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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. The Dirac Equation and C-Symmetry

#### 2.2. Radial Problem

#### Free-Particle Radial Problem

#### 2.3. Finite Basis Approximation

#### 2.3.1. Gaussian-Type Functions

#### 2.3.2. C-Symmetry in The Finite Basis Approximation

#### 2.3.3. Kinetic Balance

#### 2.3.4. Dual Kinetic Balance

## 3. Computational Details

## 4. Results and Discussion

#### 4.1. Kinetic Balance

#### 4.2. Dual Kinetic Balance

- Calculation 1: with $-e$; electronic charge.
- Calculation 2: with $+e$; positronic charge.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Matrix Eigenvalue Equations

#### Appendix A.1. Kinetic Balance

#### Appendix A.2. Dual Kinetic Balance

## References

- Kramers, H.A. The use of charge-conjugated wave-functions in the hole-theory of the electron. Proc. R. Acad. Amst.
**1937**, 40, 814–823. Available online: https://www.dwc.knaw.nl/DL/publications/PU00017118.pdf (accessed on 22 June 2020). - Schwarz, W.; Wallmeier, H. Basis set expansions of relativistic molecular wave equations. Mol. Phys.
**1982**, 46, 1045–1061. [Google Scholar] [CrossRef] - Grant, I.P. Conditions for convergence of variational solutions of Dirac’s equation in a finite basis. Phys. Rev. A
**1982**, 25, 1230–1232. [Google Scholar] [CrossRef] - Stanton, R.E.; Havriliak, S. Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations. J. Chem. Phys.
**1984**, 81, 1910–1918. [Google Scholar] [CrossRef] - Dyall, K.G.; Grant, I.P.; Wilson, S. Matrix representation of operator products. J. Phys. B
**1984**, 17, 493. [Google Scholar] [CrossRef] - Lee, Y.S.; McLean, A.D. Relativistic effects on R
_{e}and D_{e}in AgH and AuH from all-electron Dirac-Hartree- Fock calculations. J. Chem. Phys.**1982**, 76, 735–736. [Google Scholar] [CrossRef] - Ishikawa, Y.; Binning, R.; Sando, K. Features of the energy surface in Dirac-Fock discrete basis description as applied to the Be atom. Chem. Phys. Lett.
**1984**, 105, 189–193. [Google Scholar] [CrossRef] - Sucher, J. Foundations of the relativistic theory of many-electron atoms. Phys. Rev. A
**1980**, 22, 348–362. [Google Scholar] [CrossRef] - Almoukhalalati, A.; Knecht, S.; Jensen, H.J.A.; Dyall, K.G.; Saue, T. Electron correlation within the relativistic no-pair approximation. J. Chem. Phys.
**2016**, 145, 074104. [Google Scholar] [CrossRef] [Green Version] - Schwinger, J. The Theory of Quantized Fields. I. Phys. Rev.
**1951**, 82, 914–927. [Google Scholar] [CrossRef] - Hainzl, C.; Lewin, M.; Solovej, J.P. The mean-field approximation in quantum electrodynamics: The no-photon case. Commun. Pure Appl. Math.
**2007**, 60, 546–596. [Google Scholar] [CrossRef] [Green Version] - Dyall, K.G.; Fægri, K. Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set. Chem. Phys. Lett.
**1990**, 174, 25–32. [Google Scholar] [CrossRef] - Sun, Q.; Liu, W.; Kutzelnigg, W. Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations. Theor. Chem. Accounts
**2011**, 129, 423–436. [Google Scholar] [CrossRef] - Shabaev, V.M.; Tupitsyn, I.I.; Yerokhin, V.A.; Plunien, G.; Soff, G. Dual Kinetic Balance Approach to Basis-Set Expansions for the Dirac Equation. Phys. Rev. Lett.
**2004**, 93, 130405. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dirac, P.A.M. The Principles of Quantum Mechanics; International Series of Monographs on Physics; Clarendon Press: Oxford, UK, 1930; p. 255. [Google Scholar]
- Lüders, G. On the Equivalence of Invariance under Time Reversal and under Particle-Antiparticle Conjugation for Relativistic Field Theories. Mat. Fys. Medd. K. Dan. Vidensk. Selsk.
**1954**, 28, 1–17. [Google Scholar] - Bell, J.S. Time reversal in field theory. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci.
**1955**, 231, 479–495. [Google Scholar] [CrossRef] - Pauli, W. (Ed.) Niels Bohr and the Development of Physics; McGraw–Hill: New York, NY, USA, 1955; p. 30. [Google Scholar]
- Pauli, W. Contributions mathématiques à la théorie des matrices de Dirac. Ann. De L’institut Henri Poincaré
**1936**, 6, 109–136. (In French) [Google Scholar] - Rose, M.E. Relativistic wave functions in the Continuous spectrum for the Coulomb field. Phys. Rev.
**1937**, 51, 484–485. [Google Scholar] [CrossRef] - Grant, I.P. Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation; Springer Science & Business Media: Berlin, Germany, 2007; Volume 40. [Google Scholar] [CrossRef] [Green Version]
- Messiah, A. Quantum Mechanics; Dover Books on Physics; Dover Publications: New York, NY, USA, 1999; Volume 2. [Google Scholar]
- Rose, M.E. Relativistic Electron Theory; John Wiley: Hoboken, NJ, USA, 1961. [Google Scholar]
- Ishikawa, Y.; Quiney, H.M. On the use of an extended nucleus in Dirac–Fock Gaussian basis set calculations. Int. J. Quant. Chem. Quant. Chem. Symp.
**1987**, 21, 523–532. [Google Scholar] [CrossRef] - Visscher, L.; Dyall, K.G. Dirac–Fock atomic electronic structure calculations using different nuclear charge distributions. At. Data Nucl. Data Tables
**1997**, 67, 207. [Google Scholar] [CrossRef] - Visscher, L.; Aerts, P.J.C.; Visser, O.; Nieuwpoort, W.C. Kinetic balance in contracted basis sets for relativistic calculations. Int. J. Quantum Chem.
**1991**, 40, 131–139. [Google Scholar] [CrossRef] - Dyall, K.G. An exact separation of the spin-free and spin-dependent terms of the Dirac–Coulomb–Breit Hamiltonian. J. Chem. Phys.
**1994**, 100, 2118–2127. [Google Scholar] [CrossRef] - Kutzelnigg, W. Basis set expansion of the Dirac operator without variational collapse. Int. J. Quantum Chem.
**1984**, 25, 107–129. [Google Scholar] [CrossRef] - Dyall, K.G. A question of balance: Kinetic balance for electrons and positrons. Chem. Phys.
**2012**, 395, 35–43. [Google Scholar] [CrossRef] - Dyall, K.G.; Fægri, K. Optimization of Gaussian basis sets for Dirac–Hartree–Fock calculations. Theor. Chim. Acta
**1996**, 94, 39–51. [Google Scholar] [CrossRef] - Wolfram Research, Inc. Mathematica, Version 12.0; Wolfram Research, Inc.: Champaign, IL, USA, 2019. [Google Scholar]

