Supercomputer Research in Chemistry and Chemical Engineering 9780841214309, 9780841211988, 0-8412-1430-1

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ACS

SYMPOSIUM

SERIES

353

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.fw001

Supercomputer Research in Chemistry and Chemical Engineering Klavs F. Jensen, EDITOR University of Minnesota

Donald G . Truhlar, EDITOR University of Minnesota

Developed from a symposium sponsored by the Division of Industrial and Engineering Chemistry of the American Chemical Society, the Minnesota Supercomputer Institute, Cray Research, Inc., and ETA Systems, Inc., at the Industrial and Engineering Chemistry Winter Symposium, Minneapolis, Minnesota March 16-17, 1987

American Chemical Society, Washington, DC 1987

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Library of Congress Cataloging-in-Publication Data Supercomputer research in chemistry and chemical engineering. (ACS symposium series, ISSN 0097-6156; 353) "Developed from a symposium sponsored by the Division of Industrial and Engineering Chemistry of the American Chemical Society, the Minnesota Supercomputer Institute, Cray Research, Inc., and ETA Systems, Inc., at the Industrial and Engineering Chemistry Winter Symposium, Minneapolis, Minnesota, March 16-17, 1987."

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.fw001

Bibliography: p. Includes index. 1. Chemistry—Data processing—Congresses. 2. Supercomputers—Congresses. I. Jensen, KlavsF.,1952. II. Truhlar, Donald G., 1944. III. American Chemical Society. Division of Industrial and Engineering Chemistry. IV. American Chemical Society. Division of Industrial and Engineering Chemistry. Winter Symposium (1987: Minneapolis, Minn.) V. Series. QD39.3.E46S927 1987 ISBN 0-8412-1430-1

542'.8

87-24127

Copyright © 1987 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 that permitted by Sections 107 or 108 of 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 the first page 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 any right or 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 unprotected by law. PRINTED IN THE UNITED STATES OF AMERICA

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

ACS Symposium Series M. Joan Comstock, Series Editor

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.fw001

1987 Advisory Board Harvey W. Blanch University of California—Berkeley

Vincent D. McGinniss Battelle Columbus Laboratories

Alan Elzerman Clemson University

W. H . Norton J. T. Baker Chemical Company

John W. Finley Nabisco Brands, Inc.

James C . Randall Exxon Chemical Company

Marye Anne Fox The University of Texas—Austin

E . Reichmanis AT&T Bell Laboratories

Martin L . Gorbaty Exxon Research and Engineering Co.

C. M . Roland U.S. Naval Research Laboratory

Roland F. Hirsch U.S. Department of Energy

W. D. Shults Oak Ridge National Laboratory

G. Wayne Ivie USDA, Agricultural Research Service

Geoffrey K. Smith Rohm & Haas Co.

Rudolph J. Marcus Consultant, Computers & Chemistry Research

Douglas B. Walters National Institute of Environmental Health

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Foreword

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.fw001

T h e A C S S Y M P O S I U M S E R I E S was founded in 1974 to p r o v i d e a

m e d i u m for p u b l i s h i n g s y m p o s i a q u i c k l y in book form. T h e format o f the Series parallels that o f the c o n t i n u i n g A D V A N C E S IN C H E M I S T R Y SERIES 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 o f the E d i t o r s w i t h the assistance o f the Series A d v i s o r y B o a r d and are selected to m a i n t a i n the integrity o f the s y m p o s i a ; h o w e v e r , v e r b a t i m reproductions o f p r e v i o u s l y published papers are not accepted. B o t h reviews and reports o f research are acceptable, because s y m p o s i a m a y e m b r a c e both types o f presentation.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Preface THE

1987 W I N T E R S Y M P O S I U M o f the D i v i s i o n o f Industrial a n d

Engi-

neering C h e m i s t r y , hosted by the M i n n e s o t a Supercomputer Institute, was Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.pr001

supported by grants from the A m e r i c a n C h e m i c a l Society, the M i n n e s o t a S u p e r c o m p u t e r Institute, C r a y R e s e a r c h , Inc., and E T A Systems, Inc. T h e s y m p o s i u m consisted o f four half-day sessions w i t h four lectures per session and a t w o - p a r t poster session w i t h 15 poster papers. T h i s b o o k includes chapters by the lecturers plus five papers contributed by the session chairs. A l l contributions were refereed a n o n y m o u s l y a c c o r d i n g to usual procedures of the A C S S y m p o s i u m Series. T h i s v o l u m e appears w i t h a short time l a g because o f the cooperation of the participants. Fifteen o f the lecturers and both o u t - o f - t o w n session chairs brought their manuscripts to the s y m p o s i u m . We thank R o b i n G i r o u x of the A C S B o o k s Department for her c o o r d i n a t i o n o f the publisher's tasks. KLAVS F. JENSEN

Minnesota Supercomputer Institute and Department of Chemical Engineering and Materials Science University of Minnesota Minneapolis, MN 55455 DONALD G . TRUHLAR

Minnesota Supercomputer Institute and Department of Chemistry University of Minnesota Minneapolis, MN 55455 April 1, 1987

vii

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 1

Supercomputer Research in Chemistry and Chemical Engineering An Introduction 1

2

Klavs F. Jensen and Donald G. Truhlar 1

Minnesota Supercomputer Institute and Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 Minnesota Supercomputer Institute and Department of Chemistry, University of Minnesota, Minneapolis, MN 55455

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

2

This chapter gives a selected overview of the current status of supercomputing research in chemistry and chemical engineering and places the research areas discussed In the rest of the book in the context of current work. T h e first A C S S y m p o s i u m v o l u m e o n s u p e r c o m p u t l n g was p u b l i s h e d In 1981 (1.). T h a t s y m p o s i u m already I n c l u d e d mature applications o f the C r a y - 1 m a c h i n e , u s u a l l y t h o u g h t o f as the first s u p e r c o m p u t e r . A m o r e recent s u p e r c o m p u t e r s y m p o s i u m , the S e c o n d I n t e r n a t i o n a l C o n f e r e n c e o n V e c t o r P r o c e s s o r s In C o m p u t a t i o n a l Science, h e l d at O x f o r d In A u g u s t , 1984, Is p u b l i s h e d as V o l u m e 37 In C o m p u t e r P h y s i c s C o m m u n i c a t i o n s . I n t h e i r I n t r o d u c t i o n the editors o f the latter proceedings c o m m e n t e d that the m o s t notable change In the field o v e r the past few years h a d n o t been new hardware b u t rather the s o p h i s t i c a t i o n o f the hardware and the m a t u rity o f the user c o m m u n i t y . T o a large e x t e n t that o b s e r v a t i o n h o l d s for the 1984-87 p e r i o d as w e l l , especially regarding the use o f s u p e r c o m p u t e r resources In c h e m i s t r y and c h e m i c a l e n g i n e e r i n g . M u l t i p l e processors have been w i d e l y discussed as a t o o l f o r a c h i e v i n g h i g h c o m p u t e speeds, b u t m o s t o f the progress o n a c h i e v i n g h i g h e r c o m p u t a t i o n speeds f o r applications has been a c h i e v e d b y better u t i l i z a t i o n of the v e c t o r l z a t l o n strategies that were already possible e v e n In 1981. The hardware available at the t i m e o f this w r i t i n g Is a b o u t a factor o f two-tof o u r faster, o n a per processor basis, t h a n that available o n the o r i g i n a l C r a y - 1 , a n d the fastest s u p e r c o m p u t e r s have at m o s t f o u r processors. Since m o s t calculations are still r u n In the single-processor m o d e t h o u g h , we have a c h i e v e d far less t h a n an o r d e r o f m a g n i t u d e In effective top speed In five years. N e v e r t h e l e s s m a n y application fields have a d v a n c e d to systems m u c h m o r e than five times as d e m a n d i n g , n o t o n l y because of the Increases In user a n d software s o p h i s t i c a t i o n m e n t i o n e d above, but also because o f a l g o r i t h m i c advances a n d the availability o f a greater n u m b e r o f s u p e r c o m p u t e r s , a l l o w i n g m o r e processing h o u r s to be d e v o t e d 0097-6156/87/0353-0001 $06.00/0 © 1987 American Chemical Society

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

2

SUPERCOMPUTER

RESEARCH

to specific applications a n d a l l o w i n g u t i l i z a t i o n o f this k i n d o f resource at a greater n u m b e r o f sites. It Is u n f o r t u n a t e l y true t h o u g h that this k i n d o f research Is still s e v e r e l y u n d e r f u n d e d , a n d this accounts f o r the fact that Its p o t e n t i a l Is s t i l l largely u n t a p p e d . A l t h o u g h the Increase In speed o f s u p e r c o m p u t e r s has n o t b e e n d r a m a t i c In the last five years, there has b e e n a v e r y notable hardware advance In the size o f c o m p u t e r m e m o r i e s . W h e r e a s the 1 - M e g a w o r d m e m o r y o f the C r a y - 1 was once c o n s i d e r e d generous, the o p p o r t u n i t i e s created b y the C o n t r o l D a t a C o r p o r a t i o n C y b e r 205, w i t h 8 M e g a w o r d s o f real m e m o r y a n d a huge v i r t u a l address space, and the C r a y - 2 , w i t h 256 M e g a w o r d s o f real m e m o r y , are m u c h greater a n d these o p p o r t u n i t i e s are o n l y v e r y r e c e n t l y b e i n g appreciated a n d e x p l o i t e d . A n o t h e r advance that Is r a p i d l y b e c o m i n g m o r e Important Is n e t w o r k i n g . A s users get accust o m e d to t a k i n g advantage o f the best available h a r d w a r e , software, a n d databases In terms o f w h a t can be reached easily b y n e t w o r k , e v e n If It Is across the c o n t i n e n t o r the o c e a n , s u p e r c o m p u t e r capabilities m a y be e x p e c t e d to have v e r y significant effects o n larger a n d larger areas o f chemistry and engineering. I n k e e p i n g w i t h the above r e m a r k s o n the relative Importance o v e r the last few years o f hardware advances v e r s u s u t i l i z a t i o n a n d applicationm o d e advances, the present v o l u m e centers o n the latter. T h e chapters are arranged In t e r m s o f f o u r u n d e r l y i n g scientific subflelds. F i r s t c o m e s the s t u d y o f e l e c t r o n i c structure as based o n the S c h r o e d l n g e r e q u a t i o n . N e x t c o m e s the s t u d y o f e q u i l i b r i u m systems, based o n t h e r m o d y n a m i c s a n d e q u i l i b r i u m statistical m e c h a n i c s , I n c l u d i n g also s o m e aspects o f quant u m m e c h a n i c s , especially w i t h regard to q u a n t i z e d degrees o f f r e e d o m a n d spectra. T h e t h i r d application area consists o f the s t u d y o f k i n e t i c s a n d d y n a m i c s , either classical o r quantal, b u t at the m i c r o s c o p i c l e v e l w h e r e c o m p l i c a t i o n s like flow do n o t c o m e In. Q u a s l e q u l l l b r l u m theories o f rate processes, s u c h as t r a n s i t i o n state t h e o r y , t e c h n i c a l l y b e l o n g to d y n a m i c s , b u t m a y also be c o n s i d e r e d w i t h e q u i l i b r i u m properties, since the t e c h n i q u e s are s i m i l a r . T h e f o u r t h a n d final set o f chapters deals w i t h c o m p l e x i t i e s o f flow a n d transport. I n general the applications In the later sections m u s t Include at least s o m e o f those I n v o l v e d In the earlier ones; t h u s In s o m e sense the o r g a n i z a t i o n a l principle Is one o f Increasing c o m p l e x i t y . F o r e x a m p l e , the s t u d y o f d y n a m i c s always presupposes s o m e k n o w l e d g e o f Interaction energies s u c h as l n t e r m o l e c u l a r forces a n d b i n d i n g energies, a n d s u c h k n o w l e d g e m a y u s u a l l y be traced back to q u a n t u m m e c h a n i c s , spectra, o r properties o f e q u i l i b r i u m s y s t e m s . S i m i l a r l y m a n y p r o b l e m s I n v o l v i n g transport also I n v o l v e c h e m i c a l reactions a n d at a d e t a i l e d l e v e l a l l transport Is Intimately related to the m i c r o s c o p i c a t o m i c a n d m o l e c u l a r d y n a m i c s . T h i s k i n d o f c o n n e c t i o n Is discussed at greater l e n g t h b y C l e m e n t i a n d L i e ( C h a p t e r 1 4 ) , a n d It c o m e s m o r e a n d m o r e Into play In the large-scale s i m u l a t i o n s that are made possible b y superc o m p u t e r s . T h i s creates s o m e clear p r o b l e m s In assigning a few o f the chapters to one o f the f o u r sections o f the b o o k . N e v e r t h e l e s s we have

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

1.

JENSEN AND TRUHLAR

Ζ

Introduction

made choices, a n d In general a chapter Is assigned to the farthest d o w n o f the sections to w h i c h It Is d i r e c t l y relevant. A s large-scale c o m p u t a t i o n s make t h e i r presence felt In m o r e a n d m o r e areas o f c h e m i s t r y a n d c h e m i c a l e n g i n e e r i n g , a n d as the c o n t e x t o f these applications develops a significant h i s t o r y , It b e c o m e s Impossible to Include a c o m p r e h e n s i v e d i s c u s s i o n In a n y o n e v o l u m e . T h u s the present set o f chapters Is m o r e Illustrative t h a n c o m p l e t e . A n d a l t h o u g h m a n y o f the chapters c o n t a i n reasonably complete I n t r o d u c t o r y r e m a r k s , the b o o k c a n n o t serve t o p u t w h o l e subflelds Into f o c u s . T h i s Is all s y m p t o m a t i c o f a proceedings In a r a p i d l y d e v e l o p i n g field. I n the rest o f this I n t r o d u c t i o n we t r y to c o m p l e m e n t the proceedings papers b y discussing s o m e o f the w o r k In the b o o k a n d a s m a l l subset o f o t h e r Interesting a n d especially recent w o r k In the field, w i t h the goal o f slightly b r o a d e n i n g the c o n t e x t In w h i c h the proceedings and the c u r r e n t status o f s u p e r c o m p u t e r research In these fields Is v i e w e d . T h e topics w i l l be discussed In terms o f r o u g h l y the same o r d e r i n g scheme as u s e d for arranging the topics In the b o o k . E l e c t r o n i c Structure I n m a n y respects materials science Is the b r a n c h o f c h e m i s t r y that has the m o s t o b v i o u s o p p o r t u n i t i e s to gain f r o m the s u p e r c o m p u t e r r e v o l u t i o n . E x p e r i m e n t a l , e m p i r i c a l k n o w l e d g e o f materials Is difficult to organize w i t h o u t theoretical u n d e r s t a n d i n g . F o r m a n y m a t e r i a l properties s u c h u n d e r s t a n d i n g Is In principle afforded b y the analysis o f accurate electronic w a v e f u n c t l o n s a n d energies a n d In favorable cases u n k n o w n materials properties c a n e v e n be p r e d i c t e d . B u t the calculations require e n o r m o u s c o m p u t a t i o n s e v e n w h e n t h e y are feasible, hence s u p e r c o m p u t e r s are r e q u i r e d . A n e x a m p l e In the present b o o k Is afforded b y calculations o n m e t a l - c o n t a i n i n g c o m p o u n d s ( C h a p t e r 2). R e c e n t advances In m e t h o d o l ­ o g y have I m p r o v e d the r e l i a b i l i t y o f α6 initio calculations o n s u c h systems. N e w calculations give Insight Into the nature o f the b o n d i n g a n d define the t w o - a n d t h r e e - b o d y parameters u s e d f o r m o d e l l i n g larger clusters. C o m p a r i s o n o f ab initio a n d m o d e l results f o r A l clusters a n d C u B e sug­ gests that composite materials can also be m o d e l l e d based o n parameteri­ z a t i o n f r o m the ab initio calculations. T h e theoretical calculations are especially useful f o r o b t a i n i n g m o d e l Interactions parameters f o r Interac­ t i o n s o f u n l i k e a t o m s . These are m u c h harder t h a n l i k e - a t o m Interactions to o b t a i n f r o m e x p e r i m e n t a l I n f o r m a t i o n o n b u l k properties, In particular because t h e y d o n ' t e v e n o c c u r In s i n g l e - c o m p o n e n t metals. T h e theoreti­ cal studies Indicate that large clusters are r e q u i r e d before the b u l k struc­ ture b e c o m e s the m o s t stable. T h e s t u d y o f adsorbates o n B e shows that the c h e m i s t r y o f clusters c a n be v e r y different f r o m the b u l k as a r e s u l t o f the cluster's Increased f r e e d o m to e x p a n d a n d distort. T h i s leads to Interesting Insight Into the differences between b u l k m e t a l l i c systems a n d s m a l l m e t a l l i c clusters, w h i c h m a y be v e r y active In catalysis b u t w h i c h are h a r d to characterize e x p e r i m e n t a l l y . T h e n e e d f o r p o w e r f u l s u p e r c o m p u t e r resources b e c o m e s v e r y e v i d e n t w h e n we c o n s i d e r the 1 8

1 3

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

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e v e n t u a l necessity to e x t e n d the theoretical studies o f m u l t l c o m p o n e n t systems a n d clusters to Include v i b r a t i o n a l effects a n d n o n z e r o temperatures. A n o t h e r e x a m p l e f r o m the field o f materials science Is p r o v i d e d b y the chapter ( C h a p t e r 5) o n c o n d u c t i n g p o l y m e r s a n d materials w i t h n o n l i n e a r o p t i c a l properties. T h e potential u n d e r s t a n d i n g to be d e r i v e d f r o m the availability o f s u p e r c o m p u t e r s for this research Is I m m e n s e . T h e materials u n d e r s t u d y are expected to play a critical role In the future d e v e l o p m e n t o f m o l e c u l a r electronic a n d optical devices f o r I n f o r m a t i o n storage a n d c o m m u n i c a t i o n . Large-scale ab initio s i m u l a t i o n s lead to detailed unders t a n d i n g o f properties w h i c h are h a r d to extract f r o m e x p e r i m e n t a l data o r f r o m m o r e approximate a n d less d e m a n d i n g calculations. T h e m e t h o d s o f q u a n t u m c h e m i s t r y have reached a p o i n t where t h e y constitute tools o f se m l - q u a n t i t a t i v e accuracy and have significant predictive v a l u e . F u r t h e r d e v e l o p m e n t s f o r quantitative accuracy are still n e e d e d for m a n y purposes t h o u g h a n d w i l l require the application o f reliable m e t h o d s f o r d e s c r i b i n g e l e c t r o n c o r r e l a t i o n In large systems. T h e n e e d for s u p e r c o m p u t e r p o w e r w i l l be v e r y acute for s u c h correlated calculations. A n o t h e r area where electronic structure calculations can have an enorm o u s Impact o n Industrial c h e m i s t r y Is the design o f efficient catalysts. V e r y few catalytic systems have been s t u d i e d so far because o f the large c o m p u t e r t i m e r e q u i r e m e n t , w h i c h Is where s u p e r c o m p u t e r s c o m e In. The m o s t complete p r o t o s t u d y c u r r e n t l y available In this area Is the m o d e l l i n g o f the full catalytic cycle o f olefin h y d r o g é n a t i o n b y W i l k i n s o n catalyst as calculated b y M o r o k u m a , D a n l e , a n d K o g a w i t h the ab initio m o l e c u l a r o r b i t a l m e t h o d . E v e n t h o u g h these authors c o m m i t t e d about 200 h o u r s o f s u p e r c o m p u t e r t i m e to the project t h e y still h a d to make s e v e r a l s i m p l i f i c a t i o n s , s u c h as replacing s o m e p h e n y l groups b y h y d r o g e n , neglecting s o l v e n t effects, a n d Ignoring side reactions. C l e a r l y e v e n m o r e resources are r e q u i r e d to m a k e the m o d e l l i n g m o r e realistic, b u t already these authors were able to d e t e r m i n e the rate d e t e r m i n i n g step, to predict a possible Intermediate, a n d to Illustrate where a n d w h y there Is a special s e n s i t i v i t y to choice o f l l g a n d . T h i s k i n d o f c a l c u l a t i o n Is clearly v e r y stimulating. One Important aspect o f the s u p e r c o m p u t e r r e v o l u t i o n that m u s t be e m p h a s i z e d Is the hope that n o t o n l y w i l l It allow bigger calculations by e x i s t i n g m e t h o d s , b u t also that It w i l l actually stimulate the d e v e l o p m e n t o f n e w approaches. A recent e x a m p l e o f w o r k a l o n g these Unes I n v o l v e s the s o l u t i o n o f the H a r t r e e - F o c k equations b y n u m e r i c a l Integration In m o m e n t u m space rather than b y e x p a n s i o n In a basis set In coordinate space (2.). S u c h calculations require too m a n y floating p o i n t operations a n d too m u c h m e m o r y to be p e r f o r m e d In a reasonable w a y o n m i n i c o m puters, b u t once t h e y are b e g u n o n s u p e r c o m p u t e r s t h e y o p e n up several new lines o f t h i n k i n g . F i n a l l y , we m e n t i o n the v e r y e n c o u r a g i n g successes c o m b i n i n g atomic n a t u r a l orbltals a n d full C I calculations f o r s m a l l m o l e c u l e s ( A l m l ô f , B a u s c h l l c h e r , a n d T a y l o r ) . W i t h this m e t h o d the singlet-triplet splittings

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

1.

JENSEN AND TRUHLAR

Introduction

5

o f C H a n d S 1 H have been c o m p u t e d to an accuracy o f about 0.1 k c a l / m o l , a n d the dissociation energies o f C H a n d O H are w i t h i n 0.03 e V o f e x p e r i m e n t . T h e e x t e n s i o n o f these techniques to t r a n s i t i o n metals w i l l be v e r y Interesting. 2

2

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

E q u i l i b r i u m P r o p e r t i e s a n d Spectra The r a p i d advances In electronic structure applications are causing the field to be discussed u n d e r m a n y n e w names, s u c h as c o m p u t e r - a i d e d m o l e c u l a r design o r c o m p u t e r - a i d e d materials design ( b o t h a b b r e v i a t e d C A M D as a rather o b v i o u s v a r i a t i o n o n C A D / C A M ) . O n e especially p r o m i s i n g s u b f l e l d concerns the design o f bloactlve m o l e c u l a r agents (computer-aided macromolecular design). The Interaction energies Important f o r b i o l o g i c a l systems are often the w e a k n o n b o n d l n g forces that g o v e r n s u c h p h e n o m e n a as c o n f o r m a t i o n a l changes, tertiary structure o f proteins, a n d the h y d r o p h o b i c Interaction. Because o f the weakness o f these Interactions, the electronic energy cann o t be c o n s i d e r e d apart f r o m s u c h a d d i t i o n a l factors as the t h e r m a l energy a n d e n t r o p y o f the substrate a n d the s o l v e n t . B i o c h e m i c a l systems, being c o m p l e x , clearly require large calculations. Once the possibility o f these calculations Is o p e n e d u p , h o w e v e r , Interesting approaches b e c o m e available that Illustrate s o m e e x t r e m e l y Important advantages o f the c o m p u t a t i o n a l m o d e o f science o v e r the e x p e r i m e n t a l , o b s e r v a t i o n a l m o d e . F o r e x a m p l e , m a n y researchers are v e r y e x c i t e d about a n approach to calculating free energy differences In w h i c h o n e o f the c o m p a r e d systems Is Increm e n t a l l y "mutated" Into the other; this k i n d o f process Is clearly Impossible In the laboratory, b u t o f course we can nevertheless learn a l o t f r o m It theoretically. T h i s t e c h n i q u e , whose recent d e v e l o p m e n t s are due to J o r gensen a n d M c C a m m o n a n d t h e i r c o w o r k e r s , Is b e i n g a p p l i e d to the b i n d ing constants o f drugs to m a c r o m o l e c u l a r receptors, the effects o f sitespecific m u t a t i o n o n e n z y m e catalysis, a n d s o l v e n t effects ( 2 ) . T h i s technique Is discussed further In the chapter b y B e r e n d s e n ( C h a p t e r 7 ) , w h o has applied It to the b i n d i n g o f e n z y m e Inhibitors to e n z y m e s . A n o t h e r b i o c h e m i c a l e x a m p l e ( C h a p t e r 8) I n v o l v e s the c o n f o r m a t i o n a l e x p l o r a t i o n o f an octapeptlde w i t h the a i m t h e r e b y to d e v e l o p a breast cancer v a c c i n e . O t h e r w o r k b y the same author, In this case o n the 4 1 - m e r o f a tetrapeptlde, leads to constructive suggestions f o r a m a l a r i a l v a c c i n e . It Is o f course v e r y e n c o u r a g i n g to see these h u m a n i t a r i a n applications o f s u p e r c o m p u t e r s . R a t i o n a l d r u g design a n d p r o t e i n e n g i n e e r i n g are clearly fields In w h i c h m o l e c u l a r m e c h a n i c s a n d c o n f o r m a t i o n a l analysis c o m b i n e d w i t h Interactive c o m p u t e r graphics a n d m o l e c u l a r d y n a m i c s have e x c i t i n g o p p o r t u n i t i e s to l e a d to progress. E v e n t h o u g h the fields are y o u n g , the status o f c o m p u t e r - a i d e d d r u g ( a n d v a c c i n e ) design Is v e r y greatly a d v a n c e d c o m p a r e d to Its status w h e n an A C S S y m p o s i u m Series v o l u m e (4) appeared o n this subject In 1979; that v o l u m e m a y be c o n s u l t e d f o r a snapshot o f this v e r y active field Just before the availability o f supercomputers.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

β

M o s t c o m m o n approaches to s i m u l a t i o n s , w h e t h e r b i o c h e m i c a l o r o t h ­ e r w i s e , r e l y o n s o m e v a r i a n t o f the M o n t e C a r l o m e t h o d . I n general a M o n t e C a r l o m e t h o d Is a w a y to c o m p u t e a quantity b y Interpreting It as the average o f a r a n d o m s a m p l e , u s u a l l y a computer-gene rated o n e . T h e m e t h o d was p i o n e e r e d for the s t u d y o f s m a l l systems a n d Is still v e r y use­ ful for s u c h s y s t e m s . F o r e x a m p l e , one o f the poster papers at the S y m ­ p o s i u m c o n c e r n e d v e c t o r l z a t l o n strategies f o r b i n a r y c o l l i s i o n s o f A r w i t h C O a n d H e w i t h C H C N (5.). T h e M o n t e C a r l o m e t h o d b e c o m e s m o r e Important, h o w e v e r , f o r large, c o m p l e x systems w i t h m a n y degrees o f f r e e d o m , for w h i c h It m a y be the o n l y practical s i m u l a t i o n t e c h n i q u e . Since these systems are often treated classically, a v a r i a n t o f the M o n t e C a r l o m e t h o d In w h i c h the classical equations o f m o t i o n are u s e d to gen­ erate the e n s e m b l e , Is often used; this Is called m o l e c u l a r d y n a m i c s . M o n t e C a r l o calculations m a y require e n o r m o u s a m o u n t s o f c o m p u t e r t i m e , t a x i n g any conceivable c o m p u t e r s y s t e m . T h e M o n t e C a r l o m e t h o d yields Imprecise estimators w i t h a variance that m a y be r e d u c e d b y Increasing the size o f the sample set. Because the samples m a y be u n c o r r e c t e d , M o n t e C a r l o m e t h o d s are w e l l s u i t e d to the n e x t generation o f h i g h l y parallel c o m p u t e r s . A challenge to theorists w i l l be to Include q u a n t u m effects where r e q u i r e d so as n o t to r e l y o n large classical s i m u l a ­ t i o n s Just because t h e y can finally be carried o u t . S e v e r a l groups are n o w w o r k i n g o n large-scale q u a n t u m s i m u l a t i o n s . F o r e x a m p l e , In one s t u d y (&), an excess e l e c t r o n In a sample o f 300 water m o l e c u l e s at r o o m t e m ­ perature was s i m u l a t e d b y path Integral techniques I n v o l v i n g u p to 1000p o l n t discretizations o f the e l e c t r o n path. T h e h i g h l y q u a n t u m nature o f this s y s t e m Is o b v i o u s f r o m the s m a l l mass o f the e l e c t r o n , w h i c h has a t h e r m a l free-particle d e B r o g l l e w a v e l e n g t h o f a b o u t 17 À .

