Anharmonic force constant

So far we have only discussed harmonic frequencies. The effect of anhar-monicity can be treated using either a Taylor expansion of the PES in terms of normal mode coordinates or by explicitly spanning the PES on a numerical grid. The discussion of anharmonic force constants is postponed to the following section. Here, we will focus on an explicit PES generated by means of the following correlation expansion, here written up to three-mode correlations, [53]... 

Table 4 Harmonic and diagonal anharmonic force constants (in cm 1) of the relevant system modes for AT-(H20)2.

Rotational constants obtained for both the ground and the three first excited vibrational states allowed one to derive the equilibrium molecular structures of GeF2 (re = 1.7321 A, 6>e = 97.1480211) and GeCl2 (re = 2.169452 A, <9e = 99.8825°285). From measurements of the Stark effect the dipole moment of GeF2 has been determined to be 2.61 Debye283. The harmonic and anharmonic force constants up to the third order have been obtained for both molecules and reported too283,285. 

In practice, the result of the perturbation treatment may be expressed as a series of formulae for the spectroscopic constants, i.e. the coefficients in the transformed or effective hamiltonian, in terms of the parameters appearing in the original hamiltonian, i.e. the wavenumbers tor, the anharmonic force constants , the moments of inertia Ia, their derivatives eft , and the zeta constants These formulae are analogous to equations (23)—(27) for a diatomic molecule. They are too numerous and too complicated to quote all of them here, but the various spectroscopic constants are listed in Table 3, with their approximate relative orders of magnitude, an indication of which parameters occur in the formula for each spectroscopic constant, and a reference to an appropriate source for the perturbation theory formula for that constant. 

Anharmonic Force Constant Refinements.—The preceding parts of this Section 4 constitute an outline of how the vibration-rotation spectrum of a molecule may be calculated from a knowledge of the force field in some set of geometrically defined internal co-ordinates, denoted V(r) in general in this Report [but denoted V(X) in the special discussion on pp. 126—132], In practice we wish to solve the reverse problem we observe the vibration-rotation spectra, and we wish to deduce the force field. 

Table 4 gives a list of molecules on which anharmonic force constant calculations have been reported. Perhaps the first notable fact is that the list is not long although we have attempted to make the list complete as of 1973, there... 

Table 5 Dependence of spectroscopic constants on anharmonic force constants for CO2-type molecules11...

Kuchitsu and co-workers5 7 were the first to introduce what is perhaps the simplest and most generally useful model, in which they assume all anharmonic force constants in curvilinear co-ordinates to be zero with the exception of cubic and quartic bond-stretching constants. These may be estimated from the corresponding diatomics, or from a Morse function, or they may be adjusted to give the best fit to selected spectroscopic constants to which they make a major contribution. This is often called the valence-force model. It is clear from the results on general anharmonic force fields quoted above that this model is close to the truth, and in fact summarizes 80 % of all that we have learnt so far about anharmonic force fields. 

Resonant raman spectroscopy has proved to be another valuable tool for the study of the structure of the polydiacetylene chain. Due to the resonance enhancement the spectra are compared to greatly simplified, infrared spectra and show as principle feature only the in-plane modes of the polymer chain. The correlation of the CsC and C = C stretching modes and their temperature dependence have been interpreted as resonances between the mesomeric structures (I) and (II) i32) Hoy(rever, a model using simple anharmonic force constants for the acetylene structure (II) is in good agreement with the experiment, e.g, the temperature and pressure dependence of the vibration frequency and the mechanical properties... 

If an atom is iso topically substituted, the change in the effective moment can be used to calculate the coordinate of the atom.9 From the coordinates, one then obtains the so-called substitution bond length, r,.10 From Table 1 it can be seen that rs depends on both the harmonic and anharmonic force constants. The factor/in the table is given by ... 

If data from more than one isotopic species are used in an analysis, serious error may result unless the isotopic dependence is taken into account. Unfortunately, this requires knowledge of the anharmonic force constants and for most molecules can only be approximated. 

The term in Az is primarily determined by anharmonic force constants, whereas the terms in Ax2 and Ay2 are primarily functions of the harmonic force constants. For diatomic molecules where the vibrational force constants are usually well characterized, it is possible to calculate the vibrational averages and thus obtain rt from rg. For polyatomic... 

If the force constants are to be regarded as derivatives of the potential energy V with respect to the coordinates R, it is clearly important to include die coefficients 1/n and to regard the summations as unrestricted as in Eq. (1). Although this is commonly done in harmonic calculations, anharmonic force constants are often defined in a different form in which the numerical factors are omitted and the summations are restricted. 

Two kinds of basic data can be used for calculating the anharmonic force constants. The zero-order frequencies G)s and anharmonic constants x , in the second-order formula for the vibrational energy v,... 

In a critical evaluation of the results of anharmonic force constant calculations it is necessary to consider all the factors discussed above concerning the completeness, internal consistency, and accuracy of the data, the possible idiosyncrasies of the model potential used, the method of reducing the data, as well as the goodness of fit the force constants yield for the data points. It may be in order to mention that... 

TABLE 1 Anharmonic Force Constants of Some Linear Triatomics0... 

From the review of the existing information on anharmonic force constants one must conclude that our knowledge of the anharmonicity of force fields in polyatomic molecules is still very incomplete. Only the gross features of the molecular potentials have been established with some confidence. We know now that the potential opposing the stretch-... 

Buckingham asked whether ab initio calculations of anharmonic force constants are of any help at present. PLfv a felt that they are not sufficiently accurate,... 

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