Anharmonicity of Vibration in Molecules 87

ANHARMONICITY OF VlBRATION IN MOLECULES

BY K. S. VISWANATHAN (fvIemoir No. 105 of the Rarnan Research Institute, Bangalore-6) Received December 19, 1957 1. INTRODUCTION

ANHARMONICtTY Of vibration is responsible for severa1 of the finer features observed in the vibration spectra of molecules, viz., the uneven spacing of the overtone levels of the different normal modes and the splitting of le'veis that are degenerate in the harmonic oscillator approximation. The spec- trum of the vibrational energy levels, when the effect of anharmonic terms in the potential energy of the molecule is taken into account, has been the subject of several investigations, references to which can be had from the well- known books on the subject by Herzbergl and by Wilson, Decius and Cross. ~ In a detailed and elaborate piece of work, Nielson 3 has considered the problem for a general molecule taking account of one more complication, viz., the vibration-rotation interaction in the molecule. In the harmonic oscdlator approximation, the normal modes of vibration of the molecule are independent of each other and the energy of the system is the sum of the energies of the (3n -- 6) normal modes of the molecule. When anharmonicity is taken into account, product terms of the third and higher powers in the normal co-ordinates are introduced in the potential energy of the system, and as a consequence the normal modes are no longer independent but interact with each other. The standard procedure for obtaining the corrections to the energy levels due to the anharmonic terms is by the perturbatiorl method applied to N ----- (3n -- 6) varia- bles and the results of the theory indicate that the energy levels get altered by the introduction of additional quadratic terms in the vibrational quantum numbers of the normal modes of the molecule. In the present paper, we adopt a dlfferent method and follow a procedure well known in the treat- ment of electronic motiorl in atoms and mo!ecules, namely the method of the self-consistent field proposed by Hartree. Each normal mode of the molecule Ÿ assumed to be moving in the average potential field of vibration of the remaining ones, and the eigenfunctions and energy levels of each of the normal modes ate calculated accordingly. Apart from the academic interest of the fact that the Hartree method can be aoplied to the problems of vibrational motion of the molecule, the procedure deserves attention 85 86 K. S. VISWANATHAN since it enables one to regard a moleeule as ah assembly of anharmonic oscitlators and calculate the anharmonicity constants in terms of the force constants of the molecule, and also because ir enables one to evaluate a measure of the interaction exerted on each mode by the remaining ones. Thus it is shown in Section 2 that a normal mode of the anti-symmetric species does not interact with the rest even in the first order of approximation. In Section 3, the question of degeneracy has been considered and eigenfunctions and energy values for the case in which one of the modes ls doubly degenerate have been evaluated. 2. THE ENERGY LEVELS We start from the equilibrium configuration of the molecule, that is the configuration in which the forccs acting on each atom of the molecule is zero. Referred to this state, the potential enero' of the system will not contain any terms linear in the displacements of the atoms. Ir further we use normal co-ordinates, the potential and kinetic energies will not contain any quadratic cross terms and we can write therefore

N T = S ,7, z (1) i=I

1 where ~1, ~., ...... ~N(N = 3n- 6 or 3n- 5 as the case may be) are the normal co-ordinates of the system. We first consider the case in which the system is non-degenerate. The r al, A=...... AN in (2) are then all different. The wave funr237 ~ (~1 .... ~x) describing the smte of the molecule is givcn by the solution of the variational pdndple

8 [J] = 8 1 $* Hr = 0 (3~

subject to the condition i r r = J (4) where

N H=(T+V~ and dV = H d~i. s,=.t Anharmonicity of Vibration in Molecules 87

We now try to approximate ~Ÿin terms of the one normal co-ordinate ' wave functions ~i(Ti), each of which dcpending on one normal co-ordinate only. We set (5) whcre vx, v~ ...... vs are integral numbers dcnoting the dcgrce of excita- tion of the different normal modes, each moving in the field of the rest. The functions Sv~ ('/i) are all assumed to be normalised, and the best possible choice of these which approximate the physical system as closely es possiblc are obtained as the solution of the variational principle (3). This leads to the following set of differential equations which the $'s must satisfy. 4

