Analytical second derivatives in ab initio Hartree-Fock crystal orbital theory of polymers

In the framework of ab initio Hartree-Fock crystal orbital theory of polymers, the formulas for the analytical second derivatives of energy with respect to in-phase (k = 0) nuclear coordinates are derived. The coupled perturbed Hartree-Fock (CPHF) equation is iteratively solved by using the direct (recomputation of two-electron integrals) atomic-orbital-based algorithm. Frequencies of the Brillouin zone center (k = 0) vibrations of all-trans polyethylene are calculated by using the STO-3G, 3-21G and 6-31G* basis sets. The dependence of the frequencies on the number of neighbors included in the lattice summations, on the number of momentum sampling points in the first Brillouin zone, and on the convergence criterion for the CPHF solutions is examined. In our implementation, the use of analytical second derivatives is more efficient than the use of the finite differences of analytical first derivatives.