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

3

A repeated t h e m e In discussions o f s u p e r c o m p u t e r s i m u l a t i o n s Is that t h e y allow us to ask a n d answer questions that c a n n o t be a s k e d e x p e r i m e n t a l l y , especially questions about details a n d a b o u t the " w h y " o f v a r i o u s processes. ( T h i s same note was s t r u c k In the d i s c u s s i o n above o f elect r o n i c properties o f materials.) O n e s t r i k i n g e x a m p l e o f this k i n d o f e x t r a detail was p r o v i d e d b y a recent classical d y n a m i c a l s i m u l a t i o n o f D N A c o u n t e r l o n m o t i o n s In aqueous salt s o l u t i o n s (Z.). T h i s s i m u l a t i o n r e q u i r e d calculating the sequence o f c o u n t e r l o n positions o n a v e r y fine t i m e g r i d , w h i l e the total t i m e I n v o l v e d In d e t e r m i n i n g the s i m u l a t e d e x p e r i m e n t a l o b s e r v a b l e , w h i c h was an N M R signal, Is v e r y l o n g . E a c h step requires the c a l c u l a t i o n o f r o u g h l y 10,000 Interactions a m o n g the charged atoms o f the p o l y m e r a n d the s m a l l Ions. T h e c a l c u l a t i o n r e q u i r e d a b o u t 40 h o u r s o f c o m p u t e r t i m e o n the M i n n e s o t a S u p e r c o m puter C e n t e r C r a y - 2 c o m p u t e r . W i t h o u t s u p e r c o m p u t e r s the s i m u l a t i o n w o u l d have been c o m p l e t e l y lnfeaslble. C l e m e n t i a n d L i e ( C h a p t e r 14) have also c o n s i d e r e d the c o u n t e r l o n structure near D N A , a n d have c o n s i d e r e d the t i m e scale q u e s t i o n o f h o w l o n g do water m o l e c u l e s near D N A retain t h e i r l i q u i d structure as c o m p a r e d to the t i m e scale f o r those far away; this Is clearly another q u e s t i o n that w o u l d be h a r d to answer by experiment.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

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JENSEN AND TRUHLAR

7

Introduction

L e v y ( C h a p t e r 6) has also e x p l o r e d the use o f s u p e r c o m p u t e r s to s t u d y detailed properties o f biological m a c r o m o l e c u l e that are o n l y Indirectly accessible to e x p e r i m e n t , w i t h particular e m p h a s i s o n s o l v e n t effects a n d o n the Interplay between c o m p u t e r s i m u l a t i o n s a n d e x p e r i m e n t a l techniques s u c h as N M R , X - r a y structures, a n d v i b r a t i o n a l spectra. T h e chapter b y Jorgensen ( C h a p t e r 12) s u m m a r i z e s r e c e n t w o r k o n the k i n e t i c s o f s i m p l e reactions In s o l u t i o n s . T h i s k i n d o f c a l c u l a t i o n p r o v i d e s e x a m p l e s o f h o w s i m u l a t i o n s c a n address questions that are h a r d to address e x p e r i m e n t a l l y . F o r e x a m p l e J o r g e n s e n ' s s i m u l a t i o n s p r e d i c t e d the existence o f a n Intermediate f o r the reaction o f c h l o r i d e I o n w i t h m e t h y l c h l o r i d e In D M F w h i c h h a d n o t b e e n anticipated e x p e r i m e n t a l l y , a n d t h e y Indicate that the w e a k e r s o l v a t i o n o f the t r a n s i t i o n state as c o m pared to reactants f o r this reaction In aqueous s o l u t i o n Is n o t due to a decrease In the n u m b e r o f h y d r o g e n b o n d s , b u t rather due to a w e a k e n i n g o f the h y d r o g e n b o n d s . S u p e r c o m p u t e r s b e c o m e m o r e a n d m o r e useful, a n d the Insights they can generate b e c o m e m o r e a n d m o r e u n i q u e , as the c o m p l e x i t y o f the s y s t e m m o d e l l e d Is Increased. T h u s lnterfaclal p h e n o m e n a are a v e r y n a t u r a l field f o r s u p e r c o m p u t a t l o n . I n a d d i t i o n to the e x a m p l e s In this v o l u m e It m a y be useful to m e n t i o n the w o r k o f L l n s e o n l l q u l d - l l q u l d benzene-water Interfaces, w h i c h he s t u d i e d w i t h 504 H 0 m o l e c u l e s , 144 C H m o l e c u l e s , a n d 3700 Interaction sites. H e generated o v e r 5 0 m i l l i o n configurations In 50 h o u r s o n a C r a y - 1 A , a n d he was able to quantitatively assess the sharpness o f the lnterfaclal density gradient, w h i c h Is v e r y h a r d to probe e x p e r i m e n t a l l y , S i m i l a r l y S p o h r a n d H e l n z l n g e r have s t u d i e d o r l e n t a t l o n a l p o l a r i z a t i o n o f H 0 m o l e c u l e s at a m e t a l l i c Interface, w h i c h Is also h a r d to probe e x p e r i m e n t a l l y . 2

6

6

2

Microscopic Dynamics T h e present v o l u m e contains o n l y o n e chapter ( C h a p t e r 11) o n s m a l l m o l e c u l e gas-phase d y n a m i c s . I n this field the role o f the s u p e r c o m p u t e r Is d i v e r s e , b u t perhaps the m o s t critical area Is a l l o w i n g essentially exact quantal d y n a m i c s to be c a r r i e d o u t f o r p r e v i o u s l y Intractable systems. A r e c e n t e x a m p l e Is the essentially exact c a l c u l a t i o n o f the reaction thresh o l d f o r D atoms reacting w i t h v l b r a t l o n a l l y e x c i t e d H (&.). T h e same research g r o u p has c o m p l e t e d the first n u m e r i c a l l y c o n v e r g e d s o l u t i o n s o f the S c h r o e d l n g e r f o r reaction probabilities In a system w i t h a n atom h e a v i e r t h a n a n Isotope o f H , In particular 0 + H - + O H + H . B o t h calculations were c a r r i e d o u t w i t h a new basis-set approach that specifically takes advantage o f the large m e m o r y a n d h i g h v e c t o r speed o f the C r a y - 2 . A n o t h e r n e w c o m p u t a t i o n a l approach to s m a l l - m o l e c u l e d y n a m i c s that Is s t i m u l a t e d In part b y the a v a i l a b i l i t y o f fast v e c t o r m a c h i n e s Is based o n the c o m p u t a t i o n o f quantal propagators w i t h v e r y large basis sets b y recursive t r a n s f o r m a t i o n o f a large sparse H a m l l t o n l a n m a t r i x Into a m u c h s m a l l e r t r l d l a g o n a l o n e ; a recent application Is to t i m e - d e p e n d e n t energy d e p o s i t i o n In a m o l e c u l e b y a laser (fi.). A b r a h a m (1Q) has p r o v i d e d an 2

2

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

8

SUPERCOMPUTER

RESEARCH

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

e x c e l l e n t r e v i e w o f recent s i m u l a t i o n s o n t w o - d i m e n s i o n a l c o n d e n s a t i o n a n d m e l t i n g at surfaces and In t h i n films. H i s r e v i e w also p r o v i d e s s o m e r e l e v a n t b a c k g r o u n d reading for the chapter In the present v o l u m e b y G i l m e r a n d G r a b o w ( C h a p t e r 1 3 ) . T h i s r e v i e w also contains an e x c i t i n g chapter s e c t i o n e n t i t l e d "super p r o b l e m s f o r s u p e r c o m p u t e r s " (sic), In w h i c h the a u t h o r discusses s o m e o f the v e r y large c o m p u t a t i o n a l p r o b l e m s that still defy attack. W e have already m e n t i o n e d the application o f s u p e r c o m p u t e r s to b i o c h e m i c a l s i m u l a t i o n s . Internal d y n a m i c s m a y play an Important role In s u c h s i m u l a t i o n s . A n e x a m p l e w o u l d be e n z y m e blndlng-slte fluctuations that m o d u l a t e reactivity o r the d y n a m i c s o f a n t i g e n - a n t i b o d y association ( 1 1 ) . I n the specific case o f d i f f u s i o n - c o n t r o l l e d processes, m o l e c u l a r r e c o g n i t i o n m a y o c c u r because o f long-range sterlc effects w h i c h are h a r d to assess w i t h o u t v e r y e x p e n s i v e s i m u l a t i o n s (12.). In a d d i t i o n to the already m e n t i o n e d Insights Into materials properties o b t a i n e d t h r o u g h electronic structure calculations, materials science has m u c h to gain f r o m s u p e r c o m p u t e r s i m u l a t i o n s o f m i c r o s c o p i c a n d macroscopic e l e m e n t s o f materials processing. M i c r o e l e c t r o n i c c o m p o n e n t s , o p t i cal devices ( s o l i d state lasers a n d detectors), o p t i c a l fibers a n d h i g h perf o r m a n c e ceramics are artificially m i c r o s t r u c t u r e d materials made b y caref u l l y c o n t r o l l e d n u c l e a t l o n , s o l i d i f i c a t i o n , d e p o s i t i o n , a n d e t c h i n g procedures. Since the performance o f the materials s t r o n g l y depends o n the degree o f crystalline perfection a n d the nature o f the Interface, a m i c r o s copic u n d e r s t a n d i n g o f the a t o m i c scale g r o w t h a n d e t c h i n g processes Is essential. D i r e c t m o l e c u l a r d y n a m i c s i m u l a t i o n s o f crystal g r o w t h f r o m the v a p o r are discussed b y G i l m e r a n d G r a b o w ( C h a p t e r 1 3 ) . T h e difficulty In this procedure Is the large a m o u n t o f c o m p u t a t i o n r e q u i r e d to o b t a i n the a t o m i c trajectories a n d the large n u m b e r o f atoms r e q u i r e d because o f the v e r y s l o w g r o w t h rates. P r e s e n t c o m p u t a t i o n p o w e r m a y n o t be sufficient f o r a direct s i m u l a t i o n o f m o l e c u l a r b e a m e p i t a x y o f an e l e m e n t a l s e m i c o n d u c t o r (e.g. SI) a n d It l i m i t s studies o f the m a n y fundam e n t a l p r o b l e m s o f Interface f o r m a t i o n a n d g r o w t h f o u n d In m o l e c u l a r b e a m e p i t a x y o f c o m p o u n d s e m i c o n d u c t o r structures (e.g. A l G a A s / G a A s ) . Because o f the s m a l l c o r r e l a t i o n b e t w e e n samples In m o l e c u l a r d y n a m i c s i m u l a t i o n s o f crystal g r o w t h , this application s e e m s w e l l s u i t e d for n e w , special purpose, h i g h l y parallel c o m p u t e r s . M e t a l - h y d r o gen systems a n d s u p e r l o n l c c o n d u c t o r s are e x a m p l e s o f o t h e r s o l i d systems o f great t e c h n o l o g i c a l Importance o n w h i c h progress has been h a m p e r e d b y the Inability to m a k e realistic e n o u g h s i m u l a t i o n s . T h e reader Is directed to recent w o r k b y G l l l a n a n d C a t l o w a n d their c o w o r k e r s for recent progress In s t u d y i n g these k i n d s o f systems. S u p e r c o m p u t e r s can be d i r e c t e d to the s t u d y o f techniques as w e l l as materials a n d processes. F o r e x a m p l e , one can s i m u l a t e n e u t r o n scatteri n g e x p e r i m e n t s w i t h the goal o f e l u c i d a t i n g the effects o f a p p r o x i m a t i o n s u s u a l l y made In "standard" treatments o f the e x p e r i m e n t a l data. T h e u n d e r s t a n d i n g o f fluid flow Is one the areas w h e r e s u p e r c o m p u t l n g has already h a d a significant Impact. G e n e r a l fluid m e c h a n i c s falls

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

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outside the scope o f this v o l u m e , b u t applications o f fluid m e c h a n i c s to c h e m i c a l p r o b l e m s are characteristic o f the chapters g r o u p e d u n d e r the transport h e a d i n g . A s an Interesting t r a n s i t i o n b e t w e e n the m i c r o s c o p i c d y n a m i c s a n d m a c r o s c o p i c transport chapters, C l e m e n t i a n d L i e ( C h a p t e r 14) describe the s i m u l a t i o n o f a macroscopic fluid flow e x a m p l e In terms of constituent molecular motions. A n o t h e r example o f molecular simulat i o n o f fluids concerns transport and fluid properties In m i c r o porous m e d i a as d i s c u s s e d b y D a v i s et al. ( C h a p t e r 1 5 ) . Because o f the m o l e c u l a r o r n a n o m e t e r d i m e n s i o n s I n v o l v e d In these systems, e x p e r i m e n t a l characteriz a t i o n Is difficult. M o r e o v e r , fluids can be s t r o n g l y l n h o m o g e n e o u s In the c o n f i n e d pore space so that the usual m a c r o s c o p i c theories o f fluid structure a n d transport m a y n o t be applicable. T h u s , s u p e r c o m p u t e r s i m u l a tions b e c o m e an Important t o o l f o r u n d e r s t a n d i n g fluid structure a n d trans p o r t In m l c r o p o r o u s m e d i a as w e l l as f o r d e v e l o p i n g appropriate theories suitable f o r a n a l y z i n g related m a c r o s c o p i c p h e n o m e n a , s u c h as processes I n v o l v i n g porous catalysts ( e . g . h y d r o d e s u l f u r l z a t l o n ) , l u b r i c a t i o n a n d w e t t i n g , d r y i n g o f paper products a n d clay dispersions, a n d e n h a n c e d o i l r e c o v e r y . T h e s t u d y o f these practical p r o b l e m s are also natural areas f o r s u p e r c o m p u t e r research w h i c h w i l l be discussed In the s u b s e q u e n t s e c t i o n .

Transport Processes M a c r o s c o p i c analysis o f c o m p l e x c h e m i c a l processes, I n c l u d i n g materials p r o c e s s i n g , requires n u m e r i c a l s o l u t i o n o f the equations f o r local conserv a t i o n o f m o m e n t u m , energy, mass, a n d c h e m i c a l species o n Irregular d o m a i n s a n d often w i t h free b o u n d a r i e s . I n t h e i r general f o r m , the equations are n o n l i n e a r partial differential equations In space a n d t i m e , where the n o n l l n e a r l t l e s are I n t r o d u c e d b y the constitutive equations f o r fluxes (e.g. m u l t l c o m p o n e n t diffusion, n o n - N e w t o n i a n flow), reaction rates, c o n v e c t l v e c o u p l i n g between flow a n d m a s s / e n e r g y transport, a n d the dependence o f b o u n d a r y shapes o n field variables. These n o n l i n e a r Interactions s e v e r e l y complicate the n u m e r i c a l s o l u t i o n o f the c o n s e r v a t i o n equations b y causing transitions In the s o l u t i o n structure, I n c l u d i n g m u l t i p l e s o l u t i o n s , spatially a n d t e m p o r a l l y periodic s o l u t i o n s , a n d e v e n chaotic p h e n o m e n a . O t h e r c o m p l i c a t i o n s are caused b y m u l t i p l e l e n g t h a n d time scales. L e n g t h scales different t h a n those characteristic o f the d o m a i n arise f r o m the nature o f the p r o b l e m ; f o r e x a m p l e , In the case o f a catalytic reactor the active m a t e r i a l m a y be 5 n m m e t a l crystals I m b e d d e d In a 5 m m p o r o u s particle s t a c k e d a m o n g t h o u s a n d s o f particles In a 0.50 m d i a m e t e r tube. I n a d d i t i o n , different l e n g t h scales arise as a result o f b o u n d a r y a n d Internal layers caused b y rapid changes In the field variables near s o l i d b o u n d a r i e s , Interfaces, a n d reaction fronts. F o r e x a m p l e , the flame front In a c o m b u s t i o n system m a y be a f e w m m wide w h i l e the characteristic d i m e n s i o n o f the system Is In o r d e r o f meters. M u l t i p l e t i m e scales originate f r o m the m i x i n g o f transport processes a n d reaction k i n e t i c s , w h i c h have o r d e r - o f - m a g n l t u d e differences In t h e i r t i m e scales, a n d t h e y lead to stiff Integration p r o b l e m s that can tax o r e v e n e x c e e d the

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

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capabilities o f c u r r e n t s u p e r c o m p u t e r hardware. O r t e g a a n d V o l g t (12.) r e v i e w n u m e r i c a l m e t h o d s for partial differential equations o n s u p e r c o m puters a l o n g w i t h a brief d e s c r i p t i o n o f applications to fluid d y n a m i c s , r e s e r v o i r s i m u l a t i o n , a n d weather p r e d i c t i o n . T h e y specifically discuss the Influence o f parallel a n d v e c t o r c o m p u t i n g o n a l g o r i t h m d e s i g n a n d selection. T h e n o n l i n e a r nature o f detailed m o d e l s o f c o m p l e x c h e m i c a l processes Is a central Issue In t h e i r s o l u t i o n a n d one that contributes h e a v i l y to c o m p u t a t i o n a l d e m a n d s . If the n o n l l n e a r l t l e s are s t r o n g e n o u g h It m a y be essentially Impossible to find a s o l u t i o n for a particular set o f parameters f r o m a s i m p l e Initial guess. I n s u c h cases the s o l u t i o n m u s t be f o u n d b y c o n n e c t i n g It b y h o m o t o p y to a k n o w n s o l u t i o n o f the same set o f equations but w i t h a different set o f parameters o r perhaps a s o l u t i o n to a s i m p l e r , b u t analogous p r o b l e m . T h e procedure I n v o l v e s f o l l o w i n g the s o l u t i o n f a m i l y f o r v a r y i n g parameters and It Is c o m m o n l y r e f e r r e d to as c o n t i n u a t i o n . T h u s , e v e n t h o u g h a single c a l c u l a t i o n perhaps c o u l d be carr i e d o u t o n a V A X 8 6 0 0 In a few h o u r s , the large n u m b e r o f calculations I n v o l v e d In r e a c h i n g the d e s i r e d s o l u t i o n b y c o n t i n u a t i o n necessitates s u p e r c o m p u t l n g . F u r t h e r m o r e , because the n o n l l n e a r l t l e s lead to n o n u n i q u e n e s s o f the steady state a n d a m u l t i t u d e o f p e r i o d i c p h e n o m e n a , It Is necessary to u n d e r s t a n d the structure o f s o l u t i o n space, w h i c h again m e a n s t r a c k i n g s o l u t i o n f a m i l i e s f o r v a r y i n g parameters b y u s i n g speciali z e d c o n t i n u a t i o n techniques ( 1 3 - 1 6 ) . If the stability o f the s o l u t i o n Is also o f Interest, the eigenvalues o f the l i n e a r i z e d p r o b l e m m u s t be determ i n e d . F o r large-scale systems this requires extensive s u p e r c o m p u t e r calculations a n d m a n y p r o b l e m s s t i l l defy attack. T h e n o n l i n e a r b e h a v i o r o f p h y s l c o c h e m l c a l systems Is b r o u g h t up In s e v e r a l o f the application e x a m p l e s In this v o l u m e . B r o w n et al. ( C h a p t e r 17) c o n s i d e r the e v o l u t i o n o f c e l l u l a r m i c r o s t r u c t u r e s d u r i n g d i r e c t i o n a l s o l i d i f i c a t i o n , w h i c h Is a n o n l i n e a r free-surface p r o b l e m . Jensen et ai ( C h a p t e r 19) describe n o n l i n e a r flow transitions a d v e r s e l y affecting the g r o w t h o f c o m p o u n d s e m i c o n d u c t o r superlattlces b y o r g a n o m e t a l l l c c h e m ical v a p o r d e p o s i t i o n , w h i l e S m o o k e addresses a flame e x t i n c t i o n p r o b l e m ( C h a p t e r 2 0 ) . B o t h sets o f Investigators use an arclength c o n t i n u a t i o n technique due to K e l l e r (11.). K e v r e k l d l s ( C h a p t e r 16) specifically addresses c o m p u t a t i o n a l Issues In the analysis o f c o m p l e x d y n a m i c s that c a n n o t be u n d e r s t o o d t h r o u g h local stability c o n s i d e r a t i o n s . Because o f the nature o f the Instabilities u n d e r l y i n g the d y n a m i c p h e n o m e n a , It Is e x t r e m e l y difficult, If n o t Impossible, to extract an u n d e r s t a n d i n g o f the transitions b e t w e e n the v a r i o u s periodic b e h a v i o r s t h r o u g h s i m p l e s i m u l a t i o n s o f the physical e x p e r i m e n t s . T w o Illustrative e x a m p l e s based o n flame f r o n t a n d t h e r m a l c o n v e c t i o n descriptions are presented. I n the r e m a i n i n g parts o f this I n t r o d u c t o r y chapter we r e t u r n to discuss i o n o f specific applications o f s u p e r c o m p u t l n g — starting w i t h materials g r o w t h a n d p r o c e e d i n g t h r o u g h Increasingly c o m p l e x p h y s l c o c h e m l c a l syst e m s . O n e o f the application areas where large-scale s i m u l a t i o n s have already h a d an Impact o n u n d e r s t a n d i n g Is the process o f crystal g r o w t h

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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JENSEN AND TRUHLAR

Introduction

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Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

f r o m the m e l t I n c l u d i n g m o r p h o l o g i c a l solidification p h e n o m e n a . T h e latter Is a classical p r o b l e m a n d Is addressed b y B r o w n et al. ( C h a p t e r 17), w h o use large scale n u m e r i c a l finite e l e m e n t s o l u t i o n s o f t w o - d i m e n s i o n a l m o d e l s to s t u d y pattern s e l e c t i o n o f the solidification front. C e n t r a l Issues In this f u n d a m e n t a l p r o b l e m Include the cell shape, the apparent w a v e l e n g t h f o r the crystallization front, a n d the e v o l u t i o n to dendritic g r o w t h w i t h side branches. T h e s i m u l a t i o n s p r o v i d e significant n e w Insight Into the m o r p h o l o g i c a l stability p h e n o m e n a . H o w e v e r , to address the central q u e s t i o n o f w h e t h e r the o b s e r v e d d y n a m i c s are d e t e r m i n i s t i c o r a "snapshot" o f a stochastic b e h a v i o r w o u l d require the n e x t generation of supercomputers. T h e g r o w t h o f the t h i n films f r o m the gas phase b y c h e m i c a l v a p o r d e p o s i t i o n ( C V D ) I n v o l v e s a c o m p l e x m i x t u r e o f h o m o g e n e o u s reactions, surface reactions, fluid flow, heat transfer, a n d mass transfer that Is difficult to u n d e r s t a n d w i t h o u t a c o m p r e h e n s i v e m o d e l o f the process. T h e w o r k o f K e e , C o l t r l n , a n d c o w o r k e r s (1Z, C h a p t e r 18) represents a significant effort to Include detailed k i n e t i c m o d e l s In C V D reactor s i m u l a tions analogous to w h a t has b e e n done In c o m b u s t i o n m o d e l l i n g . B y u s i n g s e n s i t i v i t y analysis t h e y d e r i v e d a m e c h a n i s m o f 20 reactions f r o m a detailed pyrolysls m e c h a n i s m f o r S 1 H I n v o l v i n g 120 e l e m e n t a r y reactions. T h e i r s i m u l a t i o n s d e m o n s t r a t e d the Importance o f I n c l u d i n g detailed descriptions o f h o m o g e n e o u s a n d heterogeneous reactions In C V D reactor m o d e l s a n d they c o m p a r e d w e l l to species m e a s u r e m e n t s b y laser spect r o s c o p y ( U L ) . T h i s type w o r k c a n o n l y be r e a l i z e d b y the use o f superc o m p u t e r s . I n a d d i t i o n to treating SI d e p o s i t i o n , the authors' c o n t r i b u t i o n to the present v o l u m e ( C h a p t e r 18) also addresses the I m p l e m e n t a t i o n o f large-scale m o d e l s o f p h y s l c o c h e m l c a l processes, e.g. the c o m p u t a t i o n a n d o r g a n i z a t i o n o f t h e r m o d y n a m i c quantities, transport coefficients, a n d rate constants. T h i s Is a n Issue that transcends C V D analysis to s i m u l a t i o n o f o t h e r c o m p l e x c h e m i c a l l y reacting systems. 4

J e n s e n et al. ( C h a p t e r 19) focus o n two- a n d t h r e e - d i m e n s i o n a l trans p o r t p h e n o m e n a as w e l l as transient b e h a v i o r In the g r o w t h o f t h i n films a n d superlattlces o f c o m p o u n d s e m i c o n d u c t o r s (e.g. G a A s / A l G a A s ) . P r e v i o u s C V D m o d e l s have been based o n s i m p l i f i e d transport descriptions unable to p r o v i d e a complete e n o u g h picture o f spatial a n d t e m p o r a l v a r i a tions In the d e p o s i t i o n rate. H o w e v e r , accurate c o n t r o l o f the d e p o s i t i o n rate Is essential to the further d e v e l o p m e n t o f a d v a n c e d o p t o e l e c t r o n i c a n d m i c r o e l e c t r o n i c devices. Because o f the c o m p l e x gas flows In Irregular d o m a i n s , s u p e r c o m p u t l n g Is necessary to simulate the process m o d e l s . F u r t h e r analysis w i l l have to consider transient, t h r e e - d i m e n s i o n a l reacti n g flow p h e n o m e n a w h i c h w i l l s e v e r e l y t a x , a n d In s o m e cases e x c e e d , the capabilities o f c u r r e n t s u p e r c o m p u t e r hardware. T h e r e are m a n y o t h e r o p p o r t u n i t i e s f o r s u p e r c o m p u t e r applications In materials processing In a d d i t i o n to the crystal g r o w t h studies In this v o l u m e . F o r e x a m p l e p l a s m a a n d laser processing (19.20) c o u l d gain c o n s i d e r a b l y f r o m studies o f detailed process m o d e l s . T h e goal o f m o d e l l i n g materials processing s h o u l d be the theoretical u n d e r s t a n d i n g a n d , e v e n t u a l l y , quantitative

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

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p r e d i c t i o n o f the relationship between m a c r o s c o p i c processing c o n d i t i o n s a n d the m l c r o s t r u c t u r e o f the materials, w h i c h governs the m e c h a n i c a l , o p t i c a l , a n d electrical properties. C o m b u s t i o n o f gaseous and s o l i d fuels Is another application area that has m u c h to gain f r o m large-scale s i m u l a t i o n s o f detailed processes m o d e l s . T h e Issues are s i m i l a r to those In C V D , b u t c o m p l i c a t e d b y large e n e r g y releases and multiphase flows. T h e questions related to the organi z a t i o n o f the database o f rate constants, transport coefficients, a n d therm o d y n a m i c quantities are essentially the same as In C V D m o d e l l i n g . I n fact, the a f o r e m e n t i o n e d w o r k by K e e et al. ( C h a p t e r 18) benefits f r o m t h e i r e x t e n s i v e experience In c o m b u s t i o n m o d e l l i n g . Large-scale c o m p u tational analysis o f c o m b u s t i o n Involves s e v e r a l critical e l e m e n t s I n c l u d i n g the c h e m i s t r y code, the fluid flow treatment, a n d the r e s o l u t i o n o f sharp flame fronts. T h e latter Issue, w h i c h Is part o f the general p r o b l e m o f differing l e n g t h scales In detailed process m o d e l s , poses significant challenges to n u m e r i c a l procedures. T h e size o f large-scale p h y s l c o c h e m l c a l p r o b l e m s c o m b i n e d w i t h the n e e d to accurately resolve local structures (e.g. a flame front) necessitates the use o f d y n a m i c , se If-adaptive, local g r i d m o d i f i c a t i o n s . U n i f o r m g r i d i n g o n the basis o f the l e n g t h scale o f the local p h e n o m e n o n w o u l d lead to finite e l e m e n t / f i n i t e difference discretizations w i t h a huge n u m b e r o f equations whose s o l u t i o n w o u l d be p r o h i b i t i v e l y e x p e n s i v e o n e v e n the largest s u p e r c o m p u t e r s . T h e r e f o r e , adaptive g r i d i n g Is a r a p i d l y e v o l v i n g area In n u m e r i c a l analysis for large-scale m o d e l s . T w o e x a m p l e s f r o m gaseous a n d s o l i d fuel c o m b u s t i o n m o d e l l i n g are I n c l u d e d In the present v o l u m e ( C h a p t e r s 20 a n d 2 1 ) . A recent s u r v e y b y B a b u s k a et al. (21) shows the p r i n c i p a l directions o f w o r k adaptive g r i d i n g t e c h n i q u e s . A t m o s p h e r i c c h e m i s t r y m o d e l l i n g to predict the effect o f pollutants ( I n t e n t i o n a l l y o r u n i n t e n t i o n a l l y released) o n the e n v i r o n m e n t Is a natural application f o r s u p e r c o m p u t l n g . T h e p r o b l e m I n v o l v e s a large n u m b e r o f reactions a m o n g h y d r o c a r b o n s , fluorocarbons, nitrogen compounds, and s u l f u r c o m p o u n d s In s u n l i g h t (22*22.). I n a d d i t i o n , these reactions have rate constants that differ b y as m u c h as 14 orders o f m a g n i t u d e . S i m u l a tions o f the transport processes In the atmosphere require threed i m e n s i o n a l fluid flow s i m u l a t i o n s w i t h v e r y large grids a n d m a n y trans p o r t i n g constituents. F u r t h e r m o r e , aerosol particle n u c l e a t l o n a n d g r o w t h play Important roles In the o v e r a l l b e h a v i o r . S u p e r c o m p u t e r s i m u lations o f a t m o s p h e r i c c h e m i s t r y n o t o n l y Increase the scientific unders t a n d i n g o f s u c h c o m p l e x systems b u t also p r o v i d e a t o o l f o r regulatory agencies to s t u d y effects o f e x i s t i n g a n d p r o p o s e d p o l l u t a n t e m i s s i o n standards. T h e r e are s e v e r a l o t h e r applications where significant gains c o u l d be made t h r o u g h the use o f s u p e r c o m p u t e r s i m u l a t i o n s o f detailed physical m o d e l s . R e s e r v o i r s i m u l a t i o n s was one o f the first areas w h e r e the value o f s u p e r c o m p u t l n g was r e c o g n i z e d by Industrial c o m p a n i e s . It Is o n l y possible to measure a few properties o f Interest to e n h a n c e d o i l r e c o v e r y . F u r t h e r m o r e , field tests are e x t r e m e l y e x p e n s i v e , a n d the m o n e t a r y

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

1.

