Infra-red spectra of crystals—IV. Matrix shifts and the contribution from classical electrostatic forces

SpectrochimicaActa, 19{}7,Vol. 23A,pp. 2405 to 2419. Pergamon Press Ltd. Printed in Northern Ireland

Inira-red spectra of crystals--IV. Matrix shiRs and the contribution item classical electrostatic forces D . C . MCKEA~ Department of Chemistry, University of Aberdeen, Old Aberdeen, Scotland

(Received 5 January 1967; revi~ed 2 t"ebruary 1967)

Abstract--Two principal defects of existing treatments of vibration frequency shifts in solution are discussed. Instead of representing the vibrating molecule b y a point dipole, a more realistic description should be given by using a multipolo expansion of the electrostatic potential energy. The results of multipole calculations are given for a simplified effective charge model, in a spherical cavity, appropriate to X Y 4 molecules and the frequency shift data for Group IVB hydrides discussed. It is shown that the dipole-induced dipole term in the potential energy m a y amount to as little as 30 per cent of the total electrostatic contribution, the proportion being rather less for stretching than for bending modes. The point polarized atom model employed by Linevsky for LiF in inert gas matrices is compared with the cavity model and shown to predict much smaller shifts than the latter. Second order perturbation treatments are carried out to determine matrix elements for LiF and CO, and the very great contribution of the anharmonic terms to the frequency shift in LiF displayed. The calculated shift is even greater when the higher multipole terms appropriate to an effective charge model are included (~190 cm -1 in Xe). The chief result of the increase in calculated negative shift duo to classical electrostatic forces is to enhance the role played by repulsive forces. An alternative assignment is proposed for the two bands associated with CO in both Ar and Kr matrices. INTRODUCTIOI~ I ~ m a n y p r e v i o u s i n v e s t i g a t i o n s o f v i b r a t i o n f r e q u e n c y shifts in solution, a t t e m p t s h a v e b e e n m a d e t o d i s t i n g u i s h b e t w e e n t w o t y p e s o f i n t e r a c t i o n , a l o n g r a n g e one, a r i s i n g f r o m i n t e r a c t i o n b e t w e e n a n oscillating electric dipole m o m e n t a n d t h e s u r r o u n d i n g m e d i u m , w h i c h is a s s u m e d t o b e h a v e as a u n i f o r m dielectric, a n d a s h o r t r a n g e specific i n t e r a c t i o n b e t w e e n p a r t i c u l a r b o n d s or a t o m s o f t h e solute a n d s o l v e n t molecules.* I f t h e l o n g r a n g e i n t e r a c t i o n is i m p o r t a n t , t h e n t h e so-called K B M equation should apply: A~lv = - - C ( n 2 -- 1)/(2n ~ + 1) (1) w h e r e C i n v o l v e s t h e s q u a r e o f t h e m a g n i t u d e o f t h e dipole m o m e n t c h a n g e d u r i n g t h e v i b r a t i o n [3]. T h e i m p o r t a n c e o f specific forces b y c o n t r a s t h a s b e e n s h o w n b y t e c h n i q u e s s u c h as t h e B e l l a m y - H a l l a m - W i l l i a m s p l o t s [1], or t h e use o f m i x e d s o l v e n t s [4]. I t r e m a i n s h o w e v e r t o describe q u a n t i t a t i v e l y t h e p h y s i c a l n a t u r e o f t h e forces * eL the review by W ~ M s [1] [2] [3] [4]

[1] and later developments by I-IORAKand Pr.IVA [2].

1~. L. WIT.LIAMS,Ann. Rept. Progr. Chem. 58, 34 (1961). M. ttORhK and J. PLIVA, Speetroehim. Aeta 21, 911 (1965). D. F. BALL and ]:). C. McKEA~, Spectrochim. Aeta 18, 1029 (1962). ~¢I.L. JosYE~, I.U.P.A.C., Molecular Spectroscopy, p. 33. Butterworths (1962). 2405

2406

D . C . MoKEA~

involved, and this has been attempted by several investigators [5-7], who have tried to assess the contributions to a frequency shift from: (1) dipole-induced dipole forces (attractive), oc 1]r6 (2) quantum-mechanical dispersion forces (attractive), oc 1/r 8 (3) quantum-mechanical repulsive forces, oc 1/r 12 A distinction is thus implicitly drawn between "classical" electrostatic forces, assumed to be operating between the various smoothed-out charge clouds of each molecule, and the quantum-mechanical forces which depend on the instantaneous positions and spins of the individual electrons in these charge clouds. I f this distinction can be maintained, then clearly the role of the classical electrostatic forces in both long range and short range effects must be considered. In at least one case, that of Lr~EvsxY [6], and attempt is made to calculate the absolute magnitude of the classical electrostatic potential energy, and by subtraction from the observed shift, to assess the importance of the quantum-mechanical ones. I t is the purpose of the present paper to look more closely at the calculation of the classical electrostatic forces and to suggest that previous estimates of their magnitude m a y be misleading in two respects. The first of these is the omission of higher multipole terms. The second is the apparent inadequacy of the point-polarizable-atom model used by Linevsky compared with a cavity model, with or without multipole terms. THE :PRESENCE OF HIGHER MULTIPOLE TERMS

