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A N H A R M O N I C I T Y O F V l B R A T I O N IN
M O L E C U L E S
BY K. S. VISWANATHAN
(fvIemoir No. 105 of the Rarnan Research Institute, Bangalore-6) Received December 19, 1957
- 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 sub- ject of several investigations, references to which can be had from the well- known books on the subject by Herzberg l and by Wilson, Decius and Cross. ~ In a detailed and elaborate piece of work, Nielson 3 has considered the prob-
lem 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 vibra- tion 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 mole- cule. 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 pro- cedure for obtaining the corrections to the energy levels due to the anhar- monic 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 m o d e 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 eigen- functions and energy values for the case in which one o f the modes ls doubly degenerate have been evaluated.
- THE ENERGY LEVELS We start from the equilibrium configuration o f 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 e n e r o ' 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 = 89S ,7,z (1) i = I
where ~1, ~.,......... ~N(N = 3 n - 6 or 3 n - 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 p d n d p l e
8 [J] = 8 1 $* H r = 0 (3~
subject to the condition
i r r = J where
H = ( T + V ~ and
N dV = H d~i. s,=.t
88 K. S. VISWANATHAN
Aia~~~ = ~; ati i; and
__ 4~r 0"v m. (9)
7m -- h
The upper suffix in the above indicates the order of approximation. Further q m = J ' ( H r (1-[~m)fId~i and by substitution of the harmonic oscillator wave-funcdons in this we get N ~(1~ = S (vi + 89 h n. 1= Hence the eigenvalue of the i-th mode _(Ÿ ( q m __ 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-,/f 2 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 a s a 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 o f the mole- cule 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 mole- cule. Thus from (9) ir follows that to a first approximation a normal mode belonging to the antisymmetric species o f vibration o f 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(~
+ ai+3tol~v~+8(~ (10)
where
with
and
Anharmonicity of Vibration in Molecules
al----1 2h ~vi~ (8yi a) q- ~ (8h ~vi~-7q 2A^ ~_^ f i ~
1 f i = ~ v i (v i -- I) (v i -- 2) ai~ q-- vi (3~)iei + f/i) 2
+ (vi + 1) [3 (vi + 1) oa +/~i] 2
- ~ ( v i -~- 1) (vi +^1 2) ( v i + 3) o"
- a ~ ai-3 ----- 3h vi (87i~fi [vi (vi -- 1) (vi -- 2)] ~ o.i; - - a i ai-1 = h v i ~ - ~ ) ~ vi~ (3viai q- fli);
ai+,~ = h ~i ~~'i~) ~ (v~ + lP [3 (vi + 1) ~ + ~i]
ai+.a = 3hvi^ aito)'r~^ [(vi 4- 13..^ (vi -I- 2) (vi -a 3)]~ ni
ai - - -- ai~i 6^ and^ f l i =^ -- ~'i Z^ a,nmi (v,,~ + .}) 7 m
h z b 2 t~ 8. 2 (^) ~7/i*- -I-, Ait"%Ti q- At2'~"~7,2 q- Ai3'2',i 3 + Ai,'~'~i'~ 3 ~v,
= Wi~2~~v, ( 1 2 )
T o 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 H a m i l t o n i a n also. After considerable simplification, the wave function or the ith m o d e can be written as
The upper suffix in (10) denotes the order o f approximation. The zeroth order functions are chosen as the h a r m o n i c oscillator eigenfunctions.
Anharmonfcity o f Vibration in Molecules 91
ha the above the accent indieates that the sum excludes the term 1 o r m = i.
The quantities Rt, K4, Tz and Qz are given by
Rt = 3 (2vt + 1) ctz 4-/3~;
K t = 30az" {(vi + 89 7 } + 12~t/~l (za~t+ 89
Ti = 30r (^) t (vz + 89 + (376 + 7t 6/~z (v~ + 89 ; (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 + 89 (v~ + }) (16) i < m^ lm
Remark
We shall define W ira by
Wt,~ .~ ~ i ~ (vi + 89 Rm + o-rnmi(v,n + 89 R,
= (4)'m~'~h vm 4~'q vt
, Jgmmi{t, [ 89 (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 W91 - 89Z' 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 o f vibrations o f the molecule is doubly d e ~ n e r a t e 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• o f the overtone level of the doubly degenerate ruede is then (vs+l). Ir is well known that the eigenfunctions for the degenerate ruede m a y be written as
~bm = Nmet-V,d2)'~'Fv,~ l.~ ( x/-~;~p~,) e ~:iI.~$,, (19)
where Fr,~z,, ( x / ~ p s ) 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 v s is odd or even. Now the symmetry of an overtone level oi" d e s e e 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 o f the molecule. Thus the wave-funetions (19) will transform, under a symmetry operation, like a linear sum of the different irreducible representations o f the molecule. Of, alternatively, Che can forro linear combinafions o f 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 o f 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 o f the same
species can have different energy levels and thus the m a x i m u m number o f
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 o f functions ~s "~a (s ---- 1, 2 ,.... v s + ~ ) such
94 K. S. VISWAN&THAN
The first-order energy values and eigenfunctions may be obtained by substi-
tuting 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
Ym
For the degenerate mode, the wave equation is given by
g~2 (~~/~N_: r ~;~~~) - - W98 + =^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 func-
tions 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
Wl~ (21 ~ (~N
N ~ 2
q- 1) hvN +^ Z^ ,~sNm(v~-+ l) 4,,ra~y~hvm Rm
m = l
, ~NNNN T6~Na-{(v~., i)~ q_ ~} flNNNS48yX 2
V n + l x {S (c~.?,.) %"} (27)
Since the last term in the above may be expected to be different for functions
with different 7 a n d a (LeS for wave functions falling under different irredu-
cible 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*.
+~"~ [m
4?l?m
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 gir91y~ 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 x i k 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 eigen-
values. It is shown that, to a first arder of approximation, normal vibra-
tions 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.
- G. Herzberg
2, Wilson, E. B., DecŸ J. C. and P. C. Cross
- Nielson, H. H.
- Frr J.
- Tisza, L. A
REFERENCES .. InfraoRed and R a m a n Speetro, Van Nostrand, New York, 1945, pp. 201-226. Molecular Vibrations, McGraw Hill Book Co., 1955.
.. Physieal Re~iew, 1941, 60, 794. .. Wave Mechanics, Advanced General Theory (Oxford), 1934, 423 -28. .. Z. Physik., 1933, 82, 48.