(Hi -- q) ~v, = 0 (6) whcre Hi = f (H $v**) H (/7 $v,)/7 dvk (7)

*i is indepcndent of the sufflX i and dr the energy of the wholc moleculr To evaluatr the Hamiltonian of the i-th normal mode as well as to determine its r we adopt the procedure of variafion-iteration. Fo a first approximation, we ignore the quartic tr in (2) and substitutr for all the wave functions $vk (Vk)(kv ~ i) in (7) the harmonic oscillator :igenfuncfions of thr normal co-ordinates. Tila integrals that arise in this procr likr ~$*v~ Vk~ have becn tabulated in Wilson's ~ book and making use of thr162 wr find that equation (6) reduces to { h2 b2 8 *r~ ~VQ - (Ai~ -- qa)) + Aª + Ai~n) vq

+ Abm~iz}@v~ (YO = 0 (8) where Ajom = 27 (v~ + hv

Aid~) = "~'-2 amraiym(vm + 88 K. S. VISWANATHAN

Aia ~~~ = ~; ati i; and

__ 4~r 0" vm. (9) 7m -- h

The upper suffix in the above indicates the order of approximation. Further qm = J'(Hr (1-[~m)fId~i and by substitution of the harmonic oscillator wave-funcdons in this we get

N ~(1~ = S (vi + hn. 1=1 Hence the eigenvalue of the i-th mode (Ÿ (qm_ A0tl0, is equal to (vt-~-3) hvt. The first order approximation therfore does not introduce any corrections to the eigenvalues of the different normal modes from their harmonic oscillator values. From (8) it foUows that the Hamiltonian of the oscitlator contains an anharmonic eubic term whose coefficient is the same as the coefficient of' -0ia in the potential energy of the molecule and also a term linear in r/i. The linear term arises out of the process of avcraging terms of the type ~Ti-,/f2 and it gives a measure of tke action exerted by the other normal modes on the i-th one. It further suggests that the vibrations of the remaining modes tend to displace the equilibrium position of the i-th one from the place it oecupies in the equilibrium con¡ of" the molecule asa whole. The coefficient of the linear term is much smaller than the coefficient of the cubic one and their ratio is of the order of 1/?' ,--, 10-~6. But sinee the region wherein the displacement of the oscillator has a finite probability is of the order of 7 -~, both these temas ate of the same order of smallness. If the i-th mode belongs to an antisymmet¡ species of vibration of the moleeule, alt the coefficients of the type amrai and a¡ should be equal to zero. This is because the potential energy, includmg third and higker order terms, should be invariant under all the symmetry operations of the molecule and there will be at least one operation which will change the sign of a normal co-ordinate falling under an antisymmetric species of the molecule. Thus from (9) ir follows that to a first approximation a normal mode belonging to the antisymmetric species of vibration of the molecule does no/ interact with the rest and a[so ir suffers no anharmonicity. The first order eigenfunctions of (8) may be obtained by the perturba- tion method. The normalised eigenfunctions are given by r = at~v(O~ _z_ ai_3(ol~bv~_z(o~ -4-, a~_x(O~~bv,_lt~ -4-. a-~a(~176 + ai+3tol~v~+8(~ (10) Anharmonicity of Vibration in Molecules 89 where

al----1 A ~_ fi ~ 2h ~ vi ~ (8yi a) q- ~ (8h ~ vi~-7q 2 with

1 fi = ~ v i (v i -- I) (v i -- 2) ai~ q-- vi (3~)iei + f/i) 2

+ (vi + 1) [3 (vi + 1) oa +/~i] 2

1 + ~(vi -~- 1) (vi + 2) (vi+ 3) o"

--a~ ai-3 ----- 3h vi (87i~fi [vi (vi -- 1) (vi -- 2)] ~ o.i;

-- a i ai-1 = h vi~-~) ~ vi~ (3viai q- fli); (11)

ai+,~ = h ~i ~~'i~) ~ (v~ + lP [3 (vi + 1) ~ + ~i]

ai [(vi 4- 13. (vi -I- 2) (vi -a 3)]~ ni ai+.a = 3hvi to)'r~ . and

-- ai~i and fli= -- ~'i Z a,nmi (v,,~ + .}) ai -- 6 7m

The upper suffix in (10) denotes the order of approximation. The zeroth order functions are chosen as the harmonic oscillator eigenfunctions.

