A STUDY OF THE FUNDAMENTAL VIBRATION-ROTATION BANDS IN THE INFRARED SPECTRA OF STIBENE AND DEUTERO-STIBENE

Citation preview

A STUDY OF THE FUNDAMENTAL VIBRATION-ROTATION BANDS IN THE INFRARED SPECTRA OF ST IBENE AND DEUTERO-STIBENE

DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By William Howard Haynle, S.F., M*Sc« The Ohio State University 1961

Approved byi

«w-

Involve changes of the electric dipole moment eso sentially parallel to the symmetry axis of the moleoule. They are 1

called parallel inodes of vibration and belong to the totally symmetric species A^.

Modes to ^ and to ^ involve changes of the dipole moment

essentially perpendicular to the axis of the top.

They are called

perpendicular modes of vibration and belong to thedegenerate species E.

Modes t*j

and to

are doubly degenerate and thus the 3N-G

•»6

modes are satisfaotorily aocounted for*

normal

It may be noted from Figure 1

that modes (_,j

and t*> have in common that they are primarily "bond A £ stretching" vibrations and the two vibrations should involve essen­

tially the same foroe constants.

One might expect that these two

vibrations would have nearly the same frequencies.

Modes u# ^ and

Lu ^ have in oossnon that they are both primarily "bond angle distor­ tion" vibrations and thus they might be expected to have frequencies of the same order of magnitude.

The data on phosphlne and arsine

•Superscript numbers refer to the bibliography •*The figures are grouped together at the and of the text. ••*The notation used to designate the normal frequencies is that of Dennison. 2

beor this out »!/.,♦ andy 11s within a Tew wave numbers of saoh X 4* other in the 4 to B nlcror* region, andj/^ and>/^, although farther apart, both lie in the 10 to 12 mioron region* X atom increases from NH^ to

As the mass of the

to Ash^, the corresponding normal

frequencies of theee molecules are successively lower*

Thus the

normal frequencies of SbH^ would be expected to be lower than the corresponding frequencies of AsH^* This investigation of stibene, the last chemically stable member of the family of tri-hydrides of the fifth group, was under­ taken with the purpose of determining as many of the constants of the moleoule from a study of the infrared absorption spectra as the data would permit* Previous experimental work on stibene is described in two papersi

one published by D*C* Smith®j the other by Loomis and

Strandberg®* Smith obtained prism spectra of SbHg in the region from 2 to IB microns and found two regions of Intense absorption* the

6

The first, in

mioron region, which has the appearance of a parallel band with

a strong Q branoh at 1690 om"^, he assumed to be a superposition of V

upon a weaker y • The second region extends from 10 to 14 microns X w with two strong absorption maxima at 761*6 and 631 oin”^. Smith interpreted this absorption region as an overlapping of the two vibrations of v j and y ^*

^Throughout this paperw^ will be used to indicate the observed frequency corresponding to the normal frequenoywj, 3

Loomis and Strandberg have obtained microwave absorption spectra of SbHgD along with the data on EfcjjD and AsH~D.

These data were used

to oaloulate the molecular geometry and the results extrapolated to SbHg on the reasonable assumptions that the force oonstants and geometry of SbH^D and Sbllg are the same*

Prom their data they

determined the moleoular constants listed in Table !•

TAELE I, ro **

a * FH

A H 0

O 1.419 A o 1.623 A a 1.712 A

93*5°

3

3

SbH,

92.0° 91.6°

*** is the Y-X-Y bond angle. the X-Y bond distance.

4

Part II •

Sumntry of Theoretical Work

Tho modora lnterpretatien of moleoular band apootra i« baaod upon tho roaulta of a quantum—moohanioaI thoory developed and roflnod over tho paot twenty-five yoaro by many investigators*

Application

of group thoory prinoiploa haa aupplouantod and abottod thia interpre­ tation*

It la intondod to proaont horo a briof outline of tho

quantum-meohanioal treatment, and a imply to quote the group theory reaults that will be of uae in thia paper* In order to prediot the appearanoe of or to interpret a moleoular vibratlon-rotatlon band two aeta of information are required, namely, an expreaaion for the allowed energiea and a aet of aeleotion rules* The allowed energioa or eigexnraluea of energy are aimply the quantised vibration-retation energy atatea in whioh the partioular moleoule may exist*

