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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 sub- ject 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 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 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 eigen- functions 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 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 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 + {~.
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