Reports of the Academy of Sciences of the USSR
1966, Volume 169, No. 5

UDC 539.194+535.338.42

PHYSICS

A. Ya. Tsaune, V. P. Morozov

ALLOWANCE FOR ANHARMONICITY IN MOLECULAR VIBRATIONS USING GENERAL FORMULAS FOR TRANSITION FREQUENCIES BY SUCCESSIVE APPROXIMATIONS

(Presented by Academician I. V. Obreimov on 26 XI 1965)

The general formula for the vibrational energy levels of a polyatomic molecule, first obtained by quantum-mechanical methods in \((^{1})\) (see also \((^{2})\)), has the form

\[ (E_v/hc)=(E_0/hc)+\sum_s \omega_s (v_s+g_s/2)+ \]
\[ +\sum_{ss'} X_{ss'}(v_s+g_s/2)(v_{s'}+g_{s'}/2)+\sum_{t \leq t'} X_{l_t l_{t'}} l_t l_{t'}, \tag{1} \]

where \(E_0/hc\) is a constant quantity, which is usually omitted, since only level differences are of interest; \(\omega_s\) are zero-order frequencies; \(v_s\) are vibrational quantum numbers; \(g_s\) is a factor taking into account the degree of degeneracy of \(\omega_s\); \(l_t\) is a quantum number taking, for degenerate zero-order levels, the values \(v_t, v_t-2,\ldots,1\) or 0. If vibrational–rotational interactions are neglected, then

\[ X_{ss}=\frac{1}{4}\{6k_{ssss}-15(k_{sss}^2/\omega_s)-\sum_{s'}(k_{sss'}^2/\omega_{s'})(8\omega_s^2-3\omega_{s'}^2)/(4\omega_s^2-\omega_{s'}^2)\}, \]

\[ X_{ss'}=\frac{1}{2}\{k_{sss's'}-6(k_{sss}k_{ss's'}/\omega_s)-4k_{sss'}^2[\omega_s/(4\omega_s^2-\omega_{s'}^2)]- \]
\[ -\sum_{s''}(k_{sss''}k_{s's's''}/\omega_{s''})-\sum_{s''}(k_{ss's''}/2)\omega_{s''}(\omega_{s''}^2-\omega_s^2-\omega_{s'}^2)/[(\omega_s+\omega_{s'}+\omega_{s''}) \times \]
\[ \times(\omega_s+\omega_{s'}-\omega_{s''})(\omega_s-\omega_{s'}+\omega_{s''})(\omega_s-\omega_{s'}-\omega_{s''})]\}, \tag{2} \]

\[ X_{l_t l_t}=-\frac{1}{4}\{2k_{tttt}+\sum_{s'} k_{tts'}^2[\omega_{s'}/(4\omega_t^2-\omega_{s'}^2)]\}, \]

\[ X_{l_t X_{l_{t'}}}=0, \]

where \(k_{sss'}\) and \(k_{sss's''}\) are force constants in the anharmonic part of the potential energy (in terms of third- and fourth-degree terms), expressed in normal coordinates; \(t\) denotes values of \(s\) for which the vibrations are twice degenerate.

The usually used method for determining the vibrational constants \(\omega_s\) and \(X_{ss'}\) (see, for example, \((^{3,4})\)) by means of (1) has, in our opinion, the following shortcomings: a) to determine the quadratic force constants in natural coordinates (or symmetry coordinates), it is necessary to know in advance the transformation from natural coordinates (symmetry coordinates) to normal ones; b) to determine all vibrational constants \(\omega_s\) and \(X_{ss'}\), knowledge of a considerable number of experimental transition frequencies is required; if the number of such frequencies is limited, then, for an approximate solution of the problem, it is necessary to introduce certain relations between the \(X_{ss'}\) (or between the \(k_{sss'}\)), which leads to difficulties in choosing such relations.

