Adiabatic density-functional perturbation theory: Hessian, polarizability, NMR

Introduction

The total energy and charge density are basic quantities of density functional theory and give access to a wide range of experimental observables. Most of these quantities can be obtained as derivatives of the energy or density with respect to external perturbations, as collected in Table 7.1 for a selection of relevant response properties. As a simple example, the force exerted on a given nucleus is given by the negative derivative of the total energy of the system with respect to the position of this nucleus. The calculation of such energy derivatives can be done by finite-difference methods. The total energy is computed at slightly different values of the external perturbation and the derivative of the total energy curve with respect to the small disturbance is calculated numerically. Concerning the force, this just amounts to displacing the nucleus of interest by small amounts along the three Cartesian coordinates with respect to its equilibrium position as determined earlier from a geometry optimization. Although this is a very convenient and most straightforward method, recent practice has shown that perturbative techniques within the framework of density functional theory are much more powerful compared to numerical methods. Such techniques are very similar to the treatment of perturbations within Hartree–Fock theory, i.e. the coupled perturbed Hartree–Fock formalism [837]. These techniques were discovered and rediscovered within density functional theory many times. They are based either on the Sternheimer equation, Green's functions, sumover- states techniques, or on the Hylleras variational technique.