I n a cavity model such as the K.B.M. one, or its modification by BUCKI~GH~ [8], the change in the potential energy of the oscillator is attributed to the potential energy of a dipole moment in its own reaction field. Thus if the dipole moment is zero, the frequency shift is zero. Therefore no frequency shift can occur for an infrared inactive mode of a non-polar molecule. Moreover, the dipole, if present, is assumed to lie at the centre of the cavity. This model is a grossly inadequate representation of the real vibrating molecule, which is an extended system of charges in space, which will either fill almost completely the cavity in the liquid or crystal, or if the cavity is much larger than the molecule, will spend most of its time close to the wall of the cavity. Now in principle, the classical problem of the energy of an arbitrary extended charge system in a spherical cavity can be solved to any desired degree of precision by taking a sufficient number of terms in an expansion in terms of r, the distance from the cavity centre, whose coefficients include the monopole, dipole, quadrupole etc., moments of the system

[9]*. Two features of this expansion need mention at this point. Firstly each successive term is decreased by the quantity (r/a) ~, a being the cavity radius : thus for a :> r > * Such an expansion is indeed necessary to describe the potential energy of an ideal point dipole which is not at the centre of the cavity: see Ref. [9J, p. 101. [5] I-I. G. DRICKA~ER et al. J. Chem. Phys. 24, 548 (1956); 27, 1164 (1957); 28, 311 (1958). [6] M. J. LI~.VSK:~, J. Che~. Phys. 34, 587 (1961). [7] S. W. CHXR~S and K. O. LEE, Trans. FaradaySoe. 61, 2081 (1965). [8] A. D. BucKr~GHAM, Prec. Roy. Soc. A248, 169 (1958); A255, 32 (1960). [9] C. J. F. BOTTCHER, The Theory of Electric Polarization, p. 94. Elsevier (1952).

Infra-red spectra of crystals--IV

2407

Table 1. Contributions to the potential energy from multipole terms Pole 21

Pole moment -~.

2(8 - 1)/(28 + 1 ) . M12. l[a 3

M1 = ~, e f t " P1 (cos 0~) i

2~

--½.3(8 -- 1)/(38 + 2 ) . M22. l/a ~

M2 = ~_, 8F, ~ • -Pg. (cos 0,) i

2a

--½.4(8 -- 1)/(48 + 3) . M29" . l/a ~

M a = ~, eF~a" P3 (cos 0~)

24

etc.

etc.

i

½a, the rate at which the t e r m m a g n i t u d e s diminish is n o t particularly great. Therefore in cases where the moving atoms are far from the centre of the cavity, the higher multipole terms m a y be i m p o r t a n t even t h o u g h the dipole m o m e n t change is zero. Secondly, each t e r m involves a slightly different function of the dielectric constant, as shown in Table 1. For the 2Lth pole term, this is:

(~ + ~)(~ - 1)/[~ + (z + 1)~]

(2)

For values of 8 between 1 a n d 4, these functions have v e r y similar values, as shown in Fig. 1. Thus an experimental correlation between (e - 1)/(2e + 1) a n d A~/~ would n o t in fact distinguish between the dipole term, which is proportional to the b a n d intensity, a n d a n y of t h e higher terms. An experiment which m i g h t offer more o p p o r t u n i t y of m a k i n g this distinction would be to t a k e a molecule of sufficiently high s y m m e t r y such t h a t it possesses b o t h infra-red active a n d inactive modes of the same general kind. Thus in a s t u d y of C - - F stretching frequency shifts, comparison should be made for example between the u a n d g mode shifts in t r a n s - C 2 H ~ F ~.

F (c)

l = I

0"6

7 3

0-4

0.2

I 2"0

I 3"0

I 4"0

E

Fig. 1. Plots of 2'(e) = (1 + 1)(8 -- 1)/[/ + (l + 1)8] vs. e for l = 1, 3 and 7.

2408

D.C. McKEA~

Such an experiment would entail use of the R a m a n spectrum, which in turn prompts the reflection t h a t it is perhaps due to the historical accident t h a t frequency shifts have been studied entirely in the infra-red, t h a t attention has been paid only to the dipole term, to the exclusion of the others. Similar possibilities come to mind in tetrahedral X ¥ 4 or octahedral X¥~ molecules and these are further discussed below. A more important reason for the past omission of higher multipole terms has been t h a t since they are effective at progressively shorter ranges as I increases, the resemblance of the environment to a continuous uniform dielectric medium becomes correspondingly poorer. However the necessity of including higher terms is not limited to the cavity medium model. I n the case of a "specific" interaction the model used for calculating the classical electrostatic forces might consist of a vibrating effective charge at a distance r from a polarizable sphere of radius a. I f these represented atoms in contact, so t h a t r and a are comparable in size, a considerable expansion of the potential energy in terms of a / r would be necessary.* A final reason for the neglect of the higher multipole terms lies in the fact that, while in principle the magnitude of the dipole moment change can be measured, from the gas phase intensity, although this has very seldom been done, the information needed to calculate the change in a quadrupole or higher moment during a vibration is entirely lacking. The calculations described below suggest however t h a t these changes can be considerable. HIGHER MULTIPOLE TERMS AND DISCUSSION OF MATRIX SHIFTS IN GROUP IVb HYDRIDES For simplicity, we consider the distribution of charge within the spherical cavity to have cylindrical symmetry. Then the potential energy V of such a distribution in an arbitrarily varying electric field ¢ is given by [10]: 1

V = q¢o - - # F z - - ~ ® F =

v

1

,v

-- ~F=z

etc,

(3)

where q is the resultant charge, /~ the dipole moment, ® the quadrupole, ~ the octopole moments, etc., while F z ---- - - d ~ / d z

F ~ ' = --d2¢/dz ~,

I!

F = z -~

--da¢/dz a etc.

