Cubic and quartic force constants

Typically, the simplest parameters to detennine from experiment are ab and Dab- With these two parameters available, (Zab can be determined from Eq. (2.8), and thus the cubic and quartic force constants can also be determined from Eqs. (2.4) and (2.7). Direct measurement of cubic and quartic force constants requires more spectral data than are available for many kinds of bonds, so this derivation facilitates parameterization. We will discuss parameterization in more detail later in the chapter, but turn now to consideration of other components of tlie total molecular energy. 

For small displacements, of the order of vibrational amplitudes at room temperature, the terms in the power series expansion (1) converge fairly rapidly, and higher-order terms are related to successively smaller-order effects in the spectrum, so that they become more and more difficult to determine. Almost all calculations to this date have been restricted to determining quadratic, cubic, and quartic force constants only [the first three terms in equation (1)], and in this Report we shall not consider higher-than-quartic terms in the force field. The paper by Cihla and Chedin11 is one of the few exceptions in which force constants involving up to the sixth power have been determined for a polyatomic molecule, namely COa. 

It is common practice to present the cubic and quartic force constants f3 and /4 in a slightly different way, in terms of so-called Morse parameters.0 6 The Morse function is an empirical diatomic potential function of the form... 

The discussion so far may be summarized as follows. There are two reasons for using curvilinear co-ordinates to represent the anharmonic force field of a polyatomic molecule, despite their apparent complexity. The first is that it is only in this way that we obtain cubic and quartic force constants which are independent of isotopic substitution. The second is that in terms of curvilinear bond-stretching and angle-bending co-ordinates we obtain the simplest expression for the force field, in the sense that cubic and quartic interaction terms are minimized. The first reason is compulsive the second reason is not compulsive, but it does make the curvilinear co-ordinates very desirable. 

If the molecule has no symmetry, it is easy to see that the total number of independent quadratic, cubic, and quartic force constants is given in terms of... 

For HCN the situation is somewhat better, because the data on DCN are much more effectively independent of the HCN data. This molecule has also been the subject of much high-resolution spectroscopic study, so that the vibration-rotation energy levels are particularly well known and its vibrational spectrum is free of accidental resonances. Table 8 compares the results of three quite different calculations. The calculation by Strey and Mills is the most recent, and was based on the latest spectroscopic data the refinement was made to a and x values rather than to the vibrational levels and rotational constants as used by both the earlier workers. Strey and Mills also constrained 3 of the quartic interaction constants to zero, and refined to cubic and quartic force constants in a separate calculation to the quadratic refinement. The level of agreement between the calculations leads to conclusions rather similar to those made above for C02 in particular, standard errors should be multiplied by at... 

Figure 4.1 Composite of the GT Calculator displays for the calculations of the quadratic, cubic and quartic force constants for the examples of H2O and 50 discussed in the text.

Anharmonicity of molecular vibrations presents one of the most vexing problems in studies of molecular structure. Anharmonic corrections (of first order, involving the cubic force constants) are required in accurate determinations of the equilibrium structures of molecules from rotational spectra, as well as from electron diffraction measurements. To obtain accurate harmonic force fields, it is necessary first to correct the vibrational data for anharmonicity (using second-order corrections, involving cubic and quartic force constants). Information on anharmonic force fields obtained from experimental data is also important as a basis for comparison in quantum chemical investigations of molecular forces as well as in studies of high-temperature thermodynamic properties and of rate and dissociation processes. Yet detailed studies of anharmonic force fields have hitherto been limited to small molecules with N = 2-4 atoms (in isolated cases to N = 6). 

Empedocles [701) proposed a general semi-empirical approach using data from the united-atom and separated-atom models and calculate the quadratic (harmonic), cubic and quartic force constants kz, kz and ki from the differentiation of the virial theorem electronic kinetic energy expression Eq. (4.27), where E is the total energy... 

The fourth and final step of the transformation of the force constants from difference cartesians to dimensionless normal coordinates, i.e., the transformation from mass-scaled cartesian coordinates to dimensionless normal coordinates, is thus linear and is given for the cubic and quartic force constants appearing in eqs. (30) and (31) by... 

Another problem associated with parameterization comes from the requirement that the angle bending term be symmetric. Thus, an angle of, say, 170 must have the same energy as at 190°. In other words, the angle bending term has to have a slope of zero at 0° and 180°. To maintain this symmetry, the cubic and quartic force constant can be calculated from the quadratic force constant and the reference angle. 

A practical conclusion about force constants is that correlation effeas are relatively unimportant for the anhatmonic constants. Pulay et al. have explained that force constants need to be considered in terms of a valence electronic part and a core-plus-nuclear part. The core contribution is changing more rapidly around the equilibrium than is the valence part, since the core—core potentials are essentially a 1/R repulsion, being that they involve small, positively charged entities. Core—valence attraction will also be changing more rapidly than the valence part, and this may partly offset the core-core repulsion. The relative effect of electron correlation on the core—core potentials should be small, and if they are the leading source of anharmonidty around the equihbrium, then it fits that electron correladon is not important on the anharmonic constants. The practical result is that accurate vibrational analysis is possible by carrying out correlated calculations to obtain the harmonic force constants, but by using only SCF calculations to obtain cubic and quartic force constants. 

J. F. Gaw and N. C. Handy, Chem. Phys. Lett., 121,321 (1985). Ab initio Quadratic, Cubic and Quartic Force Constants for the Calculation of Spectroscopic Constants. 

Cubic and quartic force constants of linear radicals were estimated on the basis of a normal-coordinate analysis for NH2 [8] and ND2 [8, 9] in the (quasi-linear) A Ai state. 

Generally, only a small number of scaling constants are needed. For example, ten scaling constants and reference values were derived for hydrocarbons from comparisons of gas phase structures, conformational energies, rotational barriers, and vibrational frequencies measured by experiment and calculated by the QMFF. For the bond and bond angle energy functions in equation (1) the same scale factor is multiplied by the QMFF quadratic, cubic, and quartic force constants. Similarly, the same scale factor is used for the one-, two-, and threefold torsion force constants, and a single scale factor is used for all cross terms. 

Normal coordinates are perfectly adequate for calculating low-lying vibrational energy levels. They have been used extensively by spectroscopists to derive theoretical expressions (in terms of harmonic frequencies and cubic and quartic force constants) for the parameters of effective Hamiltonians, often used to fit experimental spectra. The simplest version of the normal coordinate kinetic energy operator was derived by Watson from the earlier work of Wilson and Howard, and Darling and Dennison. The vibrational part of the Watson kinetic energy operator is. 

F,y, ijk, and F,y have the formal meaning of harmonic, cubic, and quartic force constants, respectively. Nondiagonal terms (/ j k 1) represent couplings. Couplings other than harmonic are rarely included. One example is the angle-torsion-angle term,... 

Here, the Fijk and Fija are the cubic and quartic force constants, respectively. Second-order perturbation theory now gives the following expression for the vibrational energy... 

Information concerning unimolecular potential energy surfaces can be acquired from several sources. Thermochemical measurements provide bond dissociation energies Dq and heats of reaction AHq. Analyses of the vibrational spectra of a unimolecular reaction s reactants and products yields their quadratic force constants, and if the data is sufficiently complete, also, their cubic and quartic force constants. From kinetic measurements of the unimolecular rate constant at high pressure the phenomenological Arrhenius A factor Aee and activation energy can be derived. If one can show that that the activated complex theory is valid for a specific unimolecular reaction its threshold energy Eq and the entropy difference between the activated complex and molecule can be found from... 

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