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%\title{CONSECUTIVE ONE-PARTICLE PROJECTORS FOR EXPLICITLY CORRELATED RELATIVISTIC COMPUTATIONS}
\title{ONE-PARTICLE OPERATOR REPRESENTATION OVER TWO-PARTICLE BASIS SETS FOR RELATIVISTIC QED COMPUTATIONS}

\author{\underline{\FirstName P. \LastName HOLLÓSY},
  \FirstName P. \LastName JESZENSZKI, and \FirstName E. \LastName M{\'A}TYUS}
\address{ELTE,~E{\"o}tv{\"o}s~Lor{\'a}nd~University, Institute~of~Chemistry, Budapest, Hungary}
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\fancyhead[C]{{\footnotesize \textit{HRMS, 28th Colloquium, Dijon, France,
  Aug. 28 - Sept. 1, 2023}}}
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\noindent
This work is concerned with two(many)-spin-1/2-fermion relativistic quantum mechanics and it is about the construction of one-particle projectors and potentially, one-particle propagators, necessary for quantum-electrodynamics (QED) corrections \cite{MaFeJeMa23},  using an inherently two(many)-particle, `explicitly correlated' basis representation, necessary for good numerical convergence of the results \cite{JeFeMa21,JeFeMa22,FeJeMa22,FeJeMa22b}.
\noindent
It is demonstrated that a faithful representation of the one-particle operators, which appear in intermediate but essential computational steps, can be constructed over a many-particle basis set by accounting for the full Hilbert space, beyond the physically relevant anti-symmetric subspace.
\noindent
Applications of this development can be foreseen 
for the computation of quantum-electrodynamics corrections for a correlated relativistic reference state
and for high-precision relativistic computations of medium-to-high $Z$ helium-like systems, for which other two-particle projection techniques are unreliable.


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\bibitem{MaFeJeMa23}
E. M{\'a}tyus, D. Ferenc, P. Jeszenszki, {\'A}. Marg{\'o}csy, \href{https://doi.org/10.1021/acsphyschemau.2c00062}{\emph{ACS Phys. Chem. Au}} \textbf{3}, 222 (2023).%\href{https://arxiv.org/abs/2211.02389}{arXiv}

\bibitem{JeFeMa21}
P. Jeszenszki, D. Ferenc, and E. M{\'a}tyus, \href{https://aip.scitation.org/doi/10.1063/5.0051237}{\textit{J. Chem. Phys.}} \textbf{154}, 224110 (2021). %\href{https://arxiv.org/abs/2103.04923}{arXiv}

\bibitem{JeFeMa22}
P. Jeszenszki, D. Ferenc, and E. M{\'a}tyus, 
\href{https://aip.scitation.org/doi/abs/10.1063/5.0075096}{\textit{J. Chem. Phys.}} \textbf{156}, 084111 (2022).
%\href{https://arxiv.org/abs/2110.06638}{arXiv}

\bibitem{FeJeMa22}
D. Ferenc, P. Jeszenszki, and E. M{\'a}tyus, \href{https://aip.scitation.org/doi/abs/10.1063/5.0075097}{\textit{J. Chem. Phys.}} \textbf{156}, 084110 (2022). %\href{https://arxiv.org/abs/2110.06647}{arXiv}

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

\bibitem{JeMa23}
P. Jeszenszki and E. M{\'a}tyus, \href{https://doi.org/10.1063/5.0136360}{\textit{J. Chem. Phys.}} \textbf{158}, 054104 (2023). %\href{https://arxiv.org/abs/2211.14180}{arXiv}

%\bibitem{FeMa23}
%D. Ferenc and E. M{\'a}tyus, \href{https://doi.org/10.1103/PhysRevA.107.052803}{\textit{Phys. Rev. A.}} \textbf{107}, 052803 (2023). %\href{https://arxiv.org/abs/2301.13477}{arXiv}
 
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The high resolution of precision spectroscopy possibly provides a good test of our current understanding of the fundamental quantum theory. Our theoretical work wants to improve the relativistic description of atomic, ionic and molecular systems\cite{QEDcorr}.\par
Numerical computation were made in the no-pair variational Dirac–Coulomb(-Breit) (DC(B)) framework using an explicitly correlated Gaussian basis\cite{allorder, DiracCoulomb, Breit, Breit2, LScoupling}. In this framework one need to construct the $\Lambda_+$ operator which projects to the positive-energy states of the non-interacting Hamiltonian. There are various ways to build the $\Lambda_+$ projector. One can select in the energy list of the non-interacting system to determine the positive-energy states. An other method is the a \emph{complex coordinate rotation} (CCR). Both of the methods have their limitations. In the previous method Brown-Ravenhall (BR) states contaminate the positive-energy states while the CCR has the disadvantage that one need to work with a non-hermitian Hamiltonians and complex energies.\par 
In our recrent work we demonstrate a new method to construct the $\Lambda_+$ operator.
%The one-particle operators are built on the hole Hilbert space (Both symmetrized ant antisymmetized function) and applied consecutively.
The individual one-particle Hamiltonian are built both on the symmetric and antisymmetric space because the permutation symmetry of the system is broken, the positive-energy space is selected. The one-particle projectors are applied consecutively.
This method has the advantage that during the process no BR-states occur and the Hamiltonian is hermitian.\par
We present high-precision numerical result on the no-pair DCB ground state energies from the He isoelectronic series. Minor difference in the energies from the different methods can be seen.
%Further applications of the consecutive one-particle projectors in QED are planned. 
