Anharmonic molecular force fields determination

A major difficulty in the experimental determination of anharmonic force fields is the burgeoning number of force constants at higher-orders (Table 4, vide infra) compared with the extent of accessible data. As a consequence, reliable experimental anharmonic force fields are available for only a small number of simple molecules. Within the last decade, determination of harmonic and anharmonic molecular force fields by methods of molecular electronic structure theory has become one of the most successful applications of computational quantum chemistry. In Table 1 those species are indicated (without references given to the original publications) for which full force fields have been determined, at least at the HE level of theory and at least up to quartic force constants. 

The procedure and pertinent issues one has to consider when determining anharmonic force fields by methods of electronic structure theory may be described as follows. Once the definition of molecular force constants involving selection of an appropriate coordinate system is clear, one may need to identify all unique force (potential) constants to be determined. Then selection of the reference geometry follows, which affects precision of the force field determined and how the theoretical force field can be transformed from one coordinate system (representation) to another. Given that an appropriate basis set and level of electronic structure theory are chosen for the actual computations, the necessary quantum chemical calculations can be performed after one has carefully considered how to obtain the high-order force constants from low-order analytic information without much loss of precision. Last but not least one needs to understand the potential uses and misuses of anharmonic molecular force fields. 

A secondary motive is our general desire to verify and extend our understanding of vibration-rotation interactions in molecular spectra, and particularly to interpret data on different isotopic species in a consistent manner. Consider, for example, a constants (which measure the dependence of the rotational constant B on the vibrational quantum numbers vr) determined experimentally for several isotopic species of the same molecule. It is clear that these constants are not all independent, since they are related to the potential function which is common to all isotopic species. However, the consistency of the data and of our theoretical formulae can only be tested through a complete anharmonic force field calculation (there are at this time no known relationships between the a values analogous to the Teller-Redlich product rule). Similar comments apply to many other vibration-rotation interaction constants. 

Finally, it should be noted that with improved quality ab initio calculations it has been possible to compute the vibration-rotation parameters af in Eqs. (15) for small molecules. This requires evaluation of the molecular (vibrational) anharmonic force field as well as the harmonic portion. (See Eq. (39)). Then, with ab initio os in hand, the e qjerimental Bqs can be transformed to Bg values and thus used for the stoicture calculation. A recent example is that of Botschwina et al. [46], who determined the r structure of FC H in this manner. Such combined theory/ejqjeriment equilibrium structure determinations certainly provide an important new procedure for obtaining hi quality results. 

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). 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

Normal vibrations form a solid basis for understanding molecular vibrations. It should be remembered, however, that they are conceptual entities in that they are derived from the harmonic approximation which assumes a harmonic force field for molecular vibrations. Deviations from this approximation (i.e., deviations from Hooke s law) exist in real molecules, and the energy levels of a molecular vibration are determined by not only the harmonic term but also higher-order terms (anharmonicities) in the force field function. Although the effects of anharmonicities on molecular vibrational frequencies are relatively small in most molecules, normal (vibrational) frequencies derived in the harmonic approximation do not completely agree with observed frequencies of fundamental tones (fundamental frequencies). However, a fundamental frequency is frequently treated as a normal frequency on the assumption that the difference between them must be negligibly small. 

Since in computations of electronic structure theory derivatives of the total energy of molecular systems with respect to geometrical coordinates are best obtained in Cartesian coordinates, transformation of these derivatives to coordinate systems of more spectroscopic use, e.g., internal or normal coordinates, needs to be discussed. Furthermore, it is noted that, due to the lack of analytic higher-derivative methods at correlated levels of computational quantum chemistry, in practice higher-order force constants are usually determined first in a convenient set of internal coordinates. Then, in order to employ varia-tional or perturbational approaches utilizing anharmonic force fields they may n6ed to be expressed in normal coordinates, never known a priori to the calculation. It is thus clear that these usually nonlinear and somewhat complicated transformation equations occupy a central role in anharmonic force field studies. 

Incorporation of electron correlation in the determination of anharmonic force fields is necessary if ab initio predictions are to be made reliable under general circumstances. Since determination of even complete quartic force fields for larger molecules is not feasible without analytic derivative methods, this places a severe limitation on the size of molecular systems which can be studied. While analytic gradient techniques (see Gradient Theory) have been implemented for most state-of-the-art correlation methods, efficient analytic second-derivative... 

Accurate ab initio theoretical calculations of harmonic and anharmonic force fields of molecules as large as benzene have recently been performed. By comparing the experimentally determined molecular constants with the calculated ones, the reliability of theoretical calculations can be assessed. All the molecular constants of COF2 in Table 1 can be evaluated from the theoretical anharmonic force field. For molecules such as ammonia the computed fundamental wavenumbers are within 3 cm or better of the experiment. For anharmonic constants and centrifugal distortion constants, which are much smaller than the fundamental frequencies, the deviations are on average around 15%. The experimental determination of anharmonic constants x j that appear in the equation for the vibrational energy of a polyatomic molecule with the harmonic frequencies cop and without degenerate vibrations ... 

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