Spectroscopically determined force fields for macromolecules. Part 3. Alkene chains^☆

Abstract

A spectroscopically determined force field (SDFF) has been obtained for hydrocarbon chains with olefinic unsaturation. The parametrization of this potential energy function was based on the SDFF transformation of ab initio structures and experimentally scaled force constants, which were optimized to assigned vibrational frequencies of ethene, propene, skew- and syn-1-butene, trans- and cis-2-butene, and isobutene. The Bell torsion coordinate was used to describe torsion about the CC bond. In addition to good agreement with ab initio structures and energies, this SDFF gives an average rms error for non-CH stretch frequencies of the above molecules of 9.3 cm⁻¹, with substantially correct reproduction of ab initio eigenvectors.

Introduction

In previous papers in this series [1], [2], we demonstrated, for saturated hydrocarbon chains, the implementation of our methodology [3], [4] for producing molecular mechanics (MM) energy functions that, in addition to structures and energies, reproduce vibrational frequencies of macromolecules within criteria of spectroscopic accuracy (rms error of 5–10 cm⁻¹). We call such a function a spectroscopically determined force field (SDFF), and we obtain its parameters systematically and self-consistently from analytical transformations of ab initio structures and force fields of conformers of small molecules that are suitable models for the macromolecule. In this paper we apply this method to produce an SDFF for hydrocarbon chains with olefinic unsaturation (for which we have published some preliminary results [5]).

The goal of achieving spectroscopic accuracy in an MM function is important for at least two reasons. First, it assures the conformation dependence of the force constants and thus transferability of the force field to different structures. Secondly, not only does it guarantee the calculation of reliable normal mode frequencies and eigenvectors (and therefore, for example, infrared intensities [6]), but it also insures the high quality of other MM predictions as well as molecular dynamics (MD) simulations, which depend on the accuracy of the energy function [7]. (Thus, for example, our SDFF for linear saturated hydrocarbon chains [1] could reproduce the observed elastic modulus of crystalline polyethylene to ∼1% [8].) Most early as well as recent functions were not designed for this purpose (but rather to reproduce structures and energies), and their prediction of frequencies is spectroscopically unsatisfactory. For example, for olefinic molecules one finds rms errors of: 20 cm⁻¹ for trans- and cis-2-butene [9]; 15 cm⁻¹ for 13 alkenes [10]; 25 cm⁻¹ for 7 alkenes [11]; 63 cm⁻¹ for ethene [12]; and 51, 94, 50, 56, 61 and 44 cm⁻¹ for ethene, propene, cis-2-butene, syn-1-butene, trans-2-butene, and skew-1-butene, respectively [13]. Such inaccuracy prevents using the MM function to obtain a detailed understanding of normal mode frequencies and eigenvectors, agreement with the latter not even having received serious attention in previous developments of MM force fields.

The SDFF approach assures frequency agreement from the start, through an analytical transformation of complete (i.e. ab initio) spectroscopic force fields that are scaled to experimentally assigned bands. The subsequent reduction in the number of interaction force constants is based systematically on a pre-assigned limit on the frequency error [4]. Anharmonic force constants are obtained directly from the sampling of the varying bond lengths and angles in the different model molecules and their conformers. By starting out with a complete set of force constants, the SDFF method avoids the possibility of biasing the non-bonded parameters to compensate for the lack of inclusion of significant off-diagonal force constants. Although the parametrization of the energy function can be done in a non-redundant [1] or redundant [2] coordinate basis, in the latter case we take care that the redundant set is determinate [14]. Within this protocol, non-bonded parameters can be optimized straightforwardly [15]. An important advantage of the SDFF procedure is that significant physical contributions, such as conformation dependencies of off-diagonal terms and relevant non-bonded effects, are relatively easy to detect, with their incorporation leading to automatic adjustment of force constants. Of course, basing the parameters on a set of ab initio “data” assures an important degree of internal consistency in the resulting SDFF energy function.

As with the n-alkanes [16] and branched alkanes [17], our SDFF for alkenes is based on scaled ab initio force fields of a set of model molecules, in this case ethene, propene, skew- and syn-1-butene, trans- and cis-2-butene, and isobutene. We first discuss the development of the scaled ab initio force fields and then describe the parametrization of the SDFF.