Quartic constant

Although symmetry reduces the number of independent force constants, they still grow rapidly with the order of anharmonicity. For example, for a CH3X molecule of C.. symmetry there are 12 independent quadratic constants, 38 cubic constants, and 102 quartic constants. 

Bent Triatomic Molecules.—Calculations have been reported for many bent triatomic molecules (see Table 4). The general force field contains 2re + 4/ + 6/3 + 9A parameters, the relation to the primary spectroscopic constants being shown in Table 9. The fact that these are asymmetric top molecules, for which otj, a , and a can all be determined (generally from the microwave spectrum for the heavier molecules), means that 9 a values are available from each isotopic species to determine the 6 cubic force constants, so that the cubic force field is generally well determined. For the quartic force field the situation is much less satisfactory the experimental data on the anharmonic constants xrs are generally incomplete, and are in any case insufficient to fix all the quartic constants without good isotopic data. 

I 1 6 iS A P3 U equilibrium angle = k 6 - 6/6). A fourth-order polynomial enables an independent fit of the barrier to linearity. Such constrained polynomial fittings are rarely done. Instead the bending function is taken to be identical for all atom types, for example a fourth-order polynomial with cubic and quartic constants as a fixed fraction of the harmonic constant. These features are illustrated for H2O in Figure 2.5, where the exact form is taken ... 

Among the spectroscopic constants only depends on the quartic constants. Since the experimental data are very limited for the states with an excited stretching mode, it is impossible to obtain any closer check on these constants which are therefore left out in the following discussion. 

Symmetry and the Number of Cubic and Quartic Constants. Since the terms gumQhQiQm and h arc part of the potential energy,... 

Thus it can be said that among the cubic terms, all those which involve Qs an even number of times may occur. The followhng gki, can therefore be different from zero gm, gm, gm, gm, ffiss, g a] six cubic terms in all. An obvious extension yields the following nonvanishing quartic constants hkkkk (3) h nz, /11222 hkui (3) and hma, a total of nine quartic... 

Constants kjjk were also predicted from a van der Waals model and from a valence-force model [6,41 ]. A complete set of constants kjjkSimilar to those above and quartic constants were calculated from a model potential, containing three independent parameters for the enharmonic part, by a Monte-Carlo procedure [2]. 

Quartic constants were also derived in [1,42] from the vibrational constants XjjOf Nebgen et al. [5] and in [44] from their own Xjj. For an approximate method, see [45]. A force field was also determined by an ab initio SCF-MO calculation [53]. 

The cubic force field of pyramidal XYg-type molecules contains 14 independent parameters. Usually, however, the number of spectroscopic constants dependent on the enharmonic force field, such as rotation-vibration constants, l-type doubling constants, or anharmonicity constants, is smaller than the number of parameters to be determined. Their number thus has to be reduced by introducing model potentials and imposing certain constraints. Some of the possible routes were presented by Morino et al. [50]. Cubic force constants for NF3 pertinent to model potentials (mostly Morse potentials) have been calculated by several groups of workers [11, 12, 19, 51, 52], Some of the principal quartic constants have been estimated as well [12, 51]. 

In this chapter we will focus instead on another approach, the vibrational second-order perturbation theory (VPT2) [93-98], which provides closed expressions for most of the spectroscopic parameters required for the analysis of the experimental frequencies. VPT2 has been shown to be very effective for the study of semirigid polyatomic molecules of medium size [38-41, 99, 100] at relatively low computational cost. This level of theory requires quadratic, cubic, and semi-diagonal quartic force constants. Starting from analytical Hessians, the needed cubic i4>rst) and quartic constants can be computed by one-dimensional... 

The force constants were obtained from analytic third derivative procedures the quartic constants by finite differences. Displacement sizes 0.005 A for stretches and 0.01 rad for bends. 

In principle, therefore, it is straightforward to generate the harmonic frequencies and anharmonic constants from electronic-structure calculations although the large number of quartic constants -for example, 3060 in a system of seven atoms - makes this approach cumbersome for laige molecules. 

The (5s4p2d/3slp) Slater orbital (STO) basis has been used in a study of the quartic force field of H20. These calculations are done using SCF, SD-CI, and with various many-body models. The theoretical results agree well with the experimental force constants, with the many-body values showing better agreement than SD-CI. In fact, the most recent experimental and normal coordinate study revises several of the previously accepted force constants to be more consistent with these theoretical calculations. All of the cubic and quartic constants are determined from the calculations, while the experimental studies typically set many of these constants to zero to facilitate obtaining the force field from the normal coordinate analysis. Hence, the computed SDQ-MBPT(4) quartic force field for H2O is used as input to the local description of H2O in the 0 + H2O surface fit. This is described in detail in section IV. 

The Watson representation of the Hamiltonian for the near-symmetric asymmetric top was also used in order to obtain centrifugal distortion constants from FT-IR spectra [5 to 7], FT-MW spectra [8], and a continuous MW spectrum [2]. The resulting quartic constants are given in Table 9 see the cited papers for the values of the sextic and higher terms. 

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