\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 44, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/44\hfil Existence of minimizers]
{Existence of minimizers in restricted Hartree-Fock theory}

\author[F. Hantsch \hfil EJDE-2014/44\hfilneg]
{Fabian Hantsch}  % in alphabetical order

\address{Fabian Hantsch \newline
Universit\"at Stuttgart, Fachbereich Mathematik\\
70550 Stuttgart, Germany}
\email{Fabian.Hantsch@mathematik.uni-stuttgart.de}

\thanks{Submitted August 19, 2013. Published February 10, 2014.}
\subjclass[2000]{81V45, 49S05}
\keywords{Restricted Hartree-Fock functional; ground states;  \hfill\break\indent
variational methods}

\begin{abstract}
 In this note we establish the existence of ground states for atoms
 within several restricted Hartree-Fock theories.
 It is shown, for example, that there exists a ground state for
 closed shell atoms with $N$ electrons and nuclear charge $Z \geq N-1$.
 This has to be compared with the general Hartree-Fock theory where
 the existence of a minimizer is known for $Z >N-1$ only.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\newcommand{\sprod}[2]{\langle #1,#2\rangle}
\newcommand{\norm}[1]{\|#1\|}
\newcommand{\form}[3]{\langle #1|#2|#3\rangle}

\section{Introduction}

Computations of the electronic structure of atoms and molecules in quantum
chemistry in general rely on numerical solutions of simplified versions
of the quantum many-body problem at hand. Among those, the Hartree-Fock
approximation often serves as a starting point for more accurate approximations
such as multi-configuration methods, see for example \cite{Helgaker,SzaboOstlund}.
In the simplest version of Hartree-Fock theory the energy is minimized with
respect to antisymmetric tensor products of orthonormal one-electron orbitals,
the so-called single Slater determinants, and further restrictions are imposed
in numerical procedures implementing this variational problem \cite{Cances2003}.
In any case the question arises whether a minimizer exists.
This paper is concerned with several restricted Hartree-Fock theories
for atoms where the one-electron orbitals are products of space and spin wave
functions. For each of the considered restrictions we investigate the existence
of a minimizer both for neutral atoms and positive ions, as well as for simply
charged negative ions.

The existence of a minimizer in the \emph{general Hartree-Fock (GHF)} theory
for neutral atoms or positive ions was first established in 1977 by Lieb and
Simon \cite{LS1977}. No constraints were imposed in their work besides the
orthonormality of the one-electron orbitals. In the meantime there has been
remarkable further progress in the study of the variational problem for
the Hartree-Fock energy functional. It is known, for example, that there
exists a sequence of critical points for this functional \cite{Lions1987},
and convergence properties of various algorithms used for the approximation
of critical points were investigated in \cite{CB2000,Cances2000,Levitt}.

The main concern of this article is the minimization of the Hartree-Fock
energy functional under additional constraints.
Our general assumption is that the one-electron states are products of space
and spin functions.
First, we treat the \emph{restricted Hartree-Fock (RHF)} functional for
closed shell atoms with prescribed angular momentum quantum numbers.
Second, we drop the latter requirement, i.e.~we consider atoms with an
even number of electrons, where only pairs of spin up and spin down electrons
with the same spatial function occur. The corresponding energy functional
will be called \emph{spin-restricted Hartree-Fock (SRHF)} functional.
We prove that there exists a ground state in both cases, if $Z \geq N-1$,
where $Z$ denotes the nuclear charge and $N$ the number of electrons.
The existence of a ground state in the case $Z=N-1$ reminds of the well-known
stability of closed shell configurations in chemistry. Third, we look at
another restricted Hartree-Fock functional, which is called
\emph{unrestricted Hartree-Fock (UHF)} functional in the chemical literature,
and must not be confused with the GHF functional. In the UHF setting,
we impose that the spatial functions corresponding to spin up resp.~spin down
functions are chosen independently from each other, but still are assumed to
have prescribed angular momenta. In this case a ground state exists if $Z>N-1$,
and we provide sufficient conditions under which this is also true if $Z=N-1$.
For example, there exists a ground state for $Z=N-1$ in the spinless case
(i.e.~if all spins point in the same direction) with two angular momentum
shells $\ell_1=0$, $\ell_2 >0$.

For certain closed shell atoms (e.g.~He, Ne) it is known that the minimization
problems for  the general and restricted Hartree-Fock functionals coincide,
if $Z \gg N$ \cite{GH2011}. On the other hand there are also cases where
they differ \cite{RuskStill}, see \cite{GH2011} for an explanation of this fact.
Nevertheless, the restricted ground states are always critical points of
the GHF functional. This is due to the fact that the considered constraints
do not require additional Lagrange multipliers in the Euler--Lagrange equations.
Thus, this paper also establishes the existence of critical points for the GHF
functional in the case $Z=N-1$.
To our knowledge, the only previous result providing the existence of critical
points for the GHF functional in the case $Z=N-1$ is given in the
paper \cite{CB2000} of Canc\`es and Le Bris, which in fact even holds for
arbitrary $Z>0$. But in general, the critical points constructed in their paper
only correspond either to local (not global) minima or saddle points.

In the literature the existence of minimizers for restricted Hartree-Fock
functionals has previously been studied for special cases.
Based on Reeken's paper \cite{Reeken1970} on the solutions of the Hartree equation,
Bazley and Seydel \cite{BazleySeydel} proved the existence of a minimizer for
the spin-restricted Hartree-Fock functional of Helium $(N=2)$, which is given by
the restricted Hartree functional. For this functional it is known that there
exists a minimizer even if $Z=1=N-1$, see \cite[Theorem II.2]{Lions1987}.
In our paper we extend this result to arbitrary numbers of filled shells.
Lieb and Simon generalize their GHF existence result \cite{LS1977} to certain
restricted situations in \cite{LS1974}, but their theorem does not cover the
restrictions discussed in this paper. However, this article has been strongly
inspired by their work \cite{LS1977}.
In \cite{Lions1987}, Lions treats restricted Hartree-Fock equations, which arise
as the Euler--Lagrange equations of the RHF functional. He proves the existence
of a sequence of solutions to these equations provided $Z \geq N$. Lions' proof
relies, however, on the unproven assertion that all eigenvalues of a radial Fock
operator are simple.
His approach is motivated by the paper of Wolkowisky \cite{Wolkowisky} who shows
the existence of solutions for a system of restricted Hartree-type equations.
A numerical approach to restricted Hartree-Fock theory may be found in the book
of Froese Fischer \cite{Froese-Fischer}.
Finally, we mention the article of Solovej \cite{Solovej2003}, where he proves
the existence of a universal constant $Q>0$ so that there is no GHF minimizer
for $Z \leq N-Q$.
This establishes the \emph{ionization conjecture} within the Hartree-Fock theory.
The question whether or not there is a GHF minimizer for $Z=N-1$ is open.

