Fourier potential functions

The SHAPES force field" has been implemented in CHARMM and used to examine the structures of several square planar rhodium complexes. This force field is based on angular overlap considerations and treats angular distortions for a variety of geometries. Spherical internal coordinates and Fourier potential functions form the basis for the description of these molecular shapes. The parameters for this force field were derived from normal coordinate analysis, ab initio calculations, and structure-based optimizations. The average rms deviation for bond lengths was 0.026 A, and the average rms deviation for bond angles was 3.2°. 

In molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. 

The integral over dq now reduces to finding the Fourier transform of the potential function. 

The potential function that governs internal rotation in ethane is represented in Fig. 6. The three equivalent minima correspond to equilibrium positions, that is, three identical molecular structures. The form of this potential function for an internal rotator with three-fold symmetry can be expressed as a Fourier series,... 

The first two terms on the right-hand side of Eq. (83) are usually assumed to be harmonic, as given for example by Eq. (6-74). The third term is often developed in a Fourier series, as given by Eq. (82). The potential function appropriate to the interaction between nonbonded atoms is taken to be of the Lennard-Jones type (Section 6.7.3). In all of these cases the necessary force constants are estimated by comparing the results obtained from a large number of similar molecules. If electrostatic interactions are to be considered, effective atomic charges must be suggested and Coulomb s law applied directly [see Eq. (6-81)]. 

It is common practice to describe torsional rotations around single bonds and those around multiple bonds with the same type of potential function but with very different force constants. The function must be able to describe multiple minima. Generally, a Fourier expansion of the torsional angle with only cosine terms is used (Eq. 2.23),... 

The choice of series is not only dependent on the type of molecular motion. For example the power series may be convenient for an accurate description of the potential function close to the minima, while a Fourier series is convenient for describing potential barriers to torsional motion. 

N-methyl pyrrole77 represents an exception. Here, a potential function using V6 = -549.3 J/mol and V12 = -200 J/mol fits the microwave data just as well as a set employing = -558.6 J/mol and V12 = 167 J/mol. The lesson to be leamt from this is that even in cases where the Fourier expansion starts with a high a term as V6, rapid convergence is not always ensured. 

A similar analysis of data obtained from molecules with asymmetric end groups is more complicated. Apart from the problems connected with the separability of the torsional motion from the framework vibration, experience shows that several more terms have to be included in the Fourier series to describe the torsional potentials properly. On the other hand, the electron-diffraction data from asymmetric molecules usually contain more information about the potential function than data from the higher symmetric cases. In conformity with the results obtained for symmetric ethanes the asymmetric substituted ethanes, as a rule, exist as mixtures of two or more conformers in the gas phase. Some physical data for asymmetric molecules are given in Table 4. The electron-diffraction conformational analysis gives rather accurate information about the positions of the minima in the potential curve. Moreover, the relative abundance of the coexisting conformers may also be derived. If the ratio between the concentrations of two conformers is equal to K, one may write... 

The MD calculated DOS is, to first-order, independent of temperature and only small anharmonic effects appear at non-zero temperatures. These anharmonic effects arise from the fact that most potential functions are not parabolic. As the displacements increase in magnitude the molecules explore non-parabolic regions of the potential and the overtone frequencies with perfect integers of the base frequency C0o> such as 2coo, 3c0o..., due to the Fourier expansion of the non-parabolic potential function. 

For each fixed 0 one can obtain individual Fourier expansions F(0") with respect to rotation described by the dihedral angle 0" this corresponds to a particular section through the energy surface Ej =/( , 0"). If for " = 180° a molecule has a plane of symmetry, it suffices to consider only the first three terms of Eq. [24]. This is the case for fluoromethanol (121) (Figure 20) and dimethoxymethane (11 with 0 fixed at 180°). In addition, when one rotating alkyl group is of C3 symmetry (as is, approximately, CHj in methanol), internal rotation may be adequately described by a simple threefold potential function expressed by Eq. [26]. 

Radom et al. (234) seem to have been the first to interpret internal rotation in molecules exhibiting the anomeric effect by means of a Fourier expansion of the potential function. They performed ab initio SCF MO calculations for a wide variety of molecules but without geometry optimization. It was found that internal rotation in ethane, propane, fluoroethane, and methylamine is... 

Soon this concept was connected (234) with the magnitude of the Kj constant in a Fourier component analysis of internal rotation potential functions in molecules exhibiting the anomeric effect (see Section III.B.2). It was also used to account for bond lengths in methanediol (8) (235) and methoxymethanol (49) (244) (p. 196), but the authors claimed (235) that the shortening of the C—O bonds in the ap, ap conformers of 8 cannot be explained by the interaction pictured in Figure 26a. 

It is important to note that the results of JP R significantly influenced the thinking of subsequent researchers. In the interests of brevity I note here only the key papers of Tvaroska and Bleha [20, 21], which further demonstrated the utility of Fourier analysis, the particularly elegant paper in 1979 by Wolfe et al. [22], which refined many of the ideas laid out in JP R s prior work and extended its range of applicability, and the efforts of Thogersen et al. [23], who built upon the work of JP R in the construction of an empirical potential function for oligosaccharide conformational analysis. 

The first term represents the kinetic rotational energy. This equation is amenable to Mathieu s equation and analytical formulae for the eigenstates are known. However, with computer facilities available nowadays, it is faster to calculate numerically both eigenvalues and eigenfunctions [59]. If necessary, these calculations can be performed for potential functions including several Fourier terms. 

Figure B3.4.7. Schematic example of potential energy curves for photo-absorption for a ID problem (i.e. for diatomics). On the lower surface the nuclear wavepacket is in the ground state. Once this wavepacket has been excited to the upper surface, which has a different shape, it will propagate. The photoabsorption cross section is obtained by the Fourier transfonn of the correlation function of the initial wavefimction on tlie excited surface with the propagated wavepacket.

---