PHYSICS

D. S. BYSTROV

ON THE TRANSFERABILITY OF POTENTIAL-ENERGY COEFFICIENTS OF POLYATOMIC MOLECULES

(Presented by Academician A. N. Terenin, 2 VIII 1963)

In the existing theory of vibrational spectra \((^{1,2})\), the potential energy of molecules can be introduced in two different ways: by means of the matrix of force constants \(U\) and by means of the matrix of compliance coefficients \(C\). When linearly independent coordinates are used, these matrices are related by the relation \(C = U^{-1}\). Of great importance is the question of the transferability of the potential-energy coefficients, i.e., of the elements of the matrices \(U\) and \(C\); connected with it is also the question of the possibility of comparing, in magnitude, the values of the corresponding coefficients of different molecules.

The transferability of potential-energy coefficients under consideration means that in the matrices specifying the potential energy for a series of molecules containing an identical structural group, an identical block is singled out

\[ \left\| \begin{array}{c:c} A_0 & R_i \\ \hdashline \widetilde{R}_i & D_i \end{array} \right\|,\quad i \text{—number of the molecule} \]

(provided that the set of generalized coordinates \(q_0\) belonging to the identical group is specified in the same way in all molecules). This transferability, in the general case, can be realized only for one kind of potential-energy coefficients, since the corresponding block in a series of inverse matrices is obtained with the aid of submatrices \(R_i\) and \(D_i\), different for different molecules of the series,

\[ Z_{0i} = (A_0 - R_i D_i^{-1}\widetilde{R}_i)^{-1}. \]

To decide which kind of coefficients has the property of transferability, one should turn to the physical meaning of the quantities under consideration. It is known that

\[ F_i = -\sum_{j=1}^{n} u_{ij}q_j, \tag{1} \]

\[ q_i = -\sum_{j=1}^{n} c_{ij}F_j, \tag{2} \]

where \(u_{ij}\) and \(c_{ij}\) are, respectively, force constants and compliance coefficients (system (2) is obtained as the solution of system (1) with respect to \(q_j\)). Setting \(q_i = \delta_{jk}\) in (1), we obtain that the force constant \(u_{ik}\) has the meaning of the generalized force \(-F_i\), conjugate to the coordinate \(q_i\), and arising when the coordinate \(q_k\) is subjected to unit deformation while the remaining coordinates retain the values corresponding to the equilibrium configuration of the undistorted molecule. Similarly, from (2) we obtain that the compliance coefficient \(c_{ik}\) is the change in the coordinate \(q_i\) caused by the action of a unit generalized force \(-F_k\), which produces deformation of the molecule*.

* The forces \(F_j\) appearing in (1) and (2) are the internal forces of the system; therefore the forces \(-F_j\) are external forces—the reaction forces of the bonds holding the system in the given deformed state.

The generalized force \(F_k\) is a definite set of forces acting on the atoms that determine the coordinate \(q_k\). Indeed, by definition, \(F_k=-\partial U/\partial q_k\), but \(\|q\|=\mathbf{B}\|r\|\) and \(f_l=-\partial U/\partial r_l\), whence we obtain \(\|f\|=\mathbf{B}\|F\|\), where \(f_l\) \((l=1,2,\ldots,N)\) is the force acting on atom \(l\); \(\mathbf{B}\) is the matrix of transformation from the space of Cartesian coordinates to the space of natural vibrational coordinates (it is determined only by the geometry of the equilibrium configuration of the molecule). Thus, the generalized force \(F_k=1\) represents the system of forces \(f_{kl}=(\mathbf{B})_{kl}\) \((l=1,2,\ldots,N)\); \(f_{kl}\) is the component of \(F_k\) acting on atom \(l\), i.e., \(F_k\) is specified by row \(k\) of the matrix \(\mathbf{B}\).

