corrections are included, agreement with experiment i s excellent (errors less than 0.1 eV). This approach offers promise as a means of avoiding the excessive configuration expansions that have plagued more conventional transition metal studies.

In the l a s t ten years considerable e f f o r t has gone i n t o the study of small metal clusters. Several reviews on the subject have appeared i n the l i t e r a t u r e . Volume 156 of Surface Science, Volume 86 of Chemical Reviews and a portion of Volume 91 (especially No. 10) o f the Journal of Physical Chemistry are devoted t o t h i s topic. The small t r a n s i t i o n metal c l u s t e r s are p a r t i c u l a r l y i n t r i g u i n g as a r e s u l t of t h e i r unique structures (multiple d bonding, etc.) and as possible models f o r c a t a l y t i c processes. Furthermore, as can be seen from the compendium by Huber and Herzberg (1) and a l s o from the encyclopedic reviews by Weltner and Van Zee (2) and more r e c e n t l y by Morse Ç3) and by Salahub (4), r e l a t i v e l y l i t t l e i s known about the d e t a i l e d structures of even the simplest c l u s t e r s (diatomics) of the elements beyond the f i r s t t r a n s i t i o n row. The f i e l d would appear t o be wide open f o r computational chemists. For c l u s t e r s of only a few atoms one would expect rigorous e l e c t r o n i c structure studies (SCF plus large CI, etc.) t o y i e l d useful imformation regarding molecular geometries, d i s s o c i a t i o n energies, v i b r a t i o n a l frequencies, etc. Unfortunately, i n contrast t o recent l i g h t element work, e a r l y t r a n s i t i o n element studies proved somewhat disappointing. The chromium diatom (5-9) i s probably the best known example. However a more d i s t u r b i n g case i s SC2, the simplest t r a n s i t i o n metal diatomic. The SC2 d i s s o c i a t i o n energy (1.65 eV) i s known from the mass spectrometric work of Verhaegen e t a l . (10) although there may be some error due t o the use of rather imprecise molecular p a r t i t i o n functions (11,12). (A value of 1.22 eV was o r i g i n a l l y given but i t 0097-6156/89/0394-0309$06.00/0 ο 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

310

THE CHALLENGE OF d AND f ELECTRONS

has since been reported i n various references (1,3) as 1.65.) Resonance Raman (13), ESR (14) and MCD (15) matrix i s o l a t i o n studies suggest a 5 Σ ground state with a fundamental v i b r a t i o n a l frequency of about 239cnf 1, which i s consistent with the assignment given by the best ab i n i t i o calculation. Nevertheless, i n t h e i r f a i r l y extensive study, Walch and Bauschlicher (12) were able t o account f o r only a f r a c t i o n of the experimentally determined d i s s o c i a t i o n energy. As with diatomic chromium the d i f f i c u l t y involves c o r r e l a t i o n i n a multiple d bonded system. These discouraging r e s u l t s have prompted Morse Ç3) t o suggest that f o r t r a n s i t i o n element problems the density functional approaches (4,17-19) might be more appropriate. Of course the t r a n s i t i o n metal electron c o r r e l a t i o n problems do not necessarily begin a t the molecular l e v e l . Ab i n i t i o studies (2024) t y p i c a l l y show errors i n atomic e x c i t a t i o n energies of about 0.3 eV or more f o r t r a n s i t i o n s i n v o l v i n g the outer s and d electrons. The errors seen i n the above examples are of course the r e s u l t of the necessary incompleteness of o r b i t a l and configuration basis sets. The power of these expansion approaches i s that i f one works hard enough (uses a s u f f i c i e n t l y complete, or a t l e a s t appropriate, basis) one should get the r i g h t answer. The recent extensive t r a n s i t i o n metal hydride studies i n d i c a t e the p o s s i b i l i t i e s (25-30). Nevertheless, heavy atom electron c o r r e l a t i o n involving d and even f subshells i s such an enormous problem that every a l t e r n a t i v e should be explored. E f f e c t i v e P o t e n t i a l Quantum Monte C a r l o As configuration expansions approach the m u l t i - m i l l i o n range, alternatives such as quantum Monte Carlo (QMC) techniques begin t o appear a t t r a c t i v e . A useful overview of QMC has been given by Ceperly and Alder (31). Pioneering work i n t h i s f i e l d was done i n the mid 70 s by Anderson (32,33) as w e l l as by Kalos and coworkers (34,35). This has been followed by considerable development work as w e l l as molecular and atomic applications (36-56). The advantage of QMC i s that i t does not depend on the exhaustive configuration and o r b i t a l basis set expansions that have plagued conventional studies. As one moves down the p e r i o d i c table t o the t r a n s i t i o n elements with occupied d s h e l l s and t o the Lanthanides and Actinides with f s h e l l s , the QMC advantage becomes more apparent. Unfortunately, t o a considerable extent, what one gains i n the e l i m i n a t i o n of i n f i n i t e basis set expansions, one looses t o s t a t i s t i c a l sampling error. And furthermore the sampling e r r o r increases r a p i d l y as a function of the nuclear charge. D o l l (57), Ceperly (58), and most recently Hammond e t a l . (59) have given arguments i n d i c a t i n g that the QMC computer requirements increase with about the s i x t h power of the nuclear charge. As a r e s u l t QMC has, t o date, offered l i t t l e competition f o r conventional calculations, and we are aware of no a l l - e l e c t r o n QMC studies i n v o l v i n g elements beyond the f i r s t row. Although a l l - e l e c t r o n heavy element QMC studies are a t the present time out of the question, we have recently shown that by replacing the core electrons (and the corresponding f r a c t i o n of the nuclear charge) with an appropriate r e l a t i v i s t i c e f f e c t i v e potential (REP) the QMC domain can be quite r e a d i l y extended t o the lower portion of the p e r i o d i c table (60-62). To our knowledge, reference (60) i s the f i r s t QMC study i n v o l v i n g an element from below the f i r s t

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

22.

CHRISTIANSEN

311

Quantum Monte Carlo Studies

row and i s a l s o the f i r s t t o include r e l a t i v i t y . This work was followed c l o s e l y by a study by Hammond e t a l . (59) who used an almost i d e n t i c a l approach i n a l k a l i and a l k a l i n e earth atomic and molecular studies. As pointed out i n the review by Ceperly and Alder (_31) the d i f f u s i o n interpretation of the Schroedinger equation has an extensive history. The d i f f u s i o n analogy becomes apparent i f one writes the time-dependent Schroedinger Equation (one electron f o r s i m p l i c i t y ) i n terms o f imaginary time, t , | f = |ν2ψ - (ν - Ε )ψ

(1)

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

τ

Ψ then corresponds t o a concentration and the equation i s simulated by a combination o f random p a r t i c l e movement ( f i r s t term on the right) as w e l l as p a r t i c l e b i r t h and death according t o the f i r s t order rate constant, V-Bp (32,33). The a r b i t r a r y reference energy, Erp, can be adjusted t o maintain normalization. However a f a r more e f f i c i e n t approach involves importance sampling (34,36,37,41,42). By defining the function ί=ψΨτ# the product o f Ψ with a time-independent t r i a l wavefunction, Ψφ, Equation 1 becomes df

^

,

Η

ο

= lv2f - V · (f-νΐηψτ) -

Ψτ - E )f.

(2)

T

The middle term on the r i g h t adds a d r i f t v e l o c i t y (νψ /ψ ) t o the simulation which greatly reduces sampling i n regions of low electron density. In addition the nodes i n (resulting from e i t h e r o r b i t a l nodes or antisymmetry) can be used t o define sampling region boundaries, an assumption of the f i x e d node approximation (32,33). E l e c t r o n i c energies are u l t i m a t e l y obtained from averages of the l o c a l energies, Η Ψ / ψ . Detailed descriptions of algorithms based on Equation 2 can be found i n the l i t e r a t u r e (31,39,42,52). For QMC simulations involving atomic o r molecular systems with more than a small number of electrons, p o t e n t i a l sources of d i f f i c u l t y are f a i r l y obvious (58-60). In regions of high electron density (such as near nuclei) one sees the corresponding high density of s i n g u l a r i t i e s i n the hamiltonian r e s u l t i n g from the two-electron e l e c t r o s t a t i c interaction. A t the same time, wavefunction antisymmetry causes a high nodal density. The dense nodal structure forces one t o employ short time steps, thereby g r e a t l y increasing the computational requirements. And although the e f f e c t s of electronnucleus and two-electron s i n g u l a r i t i e s i n the p o t e n t i a l can be c o n t r o l l e d t o considerable extent using p a i r - c o r r e l a t i o n functions [see reference (37) f o r instance], unless ψ τ i s a good approximation to ψ, c o r r e l a t i o n e r r o r w i l l become overwhelmingly apparent i n the l o c a l energies, leading t o large s t a t i s t i c a l errors i n the average. The more densely packed the electrons become (this w i l l be most serious i n the core region) the more acute the d i f f i c u l t i e s w i l l be. Furthermore, with the exception of the work by Vrbik e t a l . (63) i t i s not clear how r e l a t i v i t y (essential f o r heavy element studies) would be included i n a l l - e l e c t r o n QMC work. In t h i s context the advantages i n the use o f e f f e c t i v e potentials are quite clear. The potentials eliminate the high electron density (and associated nodes) near the n u c l e i , thereby τ

τ

τ

τ

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

312

THE

CHALLENGE O F d AND f ELECTRONS

reducing sampling i n a region i n which both the wavefunction and the potential are r a p i d l y varying. This eliminates a d i f f i c u l t f r a c t i o n of the multi-electron p o t e n t i a l from the wave equation and a t the same time makes the use o f much longer time steps appropriate. Perhaps equally important/ r e l a t i v i s t i c e f f e c t i v e potentials allow one t o introduce r e l a t i v i t y i n a p a r t i c u l a r l y convenient form. The key t o the use o f conventional semi-local REPs i n QMC involves the transformation o f the PEP t o l o c a l form. In reference (60) Hurley e t a l . proposed the many-electron l o c a l p o t e n t i a l , V ^ , 1

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

V

REP

s

υΚΕΡ

ψ τ /

ψ

Τ β

(

3

)

\βΕΡ i s the conventional e f f e c t i v e potential and i s the t r i a l wavefunction (at least the determinant portion) from Equation 2. To carry out REP-QMC calculations one simply adds t o the valenceelectron hamiltonian, H, i n the importance sampling algorithm. In l i g h t o f the l o c a l energy expression, Ηφτ/ψ^/ i n Equation 2, t h i s d e f i n i t i o n o f the l o c a l potential i s rather obvious and has a l s o been used i n the work by Hammond e t a l . (59) but without r e l a t i v i t y . Equation 3 obviously adds approximations. These include the usual e f f e c t i v e potential assumptions (frozen core, etc.) i n addition to the l o c a l i z a t i o n shown i n the equation. However i n one sense i t i s a trade-off i n that the l o c a l potential e f f e c t i v e l y eliminates the f i x e d node approximation i n the core region. Atomic Studies In Table I electron a f f i n i t i e s f o r L i , Na and Κ computed using Equation 3 with e i t h e r r e l a t i v i s t i c (60) o r n o n r e l a t i v i s t i c (59) e f f e c t i v e potentials are compared with the respective experimental values (64-66). Only i n the r e l a t i v i s t i c L i c a l c u l a t i o n do we see a s i g n i f i c a n t discrepancy, and even then the e r r o r i s w e l l below 0.1 eV. In a l l o f these calculations s i n g l e determinant t r i a l wavefunctions were employed. While t h i s i s no approximation f o r the one-electron neutral atoms we might see minor problems f o r the anions, and L i could be a case i n point. Table I. A l k a l i Electron A f f i n i t i e s ( i n eV) obtained from R e l a t i v i s t i c and N o n r e l a t i v i s t i c E f f e c t i v e Potential QMC Simulations Atom Li Li Na Κ

reference 60 59 II

60

e f f e c t i v e p o t e n t i a l QMC 0.56(2) 0.61(2) 0.56(2) 0.52(1)

Expt. 0.62 0.55 0.50

Conventional shape-consistent e f f e c t i v e potentials (67-70), whether r e l a t i v i s t i c o r not, are t y p i c a l l y formulated as expansions of l o c a l potentials, Ui,(r), m u l t i p l i e d by angular projection operators. The expansions are truncated a f t e r the lowest angular function not contained i n the core. The l a s t (residual) term i n the expansion t y p i c a l l y represents l i t t l e more than the simple coulombic i n t e r a c t i o n between a valence electron and the core (electrons and corresponding f r a c t i o n o f the nuclear charge) and i s predominantly a t t r a c t i v e . The lower & terms, on the other hand., include strongly

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

22.

CHRISTIANSEN

313

Quantum Monte Carlo Studies

repulsive "Pauli" contributions. This i s i l l u s t r a t e d i n Figure 1 where we have plotted the two terms i n the L i REP (68). Note that the difference between the two curves goes r a p i d l y t o zero f o r large values of r . A s i n g l e determinant s t r i a l wavefunction f o r L i would result i n a that included only the repulsive curve. However the i n c l u s i o n of s t o ρ promotions i n the t r i a l wavefunction would introduce a small contribution from the a t t r a c t i v e term and would tend t o lower the e l e c t r o n i c energy. Fortunately, f o r L i the anion electron density i s r e l a t i v e l y d i f f u s e and the correction quite small. In some cases however serious errors can r e s u l t from the use of such a simple t r i a l wavefunction i n Equation 3. The terms i n the Be REP are quite s i m i l a r t o those of L i , but the ground state electron d i s t r i b u t i o n i s considerably more compact and the c o r r e l a t i o n correction from the p2 configuration f a r more important. In Table I I we have l i s t e d SCF and REP-QMC energies f o r the lowest ^S, P, and D states of Be along with experimental values (71) f o r comparison. Numbers i n square brackets include core p o l a r i z a t i o n corrections 2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

3

3

(22). 3

3

For the Be P and D excited states the c o r r e l a t i o n corrections are r e l a t i v e l y small and as can be seen from the table, the s i n g l e determinant approximation i n Equation 3 i s quite good. In contrast the *S state i s too high by 0.3 eV. However, as Equation 3 would suggest, a simple two-configuration ( 2 s + 2p2) wavefunction brings i n the a t t r a c t i v e "p" contribution and we get the value l i s t e d i n column four. Curiously, i n an a l l - e l e c t r o n QMC study Harrison e t a l . (74) observed a s i m i l a r d i f f i c u l t y with the Be ground state. They found that a single determinant t r i a l wavefunction gave the energy about 0.3 eV too high due t o the f i x e d node approximation. The use of a multiconfiguration t r i a l wavefunction eliminated the error. 2

Table I I . E f f e c t i v e Potential QMC Energies ( i n eV) f o r various states of Be and Mg State Be+ S Be D Be P Be i-S

reference

Z

3

3

II

Mg+ Mg Mg Mq

2

61 II II

SCF 0.0 -1.54 -6.39 -8.06

Single 0.0 -1.63(2) -6.56(2) -9.04(3)

0.0 -6.59

0.0 -7.58(1)

II

S

is

62

II

II

" "

59 73

Multi. 0.0

-9.32(1) [-9.34(1)] 0.0 -7.55(1) [-7.66(1)]

Expt. 0.0 -1.63 -6.59 -9.32 0.0 -7.65

-7.64(3) -7.57(3)

In the e f f e c t i v e p o t e n t i a l approximation Mg i s i s o e l e c t r o n i c with Be. But, as can be seen i n Figure 2, the Mg REP i s composed of t h r e e terms (s, ρ and d) w i t h the s and ρ both r e p u l s i v e . As a r e s u l t , even though the c o r r e l a t i o n correction i s almost as large as i n Be the multi-determinant correction r e s u l t i n g from Equation 3 i s only a tenth as b i g (see Table II). The discrepancy between values from references (62) and (59) i s due t o large s t a t i s t i c a l o r extrapolation error. Note that unlike Be one cannot make comparisons with experimental r e s u l t s without f i r s t taking core-valence

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

314

THE CHALLENGE OF d AND f ELECTRONS

r (a.u.) Figure

1.

Radial

consistent

plots

effective

of

the

s and ρ

potential.

(Data

terms are

of

from

a Li

shape

ref.

68.)

r(a.u.) Figure

2.

consistent

Radial

plots

effective

of

the

s,

potential.

p,

and d terms

(Data

are

from

of ref.

a Mg s h a p e 70.)

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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CHRISTIANSEN

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Quantum Monte Carlo Studies

c o r r e l a t i o n (72) i n t o account. For the heavier a l k a l i s and a l k a l i n e earths such corrections can amount t o several tenths o f an eV. The e f f e c t i v e p o t e n t i a l QMC studies discussed above have a l l employed more or less conventional shape consistent r e l a t i v i s t i c (60 61) o r n o n r e l a t i v i s t i c (59,62) e f f e c t i v e p o t e n t i a l s l o c a l i z e d according t o Equation 3. Yoshida and Iguchi (73) on the other hand have recently published Mg, Ca and Sr studies employing model potentials o f the type developed by Huzinaga e t a l . (75). By comparison with the nodeless o r b i t a l s i n the preceeding studies the model p o t e n t i a l approach employs representations o f normal HartreeFock valence o r b i t a l s . In reference (73) the p o t e n t i a l was j u s t the coulombic i n t e r a c t i o n between normal core and valence o r b i t a l s which i s e s s e n t i a l l y equivalent t o the residual term i n the shape consistent REPs. The advantage t o t h i s approach i s that the potential i s already i n simple l o c a l form and there i s no need f o r Equation 3. The dissadvantage i s that the t r i a l wavefunctions include numerous nodes i n the core region (which forces one t o employ shorter time steps) and a l s o may include a s i z a b l e amplitude near the nucleus which f o r Gaussian basis sets might require the use of nuclear cusp functions. In the QMC simulation t h i s approach almost looks more l i k e a frozen core study rather than e f f e c t i v e p o t e n t i a l . An additional (not nearly so w e l l understood) complication i s that i n the simple shielded nucleus potential a t r i a l wavefunction formed from the valence Hartree-Fock o r b i t a l s looks l i k e an excited state. One must therefore choose the t r i a l wavefunction c a r e f u l l y t o ensure orthogonality t o the f a l s e lower energy solutions. The above studies a l l involved only one and two-electron systems. And with the exception of the Be high spin excited states (61) none required the use of "Fermi s t a t i s t i c s " (wavefunction antisymmetry) i n the Monte Carlo simulations. This i s o f course a prerequisite f o r multi-electron systems. We have recently c a r r i e d out REP-QMC simulations on some three-electron systems. Aluminum i s probably the simplest. In Table I I I we show energies f o r two states o f A l and a l s o f o r A l .