JENSEN AND TRUHLAR

Introduction

13

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

decisions I n v o l v e d In the choice o f m e t h o d s f o r d r i v i n g the o i l o u t o f a particular r e s e r v o i r can equal o r perhaps e v e n e x c e e d the cost o f super­ c o m p u t e r hardware. T h u s , s u p e r c o m p u t e r s i m u l a t i o n s b e c o m e a m o r e cost effective m e t h o d than field e x p e r i m e n t s (2iL). C h e m i c a l plant design Is another area that c o u l d benefit f r o m the use o f s u p e r c o m p u t e r s . A large c h e m i c a l plant I n v o l v e s m a n y different units I n c l u d i n g separation, reactors a n d heat exchangers, w h i c h are Interconnected. S o m e o f the units m a y require the same detailed m o d e l l i n g as the above m e n t i o n e d applications. T h e r e f o r e the plant m o d e l w i l l I n v o l v e large n u m b e r s o f Interconnected equations offering considerable challenges to s u p e r c o m p u t ­ lng ( 2 & ) . T h e large-scale plant s i m u l a t i o n s c o u l d serve design, o p t i m i z a ­ t i o n , a n d c o n t r o l purposes.

Conclusions M a n y significant applications o f s u p e r c o m p u t l n g In c h e m i s t r y a n d c h e m i ­ cal e n g i n e e r i n g are e m e r g i n g as facilities f o r large-scale c o m p u t a t i o n s b e c o m e m o r e a n d m o r e accessible. T h e present v o l u m e Intends to Illus­ trate recent advances a n d applications, b u t already the field Is so b r o a d that n o single v o l u m e c a n p u t all the subflelds Into perspective. T h e pur­ pose o f c o m b i n i n g c h e m i s t r y a n d c h e m i c a l e n g i n e e r i n g applications In a single s y m p o s i u m was to emphasize their s t r o n g relationships, a n d we hope that these relationships w i l l be further s t r e n g t h e n e d b y c o n t i n u i n g Interactions b e t w e e n these fields.

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Lykos, P.; Shavitt, I. Supercomputers in Chemistry; American Chemi­ cal Society: Washington, 1981. Alexander, S. J.; Monkhorst, H. J.; to be published. Selbel, G. Chemical Design Automation News 1987, 2, 1. Olson, E. C.; Christoffersen, R. E. Computer-Assisted Drug Design; American Chemical Society: Washington, 1979. Cochrane, D.; Truhlar, D. G. Parallel Computing, In press, and addi­ tional work to be published elsewhere. Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1987, 86, 3471. Reddy, M . R.; Rossky, P. J.; Murthy, C. S. J. Phys. Chem., to be published. Haug, K.; Schwenke, D. W.; Shima, Y.; Truhlar, D. G.; Zhang, J.; Kouri, D . J . J. Phys. Chem. 1986, 90, 6757. Friesner, R. Α.; Brunet, J.-P.; Wyatt, R. E.; Leforestier, C. Int. J. Supercomputer Applications,inpress. Abraham, F. F. Advances in Physics 1986, 35, 1. McCammon, J. A. Rep. Prog. Phys. 1984, 47, 1. McCammon, J. Α.; Northrup, S. H.; Allison, J. A . J. Phys. Chem. 1986, 90, 3901.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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13. 14. 15. 16. 17. 18.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch001

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

Ortega, J. M.; Voight, R. G. SIAM Review 1985, 27 149. Keller, H . B. SIAM J. Scient. and Stat. Comp. 1983, 4, 573. Kubicek, M . ; Marek, I. Computational Methods in Bifurcation Theory and Dissipative Structures; Springer Verlag: New York, 1983. Rheinboldt, W. C. Numerical Analysis of Parameterized Nonlinear Equations; Wiley Interscience: New York, 1986. Coltrin, M. E.; Kee, R. J.; Miller J. A. J. Electrochem. Soc. 1984, 131, 425; 1986, 133, 1206. Brelland, W. G.; Coltrin, M. E.; Ho, P. J. Appl. Phys. 1986, 59, 3276; 60, 1505. Graves, D . B.; Jensen, K . F. IEEE Trans. Plasma Sci. 1986, 14, 78. Skouby, D . C.; Jensen, K . F. SPIE Proceedings 1987, 797; J. Appl. Phys. to be published. Babuska, I.; Zienkiewicz, O. C.; Gago, J.; Oliveira, E . R. Accuracy Estimates and Adaptive Refinements in Finite Element Computations; John Wiley and Sons: New York, 1986. McRae, G . J.; Goodin, W. R.; Seinfeld, J. H. J. Computational Phys. 1982, 45, 1. Rood, R. B.; Kaye, J. Α.; Nielsen, J. E.; Schoerberl, M. R.; Geller, M . A . Phys. Scr. 1987, 35, in press. Ewing, R. E.; in ref. 21, pp. 299. Levesque, J. M. Soc. Pet. Eng. J. 1985, 25, 275. Stadthere, Μ. Α.; Vegeais, J. A. Chem. Eng. Proc. 1985, 81, 21.

RECEIVED

July 6, 1987

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 2

Theoretical Approaches to Metal Chemistry 1

1

1

Charles W. Bauschlicher, Jr. , Stephen R. Langhoff , Harry Partridge , Timur Halicioglu , and Peter R. Taylor 2

3

1

Ames Research Center, National Aeronautics and Space Administration, Moffett Field, CA 94035 Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 ELORET Institute, Sunnyvale, CA 94087 2

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

3

Recent advances in methodology have made possible accurate ab initio calculations on transition metal diatomic systems as well as small (3-6 atom) simple metal clusters. These accurate calculations can in turn be used to define two- and three-body potentials for use in modelling much larger clusters. Calculations on clusters containing Be, A l and Cu atoms illustrate the accuracy of current work and the diversity of metal bonding. The structure and reactivity of small clusters vary dramatically with size, and very large clusters are required before the cluster structure approaches that of the bulk. For example, even though the bulk structure of Be is hcp, the fcc structure is still considerably more stable than hcp for a 55 atom Be cluster. Comparison of ab initio and model calculations for small Al clusters demonstrates that it is necessary to include three-body terms in the model for quantitative results. The impact of adsorbates on metal-metal bonding is studied for Be X and Al X clusters. The optimal sites for adsorption are often different for small clusters than the bulk, owing to the enhanced ability of small clusters to distort. 13

n

13

n

C o m p u t a t i o n a l chemistry is being applied at N A S A A m e s to numerous problems in chemistry, physics and materials science. One important application is to problems i n re-entry physics that are intractable to experiment, such as the extreme conditions occuring i n the bow shock wave of the aeroassisted orbital transfer vehicle ( A O T V ) (1). N o n e q u i l i b r i u m r a d i a t i o n is a significant component of the heating, owing to the large blunt heat shield of the A O T V and its trajectory through the t h i n upper atmosphere. Accurate knowledge of the chemistry of hot mixtures of nitrogen and oxygen are required for input into c o m p u t a t i o n a l fluid dynamics ( C F D ) codes involved in the heat shield design. A l s o , the chemistry of hydrogen and air mixtures is being studied to aid design of supersonic combustion ramjet engines.

0097-6156/87/0353-0016S06.00/0 © 1987 American Chemical Society

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

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BAUSCHLICHER ET AL.

Theoretical

Approaches

to Metal Chemistry

17

In a d d i t i o n to the gas-phase work, we are c o m p u t i n g (2) the vibrational spectra and rotational barriers of polymer fragments to help interpret experi­ ments. B y achieving a better understanding of polymers and their chemistry, we hope to design longer lifetime and more corrosion resistant polymers. A n o t h e r major c o m p u t a t i o n a l effort is i n the area of metals and their chem­ istry, w h i c h comprises the subject of this manuscript. T h e studies are directed towards b o t h catalysis and the development of improved materials, such as stronger m a t r i x composites. T h e materials and gas phase work have some over­ lap. For example, surface recombination affects the heating on the A O T V heat shield a n d on the walls of the scramjet. In a d d i t i o n , desorption of these molecules from the walls of the scramjet could impact the chemistry in the flow. T h e study of molecular systems containing metal atoms, particularly tran­ sition metal atoms, is more challenging than first-row chemistry from both an experimental and theoretical point of view. Therefore, we have systematically studied (3-5) the c o m p u t a t i o n a l requirements for obtaining accurate spectro­ scopic constants for diatomic and triatomic systems containing the first- and second-row t r a n s i t i o n metals. O u r goal has been to understand the diversity of mechanisms by w h i c h transition metals b o n d and to a i d i n the interpretation of experimental observations. W h i l e accurate calculations on transition metal compounds are restricted to three or fewer transition metal atoms, it is possible to consider much larger clusters of A l and B e atoms. We have considered (6^9) A l clusters of up to six atoms using correlated wave functions, and A l i , B e i 3 a n d Bess at the S C F level. These calculations give insight into how the b o n d i n g changes w i t h cluster size. Since even for these simple metals the ab initio calculations are time consum­ ing, we have interfaced (9) our ab initio methods w i t h a parameterized model approach where the potential is expanded i n two- and three-body interaction terms. For single component systems, these potentials can be determined from either b u l k d a t a or calculations. W i t h the parameterized model we can consider larger clusters a n d identify interesting clusters for further ab initio study. The parameterized model can also be used for multi-component materials, although in this case the two- and three-body parameters are not easily deduced from bulk data. T h e m o d e l approach appears well-suited for the study of alloys and m a t r i x composite materials, especially large multi-component systems not d i ­ rectly amenable to ab initio study, but it must rely on ab initio calculations to define the two- and three-body interaction potentials. T h e study of small metal clusters and their chemistry is an active area of experimental research (10). Gas phase experiments have shown ( Π ) a very large variation i n reactivity w i t h cluster size, but have been unable to determine the geometry of the cluster, or the adsorption site if the clusters have been reacted w i t h other molecules. Experiments on supported clusters have determined (12) the average metal-metal b o n d lengths, but only for a d i s t r i b u t i o n of clusters, a n d the effect of the support is u n k n o w n . T h e reactivity and metal-metal bond lengths are often considerably different from the well-studied perfect crystal faces. T h e o r e t i c a l calculations on metal clusters are therefore important for determining o p t i m a l geometries, and to explain the large changes i n reactivity w i t h cluster size. T h e cluster model is also useful for studying the chemistry of perfect crystal faces. W i t h current super computers, it has become possible to model N125X and N125X5 clusters (where X = 0 , F , S, CI), and thereby study changes in bonding w i t h coverage. These calculations (13) explain the experimental observation that oxygen shows a large shift i n v i b r a t i o n a l frequency w i t h coverage, while sulfur does not. T h e theoretical study of perfect crystals, as well as coverage dependence where experimental data is available for comparison, also helps to delineate the accuracy of the small metal cluster work. 3

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER

18

RESEARCH

Section II describes recent improvements in methodology that have signif­ icantly improved the accuracy of calculations on s m a l l metal clusters. Section III describes the calculation of some accurate dimer and trimer potentials, and the insight they give into the nature of metal chemistry. Section I V reviews the work on s m a l l metal clusters and discusses how the ab initio and parameterized model approaches are interfaced. Section V contains our conclusions.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

M e t h o d o l o g i c a l Advances In this section we give a brief overview of recent methodological advances that have significantly improved our capabilities for accurate calculations on molecules containing transition metals as well as on small clusters. Accurate results for transition metals require both large one-particle basis sets including high angular m o m e n t u m functions and a careful treatment of the correlation (or η - p a r t i c l e ) problem. Recently we have carried out (14-20) full configurationinteraction ( F C I ) calculations on molecules to assess the accuracy of correlation methods that truncate the η - p a r t i c l e expansion. T h e most important result from the F C I benchmark calculations is that a carefully designed complete-activespace self-consistent field ( C A S S C F ) calculation to optimize the orbitals, fol­ lowed by a multi-reference singles plus doubles configuration-interaction ( M R C I ) calculation from the important configurations in the C A S S C F wave function, gives consistently the best agreement w i t h the F C I . Hence this is the method of choice when the resulting configuration expansion is of manageable size (i.e. less than about 1 m i l l i o n configurations). O n e important implication of the F C I studies is that if the η - p a r t i c l e prob­ lem is treated at the C A S S C F / M R C I level, the l i m i t i n g factor i n the accuracy of the wave function becomes the one-particle basis. However, a recent develop­ ment by Almlôf and Taylor (21) has greatly increased the size of the gaussian p r i m i t i v e valence and polarization basis sets that can be used i n C I calculations. T h i s is accomplished by using general contractions w i t h coefficients determined from the n a t u r a l orbitals of C I calculations on the atoms. A t o m i c natural orbitals ( A N O s ) define a method of truncating the basis set to equal accuracy in each shell. T h e following prescription has led to extremely accurate results for excitation energies and dissociation energies of diatomic molecules (15,16). In a double-zeta plus polarization A N O basis set we find a C A S S C F / M R C I treatment that reproduces the F C I result for the η - p a r t i c l e problem, and this C A S S C F / M R C I treatment is then taken to near the one-particle l i m i t . A s shown later, this approach (17) gives a definitive prediction for the ground state of AI2, even though the lowest two triplet states are separated by less than 200 c m " . A t present, we are o p t i m i z i n g (22) A N O contractions for the first-row transition metals that are based on the average of the 3 d 4 s and 3 d 4 s occupations to satisfy the extensive basis set requirements for an accurate description of t r a n s i t i o n metal diatomics. A l t h o u g h a properly designed C A S S C F / M R C I treatment i n a large A N O basis set is expected to give quantitative results for molecular systems includ­ ing transition metals, it can be computationally very intensive. Indeed, this approach quickly becomes intractable for larger clusters, especially when a large number of electrons are correlated. We have, therefore, devoted consid­ erable effort to calibrating single reference-based correlation methods against C A S S C F / M R C I and F C I calculations. W h e n the molecular system is reason­ ably well described by a single reference configuration, we have found that the coupled pair functional ( C P F ) approach (23), a size-consistent reformulation of 1

n

2

n + 1

1

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

2.

BAUSCHLICHER ET AL.

Theoretical

Approaches

to Metal Chemistry

19

S D C I , gives an accurate representation of the molecular state. T h i s is i n con­ trast to single-reference singles-plus-doubles C I ( S D C I ) , w h i c h is often not very satisfactory for transition-metal diatomics, especially when the molecular state arises from a m i x t u r e of atomic states w i t h different d occupations. T h e C P F approach gives quantitative 'e(eV)

265 265

1.206 1.206

277 277

1.231 1.233 1.401 1.386 1.55±0.15

- 1

u

e

(cm ) 325 325

M R C I (4s3p2d) F C I (4s3p2d) M R C I (6s5p3d2f) M R C I + R e l (6s5p3d2f)

ρ

(cm *)

τ-,

%

and Σ

u

(a ) 5.241 5.240

M R C I (4s3p2d) F C I (4s3p2d) M R C I (6s5p3d2f) M R C I + R e l (6s5p3d2f) EXPT

n

3

U

21

1

Tefcm- ) 252 289 128 158

4.710 4.710

344 343

4.660

350

165 174

a

T h e M R C I calculation is a second-order C I from a l l configurations i n the C A S S C F wave function resulting from a l l arrangements of the 3s and 3p elec­ trons i n the 3s a n d 3p orbitals. T h e D a r w i n and mass velocity contributions were included using first-order perturbation theory. H u b e r and Herzberg, Ref. 42. c

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

22

Do of 1.40 e V for AI2 is w i t h i n the error bounds of the experimental value of 1 . 5 5 ± 0 . 1 5 e V determined by Stearns and K o h l (46) using a K n u d s e n cell mass spectrometric m e t h o d and assuming a Σ ~ g r o u n d state.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

3

Since accurate m o d e l l i n g of larger clusters requires the inclusion of a threeb o d y interaction function, we have devoted considerable effort to the under­ standing of t r i a t o m i c systems containing C u , A l and B e . T h i s work is also directed at understanding the nature of b o n d i n g i n alloys and composites. A l ­ though it is not possible to compute the potential energy surface of these t r i atomics to the same accuracy as for the diatomics, a good estimate of the errors in the t r i a t o m i c calculations can be obtained by performing the same level of calculations on the diatomics. L i k e the AI2 molecule, the CU3 molecule is interesting i n its own right, and has been the subject of many experimental and theoretical papers (see for exam­ ple 26-28 a n d references therein). O u r ab initio study of CU3 gives a B ground state corresponding to a C Jahn-Teller distortion away from a E equilateral triangle geometry. T h e B state was found to lie 59 c m " below the A\ state and 280 c m " below the Dzh equilateral geometry, thus confirming the pseu­ dorotation barrier and Jahn-Teller s t a b i l i z a t i o n energy deduced by T r u h l a r and T h o m p s o n (27) from an analysis of the fluorescence spectrum of Rohlfing and Valentini (29). Based on our experience (25) w i t h CU2 where b o t h higher exci­ tations and relativistic effects are important, our ab initio study of CU3 included relativistic effects v i a first-order perturbation theory and correlation effects us­ ing the C P F f o r m a l i s m . However, this level of correlation treatment required some reduction i n the one-particle basis set and yielded errors of 0.056 a i n r , 0.08 e V i n D and 8 c m " in u for C u compared w i t h the accurate experimen­ tal values. T h i s provides a good estimate of the errors in the b o n d lengths and b i n d i n g energy of the CU3 cluster. T h i s level of treatment for CU3 is expected to yield an accurate three-body interaction t e r m for use in m o d e l l i n g C u clusters. 2

2

2

f

2v

2

1

2

2

1

0

e

1

e

e

2

T h e existence of two nearly degenerate triplet states w i t h substantially different r values i n A l manifests itself in the A l (9) and C u A l (47) triatomics in terms of low-lying states w i t h considerably different geometries. For example, AI3 has three nearly degenerate states; the A a n d B\ states, which are two Jahn-Teller components of a E state, a n d the A\ state. Experiments yield conflicting d a t a as to the ground state. M a t r i x isolation E S R (48) shows a quartet state w i t h equivalent A l atoms (either an equilateral or pseudorotating triangle), while magnetic deflection experiments have been interpreted (49) as showing a doublet g r o u n d state. L i k e the n and Σ ~ states of AI2, the three states of AI3 have different geometries, w i t h a 0.39 a v a r i a t i o n i n bond length and 15° v a r i a t i o n i n b o n d angle. Since one expects the b o n d i n g i n small clusters to arise from a m i x t u r e of these low-lying states i n AI2 and AI3, we have averaged the results for the low-lying states for evaluation of the two- a n d three-body parameters. In the next section we describe our ab initio a n d parameterized model results for larger A l clusters. O u r theoretical results (47) for the Cu Be systems are summarized i n Table II. For C u B e we find two linear structures, C u - B e - B e where the bonding is very directional owing to the formation of s-p hybrids, and B e - C u - B e where the b o n d i n g is m u c h more delocalized. C u B e also has two low-lying linear structures, one of w h i c h contains delocalized metal b o n d i n g . In b o t h isomers of the C u B e and C u B e 2 linear structures, the very directional b o n d i n g implies a repulsive three-body c o n t r i b u t i o n . A s we show later, this large three-body force explains the apparently strange behavior of B e on a C u ( l l l ) surface (50). A s in the case of C u a n d C u , the ab initio calculations give more t h a n input into e

2

3

2

4

A

2

4

f

2

3

3

u

0

x

y

2

2

2

2

3

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

2.

BAUSCHLICHER ET AL.

Theoretical Approaches to Metal Chemistry

23

Table II. Spectroscopic constants for selected C u ^ B e ^ systems SDCI

CPF

2

r (a ) D (eV) e

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

CuBe Σ + (8s6p4d/4s3p) basis 4.098 0.55

0

e

4.090 0.68

(9s7p4d3flg/6s3p3dlf) basis r (a ) D (eV) e

4.022 0.77

0

e

3.999 0.92

C u 2 B e linear s y m m e t r i c * Σ + r (a ) atomization (eV) D (CuBe-Cu)(eV) e

0

4.089

4.017 2.87 2.18

e

linear asymmetric * Σ r (Cu-Cu)(a ) r (Cu-Be)(a ) T (eV) " ' c

a

0

e

0

e

+

4.462 4.402 ...

4.415 4.206 0.62

B e 2 C u linear asymmetric Σ 2

r (Be-Be)(a ) r (Cu-Be)(a ) atomization energy (eV) D (Be-BeCu)(eV) e

0

e

0

4.119 4.055

+

4.169 4.011 1.23 0.56

e

2

linear s y m m e t r i c Σ + r (a ) Te(eV) e

0

4.353 ...

4.312 0.15

I n this basis set the spectroscopic parameters for Cu2 are: r =4.148 a and D =1.77eV. a

c

e

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

0

24

SUPERCOMPUTER

RESEARCH

the m o d e l i n g approach, they yield insight into the chemistry. T h i s is especially i m p o r t a n t when the complexity of the systems precludes performing calculations on a l l s m a l l systems of interest. B e Clusters C o m p u t a t i o n a l l y the study of s m a l l B e clusters is straightforward, since struc­ tures are qualitatively correct using s m a l l basis sets and neglecting the effects of electron correlation. For example, at the S C F level using only a D Z basis, the B e - B e b o n d length in B e (of 3.97 a ) is just 0.05 a longer t h a n at the S D C I level using a much larger triple-zeta basis w i t h two sets of polarization func­ tions ( T Z 2 P ) (6,51). However, the atomization energy per a t o m ( D / a t o m ) is significantly larger (0.70 eV) at the T Z 2 P - S D C I level than at the D Z - S C F level (0.39 e V ) . Therefore, questions of structure can be answered at the S C F level, while correlation must be included to accurately compute the cohesive energy of the cluster. 4

0

0

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

e

A n i m p o r t a n t question in materials science is how large a cluster must be before its structure and chemistry is the same as that of the b u l k . The smallest strongly b o u n d B e cluster is B e , which has tetrahedral geometry. A tetrahedron can be considered the b u i l d i n g block for both the fee and hep stuctures, the latter being the structure of Be metal (52). A central a t o m i n either the hep or fee structure is surrounded by 12 nearest neighbors, w i t h trigonal symmetry for hep and octahedral symmetry for fee. S C F calculations on B e i 3 clusters, w i t h the constraint that all B e - B e b o n d lengths are equal, yield a D / a t o m that is more t h a n twice as large as B e ; 0.87 e V (hep) and 0.91 e V (fee) (6). However, the fee structure is more stable than the hep structure for the 13 atom B e cluster, whereas the bulk structure is hep. W h i l e all of the bond lengths are equal in the b u l k fee structure, there is always some distortion in the bulk hep structure. If the constraint of equivalent b o n d lengths is eliminated (except that the clusters are still required to have trigonal s y m m e t r y - Osh or D^d), both clusters show modest distortions (up to 0.27 a ). T h e fee structure is stabilized by an a d d i t i o n a l 0.24 e V due to d i s t o r t i o n . Therefore, not only is the lowest energy B e i structure different from the bulk, the 13 atom fee structure differs from the fee bulk structure by undergoing significant distortion. 4

e

4

Q

3

T h e a d d i t i o n of nearest neighbors to the twelve surface B e atoms of Β β χ results i n a 55 a t o m cluster. A t the S C F level, using a slightly smaller basis set t h a n used i n our best treatment of B e 13, the fee structure of Bess is also observed to be more stable t h a n the hep structure (7). However, the relative stability between the two structures decreases to 0.03 e V per atom (favoring fee) in Bess compared to 0.10 e V for B e i 3 . T h e D / a t o m for Be55 is significantly larger than that for B e i 3 (1.33 vs. 0.86 e V / a t o m ) , w h i c h is i n t u r n about twice that found for B e , but it is still significantly less than the bulk value of 3.38 e V / a t o m . A l t h o u g h part of this difference arises from neglect of electron correlation and basis set l i m i t a t i o n s , scaling the Bess b i n d i n g energy by 1.80 (the increase in D between the equivalent and best B e calculations) does not fully account for differences w i t h the b u l k . Note, however, that the D / b o n d increases only slightly (0.07 e V ) between B e and B e , and by even less (0.03 eV) between B e i 3 and B e . If the factor of 1.8 for basis set and correlation errors is applied to the 0.34 e V / b o n d for B e , the resulting value of 0.59 e V is i n good agreement w i t h the 0.56 e V / b o n d deduced from bulk data. T h u s , the D per bond is converging quite quickly w i t h cluster size. T h e structure, however, is probably more influenced by the number of bonds per atom, which is 3.93 for B e 5 , compared to 6 for the bulk. Hence the structure of clusters can be quite different 3

e

4

e

4

e

4

1 3

5 5

5 5

e

5

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

2.

BAUSCHLICHER ET AL.

Theoretical Approaches to Metal Chemistry

25

from the b u l k , a n d rather large clusters, p r o b a b l y between 100 a n d 300 atoms, are required before the bulk structure is o p t i m a l . A l Clusters A l t h o u g h A l is less important t h a n transition metals as a catalyst, it is a simple metal for which some experimental data is available for comparison w i t h theory (48.49.53). U n l i k e B e clusters, A l clusters are not adequately described at the S C F level i n a s m a l l one-particle basis set. Theoretical b o n d lengths (6,8,9) given i n Table III for the A l a n d A l i clusters indicate that the A l - A l distance decreases w i t h b o t h extensions of the one-particle basis a n d w i t h the inclusion of electron correlation. In the larger basis sets the A l - A l b o n d lengths show an increase w i t h increasing cluster size i n analogy w i t h B e clusters. T h e bond lengths i n b o t h cases approach that of the b u l k from below. T h e inclusion of correlation shortens the A l b o n d length by more than i n B e , but by less than the change w i t h basis set improvement. Since electron correlation increases the b i n d i n g energy by a factor of 1.5, it must be included for a quantitative determination of the cohesive energy. We have considered the larger A l - A l 6 clusters using b o t h ab initio calcula­ tions a n d the parameterized m o d e l (9). T h e results for A l a n d AI5, summarized in Table I V , show that the parameterized m o d e l a n d ab initio calculations agree well on the relative energetics i f b o t h the two- a n d three-body interactions are included. For A l it is difficult to treat all the structures at the T Z 2 P - C P F level, but for the structures considered, there is reasonable agreement between the ab initio a n d m o d e l results. W h e n the parameters deduced from the calculations o n AI2 a n d AI3 are applied to b u l k A l , the cohesive energy is too s m a l l a n d the b o n d length is too large. T h e s m a l l cohesive energy is expected because our c o m p u t e d AI2 D at the T Z 2 P - C P F level is only 7 1 % of the experimental value (42,46). T h e bulk values are i n much better agreement w i t h experiment if the model is parame­ terized using the experimental D a n d r values for the Σ ~ state. Hence, the

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

4

3

4

4

4

4

6

c

3

e

e

Table III. B o n d lengths for metal clusters of A l a n d B e Cluster/method

r (ûo) e

Be DZ S C F B e large basis set C I Beisifcc) D Z S C F Bei (hcp)DZ S C F B u l k * (hep)

3.97 3.92 4.06 4.11 4.26

Al Al Al

5.30 5.10 5.02 5.44 5.44 5.26 5.26 5.41

4

4

3

D Z - S C F rhombus large basis S C F large basis C I

4

4

4

A l i ( f c c ) large basis set S C F bulk (fcc) 3

a

a

R e f . 52 —for B e the average of the two values i n the b u l k , 4.32 and 4.21 is given.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER

26

RESEARCH

Table I V . C o m p a r i s o n of stability and structure of A l clusters between ab initio and parameterized interaction results w i t h two- and three-body terms (2+3-b) as well as using only the two-body (2-b) interaction. B i n d i n g energies ( D i n eV) per a t o m , and b o n d distances ( r in ao) are given n

e

e

D /atom

r

e

Structure Al Rhombus

e

A b initio

2+3-b

2-b

A b initio

2+3-b

2 ^

1.08

1.13

1.48

5.04

5.10

4.97

1.32 1.27 1.18 1.15 1.05

1.27 1.24 1.23 1.20 1.24

1.67 1.96 1.93 1.44 2.11

4.96 5.18 5.19 5.02/4.94 4.74

5.14 5.25 5.26 5.13/5.07 5.32

4.97 4.96 4.97 4.98/4.95 4.98

3.43°

2.75 3.70

4

Al

5

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

C2v ^4v

C

8

V2h

bulk

9.97 13.4

6

5.41

5.75 5.41

a

6

6

4.84 4.25

fc

a

E x p e r i m e n t a l value (52). F o r better comparison, these values have been calculated using the two-body potential calibrated w i t h the experimental A l data (42,46) ( D 1.55 e V and r 4.66 a for the Σ " state). 6

2

e

e

3

0

parameters needed to reproduce the bulk properties appear to be closer to those for the Σ ~ excited state, although this could be a consequence of the form of the potential, which is discussed in more detail below. It is interesting to note that if the experimental AI2 data is used, this method has errors i n the lattice constant and cohesive energy that are of the same magnitude as those found i n density functional methods developed to study solids (54). Since the calculated lowest energy structure of B e is fee, whereas the bulk structure is hep, we have carried out S C F calculations on AI13 using a large basis set to see if its structure is also different from that of the bulk. T h i s is i n fact the case since the nearly degenerate icosahedral and hep structures are b o t h about 1 e V more stable t h a n fee, w h i c h is the bulk structure. In a d d i t i o n , neither the hep structure nor the fee structure is significantly distorted from a l l bonds equal. T h i s is also opposite to the situation in the bulk where the hep structure undergoes distortion. A p p l i c a t i o n of the parameterized model (with parameters based u p o n A l and A l ) leads to a planar AI13 being about 1 e V more stable t h a n hep, fee or icosahedral, whereas this structure is 2.6 e V above the most stable structure at the ab initio level. A t present, our modelling approach uses a Lennard-Jones potential for the two-body t e r m 3

1 3

2

3

«(Γ,, ) Γ >

=

ί

( ( ^ )

1

2

- 2 ( ^ )

6

)

(1)

where ro is the e q u i l i b r i u m b o n d distance of the dimer, r y is the distance between atoms i and j and ε is the energy at r = r o . For the three-body interaction we considered the A x i l r o d - T e l l e r form: t

t J

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

2.