From B6TTCHER [9], the potential ¢ due to the reaction field in the cavity within the dielectric medium is given by:

¢=

F - - ( ~ - 1)(/--1) ,:0 L 7 T •

rl 1

• P (cos 0)

(4)

where Pz (cos 0) are the Legendre functions, and the b, coefficients represent the original charge distribution and are in fact the dipole, quadrupole etc., moments M z evaluated at the centre of the cavity.~ * el. Ref. [9], p. 150 ft. In the notation of Buckingham [10], M 1 = ~, M 2 = O~, NIs = f~z~zetc. [10] A. ]). BUCKI~COHA~,Quart. Rev. 13, 183 (1959).

Infra-red spectra of crystals--IV

2409

T h e e x t r a p o t e n t i a l e n e r g y of t h e s y s t e m in its own r e a c t i o n field is t h e n o b t a i n e d b y c o m b i n i n g (3) a n d (4), differentiating (4) t h e a p p r o p r i a t e n u m b e r o f t i m e s w i t h r e s p e c t to z a n d setting z -~ 0, since t h e q u a n t i t i e s F~ etc., are r e q u i r e d a t t h e centre o f t h e cavity.* T h e v a r i o u s t e r m s are listed in T a b l e 1. Since t h e m a g n i t u d e of e a c h successive pole involves a n e x t r a p o w e r o f r, t h e t e r m s diminish p r o g r e s s i v e l y b y t h e f a c t o r r~/a 2. T h e i r m a g n i t u d e s h o w e v e r d e p e n d on t h e values of M 1, M 2 etc., which in t u r n d e p e n d on t h e m o d e l a s s u m e d for t h e v i b r a t i n g molecule.

THE EFF~.CT~VE CHARGE ~¢[ODEL T h e o n l y m o d e l conceivable a t t h e p r e s e n t t i m e is one in which fixed effective charges are s u p p o s e d to reside on t h e nuclei a n d to m o v e w i t h t h e nuclei in all t h e i r motions. T h e size o f t h e charge should t h e n b e a s c e r t a i n a b l e f r o m t h e i n f r a - r e d intensities of t h e v i b r a t i o n s . This m o d e l of course is well k n o w n to b r e a k d o w n in n e a r l y all molecules which h a v e b e e n studied, in t h a t it implies t h a t d/~/dr f r o m a b o n d s t r e t c h i n g m o t i o n is t h e s a m e as #/r f r o m a n angle b e n d i n g one. F r e q u e n t l y , n o t e v e n t h e signs o f these t w o are t h e same. H o w e v e r in t h e molecules CF4, SiF4, SF 6, S i l l 4 a n d G e H 4 it seems likely t h a t t h e signs are identical [11-13] while in Table 2. Multipole moment changes for the effective charge model of the F 2 symmetry co-ordinates of a tetrahedral XY 4 molecule

Pole 21 22 2a 24 25 2e

(scaled on the basis) 6(dipole moment = 1) ($Mi (~Mt (stretching) (bending) r factor 1 +2 +4.13 + 2-963 +5 +5-37

1 -1 --5/3 + 50/27 --5/9 --29116

1 r

r2 ra r4 r5

SiHa t h e effective charge a p p r o x i m a t i o n is fairly g o o d [14]. Since t h e f r e q u e n c y shift d e p e n d s on t h e p o t e n t i a l e n e r g y t e r m s t h a t v a r y w i t h t h e n o r m a l co-ordinates Q, in t h e h a r m o n i c a p p r o x i m a t i o n , we shall require to calculate t h e t e r m s OM~ 2

OQ2 (~Q)2. H o w e v e r la~er on, for L i F , t h e a n h a r m o n i c t e r m s

aMz Ms" OQ aQ * A factor of 1 also enters since the charge distribution is in its own reaction field. No account is taken here of the polarizability of the charge distribution. [11] D. C. NIcKEA~, J. Chem. Phys. 24, 1002 (1956). [12] D. A. Dows and G. 1~. WrEDER, Spectroehim. Acta 18, 1567 (1962). [13] I. LEvitt, J. Chem. Phys. 42, 1244 (1965). [14] D. F. BALL and D. C. McKEA~, Spectrochim. Acta 18, 1019 (1962).

2410

D.C.

McKEA~

will also be evaluated. For the present, we treat only the molecules SiHa and GeH t, for which we assume t h a t Qa involves only the stretching s y m m e t r y co-ordinate St, Qt only the bending co-ordinate S t. I n order to use the equations (3) and (4) which depend on a cylindrical s y m m e t r y we select the s y m m e t r y co-ordinates* S 3 = (3Ar 1 -

Ar 2 -

Ar3 -

Art)

and further assume (1) t h a t the charge on atoms H~, H s and H t is evenly distributed over the circle passing through the three nuclei: (2) t h a t only the hydrogen atoms move in the vibration. The corresponding changes in the various moments are then readily calculated. The values are given in Table 2, scaled on the basis of M 1 (the T a b l e 3. C a l c u l a t e d v a l u e s o f d i e l e c t r i c c o n s t a n t s Data used:

R e f [14 l

Sill 4

~ v

2.411 2112

1.982 1598

3.528 819

2.530 675

r a 13.9

F4 43.7

SiD 4

e

1.969 2189

1.963 2112

1.859 914

1.725 819

11.1

30.9

dipole term) = 1. The high values found for the stretching motion essentially reflect the l factor when e# is differentiated with respect to z. I n calculating the various terms of Table 1 for matrices of Sill t or SiD t the radius of the cavity is assumed to be the Sill bond length, 1.48 A, plus the van der Waals radius for the H atom, 1.2 A. The dielectric constant should be the one appropriate to the frequency of the vibrating solute molecule. Those listed in Table 3 were calculated as previously [3], the contribution from atom polarization being assumed to be given by N 2 : 2"017 + ~ Adzr~(vi ~ -- v ~) for both bands i. i