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

f

+

Table I I I . Comparison of SCF, REP-QMC and Experimental Energies ( i n eV) f o r various states o f A l State A1+ J-S Al P Al P 4

2

SCF 0.0 -3.11 -5.48

Single 0.00(2) -2.42(3) -5.92(4)

Multi. 0.00(1) [-2.42(2)] -5.92(2)

Expt. 0.0 -2.38 -5.98

For the A l ground state and a l s o f o r the cation we c a r r i e d out simulations using both s i n g l e and multiple determinant t r i a l wavefunctions. (The brackets indicate a s i n g l e determinant t r i a l function f o r the P state but multiple f o r the S.) As opposed t o e i t h e r Be or Mg we see no s i g n i f i c a n t adjustment i n the energies r e s u l t i n g from the use o f the more accurate t r i a l functions. We do however see a reduction i n the s t a t i s t i c a l error. The agreement with experiment i s excellent except f o r the 0.06 eV systematic error i n the ground state. This could be due t o a s t i l l inadequate t r i a l wavefunction i n Equation 3, but based on Muller's Mg work (72) we a l s o suspect core p o l a r i z a t i o n . This remains t o be determined. As indicated i n the introduction a major motivation i n the 4

l

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

316

development of REP-QMC involves applications t o t r a n s i t i o n element problems. Although t o our knowledge there are as yet no such applications i n the l i t e r a t u r e we have recently been running preliminary three-electron simulations involving the D ground and F excited states o f Sc and Y. Our r e s u l t s so f a r have been somewhat disappointing. For both elements we see e x c i t a t i o n energy errors of about 0.5(2) eV which are two t o three times what we would have anticipated f o r the three-electron approximation. The poor r e s u l t s could be due t o the s i n g l e determinant t r i a l functions but i t might a l s o be the r e s u l t o f core p o l a r i z a t i o n or some other aspect o f the frozen-core approximation. We may be forced t o use 11-electron REPs t o achieve adequate results. I n t h i s event, a recently developed "frozen core" approach (76) could prove invaluable i n combination with the 11-electron REP. Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

2

4

Molecules To date the only molecular e f f e c t i v e p o t e n t i a l QMC r e s u l t s that we are aware o f are those o f Hammond e t a l . (59). The d i f f i c u l t y i n extending t h i s work t o diatomic or polyatomic systems involves the e f f i c i e n t evaluation o f the l o c a l p o t e n t i a l , V * ^ , i n conjunction with the projection operators i n the p o t e n t i a l and basis functions centered on d i f f e r e n t n u c l e i . In the NaH and Ν&2 work o f reference (59) the angular projection integrations were c a r r i e d out by more o r l e s s conventional means. In preliminary work however we have found a simple a l t e r n a t i v e useful. Since the r a d i a l functions i n ifiEP decay very r a p i d l y with increasing r , the product o f the Ufl,(r) with a function centered on another nucleus can be approximated quite accurately by short one-center expansions. Conveniently, the angular part o f the expansion need not go beyond the highest £ quantum number i n the core. For f i r s t row atoms ( L i , Be, C, etc.) only a small number o f s functions are required. Κ o r Mg studies would require only s and ρ functions. In preliminary studies we have found that f o r K2 the use o f s functions alone r e s u l t s i n an e r r o r o f only about a tenth o f an eV due t o the neglect o f ρ functions i n the one-center expansion. For l i n e a r molecules a p a r t i c u l a r l y e f f i c i e n t scheme (in terms of the QMC sampling) might be t o tabulate the product o f the ifi^ with each o r b i t a l on a coarse two-dimensional grid. In t h i s sense one can see that an i n t e r e s t i n g approach t o diatomic REP-QMC could be based on numerically determined SCF o r MCSCF t r i a l wavefunctions (77,78). Discussion Quantum Monte Carlo techniques have considerable p o t e n t i a l f o r a p p l i c a t i o n t o problems involving open d o r f s h e l l s where the treatment o f electron c o r r e l a t i o n has proven p a r t i c u l a r l y d i f f i c u l t . However i f QMC i s t o be a v i a b l e a l t e r n a t i v e one must be able t o l i m i t the simulations t o small numbers o f electrons and i n addition r e l a t i v e i t y must be included. R e l a t i v i s t i c e f f e c t i v e potentials o f f e r one avenue (at the present time the only avenue) f o r achieving these conditions. However, as we have indicated, REPs do introduce complications. Because o f Equation 3 one must be somewhat more c a r e f u l i n the s e l e c t i o n o f t r i a l wavefunctions and multiconfiguration algorithms are e s s e n t i a l , unfortunately the additional configurations

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

22.

CHRISTIANSEN

Quantum Monte Carlo Studies

necessarily increase the computer requirements. For the Be ground state f o r instance the addition of the p configuration j u s t about doubles the amount of processing per QMC time step. On the other hand as one can see from the Tables the sampling error i s t y p i c a l l y reduced by more than a factor of two. Noting that the sampling error varies inversely with the square root of the number of samples we see that the use of multiconfiguration t r i a l wavefunctions a c t u a l l y reduces the o v e r a l l computer requirements (for a given error level) by more than a factor of two. This i s consistent with the arguments given i n references (58,59) and with the r e s u l t s of Moskowitz e t a l . (48,49). Though not discussed above, i n a l l the studies mentioned the t r i a l wavefunctions included p a i r c o r r e l a t i o n functions, J j i , as prescribed by Reynolds e t a l . (42). Moskowitz e t a l . (48,49) have shown that the product of a r e l a t i v e l y simple multiconfiguration wavefunction with p a i r c o r r e l a t i o n functions can provide a rather accurate approximation to the exact wavefunction. In our calculations and i n those of Hammond e t a l . (59) the many-electron l o c a l p o t e n t i a l , V ^ , has been obtained by allowing the REP t o operate only on the determinantal portions of the t r i a l wavefunction. The e f f e c t s of the p a i r c o r r e l a t i o n functions have been ignored. As pointed out i n (61) the e f f e c t s of the p a i r c o r r e l a t i o n functions on the l o c a l potentials could be included by means of zeta-function expansions (79). However i n our multiconfiguration c a l c u l a t i o n s the J i j were parametrized t o correct f o r short range d i f f i c u l t i e s only, (We assumed that the configuration expansions properly accounted f o r long-range and near-degeneracy e f f e c t s ) , and we would therefore expect the J ^ j to have only a n e g l i g i b l e e f f e c t on the V^^p. A l l of the e f f e c t i v e p o t e n t i a l QMC studies that we are aware of have employed r e l a t i v e l y simple fixed-node d i f f u s i o n Monte Carlo algorithms. This i s not t o suggest that these are preferable, but rather easy t o program. One should not underestimate the advantages of Green's Function [see references (50,51) f o r instance] or other more recently developed approaches (80). E s s e n t i a l l y a l l of the e f f e c t i v e p o t e n t i a l QMC work i n the l i t e r a t u r e t o date has been, or could be, c a r r i e d out on small mini or microcomputers. For instance, using QMC algorithms s i m i l a r t o those of references (39) and (52), our Be, Mg and A l (two and three electron) studies required around 50 t o 200 MicroVAX hours f o r s i n g l e atomic state energies with standard errors of about 0.01 eV. However work on larger systems w i l l obviously require more powerful computing machinery. Although we would not a n t i c i p a t e dramatic increases i n speed due t o vectorization, (at l e a s t not f o r problems involving smaller numbers of electrons) QMC algorithms are almost t r i v i a l l y adaptable t o "massively p a r a l l e l " computing environments (81-83). By simply d i s t r i b u t i n g configurations (particles) evenly among processors one should be able t o obtain near peak p a r a l l e l e f f i c i e n c y . The only complication we forsee would be due t o the occasional renormalization required by the p a r t i c l e m u l t i p l i c a t i o n and destruction events. This would necessitate the communication of p a r t i c l e coordinates, etc. between processors t o maintain an even configuration d i s t r i b u t i o n . Fortunately the time required t o t r a n s f e r configurations between processors w i l l be proportional t o the number of electrons per configuration whereas the processing time (per time step) i s proportional to the number of electrons squared or 2

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317

1

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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THE CHALLENGE OF d AND f ELECTRONS

cubed (42). As a r e s u l t one would expect i n e f f i c i e n c i e s due t o processor imbalance t o become less important f o r bigger problems. Furthermore f o r "hypercube" machines (81-83) with a thousand o r fewer processors the transfer o f configurations could probably be l i m i t e d to adjacent nodes, thereby minimizing the transfer time. Massively p a r a l l e l (multiple instruction, m u l t i p l e data) computers with tens o r hundreds of processors are not r e a d i l y accessible t o the majority o f quantum chemists a t the present time. However the cost o f currently available hypercube machines with tens of processors (each with about the power of a VAX) i s comparable t o that o f supermini s but with up t o a hundred times the power. F o r applications of the type discussed above the performance of a machine with as few as 32 o r 64 processors would be comparable t o (or perhaps even exceed) that of a s i n g l e processor supercomputer. Although computer requirements currently l i m i t QMC applications (even with e f f e c t i v e potentials) the p r o l i f e r a t i o n of inexpensive massively p a r a l l e l machines could conceivably make the application of r e l a t i v i s t i c e f f e c t i v e potentials with QMC quite competitive with more conventional e l e c t r o n i c structure techniques. Our REP-QMC work (60-62) along with the studies by Hammond e t a l . (59) provide evidence that with the proper precautions the combination of r e l a t i v i s t i c e f f e c t i v e potentials with quantum Monte Carlo procedures may provide an a l t e r n a t i v e f o r obtaining accurate e l e c t r o n i c structure information. The possible elimination o f excessive basis set and configuration expansions f o r the t r a n s i t i o n and heavier elements i s e s p e c i a l l y appealing. And a t the same time the t r a n s i t i o n t o p a r a l l e l computing i s p a r t i c u l a r l y simple. The p o s s i b i l i t y o f carrying out d e f i n i t i v e calculations on small t r a n s i t i o n metal c l u s t e r s should not be dismissed offhand. Although we would not expect QMC (or REP-QMC) t o replace conventional approaches (for one thing an accurate conventional t r i a l wavefunction appears t o be an essential prerequisite) the electron c o r r e l a t i o n problem f o r elements containing occupied d and f subshells i s p o t e n t i a l l y so enormous that a l l possible avenues should be thoroughly researched. Acknowledgments This work has been supported i n part by the National Science Foundation under Grant No. CHE-8214665 and also by the Research Corporation.

Literature Cited 1. 2. 3. 4. 5. 6.

K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules, Van Nostrand Reinhold Company: New York, 1979. W. Weltner and R.J. Van Zee, Ann. Rev. Phys. Chem. 1984, 35, 291. M.D. Morse, Chem. Rev. 1986, 86, 1049. D.R. Salahub, Adv. Chem. Phys. 1987, 69, 447. M.M. Goodgame and W.A. Goddard, I I I , J. Phys. Chem. 1981, 86, 215. S.P. Walch, C.W. Bauschlicher, J r . , B.O. Roos and C.J. Nelin, Chem. Phys. Lett. 1983, 103, 175.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

22. CHRISTIANSEN

7. 8. 9. 10. 11. 12. 13. 14.

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15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41.

Quantum Monte Carlo Studies

319

G.P. Das and R.L. Jaffe, Chem. Phys. Lett. 1984, 109, 206. A.D. McLean and B. Liu, Chem. Phys. Lett. 1983, 101, 144. M.M. Goodgame and W.A. Goddard, III, Phys. Rev. Lett. 1985, 54, 661. G. Verhaegen, S. Smoes and J. Drowart, J. Chem. Phys. 1964, 40, 239. G. Das, Chem. Phys. Lett. 1982, 86, 482. S.P. Walch and C.W. Bauschlicher, Jr., J. Chem. Phys. 1983, 79, 3590. M. Moskovits, D.P. DiLella and W. Limm, J. Chem. Phys. 1984, 80, 626. L.B. Knight, Jr., R.J. Van Zee and W. Weltner, Jr., Chem. Phys. Lett. 1983, 94, 296. L.B. Knight, Jr., R.W. Woodward, R.J. Van Zee and W. Weltner, Jr., J. Chem. Phys. 1983, 79, 5820. R.J. Singer and R. Grinter, Chem. Phys. 1987, 113, 99. N.A. Baykara, B.N. McMaster and D.R. Salahub, Mol. Phys. 1984, 52, 891. B. Delley, A.J. Freeman and D.E. Ellis, Phys. Rev. Lett. 1983, 50, 488. J. Bernholc and N.A.W. Holzwarth, Phys. Rev. Lett. 1983, 50, 1451. C.W. Bauschlicher, Jr., S.P. Walch and H. Partridge, J. Chem. Phys. 1982, 76, 1033. K.K. Sunil and K.D. Jordan, J. Chem. Phys. 1985, 82, 873. C.M. Rohlfing and R.L. Martin, Chem. Phys. Lett. 1985, 115, 104. S.R. Langhoff and C.W. Bauschlicher, Jr., J. Chem. Phys. 1986, 84, 4485. C.W. Bauschlicher, Jr., J. Chem. Phys. 1987, 86, 5591. D.P. Chong, S.R. Langhoff, C.W. Bauschlicher, Jr., S.P. Walch and H. Partridge, J. Chem. Phys. 1986, 85, 2850. S.R. Langhoff, L.G.M. Pettersson, C.W. Bauschlicher, Jr. and H. Partridge, J. Chem. Phys. 1987, 86, 268. L.G.M. Pettersson, C.W. Bauschlicher, Jr., S.R. Langhoff and H. Partridge, J. Chem. Phys. 1987, 87, 481. A.E. Alvarado-Swaisgood, J. Allison and J.F. Harrison, J. Phys. Chem. 1985, 89, 2517. J.B. Schilling, W.A. Goddard III and J.L. Beauchamp, J. Am. Chem. Soc. 1986, 108, 582. C.W. Bauschlicher, Jr. and S.P. Walch, J. Chem. Phys. 1982, 76, 4560. D.M. Ceperly and B.J. Alder, Science 1986, 231, 555. J.B. Anderson, J. Chem. Phys. 1975, 63, 1499. J.B. Anderson, J. Chem. Phys. 1976, 65, 4121. M.H. Kalos, D. Levesque and L. Verlet, Phys. Rev. A 1974, 9, 2178. D.M. Ceperly, G.V. Chester and M.H. Kalos, Phys. Rev. Β 1977, 16, 3081. J. B. Anderson, J. Chem. Phys. 1980, 73, 3897. F. Mentch and J.B. Anderson, J. Chem. Phys. 1981, 74, 6307. F. Mentch and J.B. Anderson, J. Chem. Phys. 1984, 80, 2675. J.B. Anderson, J. Chem. Phys. 1985, 82, 2662. D.R. Garmer and J.B. Anderson, J. Chem. Phys. 1987, 86, 4025. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 1980, 45, 566.

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49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71.

72. 73. 74.

THE CHALLENGE OF d AND f ELECTRONS P.J. Reynolds, D.M. Ceperley, B.J. A l d e r and W.A. L e s t e r , J r . , J . Chem. Phys. 1982, 77, 5593. P.J. Reynolds, M. Dupuis and W.A. L e s t e r , J r . , J . Chem. Phys. 1983, 82, 1983. R.N. B a r n e t t , P.J. Reynolds and W.A. L e s t e r , J r . , J . Chem. Phys. 1985, 82, 2700. R.N. B a r n e t t , P.J. Reynolds and W.A. L e s t e r , J r . , J . Chem. Phys. 1986, 84, 4992. R.N. B a r n e t t , P.J. Reynolds and W.A. L e s t e r , J r . , J . Chem. Phys. 1987, 91, 2004. P.J. Reynolds, R.N. B a r n e t t , B.L. Hammond and W.A. L e s t e r , J r . , J . S t a t . Phys. 1986, 43, 1017. J.W. Moskowitz, K.E. Schmidt, M.A. Lee and M.H. K a l o s , J . Chem. Phys. 1982, 76, 1064. J.W. Moskowitz, K.E. Schmidt, M.A. Lee and M.H. K a l o s , J . Chem. Phys. 1982, 77, 349. D.W. Skinner, J.W. Moskowitz, M.A. Lee, P.A. W h i t l o c k and K.E. Schmidt, J . Chem. Phys. 1985, 83, 4668. K.E. Schmidt and J.W. Moskowitz, J . S t a t . Phys. 1986, 43, 1027. J . V r b i k and S.M. R o t h s t e i n , J . Comput. Phys. 1986, 63, 130. J . Vrbik, J . Phys. A 1985, 18, 1327. J . V r b i k and S.M. R o t h s t e i n . I n t . J . Quant. Chem. 1986, 29, 461. S.M.Rothstein, N. Patil and J . V r b i k , J . Comput. Chem. 1987, 8, 412. S.M. R o t h s t e i n and J . V r b i k , J . Comput. Phys. 1988, 74, 127. J.D. D o l l , Chem. Phys. L e t t . 1981, 81, 335. D.M. Ceperley, J . Stat. Phys. 1986, 43, 815. B.L. Hammond, P.J. Reynolds and W.A. L e s t e r , J r . , J . Chem. Phys. 1987, 87, 1130. M.M. H u r l e y and P.A. C h r i s t i a n s e n , J . Chem. Phys. 1987, 86, 1069. P.A. C h r i s t i a n s e n , J . Chem. Phys. 1988, 88, 4867. P.A. C h r i s t i a n s e n and L.A. LaJohn, Chem. Phys. L e t t . 1988, 146, 162. J . V r b i k , M.F. DePasquale and S.M. R o t h s t e i n , J . Chem. Phys. 1988, 88, 3784. D. Feldmann, Phys. A 1976, 277, 19. R.D. Mead, P.A. Schulz and W.C. Lineberger, Phys. Rev. A. J . S l a t e r , F.H. Read, S.E. Novick and W.C. L i n e b e r g e r , Phys. Rev. A 1978, 17, 201. P.A. C h r i s t i a n s e n , Y.S. L e e and K.S. P i t z e r , J . Chem. Phys. 1979, 71, 4445. L.F. P a c i o s and P.A. C h r i s t i a n s e n , J . Chem. Phys. 1985, 82, 2664. P.J. Hay and W.R. Wadt, J. Chem. Phys. 1985, 82, 270. W.J. Stevens, H. Basch and M. Krauss, J . Chem. Phys. 1984, 81, 6026. C.E. Moore, Atomic Energy L e v e l s , N a t l . Bur. Stand. C i r c . 467, Vols. I and I I , U.S. Government P r i n t O f f i c e : Washington, DC, 1949, 1952. W. M u l l e r , J . F l e s c h and W. Meyer, J . Chem. Phys. 1984, 80, 3297. T. Yoshida and K. I g u c h i , J . Chem. Phys. 1988, 88, 1032. R.J. H a r r i s o n and N.C. Handy, Chem. Phys. L e t t . 1985, 113, 257.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

22.

75. 76. 77. 78. 79. 80. 81. 82. 83.

CHRISTIANSEN

Quantum Monte Carlo Studies

321

S. Huzinaga, L. Seijo, Z. Barandiaran and M. Klobukowski, J. Chem. Phys. 1987, 86, 2132. W.A. Lester, Jr., private communication. E.A. McCullough, Jr., J. Chem. Phys. 1975, 62, 3991. P. Pyykko, G.H.F. Diercksen, F. Muller-Plathe and L. Laaksonen, Chem. Phys. Lett. 1987, 134, 575. M.P. Barnett, Methods Comput. Phys. 1963, 2, 95. J.B. Anderson, J. Chem. Phys. 1987, 86, 2839. R.A. Whiteside, J.S. Binkley, M.E. Colvin and H.F. Schaefer III, J. Chem. Phys. 1987, 86, 2185. Science News 1988, 133. J.L. Gustafson, G.R. Montry and R.E. Benner, SIAM J. Sci. and Stat. Computing 1988, 9.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch022

RECEIVED February 16, 1989

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 23

Relativistic Effects on Compounds Containing Heavy Elements The Influence of Kinetic Energy on Chemical Bonds 1

2

2

T. Ziegler , J . G. Snijders , and E. J . Baerends

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

1

Department of Chemistry, University of Calgary, Calgary, Alberta T2N 1N4, Canada Department of Theoretical Chemistry, The Free University, Amsterdam, Netherlands

2

It

i s shown that

relativity

will

reduce the

k i n e t i c energy of the e l e c t r o n s i n a number of compounds c o n t a i n i n g heavy elements. The

reduction o f the k i n e t i c energy leads t o

bond s t a b i l i z a t i o n and bond c o n t r a c t i o n and

i n f l u e n c e s s i g n i f i c a n t l y the chemistry

of t h i r d row t r a n s i t i o n

finite nuclei

Valence

e l e c t r o n s i n atoms

(albeit

small)

and they

instantaneous valence in

metals.

probability

c a n as

velocities.In

a

and molecules of being

this

acquire

fact,the velocities

e l e c t r o n s can approach t h a t o f l i g h t

reason

become

close to the

consequence

close proximity t o heavier n u c l e i with not too surprising

o f importance

compounds

series

5d-block

or 5f-block

as t h e y

relativistic

pass

i sfor effects

properties of

elements elements

high

for the

Ζ >72.It

f o r the chemical

containing

transition

that

have a

i n the third

i n the actinide

series. We s h a l l

here

d i s c u s s how r e l a t i v i s t i c

, r e l a t e d t o t h ehigh instantaneous v e l o c i t i e s near

heavy

involving

nuclei,

will

influence

5d- and 5f-elements.In

effects

of electrons

the chemical

particular,we

0097-6156/89/0394-0322$06.00/0 c 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

bond shall

23.