The submatrix \(A_0\), whose transferability conditions are being investigated, is determined by the set of coordinates \(q_0\) belonging to the identical group; naturally, only the internal coordinates of the group should be included in \(q_0\), i.e., those that are determined only by the configuration of the atoms of the identical group (numbered by the numbers \(N_0\)). Then the matrix \(\mathbf{B}\) of molecule \(i\) has the structure

\[ \mathbf{B}= \begin{matrix} q_0\left\{\begin{matrix} \mathbf{B}_0 & \vdots & 0\\ \hdashline \mathbf{B}_{1i} & \vdots & \mathbf{B}_{2i} \end{matrix}\right.\\[-2mm] q\left\{ \end{matrix} \begin{matrix} \\[-7mm] \underbrace{\phantom{\mathbf{B}_0}}_{N_0}\quad \underbrace{\phantom{\mathbf{B}_{2i}}}_{N} \end{matrix} \ . \]

The assumption of transferability of the force constants \(U_0\) then means that if one deforms in the same way one of the coordinates \(q_0\), keeping the remaining coordinates unchanged (with the aid of the corresponding constraints), the generalized forces \(F_0^i\) conjugate to all the coordinates \(q_0\) will be the same in all molecules of the series. However, forces \(F_k^i\) will also arise in the molecules \((i\) is the number of the molecule), conjugate to coordinates not included in \(q_0\); generally speaking, these will be different in different molecules of the series, and some of them will have components acting on the atoms of the identical groups (since \(\mathbf{B}_{1i}\ne0\)). Thus, transferability of force coefficients requires, along with the structural identity of the group, the presence of some constant component in the system of forces arising in the identical group under a standard deformation. Such a situation is quite plausible, and below the physical conditions for its realization will be discussed.

Let us now consider the case of transferability of the influence coefficients. Applying to the molecules the same generalized force \(F_k=-1\), conjugate to one of the coordinates \(q_0\), we obtain a series of deformed molecules whose nuclear configurations will be described by the vectors

\[ [q'_0:q'_i]=[c_{k1}\ldots c_{kr},\quad c_{kr+1}\ldots c^i_{kn}], \]

where the “subvector” \([q'_0]\) is the same in all molecules of the series.

The work of the external forces in a standard deformation of the identical group is

\[ p=\Delta U=\tfrac12\{q'_0:q'_i\}U^i[q'_0:q'_i]=\tfrac12 c_{kk}=\mathrm{const}. \]

But with identical submatrices \(C_0\) the submatrices \(U_{0i}\) are different and, consequently, the deformation energies of the identical group \(\Delta U_0\) are different; in this case the constancy of \(\Delta U\) can be due only to compensation effects from the changing part of the molecule. Such conditions for the transferability of potential-energy coefficients appear unacceptable.

The sometimes practiced or recommended \((^3)\) procedure of transferring influence coefficients leads to an uncontrolled change of the force constants in the group assumed to be identical; in particular, force constants that should be zero may turn out to be nonzero,

which are assumed to be zero on physical grounds—the considerable remoteness of the interacting coordinates.

The above consideration has been carried out under the assumption that the natural vibrational coordinates \(q\) are linearly independent. In reality, it is often necessary to consider potential functions in the complete system of vibrational coordinates, which includes all possible natural vibrational coordinates of the molecule and is redundant. The conclusion regarding the non-transferability of influence coefficients is also valid in this case, since the influence coefficients are invariant with respect to the operation of eliminating superfluous coordinates \((^2)\), and from their assumed transferability in the redundant system there automatically follows transferability in the system of independent coordinates. The influence coefficients retain the meaning of the deformation of individual structural elements as a result of the action on the molecule of a definite system of forces also when passing to the complete system of natural vibrational coordinates; however, in the presence of redundancy in the system, these quantities cannot characterize the properties of individual structural elements of the molecule.*