The paper, which forms a part of the author's Ph.D.~thesis, is organized as follows:
 In Section~\ref{sec:closed shell} we introduce the restricted Hartree-Fock
functional for closed shell atoms with prescribed angular momentum quantum
numbers and prove an existence theorem for minimizers of this functional.
The Section~\ref{sec:applications} is devoted to generalizations of the RHF
existence theorem to the SRHF and UHF functionals. A derivation of the RHF
functional in the closed shell case can be found in Section~\ref{app}.
Finally, there is an appendix containing technical lemmas.

\section{Minimizers for Closed Shell Atoms}\label{sec:closed shell}

The simplest Hartree-Fock approximation for atoms consists in restricting the
admissible $N$-electron states to the set of single Slater determinants,
which are of the form
\begin{equation}
\label{SD}
(\varphi_1 \wedge \dots \wedge \varphi_N)(x_1,\dots,x_N)
= \frac{1}{\sqrt{N!}} \sum_{\sigma \in S_N} \operatorname{sgn}(\sigma)
\varphi_{\sigma(1)}(x_1) \dots \varphi_{\sigma(N)}(x_N),
\end{equation}
where $S_N$ denotes the symmetric group of degree $N$,
$\operatorname{sgn}(\sigma)$ is the sign of a permutation $\sigma$,
and $\varphi_1,\dots,\varphi_N$ denote orthonormal $L^2(\mathbb{R}^3;\mathbb{C}^2)$-functions
 with $x_i=(\mathbf{x}_i, \mu_i)\in \mathbb{R}^3 \times \{\pm 1\}$ containing the
space and spin variables of the $i$-th electron. It is well-known, that the
energy of an atom with nuclear charge $Z$ and $N$ electrons in the
state \eqref{SD} is given by the \emph{general Hartree-Fock (GHF)} functional
\begin{equation} \label{HF functional}
\begin{aligned}
&\mathcal{E}^{HF}(\varphi_1,\dots,\varphi_N) \\
&= \sum_{j=1}^N \int |\nabla\varphi_j|^2
- \frac{Z}{|x|}|\varphi_j|^2 \,dx
+ \frac{1}{2} \iint \frac{\rho(x)\rho(y)-|\tau(x,y)|^2}{|x-y|} \,dx\,dy
\end{aligned}
\end{equation}
where
$$
\tau(x,y) := \sum_{j=1}^N \varphi_j(x)\overline{\varphi_j(y)},  \quad
 \rho(x) := \sum_{j=1}^N |\varphi_j(x)|^2
$$
denote the density matrix and the electronic density, respectively.
The notation $\int \,dx$ refers to integration with respect to the
 product of Lebesgue and counting measure, and $|x-y|=|\mathbf{x}-\mathbf{y}|$.

Given a closed shell atom with $s_0 \in \mathbb{N}$ shells of prescribed angular
 momentum quantum numbers $\ell_1,\dots,\ell_{s_0} \in \mathbb{N}_0$, we impose
the following form on the one-electron orbitals
\begin{equation}
\label{restr form}
\varphi_{jm\sigma}(\mathbf{x},\mu) = \frac{f_j(|\mathbf{x}|)}{|\mathbf{x}|}
Y_{\ell_j m}(\mathbf{x}) \delta_{\sigma \mu}, \quad j=1,\dots,s_0, \;
 m = -\ell_j,\dots,+\ell_j, \; \sigma = \pm 1,
\end{equation}
where the radial functions $f_j$ are in $L^2(\mathbb{R}_+)$ and
\begin{equation}\label{constraint}
\langle f_i, f_j \rangle := \int_{\mathbb{R}_+}\overline{f_i}f_j \,dr = \delta_{ij}, \quad
\text{if } \ell_i=\ell_j,
\end{equation}
 to ensure the orthonormality of the functions \eqref{restr form}.
Here $Y_{\ell m}$ denote the usual spherical harmonics.
The Hartree-Fock energy of the Slater determinant built by the orbitals
\eqref{restr form} is given by the \emph{restricted Hartree-Fock (RHF)}
functional (derived in Section~\ref{app}):
\begin{equation}
\begin{aligned}
\mathcal{E}^{RHF}(f_1,\dots,f_{s_0})
& = 2 \sum_{j=1}^{s_0} (2\ell_j + 1) \Big( \int_{\mathbb{R}_+} |f_j'|^2
+ \frac{\ell_j(\ell_j+1)}{r^2} |f_j|^2 - \frac{Z}{r} |f_j|^2 \,dr \Big)  \\
&\quad + \frac{1}{2} \sum_{j,k=1}^{s_0} (2\ell_j+1)(2\ell_k+1)
 \Big( \iint_{(\mathbb{R}_+)^2} 4 \frac{|f_j(r)|^2 |f_k(s)|^2}{\max\{r,s\}}  \\
&\quad - 2 \overline{f_j(r)f_k(s)}U_{\ell_j \ell_k}(r,s) f_k(r) f_j(s)\,dr\,ds \Big).
\end{aligned} \label{rest energy}
\end{equation}
The integral kernels $U_{\ell_j\ell_k}$ appearing in the last term on the
right-hand side are given in \eqref{Uform}. We shall only need their properties
collected in Lemma~\ref{lm:U-ell}.