The aeleotion rulea govern the tranaitiona of the moleoule

from one energy atate to another* A*

The Energy Term Value Expreaaion In eaaenoe the problem of determining the allowed energy atatea

for a vibrating-rotating moleoule ia that of solving the Sohrodinger equation HY*EY

for that particular moleoule*

(1)

H ia the ao—oalled "quantum-meohanioal

Hamiltonian" and ia a function of the masses of the moleoule, the coordinates used to deeorlbe the moleoular oonfiguration, and the momenta, expressed in operator form, oonjugate to the coordinates* Nielsen

* 0

has aet up the Hamiltonian for the general polyatomio

moleoule and has separated the extremely oomplioated expreaaion into 6

three parts, H . H , and H , whioh contribute to the energy in v X £ zero, first and saoond order of approximation respectively* He has substituted this Hamiltonian into the Sohrodinger equation and has obtained an expression for the energy term value to seoond order of approximation*

Shaffer** has solved the problem speoifioally for

the pyramidal X Y _ moleoular model and has given the energy expression o to second order of approximation as followst

The quantities

•nd Fro^ ore, respectively, the vibrational and

rotational term values in wave numbers*

These term values are given

by the following expressionss

Grw b -G.+

(3)

= B v 3(3 + l) -t(C< -B,)K"-DJ3*(3 + 0*-DJRJ(3+l)K.*-D1(K H. (4) In the expression for GT it, the coefficients Gq ,

®22* *to* are

oonstants depending essentially upon the normal frequencies,^^, and the foroe oonstants in the potential energy expression! d^ is a weight faotor which denotes the degree of degeneraoy of the mode of vibratlonj (d^s d^»

2

1

1

th normal

when the oscillation is nondegensrate,

when the osoillation is twofold degenerate, etc*) v^ ia the

vibrational quantum number associated with the ig and

5

1

th normal mode, and

^ are quantum numbers associated with vibrational or internal 6

angular momentum in the two doubly degenerate is a positive integer (v j - O* 1, 2 ,— -) and oase or the degenerate frequencies*

modes

and w ^*

v^

is sere except in the

Here it may have values v^*

v^-2, v^—4* —---- * 1, or 0*

The quantities in the rotational term

F ro^_ are defined as follows t

J is the quantum number associated with

the total angular momentum of the molecule} K is the quantum number associated with the component of angular momentum directed along the top (z) axis;

and Dg are known as the oentrifugal dis­

tortion oonstants and depend upon the atomic masses* the normal frequencies* and the force oonstants of the moleoule} B and C are proportional to the reciprocals of the moments of Inertia and are defined asi

=

“ mViI

*= r & T ?

( ) 6

in terms of an xyz coordinate system with its origin at the oenter of mass of the moleoule and its z axis along the axis of the top* The subscripts and superscripts e and v indicate that the quantity refers to the equilibrium state or to the vth vibrational state respectively* In the two perpendicular modes of vibration* w ^ and w

4

*

the motion of the X atom is isotropic in two mutually perpendioular directions* both of whioh induce electric moments perpendicular to the axis of the top*

In other words* the X atom behaves as a two

dimensional isotroplo oscillator and thus during vibration it 7

desorlbea a circular path whoaa plana la parpandloular to tha axis ol* tha top*

Teller^* has shown that tha angular momentum arising

from thia circular motion has magnitude

where -l—

Tha 'S

the so—called Taller parameters,may be thought of qualitatively as measures of tha amounts by which tha dipole moment vector leads or lags the component of tha angular momentum vector of tha moleoular framework: parallel to axis of tha top*

A Corlolia interaction exists

moreover between this vibrational or internal angular momentum and the parallel component of framework angular momentum which gives rise to a first order correction to the energy*