What has been said may be illustrated by the example of the cyclopropane molecule \((^6)\), where, on the basis of comparing the ratio of the quantities \(K^{-1}(\gamma \mathrm{CCC}) = 0.140\) and \(K^{-1}(Q \mathrm{CC}) = 0.148\) (in units of \(10^{-6}\ \mathrm{cm}^2\)), which is close to unity, with the corresponding ratio for acyclic hydrocarbons, equal to approximately 6, a conclusion was drawn about substantial strengthening of the CCC angle as a characteristic of the force field of naphthenes. In reality, the closeness of the values of the influence coefficients in the case under consideration is evidently connected with the fact that, owing to the cyclic structure of the molecule, the CC bond is stretched to approximately the same extent under the influence of the systems of forces

\[ \begin{array}{c} \includegraphics[height=1.1cm]{[[unclear: schematic force diagram]]} \end{array} \]

For the characterization of individual structural elements of a molecule one should use the quantities of the force constants in the complete system of natural vibrational coordinates; their meaning follows from the very procedure of constructing the potential function—these are the force constants of (of course, hypothetical) fragments of the molecule which have retained the internal force field by which they were characterized when entering into the composition of the molecule. The known difficulties, consisting in the mathematical ambiguity of restoring the matrix \(U\) in the complete system of natural vibrational coordinates (redundant) from the corresponding matrix in the system of independent coordinates \((^2)\), can be overcome in various ways; the most obvious is to solve the inverse vibrational problem directly in the system containing superfluous coordinates, or to carry out the processes of varying the force constants in such a way that each variation begins with specifying the potential function in the complete system of natural vibrational coordinates.

Let us turn to the physical conditions for the transferability of force constants. According to the Hellmann—Feynman theorem, the forces acting on a nucleus in a molecule at a given (any) configuration of nuclei can be obtained (in the case where the electronic state is nondegenerate) as the forces of electrostatic interaction with the other nuclei and with the spatial negative charge whose density is determined by the square of the modulus of the electronic wave function of the molecule at the given configuration of nuclei. It is natural to expect that structural identity of a group accomp—

* In this connection one cannot agree with the repeatedly made assertions that influence coefficients represent a better characteristic of the force field of a molecule than force constants \((^{4,5})\).

the identity of the electron-charge distribution in it for the equilibrium configuration of the nuclei. If one assumes that, for identical changes in the nuclear configuration, the shape of the electron cloud within the group changes in the same way, irrespective of the molecule of which the group is a part, this will ensure a constant contribution to the force system in the identical group (the variable contribution may be due to a change in the field created by the electron charge of the remaining part of the molecule, which may be redistributed in different ways upon deformation of the identical group in different molecules). Such conditions can ensure the transferability of \(U_0\). If the change in the field of the electron charge of the remaining part of the molecule at the location of the identical group is the same in all molecules of the series under consideration (which can be ensured by a considerable separation of the position at which substituents are introduced from the identical group), the systems of forces acting in the identical group are in general the same, which means the transferability of the submatrix \(U_0\) not only in the system of natural vibrational coordinates, but also in the system of Cartesian displacement coordinates.

In conclusion, it should be pointed out that the conclusions of this work do not mean that the use of influence coefficients in the theory of vibrational spectra is altogether meaningless. In fact, influence coefficients should find their own field of application, since it is precisely they that are directly related to the mean-square amplitudes of vibrations and to the constants of vibrational-rotational interaction, in which valuable additional information on the force field of molecules is contained \((^7)\).

Scientific Research Institute of Physics
of Leningrad State University
named after A. A. Zhdanov

Received
18 VII 1963

CITED LITERATURE

1. M. V. Vol'kenshtein, M. A. El'yashevich, B. I. Stepanov, Vibrations of Molecules, M.—L., 1949.
2. L. S. Mayants, Theory and Calculation of Molecular Vibrations, M., 1960.
3. S. J. Cyvin, N. S. Slater, Nature, 188, 485 (1960).
4. L. S. Mayants, Proceedings of the P. N. Lebedev Physics Institute of the USSR Academy of Sciences, 5, 63 (1950).
5. P. G. Maslov, S. A. Antipina, ZhFKh, 25, 594 (1951).
6. L. M. Sverdlov, E. P. Krainov, Optics and Spectroscopy, 3, 54 (1957).
7. J. S. Decius, J. Chem. Phys., 38, 241 (1963).