Analytic Evaluation of Third and Fourth Derivatives for Hartree-Fock Wave Functions

摘要

First and second derivatives of the energy with respect to nuclear perturbations for Hartree-Fock wave functions have provided much information regarding the curvature and character of potential energy hypersurfaces. The introduction of analytic second derivatives not only allowed the calculation of quadratic force constants for molecular species that cannot be obtained through finite differences of energy gradients, but also increased accuracy and reduced computational time required to obtain force constants. Third and fourth derivatives yield even more information concerning the potential energy surface by quantifying the anharmonicity. This work details the first implementation of the evaluation of analytic third derivatives with respect to nuclear perturbations for Hartree-Fock closed-shell molecules. The calculation of analytic third derivatives divides neatly into two calculations: the first is the contribution from the perturbation of the basis set; and the second is the contribution from the perturbation of the molecular orbital coefficients. The perturbation of the basis set yields first, second, and third derivative atomic orbital integrals. These are calculated using Hermite gaussian functions and Rys quadrature. Details of the algorithms are presented. The perturbation of the molecular orbital coefficients leads to first- and second-order coupled perturbed Hartree-Fock equations. These are reduced to first-order results. The ensuing energy equation coupling the two contributions is presented along with details concerning its evaluation. Results comparing analytic cubic force constants in normal coordinate space to those derived from experimental data for the water molecule are reported. Agreement is excellent. Initial timing information indicates that the analytic evaluation of third derivatives is computationally less expensive than the calculation through finite differences of analytic second derivatives. (ERA citation 10:015028)