Let $H^1_0(\mathbb{R}_+)$ denote the completion of $C^\infty_0(\mathbb{R}_+)$ with respect
to the $H^1(\mathbb{R}_+)$-norm.
The RHF functional \eqref{rest energy} is bounded below, if the functions
$f_1,\dots,f_{s_0}$ are in $H^1_0(\mathbb{R}_+)$ and obey the constraints
 \eqref{constraint}, see Lemma~\ref{lm:technical}.
We define the RHF ground state energy by
\begin{equation} \label{min prob}
E(N,Z) = \inf\{\mathcal{E}^{RHF}(f_1,\dots,f_{s_0}) | f_1,\dots,
f_{s_0} \in H^1_0(\mathbb{R}_+), \sprod{f_i}{f_j} = \delta_{ij} \text{ if } \ell_i= \ell_j\},
\end{equation}
where the dependence of $E(N,Z)$ on $\ell_1,\dots,\ell_{s_0}$ is omitted.
The main question of this paper is whether the infimum in \eqref{min prob}
is actually a minimum.

If there exist minimizing functions $f_1,\dots,f_{s_0}$ obeying the constraints
 of \eqref{min prob}, then they are solutions of the corresponding Euler-Lagrange
equations, which we may assume to have the form (see Remark (b) below)
\begin{equation}\label{EL eqn}
H_{\ell_i}f_i = \varepsilon_i f_i, \quad i=1,\dots,s_0,
\end{equation}
with \emph{radial Fock operators} given by
\begin{gather*}
H_{\ell_i}  =  -\partial^2_r + \frac{\ell_i(\ell_i+1)}{r^2} -\frac{Z}{r} +2U
 - K_{\ell_i}, \quad i=1,\dots,s_0, \quad \text{ where} \\
(Uf)(r)  =  \sum_{j=1}^{s_0} (2\ell_j+1) \int_{\mathbb{R}_+}
 \frac{|f_j(s)|^2}{\max\{r,s\}} \,ds f(r), \\
(K_{\ell}f)(r)  = \sum_{j=1}^{s_0} (2\ell_j+1) f_j(r) \int_{\mathbb{R}_+}
 \overline{f_j(s)}f(s) U_{\ell \ell_j}(r,s) \,ds.
\end{gather*}
We omit the dependence of the operators $U$, $K_{\ell}$ and thus $H_{\ell_i}$
on the functions $f_1,\dots,f_{s_0}$.
The Euler-Lagrange equations \eqref{EL eqn}, called \emph{Hartree-Fock equations},
form a set of $s_0$ coupled non-linear eigenvalue equations for the functions
$f_1,\dots,f_{s_0}$.
\smallskip

\noindent\textbf{Remarks.}
(a) By Lemma~\ref{lm:technical}, the operators $H_{\ell_i}$ are symmetric
semi-bounded operators on $C_0^\infty(\mathbb{R}_+)$. Therefore, minimizing functions
$f_1,\dots,f_{s_0}$ obeying the constraints of \eqref{min prob} are in the
domain $D(H_{\ell_i})$ of the Friedrichs extension of $H_{\ell_i}$,
which is contained in $H^1_0(\mathbb{R}_+)$.

(b) The Euler--Lagrange equations for minimizing functions $f_1,\dots,f_{s_0}$
obeying the constraints of \eqref{min prob} are given by
$H_{\ell_i}f_i = \sum_j \varepsilon_{ij}f_j$, where the sum runs over all indices
$j$ with $\ell_j = \ell_i$.
Since the functional $\mathcal{E}^{RHF}$ is invariant under unitary transformations
of the subspaces of $L^2(\mathbb{R}_+)$ spanned by all radial functions $f_j$ with
equal angular momentum quantum numbers, the minimizing functions
$f_1,\dots,f_{s_0}$ can always be chosen as eigenfunctions of the radial Fock
operators. This follows from standard arguments as used for the
general Hartree-Fock theory, c.~f.~\cite{Cances2003} for example.

(c) The constraints \eqref{min prob} may be relaxed without lowering the ground
state energy, more precisely $E(N,Z)=\tilde E(N,Z)$ for
\begin{equation}
\begin{aligned}
\tilde E (N,Z)  =  \inf \big\{&\mathcal{E}^{RHF}(f_1,\dots,f_{s_0}) | f_1,\dots, f_{s_0}
\in H_0^1(\mathbb{R}_+), \sprod{f_i}{f_j} = 0 \\
&\text{ if $\ell_i = \ell_j$  and $i \neq j$, }
\norm{f_i} \leq 1 \text{ for all } i \big\}.
\end{aligned} \label{generalized minimization problem}
\end{equation}
This can be seen using similar arguments as for the general Hartree-Fock
functional in \cite[section II.2]{Lions1987}.
The following theorem shows that the relaxed minimization problem always
 possesses a minimizer.

\begin{theorem}[Existence of a RHF minimizer]
\label{thm:existence}
Let $s_0\in\mathbb{N}$, $\ell_1,\dots,\ell_{s_0} \in \mathbb{N}_0$, and $Z>0$.
Then, there exist functions $f_1,\dots,f_{s_0} \in H_0^1(\mathbb{R}_+)$, which
minimize the RHF functional \eqref{rest energy} under the constraints
\begin{gather*}
\sprod{f_i}{f_j}  =  0 \quad \text{if $\ell_i = \ell_j$  and $i \neq j$,} \\
\norm{f_i}  \leq  1 \quad \text{for all } i.
\end{gather*}
Moreover, $f_i \in D(H_{\ell_i})$, $H_{\ell_i}f_i = \varepsilon_i f_i$, and:
\begin{itemize}
\item[(i)] Either $\varepsilon_i \leq 0$ or $f_i=0$. $\varepsilon_i < 0$ implies
  $\norm{f_i} = 1$.
\item[(ii)] If $Z>N- 2(2\ell_i+1)$, then $f_i \neq 0$.\\ If $Z \geq N-1$,
 then $\|f_i\|=1$ for all $i=1,\dots,s_0$.
\item[(iii)] If $Z>N-1$, then $\varepsilon_i<0$ and $\norm{f_i}=1$ for all
 $i=1,\dots,s_0$.
\end{itemize}
\end{theorem}

