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 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.
The classical dynamics for a charged spin particle is governed by the Lorentz force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation for spin precession. In static and homogeneous electromagnetic fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a spin-
1/2 particle, is consistent with the classical Hamiltonian (with both the orbital and spin parts) up to the order of
1/m14 (
m is the particle's mass) in the low-energy and weak-field limit. In this paper, we extend this correspondence to the case of inhomogeneous fields. Regardless of the field gradient (e.g. Stern-Gerlach) force, the T-BMT equation is unaltered and thus the classical Hamiltonian remains the same, but subtleties arise and need to be clarified. For the relativistic quantum theory, we apply Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.
Purpose of the study: The goal of this work is to present a comparative study between the semi-relativistic theory of Sakho and the relativistic theory of Dirac applied to calculations of the eigenvalue of the Hamiltonian fine structure of the hydrogen like systems. The three small perturbative corrections (relativistic mass correction, spin-orbit coupling, and the Darwin term) of the Hamiltonian are considered. Methodology: This research is based on correction of the semi-classical Bohr’s theory on the hydrogen like systems considering the relativistic mass correction by ignoring the electron’s spin. This approach referred to as semi-relativistic theory allows one to express the eigenvalues of the Hamiltonian fine structure to be compared to that from the relativistic wave function of Dirac Main Findings: The research gives the first expression of the quantized energy of hydrogen like systems taking only into account, the relativistic mass correction. This result provides exactly the eigenvalues of the Hamiltonian fine structure predicted from Dirac’s relativistic theory for all the non-degenerated quantum states n2Lj (j = l ± s) of the hydrogen like systems (i.e. 1 2s1/2, 2 2p3/2, 3 2d5/2, 42f7/2, and so on). Novelty/Originality of this study: Theoretical determination of the eigenvalue of the Hamiltonian fine structure predicted from Dirac’s relativistic theory for all the non-degenerated quantum states is obtained with a soft semi-relativistic theory without including in the calculations the spin-orbit coupling and the Darwin term corrections.