BAUSCHLICHER E T AL.

Approaches

to Metal Chemistry

27

where r , ry*, r^ and e*i, a , c*3 represent the sides and angles, respectively, of the triangle formed by the three particles i , j and k. T h e intensity of the three-body interaction is given by the parameter Z , w h i c h is specific to each combination of different species in the tri-atom interaction. T h e poor results for AI13 using this model may be either due to our choice of potential functions w i t h simple analytical forms or to the neglect of four-body and higher-order interac­ tions. It might be possible to avoid these higher-order terms by having effective two- and three-body parameters that vary smoothly w i t h cluster size. However, much of the problem may be that the Lennard-Jones potential rises more steeply than the ab initio potential, so that the two-body term underestimates the twob o d y contributions of second nearest neighbors. T h e three-body function is probably incapable of accounting for both the limitations of the two-body t e r m and three-body effects. Alternate forms for the potential are presently under investigation. In addition, we have developed a modified model potential, where the threeb o d y interaction has been reduced slightly to give the ab initio ordering of struc­ tures for A I 1 3 , i.e. the Ζ value in E q u a t i o n 2 is fitted to the 13-atom ab initio results. T h i s model potential was then used to study (55) the midsized clus­ ters A l 7 - A l i 5 . T h e o p t i m a l structures are summarized in Table V and shown graphically in Figure 1. T h e b i n d i n g energy per atom increases monotonically w i t h increasing cluster size, but the energy required to remove the an atom varies w i t h a pattern reminiscent of that observed (56) in mass spectroscopic experiments for the ionization potential and abundance. Figure 1 demonstrates that changes in cluster geometry w i t h increasing cluster size can be quite dra­ m a t i c . For example, the A I 9 and A l n clusters are three-dimensional while A l i o is planar. These studies must be considered qualitative because of unresolved questions about the potential, but such geometrical variations could explain the large changes in cluster properties w i t h size. t J

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

Theoretical

r

2

M i x e d Clusters of Be and C u Since the relative simplicity of C u - s i m p l e metal systems make them ideal for s t u d y i n g mixed component systems, S C F and model calculations have been car­ ried out (50) for selected C u ^ - B e x systems. T h e two- and three-body functions in the model are taken from ab initio calculations (26) on C u and CU3, and the 2

Table V . T h e b i n d i n g energy for A I 7 to AI15 based on the modified three-body potential cluster

D /atom

A I 7 planar A l g planar A I 9 3-dimensional Al planar A l n 3-dimensional AI12 planar AI13 3-dimensional Al 3-dimensional A l l 5 3-dimensional 1 0

1 4

a

2.05 1.82 1.94 2.17 2.03 2.20 2.14 2.15 2.26

1.38 1.44 1.49 1.56 1.60 1.65 1.69 1.72 1.76

T h e D has been normalized to that of A l . T h e energy required to remove one A l atom. e

b

a

e

2

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

28

cross terms are determined from calculations (47) on C u B e and C u 2 B e . We have not used B e and Be3 to define the B e - B e interaction terms, since these systems are weakly b o u n d and not indicative of larger B e clusters. Therefore we have used the parameterized model only for C u B e . T h e C u cluster was previously designed (57,58) to m o d e l chemisorption at a three-fold hollow site on the (111) surface. For this reason the three C u atoms at the adsorption site are treated using an all-electron treatment, while all of their nearest neighbors i n the first layer and the six atoms i n the second layer are modelled using an effective core potential ( E C P ) that explicitly treats only the 4s electrons. T h e geometry of the bare C u cluster was taken from bulk d a t a (52), which along w i t h the details of the C u basis set are described i n earlier studies of Ο a n d NH3 chemisorption (57,58). T h e B e a t o m was described using a (9s4p) /(4s2p) gaussian basis set (59). T h e chemisorption of one B e a t o m into the three-fold hollow is found to be repulsive at the S C F level. T h e inclusion of correlation could lead to a bound system, but it is unlikely that Be w i l l be strongly b o u n d i n the three-fold hollow of C u i g . A n a l y s i s of the parameterized m o d e l results shows that the repulsive three-body interaction overcomes the attractive two-body interaction at these geometries. T h e same three-body forces lead to the directional b o n d i n g noted for C u B e and C u B e . However, the parameterized model does predict that B e is b o u n d for the on-top site. 2

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch002

1 8

2

2

T h e interaction of B e w i t h the C u i g cluster was also considered at the S C F level. T h e B e was constrained to have T. It was found that both basis sets gave a good crystal like description o f quartz - from the M u l l i k e n population analyses. However, defect binding energies calculated using the minimal basis were higher than those calculated using the 3-21G basis. Therefore, the 3-21G basis was chosen for use in the remaining calculations. 2

(

(2

BASIS SET SUPERPOSITION E R R O R (BSSE)^>. T o test for this possible source o f error, calculations were performed on the basic S i O - cluster, using the 3-21G basis. The total energy of the cluster was calculated with and without the pres­ ence o f hydrogen atom functions. F r o m the results presented in Table III., it can be seen that the effect o f including the hydrogen functions leads to B S S E o f approximately 0.2 eV. The effect o f including the functions for S i O - cluster i n a calculation on a hydrogen atom can be seen to be about 0.1 eV. However, the inclusion of the point-ion field causes a change i n the total energy of approxi­ mately 10 eV. In this situation B S S E seems relatively unimportant. 1 2

5

l 6

1 2

5

l 6

T H E E F F E C T OF V A R Y I N G T H E V A L U E S OF T H E POINT C H A R G E S . F r o m the beginning of this investigation, calculations were carried out in two different point-ion environments. One was the fully-ionic cluster, ( S i , O ) - analogous to the classical simulation o f quartz possible with the C A S C A D E p r o g r a m ^ * . The other was the half-ionic cluster, chosen for two reasons. First, although there is still some debate over this ^>, alpha-quartz is thought to be approximately 50% ionic. Secondly, an iterative procedure, starting with the fully-ionic cluster, was carried out in which the charges calculated from the M u l l i k e n populations of the internal, S C F - M O treated cluster, were returned to the point-ions. When repeated until the calculated populations were consistent with the charges on the point-ions, this yielded values o f 2 + for the silicons and 1- for the oxygens. 4 +

2

(

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

5. S I M E T A L .

Ab Initio SCF-MO

75

Calculations

Table III. Results of the calculations to test BSSE

Cluster

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch005

H atom H atom H atom

3-21G 3-21G 3-21G plus H functions 3-21G 3-21G 3-21G plus S i 0 i functions 5

Final Total Energy (Hartrees)

Point-ions Included

Basis Set

no yes yes

-2621.37939 -3021.97588 -3021.98369

no yes yes

-0.49620 -388.92690 -388.93034

6

Later calculations showed that the defect binding energies were invariant to the values chosen for the point charges. A s those calculated for the fully-ionic system my be directly compared to those obtained using classical simulation, geometry optimizations were carried out using the fully-ionic point-ions. APPLICATION OF THE METHOD R A D I A T I O N I N D U C E D D E F E C T S I N A L P H A - Q U A R T Z . Starting geometry: The atoms and the point-ions were initially placed at the lattice sites corresponding to the geometry o f the right-handed alpha-quartz (space group P3 21). Each silicon was surrounded by four chemically equivalent oxygen atoms, forming a slightly distorted tetrahedron. These oxygens were at two pairs o f distances from the silicon, 1.616Â and 1.598Â; each oxygen has a bond o f each bond-length to a silicon atom. However, when an aluminum impurity is introduced at the silicon site, the equivalence is broken, the aluminum has two distinct types of oxygen around it. Following Mombourquette et al.® we shall use the symbol 0 ( < ) an 0 ( > ) , to indicate oxygen atoms located initially 1.598Â and 1.616Â from the central silicon site, respectively. In these calculations, 0 ( < ) corresponds to 0(2) and 0(4), and 0 ( > ) corresponds to O ( l ) and 0(3) on Figure 1. 12

Geometry optimized basic alpha-quartz structure Si O -: The central atoms of S i s 0 - cluster moved very little from their X-ray determined positions - S i ( l ) moved 0.016À; both 0 ( < ) moved 0.018Â and both 0 ( > ) moved 0.005Â. However, the relative length of the two pairs o f oxygen bonds changed. The longer one became 1.621Â, and the shorter one became 1.616Â. If the accuracy of the bond length calculated by this method is taken to be approximately 0.01À, then all Si-O bonds within the inner S i 0 cluster can be taken to be 1.62Â. This was exactly the situation calculated by Mombourquette et al.®, however, they calculated larger displacements from the X-ray determined structure and much longer bond lengths. This was probably caused by their use of a minimal basis set. 5

l6

12

16

5

4

u

Geometry optimized basic cluster minus one electron : Si O '. This defect contains an electron 'hole,' which was created by removing an electron from the calculation. This was then treated with the U H F formalism to see where it localized. It was found that the bond lengths from O ( l ) to its nearest silicon atoms increased significantly, to 1.825Â and 1.772Â, (see Figure 2.). This is about a 13% increase in bond length. The same calculations performed by classical simulation, using the program C A S C A D E , produced a 2 1 % increase in bond length (Sim, F . ; P h . D Thesis, University of L o n d o n , to be published.). Some s

l6

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch005

76

SUPERCOMPUTER RESEARCH

Figure 1. Geometry of the basic S i 0 5

12 16

- cluster.

Figure 2. Geometry optimized basic cluster with an electron hole : S i 0 5

n 1 6

-

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

5. S I M E T A L .

Ab Initio SCF-MO

77

Calculations

rearrangement of the orientation o f the inner four oxygen atoms also took place, with the angles between them either increasing or decreasing by about 7°. 13

Geometry optimized basic cluster with an aluminum substitutional: Si AlO -: This center contains an A l ion substituting for an isoelectronic S i ; this account for the formal charge of 13- given to the cluster. The introduction of the A l ion caused the bond pair order to return to its original state. However, again the difference in calculated bond length was very small, 0.08Â. A small lengthening of the bonds from the central site, to the nearest four oxygen atoms was observed, about 3%. This was accompanied by a very little alteration in the bond angles around the site - less than 4°. This is consistent with the fact that this substitution occurs so readily in quartz. 4

3 +

i6

4 +

3 +

12

Geometry optimized Al substitutional minus one electron : Si AlO -: This center consists of an electron hole, located on an adjacent oxygen atom, charge compensating for the A l ion. This hole is localized on one of the 0 ( > ) sites in the ground state, or, in a thermally accessible excited state, on one o f the 0 ( < ) sites*!^. In this calculation, the electron hole has localized on 0(4), originally a short bond oxygen. The bonds to A l and Si from this atom have increased to 1.864Â and 1.719Â respectively (see Figure 3.). This represents a 10% increase in the A l - O bond length, which is 15% shorter than that calculated using a classical simulation (Sim, F . ; P h . D Thesis, University o f L o n d o n , to be published.). In the optimized geometry there seem to be no discernible pairs of bonds (see Figure 3.). The A l - O distances are now: O ( l ) 1.662Â, 0(2) 1.654Â, 0(3) 1.638Â, 0(4) 1.864Â. This change has been accompanied by a large amount of rearrangement around 0(4).

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch005

4

]6

3 +

DISCUSSION O F RESULTS. F r o m the final total energies of the optimized configurations of these four defects, it is possible to calculate a defect binding energy, for the hole to the A l , of 5.36 eV. This is done in the following way: Defect binding energy = -

[ E ( S i AlOJ^") - E ( S i A l O ^ " ) ] 4

[E(Si Ol^-) 5

4

-E(Si OJh 5

]

This binding energy is higher than would be expected for a defect of this type. The value calculated using the classical simulation was 1.86 eV, which is much more realistic (Sim, F . ; P h . D Thesis, University of L o n d o n , to be published.). Furthermore, geometry optimization only produced a lowering of the defect binding energy of 1.0 eV. This is slightly puzzling since this method gives excellent agreement with E P R results which predict 10% and 12% relaxation*!^* around the A l substitutional, on the formation of an electron hole. Stapelbroek et al.^u concluded from calculations based on classical dipole-dipole interactions alone, that the relaxation would be about 40%. The calculations performed using C A S C A D E included polarization, which the former method did not, and this may explain why a lower estimate of the relaxation, 25%, resulted. However, even though polarization was included, the use of the fully-ionic model, in the classical simulation, might have led to an exaggeration of this effect, which caused by essentially electrostatic interactions. Mombourquette et al. calculated relaxation of 17% around the A l , this over-estimate may have been caused by the small size of the cluster that they used, or their use of an S T O - 3 G basis set. O r indeed it may have been caused by their failure to include long-range electrostatic effects, which would have had a restraining effect on the atoms which have relaxed. Our estimate of 10% relaxation of the A l - O bonds, caused by the electron hole is exactly in agreement with the 10% relaxation predicted by Adrian et al. ^>, (3

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch005

SUPERCOMPUTER RESEARCH

Figure 3. Geometry optimized aluminum substituted cluster with an electron hole :Si A10 »24

I6

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

5. S I M E T A L .

Ab Initio SCF-MO

79

Calculations

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch005

from E P R data. This confirms that the previously predicted relaxation o f around 40% was much too large. Thus to obtain results in quantitative agreement with experiment, it is necessary to use large S C F - M O clusters, a good basis set and to include long-range interactions present in a crystal. F U R T H E R APPLICATIONS. This method represent a simple and inexpensive way of including long-range effects in quantum mechanical calculations on the solidstate, or any extended system. Indeed, because it is relatively inexpensive, it is possible to use this method to develop potentials, for use in classical calculations on crystals. Previous attempts at using Hartree-Fock methods to evaluate poten­ tials, have been restricted to evaluating potentials for individual pairs o f atoms usually with no inclusion o f long-range Coulomb interactions. However, this method can be used to calculate the total energy for different configurations of the entire S C F - M O treated cluster, and these resultant energies parametrized to give two- and three-body potentials for use in classical calculations. W o r k is cur­ rently underway to produce Si-Si, 0 - 0 and S i - 0 potentials for use in the program C A S C A D E , from over 150 different configurations of the S i 0 " cluster mentioned in this paper. The advantage o f this approach is that manybody effects are included explicitly. The limiting factor in the potential's accu­ racy then becomes the method of parametrizing the energies for the classical model, and this may depend on the classical method itself. Several parameters sets may be obtained from the one set of data, for example, ones for use in both static and dynamic simulations. In this way, the S C F - M O method may be extended to use on materials, which would previously have been considered com­ pletely out with the limitations o f present day computers. 1 2

5

I 6

ACKNOWLEDGMENTS F.S. would like to thank the Department o f Chemistry, U C L and I B M Corpo­ ration for financial support, and D r . V . R . Saunders for many helpful sug­ gestions.

LITERATURE CITED 1. Gibbs, G. V., Am. Mineral., 1982, 67, 421. 2. Hagon, J. P.; Stoneham, A. M.; Jaros, M., UKAEA Report, 1986, AERE-TP.1175. 3. Jorgensen, J. D.,J.Appl.Phys. 1978, 49, 5473. 4. Saul, P.; Catlow, C. R. Α.; Kendrick, J. PHIL. MAG. 1985, Β 51, 107. 5. Vail, J. M.; Harker, A. H.; Harding, J. H.; Saul, P. J.Phys.C Solid State Phys., 1984,17,3401. 6. Weil, J. A. Radiation Effects, 1975, 26, 261; 1984, 62, 21. 7. Mombourquette, J. M.; Weil, J. A. Can. J.Phys.1985, 63, 1282. 8. Ewald, P. Ann.Phys.(Leipzig) 1921, 64, 253. 9. Dupuis, M.; Rys, J.; King, H. F. J. Chem.Phys.1976, 65, 111. 10. Dupuis, M.; Spangler, D.; Wendoloski, J. J. NRCC Software Catalogue. 1980, Vol.1, Prog. No. QC01 (Manchester National Regional Computing Center). 11. Guest, M. F.; Kendrick, J. GAMESS user manual, 1985,DaresburyTechnical Memorandum. 12. Norgett, M. J. UKAEA Report, 1974, AERE-R. 7650. 13. Clementi, E.; Chin, S.; Logan, D. in Supercomputer Simulations in Chemistry, 1986, Vol. 44 of Lecture Notes in Chemistry, Dupuis, M., Ed., SpringerVerlag (Berlin). 14. Nuttall, R. H. D.; Weil, J. A. Can. J.Phys.1981, 59, 1696. 15. Dunning, Jr., T. H. J. Chem.Phys.1970, 53, 2823.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch005

80

16. Dunning, Jr., T. H.; Hay, P. J. in Methods of Electronic Structure Theory, Schaefer, III, H. F., Ed., Plenum (New York) 1977. 17. Veillard, A. Theor. Chim. Acta 1968, 12, 405. 18. Wachters, A. J. H. J. Chem. Phys. 1970, 52, 1033. 19. Colburn, Ε. Α.; Kendrick, J. in Computer Simulations of Solids. 1982, Vol. 166 of Lecture Notes in Physics, Catlow, C. R. Α.; MacKrodt, W. C., Eds., Springer-Verlag (Berlin). 20. Pisani, C.; Dovesi, R. Int. J. Quantum Chem. 1980, 17, 501. 21. Dovesi, R.; Pisani, C.; Roetti, C.; Saunders, V. R.Phys.Rev. 1983, B 28, 5781. 22. Hehre, W. J.; Stewart, R. F.; Pople, J. A. J. Chem.Phys.1969, 51, 2657. 23. Binkley, J. S.; Pople, J. Α.; Hehre, W. J. J. Am. Chem. Soc. 1980, 102, 939. 24. Davidson, E. R.; Feller, D. Chem. Rev. 1986, 86, 681. 25. Smith, W.DaresburyLaboratory Report, DL/SCI/TM25T. 26. Leslie, M. Daresbury Laboratory Report, DL/SCI/TM31T. 27. Pauling, L. Am. Mineral 1980, 65, 321. 28. Stewart, R. F.; Whitehead, M. A. Am. Mineral. 1980, 65, 324. 29. Schnadt, R.; Schneider, J.Phys.Kondens. Mater. 1970,11,19. 30. Adrian, F. J.; Jette, A. N.; Spaeth, J. M.Phys.Rev. 1985, B 31, 3923. 31. Stapelbroek, M.; Bartram, R. H.; Gilliam, O. R.; Madacsi, D. P.Phys.Rev. 1976, B13,1960. RECEIVED

June 15, 1987

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 6

Using Computer Simulations To Probe the Structure and Dynamics of Biopolymers Ronald M. Levy, Fumio Hirata, Kwang Kim, and Peisen Zhang

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch006

Department of Chemistry, Rutgers University, New Brunswick, NJ 08903 The use of computer simulations to study internal motions and thermodynamic properties is receiving increased attention. One important use of the method is to provide a more fundamental understanding of the molecular information contained in various kinds of experiments on these complex systems. In the first part of this paper we review recent work in our laboratory concerned with the use of computer simulations for the interpretation of experimental probes of molecular structure and dynamics of proteins and nucleic acids. The interplay between computer simulations and three experimental techniques is emphasized: (1) nuclear magnetic resonance relaxation spectroscopy, (2) refinement of macromolecular x-ray structures, and (3) vibrational spectroscopy. The treatment of solvent effects in biopolymer simulations is a difficult problem. It is not possible to study systematically the effect of solvent conditions, e.g. added salt concentration, on biopolymer properties by means of simulations alone. In the last part of the paper we review a more analytical approach we have developed to study polyelectrolyte properties of solvated biopolymers. The results are compared with computer simulations.

T h e use of computer simulations to study the internal motions and thermodynamic properties of biological macromolecules is receiving increased attention as it becomes possible to simulate biologically i m p o r t a n t processes, e.g. the b i n d ing of a ligand to an e n z y m e . ' In the molecular dynamics computer s i m u l a t i o n m e t h o d , an e m p i r i c a l potential energy function is used to represent the energy of the system as a function of atomic coordinates, and the classical equations of m o t i o n for the macromolecule are solved on this potential surface. T h i s approach has its roots i n c o m p u t a t i o n a l studies of the liquid state where molecular dynamics simulations have proven to be a very powerful tool for studying l i q u i d properties. M a n y factors, however, distinguish molecular dynamics simulations of biopolymers (e.g. proteins and nucleic acids) from liquid state simulations so that it is difficult to use experience gained from molecular dynamics simulations of liquids to estimate the precision inherent i n macromolecular simulations. W h i l e b o t h l i q u i d state a n d macromolecular simulations employ e m p i r i c a l po1

2

0097-6156/87/0353-0082$07.00/0 © 1987 American Chemical Society

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch006

β. L E V Y E T A L .

Probing the Structure

and Dynamics

of Biopolymers

83

tential functions, the molecular mechanics potentials used to describe proteins a n d nucleic acids have far more adjustable parameters i n the potential than is the case for liquids. Furthermore, for l i q u i d simulations the basic system con­ tains at least 100 identical molecules so that it is possible to take advantage of considerable statistical averaging in the calculation of quantities for comparison w i t h experiment. F o r protein molecular dynamics simulations in contrast, the c o m p u t a t i o n a l effort required to evaluate the large number of interatomic inter­ actions w i t h i n a single protein molecule limits the simulated system to one or at most a very s m a l l n u m b e r of macromolecules. T h e highly anisotropic nature of the protein interior a n d intrinsic interest i n extracting site-specific information further complicates the computational p r o b l e m . A d d i t i o n a l features of macromolecular simulations that make t h e m differ­ ent from and more difficult than simulations of liquids and solids include the difficulty i n o b t a i n i n g exact results for comparison w i t h trajectory averages, the slow convergence of the macromolecular simulations, the difficulty of incorpo­ r a t i n g the crystal or liquid environment i n a realistic way, and the problem of treating q u a n t u m effects for large systems. A central feature of our research d u r i n g the past few years has been the development of procedures for comparing the results of simulations w i t h a wide variety of experiments. Such studies are necessary if the methodology is to be reliably used to study properties that are only indirectly accessible to experiment. E q u a l l y important, these studies lead to deeper insights i n t o the relationship between experimental measurements a n d u n d e r l y i n g molecular processes. In this paper we briefly review our past work concerning the use of molecular dynamics simulations to construct and interpret N M R relaxation and x-ray refinement models for macromolecules. We then discuss the development of new methods for simulating v i b r a t i o n a l spectra using detailed molecular simulations. M e t h o d s we are w o r k i n g on to incorporate q u a n t u m effects i n molecular simulations are also reviewed. T h e treatment of solvent effects i n biopolymer simulations is a difficult p r o b l e m . W h i l e a few simulations have explicitly included the solvent surround­ i n g s , most have been carried out i n vacuo. W i t h the increasing access to supercomputers it is becoming possible to include solvent explicitly i n the sim­ u l a t i o n . It is still not possible to study systematically the effect of solvent conditions e.g. salt concentration, on biopolymer properties by means of brute force simulations because of the enormous amounts of computer time required. In this regard the development of more analytical methods for s t u d y i n g the interactions of biopolymers w i t h the solvent surroundings not only makes it possible to study a range of environmental parameters not accessible to sim­ ulations but can provide l i m i t i n g results useful for c o m p a r i n g w i t h computer simulations and for focusing the computationally demanding simulations on the most interesting set of environmental variables. We have developed an approach suitable for treating the interactions of a polymeric solute molecule w i t h the solvent surroundings, based on an integral equation m e t h o d . T h e theory can be used to study polyelectrolyte properties i n solution. In the final part of this paper we review the theory and the application to simple models in w h i c h atoms are arranged along a linear chain and on a helix. We use the new theory to consider the effect of added salt on the relative free energy stablizing different forms of D N A . 3 - 7

8

N u c l e a r M a g n e t i c Resonance R e l a x a t i o n Since N M R relaxation i n proteins is determined by dynamics on the picosecond to nanosecond t i m e scale, experimental N M R relaxation parameters c a n provide i m p o r t a n t information concerning picosecond motions. T i m e correlation func-

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER

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RESEARCH

tions required for determining N M R parameters m a y be calculated directly from molecular dynamics trajectories. P r o t e i n trajectories have been used to study m o t i o n a l models used i n the analysis of N M R T i , T 2 , and Nuclear Overhauser Enhancements and as an a i d i n the interpretation of e x p e r i m e n t s . A direct c o m p a r i s o n between the results of a 96 ps molecular dynamics s i m u l a t i o n of pancreatic t r y p s i n i n h i b i t o r ( P T I ) and an experimental C N M R relaxation s t u d y of this protein has been r e p o r t e d . T h i s represented the first detailed c o m p a r i s o n of the results of molecular dynamics simulations of a protein w i t h e x p e r i m e n t a l probes of m o t i o n on a similar time scale. Order parameters, which are measures of the extent of angular motion of the bonds were calculated from a 96-ps trajectory and compared w i t h values extracted from the relaxation data. A l t h o u g h the relative flexibility of the residues studied was reasonably well de­ scribed by the s i m u l a t i o n , the theoretical order parameters were systematically larger t h a n the experimental ones, indicating t h a t there is less m o t i o n a l aver­ aging i n the 96 ps s i m u l a t i o n t h a n detected i n experiment. It was suggested t h a t this behavior occurred because the length of the trajectory was too short to statistically sample all accessible orientations. Recently, we have used a 300 ps molecular dynamics s i m u l a t i o n of m y o g l o b i n to reexamine this q u e s t i o n . F o r C N M R of protonated carbons w i t h fixed b o n d lengths, the c o n t r i b u t i o n of i n t e r n a l protein motions to N M R relaxation is determined by the angular correlation function, C ( t ) : 9 - 1 3

14

1 3

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch006

1 5

11

16

1 3

(J/.l) a=-2

2

1 3

where Y are spherical harmonics a n d (θ,φ) specifies the orientation of a C H b o n d i n a macromolecule fixed frame. Because of the restricted nature of the m o t i o n i n the protein interior, the internal correlation function ( E q . I I . l ) usually does not decay to zero. Instead a plateau value is reached for w h i c h the i n t e r n a l correlation function is equal to the e q u i l i b r i u m orientation d i s t r i b u t i o n o b t a i n e d from the entire r u n : ' a

9 - 1 1

1 7

(JJ.2) a

T h e q u a n t i t y S defined b y Eq.II-2 is the order parameter describing the re­ stricted m o t i o n of the C - H vector; for a r i g i d system S = 1, while for a completely flexible system S = 0. T o illustrate the convergence characteristics of N M R order parameters, in F i g . 1 selected order parameters for leucine residues calculated using the complete 300-ps m y o g l o b i n s i m u l a t i o n are c o m p a r e d w i t h the values calculated using the first 100-ps p o r t i o n of the trajectory. T h e results shown i n the figure are representative of the effects observed for a l l the residues studied. A s ex­ pected order parameters for b o n d vectors (C - C&) closer to the backbone are i n general larger t h a n those (C - C ) further out along the side c h a i n . F o r the C bonds, the order parameters calculated from the different portions of the trajectory agree well; for the leucine m e t h y l axis order parameters however, there are large discrepancies between the values calculated over the first 100-ps 1 3

2

2

a

1

S1

a

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch006

LEVY ET AL.