\noindent\textbf{Remarks.}
(a) Theorem~\ref{thm:existence}~(ii) shows that for $Z=N-1$ there always
 exists a normalized minimizer for $\mathcal{E}^{RHF}$. In this case we do not
know whether or not $\varepsilon_i<0$. Nevertheless, it is clear that
$Z>N-2(2\ell_i+1)$ always implies $E(N,Z)<E^{(i)}(N-2(2\ell_i+1),Z)$
for all $i=1,\dots,s_0$, where $E^{(i)}(N-2(2\ell_i+1),Z)$ denotes
the minimal energy in the case where all electrons of the $i$-th shell
are dropped. This can be seen as follows:  Theorem~\ref{thm:existence}~(iii)
is applicable to the minimization problem $E^{(i)}(N-2(2\ell_i+1),Z)$
because $Z>N-2(2\ell_i+1)$. Hence, there exist
$f_1,\dots, f_{i-1}, f_{i+1}, \dots,f_{s_0}\in H_0^1(\mathbb{R}_+)$ with $\|f_j\|=1$,
$j\neq i$, so that
\begin{equation} \label{Ungl1}
\mathcal{E}^{RHF}(f_1,\dots,f_{i-1},0,f_{i+1},\dots,f_{s_0})=E^{(i)}(N-2(2\ell_i+1),Z).
\end{equation}
It can be shown (c.f.~the proof of Theorem~\ref{thm:existence}~(ii)) that
there exists $\psi\in H_0^1(\mathbb{R}_+)$, $\|\psi\| \leq 1$,  $\psi \perp f_j$
for all $j \neq i$, $\ell_j=\ell_i$, with
\begin{equation} \label{Ungl2}
\mathcal{E}^{RHF}(f_1,\dots,f_{i-1},\psi,f_{i+1},\dots,f_{s_0})
<\mathcal{E}^{RHF}(f_1,\dots,f_{i-1},0,f_{i+1},\dots,f_{s_0}).
\end{equation}
The desired inequality now follows from $E(N,Z)=\tilde E(N,Z)$, \eqref{Ungl2}
and \eqref{Ungl1}.

(b) In general Hartree-Fock theory it is known that the minimizing functions
can be chosen as eigenfunctions to the $N$ lowest eigenvalues of the
corresponding Fock operator. Moreover, there is a gap between the occupied
and unoccupied eigenvalues \cite{BLLS1994}.
It would be interesting to know whether similar results hold also in the
restricted Hartree-Fock theory, where, unfortunately, the method
 of \cite{BLLS1994} is not applicable.
\smallskip

Before turning to the proof of Theorem~\ref{thm:existence} we introduce
the following notation that will be used throughout this paper.
$$
r_> := \max\{r,s\} , \quad r_< := \min\{r,s\} , \quad \text{for } r,s \geq 0.
$$
We write $\mathcal{E}^{RHF}(f_1,\dots, \hat f_i, \dots,f_{s_0})$ to denote the
restricted Hartree-Fock functional where the electrons of the $i$-th
shell are dropped. The following lemma exhibits the dependence of
$\mathcal{E}^{RHF}(f_1,\dots,f_i,\dots,f_{s_0})$ on $f_i$, and will be crucial
for the existence of a minimizer in the critical case $Z=N-1$.
It follows easily from the definition of $\mathcal{E}^{RHF}$ if we set
$P_i(r,s) := (2\ell_i+1)(2 r_>^{-1} - U_{\ell_i \ell_i}(r,s) )$.

\begin{lemma}[Decomposition property of the RHF functional]
\label{lm:decomp}
Let $s_0 \in \mathbb{N}$, \\
$\ell_1,\dots,\ell_{s_0} \in \mathbb{N}_0$, $Z>0$ and $f_1,\dots, f_{s_0} \in H^1_0(\mathbb{R}_+)$.
Furthermore, let $i \in \{1,\dots,s_0\}$ and let $H_{\ell_i}^{(i)}$ denote
the Fock operator where all electrons of the $i$-th shell are dropped. Then:
\begin{equation}
\begin{aligned}
\mathcal{E}^{RHF}(f_1,\dots,f_i, \dots,f_{s_0})
& =  \mathcal{E}^{RHF}(f_1,\dots,\hat f_i, \dots, f_{s_0})
 + 2(2\ell_i+1) \form{f_i}{H_{\ell_i}^{(i)}}{f_i}  \\
&\quad +  (2\ell_i+1) \form{f_i \otimes f_i}{P_i}{f_i \otimes f_i}, \label{decomp1}
\end{aligned}
\end{equation}
where $P_i(r,s)=P_i(s,r)$ and
$$
\frac{2\ell_i+1}{\max\{r,s\}} \leq P_i(r,s)
\leq \frac{4\ell_i+1}{\max\{r,s\}}, \quad r,s \geq 0.
$$
Furthermore, for all $\lambda \geq 0$, $h \in H_0^1(\mathbb{R}_+)$,
\begin{equation}
\begin{aligned}
&\mathcal{E}^{RHF}(f_1, \dots, \frac{f_i+\delta h}{\sqrt{1+\lambda\delta^2}},\dots,f_{s_0}) \\
& =  \mathcal{E}^{RHF}(f_1,\dots,f_i,\dots,f_{s_0}) + 4(2\ell_i+1)\delta\operatorname{Re}\form{h}{H_{\ell_i}}{f_i}  \\
&\quad + 2(2\ell_i+1)\delta^2 \Big( \form{h}{H_{\ell_i}^{(i)}}{h} - \lambda \form{f_i}{H_{\ell_i}}{f_i} + \operatorname{Re}\form{h \otimes h}{P_i}{f_i \otimes f_i}  \\
& \quad + \form{f_i \otimes h + h \otimes f_i}{P_i}{f_i \otimes h} \Big)
 + \mathcal{O}(\delta^3) 
\end{aligned} \label{decomp2}
\end{equation}
for $\delta \to 0$.
\end{lemma}