The purpose of this article is to present the results of work aimed at overcoming the above-mentioned difficulties in the approximate solution of the problem of allowing for anharmonicity in molecular vibrations. In doing so, we shall use the expression for the potential energy in natural coordinates (or symmetry coordinates) \(q^i\) in the following form:

\[ V=\frac{1}{2}\sum_{ij}F_{ij}q^iq^j+\sum_{ijk}F_{ijk}q^iq^jq^k+\sum_{ijkl}F_{ijkl}q^iq^jq^kq^l . \tag{3} \]

Then the following relations hold:

\[ \begin{aligned} k_{sss} &= 2.10865\cdot 10^{-2}(1/\omega_s^3)^{1/2}\sum_{ijk}x_s^ix_s^jx_s^kF_{ijk},\\ k_{sss'} &= 6.32595\cdot 10^{-2}(1/\omega_s^2\omega_{s'})^{1/2}\sum_{ijk}x_s^ix_s^jx_{s'}^kF_{ijk},\\ k_{ss's''} &= 12.65190\cdot 10^{-2}(1/\omega_s\omega_{s'}\omega_{s''})^{1/2}\sum_{ijk}x_s^ix_{s'}^jx_{s''}^kF_{ijk},\\ k_{ssss} &= 1.57766\cdot 10^{-3}(1/\omega_s^2)\sum_{ijkl}x_s^ix_s^jx_s^kx_s^lF_{ijkl},\\ k_{sss's'} &= 9.46596\cdot 10^{-3}(1/\omega_s\omega_{s'})\sum_{ijkl}x_s^ix_s^jx_{s'}^kx_{s'}^lF_{ijkl}. \end{aligned} \tag{4} \]

Here \(x_s^i\) are forms normalized according to

\[ \sum_{ij}T_{ij}x_s^jx_s^j=1 \quad \text{or} \quad \sum_{ij}F_{ij}x_s^ix_s^j=\lambda_s,\qquad \lambda_s=(2\pi c)^2\omega_s^2, \tag{5} \]

where \(T_{ij}\) are the coefficients of inertia in natural coordinates (or symmetry coordinates). In this case, the quantities used above are expressed in the following units:

\[ \begin{gathered} (2\pi c)^2\ —\ 10^{22}\ \mathrm{cm}^2/\mathrm{sec}^2,\qquad F_{ij}\ —\ 10^5\ \mathrm{dyne}/\mathrm{cm},\qquad k_{ss's''}\ —\ 10^3\ \mathrm{cm}^{-1},\\ \lambda_s\ —\ 10^{28}\ \mathrm{sec}^{-2},\qquad F_{ijk}\ —\ 10^{13}\ \mathrm{dyne}/\mathrm{cm}^2,\qquad k_{sss's''}\ —\ 10^3\ \mathrm{cm}^{-1},\\ x_s^i\ —\ 10^{11}\sqrt{10}\ \mathrm{g}^{-1/2},\qquad F_{ijkl}\ —\ 10^{21}\ \mathrm{dyne}/\mathrm{cm}^3,\qquad \omega_s\ —\ 10^3\ \mathrm{cm}^{-1}. \end{gathered} \tag{6} \]

The calculation procedure we propose is based on the following fundamental propositions:

1) the calculation is carried out by the method of successive approximations, which makes it possible to eliminate shortcoming a), since the necessary transformation from natural coordinates or symmetry coordinates to normal coordinates (determined by the normalized forms \(x_s^i\)) is obtained in the course of the successive approximations;

2) the main role in the process of successive approximations is assigned to the force constants in natural coordinates (harmonic and anharmonic) indicated in (3); this makes it possible partially to eliminate shortcoming b), since, in our opinion, there is more basis for introducing relations between anharmonic force constants in natural coordinates than in normal ones. In addition, this makes it possible simultaneously to use the transition frequencies of isotopic modifications of the molecule, since it is assumed that their force fields are identical.

The scheme of the proposed procedure is now as follows.

1) The quadratic force constants of the zeroth approximation are determined. This may be done, for example, by using the fundamental frequencies of isotopic modifications of molecules according to the commonly applied procedure. Other methods are also possible.

2) From the quadratic force constants of the zeroth approximation, the zero frequencies and the corresponding forms of the zeroth approximation are determined. The latter are normalized by one of relations (5); here \(\omega_s\) is the zero frequency of the approximation.

3) Relations are introduced between \(F_{ijk}\) and \(F_{ijkl}\), in such a way that the total number of unknowns in the final equations (which will be discussed below) does not exceed the number of equations.