The distance r of the effective charge from the cavity centre is assumed to be r, = 1.48 tlx for Sill4, 1.53 A for GeH a. Table 4 exemplifies the results for the two infra-red active modes of Sill a in an SiD a matrix. I t is seen that, in addition to the enhancement being much greater for the stretching mode t h a n for the bending one, for the latter, only the first six forms have significant values, whereas for the stretching mode, probably 3 or 4 higher ones should be considered. The final calculated frequency shift is then the Av calculated for the dipole term only, multiplied by the total multipole factor (mr). These m f values in Sill a and SiD a matrices at frequencies appropriate to the va and .va modes of Sill a, SiD a and GeH t are given in Table 5, whose 6th and 7th rows compare the calculated total shift with the observed shifts [ 15]. In general the observed shift is bigger (negatively) in every case except for vt of GeH t in Sill t, the discrepancy being always higher for * Z a x i s t h r o u g h a t o m 1,/fit ---- H i X H 1 [15] D . C. 1V£cKE.~ a n d A . A . C~AL~:ERS,

Spectrochim. Acta 23, 777 (1967).

Infra-red spectra of crystals--IV

2411

Table 4. Relative contributions to potential energy from multipole terms for Sill 4 in SiD 4 matrix va = 2189 cm -1 7 factor*

Pole 21

22 2a 24 25 26

e factor

Mode factor

~'4 = 914 cm -1 Total.

~ factor

mf

Mode factor

Total

mf

f 1

1

1

?~ 74 ~'~ 78 71°

0"3050 0"0930 0"02836 0-008650 0.002638

0"937 0.908 0"892 0.881 0.874

1

4 1"777 8-779 25.00 28.90 Total

* ? =r/a,

r e = 1-48A,

1

1

1

1

1'143 0-1501 0-2220 0.1905 0.0666

0.934 0"904 0.887 0.876 0.869

1 2"778 3"429 0.309 3.285

0"2849 0-2336 0"0863 0-0023 0-0075

Total

1-615

2.772

a =2.68/~. Table 5 Sill 4 matrix

SiD 4 matrix

1598 SiD 4 va 2.77

675 SiD 4 v4 1.70

2118 GcH 4 v3 3.03

Aucalc a~p A%~ac tot

--1"53 --4.24

--5'42 --9"19

--2"50 --7'53 --7"58 --12"9

A%bs

--8.0

Multipole factor A%alc tot

3.28 --5"1

Solute species Mode* Multipole factor

819 GeH 4 v4 1.72

2189 SiH 4

914 SiH 4

Y8

~'4

2"77

1"68

2118 GcH 4

819 era-1 GeH 4

v3

Y4

2.91

1"69

(mr)

--14-0 1.79 --7"8

--11.2 3.65 --7'6

--12.0 1.89 --9"2

--1"90 --5.27 --10.2 3.28 --6"4

--5"53 --2.01 --3.91 cm-1 --9.30 --5-86 --6-60 cm -1 --13.9

not obs --9.4

1"79 3-65 --11"0 --7"6

1.89 --9"2

t h e s t r e t c h t h a n for t h e b e n d . W e w o u l d o f c o u r s e e x p e c t d i s p e r s i o n or o t h e r q u a n t u m - m e c h a n i c a l effects t o b e g r e a t e r for t h e s t r e t c h t h a n for t h e b e n d . T h e effect o f a l t e r i n g t h e v a l u e o f a is o f c o u r s e v e r y m a r k e d a n d o n l y a s m a l l d e c r e a s e w o u l d b r i n g a b o u t e q u a l i t y o f t h e c a l c u l a t e d a n d o b s e r v e d shifts. O n e effect is s e e n i n t h e s e d a t a w h i c h t h e p r e s e n t t h e o r y p r e d i c t s . A~4 for G e H 4 is s m a l l e r i n S i D 4 (9.4 c m -1) t h a n i t is i n S i l l 4 (12.0 cm-1). T h i s is e x p l a i n e d b y t h e p r e s e n c e o f t h e i n t e n s e b e n d i n g m o d e o f t h e h o s t m o l e c u l e b e l o w 819 c m -1 i n t h e case o f SiD4, b u t a b o v e 819 c m -1 i n t h e case o f Sill4, t h e t w o d i e l e c t r i c c o n s t a n t s b e i n g r e s p e c t i v e l y 2.53 a n d 1.86.* T h e s i m i l a r s h i f t s Av 4 for S i D 4 i n SIH4(13.9 ) a n d for S i D 4 i n S i l l 4 (14 c m -1) a r e e x p l a i n e d i n t h e s a m e w a y , w i t h t h e r e s e r v a t i o n t h a t v4 i n t h e gas for S i D 4 h a s still t o b e d e t e r m i n e d a c c u r a t e l y a n d t h a t t h e e r r o r i n A~4 for S i D 4 is t h e r e f o r e c o n s i d e r able. * The calculated effect is rather greater (12.9 cm -1 in Sill4, 6.6 em -1 in SiD4) corresponding to a difference in Av of 3.3 em -1, compared with an observed difference of 2.6 em -1.

2412

D.C.