\begin{proof}[Proof of Theorem~\ref{thm:existence}]
First, we give a proof of the existence of a minimizer for the
relaxed minimization problem \eqref{generalized minimization problem},
which proceeds the same way as in the paper of Lieb and Simon \cite{LS1977}.
$\mathcal{E}^{RHF}(g_1,\dots,g_{s_0})$ is bounded below independently of
$g_1,\dots,g_{s_0} \in H^1_0(\mathbb{R}_+)$ with $\|g_i\|\leq1$, see
Lemma~\ref{lm:technical}~(ii).
Thus, let $g_1^{(n)},\dots,g_{s_0}^{(n)}$ be a minimizing sequence for
the relaxed minimization problem \eqref{generalized minimization problem}.
Again by Lemma~\ref{lm:technical}~(ii), $(g_j^{(n)})_{n \in \mathbb{N}}$, $j=1,\dots,s_0$,
is bounded in $H^1_0(\mathbb{R}_+)$.
Hence, there exist weakly-$H_0^1(\mathbb{R}_+)$ convergent subsequences
$g_j^{(n)} \rightharpoonup g_j$ $(n \to \infty)$.
Fix $i \in \{1,\dots,s_0\}$. Without loss of generality we may assume
that $g_1,\dots,g_{k_i}$ are all functions $g_j$ with $\ell_j=\ell_i$.
The matrix $M:=(\langle g_j, g_k \rangle)_{j,k=1,\dots,k_i}$ is hermitian
and obeys $0 \leq M\leq 1$ (c.f.~\cite[Lemma~2.2]{LS1977}), so there exists
a unitary $k_i \times k_i$ matrix $U$ with the property $U^* M U = D$,
where $D$ is a diagonal matrix with eigenvalues in $[0,1]$.
If we define $f_j = \sum_{k=1}^{k_i} u_{kj}g_k$, $j=1,\dots,k_i$,
then $\langle f_j, f_k \rangle = \lambda_{j}\delta_{jk}$, $0 \leq \lambda_j \leq 1$.
It is easy to see that $\mathcal{E}^{RHF}$ is invariant under such transformations.
Thus, transforming each subspace of functions with equal angular momentum
quantum numbers in this way, we obtain functions $f_1,\dots,f_{s_0}$ with
$\langle f_i, f_j \rangle =0$, if $\ell_i=\ell_j$, $i \neq j$,
$\norm{f_i} \leq 1$ for all $i$.
Furthermore, $f_1,\dots,f_{s_0}$ minimize $\mathcal{E}^{RHF}$, because
\begin{align*}
\tilde E(N,Z)
& \leq  \mathcal{E}^{RHF}(f_1,\dots,f_{s_0}) = \mathcal{E}^{RHF}(g_1,\dots,g_{s_0}) \\
& \leq  \liminf_{n \to \infty} \mathcal{E}^{RHF}(g_1^{(n)},\dots,g_{s_0}^{(n)})
= \tilde E(N,Z),
\end{align*}
where we used Lemma~\ref{lm:technical}~(v).
By further transformations we can achieve that $f_1,\dots,f_{s_0}$
are eigenfunctions of the operators $H_{\ell_i}$.

(i) Let $f_i\neq 0$ and assume that $\varepsilon_i > 0$. Then, by \eqref{decomp2}
with $\lambda=0$ and $h=f_i$, the energy decreases if we decrease the norm
of $f_i$. Let $\varepsilon_i <0$ and assume that $\norm{f_i}<1$. Then, the energy
is decreased by increasing the norm of $f_i$.

(ii) We prove the following more general statement:
Let $0 \leq \mu \leq 1$ and let \linebreak $Z \geq N-1-(1-\mu)(4\ell_i+1)$,
then $\mu \leq \|f_i\|^2 \leq 1$.

There is nothing to prove in the case $\mu=0$. Therefore, let $\mu >0$
and assume that $\|f_i\|^2 < \mu$.
We show that there exists $h \in H^1_0(\mathbb{R}_+)$ with $h \perp f_j$,
if $\ell_j=\ell_i$, such that
$$
\mathcal{E}^{RHF}(f_1,\dots, f_i+\delta h, \dots, f_{s_0})
< \mathcal{E}^{RHF}(f_1,\dots, f_i, \dots, f_{s_0})
$$
for small $\delta\neq 0$, which contradicts the minimization property of
 $f_1,\dots,f_{s_0}$.
The dependence of the left-hand side on $h \in H^1_0(\mathbb{R}_+)$ is given by
\eqref{decomp2} with $\lambda=0$. The factor of $\delta$ in \eqref{decomp2}
vanishes since $f_1,\dots,f_{s_0}$ is a minimizer. Therefore, it is sufficient
to show that there exist infinitely many normalized functions $h \in H^1_0(\mathbb{R}_+)$
with disjoint supports, such that
the factor of $\delta^2$ in \eqref{decomp2}
\begin{equation}
\label{delta2term}
\form{h}{H_{\ell_i}^{(i)}}{h} + \form{f_i \otimes h}{P_i}{f_i \otimes h}
+ \form{f_i \otimes h}{P_i}{h \otimes f_i} + \operatorname{Re}
\form{h \otimes h}{P_i}{f_i \otimes f_i}
\end{equation}
is negative.
We may drop the $\operatorname{Re}$-term because it becomes non-positive
upon a suitable choice of the phase of $h$.
Let $J \in C_0^\infty(\mathbb{R}_+)$, $\operatorname{supp}(J) \subset [1,2]$, $\norm{J} =1$. Furthermore, we define $J_R(r):= R^{-1/2} J(r/R)$ for $R>0$, then $\operatorname{supp}(J_R) \subset [R,2R]$, $\norm{J_R}=1$, $J_R \in C_0^\infty(\mathbb{R}_+)$.
Using $U(r) \leq r^{-1} \sum_{j=1}^{s_0} (2\ell_j+1)$ and $K_\ell \geq 0$ (Lemma~\ref{lm:technical}), we see that
\begin{equation}
\label{eqn inf}
\form{J_R}{H_{\ell_i}^{(i)}}{J_R}
 \leq  \form{J_R}{ -\partial_r^2 + \frac{\ell_i(\ell_i+1)}{r^2}
- \frac{Z}{r} + \frac{N-2(2\ell_i+1)}{r}}{J_R}.
\end{equation}
This inequality combined with the estimate for $P_i$ in Lemma~\ref{lm:decomp}
allows us to estimate \eqref{delta2term} with the choice $h=J_R$
\begin{gather*}
\form{J_R}{H_{\ell_i}^{(i)}}{J_R}
 \leq  \frac{1}{R^2} \form{J}{-\partial_r^2
 + \frac{\ell_i(\ell_i+1)}{r^2}}{J}
 - \frac{(4\ell_i +1)\mu}{R} \form{J}{\frac{1}{r}}{J}, \\
\form{f_i \otimes J_R}{P_i}{f_i \otimes J_R}
 \leq  \frac{(4\ell_i +1)
 \norm{f_i}^2}{R} \form{J}{\frac{1}{r}}{J}, \\
\form{f_i \otimes J_R}{P_i}{J_R \otimes f_i}
=  o\big(\frac{1}{R}\big)
\end{gather*}
for $R \to \infty$.
The sum of the three terms on the right-hand side becomes negative
for $R$ large enough, because $\norm{f_i}^2 < \mu$, by assumption. This proves (ii).

