Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation

Version 1 : Received: 30 May 2020 / Approved: 31 May 2020 / Online: 31 May 2020 (18:43:04 CEST)

A peer-reviewed article of this Preprint also exists.

Salman, M.; Saue, T. Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation. Symmetry 2020, 12, 1121. Salman, M.; Saue, T. Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation. Symmetry 2020, 12, 1121.

Journal reference: Symmetry 2020, 12, 1121
DOI: 10.3390/sym12071121

## Abstract

4-component relativistic atomic and molecular calculations are typically performed within the no-pair approximation where negative-energy solutions are discarded, hence the symmetry between electronic and positronic solutions is not considered. These states are however needed in QED calculations, where furthermore charge conjugation symmetry becomes an issue. In this work we shall discuss the realization of charge conjugation symmetry of the Dirac equation in a central field within the finite basis approximation. Three schemes for basis set construction are considered: restricted, inverse and dual kinetic balance. We find that charge conjugation symmetry can be realized within the restricted and inverse kinetic balance prescriptions, but only with a special form of basis functions that does not obey the right boundary conditions of the radial wavefunctions. The dual kinetic balance prescription is on the other hand compatible with charge conjugation symmetry without restricting the form of the radial basis functions. However, since charge conjugation relates solutions of opposite value of the quantum number κ, this requires the use of basis sets chosen according to total angular momentum j rather than orbital angular momentum ` . As a special case, we consider the free-particle Dirac equation, where the solutions of opposite sign of energy are related by charge conjugation symmetry. We note that there is additional symmetry in those solutions of the same value of κ come in pairs of opposite energy.

## Subject Areas

Dirac equation; finite basis approximation; charge conjugation symmetry; kinetic balance

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