To obtain the eigenvalues and eigenfunctions correct to the second order, we substitute the first order eigenfunctions (10) in (7). We now take into account of the quartic terms in the Hamiltonian also. After considerable simplification, the wave function or the ith mode can be written as

h z b 2 t~ 8. 2 -I-, Ait"%Ti q- At2'~"~7,2 q- Ai3'2',i 3 + Ai,'~'~i'~ ~v, ~7/i*- 3

= Wi~2~~v, (12) 90 K. S. VISWANATttAN whr

hi, '~'' = Z' ~ui(v~ + ~) 4_ ~'-~' ~u~ , 2y~ '/.~i 16lP-v-~'Yt~Q~ 1

+ Z' 2~~~-~~~fll~i T: + Z' 4q,:~m~h~~z~,~ RzRm 1 Im l "r m +.~'/h~~i (vt + {) R~ 4~,,~'W, ~,,, : lm

Ai3u') aª 1 Z' fllª Rz" (13) = -6- -f~ lŸ ~ '

and

Ki atit Ti Wi t~' - (vi + hn + (8n~) hTi + 24n3h~i

+ 3Ty~{8i~tt {2 (~i + +

+ ~-~flLlii- ~~,~ + {v~ + ,•f'l

+ {~. ~'iim r.,. +

a"ff252 (,,,, ~14) + 4-,,r ,,{ '" ~ + pq 2:lis 1 Anharmonfcity of Vibration in Molecules 91 ha the above the accent indieates that the sum excludes the term 1 orm = i. The quantities Rt, K4, Tz and Qz are given by Rt = 3 (2vt + 1) ctz 4-/3~;

Kt= 30az" {(vi + 7} + 12~t/~l (za~t +

Ti = 30r t (vz + + (3767t + 6/~z (v~ + ; (15)

Q~ --- K~ + {vi (ziz -- l ) (vt - 2) (3vz~~ + fl~) az -- vz (vi 4- I) (3vzaL -k/~~) (3vt ,.4_ 1~~ 4,- fil) + {(v~ + l) (v~ + 2) (v~ + 3) [3 (vt + 1) r + q a~ W( ~~ gives the expression for the energy of the ith normal mode. The energy of the whole molecule is Z Z ~~ 'Z ~

+ 32yZ"~ 4ym~/th vm

+Z ,&u~~4~,z~,~(vt + (v~ + }) (16) lm i

Wt,~ .~ ~i~ (vi + Rm + o-rnmi(v,n + R, = (4)'m~'~h vm 4~'q vt

, Jgmmi{t, _[_ (v i .at_ ,-- 4yrnyi ~.,,ra 07) W ira denotes the average interaction energy of the ith and mth modes eorrect to the second order of approximation. The energy of the whole moleeule is related to the energy of the individual modes by means of the relafion ~(-4 = Z W - Z' Z' W ~ra (18) 6 4 92 K. S. VlSWANATHAN

3. DEGENERACY We now consider the case of degeneracy of" vibrafions. For the sake of preciseness, we consider the case in which one of the ruedes of vibrations of the molecule is doubly de~nerate and the remaining ruedes are all non- degenerate. The more general canes can be discussed on exactly the same lines. Let us denote the ruedes that ate degenerate with che another by the suffixes (N -- 1) and N so that v~_x-~ vs. Our problem then is to evaluate the energy values and eigenfunctions of" the degenerate ruede and to find a measure of its splitting due to anharmonicity. Let us fix our attention on a state in which the different normal ruedes of" the molecule ate excited by v~, v.,,...... vN-.~ and v~ quantas respectively. The degener• of the overtone level of the doubly degenerate ruede is then (vs+l). Ir is well known that the eigenfunctions for the degenerate ruede may be written as