ZIEGLER ET AL.

Relativistic Effects on Compounds of Heavy Elements

demonstrate t h a t the in

bonds the

and

relativity

contract

k i n e t i c energy of

illustrated

how

p e r i o d i c trends The

i n many c a s e s w i l l

the

bond

the

relativistic

within

by

Pitzer

a triad

(1)

and

Slater in

method

to

some c a s e s by

method

by

studies Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

due

(£)

due

It

the

to

extend

hydrogen

retardation

atom as

1

written

as

- \

V i=l

H

D

(Zr2L) · The

relativistic and

the

will

been

results

Hartree-Fock-

Baerends

analysis

e f f e c t s . D i r a c s Hamiltonian be

be

(4.) , augmented functional

build

on

previous

Baerends.

Theories

the

neglecting

influence

r e c e n t l y proposed density .Our

is possible for

reduction further

quantum c h e m i s t r y has

t o Z i e g l e r , S n i j d e r s and

Relativistic

theory

effects

Pyykko

Snijders

the

Becke

a

of t r a n s i t i o n metals.

by

p r e s e n t e d h e r e a r e b a s e d on

strengthen

by

electrons.lt will

f i e l d of r e l a t i v i s t i c

reviewed

distances

323

1

h (i)

(7_)

relativistic

η-electron

well

as

systems

certain

by

magnetic

f o r a many e l e c t r o n s y s t e m

V (rt)+f

+

D

Dirac's to

N

i=l

Xl/|fïi*j

rj|

can

(1)

where

h (i) and

VN(r~i) In

i s the

p(i)

matrices,

electron-nuclear

a?(i),

Eq. (1)

i s the

2

c($>(±) . p ( i ) +

=

D

j = l,2,3

c p

(2)

attraction potential.

and

β

are

the

4x4

momentum o p e r a t o r , a n d c t h e v e l o c i t y

of l i g h t . T ^ e corresponding Η ° Ψ ° = E F°

D i r a c wave e q u a t i o n

reads

DV

where

the

wave

combination

of

function

solve

due

to

proposed

can

be

S l a t e r determinants

Dirac

components.

(3) Ψ°

the

presence and

as

constructed

a

from

,

linear four-

{a }.

wave e q u a t i o n

Foldy

expressed

1

component D i r a c - s p i n o r s The

Dirac

of

is

four,in

Wouthuysen

a transformation

somewhat (£_)

which a l l o w s

cumbersome

general have one

to

to

complex,

fortunately approximate

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

324

THE CHALLENGE OF d AND f ELECTRONS

the four-component Dirac equation,Eq.(3),by a twocomponent S c h r O d i n g e r e q u a t i o n i n t h e f a m i l i a r 2x2 P a u l i r e p r e s e n t a t i o n t o any g i v e n o r d e r i n t h e f i n e s t r u c t u r e constant a.The Hamiltonian H o f E q . (1) t a k e s a f t e r t h e A

D

Foldy-Wouthuysen t r a n s f o r m a t i o n t h e form sum o f t e r m s i n i n c r e a s i n g o r d e r s o f (X

o f an

infinite

2

A

H

FW

=

where

H° + a Hi + a H 2

H°

is

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

combination

the

*F

eigenfunctions

6

+ 0(a )

4

FW

(4),

non-relativistic

of

of Slater

2

F W

H

c a n be

determinants

Hamiltonian.The

written

as

a

linear

c o n s t r u c t e d from

o r b i t a l s as i n t h e n o n - r e l a t i v i s t i c c a s e . The method o f £ n i j d e r s and B a e r e n d s (4) in E q . (4) up t o OC Hi ,where H^ i s g i v e n by

spin

retains

terms

2

Hi The the

=

H

M V

+

H

first

order

2

(in a )

non-relativistic kinetic T

is

Dar

w + H

S 0

= - \

N R

(5) .

Λ

H M V Iwhich

s o - c a l l e d m a s s - v e l o c i t y term

relativistic energy

Σ V? i

represents

correction

t o the

operator

=

j

Σ i

P (i) 2

(6)

g i v e n by HMV

whereas

= - | Σ i

Vf

t h e Darwin

electrons

=

4

term,from

(ϋ_) , a f t e r

insignificant

~ 3" Σ P ( i ) i the Zitterbewegung

neglecting

two-electron

(7a)

operators

some

of the

numerically

(£a_) , t a k e s

on t h e

form Hoarw =

| Σ i

V?

(7b)

(V (ii)) N

Λ The

spin-orbit

total

energy

operator

Hso

does

of the closed s h e l l

t h e f o l l o w i n g and need n o t t h u s The energy ,with

relativistic

,is

obtained

wave

be s p e c i f i e d

considered i n here.

f u n c t i o n , and r e l a t e d

by f i r s t

the n o n - r e l a t i v i s t i c

not c o n t r i b u t e t o the

molecules

order

wave

total

perturbation theory

function

as

zero-order

solution.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

23.

Relativistic Effects on Compounds of Heavy Elements

ZIEGLER ET AL.

R e l a t i v i s t i c C a l c u l a t i o n s on M e t a l Dimers and Hydrides .Relativistic Bond Contraction Relativistic Bond Stabilization We

present

bond e n e r g i e s for

the

simple

M=Cu,Ag,and 'with

Au

MH

(4_)

Results

from M

Hg.Table

a similar

have

a

distances,as compounds

frequencies

coinage

isoelectronic I

contains

triad

series

MH

+

experimental (JUL) and

s e t o f c a l c u l a t i o n s on t h e as w e l l

are presented

follows

effects

the

H a r t r e e - F o c k - S l a t e r (HFS) c a l c u l a t i o n s .

(M=Zn,Cd,and Hg) It

of

Metal and

c a l c u l a t i o n s on

from n o n - r e l a t i v i s t i c

(M=Cu,Ag,and Au)

2

from

and v i b r a t i o n a l

as t h e

as w e l l as r e s u l t s

relativistic dimers

I results

hydrides

as w e l l

M=Zn,Cd,and

d a t a (10)

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

i n Table

,bond d i s t a n c e s

325

from

Tables

sizable

well

as

containing

M2

+ 2

i n Table I I . I and

II that

influence

on

vibrational the

metal-

as t h e d i c a t i o n s

bond

relativistic energies,bond

frequencies

heavier

5d-member

f o r the

(Au

or

Hg)

w i t h i n the t r i a d . In t h e n o n - r e l a t i v i s t i c the

wrong

order(compared

c a s e bond e n e r g i e s

with

experiment)

row>second

row>third-row.Relativistic

for

Hg

Au

and

,provide

stabilize

on t h e o t h e r

row>first

row>second

row

and

in

contract

distances proximity

the

of

by

and

0.3

third

Â

- 1

third limit,

rowsecond

compounds

.An

i s c a u s e d b y an

dfirst analysis increase

orbital

as

we

decend t h e t r i a d . c. S y n e r g i c

Metal-Ligand

Bonds.

A number o f l i g a n d s Cp" ( c y c l o p e n t a d i e n y l ) acceptors.That

is,they

s u c h as CO,O2,C2H2, C2H4, CôHg, a n d

bind have

to empty

metal

centers

π-orbitals

as

π-

capable

of

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

334

THE CHALLENGE OF d AND f ELECTRONS

accepting orbitals

electron density ,as

addition ability

from

σ-type

occupied

occupied

in la

i t i s shown f o r CO

π-type

or

metal

.The

based

d-

l i g a n d s have i n

orbitals

with

the

metal-orbitals(lb).

t o d o n a t e d e n s i t y t o empty

o ®

ô

V

υ 1c

1b

1a A number o f d e n s i t y Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

Four-electron two-orbital repulsion

Ligand metal donation

Metal-ligand (back)donatlon

f u n c t i o n a l c a l c u l a t i o n s on

carbonyls (lia)

, c o m p l e x e s of 02,C2H2, and

as

2

X ,CX,CX ,H CX 2

have

2

shown

bonds from

the

primarily the

metal

(X=0,S,Se,Te)

that

metal

is

due

center

carbonyls)

reached

strength to

in studies

the

and

same

conclusion

based

on

other

well (17d)

6

metal-ligand

(back) d o n a t i o n ligands (la

as

C6H ,Cp~

synergic

the

to

.The

(17c) of

metal-

C2H4 (17b)

of

i n the has

methods

charge case

also (12)

,see

of

been for

references. In an

attempt

found(17e) allowed

that

to

to q u a n t i f y the

operate

significantly the

acts

sum

of

Our

they

bond e n e r g y i n c r e a s e s by

investigations in

homologous row~third Figure

the

and

4

do

was are

indeed

interaction

50

% compared

π-back-bonding

the

by

from their

the

the 4d

energy and

case

M(CO)s

that

and

consequence (17a)

to

5d

when

the

of

of

the

each

the

(17b)

donor

metal

π-acceptor effects to

higher

are

first

found

row>third

as

they

on

the

in

M (CO) 6 which

ligand

orbitals

of t h i s

shown

trend,

of

orbitals

a

row>second

carbonyls to

the

,for

,as

3d

orbitals

for a qualifier

stability

first

(M=Fe,Ru,Os).This from

that

bonds

limit

metal

3 d - o r b i t a l s are

better

further

order

c o u n t e r p a r t s . The

Relativistic order

the

non-relativistic

(back) d o n a t i o n

fact

indicate

metal-ligand

,follow

i n the

in

(17.)

synergic

series row

(M=Cr,Mo,W) and

in

simultaneously

σ-bonding

the

it

π-back-bonding

alone.

strengths

set

synergic effect and

r e i n f o r c e each other.The o r b i t a l

c o n t r i b u t i o n t o the to

σ-bonding

i f the

,

energy are are

is

stems than as

a

closer

ligands(see

rationalization). (H)

to

change

row>second

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

row

the by

23.

ZIEGLER ET AL.

stabilizing

the

transition in

bonds i n v o l v i n g

series

Figure

also

Relativistic Effects on Compounds ofHeavy Elements

4.

A

,as

similar

illustrated

Figure

5.The

not

involved

in

electron

interactions

but

also

,which

rise

to

energy

particular

We

shall

effects

compounds

ligand

occupied

are

constitute

related

finally

compounds type the

for

of

of

involving

the

to

actinides

,by

Table I I I . C a l c u l a t e d and (R=CH H)

Bond

Energies

1 8

four-

in

the

metals,in electron

orbitals.

Energies

in

which

relativistic

bond

energies

representing

(Kcal

are

metal

,reduce

5p

Bond

for

a

kinetic

four

RMCI3 w i t h M=Th,U and

on

the

row

and

degree to

in

the

carbonyls

in

two-orbital 5s

is

orbitals.These

third

importance

from c a l c u l a t i o n s

empty

discussed

of

core-like

probe the

be

to

orbitals

to

increase

effects will,as

the

O2

(back)donation

metal

illustrated

Corrections Actinides.

might

stability

not

donations

substantial

involving

Relativistic Compounds o f

do

corrections

with

in

for

interactions

of

in destabilizing two-orbital

are

a

energy.Relativistic kinetic

to

favorable

(lb)

lc,give

metal

third

C2H2,C2H4,and

π-type donor o r b i t a l s . T h e s e

orbitals

interactions

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

i n the

the

metal-carbonyls

order

of

corrections

relativistic

σ-type or

only

complexes

relativistic

.Instead,the ligand

f o r the

relativistic

for

change (enhancement)

5d-elements of

illustrated

335

in

results

R=H,CH3.

1

mol" )

in

RMCI3

(M=Th,U)

3/

D(M-R) HThCl

N R

D(M-R)

30.1

3

R

a

76.0

Exp.

~ 80

CH3THCI3

35.8

79.8

~80

HUCI3

10.5

70.1

76

16.8

CH3UCI3

72.2

72

a

D(M-R) bond energies from CpM(Cl)R of Ref. It

follows

corrections

to

in

to

order

relativistic some

50-60

the

Table

actinide-R

reproduce

bond Kcal

from

energies 1

mol" .The

III

bond

19 that

energies

experimental are

seen

to

importance

be of

relativistic are

necessary

results.Non­ too

small

by

relativistic

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

336

THE CHALLENGE OF d AND f ELECTRONS

b

a 211

211 178 174

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

Cr MoW Non-Rel Figure

4 . (a)

216

216

210 178

162

162 130

Fe Ru Os

Cr MoW

Fe Ru Os

Non-Rel

Rel

R e l a t i v i s t i c and n o n - r e l a t i v i s t i c

176

Rel averaged

i n t r i n s i c bond energies f17a) i n M(CO)q (m=Cr,Mo,W) . ( b )

intrinsic 1

bond energies (17a) i n M(CO)s (m=Fe Ru Os).Energies i n kJ mol" . f

Figure

5 . (a)

f

R e l a t i v i s t i c bond energies (12h) i n (PH ) MX ,with 3

2

M=Ni,Pd, Pt and X= C2H2, C 2 H 4 , 0 . (a) R e l a t i v i s t i c bond energies (Uh) 2

i n (PH ) MX ,with M=Co,Rh,Ir and X=C2H , C2H4, O2 .Energies i n kJ mol" 3

4

2

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

1

23. ZIEGLER ET AL. corrections

Relativistic Effects on Compounds ofHeavy Elements

f o r the chemistry

of actinides

i s currently

(13.,21) .

under i n v e s t i g a t i o n

Acknowledgment All

calculations

carried

out a t t h e Cyber-205

i n Calgary(ACS) and Amsterdam(SARA).This

investigation

was s u p p o r t e d

Engineering Research

by t h e N a t u r a l

Research Council

the Netherlands

Organization

S c i e n c e and

o f Canada(NSERC)

as w e l l as

f o r t h e Advancement

o f Pure

(ZWO).

Cited

Literature

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

were

installations

1.

Pitzer,

2.

Pyykkö,Ρ.;Desclaux,J.Ρ.Acc.Chem.Res

K.S. Acc.Chem.Res.

3.

Pyykkö,Ρ.Chem.Rev.,in

4

1979,12,271. 1979,12,276.

press.

(a)Snijders,J.G.:Baerends,J.G.Mol.Phys.1978,36,1789. (b)Snijders,J.G.;Baerends,J.G.:Ros,P.Mol.Phys.1979,38, 1909.

5. 6

Becke,A.

J.Chem.Phys.

(a)

1986,84,4524.

Ziegler,Τ.;Snijders,J.G.;Baerends,Ε.J.J.Chem.Phys.

1981,74,1271. (b)Ziegler,T.;Snijders,J.G.;Baerends,E.J. Chem.Phys.Lett.

1980,75,1.

7.

Dirac,P.A.M.

Proc.R.Soc.London

8.

Foldy,

9.

Moss,R.E. Advanced

L.L.;Wouthuysen,S.A.

,Chapman and 10.

Molecular

1928 ,117,610. 1950,78,29.

Quantum

Mechanics

Hall,London,1973.

Krasnov,K.S.;Timoshinin,V.S.;Danilova,Τ.G.;Khandozhko ,S.V.Handbook of Molecular Compounds

11.

Ser.A Phys.Rev.

,Jerusalem

Baerends,E.J.;

Constants

of

Inorganic

1970.

Ellis,D.Ε.;Ros,Ρ.Chem.Phys.

1973,2,71. 12.

(a)Lee,Y.S.;Ermler,W.C.;Pitzer,K.S.;McLean,A.D. J.Chem.Phys.

1979,70,293.

(b)Hay,P.J.;Wadt,W.R;Kahn,L.R.;Bobrowicz,F.W. J.Chem.Phys. 13. Pilar

1978,69,984.

F.L. Elementary

Hill,New

Quantum Chemistry

14. Ziegler,T.;Tschinke,V.;Becke,A. 15.

Polyhedron

1987,6,685.

Baykara,N.A.;McMaster,B.N.;Salahub,D.R. Mol.Phys.1984,52

16.

,McGraw

York 1968.

(a)Ziegler,T.

891. Oragnometallics

1985,4,5675.

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337

338

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch023

17

18.

19. 20. 21.

THE CHALLENGE OF d AND f ELECTRONS

(b) Ziegler,T.;Tschinke, V.;Becke,A.J.Am.Chem.Soc. 1987,109,1351. (c) Ziegler,Τ.; Cheng,W.;Baerends,Ε.J.;Ravenek,W. Inorg.Chem.,1988,accepted. (d) Z i e g l e r , Τ . ; T s c h i n k e , V . ; V e r s l u i s , L ; B a e r e n d s , Ε . J . ; Ravenek,W.Polyhedron,in press. ( a ) Z i e g l e r , Τ . ; T s c h i n k e , V . ; U r s e n b a c h , C . J.Am.Chem.Soc. 1987,102,4825. (b) Ziegler,T. Inorg.Chem. 1985,24,1547. (c) Ziegler,T. Inorg.Chem. 1986,25,2721. (d) Ziegler,T.; Cheng,W.,unpublished work. (e) Baerends,Ε.J.;A.Rozendaal NATO AST 1986,C176, 159. Ziegler,T.;Baerends,Ε.J.;Snijders,J.G.;Ravenek,W. J.Chem.Phys,submitted.These calculations are based on a quasi-relativistic approach in which the valence density is determined variationally ,rather than by f i r s t order perturbation theory. Bruno,J.W.;Stecher,H.A.;Morss,L.R.;Sonnenberg,D.C. ;Marks,T.J. J.Am Chem.Soc. 1986,108,7275. Schwarz,W.H.F.;Chu,S.Y.;Mark,F. Mol.Phys. 1983,50,603 (a)Boerrigter,P.M.;Baerends,Ε.J.;Snijders,J.G. C h e m . P h y s . 1988,122,357. (b)Boerrigter,Ρ.Μ.,thesis,Vrije Universiteit,1987.

