\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage{natbib}




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%

\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}

\usepackage[utf8]{inputenc}
\usepackage[greek,english]{babel}



\usepackage{float}








\begin{document}

\title{Aufbau Principle and Singlet-Triplet Gap in Spherical Hooke Atoms}



\author[1]{Xabier Telleria}%
\author[2]{Jesus Ugalde}%
\author[2]{Xabier Lopez}%
\author[3]{Eduard Matito}%
\author[4]{Eloy Ramos-Cordoba}%
\author[5]{Mauricio Rodriguez-Mayorga}%
\affil[1]{Euskal Herriko Unibertsitatea (UPV/EHU)}%
\affil[2]{Euskal Herriko Unibertsitatea}%
\affil[3]{University of Girona}%
\affil[4]{Donostia International Physics Center}%
\affil[5]{Paris-Saclay University}%


\vspace{-1em}



  
  \date{May 25, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Singlet and triplet spin state energies for three-dimensionalHooke
atoms, i .e. electrons in a quadratic confinement, with even number of
electrons (2, 4, 6, 8, 10) is discussed using Full-CI and CASSCF type
wavefunctions with a variety of basis sets and considering perturbative
corrections up to second order. The effect of the screening of the
electron-electron interaction is also discussed by using a Yukawa-type
potential with different values of the Yukawa screening parameter (\selectlanguage{greek}λ\selectlanguage{english}ee
=0.2, 0.4, 0.6, 0.8, 1.0). Our results show that the singlet state is
the ground state for 2 and 8 electron Hooke atoms, whereas the triplet
is the ground spin state for 4, 6 and 10 electron systems. This suggests
the following Auf bau structure 1s \textless{} 1p \textless{} 1d with
singlet ground spin states for systems in which the generation of the
triplet implies an inter-shell one electron promotion, and triplet
ground states in cases when there is a partial filling of electrons of a
given shell. It is also observed that the screening of electronelectron
interactions has a sizable quantitative effect on the relative energies
of both spin states, specially in the case of 2 and 8 electron systems,
favouring the singlet state over the triplet. However, the screening of
the electron-electron interaction does not provoke a change in the
nature of the ground spin state of these systems. By analyzing the
different components of the energy, we have gained a deeper
understanding of the effects of the kinetic, confinement and
electron-electron interaction components of the energy.%
\end{abstract}%



\sloppy




\selectlanguage{english}
\FloatBarrier
\end{document}