McK~A~

I t m a y be noted t h a t we m ay expect the observed effect of an intense neighbouring lattice band to be less t h a n t h a t expected, in-so-far as the effect is due to atom polarization through r 4. Now the rate at which the field varies due to a high multipole moment in the cavity will be great over a distance of the order of a diameter of a neighbouring host molecule. I t seems ver y unlikely t hen t h a t an appreciable polarization of this molecule b y induction of v4 could be effected. To demonstrate the effect of omitting this atom polarization, the bot t om rows of Table 5 show the calculated multipole factors and frequency shifts assuming a uniform dielectric constant of 2.017 at every frequency, l~ot only is the shift Av4 for GeH 4 the same in Sill 4 and SiD 4, but the calculated shifts A~4 for Sill 4 in SiD 4 and for SiD 4 in Sill a now differ b y 2.2 cm -1. GENERAL DISCUSSION OF HYDRIDE M~TRIX SHIFTS The data so far available for the Group IVB hydrides are assembled in Table 6. Several consistent features are revealed: (1) For any particular mode of the molecule, the shift increases as the matrix is changed in the order SiH4-GeH4-SnH a. The only exception to this is the shift A~4 for GeH 4 in Slit4, a possible explanation for which has been advanced above. This sequence could arise either through a progressive increase in the size of the cavity, or through an increase in the polarizability of the host lattice, which will probably rise in the series Si-Ge-Sn. The increase in shift between Ge and Sn is rather bigger than t h a t between Si and Ge, as might be expected. Table 6. Matrix shifts for group IVB hydrides Solute species

S i l l 4 v3 ~4

Matrix SiDa

GeH 4

SnH 4

SiF 4

SF~

- - 10.2

- - 12.2 - - 14-4

--17"5 --17.1 --19

410"3 --10-5

-~ 9

Sill 4

SiD 4

GeD 4

SnH 4

SiF a

SF 6

- - 11.2 --12"0 --15

not inv.* --9"4

19"1 --11.7 --I8

-~- 12"9 --7-5

-~- 11.9 --1-3

--

13"9

Sill 4 S i D 4 v~ v4

G e t t 4 ~3 v4 vl

--

--8 14

--

14-8

--11.7

--

GeH 4 G e D 4 ~a va

S n H 4 va Y4

Note: obtained * Not The

--9"9 --10 Sill a

GeH a

--9,2 --11

--10'9 --12

t h e v a l u e s o f A ~ a a n d A r 4 are t a k e n f r o m t h e p h a s e o f h i g h e s t s y m m e t r y : from phase II. investigated. v a l u e o f Yl i n t h e g a s is n o t y e t k n o w n .

A y 1 f o r G o H 4 i n S i l l 4 is

Infra-red spectra of crystals--IV

2413

(2) While all the shifts f o u n d in h y d r i d e matrices are negative, b y contrast the S i l l and G e H stretching modes are raised w h e n the Sill 4 a n d G e H 4 are dissolved in SiF 4 or SF 8. This b e h a v i o u r resembles t h a t of t h e S i l l or G e H modes in films of SiH3X or GeH3X [16]*. T h e p r e d o m i n a n c e of repulsive forces here m a y be correlated with the difficulty of isolating the h y d r i d e molecules in SiF 4 or S F e. T h e repulsion is p r e s u m a b l y a specific one between h y d r o g e n a n d fluorine atoms, b u t a n additional f a c t o r could be the non-availability of the Si or S " d " orbitals due to t h e S i - F or S - F bonding. T h e fact t h a t the positive shift is greater for G e H a t h a n for Sill 4 m a y be explained b y t h e slightly greater size of the G e H 4 molecule. (3) F o r t h e bending modes of S i l l 4 a n d G e H a, in the fluoride matrices, the shifts are still negative, h u t a p p a r e n t l y r a t h e r sensitive to the shape of the hole, judging b y the two shifts f o u n d for G e H 4 in SiF a a n d SF e respectively. (4) P e r h a p s t h e most interesting f e a t u r e of Table 6 are the shifts in h y d r i d e matrices f o u n d for vl of Sill 4 a n d GeH4, which are negative, like those for v3 a n d Yd. This could arise t h r o u g h the coupling with one c o m p o n e n t of v3 which occurs with the loss of s y m m e t r y in phase I I or I I I , a l t h o u g h this is unlikely to a c c o u n t for more t h a n a small p r o p o r t i o n of the shift in view of the c o m p a r a t i v e l y small splitting in v3. I t would seem t h a t this is a case where infra-red active a n d inactive modes exhibit similar shifts, which in t e r m s of the discussion in the i n t r o d u c t i o n implies t h a t the dipole-induced dipole c o n t r i b u t i o n is negligible.~ However, as was observed in I I I [15], the high i n t e n s i t y of v1 requires some special e x p l a n a t i o n a n d it would be p r e m a t u r e t o draw a n y final conclusion a b o u t its f r e q u e n c y shift.

THE POINT-POLARIZABLE-ATOM M O D E L TREATMENT OF THE DIPOLE INDUCED-DIPoLE E N E R G Y In a recent paper in which frequency shifts of the order of 50-100 cm -I were reported for LiF in matrices of inert gases and nitrogen at low temperatures, LI~EVSKY [6] estimated the contribution from dipole-induced dipole forces using a formula which sums the separate terms for a point non-polarizable dipole interacting with each matrix atom in turn, the latter being assumed to be a point-polarizable particle : V :

~ - - ½" f~LiF " Ra

l

3 COS2 0 -k 1 2 = --½/ALiF~ ~ r6 1

(5)

(6)

where Ra is the reaction field a t t h e L i F molecule due to a n induced m o m e n t in a p a r t i c u l a r a t o m of polarizability ~ in the lattice, whose position relative to the L i F is * However some of the gas-crystal shifts quoted in Ref. [16] require revision in the light of new gas phase data for SiHsF, SiHsC/and Sill 3 Br [17]. It is of interest to note that the only treatment of the vibrations of a tetrahedral molecule in the presence of specific forces predicts a lowering of v3 but a raising or lowering of ~1 [18]. [16] D. F. BALL, M. J. BUTTLERand D. C. McKEAN, Spectrochi~n Acta. 21, 451 (1965). [17] A. G. ROBIETTE, Ph.D. Thesis, Cambridge (1964). [18] J. I-IEICK_LEI~,Spectrochira. Acta 17, 82 (1961).

