Optimal decoupling of positive- and negative-energy orbitals in relativistic electronic structure calculations beyond Hartree-Fock

Abstract

When the one-body part of the relativistic Hamiltonian H is a sum of one-electron Dirac Hamiltonians, relativistic configuration interaction (CI) calculations are carried out with an ad hoc basis of positive-energy orbitals, {uj+, j=1, 2,…,m} and, more recently, with the full bases of positive-energy and negative-energy orbitals, {uj+, uj−, j=1, 2,…,m}. The respective eigenproblems H+Ck+=Ek+Ck+, k=1, 2,…,(m), and HCk=EkCk; k=1, 2,…,(2m) are related through Ek+≤Ek+(2m−(m) [R. Jauregui et al., Phys. Rev. A 55, 1781 (1997)]. This inequality becomes an equality for the independent-particle Hartree–Fock model and some other simple multiconfiguration models, leading to an exact decoupling of positive-energy and negative-energy orbitals. Beyond Hartree–Fock, however, it is generally impossible to achieve an equality. By definition, optimal decoupling is obtained when the difference Ek+(2m)−(m)−Ek+ is a minimum, which amounts to maximize the energy Ek+ with respect to any set of m functions in the 2m-dimensional space {uj+, uj−, j=1, 2,…,m}. Straight maximization is a slowly convergent process. Fortunately, numerical calculations on high-Z atomic states show that optimally decoupled, or best positive-energy orbitals are given, to within 6 decimals in atomic units by the positive-energy natural orbitals of the full eigenfunction Ck+(2m)−(m). Best orbitals can accurately be obtained through CI-by-parts treatments for later use in large-scale relativistic CI, as illustrated with Ne ground-state calculations. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 805–812, 1998