Potentials quartic force fields

Perturbation theory has been applied to anharmonic calculations of spectroscopy from ab initio potentials in a large number of studies [19-25,115-121]. In nearly all cases so far, second-order perturbation theory was employed. The representation of the anharmonic potential generally used in these studies is a polynomial in the normal modes, most often a quartic force field. A code implementing this vibrational method was recently incorporated by V. Barone in gaussian [24]. Calculations were carried out for relatively large molecules, such as pyrrole and furan [25], uracil and thiouracil [118], and azabenzenes [119]. We note that in addition to spectroscopy, the ab initio perturbation theoretic algorithms were also applied to the calculation of thermodynamic properties... 

The procedure we followed is basically the same as that used to fit the HCN surface. Here we started with an ab initio quartic force field, calculated by Martin et al. (96). The fit from this potential proceeded in two steps. First we varied six of the quadratic force constants. Because these force constants determine Z/ 0, these constants could not be optimized using the approaches detailed above. Instead, we fit these constants to the six harmonic frequencies in H2CO. This modification leads to a mean absolute difference for the 138 observed states included in the fits of 4.9 cm. Following the procedures outlined above, we were able to further refine the potential to obtain a final mean absolute deviation of 1.5 cm 1 for these states. 

EFF (Empirical force field) [186] has been designed just for modeling hydrocarbons. It uses the quartic anharmonic potential for the bond stretching, and the cubic anharmonic for the valence angle bending. No out of plane or electrostatic terms are involved, although the cross terms, except torsion-torsion and bend-torsion ones, are included. 

The primary motive for attempting calculations of this kind is simply our desire to determine the potential function V(r) more accurately and over a wider range of co-ordinate space. Even if our immediate ambition is only to determine the equilibrium configuration and the harmonic force field, our ability to withdraw this information from spectroscopic data is limited by the need to make corrections arising from the cubic and quartic anharmonic force field. 

For large molecules, however, the computer requirements become increasingly prohibitive, especially when conformationally flexible compounds are tackled. Alternative approaches to quantum-mechanical methods are known, based on potential functions and parameters derived from detailed analysis of vibrational spectra. These so-called force field methods are now joined in what is called molecular mechanics, an empirical method that considers the molecule as a collection of spheres (possibly deformable) bound by harmonic forces (eventually corrected with cubic and quartic potentials). The energy... 

Figure 28-7 In force field calculations, different levels of approximations are used to reproduce the stretching and compression of chemical bonds The plot shows a Morse potential energy function siipenmposed with various power series approximations (quadratic, cubic, and quartic functions) Note that the bottoms of the curves, representing the bond length for most chemical bonds of interest to medicinal chemists, almost overlap exactly. This nearly perfect fit in the bonding region is the reason simple harmonic functions can be used to calculate tx>nd lengths for unstrained molecular structures in the force lield method.

Modified Hooke s law corrected with cubic (as in the MM2-based force fields [8]) and further extensions to quartic terms (as in MM3 [9], CFF [10], and MMFF [11] force fields see Eq. (3) [9]) or other expansions [12] have been developed to mimic the Morse potential and are used to speed up convergence in very distorted starting geometries, while keeping a proper description of the potential energy. 

CFF. The consistent force field (CCF) [34] developed by Biosym ° is a descendent of CVFF but differs in the specific types of potential terms. The nonbonded terms of CFF include a quartic bond-stretch term (Eq. (4.4)), a quartic... 

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]. 

The quartic bond energy function used by these three force fields closely approximates a Morse potential for bond deformations of up to 0.1 or 0.2 A. This means that as the bond length is increased, the bond energy increases less rapidly than predicted by the quadratic energy equation in equation (1). For larger bond length deviations, a Morse function will allow a bond to dissociate, while the quartic function in equation (4), as well as the quadratic function in equation (I), will not allow dissociation (since the quartic force constant is a positive number), Most current force fields for biomolecular simulations deliberately avoid the use of a Morse function, both because it is less computationally efficient to calculate and because the force fields are not calibrated to deal with dissociation effects. 

---