An adaptive numerical integrator for the exchange-correlation energy and potential is presented. It uses the diagonal elements of the exchange-correlation potential matrix as a grid generating function. The only input parameter is the requested grid tolerance. In combination with a defined cell function the adaptive grid generation scales almost linear with the number of basis functions in a system. With the adaptive numerical integrator the self-consistent field energy error, which is due to the numerical integration of the exchange-correlation energy, converges with increasing adaptive grid size to a reference value. The performance of the adaptive numerical integration is analyzed using molecules with first, second, and third row elements. Especially for transition metal systems the adaptive numerical integrator shows considerably improved performance and reliability.

An efficient method for the analytic evaluation of Fukui functions is proposed. Working equations are derived and numerical results are used to validate the method on medium size set of molecules. In addition to the obvious advantages of analytic differentiation, the proposed method is efficient enough to be considered a practical alternative to the finite difference formulation used routinely. The reliability of the approximations used here is demonstrated and discussed. Problems found in other methods for prediction of electrophilic centers are corrected automatically when using the new method.

A new approach, named auxiliary density perturbation theory, for the calculation of second energy derivatives is presented. It is based on auxiliary density functional theory in which the Coulomb and exchange-correlation potentials are expressed by auxiliary function densities. Different to conventional coupled perturbed Kohn–Sham equations the perturbed density matrix is obtained noniteratively by solving an inhomogeneous equation system with the dimension of the auxiliary function set used to expand the auxiliary function density. A prototype implementation for the analytic calculation of molecular polarizabilities is presented. It is shown that the polarizabilities obtained with the newly developed auxiliary density perturbation approach match quantitative with the ones from standard density functional theory if augmented auxiliary function sets are used. The computational advantages of auxiliary density perturbation theory are discussed, too.