New quantum number for the many-electron Dirac-Coulomb Hamiltonian

TL;DR

This paper introduces a new quantum number based on the square of a many-electron time-reversal operator for the Dirac-Coulomb Hamiltonian, providing a relativistic analogue to spin symmetry and enabling new computational and theoretical tools.

Contribution

It defines a novel quantum number from the square of a many-electron time-reversal operator, creating a relativistic counterpart to spin symmetry and leading to the concept of Kramers CSF.

Findings

The operator $\\mathcal{K}_+^2$ commutes with the Dirac-Coulomb Hamiltonian.

Introduction of Kramers CSF as a relativistic configuration state function.

Potential applications in relativistic quantum chemistry and spectroscopy.

Abstract

By breaking the spin symmetry in the relativistic domain, a powerful tool in physical sciences was lost. In this work, we examine an alternative of spin symmetry for systems described by the many-electron Dirac-Coulomb Hamiltonian. We show that the square of many-electron operator $\mathcal{K}_+$, defined as a sum of individual single-electron time-reversal (TR) operators, is a linear Hermitian operator which commutes with the Dirac-Coulomb Hamiltonian in a finite Fock subspace. In contrast to the square of a standard unitary many-electron TR operator $\mathcal{K}$, the $\mathcal{K}^2_+$ has a rich eigenspectrum having potential to substitute spin symmetry in the relativistic domain. We demonstrate that $\mathcal{K}_+$ is connected to $\mathcal{K}$ through an exponential mapping, in the same way as spin operators are mapped to the spin rotational group. Consequently, we call $\mathcal{K}_+$ the generator of the many-electron TR symmetry. By diagonalizing the operator $\mathcal{K}^2_+$ in the basis of Kramers-restricted Slater determinants, we introduce the relativistic variant of configuration state functions (CSF), denoted as Kramers CSF. A new quantum number associated with $\mathcal{K}^2_+$ has potential to be used in many areas, for instance, (a) to design effective spin Hamiltonians for electron spin resonance spectroscopy of heavy-element containing systems; (b) to increase efficiency of methods for the solution of many-electron problems in relativistic computational chemistry and physics; (c) to define Kramers contamination in unrestricted density functional and Hartree--Fock theory as a relativistic analog of the spin contamination in the nonrelativistic domain.