RECEIVED October 24, 1988

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 24

Ground-State Properties of Heme Complexes in Model Compounds and Intact Proteins 1

1

1

2

Frank U. Axe , Lek Chantranupong , Ahmad Waleh , Jack Collins , and Gilda H. Loew 2

1

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

2

The Rockefeller University, 701 Welch Road, Suite 213, Palo Alto, CA 94304 SRI International, 333 Ravenswood Avenue, Menlo Park, CA 94025 The ground and low-lying spin states of f e r r i c porphy­ r i n (heme) complexes found in model compounds and intact proteins have been studied using an INDO-RHF-SCF method parameterized to include t r a n s i t i o n metals. The results for the model compounds using known c r y s t a l structure geometries are consistent with and help explain the o r i g i n of their observed electromagnetic properties. These studies demonstrate the a b i l i t y of this method to determine with a high degree of reliability the r e l a t i v e energies of the manifold of spin states of f e r r i c heme complexes. The same method has been used to address unresolved questions regarding the resting states of four heme proteins, cytochrome P450 which is a monofunctional oxidase, cytochrome c peroxidase (CCP) and catalase (CAT) both with peroxi­ dase oxidizing a c t i v i t y , and metmyoglobin (MMB) which lacks s i g n i f i c a n t peroxidase oxidizing a c t i v i t y . The characterization of the P450 resting state leads to a possible explanation of the o r i g i n of i t s low-spin (S = 1/2) form. This result helps resolve the apparent contradiction between the presence of water as an a x i a l ligand as determined by the x-ray structure and the absence of hyperfine s p l i t t i n g s in ESR spectra with 65% O enriched water. Comparisons of the resting state calculations of CCP and MMB provides an understanding of the origins of the differences in observed e l e c t r o ­ magnetic properties for MMB and CCP in spite of the s i m i l a r i t y of their active s i t e s . Differences in function between MMB and CCP could not, however, be understood from properties of the active s i t e alone. Changing the imidazole ligand to an imidazolate in CCP makes i t s active s i t e more similar to CAT. Thus, the anionic form of the imidazole ligand of CCP, thought to be p a r t i a l l y formed by Η-bonding to a nearby Asp residue, could account at least in part for the similar a c t i v i t i e s of CCP and CAT as oxidizing enzymes. cam

17

0097-6156/89/0394-0339$06.00/0 ο 1989 American Chemical Society In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

340

THE CHALLENGE OF d AND f ELECTRONS

Heme proteins a l l share a common active s i t e or prosthetic group consisting of an iron-porphyrin (heme) unit, a nearly planar e n t i t y embedded i n the globular protein and connected to i t by one or at most two nearby amino acids which serve as a x i a l ligands. This important family of proteins performs three basic b i o l o g i c a l functions (1-5): reversible oxygen transport (globins), electron transfer (cytochromes), and metabolic oxidation of small organic molecules and peroxides (peroxidases, catalases, and cytochrome P450s). In a l l heme proteins, the b i o l o g i c a l function i s centered on the heme unit and primarily on the iron i t s e l f ( 1-5). Thus the oxidation and spin state of the iron, the nature of the a x i a l ligands, and the protein environment of the heme unit serve as subtle modulators of b i o l o g i c a l behavior. The heme group i s also the p r i n c i p a l o r i g i n of spectroscopic features of these proteins. Both electronic spectra (6-9) and ground-state electromagnetic properties such as quadrupole s p l i t t i n g s i n MOssbauer resonance spectra (10-14), anisotropic g values and hyperfine s p l i t t i n g s i n electron and nuclear spin resonance spectra (15-23) and temperature-dependent magnetic moments (24-29) originate almost e n t i r e l y on the heme u n i t . Consequently, a large f i e l d dedicated to the study o f model heme complexes has emerged i n an e f f o r t to understand the e f f e c t of changes i n the heme unit i t s e l f on these observed properties. These studies are useful i n understanding the properties o f intact heme proteins since isolated heme complexes have electromagnetic proper­ t i e s very similar to heme units embedded i n proteins. Some of these model heme systems have also been shown to mimic the b i o l o g i c a l a c t i v i t y of intact heme proteins. For instance, model oxo-iron compounds have been found to epoxidize o l e f i n s much l i k e the cytochrome P450s (34, 35). The r e l a t i v e s i m p l i c i t y o f model heme complexes makes i t possible to study the important role of the a x i a l ligands (30-33) i n modulating electronic structure and geometries without the e f f e c t of the nearby amino acid residues present i n the proteins. The insights gained from such studies can help to separately assess the r e l a t i v e importance of the heme unit i t s e l f and of i t s protein environment on the function of intact heme proteins. Up to now most quantum mechanical studies of the ground and excited states of model heme complexes have focused primarily on diamagnetic systems (36), with less frequent treatment of heme systems with unpaired spins (37-42). With the inclusion of a r e s t r i c t e d Hartree-Fock treatment (37, 38) within an INDO formalism parameterized f o r t r a n s i t i o n metals (39, 40, 42), i t i s now possible to calculate the r e l a t i v e energies of d i f f e r e n t spin states o f f e r r i c heme complexes i n an evenhanded fashion at a semiempirical l e v e l . In the work reported here we have used t h i s method i n two types of studies. The f i r s t study i s a systematic investigation of the e f f e c t of changes i n geometry and ligand type on the r e l a t i v e energies of the low-lying spin states and observable properties of eight model f e r r i c heme complexes. This study also represents a test of the c a p a b i l i t i e s of the INDO-RHF method to characterize the lowest lying doublet (S = 1/2), quartet (S = 3/2) and sextet (S = 5/2) spin states of these model f e r r i c heme complexes. The eight complexes chosen a l l have known c r y s t a l structures and include those with varying a x i a l ligands and high-, intermediate-, and low-spin ground

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

24.

A X E ET AL.

Ground-State Properties of Heme Complexes

341

states as inferred from magnetic s u s c e p t i b i l i t y measurements (43-50) and g values in electron spin resonance spectra (47-48). They also have known quadrupole s p l i t t i n g s ( A E Q ) from Mossbauer resonance spectra (44, 45, 49, 51-53), a quantity which we d i r e c t l y calculate. In the second type of study, using insights gained from the model compound studies, the active s i t e of the resting state ( i . e . , state of the enzyme when not involved in i t s biochemical cycle) of four heme proteins, cytochrome P450 , metmyoglobin (MMB), cytochrome c peroxidase (CCP), and caualase (CAT) have been characterized. These four proteins belong to d i f f e r e n t classes of heme proteins. P450, CCP, and CAT are oxidative metabolizing enzymes thought to share a similar highly oxidized b i o l o g i c a l l y active state, and MMB i s the oxidized form of an oxygen transport protein with l i t t l e or no peroxidase or monofunctional oxidase a c t i v i t y Q ) . Each of these proteins have f e r r i c resting states which have been characterized by x-ray structure determinations (54-57). Paradoxically, while a number of long-standing questions have been resolved by these structure determinations, new ones are emerging. This study addresses f i v e such s p e c i f i c questions.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

c

The f i r s t two questions involve properties of the resting state of cytochrome P450 , the only P450 with a known structure (54). The camphor-free resting state i s mostly in a low-spin (S = 1/2) form while the camphor-bound state i s a high-spin (S = 5/2) ferric complex. The x-ray structure (58) reveals, that as previously deduced, the camphor-bound state i s 5-coordinated with a cysteine residue as the single a x i a l ligand and the iron s i g n i f i c a n t l y out of the porphyrin plane. The camphor-free state retains the cysteine ligand, but s u r p r i s i n g l y , a water, and not a second amino acid as previously thought (5), binds to the iron in the d i s t a l ligand binding s i t e . There i s also evidence that t h i s water i s part of an Η-bonded network involving four more water molecules. c

With the insight gained from the x-ray structure, two puzzling aspects of the camphor-free resting state have emerged. One i s the o r i g i n of the low-spin form deduced from observed electromagnetic properties (5). This r e s u l t i s surprising since other f e r r i c heme proteins with H 0 as a s i x t h ligand such as MMB have primarily highspin (S = 5/2) ground states. The other question raised i s : If water i s an a x i a l ligand, as reported in the x-ray structure, why i s no broadening of the ESR spectra from hyperfine interactions observed in 65% enriched 0 HpO (H. Beinert, private communeiation), as i t i s in MMB (59). Since the magnitude of the hyperfine s p l i t t i n g depends d i r e c t l y on the amount of unpaired spin density on the water oxygen atom, the p o s s i b i l i t y that the negative results obtained for P450 could be a consequence of reduced spin density on the water has been investigated. 2

17

A l l the remaining questions focus on comparisons of CCP, MMB, and CAT. The f i r s t question addressed i s : Can the differences in the active s i t e of CCP, MMB, and CAT account for the differences in their observed electromagnetic properties? MMB and CCP (55, 56) have been found to have the same heme unit, a f e r r i c protoporphyrin-IX with a water and an imidazole as a x i a l ligands. CAT has a single

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

342

THE CHALLENGE OF d AND f ELECTRONS

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

phenolate group from a nearby tyrosine residue as an a x i a l ligand (57). In addition to the c r y s t a l structures, the nature of the resting states of CCP, MMB, and CAT have been probed by Mossbauer (11-13), temperature-dependent magnetic s u s c e p t i b i l i t y (27-29) and electron paramagnetic resonance (21-23) experiments. The observed properties of both CAT and MMB have been interpreted in terms of a d e f i n i t e high-spin resting state, while the properties of CCP have been interpreted in terms of a thermal d i s t r i b u t i o n of high- and lowspin states (_Π, 29). The determination of the r e l a t i v e energies of the low-lying sextet (S = 5/2), quartet (S = 3/2) and doublet (S = 1/2) states of the active s i t e s of these proteins should lead to a better understanding of the o r i g i n of these properties. The f i n a l two questions raised are the extent to which the active s i t e i t s e l f controls the function of CCP, CAT, and MMB. S p e c i f i c a l l y , we have asked: To what extent can s i m i l a r i t i e s in function between CAT and CCP be understood in l i g h t of their d i f f e r ­ ent active s i t e s ? F i n a l l y , we have asked: To what extent can differences in the function of MMB and CCP be understood in terms of their active s i t e c h a r a c t e r i s t i c s alone? Knowledge of the extent to which the active s i t e can account for function should help to under­ stand the r e l a t i v e importance of the protein environment around the heme in determining the function of each protein. Methods All calculations were carried out within the approximation of intermediate neglect of d i f f e r e n t i a l overlap (37-42) (INDO-RHF-SCF) which includes parameterization for t r a n s i t i o n metals. A restricted open-shell formalism, developed by Zerner et a l . (37,38), was employed to prevent spin contamination and to make the quantitative evaluation of the r e l a t i v e spin state energies possible. This method has been used successfully to study simple t r a n s i t i o n metal complexes l i k e [ F e C l J * (42), [CUC1J2- (42), and ferrocene (4Ί) as well as larger and more complicated systems l i k e model oxyheme (6M) and carbonylheme (6l_) and model oxyhorseradish peroxidase (62) complexes. Energies of the lowest l y i n g sextet, quartet and doublet states were calculated for each of the heme units studied. The geometries of the complexes were taken from c r y s t a l structures and s i m p l i f i e d to unsubstituted porphyrins. The orientations of the porphyrin macrocycles were such that the pyrrole nitrogens were on the x- and y-axes. The choice of the lowest energy configurations for each state was as follows: Doublet state: Quartet state: Sextet state:

(d

and d

)

3

The sextet state configuration i s unique. The choice of the lowest energy configuration for the doublet and quartet states was confirmed by comparisons of the r e l a t i v e energies of various quartet and doublet configurations, obtained by assignment of unpaired electron(s) to d i f f e r e n t iron d o r b i t a l s , in some representative complexes. It i s also corroborated by a recent detailed study of

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

24.

A X E ET AL.

343

Ground-State Properties of Heme Complexes

hemin u s i n g the INDO/RHF ( J . P h y s . Chem., i n p r e s s ) .

method

by

Edwards,

Weiner

and

Zerner

The quadrupole s p l i t t i n g ( A E Q ) o b s e r v e d i n Mossbauer resonance o f heme compounds was c a l c u l a t e * from the INDO-RHF e i g e n v e c t o r s . T h i s q u a n t i t y was determined by f i r s t c a l c u l a t i n g the n i n e components (VjJ o f the e l e c t r i c f i e l d g r a d i e n t t e n s o r , u s i n g the a p p r o p r i a t e o n e - e l e c t r o n o p e r a t o r , and c o n s i d e r i n g o n l y the c o n t r i b u t i o n o f the i r o n from a l l i t s f i l l e d o r b i t a l s . The 3> IV j j j > | V j J . These v a l u e s were then used i n the e x p r e s s i o n : k

AE

= 8 ( 1 -

R)Qq[1

+

2

η /3]*

.

( 1 )

where q = V , η = ( V ^ V , J / V * (0 < η < 1 ) , ( 1 - R ) = S t e r n h e i m e r S h i e l d i n g c o n s t a n t , ana Q = n u c l e a r quadrupole moment. The s i g n o f A E Q i s the s i g n o f the l a r g e s t component V . . . V a l u e s o f Q and ( 1 - R ) used i n these c a l c u l a t i o n s a r e 0 . 1 8 7 and O . b o , r e s p e c t i v e l y . i

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

Q

i

i

R e s u l t s and D i s c u s s i o n Model Heme Complexes. Presented i n T a b l e I a r e the c a l c u l a t e d r e l a t i v e energy d i f f e r e n c e s i n k c a l / m o l f o r the s e x t e t , q u a r t e t , and d o u b l e t s t a t e s o f the model f e r r i c heme complexes i n c l u d e d i n the present study. Also included i n Table I are the calculated quadrupole s p l i t t i n g s U E Q ) f o r the r e l e v a n t s p i n s t a t e , a l o n g w i t h the e x p e r i m e n t a l l y observed v a l u e s o f A E Q and the measured e f f e c t i v e magnetic moments. The r e s u l t s c l e a r l y demonstrate t h a t the ground s p i n s t a t e c a l c u l a t e d f o r each model complex a g r e e s w i t h the one i n f e r r e d from measured e f f e c t i v e magnetic moments. Moreover the energy s e p a r a t i o n s between these ground s t a t e s and the o t h e r two s p i n s t a t e s a r e c l e a r l y consistent with observable electromagnetic properties and help explain their origins. The observed e f f e c t i v e magnetic moments ( 4 3 - 4 6 ) f o r the 5 - and 6 - c o o r d i n a t e d complexes found t o have s e x t e t ground s t a t e s a r e a l l i n the range o f 5 . 9 - 6 y t y p i c a l o f h i g h - s p i n complexes. The c a l c u l a t e d AEQS f o r the h i g h - s p i n s t a t e o f t h e s e complexes a r e a l s o i n good agreement w i t h the e x p e r i m e n t a l v a l u e s known f o r t h r e e o f them ( 4 4 b

46,

5 0 ) .

Both 5 - and 6 - c o o r d i n a t e d h i g h - s p i n complexes have s i g n i f i c a n t s p i n d e n s i t y on the p o r p h y r i n r i n g , 6 0 ? o f which i s on the p y r r o l e nitrogens. T h i s s h o u l d be m a n i f e s t i n h y p e r f i n e s p l i t t i n g s o b s e r v ­ a b l e i n ESR o r ENDOR e x p e r i m e n t s . The u n p a i r e d s p i n d e n s i t y on the a x i a l l i g a n d s i s much l e s s than on the p o r p h y r i n r i n g and g r e a t e r on a n i o n i c than n e u t r a l l i g a n d s . The c a l c u l a t e d r e s u l t s f o r the 5 - c o o r d i n a t e d h i g h - s p i n complexes i n d i c a t e t h a t i t i s d e f i n i t i v e l y more s t a b l e than the d o u b l e t and q u a r t e t s t a t e s by - 3 0 k c a l / m o l and - 1 2 k c a l / m o l , r e s p e c t i v e l y . The 6 - c o o r d i n a t e d h i g h - s p i n complexes e x h i b i t a s i g n i f i c a n t r e d u c t i o n o f the energy s e p a r a t i o n between the s e x t e t and the q u a r t e t s t a t e s ,

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

2

5.9

5.90

5.92

experimental

6.05

1.12 1.22

0 5 18

TMS0 TMS0

4.5-5.3

3.20 3.5

2 0 25

ClO^

2.09

2.26 3.7-4.7

0 12 12

2.19 2.25

3

a

N" pyridine

0.35

2.31

0 22 19

CN" CN"

Low Spin

3.14 2.7

0 17

3

3-Clpyridine 3-Clpyridine

Intermediate Spin

value of azide complex of MMB and CCP (Reference 53, page 3)

Exp. ( y )

y

e f f

0.12

0 5 18

(NCS)" pyridine

1.01 0.76

0 12 28

2

(SpN0 r

0.44 0.46

A E Q (mm/sec)

0 12 33

Cl""

Cale. Exp.

b

(kcal/mol)

S = 5/2 S = 3/2 S = 1/2

ΔΕ

L

High Spin

"-1

1-2

Table I. Relative Energies of Different Spin States of Model Ferric-Heme Complexes

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

24.

A X E ET A K

Ground-State Properties of Heme Complexes

345

attributable mainly to the presence of the second a x i a l ligand which increases the tendency of the iron atom to move into the plane of the porphyrin.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

In both 5- and 6-coordinated complexes the energy of the doublet state i s predicted to be too high to play a s i g n i f i c a n t role in determining the observed electromagnetic properties. However, because of the smaller separation of the sextet and quartet states in the 6-coordinated high-spin compounds compared with the 5-coordinated compounds, spin mixing (\f>S) between them should be enhanced. Therefore larger zero f i e l d s p l i t t i n g s and more aniso­ tropic g values (63) in the ESR spectra should be observed. Intermediate-spin (63) heme complexes are rare and two complexes inferred to have quartet ground states have been included in our studies. As shown in Table I, the predicted ground state of each complex i s a S = 3/2 state in agreement with the spin state assign­ ment deduced from observed properties. The calculated r e l a t i v e energy of the doublet spin state i s - 20 kcal/mol above the quartet spin state, while the sextet states of [Fe(TPP)(ClO^)] and [Fe(0EP)(3-Clpy)p] are only ~ 1.8 and 2.8 kcal/mol above their respective quartet state. These results strongly suggest that observable properties can best be understood in terms of s i g n i f i c a n t spin-orbit coupling of these two low-lying states, together with the p o s s i b i l i t y of a thermal equilibrium of such spin-mixed states. +

Effective magnetic moment measurements of [Fe(TPP) (ClOi,) ] (47) have yielded values in the range of 4.5-5.3 y at 77-300°K. This temperature dependence and range of values i s consistent with c o n t r i ­ butions from sextet and quartet states with the quartet lower in energy. ESR data (47) for this complex yielded values for g and g| of 4.75 and 2.03, respectively. These results are a t y p i c a l for a high-spin complex and lend further support to the conclusion that the ground state i s a S = 3/2 or a 3/2,5/2 mixture with predominant S = 3/2 character. b

(

Magnetic susceptibility measurements (48) for [Fe(0EP)(3C l p y ) ] , the other intermediate complex studied, y i e l d a range of y between 3.7-4.7 y for the temperature range of 77-294°K. This range of magnetic moments i s also consistent with an intermediatespin or spin-mixed ground state. The EPR spectrum (48) for t h i s complex yielded values for g and g» of 4.92 and 1.97 respectively which are similar to values obtained for [Fe(TPP)(ClO^)]. +

2

e f f

b

The A E Q values calculated for the quartet state of these complexes also agree very well with the experimentally observed values (Table I) for the same complexes. A l l these r e s u l t s taken together are highly suggestive that the S = 3/2 spin state i s the p r i n c i p a l contributor to the ground electronic state of these complexes. For the dicyano and azide-pyridine complexes, our calculated results indicate that in each case a S = 1/2 spin state i s the lowest energy state with the quartet and sextet states much higher in energy. The observed e f f e c t i v e magnetic moments (Table I) of both

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

346

THE CHALLENGE OF d AND f ELECTRONS

the dicyano (49) and the azide-pyridine (50) complexes indicate that these two complexes are indeed essentially low-spin at a l l temperatures. The only experimental A E Q measured f o r either of the two lowspin complexes i s f o r the dicyano complex. Our calculated value of 2.31 mm/sec for the S = 1/2 spin state i s i n poor agreement with the experimental value (53) of 0.35 mm/sec. However, the reported experimental A E Q seems to be anomolously low for what i s considered to be a low-spin complex. There i s apparently no experimental A E Q measured for the [Fe(TPP)(N^)(py)] complex. The calculated value or 2.19 mm/sec for the doublet state i s , however, i n good agreement with the experimental values (53) of 2.45 and 2.25 mm/sec f o r CCP-N and MMB-N, respectively, which d i f f e r only by one a x i a l ligand being an imidazole rather than pyridine. This provides further evidence that [Fe(TPP)(N^)(py)] has a doublet ground state. 3

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

3

An important conclusion from t h i s study of model compounds i s the additional evidence obtained for the key role o f the S = 3/2 spin state i n the chemistry of f e r r i c heme complexes. There are no complexes for which S = 5/2 and 1/2 spin states are close enough to interact without an even greater contribution of the S = 3/2 spin state. Thus, the widely used assumption (64) of high-spin/low-spin thermal contributions to explain observable properties o f heme complexes appears to be incorrect. Explanations involving highspin/intermediate-spin interaction are much more plausible, since small energy separations between these states were found. In general, the r e l a t i v e spin state energies calculated for a l l the model heme complexes studied are consistent with and help explain their observed electromagnetic properties. Thus the INDO-RHF method used appears to be sensitive to the effect of the varying a x i a l ligands and predicts the correct energy order of spin states produced by each of them. The a b i l i t y of the method to predict the patterns of spin state behavior i n these model complexes lends credence to the use made of i t i n the second part of these studies, to further characterize the heme units i n the resting state of four heme proteins. Comparative Studies of Resting State Active Sites of Four Heme Proteins. In this second type of study reported, we have used the x-ray structure of the active s i t e of four heme proteins: cytochrome P450 (54), CCP (55), MMB (56), and CAT (57) s i m p l i f i e d to the f e r r i c porphyrin complexes, shown i n Table I I , to calculate the r e l a t i v e energies and electron and spin d i s t r i b u t i o n s i n their lowlying sextet, quartet and doublet states. çam

As shown in Table I I I , a high-spin ground state i s d e f i n i t i v e l y obtained for camphor-bound 4 5 0 i n which the single a x i a l ligand i s a mercaptide. For camphor-free 4 5 0 , with water and mercaptide as a x i a l ligands at their x-ray structure values, the sextet state i s s t i l l the lowest energy, but the energy separation to the low-spin (S = 1/2) state i s greatly diminished. In the x-ray structure deter­ mination of the resting state, the value of the Fe-water distance was p

c a m

p

c a m

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

24.