(iii) It suffices to show that $\varepsilon_j<0$, $j=1,\dots,s_0$, see (i) and (ii).
Assume that $\varepsilon_i=0$.
We show that there exists $h\in H^1_0(\mathbb{R}_+)$, $\norm{h}=1$, $h \perp f_j$,
if $\ell_i = \ell_j$, so that
$$
\mathcal{E}^{RHF}(f_1,\dots,\frac{f_i + \delta h}{\sqrt{1 + \delta^2}}, \dots, f_{s_0})
< \mathcal{E}^{RHF}(f_1,\dots, f_i, \dots, f_{s_0})
$$
for small $\delta \neq 0$. Again, the dependence on $h$ of the left-hand
side is given by \eqref{decomp2} with $\lambda=1$. Since $\varepsilon_i=0$, it
suffices to show that the factor of $\delta^2$, which is the same as
in \eqref{delta2term}, can be made negative by suitable choices of $h$.
This can be done choosing the same scaled functions as in (ii),
but now using $Z > N-1$ instead of $\norm{f_i}^2 < \mu$.
\end{proof}

\noindent\textbf{Remark.}
The crucial point in the proof of Theorem~\ref{thm:existence}~(ii)
for the case $Z=N-1$ is the fact that each radial function corresponds
 to at least two electrons (due to the closed shell condition).
Under the assumption that one of the minimizing radial functions
obeys $\|f_i\| < 1$, the attractive Coulomb interaction of the nucleus allows
one to lower the energy by a suitable variation of the radial function $f_i$.
This yields a contradiction, and the existence of a normalized minimizer
can be proved even in the case $Z=N-1$. Contrarily, the analogous estimates
for the general Hartree-Fock functional, where the single electrons are
independent, do not yield a contradiction. As mentioned in the introduction,
the question whether or not there exists a normalized GHF minimizer for
the case $Z=N-1$ is still open.

\section{Other Restricted Hartree-Fock Functionals}\label{sec:applications}

Theorem~\ref{thm:existence} can be readily generalized to other restricted
Hartree-Fock functionals which meet similar conditions as described in
the remark after the proof of Theorem~\ref{thm:existence}.
In this section we present analogous results for a spin-restricted Hartree-Fock
functional as well as for a so-called UHF functional.

The \emph{spin-restricted Hartree-Fock (SRHF)} model is frequently
used for atoms with an even number of electrons \cite{Cances2003}.
It emerges from the RHF model in Section~\ref{sec:closed shell} by
dropping the prescribed angular momentum quantum numbers.
More precisely, for an atom with atomic number $Z$ and $N=2n$ we
impose the following form on the one-electron orbitals
$$
\varphi_{i\sigma}(\mathbf{x},\mu) = \varphi_i(\mathbf{x})\delta_{\sigma\mu}, \quad
 i=1,\dots,n, \ \sigma = \pm 1,
$$
where $\varphi_i \in H^1(\mathbb{R}^3)$ and
$\langle \varphi_i, \varphi_j \rangle := \int_{\mathbb{R}^3} \overline{\varphi_i}\varphi_j \,d\mathbf{x}
= \delta_{ij}$. Then the restricted Hartree-Fock functional reads
\begin{equation}
\begin{aligned}
&\mathcal{E}^{SRHF}(\varphi_1,\dots,\varphi_n) \\
&= 2 \sum_{i=1}^n \int |\nabla \varphi_i(\mathbf{x})|^2
  - \frac{Z}{|\mathbf{x}|}|\varphi_i(\mathbf{x})|^2  \,d\mathbf{x}
  + \frac{1}{2} \int \!\!\!\int 4 \frac{\rho(\mathbf{x})
  \rho(\mathbf{y})}{|\mathbf{x}-\mathbf{y}|}
  - 2 \frac{|\tau(\mathbf{x},\mathbf{y})|^2}{|\mathbf{x}-\mathbf{y}|}
 \,d\mathbf{x}\,d\mathbf{y}.
\end{aligned} \label{srhf energy}
\end{equation}
Here the electronic density matrix and the electronic density are given by
$$
\tau(\mathbf{x},\mathbf{y}) = \sum_{i=1}^n \varphi_i(\mathbf{x})
\overline{\varphi_i(\mathbf{y})}, \quad \rho(\mathbf{x})
= \sum_{i=1}^n |\varphi_i(\mathbf{x})|^2.
$$
The corresponding Fock operator is given by
$$
H = -\Delta - \frac{Z}{|\mathbf{x}|}
+ 2 \int \frac{\rho(\mathbf{y})}{|\mathbf{x}-\mathbf{y}|} \,dy - K,
$$
where $(K\varphi)(\mathbf{x})
:=\int \frac{\tau(\mathbf{x},\mathbf{y})\varphi(\mathbf{y})}{|\mathbf{x}-\mathbf{y}|}
 \,d\mathbf{y}$.
Using similar ideas as in the proof of Theorem~\ref{thm:existence} the
following existence theorem holds true for the spin-restricted Hartree-Fock
functional.