This contribution to the

energy is represented by the last quantity in the energy expression* - -Vf $1 )£•

The 5 ^ may also be considered as measures of

the magnitudes of the Corlolls couplings* In the expression for the rotational term D j J2(J

♦ 1)2, DJKJ (J + 1)k2* and

*r# sin* H

the quantities corrections to the

energy whioh arise because rotation of the moleoular framework tends to alter the effective moments of inertia*

These centrifugal distor­

tion terms contribute to the energy in an order higher than was experi­ mentally detectable in this investigation and therefore they shall be omitted from the rotational term*

The energy term value expression*

to the approximation that will apply to the data* is then & . aB t B*

< ) 6

The Selection Rules Selection rules are obtained by considering the following type

of integrals Involving components of the dipole momenta 8

JV'rW'dt

^

(?)

whereY" •BdY* ar# 'th* eigenfunctions aesooiated with tha lowar and tha uppar energy atatas respectivelyj

ia a oomponant of tha dipola

momarrt; and dx ia tha volume element of tha configuration apaoa uaad to daaoriha tha moleoule.

A tranaition from tha lowar (doubla prima)

anargy atata to tha uppar (aingla prima) anargy atata is allowad only whan tho intagral doaa not vanish*

Thaaa integrals or matrix alamanta

have baan thoroughly investigated and the results are summarised below. Vibrational transitions are governed by tha rule AVi. -

.

( ) 8

Rotational transitions are governed by two sets of aeleotion rules, depending upon the nature of tha normal mode of vibration. Dennison^"® has derived the rotational selaotion rules and states that for modes of vibration in whioh the dipole moment is oscillating essentially parallel to tho top axis tha rules are

A. K = O,

A l = il

A K

A3

=o ,

;

- o,il

JJ^iC=0

j

.

An anargy transition from tha ground state to an exoited state of one of tho parallel modes of vibration ooupled with a simultaneous ohange of tha rotational anargy aooording to the above rules gives rise to a parallel vibration-rotation band. 9

for modas of vibration in whioh tho dipolo moment io oscillat­ ing essentially perpendicular to the axis of tho top tho selection rules are

j As-Ojti

do)

T«ll«r ^ h*a shown that tho solootlon rulo for tho lntornal angular momentum quantum number £. depends upon the K aeleotion rule as follows AS--M

*,

Jf»vAV

AS=~\

.

\

J

(ii)

Bands arising from this type of transition are oalled perpendicular bands• The energy term value expression together with the aeleotion rules and aqualitative cient toallow

idea of the relative intensities are suffi­

us topredict

the appearanoe of thevibratlon-rota-

tion bands of a pyramidal XY_ molecule* confined to the fundamental bands ( v

0

of bands will be considered separately*

The dlsoussion will be to t^I), and the two types Quantities identified by a

double prime are associated with the lower state and those with a single prime refer to the upper state of the particular transition being considered* C*

The Parallel Bands The final energy term value expression for a pyramidal XYj 10

molecule was found to bo

For tho parallel modtiw

and cj

value is zoroi i.e., tho intornal angular momentum Is sera* consider y^ as tho example of a parallel band*

Lot us

In the lower state

the term value has the form

(12 ) and in the upper state it is

(13) where G«

VJ represents the vibrational term value for v — 1*

For the transition

a v « + l and A K = 0, A«J=- 0, we obtain

(14) This transition gives rise to tho Q branoh of tho band*

It should

bo noted that If tho rotational oonstants have tho same values in tho ground state and in the first exalted state, i.e., if B’VB* and C s C 1, then the Q branoh should consist of a group of lines having tho same frequency y^»

In phosphino, arsine, and, as will be shown,

in stibene, B” is slightly greater than B* and also tho quantity [(C —C")-(B*-BH)3 1

is very nearly coro.

Thus tho Q branoh oonaista

of a sot of lines very closely spaced with respeot to oaoh other 11

(usually spsotrosooploally unresolvable)• For tha transition* A v »-■*■1 a nd A K- rAJ*11, tha frequencies 0

are given by V «

X ± 3"(B + B"i + 3*V6'-B"»+{(C,-C»)

K*s .

