The change of picture of the Hellmann–Feynman force operator in approximate relativistic methods

Abstract

The calculation of energy gradients in one- and two-component approximate methods of relativistic quantum chemistry which follow from the block-diagonalisation of the Dirac hamiltonian, is considered. It is indicated that the transition to these approximate methods involves the so-called change of picture for all operators. In particular this applies to the Hellmann–Feynman force operator. To avoid explicit transformation of this operator to the new picture two schemes for its modelling in terms of the usual Coulomb attraction operators are proposed. One of them is based on the point charge nuclear dipole moment model. The other one, the so-called shifted nucleus model, involves a parametrised shift of the nucleus. Both these models lead to a semianalytical method for the evaluation of energy gradients in approximate relativistic calculations, in which the relaxation terms is obtained analytically whereas the matrix elements of the Hellmann–Feynman force are calculated by using the finite difference method. The accuracy and relative merits of the two models are analysed. The two models are tested within the one-component Douglas–Kroll method. A large change of picture contribution is found in calculations of intramolecular forces in the coinage metal hydrides. This shows that any approximate relativistic technique for calculations of the energy gradients and for the relativistic geometry optimisation must take into account the change of picture contribution to the evaluated Hellmann–Feynman force.