Probing the Structure and Dynamics

of

Biopolymers

.3 .2 -

20

40

60

80

100

120

140

I.Or .9 r

Residue a

Figure 1. (Top) Order parameters for each of the Leucine C - C ^ bond vectors. , order parameters calculated using the 0-100 picosecond portion of the trajectory; , order parameters calculated using the entire 300 picosecond simulation. (Bottom) Same as top figure except that the calculations are for the leucine 0 -Ο bond vectors. (Reproduced with permission from Ref. 16. Copyright 1985 Rockefeller University Press.) Ί

β

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch006

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SUPERCOMPUTER RESEARCH

of the trajectory as compared w i t h the complete s i m u l a t i o n . For fourteen of the eighteen leucines studied, the order parameters are smaller when the entire trajectory is used to evaluate the correlation functions as compared w i t h the first 100-ps; for seven of the eighteen residues the order parameters decrease by more t h a n 50% when the entire 300-ps d i s t r i b u t i o n of b o n d vector orientations is used to evaluate the order parameters. It is clear from these results that the order parameter calculations have not converged for this 300-ps trajectory. T h e order parameter depends on the longtime behavior of the N M R correlation function w h i c h is determined by b o t h the higher frequency local motions and more extensive conformational changes. We have examined the extent to w h i c h the decay of the N M R angular correlation function at short times varies for dif­ ferent portions of the s i m u l a t i o n . We obtained results consistent w i t h a model of protein m o t i o n i n w h i c h groups of atoms (e.g. leucine side chains) oscillate about a mean conformation for tens of picoseconds and then move r a p i d l y to a new c o n f o r m a t i o n . T h e conformational change has a large effect on the order parameter. In a d d i t i o n , the effective potential i n w h i c h the atoms move are somewhat different i n the different conformations. F r o m our study of N M R order paramters as well as atomic fluctuations i n m y o g l o b i n a qualitative pic­ ture emerged w h i c h suggests that the longer time motions of proteins involves m u l t i p l e m i n i m a . F o r m y o g l o b i n , temperature dependent kinetic experiments on oxygen r e b i n d i n g a n d studies of x-ray temperature f a c t o r s have provided experimental evidence for the existence of m u l t i p l e conformational states of this protein. In the following section we review our use of computer simulations of biopolymers to analyze molecular information concerning macromolecular flex­ i b i l i t y contained i n crystallographic Debye-Waller temperature factors. 16

1 8

19

Restrained X-Ray Refinement of Biopolymers M o l e c u l a r dynamics simulations of proteins and nucleic acids provide a very powerful m e t h o d for testing crystallographic refinement models. T h e sim­ ulations constitute the most detailed theoretical approach available for study­ ing the internal motions of these macromolecules. F r o m the time evolution of the atomic positions, time averaged X - r a y intensities c a n be calculated and treated as d a t a for crystallographic refinement. T h e final structure a n d tem­ perature factors obtained from the refinement can then be compared w i t h the "exact results" obtained directly from the trajectory. In this review, we discuss the results of an a n a l y s i s of the temperature dependent molecular dynamics a n d X - r a y refinement of a Z - D N A hexamer 5 B r d C - d G - 5 B r d C - d G - 5 B r d C - d G for w h i c h the experimental X - r a y d a t a are available and whose crystal structure has been refined to h i g h r e s o l u t i o n . T h i s hexamer crystallizes i n the lefthanded Z-conformation w i t h the cytosine bases i n the a n t i conformation and C i 2 ' } - e n d o sugar puckers, and the guanine bases i n the s y n orientation w i t h C(3')-endo sugar puckers, except at the 3'-terminal guanine bases. T h e phosphodiester conformations are (gauche-plus, gauche-plus) at the C p G phosphates a n d (gauche-minus, trans) at the G p C phosphates. T h e refinement p r o g r a m used was N U C L S Q , w h i c h is the restrained least squares refinement p r o g r a m of H e n d r i c k s o n a n d K o n n e r t adapted to nucleic a c i d s . T h e section is divided i n two parts. F i r s t , the molecular dynamics calculations are described. T h e simulations were carried out at a series of temperatures between 100 Κ and 300 K . Second, the restrained refinements of the time averaged structure factors obtained from the molecular d y n a m i c s simulations at the various temperatures are discussed a n d the results compared w i t h "exact" values calculated directly from the simulations a n d w i t h experimental X - r a y results. D u r i n g the course of the w o r k , it was found that low temperature molecular dynamics simulations 20

21

2 1 , 2 2

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

β.

LEVY ET AL.

Probing the Structure

and Dynamics

of

87

Biopolymers

may be used advantageously i n the refinement process against experimental data. 2 0

M e t h o d s for s i m u l a t i n g restrained x - r a y refinement data from molecular dy­ namics trajectories. M o l e c u l a r d y n a m i c s simulations were carried out o n the 248 a t o m Z - D N A hex­ amer, using the A M B E R nucleic acid force field w i t h a distance dependent dielectric and e x c l u d i n g c o u n t e r i o n s . A l t h o u g h the model treats electrostatic effects only i n a qualitative way, recent molecular dynamics simulations for b o t h p e p t i d e s a n d nucleic a c i d s have demonstrated that for localized conforma­ tions sampled d u r i n g short molecular dynamics simulations, average properties are not very sensitive to the electrostatic model; it is the p a c k i n g and hydro­ gen b o n d i n g terms w h i c h together w i t h the v i b r a t i o n a l potential (bond, b o n d angle a n d torsional stretching) w h i c h dominate the calculated e q u i l i b r i u m and d y n a m i c a l properties. T h e crystal structure of the Z - D N A hexamer was first energy m i n i m i z e d w i t h 200 conjugate gradient steps to relieve any i n i t i a l strain in the structure before the molecular dynamics simulations were begun. T h e rms displacement betweeen the c r y s t a l and energy m i n i m i z e d coordinates are less t h a n 0 . 1 Â . Simulations were performed at a series of temperatures defined by the mean kinetic energy of the system between 100 Κ a n d 300 K . F o r each temperature, 10 trajectories, each 2 ps i n length, were calculated by solving simultaneously the classical equations of m o t i o n for the atoms of the helix. T h e use of m u l t i p l e short trajectories instead of a single long trajectory has been found to be a more efficient m e t h o d for s a m p l i n g conformations for macro­ molecular systems containing m a n y h a r m o n i c degrees of f r e e d o m . C r y s t a l l o g r a p h i c refinement is a procedure w h i c h iteratively improves the agreement between structure factors derived from X - r a y intensities and those derived from a m o d e l structure. F o r macromolecular refinement, the limited diffraction d a t a have to be complemented b y a d d i t i o n a l information i n order to i m p r o v e the parameter-to-observation ratio. T h i s a d d i t i o n a l information consists of restraints on b o n d lengths, b o n d angles, aromatic planes, chiralities, a n d temperature factors. In the restrained refinement procedure a function of the form: 23

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch006

24

25

26

(7//.1)

is m i n i m i z e d . W Q is the weight assigned to the structure factors and it varies linearly w i t h Q w i t h coefficients adjusted so that low resolution structures are weighted more strongly t h a n h i g h resolution ones. F (Q) and F (Q) are respectively the observed and calculated structure factors. T h e second t e r m i n equation 1 contains the stereochemical restraint information, Δ is the deviation of a restrained parameter (bond lengths, b o n d angles, volumes, non bonded contacts, and temperature factors) from its ideal value and W , is the weight assigned to the restraint. T h e form o f equation 1 is such that the weights W , correspond to the inverse of the variance Δ for each set of observations. The structure factor F ( Q ) i n X - r a y crystallography is the fourier transform of the electron density for the molecule: D

c

2

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

88

SUPERCOMPUTER RESEARCH

where ρ(τ) is the electron density at r. In a crystallography experiment the elec­ t r o n density varies w i t h time due to t h e r m a l motion and the observed structure factor a m p l i t u d e is the time average of equation III.2:

F „ ( Q ) = < F ( Q ) > = J dr
S**

(I II.3)

In order to generate a set of calculated structure factors F ( Q ) from a set of coordinates, it is necessary to introduce a model for the time variation of the electron density. T h e usual assumptions in macromolecular crystallography include harmonic isotropic motion of the atoms and i n addition, the molecular scattering factor is expressed as a superposition of atomic scattering factors. W i t h these assumptions the calculated structure factor (equation III.2) is given by:

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch006

C

2 7

Ν F (Q) C

= ^ V ^ ^ - e ^ ^ . F ^ Q )

(777.4)

3=1 where F j ( Q ) is the atomic scattering factor for a t o m j and Tj is the position of a t o m j i n the m o d e l structure. T h e t h e r m a l averaging of atomic m o t i o n is contained i n the atomic Debye-Waller factor exp ( W y ( Q ) . ) W is given b y : 7

2

Wj'(Q) — ~&3'IQI atomic

where Bj is the atomic temperature factor. The mean square (ΔΓ^)

fluctuation

2

for atom j is obtained from the refined temperature

factors t h r o u g h the r e l a t i o n :

42

< (ΔΓ,)

2

>=

^ B

(7/7.5)

3

There are therefore four adjustable parameters per a t o m in the refinement ( x j , yy, Z j , B y ) . In the computer experiments we have carried out to test the assumptions of the nucleic acid refinement model we have generated sets of "observed" structure factors F ( Q ) , from the Z - D N A molecular dynamics trajectories. T h e t h e r m a l averaging i m p l i c i t i n E q u a t i o n III.3 is accomplished by averaging the atomic structure factors obtained from coordinate sets sampled along the molecular dynamics trajectories at each temperature: D

Μ F (Q) 0

where

= < F(Q)

Ν

> = ^ Σ Σ

,

* i(

H

w

r w


500, i f the number of terms i n t h e e x p a n s i o n (3) o f the p o t e n t i a l i s l a r g e , the V c a l c u l a t i o n may i n v o l v e c o n s i d e r a b l e CPU time, e.g., one t o s e v e r a l h o u r s o n a supercomputer. The ΓΓ^ s t e p s take up t o 17 h o u r s o f supercomputer time f o r t h e l a r g e s t s i n g l e r u n ( a 3-energy r u n w i t h Ν = 948) i n v o l v e d i n the p r e s e n t s t u d y . A l l calculations are v e c t o r i z e d , a n d p r o d u c t i o n runs were c a r r i e d o u t o n Cray-1 a n d C y b e r 205 computers. R i g i d R o t a t o r C a l c u l a t i o n s (14.16.23.24) I n the p r e s e n t o v e r v i e w we c o n c e n t r a t e o u r a t t e n t i o n o n c a l c u l a t i o n s w i t h J=jyz^=^^=e=0. J i s a conserved q u a n t i t y and the f i n a l v a l u e s o f t h e o t h e r quantum numbers a r e d e n o t e d j j , €'.

j^,

We w i l l a l s o u s e the n o t a t i o n

(Do n o t c o n f u s e

t h i s s i m p l e sum w i t h the magnitude j j ^ o f t h e

v e c t o r sum.) Because the m o l e c u l e s a r e i d e n t i c a l t h e f i n a l energy s t a t e s a r e l a b e l l e d by a n u n - o r d e r e d p a i r o f r o t a t i o n a l quantum numbers a n d j£ ( 4 0 ) . S i n c e t h e o r d e r i s n o t s i g n i f i c a n t we u s e the c o n v e n t i o n j j < j ^ . F o r g i v e n v a l u e s o f J , j j , a n d j ^ . quantum numbers i' triangle relations.

the

o r j j ^ n^Y take o n a l l v a l u e s a l l o w e d b y t h e S i n c e J = 0 though, j j ^ =

· Summing t h e

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

192

t r a n s i t i o n p r o b a b i l i t i e s over

(

o r

equivalently over i')

yields

values c a l l e d P^ .,, and summing these over a l l p a i r s of j i < J J r

1

1

Δ

2

consistent with a given value of j '

y i e l d s values c a l l e d P?, sum The AD and RBST potentials are defined i n terms of Equation (3) and have f i n i t e M values of 9 and 23, respectively. The BM and RB p o t e n t i a l s must be re-expressed as i n (3) — by using Equation (11) — and i n these cases we converged the dynamics r e s u l t s with respect to increasing M, y i e l d i n g the (unexpectedly large) values of 525 and 825, respectively. The c a l c u l a t i o n s must a l s o be converged with respect to increasing the number of channels N. We d i d t h i s , and the f i n a l calculations involve Ν = 285 for the AD p o t e n t i a l and Ν = 440 for the other p o t e n t i a l s .

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

s u m

J

Table 1 compares P? values for a l l four p o t e n t i a l s a t a sum r e l a t i v e t r a n s l a t i o n a l energy, ^ γ» of 76 meV. We see a great r

ΤΒ

q u a l i t a t i v e difference between the r e s u l t s f o r the AD p o t e n t i a l and the others, with smaller differences between the BM, RB, and RBST p o t e n t i a l s . Evidently the r e s t r i c t i o n M = 9 i s a serious l i m i t a t i o n , g r e a t l y decreasing the r o t a t i o n a l i n e l a s t i c i t y , but the s i m p l i f i c a t i o n s of the angular anisotropy i n the M = 23 p o t e n t i a l are less serious.

Table 1.

sum 0 1 2 3 4 5 6

Rigid-rotator t r a n s i t i o n p r o b a b i l i t i e s f o r four potentials for Ε , = 76 meV AD 0.934 0.004 0.047

BM 0.248 0.025 0.152 0.105 0.228 0.051 0.191

RB 0.031 0.077 0.120 0.030 0.228 0.119 0.394

RBST 0.211 0.055 0.074 0.082 0.151 0.141 0.286

Under some circumstances the r o t a t i o n a l l y anisotropy may be even further s i m p l i f i e d for T-R energy transfer of polar molecules l i k e HF (41). To explore t h i s q u a n t i t a t i v e l y we performed a d d i t i o n a l r i g i d - r o t a t o r c a l c u l a t i o n s i n which we retained only the s p h e r i c a l l y symmetric and dipole-dipole terms of the AD p o t e n t i a l , which y i e l d s M = 3 (see Figures 1, 3, and 4). These c a l c u l a t i o n s converge more r a p i d l y with increasing Ν and usually y i e l d even less r o t a t i o n a l l y i n e l a s t i c scattering. For example Table 2 compares the converged i n e l a s t i c t r a n s i t i o n p r o b a b i l i t i e s

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

11.

SCHWENKE & TRUHLAR Table 2.

Ε

Vibrational β Rotational Energy Transfer 193

I n e l a s t i c Τ -* for f u l l (M = potentials as translational

R transition probabilities 9) and truncated (M = 3) AD functions of r e l a t i v e energy

, (meV) rel ' v

M = 9

M = 3

0.07 0.59 0.90

0.15 0.28 0.82

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

76 657 1550

^sum^ for the two surfaces a t three energies. p o t e n t i a l , for which P

i n e

sum In comparison to the RB

^ t 0 . 9 7 a t a l l three energies, the t o t a l

i n e l a s t i c i t y i s low a t a l l energies studied for both surfaces i n Table 2. C l e a r l y the dipole-dipole term or even a small subset of low-order anisotropic terms do not account for most of the T-R energy transfer a t J = 0 . The dramatic difference between the AD and the other potentials p e r s i s t s up to higher translational energies. For example, PQ for the AD potential

i s 0 . 4 1 4 a t 6 5 7 meV, and thesum

of P£, p j , and P * i s 0.713 (14), whereas, for example, f o r the BM potential

these same quantities are 0.005 and 0.035, respectively, •n and the three largest values of P . , occur f o r j ' = 12-14 (23). J'sum sum —' A more d e t a i l e d comparison of two of the surfaces a t a higher energy, Ε j = 322 meV, i s given i n Table 3. This table compares v

P., for a l l energetically accessible values of j ' sum

at this

energy and also twelve selected value of P*f ., (the transitions 1 2 r

J

J

included are those for which P*f, > 0.035 f o r the BM p o t e n t i a l ) . 1 2 J

J

We see that both Ρ*?. d i s t r i b u t i o n s peak for j ' = 6-9 a t this sum energy with a "shoulder" a t j ' = 4-5. Individual t r a n s i t i o n sum p r o b a b i l i t i e s t y p i c a l l y , but not always, agree within a factor of two and for the most part the same individual f i n a l states have large t r a n s i t i o n p r o b a b i l i t i e s f o r the two potentials. (Three other f i n a l states have larger p r o b a b i l i t i e s f o r the RB potential — i n p a r t i c u l a r , P ^ = 0.055, P^ = 0.061, and P^ = 0.073.) J

4

6

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

194

SUPERCOMPUTER Table 3.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

sum 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

RESEARCH

Rigid-rotator t r a n s i t i o n p r o b a b i l i t i e s f o r two potentials for E ^ ^ = 322 meV

BM 0.055 0.001 0.043 0.020 0.048 0.100 0.173 0.222 0.140 0.125 0.047 0.013 0.011 0.002 0.001

RB

J2

Ji

0.021 0.003 0.026 0.033 0.079 0.061 0.135 0.159 0.259 0.165 0.052 0.007 0.001 0.000 0.000

0 1 3 4 4 3 5 5 6 5 5 6

0 1 2 1 2 3 1 2 1 3 4 3

RB

BM 0.055 0.036 0.054 0.042 0.044 0.039 0.068 0.122 0.065 0.065 0.069 0.038

0.021 0.026 0.037 0.021 0.059 0.026 0.046 0.075 0.016 0.140 0.057 0.078

Another perspective on the comparison of the BM and RB surfaces i s provided i n Figures 12 and 13. This figure shows quantum mechanical values (23.24) of P., for both p o t e n t i a l s and sum J

q u a s i c l a s s i c a l trajectory values (24) of P^ f o r the RB surface sum for two r e l a t i v e t r a n s l a t i o n a l energies. The two sets of quantal r e s u l t s agree with each other much better than either agrees with the c l a s s i c a l simulation. Thus, e s p e c i a l l y when we consider that the approaches used (19.21) to construct the two p o t e n t i a l s were very d i f f e r e n t , we gain some confidence that the dynamically important features of the anisotropy of the p o t e n t i a l are probably reasonably accurate for both a n a l y t i c a l representations. We thus f e e l that i t i s very worthwhile to t r y to converge the V-V and V-V.R energy transfer p r o b a b i l i t i e s f o r the RB p o t e n t i a l or f o r the more convenient RBST modification of the RB p o t e n t i a l . r

Vibrating Rotator Calculations (13-16.22) We consider the V-V.R process (1), 2HF(v=l,j=0)

H F ( v J = 2 , j p + HF(v£=0,j£)

with t o t a l angular momentum J = 0. = 0, and we again sum over j ' or ê'

Therefore we again have j ^ = ^ for a given j '

defined by

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SCHWENKE & TRUHLAR

Vibrational

& Rotational

Energy Transfer

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

11.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

195

SUPERCOMPUTER RESEARCH

196

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

0.257 ι—ι—j—ι

1—ι—ι—ι—τ—ι—ι—ι—I—r

Figure 13. Same as F i g . 12 except for Ε

, = 322 meV.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

11.

SCHWENKE & TRUHLAR

Vibrational

β Rotational

Energy Transfer

197

Equation (12); i n t h i s case the r e s u l t i s c a l l e d P., . Summing ^sum W P., over j ' y i e l d s the t o t a l V-V.R t r a n s i t i o n p r o b a b i l i t y , sum W which i s c a l l e d Ρ Selected r e s u l t s are given i n Table 4, where W the rows l a b e l l e d 0-7 are P., , and the rows l a b e l l e d "sum" are sum W P . The largest Ν f o r which calculations were performed f o r the MAD p o t e n t i a l i s 948, and comparison (15) of these c a l c u l a t i o n s to others with Ν = 694 , 824. and 880 shows that the Ν = 948 c a l c u l a t i o n s are well converged a t E j = 2.455 meV and approximately converged a t 76 meV. The largest Ν f o r which c a l c u l a t i o n s have been performed on the RB p o t e n t i a l i s Ν = 694, which i s not enough f o r quantitative convergence, but which i s adequate f o r a discussion of trends. [For these c a l c u l a t i o n s the V matrix was not calculated as accurately as f o r the c a l c u l a t i o n s with the MAD and RBST potentials, and the sum of Equation (3) was truncated a t M = 161, but these approximations should not matter r e

Table 4.

sum

V-V.R energy transfer p r o b a b i l i t i e s

MAD

Ν = 694 RB

Ν = 948 RBST

MAD

RBST

_ = 2.455 meV rel 0.85 0.10 0.04 0.03 0.00 0.19

0.02 0.17 0.38

0.90 0.04 0.00

0.03 0.24 0.35

3 sum

0.00 0.89

0.42 0.99

0.00 0.94

0.37 0.99

0 1 2 3 4 5 6

rel 0.81 0.05 0.06 0.01 0.00 0.00 0.00

0.20 0.42 = 76 meV 0.00 0.01 0.01 0.01 0.02 0.02 0.02

0.00 0.05 0.02 0.05 0.06 0.06 0.09

0.73 0.03 0.09 0.01 0.00 0.00 0.00

0.00 0.05 0.02 0.05 0.06 0.06 0.09

0.02 0.12

0.12 0.45

0.00 0.87

0.12 0.45

Ε 0 1 2 a

s

7 sum

0.00 0.93

h i g h e s t value allowed by conservation of t o t a l energy at t h i s Ε relτ

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

198

SUPERCOMPUTER RESEARCH

for the present q u a l i t a t i v e discussion.] To f a c i l i t a t e comparison of the r e s u l t s f o r the RB potential to those f o r the other potentials, the l a t t e r are tabulated for both Ν = 694 and Ν = 948. There are two major differences between the r e s u l t s f o r the RB and the MAD p o t e n t i a l . The MAD potential predicts more V-V.R energy transfer, and i t also predicts that the V-V.R process involves very l i t t l e r o t a t i o n a l e x c i t a t i o n whereas the RB p o t e n t i a l predicts a wide j d i s t r i b u t i o n peaking for the higher energetically allowed g

u

m

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

values.

We believe that the j ' d i s t r i b u t i o n for the MAD sum p o t e n t i a l i s l i k e l y to be an a r t i f a c t of the r e s t r i c t e d r o t a t i o n a l anisotropy of the AD p o t e n t i a l . Table 4 also shows that the RBST p o t e n t i a l , despite the r e s t r i c t i o n to M = 23, predicts a broad j ' d i s t r i b u t i o n similar sum to that obtained for the RB p o t e n t i a l , but i t predicts a higher value f o r the t o t a l V-V.R t r a n s i t i o n p r o b a b i l i t y . I t i s not known at t h i s time which potential i s more accurate. Further work i s underway. Conclusion K

The use of supercomputers has allowed us to test the s e n s i t i v i t y of accurate quantal molecular energy transfer p r o b a b i l i t i e s i n diatom-diatom c o l l i s i o n s to the choice of p o t e n t i a l energy surface, even a t t o t a l energies great enough to allow both diatoms to be v i b r a t i o n a l l y excited. Acknow1edgment This work was supported i n part by the National Science Foundation, the Minnesota Supercomputer Institute, and the Control Data Corporation.

Literature Cited 1. Kondratiev, V. N.: Nikitin, Ε. E. Gas Phase Reactions: Springer-Verlag; Berlin, 1981. 2. Yardley, J. T. Introduction to Molecular Energy Transfer; Academic Press: New York, 1980. 3. Secrest, D. In Atom-Molecule Collision Theory; Bernstein, R. B., Ed., Plenum: New York, 1979; p. 377. 4. Slater, J. C. Quantum Theory of Molecules and Solids, Vol. 1; McGraw-Hill. New York, 1963, Appendix 2. 5. Arthurs, A. M.; Dalgarno, A. Proc. Rov. Soc. Lond. Ser. A 1960, 256, 540. 6. Davison, W. D. Discussions Faraday Soc. 1962, 33, 71. 7. Green, S. J. Chem. Phys. 1975, 62, 2271. 8. Bellman, R.; Kalaba. R. Proc. Natl. Acad. Sci. 1956, 42, 629. 9. Bellman, R.; Kalaba. R.; Wing. G. M. J. Math. Phys. 1960, 1, 280. 10. Bellman, R.; Kalaba, R.; Wing, G. M. Proc. Natl. Acad. Sci. 1960, 46, 1646. 11. Light, J. C.; Walker, R. B. J. Chem. Phys. 1976, 65, 4272.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch011

11.

SCHWENKE & TRUHLAR

Vibrational

& Rotational

Energy Transfer

199

12. Truhlar, D. G.; Harvey, N. M.; Onda, K.; Brandt, M. A. In: Algorithms and Computer Codes for Atomic and Molecular Quantum Scattering Theory. Vol. 1; Lawrence Berkeley Laboratory: Berkeley, 1979, p. 220. 13. Schwenke, D. W.; Truhlar, D. G. Theor. Chim. Acta 1986, 69, 175. 14. Schwenke, D. W.; Truhlar, D. G. In: Supercomputer Simulations in Chemistry. Dupuis, Μ., Ed. Springer-Verlag: Berlin, 1986, p. 165. 15. Schwenke, D. W.; Truhlar, D. G. Theor. Chim. Acta, in press. 16. Schwenke, D. W.; Truhlar, D. G. unpublished. 17. Alexander, M. H; DePristo, A. E. J. Chem. Phys. 1976, 65, 5009. 18. Poulsen, L. L . ; Billing, G. D.; Steinfeld, J. I. J. Chem. Phys. 1978, 68, 5121. 19. Brobjer, J. T.; Murrell, J. N. Mol. Phys. 1983, 50, 885. 20. Cournoyer, M. E.; Jorgensen, W. L. Mol. Phys. 1984, 51, 119. 21. Redmon, M. J.; Binkley, J. S., J. Chem. Phys., in press. 22. Schwenke, D. W.; Truhlar, D. G. In: Supercomputer Applications. Numrich, R. W., Ed. Plenum: New York, 1985, p. 295. 23. Schwenke, D. W.; Truhlar, D. G. J. Comp. Chem., in press. 24. Schwenke, D. W.; Truhlar, D. G.; Coltrin, Μ. Ε. J. Chem. Phys., in press. 25. Poulsen, L. L . ; Billing, G. D. Chem. Phys. 1979, 36, 271. 26. Poulsen, L. L . ; Billing, G. D. Chem. Phys. 1980, 46, 287. 27. Poulsen, L. L . ; Billing, G. D. Chem. Phys. 1980, 53, 389. 28. Coltrin, M. E.; Marcus, R. A. J. Chem. Phys. 1980, 73, 2179. 29. Coltrin, M. E.; Koszykowski, M. L.; Marcus, R. A. J. Chem. Phys. 1980. 73. 3643. 30. Coltrin, M. E.; Marcus, R. A. J. Chem. Phys. 1980, 73, 4390. 31. Coltrin, M. E.; Marcus, R. A. J. Chem. Phys. 1982, 76, 2379. 32. Sileo, R. N.; Cool, T. A. J. Chem. Phys. 1976, 65, 117. 33. Maillard, D.; S i l v i , B. Mol. Phys. 1980, 40, 933. 34. Kollman, P. A. In Chemical Applications of Atomic and Molecular Electrostatic Potentials; Truhlar, D. G., Ed. Plenum Press: New York, 1981, p. 243. 35. Gianturco, F. Α.; Lamanna. U. T.; Battiglia, F. Int. J. Quantum Chem. 1981, 19, 217. 36. Truhlar, D. G.; Brown. F. B; Schwenke, D. W.; Steckler, R.; Garrett, B. C. In Comparison of Ab Initio Quantum Chemistry with Experiment for Small Molecules, Bartlett, R. B., Ed. Reidel: Dordrecht, Holland, 1985, p. 95. 37. Hancock, G. C.; Rejto, P.; Steckler, R.; Brown, F. B.; Schwenke, D. W.; and Truhlar. D. G. J. Chem. Phys. 1985, 85, 4997. 38. Hancock, G. C.; Truhlar. D. G.; Dykstra, C. E., to be published. 39. Schwenke, D. W.; Truhlar. D. G. Computer Phys. Commun. 1984, 34, 57. 40. Alexander, M. H.; DePristo, A. E. J. Chem. Phys. 1977, 66, 2166. 41. Alexander, M. H. J. Chem. Phys. 1980, 73, 5135. RECEIVED June 15, 1987

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 12

Computational Investigations of Organic Reaction Mechanisms 1

William L. Jorgensen, James F. Blake, Jeffry D. Madura , and Scott D. Wierschke

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

Department of Chemistry, Purdue University, West Lafayette, IN 47907

Ab initio molecular orbital calculations are being used to study the reactions of anionic nucleophiles with carbonyl compounds in the gas phase. A rich variety of energy surfaces is found as shown here for reactions of hydroxide ion with methyl formate and formaldehyde, chloride ion with formyl and acetyl chloride, and fluoride ion with formyl fluoride. Extension of these investigations to determine the influence of solvation on the energy profiles is also underway; the statistical mechanics approach is outlined and illustrated by results from Monte Carlo simulations for the addition of hydroxide ion to formaldehyde in water. The i m p o r t a n c e o f d i s p l a c e m e n t r e a c t i o n s on c a r b o n y l compounds i n c h e m i s t r y and b i o c h e m i s t r y has r e s u l t e d i n numerous m e c h a n i s t i c studies. In solution, there is general acceptance of the f o l l o w i n g mechanism f o r a d d i t i o n o f a n i o n i c n u c l e o p h i l e s which features a tetrahedral intermediate, 1, and i s d e s i g n a t e d B 2 (1). However, r e c e n t e x p e r i m e n t a l (2-10) and t h e o r e t i c a l (11-17) A C

X"

+

R—C

studies have found the situation in the gas phase to be intriguingly complex w i t h the possibility of the tetrahedral s p e c i e s as a t r a n s i t i o n s t a t e , the i n t e r v e n t i o n o f i o n - d i p o l e c o m p l e x e s , 2, as i n t e r m e d i a t e s , and energy p r o f i l e s f e a t u r i n g one, 1Current address: Department of Chemistry, University of Houston, Houston, TX 77004

0097-6156/87/0353-0200506.00/0 © 1987 American Chemical Society

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

12.

