Relativistic kinetic-balance condition for explicitly correlated basis functions

Abstract

This paper presents the derivation of a kinetic-balance condition for explicitly correlated basis functions employed in semi-classical relativistic calculations. Such a condition is important to ensure variational stability in algorithms based on the first-quantized Dirac theory of 1/2-fermions. We demonstrate that the kinetic-balance condition can be obtained from the row reduction process commonly applied to solve systems of linear equations. The resulting form of kinetic balance establishes a relation for the 4 N components of the spinor of an N -fermion system to the non-relativistic limit, which is in accordance with recent developments in the field of exact decoupling in relativistic orbital-based many-electron theory.

Perspective: Explicitly correlated electronic structure theory for complex systems

“…The essential ingredients would appear to be present for at least mediumsized molecules within the standard ab initio framework represented by the F12 approach, yet several challenges remain, e.g., efficient algorithms for many-electron integrals with general correlation factors, 170 relativistic treatment of explicit electron correlation, [171][172][173][174] especially beyond the Dirac-Coulomb Hamiltonian, composite schemes, 175 and three-body correlation factors for the triple-point coalescence. 176 Above all, the requirement of manifold integrals with CABS indices for nuclear derivatives has complicated the implementation of analytical energy gradients even for MP2-F12.…”

Microsolvation of Sr2+, Ba2+: Structures, energies, bonding, and nuclear magnetic shieldings

“…two-component and one-component Hamiltonians [31][32][33][34][35][36]. NMR magnetic shieldings have additional constraints regarding the consideration of gauge origin and the way small components are related with large components in four-component procedures [37][38][39]. Among the different models and theories that have been applied during more than 20 years [40][41][42][43][44], there is the four-component polarization propagator approach, which has the advantage of getting the NR values of the properties straightforwardly making c (the speed of light in vacuum) go to infinity [45,46].…”

On the Breit interaction in an explicitly correlated variational Dirac-Coulomb framework

“…At the same, we have not noticed any major sign of prolapse during the no-pair DC computations (Paper I). Nevertheless, we will consider testing other types of (approximate) kinetic balance conditions [18,39] in future work. On the other hand, we can compare our variational results only with finite-order, up to O(α 3 ), perturbative energies.…”

“…Over the past two decades, explicitly correlated methods have been developed and used for solving the DC equation of (helium-like) atoms [15][16][17][18]. Explicitly correlated, non-separable basis functions represent a departure from the single-particle picture, and the construction of an E + projection operator is not immediately obvious in this framework.…”