Long-range effects in optimizing the geometry of stereoregular polymers. I. Formalism

We develop an original methodology to calculate analytically the long-range (LR) Coulombic effects to be included in Hartree-Fock forces computed on stereoregular polymers. The technique presented, based on multiple Taylor series expansions, is completely general and can be easily extended to all orders of expansion and to geometrical derivatives of higher order. In the McMurchie Davidson or similar schemes, the LR terms are added directly to Hermite integrals and LR effects are naturally considered during the computation of the energy and its derivatives. Each type of derivative (nuclear-repulsion energy, overlap and kinetic energy integrals, electron-nuclear attraction and two-electron integrals) is examined to quantify the impact of LR corrections. It turns out that the lattice sums of the gradients evaluated on a model macromolecule converge much faster when including those corrections. In addition, the dependence of the energy with respect to the unit cell length is for the first time considered in full details.