2414

D.C. McKEAlV

given by 0 and r. The summation ~ is over all lattice atoms. For the face-centred cubic lattice, ~ ( 3 cos 2 0 + 1)/r 6 = 2 × 14.45/d6 e where d o is the lattice spacing. We examine at a later point Linevsky's subsequent handling of/~*, but must observe here t h a t the cavity model replaces (5) by v= where ~ is the total reaction field arising from the surrounding medium, which is given by [19] ~ , , : f / ( 1 -- f ~ ' ) PLiF

(7)

where f is 2 ( ~ - - 1)/(2e + 1)a 8 and ~' is the polarizability of the oscillator itself. In view of the assumption of LI~]~VSK¥ t h a t the LiF occupies a'single site in the host lattice, and t h a t all of the sites are normally occupied, it seems fair to take for the cavity radius a the value ½de. With the aid of dielectric constants recently published by AMv+Yand COT,V. [20], values o f f are calculated and displayed in Table 7. Ignoring the term in f~', which is unlikely to exceed a few per cent, we see from the last column of Table 7, t h a t the potential energy given by the cavity model is more t h a n twice t h a t of the Linevsky one. Two explanations m a y be advanced for this difference. On the one hand, the Linevsky model takes no account of any further polarization arising from the induced moments themselves. I t cannot be true t h a t the total energy can be expressed as the sum of a series of terms as in (5). Table 7. Comparison of "cavity" and "point polarizable atom" models

Ar Kr Xe

do

e*

f x l0 -22

L(~, 0, r)# × 10~22

3.83 3.95 4.34

1-599 1.784 2.033

4.06 4.46 3.99

1.49 1.80 1-73

A%av/A~e.P.S. 2.7 2.5 2.3

* Ref. [20].# L(~, 0, r) = ~¢Z(3 cosS0 + 1)/r s. t / = 2(8 -- 1)/(28 + 1)a3. On the other hand, the Linevsky model effectively ignores the contributions from those outer parts of the host atoms immediately touching the LiF, by treating the polarizability as located at the centre8 of these atoms. The dependence on I/r e means t h a t these near regions are of much greater importance t h a n those further out. The formula (6) is only applicable when the diameter of the particle considered is small compared with the distance from the primary dipole, as for example, in a gas. In the cavity model, b y setting a ---- ½d0, due weight is being given to the effect on the region immediately surrounding the solute molecule. While this m a y lead to a [19] Ref. [9], p. 68. [20] R. L. A~E~ and R. H. COT.~,J. Chem. Phys. 40, 146 (1964).

Infra-red spectra of crystals--IV

2415

slightly overestimation of the energy it seems likely that the point-polarizable a t o m model underestimates b y a factor of two the true potential energy from this particular source. Proceeding now with the further calculation of the frequency shift, Linevsky uses the result of first order perturbation theory, that

=

{[v]~

-

[¢],}/~

where [ V]~ = ~v2~*. V ~ dt and x and g refer to the first excited and ground vibrational states respectively. From (6) [V], = --½a~{(3 cos ~ 0 + 1)]r~}. IW,*tu*~, dt d

and thus AVv,b = constant. [/,2]® _ [/~2],

(8)

all these averages being over each vibrational level. For the values of [/,~]~ and [/x~lu, Linevsky takes the data of BRAU~STEn~ and TRISCm~A [21] for /,~I from the Stark effect on the pure rotational spectrum of LiF in different vibrational levels. However, what is measured in the Stark effect is not [/~2]vm b u t [ # ] ~ , the period of a vibration being so much less than that of a rotation. The effect of this substitution is seen b y making the usual expansion in terms of ~ = (r -- r,)/ro /x =/z~ + / ~

+/z~ ~ + ...

from which [g~]~ -- [/t]~ = gl~{[~]~ -- [~]g~} + t g ~ { [ ~ ] ~ -- [~] ~} + 2/~/t~{[~]~ -- [~]~}

+ ~g~{[~]~. [~]~

-

[~]~[~%} + . . .

whereas

T h u s i n the double electrical-mechanical h a r m o n i c a p p r o x i m a t i o n , i n which/~9 : 0, [~1~ = 0, [#]x ~ -- [/~]g2 is zero, whereas [#~]~ -- [/~21ais finite. W h a t m i g h t be expected

to be the major term in the frequency shift, arising from/~1 ~ (which is proportional to (d/~/dQ) ~) is in fact missing from [/~]2 _ ~]g2. Surprisingly enough the effects of anharmonicity in LiF are so great that this omission makes only a small difference.* This m a y be demonstrated b y a specimen calculation based in the data of W~AR,TON et al. [221 for the dipole moment of LiF as a function of vibrational quantum number v. The term which is linear in v depends on/~1 and ~u~and the cubic anharmonicity coefficient a 1 [231. /~1 was assumed to be equal to ~u~ (the effective charge model), * I am indebted to Dr. W h a r t o n for an alternative demonstration of this point. [21] R. BRAU~STEI~ and J. W. TRISCHKA, Phys. Rev. 98, 1092 (1955). [22] L. WHArTOn, W. K~EMPERER, L. P. GOLD, R. STRAUCH,J. J. GAT~T.AGHERand V. E. DERR,

J. Chem. Phys. 88, 1203 (1963).

2416

D . C . McK~,-¢ Table 8 (a) Calculation of [/~]2 for LiF Term

v = 0

1

A1-°

/Xe2 2/~e~tl[~] /~e/x,,[~~] /~1~[~2] ~tl#~[~a]