A X E ET AL.

347

Ground-State Properties of Heme Complexes

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

Table I I . Ligand Distances (Â, X-Ray) Used for Oxidized Resting State of Four Model Heme Proteins

[MMB] L

[CCP]

H0

2

2

L

1 Fe-0 Fe-L 7 a *Fe

H0

--

Imidazole

Phenolate

2

Imidazole

[CAT]

[P450] H0 2

(SCH3)"

2.40

--

2.02

1.93

1.76

2.32

0.25

0.13

0.13

0.24

1.90

2.24

Extent of out-of-planarity of the iron atom from the mean porphyrin ring plane.

constrained and the exact position of the water was not e x p l i c i t l y optimized. Thus we have considered the consequences of a shorter iron-water distance (2.0 Â) and movement of the iron into the heme plane. In this geometry, a low-spin state i s calculated to predominate. An alternative o r i g i n of the s t a b i l i z a t i o n of the lowspin state comes from the p o s s i b i l i t y that some anionic character i s imparted to the water ligand by i t s postulated interaction with the network of Η-bonded water molecules, seen in the x-ray structure (54). This e f f e c t was simulated by using an OH" as an a x i a l ligand with an Fe-0 bond length of 1.75 A. As seen in Table I I I , in t h i s model of the resting state, the low-spin (S = 1/2) state i s favored by 16 kcal/mol over the high-spin state. While this i s an extreme model for the e f f e c t of Η-bonding, i t does demonstrate that p a r t i a l anionic character of the water ligand could account for the predominant low-spin ground state observed. Table IV gives the spin densities calculated on the water oxygen for P 4 5 0 and for comparison, in MMB. Experimental values of y f f and our calculated results (Table V) indicate MMB i s in a high-spin cam

e

American Chemical Society Library 1155 16th St, N.W. Washington, D.C. Salahub, 20036 D., el al.; In The Challenge of d and f Electrons; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

348

THE CHALLENGE OF d AND f ELECTRONS Table I I I . Origin of the Low-Spin Form of the Resting State of Cytochrome P450

Models for Resting State a

1

(SCH )~

L

H0

H0

S = 1/2

6

S = 5/2

0

L

3

2

2

(SCH )"

Substrate Bound State (SCH )~

b

C

3

3

(SCH )"

(0ΗΓ

—

0

0

15

1.5

16

0

2

d

3

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

ΔΕ (kcal/mol)

a

Geometry from x-ray structure as shown in Table II (Ref. 54)

b

X-ray structure with Fe-Water distance of 2.00 Â and iron moved into the porphyrin plane

c

X-ray structure with Fe-0H~ distance at 1.75 Â as i t i s in model Fe-0 complexes.

d

Geometry form x-ray structure as in Reference 58.

2

ground state. In i t s c r y s t a l structure geometry, the water oxygen of the high-spin f e r r i c MMB i s calculated to have 0.057e or about 1.1? of the t o t a l spin. For this protein, a barely detectible amount of broadening of the g=2 signal was observed in the ESR spectra in the presence of 0 enriched H 0 (59). By contrast, in both the highspin state and the low-spin state of P450, in i t s c r y s t a l structure geometry, the spin density on the a x i a l water ligand i s much lower than in the corresponding state of MMB. Allowing the Fe-0 distance of the water ligand to decrease, or simulating i t s ionic character by an 0H~, both of which favor a low-spin ground state, somewhat increases the spin density on the oxygen. However i t remains at most about 1/6 that of MMB. Since the broadening in MMB was barely detectable, no measurable broadening of the ESR spectra in 0 enriched water would be expected for either low-spin model of 4 5 0 currently proposed here. These results then account for the absence of such broadening in a manner consistent with the presence of water as an a x i a l ligand in the resting state of 4 5 0 as observed in i t s x-ray structure. 17

2

17

p

c a m

p

c a m

Turning now to comparisons of CCP, MMB, and CAT, the r e l a t i v e energies of the germane doublet, quartet, and sextet spin states have been calculated using the same INDO-RHF-SCF method as for the model complexes and the results are presented in Tables V and VI. The geometries for the resting states of CCP, MMB, and used here were takem dire'ctly from their respective x-ray c r y s t a l

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

CAT

24.

A X E ET AL.

349

Ground-State Properties of Heme Complexes

Table IV. Calculated Spin Densities of Oxygen of H 0 Ligands in P 4 5 0 and MMB

a

a

2

cam

MMB

P450 OH"

H0 2

S = 5/2

s

0.01

= 3/2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

B

0.16

0.057

0.01

0.030

0.01

0.004

(0.03)

s = 1/2 a

(0.03)

0.0005 (0.004)

Values underlined are for lowest lying state.

^Values in parenthesis calculated for H 0 at 2.00 Â and Fe in the porphyrin plane. 2

Table V.

Effect of Geometry on Resting States of CCP, MMB, and CAT

ΔΕ

D

4h

X-Ray

4h

X-Ray

D

0

0

0

1.5

2.1

2.8

5.7

5.8

6.3

D

4h

(kcal/mol)

S = 5/2

0 (0)

S = 3/2

-1.2 (-2.3)

S = 1/2 AEQ

e f f

a

8.0 (13.0)

b

0

0 (0)

-1.1

2.7 (5.6)

7.7

7.5 (11.4)

(mm/sec)

Cale. Exp. y

CAT

MMB

CCP X-Ray

Exp. ( y ) b

3.20

0.76 1.33

0.72 0.84

4.86

6.00

5.92

a

Values

b

Values in parenthesis with iron moved 0.1 Â further out of mean plane of porphyrin ring.

in parenthesis without d i s t a l water

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

350

THE CHALLENGE OF d AND f ELECTRONS Table VI. E f f e c t of Ionization State of Axial Ligand i n CCP and Catalase

CCP L L

1

Catalase

H0 Im

H0 Im"

2

2

2

Phenolate

Phenol

ΔΕ (kcal/mol)

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

S = 5/2 S = 3/2 S = 1/2

1.2 0.0 9.2

0 2.4 9.5

Net Charge Fe 1.25 0.11 1 0.18 L •0.55 Porph. Spin Fe 2.79 L1 0.02 L2 0.07 0.12 Porph. a

t

2

2

0 2.1 5.8

b

b

1.35 0.10 -0.63 -0.83

4.26 0.02 0.16 0.56

a

F o r S = 3/2 spin state

b

F o r S = 5/2 spin state

1.37 —

b

4.7 0 12.2

1.28* —

-0.55 -0.82

0.14 -0.42

4.24

2.74

—

0.25 0.51

—

0.11 0.15

coordinates (55-57). However, i n order to examine various geometric e f f e c t s on the spin states of each heme unit, calculations were also carried out at several step-wise regularized geometries, s t a r t i n g from the c r y s t a l geometry of each protein. The e f f e c t s of porphyrin r u f f l i n g and doming were examined by regularizing the porphyrin c r y s t a l geometry to symmetry f o r CCP, MMB, and CAT, while leaving the a x i a l ligands at the same geometry as i n their c r y s t a l structure. Further differences i n the geometries of CCP and MMB were examined by removal of the a x i a l water in CCP and by increasing the out of plane distance of the iron in MMB by 0.1 Â. The calculated r e l a t i v e energies of CCP (Table V) indicate that the S = 3/2 state i s the lowest energy spin state in CCP, with the S = 5/2 and S = 1/2 spin states being -1 kcal/mol and ~9 kcal/mol higher i n energy. Furthermore, the energy ordering and separation of the spin states are rather insensitive to regularization of the porphyrin and a x i a l ligand geometries. The predominance of the quartet state i n CCP appears to be due to a combination of near planarity of the iron and a weak a x i a l ligand. The Fe-water distance in CCP at 2.4 Â i s considerably longer than that i n MMB. Indeed, calculations at the c r y s t a l geometry i n which the water ligand i s

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

24. A X E ET AL.

351

Ground-State Properties of Heme Complexes

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch024

removed r e s u l t s i n the enhanced s t a b i l i t y o f the q u a r t e t s t a t e o f the complex i n s p i t e o f the 5 - c o o r d i n a t e n a t u r e o f the i r o n ( T a b l e V ) . For MMB ( T a b l e V ) , the s e x t e t s t a t e i s the l o w e s t energy s t a t e f o r the n e u t r a l i m i d a z o l e l i g a n d w i t h the q u a r t e t and d o u b l e t s t a t e s ~3 and ~8 k c a l / m o l h i g h e r i n energy. In c o n t r a s t t o CCP where r e g u l a r i z a t i o n o f the p o r p h y r i n r i n g t o symmetry had l i t t l e e f f e c t ; f o r MMB i t l o w e r s the energy o f b o t h the q u a r t e t and d o u b l e t s t a t e s r e l a t i v e t o the s e x t e t . These r e s u l t s suggest t h a t the enhanced doming o f the p o r p h y r i n r i n g observed i n MMB r e l a t i v e t o CCP i s a f a c t o r i n s t a b i l i z i n g the s e x t e t s p i n s t a t e i n MMB. However, s i n c e the s e x t e t s t a t e i s s t i l l l o w e s t i n energy even when the p o r p h y r i n i s r e g u l a r i z e d t o D j ^ symmetry, the enhanced o u t - o f - p l a n e d i s t a n c e o f the i r o n must be the main c o n t r i b u t o r t o the s t a b i l i z a t i o n o f the s e x t e t s t a t e . T h i s e f f e c t i s v e r i f i e d by the f u r t h e r s t a b i l i z a t i o n o f the s e x t e t r e l a t i v e t o the q u a r t e t s t a t e when the i r o n atom i n MMB i s moved by an a d d i t i o n a l 0.1 Â out o f the mean porphyrin plane (Table V ) . The c a l c u l a t e d r e l a t i v e s p i n s t a t e e n e r g i e s f o r CAT ( T a b l e V) a t the c r y s t a l geometry shows t h a t the the s e x t e t s t a t e i s the most s t a b l e s t a t e w i t h the S = 3/2 and S = 1/2 s t a t e s , r e s p e c t i v e l y , ~2 k c a l / m o l and ~6 k c a l / m o l h i g h e r i n e n e r g y . The c l o s e e n e r g e t i c p r o x i m i t y o f the S = 3/2 s p i n s t a t e i s a r e s u l t o f the s m a l l d i s p l a c e m e n t o f the i r o n atom from the p y r r o l e n i t r o g e n plane. Changing the h i g h l y r u f f l e d p o r p h y r i n m a c r o c y c l e o f CAT t o one o f pure Djj symmetry l e a d s o n l y t o a v e r y s m a l l s t a b i l i z a t i o n o f the s e x t e t s p i n s t a t e o f - 0 . 5 k c a l / m o l r e l a t i v e t o the q u a r t e t and doublet s p i n s t a t e s . Thus, the h i g h l y i r r e g u l a r p o r p h y r i n m a c r o c y c l e i n the c r y s t a l s t r u c t u r e has v e r y l i t t l e e f f e c t upon the r e l a t i v e s p i n s t a t e o r d e r i n g s i n t h i s system. n

In a d d i t i o n t o geometry v a r i a t i o n s , the e f f e c t s o f hydrogen bonding and the r e s u l t i n g i o n i c i t y o f the p r o x i m a l l i g a n d s i n CCP and CAT were s i m u l a t e d by d e p r o t o n a t i o n o f the i m i d a z o l e Ν i n CCP and p r o t o n a t i o n o f the t y r o s i n e oxygen i n CAT. D e p r o t o n a t i o n t o form an Im" l i g a n d i n CCP r e v e r s e s the o r d e r o f the s e x t e t and q u a r t e t s t a t e e n e r g i e s ( T a b l e V I ) . S i n c e t h i s i s an extreme model f o r the p a r t i a l p r o t o n t r a n s f e r t h a t c o u l d o c c u r as a r e s u l t o f the i m i d a z o l e b i n d i n g t o a nearby a s p a r t a t e r e s i d u e i n CCP, the p a r t i a l a n i o n i c n a t u r e c o u l d r e s u l t i n near degeneracy o f the q u a r t e t and s e x t e t s t a t e s . δ

Both the q u a l i t a t i v e and q u a n t i t a t i v e r e s u l t s o b t a i n e d f o r the a c t i v e s i t e s o f the t h r e e p r o t e i n s p r o v i d e an improved b a s i s f o r u n d e r s t a n d i n g the observed e l e c t r o m a g n e t i c p r o p e r t i e s o f the r e s t i n g s t a t e s o f CCP, MMB, and CAT. An i m p o r t a n t a s p e c t o f the p r e s e n t results i s t h a t f o r t h e s e p r o t e i n a c t i v e s i t e s , the s e x t e t and q u a r t e t s t a t e s a r e c l o s e i n energy and the d o u b l e t s t a t e i s s i g n i f i ­ cantly higher. Thus, the dominant contributions to observed p r o p e r t i e s i n these p r o t e i n s a r e expected t o come from the S = 5/2 and S = 3/2 s p i n s t a t e s , which can mix by s p i n - o r b i t c o u p l i n g (63) as w e l l as be i n thermal e q u i l i b r i u m . These r e s u l t s provide a consistent e x p l a n a t i o n o f the e l e c t r o m a g n e t i c p r o p e r t i e s o f the r e s t i n g s t a t e s o f CCP, MMB, and CAT. The a l t e r n a t i v e e x p l a n a t i o n , a thermal e q u i l i b r i u m between s e x t e t and d o u b l e t s t a t e s , w i t h o u t a

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

352

THE CHALLENGE OF d AND f ELECTRONS

contribution from the quartet proteins.

state, does not seem possible i n these

These results are p a r t i c u l a r l y important for a correct q u a l i t a t i v e understanding of the the observed properties of CCP. For CCP the experimentally observed magnetic s u s c e p t i b i l i t y ( 2 9 ) and MOssbauer spectra (JM) have been interpreted in terms of a thermal mixing between high- and low-spin states, ignoring any contribution from the intermediate-spin state. This explanation i s contrary to our findings of E ç ^ ~ E3/2 1 / 2 * Measured values of y f f ( 2 9 ) for CCP that are i n the range of 3 . 7 to 4 . 0 y over a temperature range of 7 7 - 2 5 0 ° K can more correctly be understood in terms of a thermal contribution from heavily spin-mixed sextet and quartet spin states. These r e s u l t s also strongly indicate that a re-analysis of the Mossbauer resonance spectra of CCP (JM) as a mixture of quartet and sextet states would also be more appropriate.
-1.0 V (SSCE). Reduction potentials for alkyl halides of interest are generally more negative than -1.5 V (SSCE) (17). Alkyl halide photoreduction is observed for binuclear d& complexes whose excited-state reduction potentials are more positive than -1.0 V (SSCE) in CH3CN. An alternative pathway to outer-sphere electron transfer, which yields similar photoredox products with alkyl halides, is excited-state atom transfer (Figure 3b). Data obtained for Pt2(P20sH2)4 ~ indicate that alkyl Z

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch025

357

+

Î

0

+

2

2

+

2

2

4

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

z

THE CHALLENGE OF d AND f ELECTRONS

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch025

358

Figure 2. P i c t o r i a l representation of the M2-localized hole in a 3(da*pa) state.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch025

25.

8

SMITH & GRAY

Atom-Transfer Reactivity ofBinuclear d Complexes

b.

RX + M-M

°M-M

•

+ RX

•M-M-X + R.

RX + . M - M . ·

•M-M-X + R. R. =

/

X-M-M-X + C H C H 2

2

\

R. = .CR'X (R* = alkyl, aryl) R-M-M-X

Figure 3. a. S R ^ I mechanistic scheme for halocarbon photooxida tive addition to binuclear d^ complexes, b. Atom-transfer mechanism for halocarbon photooxidative addition.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

359

360

THE CHALLENGE OF d AND f ELECTRONS

and aryl halides react with the 3(0) and t h e c l u s t e r s

Fe(II)

make

spins

5/2

identifiable

that

c a n be

break

describe

2

magnitudes

transparently

that

iron

b y Gibson and

approximated as

Β i s a

splitting

i n the

to

2

and S

1

with

These

Ε -

(S'+h)

e

i s favored

to

models

application

S ' = 9 / 2 , and i n which

(4).

by just

closely

Here

S

l

by recognizing

derealization more

Here

i n which

strong

indistinguishable rationalized given

JS

andfour-iron

occurs,

few y e a r s

Hamiltonian

cluster.

character,

a

not

theclassic

a n d have

coupling

valence

i n the past

or spin Hamiltonian

H -

Fe(III)/Fe(II)

antiferromagnetic sites.

follow

clusters.