\begin{theorem}[Existence of a SRHF minimizer]
\label{thm:existence spin}
Let $Z>0$ and $N=2n$. Then, there exist functions $\varphi_1,\dots,\varphi_n \in H^1(\mathbb{R}^3)$,
which minimize the SRHF functional \eqref{srhf energy} under the constraints
\begin{gather*}
\sprod{\varphi_i}{\varphi_j}  =  0 \quad \text{if } i \neq j, \\
\norm{\varphi_i}  \leq  1 \quad \text{for all } i.
\end{gather*}
Moreover, $\varphi_i \in D(H)=H^2(\mathbb{R}^3)$, $H \varphi_i = \varepsilon_i \varphi_i$, and:
\begin{itemize}
\item[(i)] Either $\varepsilon_i \leq 0$ or $\varphi_i=0$. $\varepsilon_i < 0$ implies
$\norm{\varphi_i} = 1$.
\item[(ii)]
If $Z>N-2$, then $\varphi_i \neq 0$ for all $i=1,\dots,n$. \\
If $Z\geq N-1$, then $\| \varphi_i \|=1$ for all $i=1\dots,n$.
\item[(iii)] If $Z>N-1$, then $\varepsilon_i<0$ and $\norm{\varphi_i}=1$ for all $i=1,\dots,n$.
\end{itemize}
\end{theorem}

\noindent\textbf{Remark.}
For this spin-restricted Hartree-Fock functional the minimizer exists
for all $Z \geq N-1$. Again we do not know whether or not $\varepsilon_j$ are
the $n$ lowest eigenvalues of $H$, although there seem to be no numerical
counterexamples \cite{Cances2003}.

The second generalization of Theorem~\ref{thm:existence} concerns the UHF functional.
Here we continue assuming that the electrons are in product states of space
and spin but we drop the condition that the spatial wave functions for
spin up resp.~spin down electrons are equal in each shell with fixed
angular momentum quantum numbers. More precisely, we consider electrons
that are in states of the form
\begin{gather}
\label{restr form2 filled}
\varphi_{jm\uparrow}(\mathbf{x},\mu)
=  \frac{f_j^\alpha(|\mathbf{x}|)}{|\mathbf{x}|} Y_{\ell_j^\alpha m}(\mathbf{x})
 \delta_{\mu,+1}, \quad j=1,\dots,s_0^\alpha, \;
 m = -\ell_j^\alpha,\dots,+\ell_j^\alpha, \\
\label{restr form2 half-filled}
\varphi_{jm\downarrow}(\mathbf{x},\mu)
=  \frac{f_j^\beta(|\mathbf{x}|)}{|\mathbf{x}|} Y_{\ell_j^\beta m}(\mathbf{x})
 \delta_{\mu,-1}, \quad j=1,\dots,s_0^\beta, \;
 m = -\ell_j^\beta,\dots,+\ell_j^\beta,
\end{gather}
where $s_0^\alpha, s_0^\beta \in \mathbb{N}_0$,
$\ell_1^\alpha,\dots,\ell_{s_0^\alpha}^\alpha,\ell_1^\beta,
\dots,\ell_{s_0^\beta}^\beta \in \mathbb{N}_0$, and for all
$\nu \in \{\alpha,\beta\}$, $i,j \in \{1,\dots,s_0^\nu\}$
$$
f_i^\nu \in H^1_0(\mathbb{R}_+), \quad
\langle f_i^\nu, f_j^\nu \rangle = \delta_{ij},\quad
 \text{if } \ell_i^\nu = \ell_j^\nu.
$$
The corresponding Hartree-Fock functional, which is called
\emph{unrestricted Hartree-Fock (UHF)} functional, takes the form
\begin{equation}
\begin{aligned}
&\mathcal{E}^{UHF}(f_1^\alpha,\dots,f_{s_0^\alpha}^\alpha;f_1^\beta,
 \dots,f_{s_0^\beta}^\beta) \\
& =  \sum_{\nu \in \{\alpha,\beta\}} \sum_{j=1}^{s_0^\nu} (2\ell_j^\nu + 1)
 \form{f_j^\nu}{ -\partial_r^2 + \frac{\ell_j^\nu(\ell_j^\nu+1)}{r^2}
 - \frac{Z}{r}}{f_j^\nu}  \\
&\quad + \frac{1}{2} \sum_{\nu \in \{\alpha,\beta\}} \sum_{j,k=1}^{s_0^\nu}
 \left( D[f_j^\nu,f_k^\nu]- E[f_j^\nu,f_k^\nu]\right)
 + \sum_{j=1}^{s_0^\alpha}\sum_{k=1}^{s_0^\beta} D[f_j^\alpha,f_k^\beta].
\end{aligned} \label{uhf energy}
\end{equation}
Here we use the shorthand notation
\begin{gather*}
D[f_j^\nu,f_k^\mu]  :=  (2\ell_j^\nu+1)(2\ell_k^\mu+1)\form{f_j^\nu
  \otimes f_k^\mu}{\frac{1}{r_>}}{f_j^\nu \otimes f_k^\mu}, \\
E[f_j^\nu,f_k^\mu]  :=  (2\ell_j^\nu+1)(2\ell_k^\mu+1)\form{f_j^\nu
  \otimes f_k^\mu}{U_{\ell_j^\nu \ell_k^\mu}}{f_k^\mu \otimes f_j^\nu}.
\end{gather*}
Given $\nu\in\{\alpha,\beta\}$ and $\ell \in \mathbb{N}_0$ we introduce Fock operators
\[
H_\ell^\nu  :=  -\partial_r^2 + \ell(\ell+1)r^{-2} - Zr^{-1} + U - K^\nu_\ell,
\]
where
\begin{gather*}
(Uf)(r)  =  \sum_{\kappa\in\{\alpha,\beta\}}\sum_{j=1}^{s_0^\kappa}
(2\ell_j^\kappa  + 1) \int_{\mathbb{R}_+} \frac{|f_j^\kappa(s)|^2}{\max\{r,s\}}\,ds f(r), \\
(K_\ell^\nu f)(r)  =  \sum_{j=1}^{s_0^\nu} (2\ell_j^\nu + 1)f_j^\nu(r)
 \int_{\mathbb{R}_+} \overline{f_j^\nu(s)} U_{\ell\ell_j^\nu}(r,s) f(s) \,ds
\end{gather*}
for $f\in L^2(\mathbb{R}_+)$. Again these operators depend on the functions
$f_1^\alpha,\dots,f_{s_0^\beta}^\beta$.
Using the same methods as in the proof of Theorem~\ref{thm:existence},
the following existence theorem can be proved.