(16)

Ths transitions AJ - ^ 1 and A J- - 1 give rise to ths P and R branohss respectively, and these branohss frequency sidss of ths Q branoh*

1 1

s on ths low and ths high

Eaoh branoh consists of a sst of

lints, on« for saah value of Jn and saoh 11ns has J components corresponding to all ths possible values of K. greater than B 1, the term

Slnoe BH is usually

oauses tha P branoh lines to

diverge and tha R branoh lines to oonvsrge as J Increases.

If tho

difference B'-B” is negligibly small the K components will coincide for eaoh line and ths P and R branohss will oonslst eaoh of a set of single, equally spaoed lines*

Another mors oommon way of looking

at the fine structure of a parallel band la to say that the band con­ sists of a superposed

set of sub-bands, one sub-band for eaoh ofthe

possible values of K*

Bach sub-band is comprised of a P, Q, and R

branoh exoept for the sub-band where K«0, in whioh the Q branoh is missing. In the previous paragraph it is pointed out that lines in tho R branoh arise from a

transition A

vl or J” to J’S 1 and in the P

branoh the lines arise from t ransition A J* - 1 or J* to JB—1* If we subtract the term value for the P branoh line P(JMvl) from the term value for the R branoh line R(JM-l) the resulting expression oontalns 18

only tho rotational constant Tor the lower state*

To simplify the

notation let us denote J” simply by J and let F"(J) and F*(J) denote ths rotational term values for the lower and the upper state respectively*

Then the frequenoies of the P(J+1) and the R(J-l)

lines may be expressed as

R ( 3 - 0 = V, t F '(3) - F"( 3 -O (16)

Taking the differenoe we obtain

- P C 3 t o = F"(.3 t o - F"C3-i') * A l F"Ci)

y

(17)

where F"(3-»») = B " C 3 + 0 ( 3 +2.1 (18) F"f3-l)

t LCc'-c")-CB -

and thus

.

A t F"(3)

(19)

If instead, we take the differenoe between R(J) and P(J) the result involves the rotational oonstant of the upper state only and Is (20)

13

(2 1 )

R C3) - P(.3) - f 'C3 i-n - F fC3 - o = A t F *(3^ ,

(2 2 )

.

A t F'(cO - a B ' U l i

The two relatione

A d F " 0 ) - R ( 3 - U - PCs ii) - ^ " U s - n ) A^F'C3)= are the

1

0

K(3)

- PCJ)

,

(IS)

(2 2 )

=20'(Z3ti) ,

-oalled combination relations*

If* A g F(J) is plotted

against 2Jvl the slope of the

resulting line gives the value of* ths

rota ional constant B.

is one more relationthat is useful*

1

There

It is obtained by adding the term value

of P(J) to the term value of*

R(J-l) and the result is

ft('s-i) i p (3) = a w t -» ale' - b ")t * . A plot of ^ [r(J-1)v P(J)]

against J

2

(23)

yields the band oenter ^

as the Intercept* and the differenoe B*—Bn as the slope* D•

The Perpendicular Bands In order to discuss the perpendioular bands of a pyramidal

XYj moleoule we need to reoall the energy term value ( ) and the 6

rotational selection rules associated with an oscillation of the dipole moment perpendioular to the top axis (

1

14

0

)# and (

1

1

)•

A K - — I

,

AS

■,

m i

A

3

-

0

,

1

(

1

1

0

)

a kK - -t>

(11 ) A S - -I

>

A

We shall oonsider V g as an example of the perpendioular bands. In the ground state (vg-jfg-O) the energy term value is F“ = 0ro te.Vc5''-H> ^ c c "- b *) k "* ,

-+(c'-B')K'*-aCe C^ S J k ' . For the transit ions A J- #A K- t 0

1

(25)

*Ajwe obtain the following

expressions for the frequencies of the lines in the ^Q(AK*tl) and PQ(AK- -1) branohes of the Kth sub-band

+ (B"- C* )