2

iron

mixed-valence trapped

coupling"

(3) o f t h e H e i s e n b e r g

describe

with

p r o g r e s s h a s b e e n made

o f "vector

co-workers

367

Spin Coupling and Electron Derealization

calculations

picture

included an alternate derivation

of

work,

(6,8)and had

many

can be used of

their

o f t h e (S'+H)

factor

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

368

THE C H A L L E N G E OF d AND f ELECTRONS

relating this

resonance e n e r g i e s t o c l u s t e r s p i n (6) .

a n a l y s i s t o t h e o x i d i z e d forms

linear

a n dc u b a n e - l i k e

forms o f t h e three reduced The

systems

.

(11) the

.

these

iron

to create

of three-iron

Broken

since

i t

crux

o f our computational

broken

metal

sites

symmetry have

compute a n d i n t e r p r e t , used.

cobalt

interest

because i t

approach a r i s e s

the

from

( i n which

broken

symmetry

the recognition

the

pure-spin

ground

three

S=5/2

extend for in

locations

We

Heisenberg

the

pure

We h a v e

spin

earlier

applied

H e n c e , we

computed

from

including

this

clusters

to the reduced and doubly

are

that are

parameters

states,

three-iron

t i m e we c o n s i d e r t h e e f f e c t s

assume

spins

andunderstand. to energies

methods

(which

states

anduse the resulting

as i n oxidized

the analysis

polynuclear

iron

state.

spins,

the f i r s t

to approximate

easy to

functional

wavefunctions

spin Hamiltonian of

equivalent

are relatively

density

"correct"

wavefunctions,

estimate

can be systems

stabilizes

otherwise

2

t o f i t a n (assumed)

four-iron

mixed-metal

eigenfunctions of S ) aregenerally multiconfiguration choose

to

a n dz i n c

Cluster.

especially i f local

c o n s i d e r a b l y more d i f f i c u l t

procedure to ( £ ) .

H e r e we

reduced species,

of electron

where

derealization

clusters. that

the true

together

electrostatic

can be

replaced

interactions

by

an

that

interaction

couple of the

type:

2

1 2

that

the off-diagonal

electron

derealization

discussed then

above.

consists

matrix

The essence of

choosing

eigenvalues this

important may

be We

we w i l l

basis

spin matrices,

for

first

state.

In

on each

a

three

where

Hamiltonian to

be

model

included,

andcomparing the r e s u l t i n g

i n order

to estimate

interpretations

the high site

three

iron spin

Β and J .

of

models

a l l

We f o r m

basis

three

configurations, in

a

configurations

worked

with

the

equivalent

Fe(II)/Fe(III)/Fe(III)

are aligned

(which must b e spin-down) We h a v e

cluster

reduced",

spin-up.

sites.

states

more c o m p l e x s p i n H a m i l t o n i a n

quantitative

d-electron three

states

c o n s i d e r o n l y the s i m p l e s t models t h a t have t h e

i n the "singly

d-electrons

connecting

be o f the form B(S'+H), as

data.

consider

sites

oxidation

will

of choosing a spin the

physical interactions; necessary

experimental metal

elements

t o t h e computed Χα e n e r g i e s

paper,

(3)

1

i s "allowed"

diagonalizing theresultant In

to the

H — J SiS +Ji3S 'S3+J23S2S3 and

of

certain

i s diamagnetic.

spin populations)

By c o n t r a s t ,

in

c l u s t e r s b u t does n o t a f f e c t t h e

wavefunctions

different

i nt h e

andi n aconitase

converted

novel

Symmetry A n a l y s i s f o r a T h r e e - I r o n

that

sites

vinlandii)

shown t h a t

TheZn complex i s o f p a r t i c u l a r coupling problem

are

been

clusters,

3

a r e themselves

the active

c a n be r e a d i l y

of the fourth

more r e d u c e d forms

The

for

a n d Azobacter

sites

both

states.

chosen f o r i l l u s t r a t i o n

andi t has recently

i n place

spin

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch026

of

extended

(with

we c o n s i d e r t h e r e d u c e d

and analogous ZnFe

as models

( f r o m D. gigas

Many

clusters,

we h a v e

interest

ferredoxins

used

iron clusters,

We r e c e n t l y

iron clusters

( 9 ) . ) Here

a n dd o u b l y - r e d u c e d o x i d a t i o n

considerable (10)

geometries

o f three

to reside

out the

parallel

five

fashion, say

by allowing i n turn

matrix

formal

the f i r s t the

final

on each o f the

elements

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

of the

26. SONTUM ET AL· Heisenberg

portion

delocalization Hence,

of

the Hamiltonian

terms

characterizes

we

resonance

assume matrix

and below,

between

each

the diagonal

5 for parallel

equivalent

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch026

(65/4)J these

+

elements

clusters, symmetry

atoms

(which

equivalent

we

atoms

configurations, the

2

3

find

state,

we

J =

2

represents

(65/4)J

"a") the

dimers. -

5B'

five of

pair

d-electron.

Following

the

λ

opposite

Papaefthymiou

only between

t h e two i r o n s

x

lowest.

For

For the

are

to

that

both

three

basis of the

et

s t i l l

iron

of

possible locations

that

(7),

al.

we w i l l

adopt

resonance i n t e r a c t i o n

o f t h e same s p i n ,

pair

"b" .

is

Here

spin matrix i s :

Here

we

have

symmetry results the

allowed

state,

0

0

•(15/4)J

0

0

•(25/4)J

the

The

symmetry

from t h a t

case

are E

1

J ' s and B ' s c a n , thus,

differences

arising

from

state

value)

energies

are

then

Hamiltonian. various

orbital

-

the model

o f Β and J are given

the

broken the Χα

Eigenvalues f o r 3

=

-(15/4)J.

by comparing those

state

the

computed

and estimates

resulting used

in

+ 5B a n d E

with

approach,

(5)

spin state;

indeed happens.

-(25/4)J

the ground

from

(For the simple

values

2

formulas

(including made

>

parameter

i n the high

be e s t i m a t e d

these

b r o k e n symmetry m o l e c u l a r spin

delocalization

B, to d i f f e r

r e p o r t e d b e l o w show t h a t t h i s

broken

5B

(25/4)J 5B

energy

carry

out an approximate

spin

and i t s

parametrized

here,

i n R e f . 7.)

projection

energy from

spin spin

the eigenstates In

a

o f the pure

for

the language

o u r p r e v i o u s p a p e r s , we a r e u s i n g t h e H e i s e n b e r g H a m i l t o n i a n to

three

are E =

o f one o f t h e

spin

There

simplest delocalization hypothesis,

important

lies

Ε

corresponding to the three

give

(S'+^s)

degenerate).

d-electrons

"b".

energy

elements

We a r e a s s u m i n g

(doubly

and, hence

is

the system

The e i g e n v a l u e s

2

B'2.0

3 shows t h e m o l e c u l a r

orbitals

h a s some

the Fermi

corresponds to antiferromagnetic and a

observed irons is

S-2 Fe(II)

(11).

monomer,

are reduced to Fe(II) low-energy

electrons. there

yielding

net

for

and the other 55a').

This

Fe(II)/Fe(III)

spin

"a" iron 1

of

5/2,

remains

Fe(III) several

a r e now p o s s i b l e a c c e p t e r s low-lying

orbitals

"a" site

orbitals

o f many m o r e

the existence

as

However,

o f these

to the u combination o f the "b" pair

a r e now a d d i t i o n a l

this

i n which both o f the "b"

and the unique

and 37a") and on the u n i q u e , imply

57a'),

a

occupation,

features,

level

orbital

g

i n energy by 1,300 cm" ).

orbitals

In a d d i t i o n

the

parameters.

w i t h one e x t r a

coupling o f an S ' - 9 / 2

(An a l t e r n a t i v e

computed to be h i g h e r

other

(in orbital (in

the

knowledge

by the zinc i o n .

l o c a l i z e d on the " a " i r o n "b" pair

worth

of

special

d-orbitals near

is

much

electron

dimer

cubane-like It

f o r o u r l o w e s t - e n e r g y b r o k e n symmetry s t a t e , the

likewise

geometries

n o r d o we h a v e

of the iron

(for

since,

i s predicted

(7).

cluster,

over

we

the ground s t a t e

cluster

3

o f the f o r the

t h a n t h o s e shown i n t h e

o f J and Β t o changes i n s t r u c t u r a l

reduced

stabilization

delocalized

than

discussed below);

experimental

a r e n o t known,

As with

| B | / J a r e around 1.5

|B|/J to be l a r g e r

with a l l J ' s equal,

though,

doubly

of

the

the doubly Zn complex,

about the s e n s i t i v i t y The

for

combination. t o the geometry

the computed J ' s t o be t o o l a r g e ,

cluster,

remembering,

values

Computed magnitudes

except

we e x p e c t

three-iron

to be r e l a t e d

absolute

the true values of

Table, to

below the antisymmetric

o f Β appear

both

there

(38a" and 6 0 a ' ) . states

than

extra

(36a"),

(e.g.,

56a'

These

extra

i n our model

o f E q . (7) , a n d s u g g e s t t h a t more c o m p l i c a t e d s p i n H a m i l t o n i a n m o d e l s may

be necessary

to

Hence,

the estimate

rather

approximate;

much s m a l l e r The

nevertheless,

reported

that

gas often

electron (19).

species

correlation

the q u a l i t a t i v e use

include

than

effects

(9).

effects

of

the values

past

found

in

a

spin-dependent

generally

studies

a

approximation,

effects

(20),

on the reduced

exchange-correlation potentials

of

reliable.

exchange-correlation

account

states

to increase J over

Calculations

Χα

density

a better

high-spin

conclusion

should be

the c o r r e l a t i o n

give

was i n d e e d f o u n d i n o u r e a r l i e r

clusters

calculations.

and c a n be compared t o

local

Since correlation

more

the

interest

the

uniform

this

here

Within

parametrizations properties

orbital

f o r J i n the doubly reduced complex i s

which has h i s t o r i c a l

calculations.

spin

the molecular

1

J f o r the doubly reduced complex

results

function,

describe

o f 45 cm"

stabilize

one would

low

expect

reported here, and

of oxidized species

three-iron

using

improved

give

serious

are i n progress.

Conclusion The

calculations

present

here

among

c o n s i d e r a t i o n to the competing e f f e c t s delocalization

in

polynuclear

qualitative

features

to

experimental

a

describe particular

solely

support

mechanism

from a n a l y s i s

transition

the "double

spectra in

the

these

(7),

first

to

o f s p i n coupling and e l e c t r o n metal exchange"

although

we d o n o t

calculations--the

o f computed t o t a l

energies

clusters. model

put

The forth

postulate

results

for various

arise states.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

26.

SONTUM E T A L .

-0.

80

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch026

-0.

Iron Β

Iron A

05k

-0.10

375

Spin Coupling and Electron Delocalization

α'

38

a"

57

α',

58%

85%

1

86%

39

α "_

36%

59

α*

25%

5b

α'

33%

37

a".

37%

58

α'

39%

38

α"

45%

15

CO

55 a'...! — 44%

cn 0) JO

>^ - 0 .

20

54

c LU

.22%

α ' 3 9 %

53

-0.

ο*.

.26%

25

Figure 3. As i n Figure 1, for the lowest energy broken symmetry state of the doubly-reduced ZnFe cluster. Valence o r b i t a l s are numbered within each symmetry and have been placed i n three columns depicting the location of their primary charge d i s t r i b u t i o n . V e r t i c a l bars indicate occupied o r b i t a l s . 3

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

376

THE CHALLENGE OF d AND f ELECTRONS

Even

though

general

the magnitudes

trends

qualitative elsewhere

information (21)

that

spin coupling i n Fe S 4

at the

first

sight

+ 3

delocalization others

lines

(23).

of

Although there

wave m o d e l s , to

they

obtaining

ideas

Furthermore, computers)

are clear

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch026

for

a wide

range

can

a clear

to

relatively

that

incorporate

and

electron

a n d t h e method

i s being

to

t o ab

local

exchange

the method, initio

t h e scheme o u t l i n e d h e r e localized

are

interactions

the

models

shows

explain

results

and

the such

limitations

and

along

should enable

and d e l o c a l i z e d

adapted

muffin-tin

and one c a n

studies

t o b e made b e t w e e n s p i n H a m i l t o n i a n m o d e l s a n d p r a c t i c a l metal

Even

one o f u s ( L . N . ) these

offer

useful

exchange

are not essential

Thus,

too high,

c l u s t e r s , where t h e e x p e r i m e n t a l

of

(with faster

are often

to be r e l i a b l e .

useful:

extension

i n complex c l u s t e r s ,

(22).

approximations forward

path

are l i k e l y

c a n be quite

A

physics

I

simple

perplexing.

straightforward

by

a

Χα and scattered

essential

o f J we p r e d i c t

seen i n Table

look

t h e same

connections calculations

polynuclear

transition

complexes.

Acknowledgments We t h a n k

E c k a r d Munck

Institutes

of Health

f o r many u s e f u l (GM39914)

discussions,

for financial

and The N a t i o n a l

support.

Literature Cited 1.

Spiro. T.G., ed. "Iron-Sulfur Proteins", Vol. 4; New York, John Wiley, 1982. 2. a. Anderson, P.W. Phys. Rev. 1959, 115; 2. b. Hay, P.J.; Thibeault, J.C.; Hoffman, R. J. Am. Chem. Soc. 1975, 97, 4884; c. Ballhausen, C.J. "Molecular Electronic Structures of Transition Metal Complexes", New York, McGraw-Hill, 1979; Section 3-6. 3. Gibson, J.F.; Hall, D.O.; Thornley, J.H.M.; Whatley, F.R. Proc. Natl. Acad. Sci. USA 1966, 56, 987. 4. For a review, see Munck, E.; Kent, T.A. Hyp. Int. 1986, 27, 161. 5. Anderson, P.W.; Hasegawa, H. Phys. Rev. 1955, 100, 675. See also Borshch, S.A.; Kotov, I.N.; Bersuker, I.B. Sov. J. Chem. Phys. 1985, 3, 1009. Borshch, S.A. Sov. Phys. Solid State 1984, 26, 1142. 6. Noodleman, L.; Baerends, E.J. J. Am. Chem. Soc. 1984, 106, 2316. 7. Papaefthymiou, V.; Girerd, J.-J.; Moura, I.; Moura, J.J.G.; Münck, E. J. Am. Chem. Soc. 1987, 109, 4703; Münck, E.; Papaefthymiou, V; Surerus, K.K.; Girerd, J.-J., ACS Symposium Series, L. Que, ed. (in press). 8. a. Noodleman, L. J. Chem. Phys. 1981, 74, 5737; b. Norman, J.G., Jr.; Ryan, P.B.; Noodleman, L. J. Am. Chem. Soc. 1980, 102, 4279; c. Aizman, Α.; Case, D.A. ibid. 1982, 104, 3269. d. Noodleman, L.; NormanJ.G., Jr.; Osborne, J.H.; Aizman, Α.; Case, D.A. ibid. 1985, 107, 3418; e. Noodleman, L.; Davidson, E.R. Chem. Phys. 1986, 109, 131. 9. Noodleman, L.; Case, D.A.; Aizman, A. J. Am. Chem. Soc. 1988, 110, 1001. 10. a. Beinert, H.; Thomsom, A.J. Arch. Biochem. Biophys. 1983, 222. 333, and references therein. See also Ref. 4. 11. Moura, I.; Moura, J.J.G.; Münck, E.; Papaefthymiou, V.; LeGall,

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

26.

12. 13. 14.

15.

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch026

16. 17. 18. 19. 20. 21. 22. 23.

SONTUMETAL.

Spin Coupling and Electron Delocalization

377

J. J. Am. Chem. Soc. 1986, 108, 349; Surerus, K.K.; Münck, E.; Moura, I.; Moura, J.J.G.; LeGall, J. ibid. 1987, 109, 3805. a. Johnson, K.H. Annu. Rev. Phys. Chem. 1975, 26, 39. Case, D.A. ibid. 1982, 33.151. Cook, M.; Karplus, M. J. Chem. Phys. 1985, 83, 6344. Antonio, M.R.; Averill, B.A.; Moura, I.; Moura, J.J.G.; Orme-Johnson, W.H.; Teo, B.-K.; Xavier, A.V. J. Biol. Chem. 1982, 257, 6646. Beinert, H.; Emptage, M.H.; Dreyer, J.-L.; Scott, R.A.; Hahn, J.E.; Hodgson, K.O.; Thomson, A.J. Proc. Natl. Acad. Sci. USA 1983, 80, 393. Stephens, P.J.; Morgan, T.V.; Devlin, F.; Penner-Hahn, J.E.; Hodgson, K.O.; Scott, R.A.; Stout, C.D.; Burgess, B.K. ibid. 1985, 82, 5661. Robbins, A.H.; Stout, C.D. "Iron-Sulfur Cluster in Aconitase at 3.0 ÅResolution", (submitted for publication). Hagen, K.S.; Holm, R.H. J. Am. Chem. Soc. 1982, 104, 5496. Cook, M.; Case, D.A. Quantum Chemistry Program Exchange #465, Bloomington, Indiana. Girerd, J.J.; Papaefthymiou, G.C.; Watson, A.D.; Gamp, E.; Hagen, K.S.; Edelstein, N; Frankel, R.B.; Holm, R.H. J. Am. Chem. Soc. 1984, 106, 5941. Vosko, SN.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58. 1200. Salahub, D.R. In: other ab initio Methods in Quantum ChemistryII. K.P. Lawley, ed. (John Wiley, 1987), p. 447. Noodleman, L., Inorg. Chem. (in press). Bencini, Α.; Gatteschi, D. J. Am. Chem. Soc. 1988, 108, 5763; S. Mattar, personal communication. Yamaguchi, K.; Tsunekawa, T.; Toyoda, Y.; Fueno, T. Chem. Phys. Lett. 1988, 143, 371. For a recent overview of Ab initio calculations, see de Loth, P; Karafiloglou, P.; Daudey, J.-P.; Kahn, O. J. Am. Chem. Soc. 1988, 110, 5676. See also Ref. 6.

RECEIVED October 24, 1988

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

Chapter 27

Role of High- and Low-Spin Electronic States in the Co(NH )6 Exchange Reaction 2+/3+

3

Marshall D . Newton

1

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

Institute for Molecular Science, Myodaiji, Okazaki 444, Japan

Ab initio SCF and Möller Plesset calculations with flexible valence basis sets including 4f orbitals are carried out for the ground and first excited spin states of the Co(NH )6 and Co(NH3)6 complexes. The results of the calculations in conjunction with a first-order spin-orbit coupling model yield an estimate of 10 for the electronic transmission factor in the Co(NH )6 exchange reaction using an apex-to-apex approach of reactants, thus providing a mechanism characterized by only a modest degree of non-adiabaticity, consistent with the experimental kinetic data. 2+

3+

3

-2

2+/3+

3

The mechanistic analysis of the kinetics of electron transfer processes involving transition metal complexes in solution continues to stimulate intense theoretical activity (1-17). In terms of the conventional transition state expression for the rate constant for activated electron transfer,

it is of particular importance to assess the various contributions to the activation energy (Ε ) and to the electronic transmission factor (κ ), which is a measure of the probability of successful reaction once the reactants have been activated (v and Γ are, respectively, the effective harmonic frequency associated with the reaction coordinate, and the nuclear tunnelling factor). In the context of the present volume, it is to be emphasized that the techniques of computational quantum chemistry have proven to be valuable tools for estimating and analyzing the relevant activation energies and electronic transmission factor (2-11). Since electron transfer between transition metal complexes generally involves the nominal exchange of valence d-electrons, it is seen that the particular challenge to quantum chemistry is that of treating "d-electrons" -- including the energetics of the various possible electronic states associated with d-electron manifolds, and the †

el

n

1

η

Permanent address: Department of Chemistry, Brookhaven National Laboratory, Upton, NY 11973

0097-6156/89/0394-0378$06.00/0 c 1989 American Chemical Society

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

27.