J O R G E N S E N ET A L .

Organic Reaction

X"

· · · ·

201

Mechanisms

R— Y 2

two o r t h r e e minima. The gas-phase d i s p l a c e m e n t r e a c t i o n s may also be accompanied by competing proton transfer and SJJ2 (bimolecular s u b s t i t u t i o n ) processes that are not observed i n solution. These r e s u l t s have, i n t u r n , l e d t o doubts about the u n i v e r s a l i t y o f the t r a d i t i o n a l B 2 mechanism i n condensed phases and have r a i s e d many q u e s t i o n s c o n c e r n i n g the e n e r g y s u r f a c e s f o r the r e a c t i o n s i n the gas phase, i n s o l u t i o n , and i n b i o m o l e c u l a r environments (12). The c o n t r i b u t i o n o f modern t h e o r e t i c a l methods t o e l u c i d a t i n g organic r e a c t i o n mechanisms i n c l u d e s the a b i l i t y o f ab initio molecular orbital calculations to provide quantitative c h a r a c t e r i z a t i o n o f the gas-phase energy s u r f a c e s and of the s t r u c t u r e s o f any i n t e r m e d i a t e s and t r a n s i t i o n s t a t e s . For the reactions o f c a r b o n y l compounds w i t h a n i o n s , the s i z e o f the systems, t h e i r a n i o n i c n a t u r e , and the need f o r e x t e n s i v e geometry optimizations make reliable calculations taxing on computer resources. N e v e r t h e l e s s , the a v a i l a b i l i t y o f supercomputers s u c h as the Cyber 2 0 5 a t Purdue and s u p e r m i n i c o m p u t e r s such as the G o u l d 3 2 / 8 7 5 0 i n our l a b o r a t o r y now a l l o w s i g n i f i c a n t p r o g r e s s i n t h i s area. Some r e c e n t r e s u l t s a r e r e v i e w e d h e r e t h a t show the u t i l i t y o f the methodology as w e l l as the v a r i e t y o f energy s u r f a c e s and r e a c t i o n p a t h s t h a t may o c c u r f o r even r e l a t i v e l y s i m p l e systems; s p e c i f i c a l l y , the r e a c t i o n s t h a t a r e c o n s i d e r e d are f o r hydroxide i o n w i t h methyl formate and formaldehyde, c h l o r i d e i o n w i t h f o r m y l and a c e t y l c h l o r i d e , and f l u o r i d e i o n with formyl f l u o r i d e . I n a d d i t i o n , the e f f e c t o f s o l v a t i o n on the gas-phase e n e r g y s u r f a c e s i s b e g i n n i n g to be s t u d i e d v i a Monte C a r l o s t a t i s t i c a l mechanics and m o l e c u l a r dynamics c a l c u l a t i o n s for the reacting system surrounded by hundreds of solvent molecules ( 1 8 ) . Our i n i t i a l e f f o r t s a l o n g t h e s e l i n e s on the SJJ2 r e a c t i o n o f C i " + C ^ C i were o n l y made p r a c t i c a l by the advent o f the Cyber 2 0 5 a t Purdue i n 1 9 8 3 ( 1 9 ) . More r e c e n t r e s u l t s ( 1 1 ) on the r e a c t i o n o f OH" + H2C-O i n w a t e r a r e a l s o summarized i n the f o l l o w i n g and i l l u s t r a t e the d e t a i l s t h a t a r e now accessible on the v a r i a t i o n o f s o l v a t i o n a l o n g r e a c t i o n p a t h s , the o r i g i n o f solvent-induced activation barriers, and the location of t r a n s i t i o n states i n solution.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

A C

Energy S u r f a c e s from the Ab

Initio

Calculations

(a) OH" 4- HCOOCH3. The r e a c t i o n s o f h y d r o x i d e i o n w i t h m e t h y l f o r m a t e have b e e n s t u d i e d by s e v e r a l groups i n the gas phase and nicely illustrate the variety of available reaction paths (2,9,10). F l o w i n g a f t e r g l o w e x p e r i m e n t s f o u n d the f o l l o w i n g B ^2, SJJ2, and p r o t o n t r a n s f e r pathways to account for 34%, 5% and 61% of the product distribution, respectively (9). These interesting results leave open numerous q u e s t i o n s about the c o r r e s p o n d i n g e n e r g y s u r f a c e s t h a t we s e t out t o e x p l o r e w i t h ab i n i t i o molecular o r b i t a l c a l c u l a t i o n s . A

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

202

SUPERCOMPUTER

li5

18,OH"

HC0 0-

+

CH3OH

HCOO"

+

CH

RESEARCH

(1)

SM2

HCOOCH3

1 8 3

OH

(2)

p.t. _

CO + C H 0 - · · Η

1 8

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

3

ΟΗ

(3)

The computations were c a r r i e d o u t w i t h t h e GAUSSIAN/82 program (20) and t h e split-valence 4-31+G b a s i s s e t which i n c l u d e s a s e t o f d i f f u s e s and ρ o r b i t a l s on a l l atoms e x c e p t hydrogen (21). D i f f u s e f u n c t i o n s a r e well-known t o be i m p o r t a n t for describing the e l e c t r o n i c s t r u c t u r e o f anions c o n t a i n i n g f i r s t - r o w elements (21-23). The 4-31+G b a s i s s e t was a l s o chosen b e c a u s e i t has been f o u n d t o y i e l d e x c e l l e n t p r o t o n a f f i n i t i e s f o r o r g a n i c a n i o n s ( 2 1 ) . Furthermore, t h e computed e n e r g e t i c s f o r t h e three r e a c t i o n channels are i n acceptable accord with experimental data. Specifically, t h e 4-31+G AE's f o r r e a c t i o n s 1 and 3 a r e -43.3 a n d -19.1 k c a l / m o l , w h i l e t h e e x p e r i m e n t a l e n t h a l p y changes a r e -44.0 and -27.6 k c a l / m o l (10,24). F o r comparison, t h e ΔΕ f o r r e a c t i o n 1 computed w i t h t h e 4-31G b a s i s s e t i s -62.9 kcal/mol (16). Clearly, the d i f f u s e functions p r e f e r e n t i a l l y improve t h e d e s c r i p t i o n o f t h e charge l o c a l i z e d h y d r o x i d e i o n . I t may a l s o be n o t e d t h a t t h e 4-31+G d i s s o c i a t i o n energy f o r t h e methoxide-water complex from t h e R i v e r o s r e a c t i o n (3) i s 28.4 k c a l / m o l which compares w e l l w i t h t h e l a t e s t e x p e r i m e n t a l Δ Η o f 23.9 k c a l / m o l ( 2 4 ) . The r e l a t i v e l y g r e a t e r importance o f t h e d i f f u s e atomic o r b i t a l s t h a n c o r r e l a t i o n energy o r z e r o - p o i n t e f f e c t s i n t h i s c o n t e x t h a s b e e n d i s c u s s e d elsewhere ( 2 2 ) . The key s t r u c t u r a l and e n e r g e t i c r e s u l t s f o r the three r e a c t i o n s a r e summarized i n F i g u r e s 1-3. F o r t h e B ^ 2 p r o c e s s , the t e t r a h e d r a l s p e c i e s , 3, i n F i g u r e 1 was f o u n d t o b e a n energy minimum w i t h no symmetry c o n s t r a i n t s . The i l l u s t r a t e d conformer with t h e h y d r o x y l hydrogen e c l i p s i n g the alkoxy oxygen was p r e v i o u s l y shown t o be t h e l o w e s t energy form a t t h e 4-31G l e v e l (16). W i t h t h e 4-31+G c a l c u l a t i o n s , i t i s 31.1 k c a l / m o l lower i n energy t h a n t h e r e a c t a n t s . T h i s s p e c i e s may r e a r r a n g e t o t h e formate-methanol complex, 6, shown a t t h e bottom o f F i g u r e 2 which i s 30.1 k c a l / m o l lower i n energy t h a n 3 a n d 17.8 k c a l / m o l below the s e p a r a t e d p r o d u c t s , HCOO' + CH3OH. The hydrogen-bond e n t h a l p y for t h e formate-methanol complex has a l s o been determined e x p e r i m e n t a l l y a s 17.6 k c a l / m o l (24). Though a n i o n - d i p o l e complex l i k e 2 was sought p r e c e d i n g 3, none was f o u n d as a n energy minimum w i t h t h e 4-31+G c a l c u l a t i o n s . However, a t r a n s i t i o n s t a t e was l o c a t e d b y g r a d i e n t methods f o r t h e e l i m i n a t i o n o f methanol from 3 on t h e way t o 6; i t i s l a b e l l e d TS i n F i g u r e 1 a n d i s 10.4 k c a l / m o l h i g h e r i n energy t h a n 3. The h y d r o x y l h y d r o g e n h a s r o t a t e d i n t h i s s t r u c t u r e t o h e l p form t h e i n c i p i e n t 0-H bond o f methanol and t h e C-OCH3 bond h a s l e n g t h e n e d from 1.48 t o 2.00 À . Thus, t h e r e a c t i o n p r o f i l e f o r t h e B ^ 2 r e a c t i o n 1 h a s a doublewell form with 3 and 6 as i n t e r m e d i a t e s s e p a r a t e d b y t h e t r a n s i t i o n state f o r the elimination. The e x i s t e n c e o f 3 as a n energy minimum i s c o n s i s t e n t w i t h l o w e r - l e v e l t h e o r e t i c a l r e s u l t s (15,16) a n d r e c e n t o b s e r v a t i o n s o f p r o t o n exchange i n such A

A

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

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F i g u r e 1. 4-31+G r e s u l t s f o r r e a c t i o n 1. k c a l / m o l and l e n g t h s i n angstroms t h r o u g h o u t .

Energies

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In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

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204

F i g u r e 2.

4-31+G r e s u l t s f o r r e a c t i o n

2.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

12.

JORGENSEN ET AL.

Organic Reaction

Mechanisms

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

Proton T r a n s f e r

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4-31+G r e s u l t s f o r r e a c t i o n 3.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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i n t e r m e d i a t e s ( 2 5 ) , though i t c o n t r a s t s the i n t e r p r e t a t i o n o f some e a r l i e r experimental findings (2c). The Sfl2 r e a c t i o n was s t u d i e d i n C symmetry and f o u n d t o have the double-well form typical of gas-phase S^2 reactions (12,25,27), as summarized i n F i g u r e 2. The i n i t i a l i o n - d i p o l e complex 4 i s 17.5 k c a l / m o l below the energy o f the r e a c t a n t s . The t r a n s i t i o n s t a t e 5 was l o c a t e d w i t h g r a d i e n t methods, 8.1 k c a l / m o l above 4, but still 9.4 kcal/mol below the reactants. Rearrangement o f 5 t o the e x i t complex, 6, i s t h e n accompanied by r e l e a s e o f 51.8 k c a l / m o l . S e p a r a t i o n t o the p r o d u c t s , HCOO" + CH3OH, r e q u i r e s 17.8 k c a l / m o l , as d i s c u s s e d above, and y i e l d s the o v e r a l l 4-31+G ΔΕ o f -43.3 k c a l / m o l . These e n e r g e t i c r e s u l t s a r e all similar t o our p r e v i o u s ab i n i t i o f i n d i n g s f o r the S2 r e a c t i o n o f HO" + C ^ C i (22); however, the b e n t arrangement o f the n u c l e o p h i l i c , e l e c t r o p h i l i c and l e a v i n g atoms i n 4 i s u n u s u a l . s

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

N

The p r o t o n t r a n s f e r p r o c e s s was a l s o s t u d i e d i n C symmetry as summarized i n Figure 3. The proton transfer occurred s p o n t a n e o u s l y as the geometry o p t i m i z a t i o n b r o u g h t the r e a c t a n t s t o g e t h e r t o y i e l d the w a t e r - a c y l a n i o n complex, 7. Upon f l i p p i n g the f a r h y d r o g e n o f water, f u r t h e r o p t i m i z a t i o n y i e l d e d 8 w h i c h may be c a l l e d the " R i v e r o s complex" and i s 26.6 k c a l / m o l lower i n energy t h a n the r e a c t a n t s . The complex may be v i e w e d as h y d r a t e d methoxide i o n c o o r d i n a t e d w i t h c a r b o n monoxide. L o s s o f CO t o y i e l d C ^ O ' ^ ' ^ O r e q u i r e s o n l y 7.5 k c a l / m o l , whereas l o s s o f water t o g i v e the methyl f o r m y l a n i o n r e q u i r e s 19.9 kcal/mol. O v e r a l l , t h e s e r e s u l t s s u g g e s t t h a t the R i v e r o s r e a c t i o n (3) has a s i n g l e - w e l l energy s u r f a c e w i t h the i n t e r e s t i n g s t r u c t u r e 8 as the only intermediate. In summary, the 4-31+G c a l c u l a t i o n s f i n d the energy s u r f a c e s for the t h r e e o b s e r v e d r e a c t i o n s t o a l l c o n t a i n s i g n i f i c a n t l y s t a b i l i z e d i n t e r m e d i a t e s whose s t r u c t u r e s and r e l a t i v e e n e r g i e s have b e e n c h a r a c t e r i z e d . The formate-methanol complex 6 i s the g l o b a l energy minimum, 61 k c a l / m o l lower i n energy t h a n the reactants. I t i s a c c e s s i b l e t h r o u g h b o t h the S 2 and B 2 channels. The t r a n s i t i o n s t a t e s f o r t h e s e p r o c e s s e s have a l s o b e e n l o c a t e d and a r e s i g n i f i c a n t l y lower i n energy than the reactants. Thus, none o f the r e a c t i o n s has a n e t p o s i t i v e a c t i v a t i o n energy which i s c o n s i s t e n t w i t h t h e i r o b s e r v e d f a c i l i t y (2,9,10). (b) C i " + HCOCi and C l " + CH3COCI. A s u b i o j o and Brauman made the p r o v o c a t i v e p r o p o s a l t h a t the gas-phase displacement reactions of nucleophiles i n c l u d i n g h a l i d e ions with a c y l h a l i d e s have d o u b l e - w e l l energy s u r f a c e s ; the i n t e r m e d i a t e s were s u g g e s t e d to be i o n - d i p o l e complexes and the t e t r a h e d r a l s p e c i e s was l i k e l y to be a t r a n s i t i o n s t a t e ( 4 ) . In order to address t h i s p r o p o s a l , we expanded our s t u d y o f s u b s t i t u t i o n r e a c t i o n s t o the d e g e n e r a t e exchange reactions of chloride ion with formyl and acetyl c h l o r i d e , and o f f l u o r i d e i o n w i t h f o r m y l f l u o r i d e ( 1 2 ) . Key i s s u e s a r e the number o f minima on the energy s u r f a c e s and the s t r u c t u r e s o f any i n t e r m e d i a t e s and t r a n s i t i o n s t a t e s . Han and Brauman a l s o r e c e n t l y extended t h e i r ICR i n v e s t i g a t i o n s o f C i " w i t h CF3C0Ci and C H 0 C 0 C i ( 5 ) . They were a b l e t o show t h r o u g h l a b e l i n g s t u d i e s t h a t the two c h l o r i n e s i n the adducts a r e none q u i v a l e n t w h i c h s u p p o r t s the d o u b l e - w e l l p i c t u r e and the p r o b a b l e s

N

3

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

A C

12.

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Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

e x i s t e n c e o f a t e t r a v a l e n t s p e c i e s as a t r a n s i t i o n s t a t e ( 5 ) . At the same time, Howard and K o l l m a n have c a r r i e d out a r e l a t e d ab initio s t u d y o f the r e a c t i o n o f HS" and formamide ( 1 4 ) . In c o n t r a s t t o t h e i r r e s u l t s f o r HO" w i t h formamide (13), an i o n d i p o l e complex r a t h e r t h a n the t e t r a h e d r a l a d d u c t was f o u n d as an e n e r g y minimum. The geometry optimizations for reactions 4 and 5 were p r i m a r i l y c a r r i e d out w i t h the 3-21+G b a s i s s e t (23). This alternative typically g i v e s r e s u l t s s i m i l a r t o 4-31+G, b u t i t i s Ci"

+

HCOCi

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HC0Ci "

Ci"

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3

(4)

2

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n o t a b l y f a s t e r f o r the g r a d i e n t c a l c u l a t i o n s w h i c h i s p a r t i c u l a r l y desirable given the five non-hydrogen atoms i n reaction 5. N e v e r t h e l e s s , the e f f e c t o f b a s i s s e t e x t e n s i o n f o r the s t a t i o n a r y p o i n t s i n r e a c t i o n 4 was shown t o be s l i g h t t h r o u g h o p t i m i z a t i o n s w i t h the 6-31+G(d) b a s i s s e t ( 1 2 ) . The 3-21+G r e s u l t s t u r n e d out t o be f a s c i n a t i n g . The energy s u r f a c e s f o r r e a c t i o n s 4 and 5 i n d e e d have the d o u b l e - w e l l form. The i o n - d i p o l e complex, 9, shown i n F i g u r e 4 i s the o n l y minimum f o r r e a c t i o n 4 and the t e t r a h e d r a l adduct, 10, i s a t r a n s i t i o n state. Computation o f the v i b r a t i o n a l f r e q u e n c i e s f o r 9 and 10 u n e q u i v o c a l l y e s t a b l i s h e d these designations. As shown i n F i g u r e 5, 9 and 10 a r e c a l c u l a t e d t o be 21.7 and 7.1 k c a l / m o l below the e n e r g y o f the r e a c t a n t s a t the 3-21+G l e v e l . More s u r p r i s i n g l y , f u r t h e r s e a r c h f o r a n o t h e r t e t r a v a l e n t s p e c i e s f o u n d the p l a n a r a d d u c t 11 w i t h C symmetry ( F i g u r e 6) as a s e c o n d t r a n s i t i o n s t a t e , a g a i n v e r i f i e d by the v i b r a t i o n a l a n a l y s e s . I n f a c t , 11 i s 1.6 k c a l / m o l lower i n e n e r g y t h a n 10; the two s t r u c t u r e s s i t on a r i d g e between the r e a c t a n t s and p r o d u c t s and were f o u n d t o be s e p a r a t e d by a b a r r i e r o f o n l y 0.7 k c a l / m o l ( 1 2 ) . The v a l l e y s t o 10 and 11 b o t h emanate from the i o n - d i p o l e complex 9 and are i l l u s t r a t e d by the s t r u c t u r e s i n F i g u r e s 4 and 6, and by the e n e r g y p r o f i l e s i n F i g u r e 5. 2 v

However, the s t r u c t u r e f o r 11 i s u n u s u a l w i t h l o n g C - C i bond l e n g t h s o f 2.62 Â as opposed t o 2.24 À f o r 10. T h i s suggests that the more compact structure 10 might become r e l a t i v e l y more f a v o r a b l e when e l e c t r o n c o r r e l a t i o n i s i n c l u d e d . Consequently, Miller-Plesset perturbation theory t o t h i r d - o r d e r was used to compute the c o r r e l a t i o n e n e r g i e s w i t h the 6-31+G(d) b a s i s s e t on the 3-21+G o p t i m i z e d g e o m e t r i e s f o r the stationary points in r e a c t i o n 4. I n the s t a n d a r d n o t a t i o n (23), t h e s e c a l c u l a t i o n s a r e designated MP3/MP2/6-31+G(d)//3-21+G. The effects on the e n e r g e t i c s f o r the pathway l e a d i n g t o the t e t r a h e d r a l a d d u c t 10 with C symmetry are n o t g r e a t ; 9 and 10 a r e now 16.7 and 1.4 k c a l / m o l below the r e a c t a n t s . However, the C s t r u c t u r e 11 i s d i f f e r e n t i a l l y d e s t a b i l i z e d so t h a t i t i s 11.6 k c a l / m o l above 10 and i s u n d o u b t e d l y no l o n g e r a t r a n s i t i o n s t a t e . Nevertheless, the b a r r i e r t o i n v e r s i o n o f 10 t h r o u g h 11 i s r e m a r k a b l y low. Though i t c o u l d be a r g u e d t h a t a p l a n a r s t r u c t u r e a n a l o g o u s t o 11 f o r a c e t y l c h l o r i d e would be d i s f a v o r e d by s t e r i c c r o w d i n g between the c h l o r i n e s and the m e t h y l group, t h i s was n o t f o u n d t o s

2 v

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

208

F i g u r e 5. 3-21+G energy p r o f i l e s f o r r e a c t i o n s 4 and 5. The reaction coordinate i s d e f i n e d as t h e d i f f e r e n c e i n t h e two C-Ci d i s t a n c e s .

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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be a dominant e f f e c t w i t h t h e 3-21+G c a l c u l a t i o n s . I n s t e a d adduct 13 i n F i g u r e 7 was f o u n d t o be the o n l y t r a n s i t i o n s t a t e ; i t i s 5.2 k c a l / m o l lower i n energy t h a n a s t r u c t u r e a n a l o g o u s t o 10 w i t h a CCO a n g l e o f 140°. The r e a s o n f o r the i n c r e a s e d p r e f e r e n c e f o r the C 2 - l i k e structure c a n be attributed to the substantial p o s i t i v e c h a r g e on the c a r b o n y l c a r b o n t h a t i s a p p a r e n t from population analyses. Thus, t h e R-C-0 fragment i n 11 and 13 has s i g n i f i c a n t a c y l c a t i o n c h a r a c t e r t h a t i s s t a b i l i z e d by the m e t h y l substituent.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

v

The 3-21+G r e a c t i o n p a t h t o 13 i s i l l u s t r a t e d i n F i g u r e 7 and i n v o l v e s the i n t e r m e d i a c y o f t h e i o n - d i p o l e complex 12. As shown i n F i g u r e 5, t h e w e l l - d e p t h f o r 12 i s r e d u c e d t o 14.3 k c a l / m o l s i n c e t h e c h l o r i d e i o n i s k e p t by the m e t h y l group about 1.5 Â f a r t h e r from the c a r b o n y l c a r b o n t h a n i n 9. This interaction e n e r g y f o r C i " · · - C ^ C O C i compares f a v o r a b l y w i t h t h e v a l u e o f 11 kcal/mol estimated by Asubiojo and Brauman from their ICR experiments ( 4 ) . Of c o u r s e , t h e l o n g C - C i bonds i n 13 a g a i n s u g g e s t that electron correlation should s i g n i f i c a n t l y a f f e c t the results. Assuming t h e e f f e c t s a r e q u a n t i t a t i v e l y s i m i l a r t o t h o s e f o r r e a c t i o n 4, t h e t e t r a h e d r a l t r a n s i t i o n s t a t e w o u l d end up about 8 k c a l / m o l lower i n energy t h a n 13 and r o u g h l y 4 k c a l / m o l above the energy o f the r e a c t a n t s . A s u b i o j o and Brauman a l s o a d d r e s s e d t h i s point. Though t h e y c o u l d e x p l a i n the o b s e r v e d e f f i c i e n c y f o r r e a c t i o n 5 t h r o u g h RRKM c a l c u l a t i o n s i n w h i c h t h e t r a n s i t i o n s t a t e i s c a . 7 k c a l / m o l below t h e energy o f t h e r e a c t a n t s , a r e s u l t a n t p r o b l e m i s an i m p l i e d , u n r e a s o n a b l y h i g h e l e c t r o n a f f i n i t y f o r C h ^ C C ^ O r a d i c a l ( 4 ) . The h i g h e r energy f o r t h e t r a n s i t i o n s t a t e s u g g e s t e d h e r e would h e l p a l l e v i a t e the l a t t e r dilemma. ( c ) F" + HCOF. The r e a c t i o n o f f l u o r i d e i o n w i t h f o r m y l f l u o r i d e was a l s o s t u d i e d w i t h 3-21+G c a l c u l a t i o n s f o r comparison. Briefly, t h i s system i s f o u n d t o have the t r i p l e - w e l l energy p r o f i l e shown i n F i g u r e 8. Two e q u i v a l e n t i o n - d i p o l e complexes a n a l o g o u s t o 9 now f l a n k the t e t r a h e d r a l i n t e r m e d i a t e , 14, w h i c h i s a l s o an e n e r g y minimum. I n t h i s c a s e , i n v e r s i o n t h r o u g h the F

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14

15

p l a n a r form 15 i s h i g h l y u n f a v o r a b l e w i t h 15 as a maximum 40 k c a l / m o l above 14. The r e q u i r e d s t r e t c h i n g o f the C-F bonds i s f a r more c o s t l y i n energy t h a n f o r the weaker C - C i bonds i n 11. T a k i n g a l l o f t h e s e r e s u l t s t o g e t h e r , some g e n e r a l p a t t e r n s emerge. Foremost, the t e t r a h e d r a l a d d u c t s 1 a r e f o u n d t o be e n e r g y minima when the s u b s t i t u e n t s X and Y a r e b o t h f i r s t - r o w elements. However, when X and Y a r e b o t h second-row e l e m e n t s , the t e t r a h e d r a l s p e c i e s i s a t r a n s i t i o n s t a t e and the o n l y minima a r e ion-dipole complexes, 2. Clearly, two key factors i n the f o r m a t i o n o f the t e t r a h e d r a l adduct 1 a r e the d i f f e r e n c e i n gasphase b a s i c i t i e s f o r the two a n i o n s (X" and 1) and the d i f f e r e n c e

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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F i g u r e 7. 3-21+G o p t i m i z e d s t r u c t u r e s f o r r e a c t i o n 5.

along the r e a c t i o n

path

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i n bond e n e r g i e s between a C-0 π-bond and a C-X s i n g l e bond. The former is remarkably strong a t about 90 k c a l / m o l (28), w h i l e some C-X s i n g l e bond e n e r g i e s a r e f o r X - Ν (69-75), 0 (85-91), F (116), S ( 6 6 ) , C i (79), Br (66), and I ( 5 2 ) . Furthermore, s i n c e t h i o l a t e , c h l o r i d e , bromide, and i o d i d e i o n s a r e much weaker b a s e s i n the gas phase t h a n a l k o x i d e s (29), t h e s e n u c l e o p h i l e s should a l l f a v o r i o n - d i p o l e complexation over formation o f t e t r a h e d r a l adducts f o r most acyl electrophiles including acid halides, amides, and e s t e r s . Thus, the t e t r a h e d r a l a d d u c t w i l l o f t e n n o t be an e n e r g y minimum i n t h e s e c a s e s . I n c o n t r a s t , C-F bonds a r e unusually s t r o n g and f l u o r i d e i o n i s comparable i n b a s i c i t y t o alkoxides i n the gas phase ( 2 9 ) . Therefore, formation of the tetrahedral adduct should be favored by fluoride as the nucleophile and s i n g l e or t r i p l e - w e l l energy s u r f a c e s can be expected. S i m i l a r analyses propose t h a t l e s s s t a b l e a l k o x i d e s and OH" should yield tetrahedral intermediates with acyl e l e c t r o p h i l e s , w h i l e s t a b i l i z e d a l k o x i d e s may prefer ion-dipole complexation. These notions are fully supported by the t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s r e v i e w e d h e r e , and p r o v i d e a r i c h v a r i e t y o f energy s u r f a c e s f o r n u c l e o p h i l i c r e a c t i o n s w i t h a c y l e l e c t r o p h i l e s i n the gas phase. The

E f f e c t o f H y d r a t i o n on

the R e a c t i o n o f OH"

