{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T13:01:25Z","timestamp":1774357285461,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T00:00:00Z","timestamp":1593993600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR- 17-CE29-0004-01"],"award-info":[{"award-number":["ANR- 17-CE29-0004-01"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Four-component relativistic atomic and molecular calculations are typically performed within the no-pair approximation where negative-energy solutions are discarded. These states are, however, needed in QED calculations, wherein, furthermore, charge conjugation symmetry, which connects electronic and positronic solutions, 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    \u03ba   , this requires the use of basis sets chosen according to total angular momentum j rather than orbital angular momentum \u2113. As a special case, we consider the free-particle Dirac equation, where opposite energy solutions are related by charge conjugation symmetry. We show that there is additional symmetry in that solutions of the same value of    \u03ba    come in pairs of opposite energy.<\/jats:p>","DOI":"10.3390\/sym12071121","type":"journal-article","created":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T11:07:42Z","timestamp":1594033662000},"page":"1121","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Charge Conjugation Symmetry in the Finite Basis Approximation of the Dirac Equation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8214-9524","authenticated-orcid":false,"given":"Maen","family":"Salman","sequence":"first","affiliation":[{"name":"Laboratoire de Chimie et Physique Quantique, UMR 5626 CNRS, Universit\u00e9 Toulouse III-Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6407-0305","authenticated-orcid":false,"given":"Trond","family":"Saue","sequence":"additional","affiliation":[{"name":"Laboratoire de Chimie et Physique Quantique, UMR 5626 CNRS, Universit\u00e9 Toulouse III-Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,6]]},"reference":[{"key":"ref_1","first-page":"814","article-title":"The use of charge-conjugated wave-functions in the hole-theory of the electron","volume":"40","author":"Kramers","year":"1937","journal-title":"Proc. R. Acad. Amst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1080\/00268978200101771","article-title":"Basis set expansions of relativistic molecular wave equations","volume":"46","author":"Schwarz","year":"1982","journal-title":"Mol. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1230","DOI":"10.1103\/PhysRevA.25.1230","article-title":"Conditions for convergence of variational solutions of Dirac\u2019s equation in a finite basis","volume":"25","author":"Grant","year":"1982","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1910","DOI":"10.1063\/1.447865","article-title":"Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations","volume":"81","author":"Stanton","year":"1984","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1088\/0022-3700\/17\/4\/006","article-title":"Matrix representation of operator products","volume":"17","author":"Dyall","year":"1984","journal-title":"J. Phys. B"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1063\/1.442680","article-title":"Relativistic effects on Re and De in AgH and AuH from all-electron Dirac-Hartree- Fock calculations","volume":"76","author":"Lee","year":"1982","journal-title":"J. Chem. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/0009-2614(84)85647-X","article-title":"Features of the energy surface in Dirac-Fock discrete basis description as applied to the Be atom","volume":"105","author":"Ishikawa","year":"1984","journal-title":"Chem. Phys. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1103\/PhysRevA.22.348","article-title":"Foundations of the relativistic theory of many-electron atoms","volume":"22","author":"Sucher","year":"1980","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"074104","DOI":"10.1063\/1.4959452","article-title":"Electron correlation within the relativistic no-pair approximation","volume":"145","author":"Almoukhalalati","year":"2016","journal-title":"J. Chem. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1103\/PhysRev.82.914","article-title":"The Theory of Quantized Fields. I","volume":"82","author":"Schwinger","year":"1951","journal-title":"Phys. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1002\/cpa.20145","article-title":"The mean-field approximation in quantum electrodynamics: The no-photon case","volume":"60","author":"Hainzl","year":"2007","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/0009-2614(90)85321-3","article-title":"Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set","volume":"174","author":"Dyall","year":"1990","journal-title":"Chem. Phys. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/s00214-010-0876-6","article-title":"Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations","volume":"129","author":"Sun","year":"2011","journal-title":"Theor. Chem. Accounts"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"130405","DOI":"10.1103\/PhysRevLett.93.130405","article-title":"Dual Kinetic Balance Approach to Basis-Set Expansions for the Dirac Equation","volume":"93","author":"Shabaev","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","unstructured":"Dirac, P.A.M. (1930). The Principles of Quantum Mechanics, Clarendon Press."},{"key":"ref_16","first-page":"1","article-title":"On the Equivalence of Invariance under Time Reversal and under Particle-Antiparticle Conjugation for Relativistic Field Theories","volume":"28","year":"1954","journal-title":"Mat. Fys. Medd. K. Dan. Vidensk. Selsk."},{"key":"ref_17","first-page":"479","article-title":"Time reversal in field theory","volume":"231","author":"Bell","year":"1955","journal-title":"Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci."},{"key":"ref_18","unstructured":"Pauli, W. (1955). Niels Bohr and the Development of Physics, McGraw\u2013Hill."},{"key":"ref_19","first-page":"109","article-title":"Contributions math\u00e9matiques \u00e0 la th\u00e9orie des matrices de Dirac","volume":"6","author":"Pauli","year":"1936","journal-title":"Ann. De L\u2019institut Henri Poincar\u00e9"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1103\/PhysRev.51.484","article-title":"Relativistic wave functions in the Continuous spectrum for the Coulomb field","volume":"51","author":"Rose","year":"1937","journal-title":"Phys. Rev."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Grant, I.P. (2007). Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation, Springer Science & Business Media.","DOI":"10.1007\/978-0-387-35069-1"},{"key":"ref_22","unstructured":"Messiah, A. (1999). Quantum Mechanics; Dover Books on Physics, Dover Publications."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Rose, M.E. (1961). Relativistic Electron Theory, John Wiley.","DOI":"10.1063\/1.3057239"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1002\/qua.560320751","article-title":"On the use of an extended nucleus in Dirac\u2013Fock Gaussian basis set calculations","volume":"21","author":"Ishikawa","year":"1987","journal-title":"Int. J. Quant. Chem. Quant. Chem. Symp."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1006\/adnd.1997.0751","article-title":"Dirac\u2013Fock atomic electronic structure calculations using different nuclear charge distributions","volume":"67","author":"Visscher","year":"1997","journal-title":"At. Data Nucl. Data Tables"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1002\/qua.560400816","article-title":"Kinetic balance in contracted basis sets for relativistic calculations","volume":"40","author":"Visscher","year":"1991","journal-title":"Int. J. Quantum Chem."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2118","DOI":"10.1063\/1.466508","article-title":"An exact separation of the spin-free and spin-dependent terms of the Dirac\u2013Coulomb\u2013Breit Hamiltonian","volume":"100","author":"Dyall","year":"1994","journal-title":"J. Chem. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1002\/qua.560250112","article-title":"Basis set expansion of the Dirac operator without variational collapse","volume":"25","author":"Kutzelnigg","year":"1984","journal-title":"Int. J. Quantum Chem."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.chemphys.2011.07.009","article-title":"A question of balance: Kinetic balance for electrons and positrons","volume":"395","author":"Dyall","year":"2012","journal-title":"Chem. Phys."},{"key":"ref_30","first-page":"39","article-title":"Optimization of Gaussian basis sets for Dirac\u2013Hartree\u2013Fock calculations","volume":"94","author":"Dyall","year":"1996","journal-title":"Theor. Chim. Acta"},{"key":"ref_31","unstructured":"Wolfram Research, Inc (2019). Mathematica, Version 12.0, Wolfram Research, Inc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1121\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:47:50Z","timestamp":1760176070000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1121"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,6]]},"references-count":31,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071121"],"URL":"https:\/\/doi.org\/10.3390\/sym12071121","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202005.0492.v1","asserted-by":"object"}]},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,6]]}}}