39"49444 0"5048 0"0210 0"0642 0"0005

39"49444 1"5682 0"0667 0"2038 0"0033

0 1"0634 0"0457 0" 1396 0"0028

¼/~22[~ 4]

8'6 × 10 - e

5"0 × 10 - 5

Total

40.0849

0

41.3364

1.2515

6.41453 6.41511 41-14620 41.15364

1.1243 1 "1146

(b) Calculation of [/~]2 for LiF [P,]caxc [/~]obs [/~]~aae [H]~obs

6.32629 6.32764 40"02195 40'03903

Data: a I ~ --2.70072, a 2 ~ -4-5"1868, #e = 6.28446 D,/~1 ~ ;% (assumed), #2 = -t-2-056 D. w h e n c e ;ue w a s f o u n d t o be + 2 - 0 5 6 D units. A s e c o n d o r d e r p e r t u r b a t i o n t r e a t m e n t b a s e d o n t h e v a l u e s o f a I a n d a 2 o f reference [22] w a s t h e n c a r r i e d o u t t o d e t e r m i n e [~], [~2], [Ca] a n d [~4] in t h e s t a t e s v : 0 a n d 1, t h e results o f w h i c h are s h o w n in T a b l e 8.* T h e difference [/x~]~ -- [#2]g is seen t o be o n l y a b o u t 14 p e r c e n t g r e a t e r t h a n I/x], 2 -- [;u]g3. This r e s u l t is u n l i k e l y t o be sensitive t o t h e v a l u e o f #1 c h o s e n a b o v e . H o w e v e r if t h e s a m e c a l c u l a t i o n is r e p e a t e d for a m o l e c u l e w i t h a small p e r m a n e n t dipole m o m e n t , s u c h as CO, t h e d i s c r e p a n c y is, as e x p e c t e d , m u c h g r e a t e r . F o r t h e l a t t e r molecule, YOUNO a n d FAeriES [24] h a v e d e r i v e d v a l u e s a n d r e l a t i v e signs for /Xl, /x2 a n d / x a : t h e i r signs r e l a t i v e t o t h e sign of/t~ h o w e v e r r e m a i n u n k n o w n . T a b l e 9 s h o w s t h e results o f a c a l c u l a t i o n for CO similar t o t h e a b o v e for L i F . Clearly A[/x~] is less t h a n 25 p e r c e n t o f A[/~2]. MULTIPOLE FORCES IN THE CASE OF L i F W e m a y f u r t h e r a s k if t h e m u l t i p o l e t e r m s will p l a y a n i m p o r t a n t role in t h e ease o f d i a t o m i e species s u c h as L i F or CO. L i F s h o u l d be well r e p r e s e n t e d b y t h e effective charge model. Since t h e ionic r a d i u s o f t h e Li+ is v e r y m u c h less (0-60 A [25]) t h a n t h a t o f t h e F - ion (1.36 A [25]) t h e n if t h e o u t e r p a r t o f e a c h ion is e q u i d i s t a n t f r o m t h e wall o f t h e c a v i t y , z+° will be + 1 . 1 6 2 A, z_ ° --0-402 A, for a c a v i t y r a d i u s o f 2.17 A (Xe). M o r e o v e r , d u e t o t h e difference in m a s s o f t h e t w o ions, dz+ = - - 2 . 7 1 4 dz_ * The values of/t 1 and ~o calculated reproduce the experimental ones satisfactorily. [23] C. SCHLIER, Z. Physik, 154, 460 (1959). [24] L. A. You~cG and W. J. EAclrus, J. Chem. Phys. 44, 4195 (1966). [25] L. PAur~I~G, The Nature o]the Chemical Bond. Cornell University (1940).

Infra-red spectra of crystals--IV Table

2417

9

(a) C a l c u l a t i o n o f [p2J f o r C O Term

v = 0

(1) (2) (3) (4)

Ps 2/x~ui[~ ] p~u~C~J pl~[~ ~] (5) fllp2[~ 3] (2) (~s + ve~* and (4) e -- re]

v = 1

Ai-0

+0"01254 -4-0.00284 ±0.00004 +0.01122 1"6 X 10 - 5 +0.01406

+0"01254 ±0.00873 ±0.00012 +0.03480 10.6 x 10 - 5 +0.04353

0 ±0.00589 ±0-00008 +0.02358 +0-00009 +0.02947

+0-00838

+0-02607

+0.01769

(b) C a l c u l a t i o n o f [p2] f o r C O [P] {Pe + r e } * /x e - - v e

[p] lp . + v e I * p

-- veJ

+0.12449 --0.09951

+0.15003 / --0-07397J

+0.02554

+0.01550 +0.00990

+0.02251 +0.00547

+0.0070 +0-0044

* R e l a t i v e t o Pi" Data: a 1 = --2-6954, a 2 = +4.4532, p2 = - - 0 - 1 9 1 0 . T a b l e 10. C o n t r i b u t i o n s

P6 =

±0-112,

Pl = +3.509,

to P.E. from multipole terms for LiF in Xe

Coordinates of atoms with respect to cavity centre: z + ° -~-+ 1 . 1 6 2 ~ z_°=--0.402.~ dz+ ~ --2-714 d z Mode factor Pole 21 22 2s 24 25 26 2~ 2s 22

(s, ~) factors 1 0"12185 1-535 X 10-2 1-9565 X 10-a 2-5135 × 10-4 3.2373 × 10-5 4-1785 × 10-6 5.4005 × 10-~ 6.9869 X 10-s

(harmonic) 1 3"5893 25-543 89-187 320.65 1008.9 3031.3 8742.9 24,436

Mode factor Total 1 0"4374 0-3921 0-1745 0.0806 0-0327 0-0127 0.0047 0"0017 Total 2.136

anharmonic 1 1"8412 11.157 22.690 65.179 170.56

Total

1 0"2244 0.1713 0.0444 0-0164 0-0055 Total 1.463

a ~ 2.17 ~ , e = 2.033.