K(Vt'"£,C€ - G * J

1

3C3VIXI3'- B")

+ K ’ Lfc'-C*) - f B ' - B " ) ]

For the transitions A J — v 1# A K

(2 6 )

.

il and A J - - 1, A K - ± 1 D p we obtain the following expression for the lines in the R, R, and 1

®P, PP branohes of the Kth sub-bandi

v --

* C b “-c") t a K L < C ' - ^ c € )-D*'J t 3(o^d")

15

(27)

In order to simplify these expressions somanhat, tha assumption is usually made that 0^*3: C". K,

£»

“■ IVfc35 * IO

—*40

1

cjvw

_

(54)

1

Ths momsnts of insrtia sxprsassd in tsrms of ths atondo masses, ths pyramid height, h, and / 3 , ths angle bstwssn ons of ths XY bonds and ths axis of ths top, ars 1*

- 3 rw, V

i c = 3 r m -l\

(i U ^ " /3 +

£a/w*/3

j

)

,

(35)

(36)

whsrs it is assumed that ths differences bstwssn ths values of ths 31

moment■ of inertia in the ground state* I"* and the equilibrium values*

* are small so that I" may be used for I# *

M is the mass

of the X atom (antimony) and m represents the mass of the Y atom (hydrogen or deuterium)* and IQ is the moment of inertia about the axis of the top*

If the expression for 1B is divided by the oorres*

ponding expression for Ig** the resulting equation is

°

where it

Isassumed that h » h * and B s B * »

.

(37)

Letus simplify the

expressionby making the following substitution! M

«H*

M

m

» R*

B*

following values for stibenei

where the oonstants have the

H » 0*97576* 0 * 0.95271* and R * 0*60037.

Then (37) beoomes

L'xi^.^/3

—

~l

-t- D

J

-

5

(se)

If this equation is solved for & tan^/3* the result la t ^ / 3

Z

_ ~

KH - SP S~R

(39)

The quantities H* D* and R are all positive with H > D and* therefore* in order that the angle

be a real angle* the quantity S must lie DU within the limits R W (

S?

*

J

+

XA J

and

•

>

f

r- V

L

S' ^

I-

r ^ f

+

s«

J -*-

^

^ r J m + Al

1

J

)

,

~ -*_?*

*1

^ ^

If the values of these constants are substituted into

(53) the result may be written in the formt

m

-r. x ;

:us - « * z , z ;

( % , ,■ •

(55)

xi,

* 1

9l Shaffer^ has obtained expressions for the perpendicular normal frequencies of a pyramidal XY^ molecule as followsi

■■i

{ ( j . ' S r l a - ’s r - r i *

F

f

M >

where K,» >i-* and are generalized foroe constants associated with 1 2 5 ii. the perpendicular modes of vibration* Uj ^ and and^JLA JJ % 55

The 'transformation oosfflolants from the intermediate symmetry coordinates to the normal coordinates are expressed in terms of the force constants as

f

f

l

(57) J

where

2

3,

and ac 2 are related to the 3 ^ as follows*

(68 )

From Equation 56 we obtain

£

*

(B9>

and

v

j

£

]

\

co)

and from Equation 57 we get =

/

(si)

and

* - *

=

~T— ^

^

56

Substituting (60) into (62) yislds H ,

5

_

>V*

X "=

!=. .

(63)

Equations 68 and 61 may b« solvsd simultansously to giT* I + lu X* .

(64)

and

S

**

J — 1 I X?. - '

L

=

-

(65)

#

Using (64) and (66) Equation 63 bsoomss ✓ ^ i _ bj\ _

f(

*J= j

.tin

t ( ,f

*■

^

1

J

J■»IT a.