~bm = Nmet-V,d2)'~'Fv,~ l.~ ( x/-~;~p~,) e ~:iI.~$,, (19) where Fr,~z,, (x/~ps) is a polynomial of" degree v~ in p~ [= (~N_~~ § ~~2)t], and IN is ah integer whicla can assume the values vN, v_,r v~r .... 1 of 0 depencL/,ng on whether vs is odd or even. Now the symmetry of an overtone level oi" desee v~ of" a ruede which fa]ls under a species ,P of the molecule is given by (F) v'~ and this is in general, a linear sum of the irreducible representations of the point group of the molecule. Thus the wave-funetions (19) will transform, under a symmetry operation, like a linear sum of the different irreducible representations of the molecule. Of, alternatively, Che can forro linear combinafions of the above tvs-Ÿ 1) wave-functions in such a way that the resulting functions f'all exclusively each under any one of the irreducible representations of the symmetry group of the molccule. Let us suppose that the structure of the level under consideration is given by Znr 'c~~ where Pr stands for the yth species and nr is the number of" times this species occurs in the reduced representation of the level. Ir has been shown by Tisza s that only wave-functions which either belong~to different irreducible representations or to different matrix representafions of the same species can have different energy levels and thus the maximum number of components into which the level can be split up by the anharmonie terms is 2'n c'n. Let us now subject the set of wave-functions (19) to an orthogonal trans- formation and obtain a new set of functions ~s "~a (s ---- 1, 2,....vs+~) such Anharmonicity of Vibration in Molecules 93 that each one of these faU exclusively under one of the irreducible representations of the symmetry group of the motecule. In symbols,

VN+I es ~a = Z Csrd~a‰ (s = 1, 2,....VN+3 (20) where

(~N+I) Z' (CsmTa) ~'== I (21) f'A~l and

Z' C~r "t'a" = 0 (s ~ s')

In the above the upper suffix y denotes the irreducible representation F~~~ under which ~s'ra falls, anda denotes the matrix representation of the irreducible rcpresentation. Al1 functions with different y anda may therefore be expected to have different energies. The wave function for the ith ruede is now given by

(Hi-- q) Cv, = 0 (22) whr Hi = [ (II~*v~) H ( II ~bv,) II &?k (22 a)

In evaluating the above expression, we may notice that the potential energy of the system does not contain the factors Ÿ or ~~ individually, but involves these only through the sum of their squares (i.e.) through

07s-t e + ~?s') = ,o~e.

Thus, we have

atjt~-t = ~jN = 0

(i, j = 1, 2 ..... N -- 2) ; ~N--1 N--1 N--1 = ~Ir = 0 ;

#ljk~-~= #~~k~ = ..... /h~~ = 0 .... 0,1, k = 1, 2, .... N -- 2) and

~is-t s-t = a~~ ; ~ij~-z ~r-~ = ~/,~.,or, etc. 94 K. S. VISWAN&THAN

The first-order energy values and eigenfunctions may be obtained by substituting the harmonic oscillator wave functions for the non-degenerate modes and the function 4,s"~a for the degenerate one. We get qm = Z (vi + d~/2) hvi where the sum is over all modes with distinct frequencies and di denotes the degeneracy of the ith mode. The wave equation is given by the same formula~ (8-) and (9) with this difference that the constants Aiom and Ait m get modi¡ in this by

and (Zrami(Vm-+-d-m-) Ail TM = {- ~" -- (23)

The first order wave equation is therefore still given by (10) with this change that fl~ should be defined here by

(24) Ym

For the degenerate mode, the wave equation is given by

g~2 (~~/~N_: r ~;~~~) -- W + = 0 (25) where WCn = (oN +_1) hv~.

Wc set that in the first order approximation, anharmonicity does not affect the degenerate modo at ail. The Hamiltonian as weIl as the eigenvaIue for the degeneratc mode correet to the seeond ordcr may be obtained by substituting the wave functions given by (10) and (24) in the variational pt• (6). One gets

h_~" b"- b '~

-- Ws c~ + AN(~~ p.~~ + A~,-(*~p,~a} ~bvs = 0 (26) Anharmonicity of Vibration in Molecules 95 where

