Methods for the Fourier‐series expansion of torsional energies

Abstract

AbstractSeveral general procedures for the derivation and analysis of a Fourier‐series expansion V(ϕ) over computed torsional energies E(ϕi) are formulated. STO‐3G energy values in n‐butane, 1‐chloropropane, and 1,2‐dichloroethane are used as test data for deriving V(ϕ) with the numerical methods of interpolation and least squares. The accuracy of each derived V(ϕ) is assessed on the basis of calculated conformational properties, mean and rms deviations, and an error curve, V(ϕ)–V(ϕ)ref, where V(ϕ)ref represents a reference set of E(ϕi). Results indicate that given the same number of expansion terms, interpolation and least squares yield functions of comparable accuracy; however, interpolation is a more efficient procedure for monitoring the accuracy of a function in regions of interest. In cases where there are too few input energies to achieve the desired accuracy, energy derivatives can be employed effectively for expanding the input set. In designing special‐purpose functions, the error curve can be used meaningfully as a guide; an example for producing functions that are especially well behaved in regions for gauche conformations is provided. The present study continues to add systematics and rigor to the fitting of an internal rotation potential function from energy data.