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\documentclass[11pt]{article}

\usepackage{hrms}  %% Page settings, style ... for the HRMS abstract book.

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\title{PRE-BORN--OPPENHEIMER DIRAC--COULOMB--BREIT COMPUTATIONS FOR TWO-SPIN-1/2-FERMION SYSTEMS}

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\author{\underline{\FirstName D. \LastName FERENC},
  \FirstName E. \LastName M\'ATYUS}
\address{ELTE, E\"otv\"os Lor\'and University, Institute of Chemistry, P\'azm\'any P\'eter s\'et\'any 1/A, Budapest, H-1117, Hungary}

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%\author{\underline{\FirstName G. \LastName HHHHH-IIIIIII Jr},
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  %\underline{\FirstName L. \LastName MM NN PPPPPP}}
%\address{Address of the institution for the second author(s)}

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\usepackage{fancyhdr}
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\fancyhead[C]{{\footnotesize \textit{HRMS, 28th Colloquium, Dijon, France,
  Aug. 28 - Sept. 1, 2023}}}
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\usepackage[colorlinks=true,urlcolor=blue,linkcolor=blue,citecolor=blue]{hyperref}
\voffset -.65 cm

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\category {1,14}
\begin{document}

\maketitle  %% This will also build the list of author(s) and addresse(s).

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\noindent
Positronium,  muonium, hydrogen atom, and muonic-hydrogen are the simplest, yet some of the most extensively studied bound-state systems. High-precision spectroscopy experiments in combination with theoretical computations are proposed to test our fundamental understanding of ordinary matter in the low-energy range and to probe physics beyond the Standard Model. 
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So far almost exquisitely relativistic and QED effects are treated within the non-relativistic quantum electrodynamics framework, which yields excellent agreement with experimentally measured transitions \cite{AdCaPe22}. An alternative approach based on the relativistic Bethe--Salpeter wave equation was developed for the simplest two-fermion case in an external Coulomb field \cite{JeFeMa21,JeFeMa22,FeJeMa22a,FeJeMa22b,JeMa23,MaFeJeMa23}. The introduction of the equal-time wave function and separating a non-retarded interaction kernel gives rise to the no-pair Dirac--Coulomb(--Breit) equation, which is solved variationally to high precision. The advantage of the approach is that both correlation and special relativity are accounted for already in zeroth order.
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In this contribution, the extension of the Dirac--Coulomb(--Breit) framework is presented to pure two-body systems: positronium,  muonium, hydrogen atom, and muonic-hydrogen as the first applications without the introduction of the Born--Oppenheimer approximation \cite{FeMa23}. The numerical results show excellent agreement with the corresponding analytic relativistic perturbative energy contributions. 

\vfill
\begin{thebibliography}{9.}
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%\bibitem{FeKoMa20}
%D. Ferenc, V. I. Korobov, and E. M\'atyus, \href{https://doi.org/10.1103/physrevlett.125.213001}{\textit{Phys. Rev. Lett.}} \textbf{125}, 213001 (2020).

\bibitem{AdCaPe22}
G.S. Adkins, D.B. Cassidy and J. P\'erez-R\'ios, \href{https://doi.org/10.1016/j.physrep.2022.05.002}{\textit{Phys. Rep.}}
\textbf{975}, 1 (2022).

\bibitem{JeFeMa21}
P. Jeszenszki, D. Ferenc, and E. M\'atyus, \href{https://doi.org/10.1063/5.0051237}{\textit{J. Chem. Phys.}} \textbf{154}, 224110 (2021).

\bibitem{JeFeMa22}
P. Jeszenszki, D. Ferenc, and E. M\'atyus, \href{https://doi.org/10.1063/5.0075096}{\textit{J. Chem. Phys.}} \textbf{156}, 084111 (2022).

\bibitem{FeJeMa22a}
D. Ferenc, P. Jeszenszki, and E. M\'atyus, \href{https://doi.org/10.1063/5.0075097}{\textit{J. Chem. Phys.}}  \textbf{156}, 084110 (2022).

\bibitem{FeJeMa22b}
D. Ferenc, P. Jeszenszki, and E. M\'atyus, \href{https://doi.org/10.1063/5.0105355}{\textit{J. Chem. Phys.}} \textbf{157}, 094113 (2022).

\bibitem{JeMa23}
P. Jeszenszki and E. M\'atyus, \href{https://doi.org/10.1063/5.0136360}{\textit{J. Chem. Phys.}} \textbf{158}, 054104 (2023).

\bibitem{MaFeJeMa23}
E. M\'atyus, D. Ferenc, P. Jeszenszki, and \'A. Marg\'ocsy, \href{https://doi.org/10.1021/acsphyschemau.2c00062}{\emph{ACS Phys. Chem. Au}} \textbf{3}, 222 (2023).

\bibitem{FeMa23}
D. Ferenc and E. M\'atyus,  \href{https://doi.org/10.1103/PhysRevA.107.052803}{\emph{Phys. Rev. A}} \textbf{107}, 052803 (2023).



 
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