+ HoC-0

The i m p o r t a n c e o f s o l v a t i o n on r e a c t i o n s u r f a c e s i s e v i d e n t in s t r i k i n g medium dependence o f r e a c t i o n r a t e s , p a r t i c u l a r l y f o r p o l a r r e a c t i o n s , and i n v a r i a t i o n s o f p r o d u c t d i s t r i b u t i o n s as f o r methyl formate discussed above and of relative reactivities (18,26). Thus, i n o r d e r t o o b t a i n a m o l e c u l a r l e v e l u n d e r s t a n d i n g o f the i n f l u e n c e o f s o l v a t i o n on the e n e r g e t i c s and c o u r s e s o f r e a c t i o n s , we have c a r r i e d out s t a t i s t i c a l mechanics s i m u l a t i o n s t h a t have y i e l d e d f r e e e n e r g y o f a c t i v a t i o n p r o f i l e s (30) for s e v e r a l o r g a n i c r e a c t i o n s i n s o l u t i o n (11.18.19.31). The computational procedure t y p i c a l l y involves three main s t e p s . F i r s t , the minimum energy r e a c t i o n p a t h i s d e t e r m i n e d f o r the gas phase u s i n g ab i n i t i o c a l c u l a t i o n s . The e n e r g e t i c and g e o m e t r i c v a r i a t i o n s a l o n g the r e a c t i o n path are then f i t to continuous functions of the reaction coordinate. Then, i n t e r m o l e c u l a r p o t e n t i a l f u n c t i o n s are obtained to d e s c r i b e the i n t e r a c t i o n s between the r e a c t i n g system and a s o l v e n t m o l e c u l e . These usually vary with the reaction coordinate and are r e p r e s e n t e d t h r o u g h Coulomb and Lennard-Jones i n t e r a c t i o n s between s i t e s n o r m a l l y s i t u a t e d on the atoms. F o r aqueous s o l u t i o n s , the solute-water p o t e n t i a l functions are d e r i v e d from numerous ab i n i t i o r e s u l t s f o r complexes o f the r e a c t i n g system and a water m o l e c u l e , w h i l e the w a t e r - w a t e r i n t e r a c t i o n s a r e d e s c r i b e d by the well-proven TIP4P model (32)· Finally, with analytical descriptions of the gas-phase reaction path and of the i n t e r m o l e c u l a r p o t e n t i a l f u n c t i o n s , Monte C a r l o s i m u l a t i o n s are c a r r i e d out t o c a l c u l a t e the f r e e e n e r g y p r o f i l e f o r the r e a c t i o n path i n s o l u t i o n . A c t u a l l y a series of simulations a r e needed w i t h "importance s a m p l i n g " t o c o v e r the f u l l range o f the r e a c t i o n coordinate (18). The q u a n t i t y t h a t i s u l t i m a t e l y computed i s the p r o b a b i l i t y o f o c c u r r e n c e , g ( r ) , o f each v a l u e o f the r e a c t i o n c

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coordinate, r . I n t u r n , t h i s i s r e l a t e d t o the f r e e energy change a l o n g the r e a c t i o n c o o r d i n a t e o r " p o t e n t i a l o f mean f o r c e " by w ( r ) - -kgT In g ( r ) + const. B e s i d e s the thermodynamic results, the simulations also yield a d e t a i l e d view o f the s t r u c t u r a l and e n e r g e t i c changes i n s o l v a t i o n a l o n g the r e a c t i o n path. I t s h o u l d be emphasized t h a t we have o n l y b e e n computing the e f f e c t s o f s o l v a t i o n on the gas-phase r e a c t i o n p a t h s ; changes i n mechanism i n s o l u t i o n a r e n o t p r o v i d e d f o r so f a r . A f t e r the i n i t i a l work on SJJ2 r e a c t i o n s , the methodology was a p p l i e d t o the a d d i t i o n o f h y d r o x i d e i o n t o f o r m a l d e h y d e . The ab initio c a l c u l a t i o n s f o r the gas-phase MERP and the potential f u n c t i o n s were a l l c a r r i e d out w i t h the 6-31+G(d) b a s i s s e t ( 1 1 ) . As i l l u s t r a t e d i n F i g u r e 9, the a p p r o a c h a t l a r g e s e p a r a t i o n i s coplanar with the hydroxide ion on the dipole axis of formaldehyde. An a p p a r e n t i o n - d i p o l e minimum o c c u r s a t a C-0 separation o f 2.74 À w i t h a b i n d i n g e n e r g y o f 19 k c a l / m o l , as shown by the s o l i d c u r v e i n the l o w e r p a r t o f F i g u r e 10. However, an a c t i v a t i o n e n e r g y o f o n l y 1 k c a l / m o l i s needed to reach the transition s t a t e w i t h r(C-O) - 2.39 Â a t w h i c h p o i n t the h y d r o x y l fragment has lifted out of the plane to assume the more t e t r a h e d r a l , f i n a l approach. The e n e r g y change i s t h e n r a p i d as covalent b o n d i n g s e t s i n between the t r a n s i t i o n s t a t e and the t e t r a h e d r a l p r o d u c t a t r ( C 0 ) - 1.47 Â. The o v e r a l l e n e r g y change for the reaction is calculated t o be -35.2 kcal/mol. The e x i s t e n c e o f the i o n - d i p o l e minimum i s c l e a r l y t e n t a t i v e and may n o t s u r v i v e f u r t h e r i n c r e a s e s i n the l e v e l o f t h e o r y . Thus, the e n e r g y s u r f a c e c o u l d be a s i n g l e - w e l l , though the a p p r o a c h has two s t a g e s dominated r e s p e c t i v e l y by i o n - d i p o l e and c o v a l e n t forces s e p a r a t e d n e a r 2.5 Â. c

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

c

c

A s e c o n d t r a j e c t o r y was a l s o s t u d i e d as i n d i c a t e d by the dashed c u r v e s i n F i g u r e 10. I n t h i s c a s e , the 0C0 a n g l e was f i x e d a t i t s v a l u e o f 127° a t the t r a n s i t i o n s t a t e f o r a l l s e p a r a t i o n s beyond r ( C 0 ) - 2.39 Â. The more t e t r a h e d r a l a p p r o a c h c o r r e s p o n d s t o t r a d i t i o n a l i d e a s about a p p r o a c h v e c t o r s i n a d d i t i o n r e a c t i o n s (33). The ion-dipole minimum no longer occurs for this t r a j e c t o r y , though the e n e r g y s u r f a c e has a r e l a t i v e l y f l a t r e g i o n between r ( C 0 ) - 1.9 and 2.4 Â. The Monte C a r l o c a l c u l a t i o n s were s u b s e q u e n t l y e x e c u t e d f o r the r e a c t i n g system s u r r o u n d e d by 269 TIP4P w a t e r m o l e c u l e s i n a r e c t a n g u l a r box w i t h p e r i o d i c boundary c o n d i t i o n s a t 25°C and 1 atm. The d e t a i l s are r e p o r t e d e l s e w h e r e (11), though the key r e s u l t s a r e i n the top p a r t o f F i g u r e 10. The f r e e e n e r g y c u r v e s f o r the two t r a j e c t o r i e s i n w a t e r r i s e o n l y g r a d u a l l y f r o m the reactants to r(C0) ca. 3 Â. Loss in hydroxide-water i n t e r a c t i o n s i s l a r g e l y o f f s e t by i n c r e a s e i n the i o n - f o r m a l d e h y d e a t t r a c t i o n i n t h i s region. Then, the c h a r g e d e l o c a l i z a t i o n s e t s i n , w h i l e the gas-phase e n e r g y i s r e l a t i v e l y c o n s t a n t between 2 and 3 Â. C o n s e q u e n t l y , the weakening s o l v a t i o n i s no longer b a l a n c e d and the f r e e e n e r g y o f a c t i v a t i o n c u r v e s r i s e r a p i d l y t o the t r a n s i t i o n s t a t e w h i c h i s p r e d i c t e d t o o c c u r a t r ( C 0 ) 2.05 Â. From t h e r e , the gas-phase e n e r g y descends q u i c k l y , t o the t e t r a h e d r a l a d d u c t and the p o t e n t i a l o f mean f o r c e f o l l o w s s u i t . I t i s a l s o n o t a b l e t h a t no e v i d e n c e i s f o u n d f o r any i n t e r m e d i a t e s o t h e r t h a n the p r o d u c t i n water, and t h a t the a c t i v a t i o n b a r r i e r

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

12.

JORGENSEN ET AL.

Organic Reaction

Mechanisms

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

0

-

ι

7

1 1 1 1 1 1 -

5

3 1 1 3 REACTION COORDINATE (ft)

5

1 7

F i g u r e 8. 3-21+G energy p r o f i l e f o r t h e r e a c t i o n o f F" + HCOF. The r e a c t i o n c o o r d i n a t e i s t h e d i f f e r e n c e i n t h e two CF distances.

F i g u r e 9. 6-31+G(d) o p t i m i z e d s t r u c t u r e s p a t h f o r t h e r e a c t i o n o f OH" + H C-0.

along

the r e a c t i o n

2

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30

i 1

1 1 1 1 1 1 2

1

3

4 5 6 7 8 REACTION COORDINATE (A) F i g u r e 10. C a l c u l a t e d p o t e n t i a l e n e r g i e s i n t h e gas phase (bottom) and p o t e n t i a l s o f mean f o r c e i n aqueous solution (top) f o r the A d r e a c t i o n o f OH" + h^C-O. Solid lines r e p r e s e n t t h e c o l l i n e a r MERP w i t h C symmetry i n t h e gas phase, w h i l e t h e dashed l i n e s a r e f o r a more p e r p e n d i c u l a r initial approach. The r e a c t i o n coordinate i s the C-0 distance. N

s

H

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JORGENSEN ET AL.

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215

for the addition reaction is entirely solvent-induced. The s o l v e n t e f f e c t s a r e c l e a r l y p r o f o u n d ; the o r i g i n o f the b a r r i e r was c a r e f u l l y s t u d i e d and i s a t t r i b u t a b l e t o the weakening o f h y d r o g e n bonds to the substrates t h a t accompanies the charge d e l o c a l i z a t i o n i n s i d e r(CO) - 3 À . The average number o f s t r o n g h y d r o g e n bonds i s c o n s t a n t a t 6-7 a l o n g the e n t i r e r e a c t i o n p a t h ; however, the average w a t e r - i o n i n t e r a c t i o n weakens from c a . 20 k c a l / m o l f o r the h y d r o x i d e i o n to 13 k c a l / m o l f o r the p r o d u c t

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(11). The energetic results a r e s i m i l a r to e x p e r i m e n t a l d a t a o f G u t h r i e (34) and the t h e o r e t i c a l p r e d i c t i o n s o f Weiner e t a l . (13) for a l k a l i n e hydrolyses of amides. For t h e i r systems, the a c t i v a t i o n energy f o r the a d d i t i o n s t e p i s about 22 k c a l / m o l and the t e t r a h e d r a l i n t e r m e d i a t e i s 9-18 k c a l / m o l above the r e a c t a n t s . Ester hydrolyses are typically more facile with activation energies of 15-20 kcal/mol (34)· The combined quantum and m o l e c u l a r mechanics a p p r o a c h o f Weiner e t al. a l s o l e d to a p r e d i c t e d CO d i s t a n c e o f about 2.0 Â i n the aqueous transition s t a t e f o r the a d d i t i o n o f OH" to formamide. T h u s , the t r a n s i t i o n states for these endoergic processes are geometrically very product-like. For formaldehyde, the computed endoergicity o v e r e s t i m a t e s the a v a i l a b l e e x p e r i m e n t a l d a t a by c a . 15 k c a l / m o l (34) Î the computed r e s u l t i s more i n l i n e w i t h d a t a f o r k e t o n e s , where f o r m a t i o n o f h y d r a t e s i s l e s s f a v o r a b l e . The d i s c R r e p a n c y l i k e l y comes from o v e r l y e x o t h e r m i c h y d r a t i o n o f the h y d r o x i d e i o n w h i c h lowers the r e a c t a n t end o f the f r e e energy c u r v e s . This results from use o f two-body p o t e n t i a l functions t h a t do n o t adequately t r e a t p o l a r i z a t i o n . Nevertheless, the f e a s i b i l i t y o f p e r f o r m i n g such c a l c u l a t i o n s i n s o l u t i o n has been e s t a b l i s h e d and the i n s i g h t s on the v a r i a t i o n i n s o l v a t i o n a l o n g the r e a c t i o n p a t h a r e most l i k e l y v a l i d , though somewhat a m p l i f i e d . O v e r a l l , an e x c i t i n g p e r i o d has c l e a r l y been e n t e r e d i n w h i c h theoretical c a l c u l a t i o n s c a n p r o v i d e extreme d e t a i l s on the c o u r s e o f o r g a n i c r e a c t i o n s b o t h i n the gas phase and i n s o l u t i o n . Acknowledgments Gratitude National programs.

is e x p r e s s e d t o the N a t i o n a l S c i e n c e F o u n d a t i o n and Institutes of Health for support of our research

Literature Cited 1. March, J. Advanced Organic Chemistry: Wiley: New York, 1985; Chapters 10 and 16. 2. (a) Faigle, J. F. G.; Isolani, P.C.; Riveros, J. M. J. Am. Chem. Soc. 1976, 98, 2049. (b) Takashima, K.; Riveros, J. M. ibid. 1978, 100, 6128. (c) Takashima, K.; Jose, S. M.; do Amaral, A. T.; Riveros, J. M. J. Chem. Soc. Chem Commun. 1983, 1255. 3. Comisarow, M. Can. J. Chem. 1977, 55, 171. 4. Asubiojo, O. I.; Brauman, J. I. J. Am. Chem. Soc. 1979, 101, 3715. 5. Han, C.-C.; Brauman, J. I. J. Am. Chem. Soc. 1987, 109, 589.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

216

6. 7. 8. 9. 10. 11. Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch012

12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

SUPERCOMPUTER RESEARCH

Fukuda, Ε. K.; McIver, R. T., Jr. J. Am. Chem. Soc. 1979, 101, 2498. Bartmess, J. E.; Hays, R. L.; Caldwell, G. J. Am. Chem. Soc. 1981, 103, 1338. McDonald, R. N.; Chowdhury, A. K. J. Am. Chem. Soc. 1982, 104, 901. DePuy, C. H.; Della, E. W.; Filley, J . ; Grabowski, J. J . ; Bierbaum, V. M. J. Am. Chem. Soc. 1983, 105, 2481. Johlman, C. L.; Wilkins, C. L. J. Am. Chem. Soc. 1985, 107, 327. Madura, J. D.; Jorgensen, W. L. J. Am. Chem Soc. 1986, 108, 2517. Blake, J. F.; Jorgensen, W. L. J. Am. Chem. Soc. 1987, 109, 0000. Weiner, S. J . ; Singh, U. C.; Kollman, P. A. J. Am. Chem. Soc. 1985, 107, 2219. Howard, A. E.; Kollman, P. A. J. Am. Chem. Soc. 1987, 109, 0000. Dewar, M. J. S.; Storch, D. M. J. Chem. Soc. Chem. Commun. 1985, 94. Ewig, C. S.; Van Wazer, J. R. J. Am. Chem. Soc. 1986, 108, 4774. Yamabe, S.; Minato, T. J. Org. Chem. 1983, 48, 2972. Jorgensen, W. L. Adv. Chem. Phys. 1987, 00, 0000. (a) Chandrasekhar, J . ; Smith, S. F.; Jorgensen, W. L. J. Am. Chem. Soc. 1985, 107, 154. (b) Chandrasekhar, J . ; Jorgensen, W. L. ibid. 1985, 107, 2974. Binkley, J. S.; Whiteside, R. Α.; Raghavachari, K.; Seeger, R.; DeFrees, D. J . ; Schlegel, Η. B.; Frisch, M. J.; Pople, J. A.; Kahn, L. R. Gaussian 82 Release H; Carnegie-Mellon University: Pittsburgh, 1982. Chandrasekhar, J . ; Andrade, J. G.; Schleyer, P.v.R. J. Am. Chem. Soc. 1981, 103, 5609, 5612. Gao, J . ; Garner, D. S.; Jorgensen, W. L. J. Am. Chem. Soc. 1986, 108, 4784. Hehre, W. J . ; Radom, L., Schleyer, P.v.R.; Pople, J. A. Ab Initio Molecular Orbital Theory: Wiley: New York, 1986. Meot-Ner, M.; Sieck, L. W. J. Am. Chem. Soc. 1986, 108. 7525. Nibbering, N. M. W., to be published. Olmstead, W. N.; Brauman, J. I. J. Am. Chem. Soc. 1977, 99, 4219. Evanseck, J. D.; Blake, J. F.; Jorgensen, W. L. J. Am. Chem. Soc. 1987, 109, 0000. Reference 1, p. 23 Bartmess, J. E.; McIver, R. T., Jr. in Gas Phase Ion Chemistry, Volume 2, Bowers, M. T., Ed.; Academic Press: New York, 1979, p. 87 Kreevoy, M. M.; Truhlar, D. G. In Investigation of Rates and Mechanisms of Reactions, 4th edition; Bernasconi, C., Ed.; Wiley: New York, 1986, Part 1, chapter 1. Jorgensen, W. L.; Buckner, J. K.; Huston, S. E.; Rossky, P. J. J. Am. Chem. Soc. 1987, 109, 1891.

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32. Jorgensen, W. L.; Chandrasekhar, J . ; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926. Jorgensen, W. L.; Madura, J. D. Mol.Phys.1985, 56, 1381. 33. Burgi, H. B.; Dunitz, J. D. Accts. Chem. Res. 1983, 16, 153. 34. Guthrie, J. P. J. Am. Chem. Soc. 1978, 100, 5892; ibid. 1973, 95, 6999; ibid. 1974, 96, 3608. 15, 1987

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RECEIVED June

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 13

Molecular Dynamics Studies of Crystal Growth and Thin Films George H. Gilmer and Marcia H. Grabow

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

AT&T Bell Laboratories, Murray Hill, NJ 07974 We discuss the application of atomic scale computer models to bulk crystal growth and the formation of thin films. The structure of the crystal-fluid interface and the mobility of the material at this interface are discussed in some detail. The influence of strain on thin film perfection and stability is also examined. A n u n d e r s t a n d i n g of the a t o m i c scale processes t h a t o c c u r d u r i n g c r y s t a l g r o w t h is essential to the development of technologies t h a t u t i l i z e h i g h l y perfect c r y s t a l s .

T h e s t r u c t u r e of the interface between the c r y s t a l a n d the

s u r r o u n d i n g l i q u i d or v a p o r phase is of great i m p o r t a n c e since the

interface

serves t o order a n d s t a b i l i z e the adjacent molecules i n the fluid phase, t h u s f a c i l i t a t i n g t h e i r i n c o r p o r a t i o n i n t o the c r y s t a l l a t t i c e .

In this article some

of the basic mechanisms of c r y s t a l g r o w t h are considered, together w i t h the i m p a c t of c o m p u t e r s i m u l a t i o n s o n our perception of these processes. i n c l u d e d are m o l e c u l a r d y n a m i c s ( M D ) s i m u l a t i o n s of t h i n films.

Also

These pro-

vide i n f o r m a t i o n o n the s t a b i l i t y of s t r a i n e d films against the spontaneous generation of misfit dislocations or a b r e a k u p i n t o islands. T h e rate of c r y s t a l g r o w t h c a n be extremely sensitive to the b i n d i n g energy of atoms at different sites o n the surface. array

of energetic

efficiently.

A surface w i t h a dense

b i n d i n g sites condenses atoms from the

vapor

most

T h e density of these active sites depends o n the surface

tem-

p e r a t u r e , c r y s t a l l o g r a p h i c o r i e n t a t i o n a n d i m p u r i t y content. A surface near a close-packed o r i e n t a t i o n is i l l u s t r a t e d i n F i g . 1. Here the active sites are l o c a t e d at the edges of steps, where molecules condensing from the v a p o r c a n i n t e r a c t w i t h a large n u m b e r of neighbors.

In the presence of a super-

s a t u r a t e d v a p o r , these steps advance as the edge sites are filled, a n d event u a l l y the steps a n n i h i l a t e at the edge of the c r y s t a l . W h e n a l l of the existing steps are a n n i h i l a t e d i n this w a y , the c r y s t a l is b o u n d e d b y close-packed layers

and

growth

terminates,

unless

there

is

a

mechanism

for

generation of new steps. 0097-6156/87/0353-0218$06.00/0 © 1987 American Chemical Society

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

the

13.

GILMER AND GRABOW

Crystal Growth and Thin

219

Films

Surface Roughening and Crystal Growth E a r l y a t t e m p t s to c a l c u l a t e g r o w t h rates were based o n the n u c l e a t i o n of clusters o n the surfaces of a perfect c r y s t a l . A c c o r d i n g t o these theories, clusters are generated b y a fortuitous series of i m p i n g e m e n t events t h a t o c c u r o n neighboring sites.

V e r y s m a l l clusters are l i k e l y to

since few neighbors are present t o s t a b i l i z e the s y s t e m .

disintegrate

B u t occasionally a

large stable cluster m a y appear, a n d its p e r i p h e r y t h e n provides the active sites for c r y s t a l g r o w t h .

T h i s cluster c o u l d t h e n e x p a n d a n d cover the sur­

face, or merge w i t h neighboring clusters t o complete the l a y e r . Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

A

difficulty

with

this

mechanism

is

the

small

nucleation

rate

p r e d i c t e d (1).

Surfaces of a c r y s t a l w i t h low v a p o r pressure have v e r y few

clusters

two-dimensional

and

nucleation

is

almost

impossible.

Indeed,

dislocation-free crystals c a n often r e m a i n i n a m e t a s t a b l e e q u i l i b r i u m w i t h a s u p e r s a t u r a t e d v a p o r for long periods of t i m e .

N u c l e a t i o n c a n be i n d u c e d

by resorting to a v a p o r w i t h a v e r y large s u p e r s a t u r a t i o n , but this often has undesirable

side effects.

Instabilities i n the

interface

shape result

in a

d e g r a d a t i o n of the q u a l i t y a n d u n i f o r m i t y of c r y s t a l l i n e m a t e r i a l . O n m e t h o d t o f a c i l i t a t e c r y s t a l g r o w t h i n a c r y s t a l - v a p o r system is to grow at h i g h t e m p e r a t u r e s .

T h e large e q u i l i b r i u m v a p o r pressure causes

more atoms t o adsorb o n the surface, a n d the p r o b a b i l i t y of finding large clusters is increased. that

a surface

In fact, B u r t o n et a l . (1) a n d J a c k s o n (2) p r e d i c t e d

phase t r a n s i t i o n occurs at

h i g h t e m p e r a t u r e s where

the

adsorbed atoms o c c u p y a b o u t 5 0 % of the a v a i l a b l e sites, p r o v i d e d t h a t the c r y s t a l does not melt at a lower t e m p e r a t u r e .

A l t h o u g h their calculations

were m a i n l y derived from a model l i m i t e d t o a single l a y e r of a d a t o m s on the surface

of a perfect

c r y s t a l , later w o r k confirmed the existence of a

roughening t r a n s i t i o n i n m u l t i l e v e l surfaces (3,4). T y p i c a l surfaces observed i n Ising model s i m u l a t i o n s are i l l u s t r a t e d i n F i g . 2.

T h e size a n d extent of a d a t o m a n d v a c a n c y clusters increases w i t h

the t e m p e r a t u r e . A b o v e a t r a n s i t i o n t e m p e r a t u r e T

R

face i l l u s t r a t e d ) , the clusters percolate.

( Τ « 0 . 6 2 for the sur­ Λ

T h a t is, some of the clusters l i n k Above T , R

cry­

s t a l g r o w t h c a n proceed w i t h o u t t w o - d i m e n s i o n a l n u c l e a t i o n , since

up t o produce a connected n e t w o r k over the entire surface.

large

clusters are a n inherent p a r t of the interface s t r u c t u r e . are expected at a r b i t r a r i l y s m a l l values of the M o d e l c a l c u l a t i o n s of the g r o w t h rate R are

plotted

as

Δ μ ~ \n(p /p ), e

a

function

where ρ a n d p

of e

the

F i n i t e g r o w t h rates

supersaturation. are s h o w n i n F i g . 3.

driving

force

for

These

crystallization,

are the a c t u a l a n d e q u i l i b r i u m v a p o r pres­

sures, respectively. A t v e r y low temperatures, the g r o w t h rate is essentially

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

220

Fig. 1

S c h e m a t i c representation of a c r y s t a l surface i n c l i n e d at a s m a l l

angle t o a low-index c r y s t a l l o g r a p h i c o r i e n t a t i o n .

Fig. 2 In

T y p i c a l Ising model surfaces p r o d u c e d b y c o m p u t e r s i m u l a t i o n s .

this system

s h o w n o n the

T#=0.62, i n terms of the

dimensionless

temperature

figure.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

13.

GILMER AND GRABOW

221

Crystal Growth and Thin Films

zero for s m a l l Δ μ , as p r e d i c t e d b y n u c l e a t i o n t h e o r y .

A t higher tempera­

tures measurable g r o w t h occurs at s m a l l e r values of Δ μ , a n d above T

R

the

m e t a s t a b l e region is absent (5). T h e m o r p h o l o g y of crystals is affected b y the t e m p e r a t u r e .

A t low

t e m p e r a t u r e s a growing c r y s t a l is b o u n d e d b y the close-packed planes t h a t move most s l o w l y .

F a s t - m o v i n g orientations t h a t m a y be present o n the

i n i t i a l c r y s t a l surface

move t o the

c r y s t a l edges, d i s a p p e a r i n g from

the

g r o w t h f o r m . A t moderate t e m p e r a t u r e s the d i s p a r i t y between the kinetics o n different faces is reduced; the c r y s t a l assumes a more c o m p a c t shape a n d Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

r o u n d e d edges are present.

A t h i g h t e m p e r a t u r e s some of the

bounding

faces m a y disappear because of surface roughening, a n d i f a l l of the faces are r o u g h the c r y s t a l assumes a shape t h a t is n e a r l y s p h e r i c a l , a c c o r d i n g to this m o d e l (see below). morphology w i t h

U s i n g this a p p r o a c h , J a c k s o n (2) has

a n o r m a l i z e d interface

correlated

t e m p e r a t u r e for crystals grown

from the m e l t .

Molecular Dynamics Studies of Interfaces Ising models w i t h e l e m e n t a r y l a t t i c e structures are not a p p r o p r i a t e for c a l c u l a t i o n s of the

influence of surface

other complex surface structures.

stress, surface

reconstruction

In most s i m u l a t i o n s , the surface

or

structure

is represented o n l y b y the presence or absence of atoms at b u l k l a t t i c e sites, a l t h o u g h more general structures c a n be i n c l u d e d b y the use of a fine grid lattice.

A n i m p o r t a n t factor i n v a p o r g r o w t h systems is the rate of mass

transport

along the surface t o the active g r o w t h sites.

T h e m i g r a t i o n of

atoms along the surface c a n be i n c l u d e d as a n a d d i t i o n a l M o n t e C a r l o event i n Ising model s i m u l a t i o n s . H o w e v e r , the rate constants for this process a n d t h e i r dependence o n the local surface configuration must be assigned i n a somewhat

arbitrary

manner.

Molecular

dynamics

calculations

permit

u n a m b i g u o u s measurements of the surface t r a n s p o r t of atoms, a l t h o u g h the a p p l i c a b i l i t y of the results depends o n the v a l i d i t y of the i n t e r a t o m i c poten­ tial employed. We

now describe a r e l a t i v e l y simple M D model of a low-index c r y s t a l

surface, w h i c h was conceived for the purpose of s t u d y i n g the rate of mass t r a n s p o r t (8).

The

effect

several c o m p e t i n g processes. jectories s o m e w h a t , sidered.

of t e m p e r a t u r e

o n surface

transport

involves

A rough surface s t r u c t u r e complicates the t r a ­

a n d the diffusion of clusters of atoms must be con­

In order to simplify the model as m u c h as possible, but r e t a i n the

essential d y n a m i c s of the mobile atoms, we w i l l consider a model i n w h i c h the

atoms

move o n a "substrate" represented

by an analytic

potential

energy f u n c t i o n t h a t is adjusted t o m a t c h t h a t of a surface of a (100) facecentered cubic c r y s t a l composed of atoms i n t e r a c t i n g w i t h a L e n n a r d - J o n e s

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

222

SUPERCOMPUTER RESEARCH

( L J ) p o t e n t i a l (6).

T h e diffusing atoms also have L J forces between t h e m .

A t o m s i n t e r a c t w i t h a ghost a t o m i n the substrate t h a t is subjected to r a n dom a n d dissipative forces t h a t closely m a t c h the forces exerted b y a neighb o r i n g shell of atoms i n the c r y s t a l . In this w a y the M D c o m p u t a t i o n is l i m i t e d to a r e l a t i v e l y s m a l l n u m b e r of mobile atoms a n d t h e i r ghost atoms, a n d the influence of the large n u m b e r of atoms i n the c r y s t a l is represented by the forces a p p l i e d to the ghost a t o m . T u l l y el a l (7) have s t u d i e d the m o t i o n of single atoms a n d s m a l l clus-

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

ters i n s u c h a system, a n d found t h a t the diffusion rates have a n A r r h e n i u s temperature

dependence.