~Ÿ N~'~

47ffh vg 4yt I=~ |=I

N~2 Wl~ (21 ~ (~N q- 1) hvN + Z ,~sNm(v~-+4,,ra~y~hvm l) Rm m=l

+ Z #~~~= m

, ~NNNNT6~Na- {(v~., i)~ q_ ~} flNNNS48yX2

Vn+l x { S (c~.?,.) %"} (27)

Since the last term in the above may be expected to be different for functions with different 7 anda (Le S for wave functions falling under different irreducible representations or different matrix representations of the same irreducible representation, each level of the degenerate mode will get split into not more than (In~') subtevels. The energy of the whole molecule is .,,.,__Z(o,§247 Z i~,

|~1 |M ,n~N

N~2 Z I*.1 +~"~ 4?l?m [m 96 K. S. VISWANATHAN

T ...... 7r ]~NNNN 16~,,~-

VN+ 1 x { S (Csd"~)~lm "} (28) tllat I

Nielson has given explicit expressions for the vibrational energy levels of the molecule. His formula for the energy of the molecule may be written as G (v~, %, .... )

Xik i k>i

(29) i k>i In the above, the cross coefficient gi~ of the azimuthal quantum numbers l~ and lk depends on the vibration-rotation interaction of the molecule which is a feature considered in Nielson's work. This explains the absence of the cross terms in the quantum number 'l' in the equation (28). Further, as has been pointed out by Herzberg, the formula (29) gives in general a spIitting into fewer levels than what a group theoretical theory would indicate. This is because the wavefunctions used in the evaluation of energy in (29) are the functions (19) and not linear combinations of these faUing under different representations of the point group of the molecule. A comparison of (28) with (29) however indicates that the difference is only slight and consists in the replacement of gzt by a coefficient gffya depending on the symmetry of the state of the sublevel. For a molecule with severa1 degenerate modes of vibration, the symmetry of a general vibmtional level is given by the formula

r' = (G~ TM x (G) ~' x q'b v, (30) where the 'X' denotes the direct produet multiplication of groups and vi, v,. .... vj, are the degrees ofexcitation of the modes 1,2,....frespectively. /" is a linear sum of the different irreducible representations of the molecule, and thus by forming linear sums of products of the wavefunctions of the type (19) which faU exclusively under one of the matrix representation of ah irreducible representation of the symmetry group of the molecule, one can evaluate the energy levels of the moleeule. If one adopts the standard Anharmonicity of Vibration in Molecules 97

perturbarŸ method involving N va¡ the expeeted energy levels may be written in the forro

G ~ (v t v 2 .... v~) :Z(o,+~,)~~~ + S S xm "~a

2- vk + -1- Z' Z gir y~ li lr, (31) i k>i In the above, all coefficients xik~ ~ pertaining to non-degenerate modes wiU be independent of >, and a, but the coefficients xik Ta pertaining to degenerate modes and the coefficients gik "ta wilJ, in general, be different for levels of different symmetries. The author's thanks are due to Professor Sir C. V. Raman for his kind interest in this work.

SLrb~ARY The presence of anharmonicity entails the interactions of the normal modes of vibrations, which are independent in the harmonic oscillator approximation. The method of Hartree has been applied to study the mutual interaction of the normal modes, each assumed to be moving in the average potential field of the rest, and to evaluate their wave funetions and eigenvalues. It is shown that, to a first arder of approximation, normal vibrations belonging to the antisymmetrie species do not interact with the rest and suffer no anharmonicity at all. The wave functions and eigenvalues of the different normal modes have been evaluated correct to the s~ebnd arder. The question of degeneracy has been considered and expressions have been given for the energy values of the different sublevels into which ah overtone level of a degenerate system may be expected to split up aecording to group-theoretical considerations.

REFERENCES

1. G. Herzberg .. InfraoRed and Raman Speetro, Van Nostrand, New York, 1945, pp. 201-226. 2, Wilson, E. B., DecŸ J. C. Molecular Vibrations, McGraw Hill Book Co., 1955. and P. C. Cross 3. Nielson, H. H. .. Physieal Re~iew, 1941, 60, 794. 4. Frr J. .. Wave Mechanics, Advanced General Theory (Oxford), 1934, 423 -28. 5. Tisza, L. .. Z. Physik., 1933, 82, 48. A4