NEWTON

nature

o f the

orbitals not

states

always

either

from

for

be h i g h o f view

electronic

latter

quantity

since

the

reliable

complexes Electron required states

i n general

with

pairing

schemes

(via

available

d-orbitals)

can y i e l d

reliable

(18-25)»

excitation

Among t h e

arising

energy o r treatment

determining (7-10,16).

valuable

various

as w e l l

information,

types

metal

correlation.

t o obtain

i s certainly

energies,

especially

for

electron

of electron

metal

asradial

of d-electrons

even

and

the

intermediate

ordifferent

intransition

4f orbitals)

correlation

activation for

i n some c a s e s

and i t —

A balanced

inclusion of electron

lengths

state

of transition

spin multiplicities

( 1_1 ) .

effects

complexes,

by v a r i o u s

properties

the

bond

for

different

correlation

o f the

i s necessary

metal-ligand

the

dominated

valence

dominant

i s important

Fock model

necessitates

correlation

reasonable

states

the

of low-lying

super-exchange mechanisms

Hartree

calculation

often

metal

kinetically

of minimizing

i t i s often

through

the

and

density

transmission factor.

of different

states

The

transition

i s the

energies

Although

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

may

the

"virtual"

d-orbitals

ligands.

point

the

metal

coordinated

obvious which

the

maximizing of

c o u p l i n g between

of their

electronic is

379

High-and Low-Spin Electronic States

complexes,

(via

angular

relatively

i s expected

diffuse

t o be important

(26-33). With

this

interest,

b a c k g r o u n d we

the

Co(NH )£

Co(NH ) 2+ 3

r

[t

5 2

(the

e

g

/

,

m

T

lowest

species

+

6

2. 4 g

l

reaction

g

r

]

6 [ t

formally

provide

2 + 6

[t2 e /

2

g

2 +

3

4

[t2ge / g

the

two

Co(NH ) 3

6 2

g

T

e

i

low

2 +

Ί

-

3 +

3

g

-

T

l

g

]

3

t

^

3 +

3

l

g

l

g

]

each the

nonadiabatic as written (a t

2

*+ e

g

g

inter-complex e low

( 1_6).

g

probability

Accordingly,

states

s oa s t o

2 +

Co(NH ) 3

+

]

2

E ] g

2 +

Co(NH ) 3

4

2

[t\ e / g

(3)

6

g

6

/ T

for

[tf e{/

ig

Co(NH )

[

+

A]

2g

,

g

(11,17) :

l

6

[t /

6

3 + 6

2 , 4 e / T

While

excited

exchange

3

ig

[ti e{/

g

g

a very

process

(2)

6

m

2

state

process

t o have

Co(NH )

A]

3

g

(4)

6

T

l

g

]

reactions",

+

1 , 2^ / E]

g

e

X

Co(NH )

"cross

6

3 + 6

and

as the

ligand-field for

5 [ t

r

]

reaction

"1-electron"

pathways

g

3

a strongly

as w e l l

be expected

lying

g

]

g

l

< 1 ) , the

e

would

l4 /

+

6

g

2 +

Co(NH )

i n brackets).

indicate

K ^

3

g

4

2

configuration

do n o t

+

6

6 , 1 , [ t / A

i s indicated

Co(NH )

E ]

g

Co(NH )

[t

thus

+

r

]

g

3 +

3

to a "three-electron"

"1-electron"

3

r

,

reaction:

Co(NH )

each reactant,

t o invoke

Co(NH )

-

l

t o a conventional

seeks

and

. 1 / A

g

(34)

within

relative

3 + 6

one where

interchange

one

now c o n s i d e r a s p e c i f i c p r o c e s s o f exchange

electronic

corresponds

and

+

A

2

data

(i.e.,

3

/

3

energy

exchange),

+

Co(NH )

at equilibrium

experimental

2

3

Co(NH ) 3

r

[t

5 2

g

e

3 +

-

6

1, 3 / T

ι

m

g

l

g

]

Co(NH ) 3

r

6 [ t

3 +

, 1 / A A

2

g

+

6

Co(NH ) 3

-ι ] «[ t. 5 e2. / 4T_

2 +

r

l

g

2

g

g

(5)

6

Ί 2

g

]

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

THE CHALLENGE OF d AND f ELECTRONS

380 2 +

Co(NH ) 3

4

[t'geg/

T

These

l

separations

through

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

3

+

l g

where

2 +

2

c

E

1

degree

first-order

where

equilibrium states due

T

g

mixing

theory),

ligand-field

experiment

are available

e t a l . (~ 3 3 0 0 - 3 8 0 0

7000

1

(17).

a modified

INDO m o d e l ,

associated

with

than

In orbital

theory

perturbation

"1-electron o f H^f i f

v i a the coefficients

thus

f o r these be known

vary

an i n v e r s e underscoring

quantities.

at the transition smaller

arises

than f o r

because

low-spin

C o - N bond

A variety

ranging

from

ligand-field

reactants

lengths

i s known

of

estimates

i n t h e 2+

the recent

model

estimate

estimate

yields

an estimate

out CI c a l c u l a t i o n s the transition

may b e n o h i g h e r

of

of using

state i n energy

state.

we e v a l u a t e

C o - N bond

the high-spin/low-spin

lengths,

T h e known e n e r g y

geometry

sides of

β

separation

(RHF a n d U H F ) , i n c l u d i n g

theory.

i t s equilibrium

calculated

1

work,

matrix

f o r t h e 3+ c o m p l e x cm"*.

et a l . carried

g

f o r various

f o r t h e 2+ a n d

κ ^ t o have

at larger

and concluded that

2

(7) ]

g

separations ( v i a

-

transition

the present

separations

(e.g.,

c m * ) (1_1) t o a n e a r l i e r

the E / A ^ g

t h e ^"T^g/^A^g

c E

(17)

strength.

(11.17.36).

Larsson

2 + 6

2

+

t o the square

must

favorable

A conventional

cm~T~(36).

case.

the spin-orbit

situation

* A ^ g •+ ^ T j g e n e r g y

complex

l g

hand

may be s u b s t a n t i a l l y

(35) t o be ~ 13,700

Larsson

[ T

separations,

t h e a n a l o g o u s " ~ ^ T i g •+ E g v e r t i c a l

9000 c m "

with

estimates

This

less

4

]

g

the spin-orbit

separations

become r e l a t i v e l y

3

and r i g h t

we e x p e c t

reliable

( U ) .

Co(NH )

b e a sum o f t h e v a r i o u s

and s i n c e

magnitudes

l

spin-orbit

which couples t h e i n i t i a l and

i s not too great

their

+

T

of

2-6, to yield

coefficients

e

geometries

states. energy

of the

electron-transfer

to the left

energy

either

Boltzmann

i n the latter

3 +

3

C

quadratically

(.16),

i n previous

estimation

3

ig

will

]

g

in the reaction

theory

lA +

]

E

by appropriate

Co(NH )

2

for

at

l

2

g

(6)

6

the ground e l e c t r o n i c

l

(1-17).

of having

[tf e{/

analyzed

the h i g h - s p i n / l o w - s p i n energy

The v e r t i c a l

of

->

these

t o reduced

from

3 +

dependence on t h e energy

importance

Furthermore,

]

3

and f o r estimates

The o v e r a l l

perturbation

fourth-order

g

2 +

Co(NH )

of the high-spin/low-spin

K ^ i s proportional

model)

with

into

case,

6

of adiabaticity

Landau-Zener inversely

Since

been

perturbation

(corresponding

coefficients.

state,

mixing

3

C

vary

l

may p a r t i c i p a t e

are the spin-orbit

and w i l l

T

(as determined

7, r e s p e c t i v e l y ) ,

pathways"

3

+

we m a y c o m b i n e E q u a t i o n s

3

ig

6

i . e . , for direct

Co(NH )

[A +

3 +

3

g

estimates

respectively.

states

[t| .J/

w h i c h have

coupling,

l

Co(NH )

]

(11.17).

H^f = J $ i H $ f

Equation

the

states,

first-order

]

g

c and c

element,

the

l

-

i n the former

+

6

3+ s t a t e s , final

i g

/

are required:

spin-orbit

4

A

g

reliable

factors

Co(NH ) [ T

[t|

population

case,

Boltzmann With

6

or v i a spin-orbit

either

mixing

3

studies

thermal

factors) In

]

g

3 +

Co(NH )

excited

theoretical through

+

6

using

electron

separation

i s used t o c a l i b r a t e

ab i n i t i o

energy

molecular

correlation v i a f o r t h e 3+ c o m p l e x and adjust

t h e raw

results.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

27. NEWTON

High-and Low-Spin Electronic States

Computational

Details

The

ab

initio

calculations

second-order Gaussian the

82

computer

Institute

basis

(.39)

Molecular

(14s,

9p,

and

p-GTO's

4p

orbital

with

orbital

GTO

(after

Hay)

(40),

2.58

a^ ,

level,

consists

of

no.

Table

an

a

in

Wachters VI

instead

(8/5/3/1)

1

of

of

The

original

of

basis.

[3/2]),

For

the

the

full

39,

Co

a

2

+

with fit

additional,

and C o

that

as

ions

two

to

a n STO 3d

exponent

at

for

the

MP2

Co

(contraction

employed above,

4-31G

at

diffuse

basis

we

noted the

+

basis

contracted

except

3

the

Wachters

supplemented an

of

computers

least-squares

ligands,

3

Co a t o m s ,

contracted

extended

NH

HITAC

GTO ( w i t h GTO o r b i t a l

(8/4/2)

Reference

a version

the

aQ ),

4f

SCF and

the

1

1.0

the using

for

was

a 2-GTO

single

u s i n g UHF o r b i t a l s ) .

1

as

exponent

and

For

primitives)

(contracted

at

(37,),

adapted

Science.

5d

out

(MP2)

(38)

based on o p t i m i z a t i o n

contraction

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

for

carried

level

program

additional

2

were

Môller-Plesset

381

a

[4/1]

thus

basis

3d

yielding

(41)

was

employed. The were using The

parameters

optimized the

computer

parameters

parameters. the

outer

above

functions

for

Co(NH )$

of

For *Ag

Cr

subject

2

^Tjg to

(2+),

the

is

the

ascertained cautions

the

the

for

SCF

(42). in

all

Co s - f u n c t i o n s

employed

and

scaled

of

Co

level,

original

were

two

were

the

a total

contracted

state

hole

in

the

xy

hole

was

the

on the at

for

results

3

+

2

E

by

2.0

in

SCF energy

of

45

the for

contracted

functions

of

also

the

g

in a

for

the

core,

for were

and

2

z

2

for

43,).

Representation

t geg/^Ti 2

to

g

ls-2p(Co) to

yield the

the eg

provide and

Extension of

some o f

the

state,

within

2

small

octahedral

10~ of

5

au

the

use

of

an

orbital.

An

state.

associated with

found

quite

the

energies

required

the

reflect

e

total

intended the

is

t§g g/^Eg

or

the

departure

calculations the

of

C0N5

in

Accordingly,

2

the

energy

was

2

the

the

protons

3

an o x c u p i e d x - y

calculations.

shell

actual

determinant and

Accordingly,

ls-2p this

yield

Reference

correlation

3s/3p

the

lengths

for

NH

au).

a

exponent

(26).

employed

the 3

x -y

employed

MP2

10""

employed

a 4f

optimized

symmetry,

choices of

single

al.

C o - N bond

symmetry

3 (

section

with

D j

For

calculations

the

to

symmetry.

manifold

g

the

structures

i n d u c e d by

orbital

next

Botch et

octahedral 3

n

the

Co a t o m ,

complexes were

Co(NH )$

though 0

of

(2+)

g

in

the

complexes,

x

even

as

t|

configurations. in

reported

levels


atom f2 very

these

for

find

~

20%

the

the

similarity

is

(positive) angular

calculated

2

2

•+ f )

correlation

MP c a l c u l a t i o n energies

in

3s/3p rather

(MP2

small,

is

I

do

II

were

with

only

differences in

by

the

MP

correspond

So a s

not

MP4).

typically

can

a frozen

diffuse

and

included

they

10-20%).

employed

-

energy

can change

shell

of

energies

Tables

are

(both

with

calculations

correlation

in al.

energies

importance

unimportance

Additional

et

series.

increasing (d

UHF

increase

correlation

relative

that

of

the the

seen to

correlation

the

of

o b s e r v e d by J a n k o w s k i

the

of

associated

increasing

isoelectronic

the

indications

and

magnitude

with

calculated

have

Cr

of

account

the

(26)»

the

all

in

to

employing

In

in

MP-level

a n d d2

configuration),

(typically we

entry

those

diffuse

similar

of

in

the

been

the

eV)

as

to

wavefunction

The

(3.4

found

configurations of

more

UHF

calculations

has

indicate

3d

(similar

the

in

the

viewed

study

f-orbital

UHF c o r e .

be

for

the

correlation

d-electronic these

while

of

in

"tight"

energy.

ion,

decrease

The

and

can

d orbital

d-orbitals

angular

electrons

final

(manifested

and

correlation

energy.

Co3+

again,

charge,

on the

is

in

the

calculations

five

expansion.

electronic

the

of

variational

limit

orbital

diffuse

behavior

II.

appreciable source

the

a

to

the

Cr(d^)

orbital

energy

The

correlation

observed.

frozen

ls-2p

of

seen to

relative)

valence a

is

positive are

3d

previous set.

correlation of

to

correlation

the

comparisons of

strongly

The based

2

a given

studies

Table

present

RHF r e f e r e n c e ,

isoelectronic

Similar

provided

in

basis

the

β spin which

a correlation

associated with

their

the

an o r t h o g o n a l

Furthermore,

(for

Additional

of

as

high-spin

their

of

a tight

the

correlation

energy

charge

and to

a natural -» f

the

a n d RHF r e s u l t s ) . correlation

in

atom

and

for

one

b a s e d on v a r i a t i o n a l

importance

d-electron

of

1 eV o f

calculated

MCSCF

the

has

energy

UHF a n d RHF e n e r g i e s ,

orbital

estimate

and d

least

the

configurations of

Co

demonstrated

accounting

at the

mixture

similar

angular

total

in

in

one so

correlation

corresponds roughly

eV o b t a i n e d an

in

in

utility

manifested

one

yields

sets

the

is

nearly

energy,

studies,

the

consider

radial

elements

place

relative

large of

to

electrons)

correlation basis

and

we

the

However,

previously

for

calculations

50-50

offers

have

latter

state.

calculations,

row

and

a

ground

d-orbitals

type,

study

the

(2£)

which

captures

present

in

other al.

6.5,

Thus

which in

case

spin

α-spin

character. the

of

difference

a

states

in

the

the

various

configuration

configuration

d-orbital

of

of

d^

course,

CI

appreciable

The

nominal

tendency

a diffuse

orbitals

2

electronic

I).

Of

transition

3 d

3d^/^D

(Table

consider

examining

the

diffuse

first

the

species

we

by

is

Botch et

multi-reference

the

it

+

complexes,

correlation

complex.

comparison with

neutral

in

compare 3

Co

d^ c o n f i g u r a t i o n

states.

n + 1

and

1

importance 3d

Co(NH3)5 +/3+

on e l e c t r o n

We f i r s t

Cr^

CoiN^)^" "

low-spin

2

the

charge

interest

allow

to

Ions

to

to

eliminate

ls-2p

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

core

for

27. NEWTON

383

High- and Low-Spin Electronic States

Table

I.

Valence

Shell

6

5

3d / D

Species

Correlation

States

Correlated Wave

Energy

o f C r a n d Co3+

/

(eV)

f o r the

a

SCF

Basis

/Reference

Set

a

(8/5/3/1)

(8/5/3)

2.6

1.9

3.4

2.7

Function

Cr

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

present r e s u l t s /

UHF 2

(=6.5)

UMP3 previous results

m

7

+

l

+

/ R H F

2

RHF+1+2 Co3

present results

+

/

RHF

d

3.7

3.1

/

RHF

6

4.6

3.5

2.0

0.9

2.2

1.1

^ /

UHF 2

(=6.0)

XJMP3

a)

The c o r r e l a t i o n

wavefunction wavefunction. energy

In

relative

constraints

of

calculations. those

b)

ls-3p

section,

The e x p e c t a t i o n compared w i t h

d)

Variational

(26),

using

multi

reference

CI

correlated

this

c o r r e s p o n d s t o t h e UMP3

were

Both

Table

the spin

relaxed

(to within

II).

and

spatial

i n t h e UHF 0.2

eV)

to

T h e MP c a l c u l a t i o n s

core.

results,

the basis

with

values

the exact

singles

a basis

sets

or without

singles

energy

calculation

Sekiya,

the

value

are as d e f i n e d

the 4f

using

operator of

and d o u b l e s

function

i n the (see

also

and w i t h

CI

relative

CI, a 4f

eV ( r e l a t i v e

a very

at

t h e UHF l e v e l

are to

6.0.

to that

and double

of 4.2

private

of

set similar

a MCSFC c a l c u l a t i o n ,

correlation

(M.

the

d and e ) .

be

e)

of

the r e f e r e n c e SCF

are s i m i l a r

(see also

either

c)

from

of

work,

results

a n d UMP4

F o r the present

footnotes

i s the energy

t h e UHF o r RHF e n e r g y .

T h e UMP3

C

RHF

t h e RHF w a v e f u n c t i o n

a frozen

previous

/

to the energy

the present

to

f r o m UMP2

employed

energy

relative

C

large

to

a RHF r e f e r e n c e

used

i n the present

using

configurations

virtual,

yielded

work.

a

t o t h e RHF e n e r g y ) .

atomic

orbital

basis

set

communication).

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

A

obtained

384 Co

THE CHALLENGE OF d AND f ELECTRONS in

the

orbitals the

Co(NH3>6 were

Ν atom

Is

the

excluded

orbitals,

From t h i s conclude

calculations.

also

that

brief for

set

model.

of

This

zero-valent reduced the

4f is

MP2

to

be

s-level

multi-configuration

ls-2p

of

with

a

provides with

transition

systems

nucleus

increase

the

importance

diffuse

state

did

not

and

the of

and more

appear

Co d - o r b i t a l

(40)

with

energies,

interest basis,

cases

formal

diffuse be

in

the

a

structure

of

(26-32)

where

t

the

occupation

including

to

we

here,

and

electronic

some n o t a b l e

the

it

of

d-level

suitable

metal

along

correlation

complexes

from

reference

unoccupied

space.

split a

five

orbitals),

atomic

charged

contrasted

Even though

additional

level,

state

highest

MP a c t i v e

positively

Coulombic f i e l d

orbitals.

the

primitives,

ground

valence

the

from

consideration

the

single-reference

single

The

("virtual"

d

of

a

and

crucial,

f

we

included

Co(NH )$ 3

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

calculations. Properties As

of

a partial

compare

in

for

the

the

3+

the

bond

Co(NH^)^ measure

Table

*Aig and

of

III

of

equilibrium

bond

gravity,

as

noted

expected

(18-25)

the in

has

been

lengths

noted for

agreement In

the

the

oxidized

and give

experimental

data.

geometry

length

rt

for

the

*Ai

arithmetic

constant

species

(.14),

equilibrium thus

activation The

is and

total

symmetric (2+)

a value

~

shift

of

0.10

(14.15).

the

features

(the Â),

CoN bond

the

one

of

bond

lengths

of

find

that

the

(0.22 the

both

simply

Â)

about

of

2.15

twice  for

as

by

that

CoN bond

reaction (.16).

constant, be

that

(SCF)

value

Since for

and ~

16

energy

Thus

from

aside give

potential

the

the

the value

(MP2).

estimated uniform

encouraging

energy

surface

2+

calculated

kcal/mole is

the

the

r t , b a s e d on of

If

then

the

corresponding experimental the

the

transition

species

force

activation

calculations

of

reactions,

Co-N distances.

is

with

kcal/mole

inner-shell

kcal/mole Â,

2.05

of

better

in

of

it

bond

parents.

reduced

breathing

equilibrium

lengths 15

energy

Perhaps states,

significantly

defined

system would

estimate

17

in

g

the

each

major

oxidation

quantity

reaction

As the

 for

of

metal-ligand

component

oxidized

species

energy

this

( 1_1 ) . than

(0.10

neutral

a n d we

exchange

3+

~

in the

both

the

about

variations

for

"inner-shell"

al.)