4) All the quantities and expressions obtained above are substituted into (4), which, in turn, are substituted into (2). In both (4) and (2), \(\omega_s\) are the zero frequencies of the zero approximation indicated in the second point of the scheme.

Table 1

Quadratic force constants
\((10^5\) dyn/cm)

Table 2

Zero frequencies \((\mathrm{cm}^{-1})\)

5) According to \(\nu = (E_\nu - E_{\nu'})/hc\), from (1) one finds expressions for those transition frequencies for which experimental values are known. Substitution into them of the expressions for \(X_{ss'}\) obtained above leads to equations that we call final equations.

Table 3

Some overtones and combination frequencies

Further solution is possible in two variants:

a) In the final equations the unknowns are \(\omega_s\), obtained from (1), and \(F_{ijk}\) and \(F_{ijkl}\), left as unknowns. If, in doing this, the transition frequencies of different isotopic modifications of the molecule are used, then product rules may be added to the final equations. Solving all these equations gives zero frequencies of the first approximation, from which the quadratic force constants of the first approximation are found, and the calculation is repeated until complete agreement is obtained.

6) In the final equations, the following expressions are substituted for the corrections to the zero frequencies:

\[ \Delta \omega_s = [\,^{1}/_{2}\,(2\pi c)^2\omega_s]\sum_{ij}\Delta F_{ij}x_s^i x_s^j \tag{7} \]

(here \(\omega_s\) in the denominator is the zero frequency of the zero approximation, and \(x_s^i\) are the normalized forms of the zero approximation). Now, in the final equations, the unknowns are the corrections to the quadratic force constants \(\Delta F_{ij}\) and the previously selected \(F_{ijk}\) and \(F_{ijkl}\). Solving these equations leads to the first approximation of the quadratic force constants, which makes it possible to repeat the calculation.

It is quite obvious that the method proposed by us makes it possible to obtain a result that is the more accurate, the greater the number of experimental transition frequencies that can be used in the calculation and the closer to the true values the chosen relations between \(F_{ijk}\) and between \(F_{ijkl}\) are.

As an illustration, we present the results of a calculation for a pair of molecules H\(_2\)O and D\(_2\)O, carried out according to variant a). As the zero approximation of the zero frequencies, the fundamental frequencies for these molecules \((\mathrm{cm}^{-1})\) were taken from \((^3)\): \(\nu_1 = 3656.65\); \(\nu_2 = 1594.59\); \(\nu_3 = 3755.79\); \(\nu'_1 = 2671.46\); \(\nu'_2 = 1178.33\); \(\nu'_3 = 2788.05\). The overtones and combination frequencies were not used as initial data for the calculation.

The following relations were adopted between the anharmonic force constants in natural coordinates:

\[ F_{111}=F_{222}, \qquad F_{1111}=F_{2222}\text{—by virtue of symmetry;} \]

\[ F_{123}=F_{333}=0,\qquad \text{the remaining }F_{ijk}=0.05F_{111},\ \text{the remaining }F_{ijkl}=0 \tag{8} \]

(indices 1 and 2 refer to changes in bond lengths, and index 3 to a change in the valence angle).

After 4 approximations the results given in Tables 1–3 were obtained; for comparison, data taken from \((^3,^4)\) are also indicated.

It is seen that the calculation results may be regarded as satisfactory, despite the small amount of initial data and the rather simple assumptions (8).

In conclusion, we note that one can show the connection between the method proposed in the present communication and the method of effective parameters described in \((^5)\).

Dnepropetrovsk Institute of Chemical Technology

Received
20 XI 1965

CITED LITERATURE

1. H. H. Nielsen, Phys. Rev., 60, 794 (1941).
2. H. H. Nielsen, Rev. Modern Phys., 23, No. 2, 90 (1951).
3. W. S. Benedict, N. Gailar, E. K. Plyler, J. Chem. Phys., 24, No. 6, 1139 (1956).
4. G. A. Khachkuryuzov, Collection of Works of the Institute of Applied Chemistry, issue 42, 51 (1959).
5. A. Ya. Tsaune, V. P. Morozov, Optics and Spectroscopy, 19, issue 2, 186 (1965).