Although

longer j u m p

distances

h i g h temperatures,

at

it is true t h a t

adatoms

with

an

experience

average

j u m p of

a p p r o x i m a t e l y four a t o m i c diameters at the m e l t i n g point T , there is no M

a n o m a l y i n the t e m p e r a t u r e dependence.

C l u s t e r s of t w o to six atoms were

found t o diffuse at a slower rate, as m i g h t be expected, b u t c o u l d alter the t o t a l mass t r a n s p o r t i f present i n large quantities. The model.

essential influence of surface roughening is also present

i n this

G r a n d c a n o n i c a l M o n t e C a r l o c a l c u l a t i o n s were used t o generate

a d a t o m p o p u l a t i o n s at v a r i o u s temperatures up t o T . m

C h e m i c a l potentials

corresponding to those i n the b u l k L J c r y s t a l were used, a n d these p r o d u c e d adatom

densities t h a t increased w i t h t e m p e r a t u r e

and roughly approxi-

m a t e d the values observed i n Ising model s i m u l a t i o n s below A

plot of the

adatom

density versus

L

T~

Tr.

is s h o w n i n F i g . 4.

anomalous increase i n the density is observed at h i g h temperatures.

An The

dashed line represents the a d a t o m p o p u l a t i o n t h a t w o u l d be p r e d i c t e d i f there were no l a t e r a l i n t e r a c t i o n s . H o w e v e r , the L J p o t e n t i a l between a d a toms tends t o s t a b i l i z e t h e m at the higher coverages, a n d i t is this effect t h a t causes the d e v i a t i o n from A r r h e n i u s b e h a v i o r at h i g h temperatures.

A

s i m i l a r t e m p e r a t u r e dependence is observed i n the rate of mass t r a n s p o r t o n some

m e t a l surfaces (8,9), a n d

it is possible t h a t

it is caused

by

the

H o w e v e r , the increased n u m b e r of a d a t o m s at h i g h temperatures

can

e n h a n c e d p o p u l a t i o n of the superlayer at h i g h temperatures.

influence t h e i r m o b i l i t y , since clusters of L J atoms were observed to have s m a l l e r diffusion coefficients t h a n isolated atoms.

F i g u r e 5 shows the aver1

age diffusion coefficients of a d a t o m s , also as a f u n c t i o n of T"" ; here the d e v i a t i o n from A r r h e i n u s b e h a v i o r is i n the other d i r e c t i o n . T h e rate of mass t r a n s p o r t is the p r o d u c t of these t w o factors, the dens i t y of atoms a n d the diffusion coefficient per a t o m , as s h o w n i n F i g . 6. O v e r a large t e m p e r a t u r e i n t e r v a l up to T

M

almost perfectly A r r h e n i u s i n n a t u r e . at h i g h temperatures atoms.

the mass t r a n s p o r t coefficient is

T h e e n h a n c e d a d a t o m concentrations

are offset b y the lower m o b i l i t y of the

interacting

T h u s , surface roughening does not appear to cause anomalies i n the

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

13.

GILMER AND GRABOW

Fig. 3

Crystal

Growth and Thin Films

223

Ising model c a l c u l a t i o n s of the n o r m a l i z e d g r o w t h rate R as a

f u n c t i o n of the

d r i v i n g force.

closed circles, 0.54 T , R

T h e surface

t e m p e r a t u r e s are 0 . 4 0 Τ , Λ

open circles, a n d 1.08 T , R

squares.

-I

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

224

ι

ι

ι

1.5

2

23

ι

I

3

Φ/kT

Fig. 5

f

Diffusion coefficient D for atoms o n surface.

1.5

(T ) M

Fig. 6

2

2.5

3

Φ/kT

M a s s t r a n s p o r t coefficient of surface.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

13.

GILMER ANDGRABOW

Crystal Growth and Thin

225

Films

mass t r a n s p o r t rate, a l t h o u g h it is possible t h a t one of the t w o effects could d o m i n a t e i f a different i n t e r a t o m i c p o t e n t i a l were present.

Crystal Growth T h e k i n e t i c s of c r y s t a l g r o w t h have not been o b t a i n e d b y M D techniques for c r y s t a l - v a p o r systems, because of the v e r y slow g r o w t h rate a n d the extensive c o m p u t a t i o n required.

T h e r e l a t i o n s h i p between the g r o w t h

rate a n d d r i v i n g force as s h o w n i n F i g . 2 requires g r o w t h s i m u l a t i o n s at s m a l l values of the d r i v i n g force.

T h e difficulty of a direct M D c a l c u l a t i o n

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

is the large a m o u n t of c o m p u t a t i o n required t o o b t a i n the a t o m i c tories.

trajec-

T h e complex m o t i o n of a n a t o m t h a t impinges from the v a p o r a n d is

i n c o r p o r a t e d i n t o the growing c r y s t a l is represented i n the Ising model b y a simple s p i n

flip.

T h e M D s i m u l a t i o n s of v a p o r g r o w t h t h a t have

been

a t t e m p t e d t o date were performed w i t h a n i n c i d e n t flux eight t o t e n orders of

magnitude

larger

than

that

of the

equilibrium vapor.

Under

such

extreme conditions the e v a p o r a t i o n flux is essentially zero, corresponding to a dimensionless g r o w t h rate of u n i t y .

S u r p r i s i n g l y , L J systems were found

t o grow w i t h r e l a t i v e l y w e l l ordered layers under these conditions, a l t h o u g h the s t a c k i n g of adjacent

layers d i d not correspond to a regular space lat-

tice (10). Different results were o b t a i n e d i n the case of the S t i l l i n g e r - W e b e r ( S W ) p o t e n t i a l for s i l i c o n (10). I n this case, m a t e r i a l deposited o n the (111) orient a t i o n was disordered, w i t h o u t d i s t i n c t layers of atoms.

G r o w t h o n the

(100) face d i d produce about t e n d i s t i n c t layers at sufficiently h i g h temperatures, a l t h o u g h i t is not clear whether t h i c k e r deposits w o u l d r e t a i n this order since some degeneration was observed as successive layers were deposited. Molecular

dynamics

c a l c u l a t i o n s of solidification

at

a

crystal-melt

interface have been performed; the faster g r o w t h k i n e t i c s of this system make

it

possible t o

c r y s t a l l i z e a significant a m o u n t

presently a v a i l a b l e c o m p u t e r technology.

of m a t e r i a l

using

L a n d m a n et a l (11) have s i m u -

l a t e d the m o t i o n of atoms i n a slab of supercooled l i q u i d t h a t was p l a c e d i n contact

w i t h a c r y s t a l surface.

O r d e r i n g of the

atoms

i n t o layers was

observed first, a n d t h e n the l o c a l i z a t i o n of the atoms at l a t t i c e sites w i t h i n the layers. A n interface speed of « 1 0 0 m /s was e s t i m a t e d d u r i n g the early stages of o r d e r i n g w h e n p a r a m e t e r s a p p r o p r i a t e

for argon were

inserted.

T h e m e l t i n g a n d resolidification of a two-component s y s t e m has also been s i m u l a t e d (12). Steady-state

c r y s t a l l i z a t i o n rates were measured for a range of tem-

peratures below the m e l t i n g point b y B r o u g h t o n et a l (13). A face-centered cubic (100) c r y s t a l - m e l t interface was e q u i l i b r a t e d i n a box elongated i n the

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

226

SUPERCOMPUTER RESEARCH

direction

normal

to

the

interface.

Periodic boundary

conditions

were

a p p l i e d i n the p a r a l l e l directions. P a r t i c l e s at the t w o ends of the box were coupled to a heat b a t h at a fixed t e m p e r a t u r e T dissipative forces. below T .

0

by means of r a n d o m a n d

C r y s t a l g r o w t h was observed

when

T

was

0

reduced

N e w l i q u i d particles were s u p p l i e d at the lower end of the box

m

at a rate t h a t was adjusted to keep the interface roughly at the center of the

box.

Crystalline material

extruding

from the

top

of the

box

was

T h e measured g r o w t h rates are i l l u s t r a t e d by the circles i n F i g . 7.

The

removed.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

interface v e l o c i t y is p l o t t e d versus the interface t e m p e r a t u r e T . of Τ is a l w a y s greater t h a n T

0

the interface. The

T h e value

because of the release of the latent heat at

Dimensionless units for Τ a n d the v e l o c i t y are used here.

m a x i m u m v e l o c i t y corresponds

s u r p r i s i n g aspect is the

The

most

r a p i d c r y s t a l l i z a t i o n at low t e m p e r a t u r e s .

to

~ 8 0 r a /s

for

argon.

Most

m a t e r i a l s e x h i b i t s h a r p l y reduced rates at low t e m p e r a t u r e s , as expected for a n a c t i v a t e d g r o w t h process.

T h a t is, the k i n e t i c s c a n be represented as

the p r o d u c t of a n A r r h e n i u s factor F(T)

a n d a t e r m t h a t accounts for the

net p r o d u c t i o n of c r y s t a l l i n e m a t e r i a l as a result of the atoms ordering a n d disordering at the interface, R =^(Γ)[1-βχρ(-Δμ/*Γ)]

(1)

T h e A r r h e n i u s factor is often represented b y a n expression of the type

F(T) = Z>«/ /A

2

(2)

0

where D is the diffusion coefficient i n the l i q u i d a n d Λ is the m e a n path.

It

is assumed

that

atoms i n the

adjacent

free

l i q u i d of thickness 2

impinge o n the c r y s t a l surface at a rate p r o p o r t i o n a l to D/A .

a

A site fac­

tor / o < l is i n c l u d e d to account for thee fact t h a t some of these collisions do not c o n t r i b u t e t o c r y s t a l g r o w t h , either because t h e y are not sufficiently close t o a lattice site or because the region neighboring the site is r e l a t i v e l y disordered.

T h e low e n t r o p y of fusion Δ 5 = 1 . 6 2 (13) insures t h a t the sur­

face t e m p e r a t u r e is well above the n o r m a l i z e d roughening t e m p e r a t u r e (2), a n d therefore the g r o w t h sites are not l i m i t e d to the edges of steps s u p p l i e d by a l a t e r a l g r o w t h m e c h a n i s m s u c h a t w o - d i m e n s i o n a l n u c l e a t i o n or s p i r a l growth.

T h e diffusion coefficient i n the l i q u i d has been measured over the

range from T

d o w n to 0.4 T , where the h i g h v i s c o s i t y prevents a c c u r a t e

measurements.

A n A r r h e n i u s expression fits the d a t a , a n d c a n be used to

m

m

e x t r a p o l a t e to lower t e m p e r a t u r e s .

T h e d r i v i n g force Δ μ c a n be c a l c u l a t e d

a c c u r a t e l y using t h e r m o d y n a m i c d a t a for the b u l k c r y s t a l a n d the super­ cooled l i q u i d . T h e solid curve i n F i g . 7 is a plot of eqs. ( l ) a n d (2), w i t h the values of Δ μ a n d D o b t a i n e d as described above.

T h e supercooled L J l i q u i d becomes

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

13.

GILMER AND GRABOW

Crystal Growth and Thin

227

Films

>-

u ο

_J

lxl >

0

0.2

0.4

0.6

TEMPERATURE

Fig. 7

M o l e c u l a r d y n a m i c s c a l c u l a t i o n s , open circles, for the v e l o c i t y of

the c r y s t a l - m e l t interface versus the t e m p e r a t u r e of the interface. s o l i d curve corresponds to eq. (1) a n d the dashed curve to eq. (3).

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

The

SUPERCOMPUTER RESEARCH

228

h i g h l y viscous at low temperatures whereas the measured interface veloci­ ties i n t h i s region are quite large.

C r y s t a l i z a t i o n is a p p a r e n t l y not l i m i t e d

by a n a c t i v a t e d process. E v e n a s m a l l a c t i v a t i o n b a r r i e r w o u l d reduce the g r o w t h rate significantly at low temperatures.

T u r n b u l l a n d B a g l e y (14)

h a d a r g u e d t h a t c r y s t a l l i z a t i o n of simple melts s h o u l d not be l i m i t e d b y the liquid

diffusion

rates,

since the

movement

across

the

interface

is less

impeded by backscattering. T h e large m o b i l i t y i n the interface is a p p a r e n t l y the result of a density deficit i n this region.

A t the m e l t i n g point the i n t e r f a c i a l densities are

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

i n t e r m e d i a t e between the t w o b u l k phases.

T h e r e is no i n d i c a t i o n of voids,

a n d the m o b i l i t y of the i n t e r f a c i a l atoms is a p p r o x i m a t e l y the same as t h a t of the b u l k l i q u i d .

H o w e v e r , as the temperature is reduced below T , the m

density of the interface region drops, a n d at Τ =0.15 it is 5 % lower t h a n t h a t of the l i q u i d phase.

T h u s , the a m o u n t of free v o l u m e a v a i l a b l e for

a t o m i c m o t i o n increases at low temperatures.

T h i s e x t r a free v o l u m e could

be caused b y the large v i s c o s i t y o f the l i q u i d phase.

A t large g r o w t h rates

there is l i t t l e t i m e for the l i q u i d t o accomodate t o c r y s t a l l i n e m a t e r i a l , a n d hence the interface energy is h i g h .

E n h a n c e d diffusion of atoms at

the

interface between t w o solid phases is c o m m o n l y observed i n experiments o n g r a i n b o u n d a r y diffusion (15). In the absence of a p o t e n t i a l b a r r i e r , the rate at w h i c h l i q u i d atoms i n the interface c o u l d move t o l a t t i c e sites is d e t e r m i n e d b y the average ther­ m a l v e l o c i t y , (ZkT/πιγ.

If t h e y t r a v e l a distance λ, the interface v e l o c i t y

is e

A

r

3

R = (a A ) ( 3 * r / m ) V o [ l - x p ( - M / * ) ]

( )

T h e dashed curve i n F i g . 7 is o b t a i n e d w h e n λ = 0 . 4 α , the average distance from the center of points u n i f o r m l y d i s t r i b u t e d i n a sphere, a n d

/ =0.27. 0

T h i s expression is i n good agreement w i t h the d a t a over the full range of T , and

has no a c t i v a t i o n b a r r i e r whatsoever.

A p p a r e n t l y the atoms i n the

l i q u i d c a n rearrange i n t o a c r y s t a l l a t t i c e along a p a t h i n configuration space t h a t involves a monotonie r e d u c t i o n i n energy.

T h i s unexpected con­

clusion applies o n l y t o simple atoms a n d molecules. T h e c r y s t a l l i z a t i o n of ordered alloys w o u l d involve the diffusion of atoms t o the correct s u b l a t t i c e sites, a n d m a y involve a n a c t i v a t i o n energy. S i m i l a r l y , the c r y s t a l l i z a t i o n of most m o l e c u l a r crystals requires a r e o r i e n t a t i o n of the molecules, a n d w o u l d also be i n h i b i t e d at low temperatures. If eq. (1) were a p p l i c a b l e to other m a t e r i a l s , a p p r o x i m a t e values of the maximum

g r o w t h rates

could

be

obtained

by scaling w i t h

%

(T /m) . m

A c c o r d i n g l y , we estimate m a x i m u m rates of 400 m / s for n i c k e l a n d 430 m / s for

silicon.

Interface

velocities of 50 m / s

have been measured

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

for N i

13.

GILMER AND GRABOW

Crystal Growth and Thin

dendrites growing i n t o a supercooled melt at 0.9 T

m

229

Films

(16). T h e v e l o c i t y d a t a

of F i g . 7 i m p l y t h a t the interface t e m p e r a t u r e is « 0 . 9 6 T . m

0

T h i s is a 70 C

undercooling at the interface, a s u r p r i s i n g l y large v a l u e for s u c h a simple system.

T h e m a x i m u m v e l o c i t y measured for silicon is o n l y 18 m / s , a n d

this indicates t h a t the m o b i l i t y of interfaces i n covalent m a t e r i a l s is m u c h s m a l l e r t h a n i n simple metals a n d the noble gases. An

increase i n the

n u m b e r of l a t t i c e defects was noted d u r i n g the

g r o w t h of the L J c r y s t a l at low t e m p e r a t u r e s .

The crystallizing material

formed at Τ =0.05 c o n t a i n e d 0 . 5 % vacancies, whereas the e q u i l i b r i u m con­ Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

centration observed.

at

this t e m p e r a t u r e

is less t h a n

3 0

10~ .

T w i n n i n g was

also

In every instance this process began as a defect i n v o l v i n g a single

row of atoms i n a p a r t i a l l y ordered (100) layer.

T h e atoms were displaced

by a distance a / 2 i n a d i r e c t i o n p a r a l l e l to the row. T h e next layer to crys­ t a l l i z e u s u a l l y c o n t a i n e d t w o adjacent

rows d i s p l a c e d , a n d so o n u n t i l the

entire layer was i n the new p o s i t i o n a n d the c r y s t a l was a g a i n perfect. Significant

reductions

i n the

defects were i n the interface.

g r o w t h rate were

T h i s contrasts

observed while

w i t h the usual

the

assumption

t h a t defects enhance the g r o w t h b y p r o v i d i n g sources of steps or preferred n u c l e a t i o n sites.

In this case the i n t r i n s i c a l l y rough interface a l r e a d y con­

t a i n s a h i g h density of good g r o w t h sites.

T h e s t r a i n a n d reduced bonding

i n the defective c r y s t a l reduce the difference i n the free energy between the c r y s t a l a n d the melt, a n d therefore the effective d r i v i n g force for c r y s t a l l i z a ­ t i o n is reduced.

A n o t h e r factor is the presence of new sites at the g r o w t h

interface where the p o t e n t i a l energy is a m i n i m u m , but w h i c h are not con­ sistent

with

closely spaced

neighboring sites.

C o m p e t i t i o n between

the

different sites for l i q u i d atoms a p p a r e n t l y retards the c r y s t a l l i z a t i o n process. R e c e n t u n p u b l i s h e d d a t a for the (111) interface demonstrates this T h e close p a c k e d layers m a y s t a c k i n the face-centered sequence, or as ababab...

cubic

effect.

abcabc...

to f o r m hexagonal close p a c k e d m a t e r i a l .

The

energy difference between these t w o lattices is extremely s m a l l , a n d indeed the

crystallized material

does

not

have

a

regular

stacking

sequence,

a l t h o u g h the i n d i v i d u a l close p a c k e d planes are essentially perfect. T

m

Near

the g r o w t h rate on this face is 5 0 % of t h a t o n the (100), a n d at low

t e m p e r a t u r e s it has the f o r m expected for a n a c t i v a t e d process.

T h u s , the

(100) a n d (111) faces have q u a l i t a t i v e l y different b e h a v i o r : the (100) g r o w t h is l i m i t e d o n l y b y the rate at w h i c h l i q u i d atoms c a n a r r i v e at g r o w t h sites, whereas the (111) g r o w t h requires t h a t a n a c t i v a t i o n energy b a r r i e r be sur­ mounted.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

230

Thin Films E p i t a x i a l g r o w t h of t h i n films u s u a l l y involves the f o r m a t i o n of s t r a i n e d material

as

a result

o f m i s m a t c h between

the

film

and substrate

because of the large surface to v o l u m e r a t i o i n the film.

and

Surface stress c a n

be a major factor, even w h e n the l a t t i c e constants of film a n d s u b s t r a t e are perfectly m a t c h e d .

A l t h o u g h it appears t o be difficult

to eliminate

the

stress t o t a l l y , it is i m p o r t a n t to be able to c o n t r o l it a n d even use i t t o produce desired qualities. W e have seen t h a t the deposition of crystals from the v a p o r is m u c h

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch013

too slow t o model b y M D techniques.

M o s t l a b o r a t o r y equipment for pro-

d u c i n g t h i n films involves r e l a t i v e l y slow c r y s t a l g r o w t h processes, a n d is not s u i t a b l e for direct s i m u l a t i o n . I n f o r m a t i o n o n the s t a b i l i t y a n d properties of t h i n films c a n be o b t a i n e d b y s i m i l a r m o d e l i n g techniques, however. W e describe below some of our results t h a t provide necessary d a t a to find the e q u i l i b r i u m configuration of t h i n films at low t e m p e r a t u r e s . T h e m o t i o n of particles of the film a n d s u b s t r a t e were c a l c u l a t e d b y s t a n d a r d m o l e c u l a r d y n a m i c s techniques.

In the s i m u l a t i o n s discussed here,

our purpose is to c a l c u l a t e e q u i l i b r i u m or m e t a s t a b l e configurations of the s y s t e m at zero K e l v i n .

F o r this purpose, we have a p p l i e d r a n d o m a n d dissi-

p a t i v e forces t o the p a r t i c l e s .

F i n i t e r a n d o m forces provide the

thermal

m o t i o n w h i c h allows the system t o explore different configurations, a n d the d i s s i p a t i o n serves

t o s t a b i l i z e the

system

at

a

fixed

temperature.

The

p o t e n t i a l energy m i n i m a are p o p u l a t e d b y r e d u c i n g the r a n d o m forces to zero, t h u s p e r m i t t i n g the d i s s i p a t i o n to absorb the k i n e t i c energy. A l l particles of the film a n d s u b s t r a t e i n t e r a c t w i t h L J potentials, a n d for p a r t i c l e s t a n d j w i t h s e p a r a t i o n r - this p o t e n t i a l is tJ

12

6

M'y) =4 ^[(Wr,y) -( defined as the average o f the l Legendre polynomial o f cos Θ : ,h

β

C?(t) =

< Pi ( cos

θ )> α

h

where 0 is the angle between the a' principal axis at some time t = 0 and at a later time t. Those are given in Table I for /=1 and 2. The decay o f the a

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

SUPERCOMPUTER RESEARCH

246

orientational correlation is significantly slowed by the three-body forces, and the relaxation times more than doubled in the three-body liquid. The time integral of Q W gives an estimate of the N M R relaxation time, x , associated with intermolecular dipolar coupling. Those are also given in Table I where it can be seen that the three-body forces significantly improve agreement with the experiment. Let us now consider the velocity autocorrelation function ( V A C F ) obtained from the M C Y L potential, (namely, with the inclusion of vibrations). Figure 3 shows the velocity autocorrelation function for the oxygen and hydrogen atoms calculated for a temperature of about 300 K . The global shape of the V A C F for the oxygen is very similar to what was previously determined for the M C Y model. Very notable are the fast oscillations for the hydrogens relative to the oxygen. The Fourier transform of the V A C F , namely the spectral density, are also given in Figure 3. A s known, those are related to the infrared spectrum o f the liquid water. The band centered at about 1,740 cm-' is the intramolecular bending mode while those at 3,648 cm- and 3,752 cm* are associated with the intramolecular O - H bond stretches. In going from gas to liquid phase there is an upshift of 55 cm- in the bending frequency and downshifts o f 198 and 203 c n r in the stretching frequencies. These shifts are all in good agreement with experimental I R and Raman results of 50, 167, and 266 c m , respectively.^ They are also better than shifts obtained from other semi-empirical potentials,* * such as ST2. Notice that, since the center of mass o f the water molecule is very close to the oxygen atom, the drastic intensity difference between the Fourier transform of hydrogen and oxygen in Figure 3 allows us to identify immediately that the broad band centered about 500 c n r is due mainly to the rotational motion of the molecules, whereas the bands centered around 40 and 190 c n r arise from the hindered translational motions.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch014

NMR

1

1

1

1

1

21

1

1

T O W A R D S T H E H Y D R O D Y N A M I C LIMIT: S T R U C T U R E FACTORS A N D SOUND DISPERSION. The collective motions of water molecules give rise to many hydrodynamical phenomena observable in the laboratories. They are most conveniently studied in terms o f the spatial Fourier (k) components o f the density, particle currents, stress, and energy fluxes. The time correlation function o f those Fourier components detail the decay o f density, current, and fluctuation on the length scale of the \/k. Figure 4 (left) shows the spatial Fourier transform of the density correlation function, F(fc,r)/S(fc), for the two- and three-body models of water, for the first few wavevectors k = 0.253, 0.358, 0.438, 0.506, 0.566À-' allowed due to the use of the periodic boundary conditions in the simulation. In this range o f k values both systems exhibit an initial rapid decay followed by a slower oscillatory decay. The rapid decay and oscillations are associated with the compressive elasticity o f the fluid while the slower decay at longer times is related to the diffusive mixing o f the molecules. The most apparent difference between the twoand three-body liquids is the overall slower decay of spatial order in the latter system. The first minima and maxima in F(fc, /) are shifted to somewhat smaller times, indicating that at these wavelengths the three-body liquid is less compressible. The oscillations in F(fc, /) are generally more damped in the threebody liquid. Thus, sound waves will travel faster and will be more strongly damped than in the two-body liquid.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

CLEMENTI A N D LIE

New Horizons for Computational Chemistry

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch014

14.

Figure 3. Hydrogen and oxygen velocity autocorrelation function from twobody M C Y with vibrations allowed ( M C Y L ) , and computed infrared spectrum for intramolecular bending modes and bond stretching.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

247

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Figure 4. Intermediate scattering function (left), F(k,t) and dynamic structure factor (right), S(k,(o), computed from M C Y with and without three-body corrections.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch014

w >

W

H W

8

M

to 00

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch014

14.

CLEMENTI AND LIE

New Horizons for Computational

Chemistry

249

Figure 4 (right) shows F(k, t) for values o f k near the first peak i n the x-ray and neutron scattering spectra. The rapid and slow decay processes merge into a more gradual decay o f quite long duration. This long decay is evidence o f the persistence o f the first shell o f neighbors for the smaller k shown. The even slower decay at larger k indicates that, on average, pairs o f molecules remain together for quite long times. The slower decay for the three-body liquid is again explained by the stronger intermolecular binding i n that system. A t still larger k values the F(k, r) begin to look the same for the two liquids, as is shown in Figure 4 (left, c). This is as expected because on very small length scales all systems asymptotically exhibit ideal gas behavior, irrespective o f the intermolecular potential. The temporal Fourier transform o f the F(k,t) shows the propagation o f normal sound modes at k 0 and t -* 0. A t higher frequencies and shorter wave lengths the fluid no longer behaves as a continuum and the effects o f local structure become important, which could then give rise to the phenomena o f sound dispersion. The sound speed calculated from the M C Y potential at k~0Ak~\ about 3000 m/sec, was originally thought o f as a defect o f the p o t e n t i a l ^ when compared to the experimental value o f 1500 m/sec. However, i n view o f the positive dispersion observed in the two body M C Y liquid for 0.25 V

n u n m i 11 4 4 4 III l 11 n u n 111111

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1

jiitmtm

1V 1*

i (UM« UUM «> ι

ιιν*·*·»···»*]

i4 Π M M H

Figure 6. Central region showing eddy pair and location of the circular obstacle at t « 230 psec (left) and a single eddy and portion of the oscillatory wake at t = 740 psec (right). The flow is from top to bottom.

In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., el al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch014

14.

CLEMENTI AND LIE

New Horizons for Computational

Chemistry

253

It should be noted that in the spirit of the global simulation, all the interaction potentials used are obtained from ab initio computations, although at the present time only at the S C F level with minimum basis sets. The systems reported here are a single turn o f B - D N A with G - C , A - T base pair sequence and the left handed Z - D N A with G - C base pair sequence. The B - D N A system is simulated for 4.0 psec and Z - D N A is simulated for 3.5 psec after equilibration. The simulation results are then analyzed for structural and dynamical properties.*^ F r o m a global analysis o f the distribution o f water molecules within 3.0À from the D N A segment, it has been found that there are 241 and 292 oxygen atoms of the water molecules surrounding B - and Z - D N A respectively. The phosphate-water radial distribution functions indicated a layer o f hydrogen atoms in the region between the counterion and the phosphate group, which shows up clearly as a peak at r~2.8 Â in the radial distribution functions of hydrogen atoms around the phosphorous atom. The analysis o f the ion-water radial distribution functions suggest 3.8 and 4.9 for the number of water molecules within the first hydration shell of the ion in B - and Z - D N A respectively. The corresponding number is 7.5 in the case o f an ionic solution. A n analysis o f the hydration structure o f water molecules in the major and minor grooves i n B - D N A has shown that there is a filament o f water molecules connecting both the inter and the intra phosphate groups of the two strands of B - D N A . However, such a connectivity is absent in the case o f Z - D N A confirming earlier M C simulation results. The probability density distributions of the counterions around D N A shows deep penetration o f the counterions in Z - D N A compared to B - D N A . Further, these distributions suggest very limited mobility for the counterions and show well defined counter-ion pattern as originally suggested in the M C study. ^> For the analysis of the dynamical properties o f the water and ions, the simulation cell is divided into eight subshells o f thickness 3.0Â and of height equal to the height o f one turn o f D N A . The dynamical properties, such as diffusion coefficients and velocity autocorrelation functions, of the water molecules and the ions are computed in various shells. F r o m the study of the dipole orientational correlation function (

C(t) =