Interfacing relativistic and nonrelativistic methods. I. Normalized elimination of the small component in the modified Dirac equation

Abstract

The introduction of relativistic terms into the nonrelativistic all-electron Schrödinger equation is achieved by the method of normalized elimination of the small component (ESC) within the matrix representation of the modified Dirac equation. In contrast to the usual method of ESC, the method presented retains the correct relativistic normalization, and permits the construction of a single matrix relating the large and small component coefficient matrices for an entire set of positive energy one-particle states, thus enabling the whole set to be obtained with a single diagonalization. This matrix is used to define a modified set of one- and two-electron integrals which have the same appearance as the nonrelativistic integrals, and to which they reduce in the limit α→0. The normalized method corresponds to a projection of the Dirac–Fock matrix onto the positive energy states. Inclusion of the normalization reduces the discrepancy between the eigenvalues of the ESC approach and the Dirac eigenvalues for a model problem from order α2 to order α4, providing a closer approximation to the original, uneliminated solutions. The transition between the nonrelativistic and relativistic limits is achieved by simply scaling the fine structure constant α.