-I

(66 ) r - (* * w *

1

—

^ 2

- (jx* - £.) ^

^

i 1

-

Lot us now writs Equation 69 for two isotopio forms of ths XYj molsouls* >». >*1 67

A-

>■»

(67)

>*>*

(68)

In general the force constants,

will b« different from

th« force oonstarrts for tha isot ops,

however, ths poten—

tial energy of the molecule is expressed in valence coordinates, then to a good approximation, the resulting valence force constants will be the same for both isotopio forms of the molecule*

A compari­

son of the two potential energy expressions shows that the generalised force constants are related as followsi

>

/

x *

V

*. = ( i ^ i „ ;

*■

j

=

n>

•

Equation 68 may then be expressed in terms of >i_ and X

£

-

=

11 >

*

^

/*_ as t 3

>

(69)

X- - ( § f § ; y

.tar.

Equation 67 and 69 are solved simultaneously for

»■

I*"
** V*1

1 Y k * •>•,) - X ( *1 *

.

(7i)

Substituting these values into (66) yields

[

M

I *

( *..»

*■'•»*>] j

^

(72)

^

where

(73) -

[^-o^o-or

+ > * )

]

.

A snail amount of algebraic manipulation shows that R is Identically zero and therefore Equation 72 is equivalent to ths original expression derived by Dennison 19 •

69

BIBLIOGRAPHY 1.

Robertson, R., and Fox, J.J., Proo. Roy. Soo. A120, (1928), p. 128.

2.

Barker, K.F., Phys. Rev., 33_, (1929), p. 684*

3*

Dennison, D.M., and Hardy, J.D., Phys. Rev., 39, (1932), p. 938.

4*

Lee, B. and Yfu, O.K., Trans. Faraday Soo., 36, (1939), p. 1366*

5.

MoConaghle, V.M., and Nielsen, H.H., Proo. Nat. Aoad. Sol., 34, (1948), p. 456.

6.

MoConaghle, V.M., and Nielsen, H.H., Phys. Rev., 76, (1949), p. 633*

7.

Nielsen, H.H., Diso. Faraday Soo., 9, (1950), p. 85.

8.

Smith, D.C., J. Chem. Phys., 18, (1951), p. 384.

9.

Loomis, C.C., and Strandberg, M.W.P., Phys. Rev., 61, (1951), p. 798*

10.

Nielsen, H.H., Phys. Rev., 60, (1941), p.794.

11.

Shaffer, W.H., J. Chem. Phys., JJ, (1941),p. 607.

12.

Teller, E., Hand-und Jahrb* d. Chem. Phys., vol. 9, II, (1934), p. 43.

13. 14.

Dennison, D.M., Phys. Rev., 28, (1926), p. 318* lisllor, J.W., A Comprehensive Treatise on

Inorganloand

Chemistry, Vol. VIII, Longmans, Green and Co., NevrYork 15.

Theoretioal

(1928).

Thorneyoroft, W.E., Textbook of Inorganlo Chemistry. Vol. VI, Part 5, J.B. Lippinoott, Philadelphia (1936).

16.

Bell, E.E., Noble, R.H., and Nielsen, H.H., Rev. Soi. Inst., 18. (1947), p. 48.

17.

DuMond, J.W.M., and Cohen, E.R., Rev. Mod. Phys., 20, (1948), p. 62.

18.

Leohner, F., Site, ber Akad. Wiss. Wien, 141. (1932), p. 633.

19.

Dennison, D.M., Phys. Rev., 12. (1940), p. 175. 60

AUTOBIOGRAPHY I* William Howard Haynie* was born in Royal Oak* Michigan, July 24* 1923*

I obtained my sooondary education at Savannah

High Sohool* Savannah* New York* and received a Baohslor of Soienoe degree with a major in physios from the Univarsity of Chioago, Chicago* Illinois* in Juna* 1944*

From Juna* 1944* to

Ootobar* 1946* I ha Id a position as physicist with the National Advisory Committee for Aeronaut!os in the instrument resaaroh seotion of their Cleveland* Ohio* laboratory*

From Ootobar* 1946*

to Juna* 1951* I was a resaaroh fallow on an Air Material Command projaot under the supervision of Professor R* A* Oatjan*

In

June* 1948* I reoeived a degree of Master of Soienoe with a major in physioa from the Ohio State University.

Slnoe Ootobar* 1951*

I have held a graduate assistantship in the Department of Physios*

61