T a b l e 10 shows t h a t t e r m s b e y o n d t h e 27 pole are insignificant a n d t h a t t h e t o t a l increase o v e r the dipole t e r m alone is N 1 1 4 per cent. This increase of course refers t o t h e h a r m o n i c c o n t r i b u t i o n or t e r m in/xi2~ 2 in Table 8. T h e corresponding t e r m s t o p ~ . / x 1 . ~ m a y also be calculated in like m a n n e r . These fall off more r a p i d l y t h a n t h e h a r m o n i c ones, and, u p t o the 25 pole, a d d a n o t h e r 46 per cent. T h e t o t a l shift in t h e x e n o n m a t r i x , shown in Table 11, is t h e n increased from --126 cm - i to --192 cm -l*. * The magnitude of these shifts means that first order perturbation quate in this situation. 13

theory would be inade-

2418

D . C . McKEA~

The m a g n i t u d e o f these calculated shifts, with or w i t h o u t the multipole terms, relative to the observed shifts, can i m p l y only one thing. T h e actual shifts are largely d e t e r m i n e d b y repulsion forces in addition to the classical a t t r a c t i v e ones, and there is no question of being able to estimate contributions from dispersion forces in the m a n n e r o f Linevsky. I t m a y be a d d e d t h a t because of the presence o f repulsion forces, one c a n n o t argue f r o m a poor correlation between Av a n d (8 -- 1)/(28 + 1) for example t h a t the dipole-induced dipole forces are negligible, or t h a t the c a v i t y model m a y be dispensed with. Table 11. Comparison of observed and calculated matrix shifts for LiF (cm-1) Matrix Ar Kr Xo

A~oba --55,60 --65 --71

Xo

AVealc(eavity model)

A%~c (p.p.a. model)

-128 --140 --126 with multipo~ terms --192

--41 --50 --48

FREQUENCY SHIFTS FOR CO Ilg MATRICES F o r CO there is no simple model to enable t h e multipole t e r m s to be calculated. Accordingly only t h e dipole one has been used to calculate the shifts in Table 12, which shows also the e x p e r i m e n t a l values [26, 27] a n d also the calculated shifts of CHARLES a n d LEE [7]. A c o m m e n t is first n e e d e d on t h e assignments of the two bands observed b y the l a t t e r a u t h o r s in K r a n d X e [27] a n d b y n u m e r o u s a u t h o r s [26] in Ar. Table 12. Frequencies of CO in inert gas matrices and quinol-clathrate (cm-1) Gas Ar Kr Xe Clathrate

2143.3 2148.8" 2145.2~ 2141.4~

v

d0(A)

A%alc (car)

Areale (p.p.a.) +

2138.0" 2135-3~ 2133.4~ 2133 §

3.82 3-95 4.34 4.0

--(3.0 or 1"8) --(3.3 or 2.0) --(3.0 or 1.8) --4§

--0.9 --1.0

--1.0

* R o L [26].

t Ref. [27]. ++ Rof. [7]. § Ref. [28]. I n t h e Ar m a t r i x , LEROI et al. [26] assign the lower b r o a d e r b a n d to aggregates of CO molecules, t h e higher, n a r r o w e r b a n d t o isolated monomers. I n K r , Charles a n d Lee a t t r i b u t e the lower b a n d to m o n o m e r s on a n interstitial site, the higher b a n d [26] G. E. L~.ROI, G. E. EWING and G. C. P ~ ¢ T E L , J. Chem. Phys. 40, 2298 (1964). [27] S. W. CHARLESand K. O. LEE, Trans. FaradaySoo. 61, 614 (1965). [28] D. F. BAT,T.and ]:). C. MoKEAN, Spectrochim. Acta 18, 933 (1962).

Infra-red spectra of crystals--IV

2419

to monomers on substitutional sites. However they reverse this order for Xe on the grounds of the increase in size of the lattice hole from Kr to Xe. I t seems relevant to point out that the shift for CO in a quinol-clathrate cage, which is considered to have a diameter of 4.0 A, is -- 10 cm -1 [28] and that if in all three inert gas matrices the lower band is taken to be that of the monomer on a substitutional site, with a modicum of translational or vibrational freedom, then the shift, which is always negative, increases smoothly with the size of the cavity up to about 4.0 A, beyond which it remains roughly constant. Similarly the frequency of the higher band decreases uniformly in the series Ar, Kr, Xe, and it would be natural to assign this to the same species on an interstitial site throughout. Whatever the assignments, the decreases in frequency in the series A r - K r - X e evidently result from a decrease in the magnitude of the repulsion forces, as concluded b y C~ARLES and L~.E [7]. The dipole-induced dipole contribution calculated b y the latter however is likely to be too small. CONCLUSION The present results show that the cavity model with dipole terms only gives a higher frequency shift than the point-polarizable-atom model, while addition of higher multipole terms gives a further considerable increase. Both increases result from the inclusion of short range electrostatic effects. The present calculations require to be improved, firstly in the model assumed for the vibrating molecule and secondly in the description given of the environment. I t seems unlikely however that the general nature of the results will be affected.* The net result of increasing the size of the attractive forces is to emphasize the importance of the role played b y repulsive ones. Acbnowledgement~I am indebted to Dr. A. A. CHA:LM:ERSfor checking part of the calculations. * Note added in proof

The implicit assumption that "classical" and "quantum-mechanical" effects are additive also requires a close scrutiny.