**Figure 1.**Schematic spectrum of the Dirac equation for (

**a**) an electron in an attractive potential $\varphi (r)$, (

**c**) a positron in the same potential and (

**b**) a free particle, $\varphi (r)$ = 0.

**Table 1.**Eigenvalues of the free-particle restricted kinetic balance (RKB) calculation (in ${E}_{h}$).

Eigenvalue No. | 1 | 2 | 3 | 4 | |
---|---|---|---|---|---|

$\kappa $ | $-1$ | 18,784.744 | −18,784.744 | 18,780.067 | −18,780.067 |

$+1$ | 18,786.676 | −18,786.676 | 18,780.981 | 18,780.981 |

Coefficients | ||||||
---|---|---|---|---|---|---|

$\mathit{\kappa}$ | No. | $\mathit{\u03f5}({\mathit{E}}_{\mathit{h}})$ | ${\mathit{c}}_{\mathit{1}}^{\mathit{L}}$ | ${\mathit{c}}_{\mathit{2}}^{\mathit{L}}$ | ${\mathit{c}}_{\mathit{1}}^{\mathit{S}}$ | ${\mathit{c}}_{\mathit{2}}^{\mathit{S}}$ |

$-1$ | 1 | +18,784.744 | 4.9279 | −10.2190 | 4.9271 | −10.2174 |

2 | −18,784.744 | 0.0616 | −0.1278 | −393.7590 | 816.5380 | |

$+1$ | 1 | +18,786.676 | −4.0603 | 13.2692 | −4.0594 | 13.2665 |

2 | −18,786.676 | 0.0585 | −0.1913 | −281.4726 | 919.8612 |

Eigenvalue No. | 1 | 2 | 3 | 4 | |
---|---|---|---|---|---|

$\kappa $ | $-1$ | −18,786.676 | 18,786.676 | −18,780.981 | 18,780.981 |

$+1$ | −18,784.744 | 18,784.744 | −18,780.067 | 18,780.067 |

Coefficients | ||||||
---|---|---|---|---|---|---|

$\mathit{\kappa}$ | No. | $\mathit{\u03f5}({\mathit{E}}_{\mathit{h}})$ | ${\mathit{c}}_{\mathit{1}}^{\mathit{L}}$ | ${\mathit{c}}_{\mathit{2}}^{\mathit{L}}$ | ${\mathit{c}}_{\mathit{1}}^{\mathit{S}}$ | ${\mathit{c}}_{\mathit{2}}^{\mathit{S}}$ |

$-1$ | 1 | −18,786.676 | −4.0594 | 13.2665 | −4.0603 | 13.2692 |

2 | 18,786.676 | −281.4726 | 919.8612 | 0.0585 | −0.1913 | |

$+1$ | 1 | −18,784.744 | 4.9271 | −10.2174 | 4.9279 | −10.2190 |

2 | 18,784.744 | −393.7590 | 816.5380 | 0.0616 | −0.1278 |

Charge | $\mathit{\kappa}$ | Eigenvalues | |||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | ||

$-e$ | $-1$ | −18,788.264 | 18,782.511 | −18,781.851 | 18,778.739 |

$+1$ | −18,787.149 | 18,785.113 | −18,781.223 | 18,780.084 | |

$+e$ | $-1$ | 18,787.149 | −18,785.113 | 18,781.223 | −18,780.084 |

$+1$ | 18,788.264 | −18,782.511 | 18,781.851 | −18,778.739 |

Coefficients | ||||||
---|---|---|---|---|---|---|

Charge | $\mathit{\kappa}$ | $\mathit{\u03f5}({\mathit{E}}_{\mathit{h}})$ | ${\mathit{c}}_{\mathit{1}}^{[+]}$ | ${\mathit{c}}_{\mathit{2}}^{[+]}$ | ${\mathit{c}}_{\mathit{1}}^{[-]}$ | ${\mathit{c}}_{\mathit{2}}^{[-]}$ |

$-e$ | $-1$ | −18,788.264 | $1.235\times {10}^{-5}$ | $-2.295\times {10}^{-5}$ | $-3.882$ | $13.106$ |

$+1$ | −18,787.149 | $1.069\times {10}^{-5}$ | $-3.745\times {10}^{-5}$ | $4.489$ | $-9.881$ | |

$+e$ | $-1$ | 18,787.149 | $4.489$ | $-9.881$ | $1.069\times {10}^{-5}$ | $-3.745\times {10}^{-5}$ |

$+1$ | 18,788.264 | $-3.882$ | $13.106$ | $1.235\times {10}^{-5}$ | $-2.295\times {10}^{-5}$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Salman, M.; Saue, T.
Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation. *Symmetry* **2020**, *12*, 1121.
https://doi.org/10.3390/sym12071121

**AMA Style**

Salman M, Saue T.
Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation. *Symmetry*. 2020; 12(7):1121.
https://doi.org/10.3390/sym12071121

**Chicago/Turabian Style**

Salman, Maen, and Trond Saue.
2020. "Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation" *Symmetry* 12, no. 7: 1121.
https://doi.org/10.3390/sym12071121