(the

of

complexes.

difference

(.16),

for

be

metal

are

center

cases

positive

for

the

species

the

to

obtained

complex

longer

uniform

charge-transfer

is

two

to

is

are

calculated

ions

the

we

for

those

of

experimental

important

that

than

same

SCF bond

estimated

compared

activated

(3+)/^Ti

g

mean

bias

results

(2V)

which minimizes the

this

same v a l u e

the

et

when

positive

The

for

s p e c i e s had

force

of

the

Larsson

for

have

(2+)

this

some t r a n s i t i o n

quantities

( r t ) , common t o

partners, each

for

reduced

calculations state

modest

E

about

lengths

experiment

important

2

we

we

lengths

states

addition,

of

present

context

most

C o - N bond

electronic

In

bond

metallocene

the

ground

work

previously

with

SCF c a l c u l a t i o n s ,

experimental

disproportionate

However,

observed

the

SCF L e v e l

earlier

relatively the

the

the

calculated

values.

with

of

and

respectively.

the

the

species)

consistent

quality

at

octahedrally-constrained

experimental exaggeration

Complexes

species,

lengths

t

3 +

the

complexes,

length

and

/

calculated

and ^ T j g

2+

2 +

to

be

horizontal

account

of

pertaining

length.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

to

is

27. NEWTON

Table

II.

Valence

Ion

Shell Correlation

Level

Energy

a

2 +

7

3 +

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

a)

2

+

and

Co

3

+

Ions

Basis (8/5/3)

4

MP 2

3.24

3.19

1.86

MP 3

3.26

3.22

1.84

6

b

3.26-3.35

5

(3d / D) MP 2

1.91

1.90

0.83

MP 3

2.01

2.00

0.87

MP4

b

2.01-2.04

B a s e d on UHF o r b i t a l s .

the

Co

(3d / F)

MP4 Co

for

(8/5/2/1)

(8/5/3/1) Co

385

High-and Low-Spin Electronic States

fourth

significant

UHF s p i n

figure

contamination 2

the

calculations

employed

b)

corresponds to presence

The range

frozen

of

ls-3p

values.

shows up o n l y

in

T h e MP

UHF c o r e . or

absence

of

single

and

triple

excitations.

Table

III.

Equilibrium

C o - N Bond L e n g t h s

(Â) Results

Species

Present

Experiment

Results*

N e 4

T

l

2

E

g

< g

3 +

>

(2+) (2+)

ab 1.97

2.29

2.198

2.23

[2.13]

a)

Based on an o c t a h e d r a l RHF o r U H F .

c)

Taken

d)

Calculated

e)

Reference

f)

References

14 a n d 4 5 .

g)

References

14 a n d 4 4 .

h)

Estimate

all

three

f

2.07

crystal minus

Larsson

d

5

b)

from

Ar

0

C0N5

a l .

e

INDO

2.03

2.02

0.10

2.27

2.14

2.20

2.09

h

framework.

and aqueous

experimental

of

0.10 [0.10]

h

initio

et

solution

diffraction

data.

value.

11.

based

on the

assumption

that

the

same

Ar value

applies

species.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

to

386

THE CHALLENGE OF d AND f ELECTRONS

Energy The

Splittings

salient IV.

state

separations.

The

charge

t|g

pair

(MP2)

state

is that

the of

it

which

calculated

known v e r t i c a l

for

brings

T

•* ^ l g

T

E

"* ^ g

amount.

excitation

A similar

calculations

the

raw has

been

energy

Â,

and

2.15

to

the

length

is

state,

differing

 and

2.25

Â,

that

the

length.

In

fact,

expected

to

be

due

to

of

state

the

force

linear

factor,

section,

splittings

interest. states (ΔΕ the

As

slightly 2 +

4

for

~ 4200

All

MP2

T i

2

g

/ E g

splitting 4f

by

virtual

splitting The are

initial

by

the lg/^

final

lg

spin

spin

is

state,

electrons. useful the

detail of

estimates

electronic

in

the

next

greatest was

estimated

find

the

two

rt to

be v e r y

close

in

larger

rt, although

still

to

spin

energy

state

appreciably

IV

990

include and

= 2.15

the

splitting

at

(by well

au,

Co 3 s / 3 p

of

the

reduces

Â.

below

the

of

the the

^Aig/^Tjg 4

the

T^g 4

on t h e

reduction

in

increases

excitations

both

effect

a net

shell

UHF c o r e

the

While

magnitude

(e.g.,

the

as p a r t

1

cm"

0

2-7),

by

the

exchange

aside

above

from

non-interacting.

we in

discussed

which,

states note the

that

the

If

of

to

and

the 2

Eg

2

Tjg/ Eg

only

initial

While

(Equation

H^f 3)

for is

rt are

bond

we

440

cnT*

exchange

energies

at

final

matrix the

on

constraint

state

full

reaction

of

transition-state and

based

length

now c o n s i d e r t h e

the

relative

bimolecular in

for

the

associated with

electron-transfer

introduction. T

higher

state

shell

rc jj

splittings

by d i f f e r e n c e s

A

of

Â).

affected

l

spin bond

(17).

by ~ 0.06

states

in

this

at

low-spin

represented

E

procedure

each

lower

and

C o - N bond

r t ) , we

at

Table

much s m a l l e r

otherwise

(Equations

- 1

for

are

Â

at

is

cm"^

increase

reactants

and

in

splitting

state

individual

found

Freezing cm

at

2

the

2.15

equilibrium

corresponding calculated

9000

results

510

(where

lengths

for with

the

the

2+

linear

state

are

and

This

(eg)

1

cm"" )

same

energy

for

more

Â)

,

C o - N bond

estimating

in

2.15

state

geometry

that

energies

is

= 2.25

(rt),

The

of

space

correlation rCo-N

than

space.

11 g

complexes

1

estimate

active 4

2+

Tj^

with

r t (~

Reference

cm" ).

cm"*)

at 4

the

that

with

(13,700

11.

and

obtained

the

anti-bonding

dealt

e

the

3+

constant)

for

than

1

the

expression should provide

K ^,

equilibrium

earlier

the

the

= 2200

Tj[g

in

below

constant

-

cm

Reference

of

of

For purposes of

transmission

lie

variation

—

vertical

6000

of

term

at

coincidence

complex

the

results

magnitude

of

in

for

a

calculated

separation

the

respectively.

smaller

population

the

separations. the

the

MP2

which

the

into 3+

for

a range

of

a common f o r c e

in

somewhat

a higher

Nevertheless,

(with

only

Â)

the

adopted

in

same c o n s t a n t

energy

estimating

over

extent

in

been d e c r e a s e d by

p r o c e d u r e was

harmonic

T^g

i n c r e a s e d by

has

Level

corrected

correct

by the

splitting

interpolation

exact

to

(2.29

expressions for

linear

3

MP2

displayed the

eg p r o c e s s

2

for

MP2

the

high-spin transition

length

splittings

is

a tg

are are

to

energy

provided 2.05

interest

* A ^ g •+

high-spin/low-spin by

Complexes at

each process

transition

excitation

Convenient

3 +

low-spin

*A^g C o - N bond

A c c o r d i n g l y , the

^ l g

^

seems r e a s o n a b l e

energy

the

^ lg

MP2

primary

Since the

broken,

2 +

Co(NH3)^

corresponds to

the (35).

of

results

transition

namely,

the

results

Table each

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

of

high-spin

c o m p l e x may coupling,

element,

H^f,

and be

as

discussed

"three-electron"

expected

to

be

quite

small,

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

as

27. NEWTON

Table

IV.

High-Spin/Low-Spin Energy for

Δ Ε

Co(NH 3+

ΔΕ A)

Results

at

Ξ

2 +

E

3

(

) 3

2

+

/

3

+

l

)

g

-

2

Ξ

E( E )

l

E( A

(2.15 Â)

=

E( T

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

2 +

(2.15 Â)

C)

Dependence

A

E

Results

i

3

2

ΔΕ

shift

2 +

=

on C o - N Bond

+

+

Length

r

(

r

at

< C0N>

"

CoN>

"

3 +

(2.07 Â)

=

ΔΕ

2 +

(2.29 Â)

=

is shifted

b) X

The t o t a l A

l

g

value

separation CoN bond

(uncorrected

2.2

(corrected

8.1

4

-0.079181,

-1.840157

(

c)

interpolation

Linear

2.05 d)

 (3+)

Based

e)

T

l

g

(2+));

or 2.25 Â

on c a l c u l a t e d

splittings

have

Reference

3+

MP2 r e s u l t ) < »

D

j

MP2

(uncorrected

a

result)

MP2

3

result)^ J

0

1

0

9

6

-

3

· 1

"

4

6+

·

6

2

9

,

·

2

"CON

7

r

CoN

13.7 (present 6.4

) upward

(present

been

results)

(IND0)

amount

cm" ) at

MP2 ( l s - 3 p

-1.241193

(^T

results

1

cm"" )

the calculated

(2.07 Â ) , as d e s c r i b e d

and - 0 . 6 4 4 1 7 9 ,

e

(5900

core)

to -1717 a u ) . -0.068346,

of

results)

needed

so as t o match the

1

(13,700

length

(relative

(3+));

result)

(SCF r e s u l t )

b y t h e same

(ΔΕ

SCF and u n c o r r e c t e d

respectively (

downward

the ^ T i g - ^ ^ g

experimental

MP2

Equilibrium^

ΔΕ

equilibrium

1

cm"" )

)

(SCF r e s u l t )

3.3-4.7 a)

g

2.4

^13.8

I

l

10.0 (corrected

r

A

3

)

i g

4

-

g

r 4.0

ΔΕ

(10

Complexes

6

T

Separations

rt

3 +

ΔΕ

B)

387

High-and Low-Spin Electronic States

l g

(3+));

text.

energies are,

-1.259428 -0.707071,

-1.803265

based

^ i g

i n the

(

2

E

g

(2+)).

on 2.15 Â and

either

(2+). (SCF) e q u i l i b r i u m

corrected

r£ jj 0

(see footnote

values. a and t h e

T h e MP2 text).

11.

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

to

388

THE CHALLENGE OF d AND f ELECTRONS

noted

above,

(Equation

4)

orientation (16)

that

crossing l

H

ifl

the

coupling

may

be

of

the

reactants

the

diabatic

the

2-state

is

avoided).

reactants

to

attain

octahedral provided

an

contact of

detailed

suppression expected energy ~

of

of

the

1

2

is

less

Â),

at

in

the

evaluate

at

K p, e

Electronic cases

of

expression Equation

^Ajg/^Tig

electron 2-state

for

the

E^ i s

effective state in v

409

n

«

1

E^ in

(the

the

e

the

for

are -

each

relevant

clear

require

that

as w e l l

as

an

below

that

energetics

/ T i g couple.

g

required 16

to

kcal/mole the

the Further

ascertain

appropriate

in

of

4

*Ai

the

of

state

the

on t h e

of

rt

(see

next

if

the

value) previous

section

to

state.

( κ ^ ) between

relatively leads

weakly

coupled

to

the

following

transmission factor,

K p,

(as

to

( ^ / E

f

-

E^

n

better

+ E

o

2

+

u

t

3

/

to

1

2

/

energy

/ h v

e

in

we m u s t

the

-

-

97.7

a

by

7.

e

we

effect 87.6

To complete H^f

or

final

is

implicit less

take

(for

small

replaced

evaluate

K ^

reaction

kcal/mol

include

the

factors

10% w h e n

exchange

by E q u a t i o n

is

n

initial

exponential

than

(6.15).

(8)

n

and v

+

s h o u l d be

E q u a t i o n 8)

and c *

R T )

K

associated with

E^ ~ 4Et;

kcal/mol

c

i

expansion of

represented

express

2

2 H

Co(NH3)é

room t e m p e r a t u r e w

is

may p o s s i b l y p l a c e

that

reorganization

accurate

a n d E\

97.7

schematically (iZ)»

is

*

frequency

reactions,

evaluating at

H^f

weight

the

|H^f|,

Landau-Zener model

efc

total

For

cm"

tunnelling,

e

8

0.2).

exchange

K g,

the

harmonic

Equation -

have

other

(II),

transition

electronic

system (15.16)

than

to

1

1): K

where

due

cm"

two

their

of

nevertheless

proceed

transfer

the

1000

complex would

(based

Transmission Factor

reactants,

rms v a l u e

encounter

is

underway)

we

-

the

fact

4

for

by

the

the

of

recall

the

reduced

of

somewhat

g

axis

assuming equal

favors

meantime,

the

the

4

energy

of

which

magnitudes

energy

A x / T g value

the

reduction in

be

(i.e.,

determination it

the

will

solved

*Aig/ Tig transition of

reactants

1

the

In

H^f

barrier

couple

(currently

than

in

relaxation

spite

activation

section).

For

2

A^g/ Eg

^Aig/^Eg

calculations A^g/ Eg

accurate

We

s e m i e m p i r i c a l methods

(2),

simulations,

the

couple

2.15

value

orientations

"non-interacting" 1

an

Jahn-Teller

^ig/^Tig (r*

of

smaller

the

(6.7.11).

states

is

a common 4 - f o l d

suggesting that

While

computer

final

Various

the

(7.11).

distribution

and

process

on

associated with

configuration

along

(2).

apex-to-apex

orientation.

state

energy

equation

*Aig/ Eg

depending

T h e maximum b a r r i e r

frameworks

the

activation

initial

orientations of

cm"" ,

transition

apex-to-apex

estimates

1

1000

the

2

"1-electron"

-

secular

-

25%

the

as

in

zeroth-order

of

corresponds

for

high

when

crossing

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

as

for

the the

Following

from

kcal/mol

when

evaluation

of

process Buhks et

al.

as,

c

-

- ( J f

C

-

-(J

4

T

H

l

**3τ l

H l

g

g

Q

*

* SO

2

l

)/AE

E

2 +

)/AE

A

A

l

(rt)

3 +

(rt)

g

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

(9)

(10)

27.

NEWTON In

mixes 4

Tig

the

the

0^

which mix

perturbation to

coupling

usual

E

, and

g

both

one

and

these

for

two

the

rate

two

initial and

2

where

+

3d

and

one

model

the

/ 2

o

The

states

Γ3

two

distinct

2

and 4 ξ

+

/ 3

effective

with

spin-orbit

(17). of

must

for

the

activated

state)

s

first-order

available

states

for

final

+

ξ

is

+

are

final

2

at

2

orbital

levels

of

H

*A|g.

9 yields

level

lie ξ

with

g

degenerate

the

states

Γ3

constant

initial

2

coupling via ^Tj

Equation

At

c.

Co

of

four-fold

state,

low-lying the

spin orbit

component

accordingly,

c^

Γ$ ground

integral in

(46),

different

theory,

2-state

(i.e.,

two

denoted

the

Because species

2

with

group

symmetric

yields

coefficients, respect

double

totally

state

389

High- and Low-Spin Electronic States

the

electron

be

Co(NH3)^

reaction,

2 +

the

transfer

generalized

as

follows : 2

Publication Date: June 8, 1989 | doi: 10.1021/bk-1989-0394.ch027

, act

Σ

et

Y

(11)

k

J ik

j,k-l where

the

normalized

Boltzmann

factor

yj

is

given

by

3 y

-

g.

exp

-

(βΕ

)/(

J

J

In

Equation

Epir^) -

12,

doubly-degenerate state

Γ3 the

(through

degeneracy

first

with

Equation

the

11

four

v ,

and E t

T,

n

n

"paths"

or

in

and

of

e

«

i

e f i f

We h a v e the

Co(NH3)5 with

Υ,

j - i

J

a

-

7

= E^/4 in

ν

ell

are

l

c'

AE

3 +

employed

(rt)

are

1

1/5 /

2

(

For using

C J L

),

1,

we

second

gj

are

. find

it

convenient

to

(13)

r

to

be

common t o

and where

all

(using

2

ΥΛΗ|ϊ) ](π3/Ι J

j k-i transfer

assuming and

l

f

Equation

K

RT)

1

/

2

/hv

x

8),

(14) n

in

ξ

2

+

»

Reference

H^f

(where

and

final

initial

an e f f e c t i v e (2+)

1_7).

2

simplicity,

a one-electron model

numerators (c ),

and

2

we

of

3d

ξ

3

+

the -

cm"

1

(c») the

in

wavefunctions

AE

1

9 and

10 of

many-electron

2 +

the h,

c^,

c, 2

same

(rt)

and 1

cm" ,

have

magnitudes

ξ. wavefunctions

on e a c h Co

contain

of

appropriate

(the

10,000

units

nominally

by

cm""

for

spin-orbitals

denote

coefficients

600 and

Equations 2

construct

basis

,

(2200

in 1

i

The v a l u e s

IV 6 '

-

j,k

states

Hamiltonian,

represented

and w i t h

cm

the Γ3

1-electron

states

515

from Table

The 1

Γ3

integrals, of

wavefunctions,

using

3/5 '

Although these

1

the

The

βχρ(-βΙ*)

n

11,

Σ

combinations

taken

respectively).

2[

j

(3+)

rj

evaluated

values

«

the

multi-determinantal and

Γ

η

taken

Equation

κ

complex)

relevant

coupling).

f

possible 2 +

the with

2

Σ

evaluated

four

since

^ β Τ ) "

Equation

2

κ

β Ξ

(12)

degenerate

spin-orbit

4),

) K

follows:

^ et

involved

accidentally

form

as

k

where

(2

βχρ(-βΕ

K

E$(Tj) « £ 2 ^ 3 )

and is

order

factors

Consistent reexpress

0,

state

Γ7

g

k«0

only

site. the

In The Challenge of d and f Electrons; Salahub, D., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

390

THE CHALLENGE OF d AND f ELECTRONS

thirteen 6

d /d

7

electrons

final

states),

the

influence

the

various

also

in

the

L7,

To

along

z-axis,

matrix

element

n3d

z

23d 2»

*

"pathways"

as

between t

l

i

e

s

a

m

all

other

[. j Equation

14

which

an e

in

This spin-orbit

3d

by

with

calculated

for

limit,

to The

this

two

axis

Co

possible

matrix

achieve be

atoms,

1-electron

of

1000

cm"

(6,7),

and

elements

We f i n a l l y

to

Hamiltonian

calculations

elements.

the

especially

4-fold

on the

four

off-diagonal

of

obtain

h,

1

to

as

for

well

the

14,

VjKÏ) ]

ev, ft

= (0.030 h

is

-

1.0

which

at

least

recent

3 d z 2 3 d 2 2

2

to

the

in

)

2

(15)

degree

K ^ e

mean v a l u e

of of

an o r d e r