\begin{theorem}[Existence of a UHF minimizer] \label{thm:existence general}
Let $s_0^\alpha, s_0^\beta \in \mathbb{N}_0$, 
$\ell_1^\alpha,\dots,\ell_{s_0^\alpha}^\alpha,\ell_1^\beta,\\ \dots,
\ell_{s_0^\beta}^\beta \in \mathbb{N}_0$, and $Z>0$. Then, there exist functions
$f_1^\alpha,\dots,f_{s_0^\alpha}^\alpha,f_1^\beta,\dots,f_{s_0^\beta}^\beta 
\in H^1_0(\mathbb{R}_+)$, which minimize the UHF functional \eqref{uhf energy} 
under the constraints: for all $\nu\in\{\alpha,\beta\}$ and 
$i,j \in \{1,\dots,s_0^\nu\}$
\begin{gather*}
\langle f_i^\nu, f_j^\nu \rangle  
=  0 \quad \text{if } \ell_i^\nu = \ell_j^\nu, \; i \neq j, \\
\norm{f_i^\nu}  \leq  1.
\end{gather*}
Moreover, $f_i^\nu \in D(H_{\ell_i^\nu}^\nu)$, 
$H_{\ell_{i}^\nu}^\nu f_i^\nu = \varepsilon^\nu_i f_i^\nu$.
\begin{itemize}
\item[(i)] Either $\varepsilon_i^\nu \leq 0$ or $f_i^\nu =0$. 
 $\varepsilon_i^\nu <0$ implies $\|f_i^\nu\| =1$.
\item[(ii)]
If $Z > N-(2\ell_i^\nu+1)$, then $f_i^\nu \neq 0$. \\ 
If $Z\geq N-1$ and $\ell_i^\nu \neq 0$, then $\| f_i^\nu \| =1$.
\item[(iii)] If $Z>N-1$, then $\varepsilon_i^\nu <0$ and $\norm{f_i^\nu}=1$ 
for all $\nu \in \{\alpha,\beta\}$, $i=1,\dots,s_0^\nu$.
\end{itemize}
\end{theorem}

\noindent\textbf{Remarks}.
(a) We do not know, except for the case where $\ell=0$, whether 
the occupied eigenvalues of the corresponding Fock operator are the 
lowest eigenvalues or whether there is a gap between occupied 
and unoccupied eigenvalues.

(b) In general, Theorem~\ref{thm:existence general} does not imply 
the existence of UHF minimizers in the case of $Z=N-1$. Nevertheless, 
in the special case where all spins point in the same direction 
(i.e.~the spinless case) the following existence result holds true.


\begin{corollary}[UHF minimizers in the case $Z=N-1$]
\label{existence of negative ions}
Let $s_0^\alpha \in \mathbb{N}$, $s_0^\beta=0$, and let $\ell_1^\alpha=0$, 
$\ell_2^\alpha,\dots,\ell_{s_0^\alpha}^\alpha>0$ with
$$
s_0^\alpha < 2 + \sum_{i=2}^{s_0^\alpha} 
\Big( \frac{\ell_i^\alpha}{\ell_i^\alpha+1} \Big)^2.
$$
If $Z=\sum_{i=2}^{s_0^\alpha} (2\ell_i^\alpha+1)$ and $N= Z+1$, 
then the UHF functional \eqref{uhf energy} has a minimizer under the
 constraints $\langle f_i^\alpha, f^\alpha_j \rangle = \delta_{ij}$ 
for all $i,j = 1,\dots, s_0^\alpha$ with $\ell_i=\ell_j$.
\end{corollary}

\noindent\textbf{Remark.} 
The condition of Corollary~\ref{existence of negative ions} 
always holds in the case of two shells $s_0^\alpha=2$, $\ell_1^\alpha=0$, 
$\ell_2^\alpha>0$.

\begin{proof}[Proof of Corollary~\ref{existence of negative ions}]
Theorem~\ref{thm:existence general} yields the existence of 
$f_1^\alpha,\dots, f_{s_0^\alpha}^\alpha \in H^1_0(\mathbb{R}_+)$, which minimize 
\eqref{uhf energy} under the constraints 
$\langle f_i^\alpha, f_j^\alpha \rangle = 0$ if 
$\ell_i^\alpha = \ell_j^\alpha$ and $i \neq j$, $\| f_i^\alpha \| \leq 1$ 
for all $i$. Clearly, $\| f_2^\alpha \|=\dots=\| f_{s_0^\alpha}^\alpha\|=1$ by (ii). 
Observe that
\begin{equation}\label{cor:eqn1}
\mathcal{E}^{UHF}(f_1^\alpha,\dots,f_{s_0^\alpha}^\alpha) 
\leq \inf_{g \in H^1_0(\mathbb{R}_+), \,  \|g\| \leq 1}
 \mathcal{E}^{UHF}(g,0,\dots,0) = - \frac{Z^2}{4},
\end{equation}
and on the other hand
\begin{equation}
\begin{aligned}
\mathcal{E}^{UHF}(0,f_2^\alpha,\dots,f_{s_0^\alpha}^\alpha)
&\geq  - \frac{Z^2}{4} \sum_{i=2}^{s_0^\alpha} 
 \frac{2\ell_i^\alpha+1}{(\ell_i^\alpha+1)^2}\\
&= -\frac{Z^2}{4} \Big( s_0^\alpha-1 - \sum_{i=2}^{s_0^\alpha}
\Big(\frac{\ell_i^\alpha}{\ell_i^\alpha+1}\Big)^2 \Big)  
 >  -\frac{Z^2}{4},
\end{aligned}\label{cor:eqn2}
\end{equation}
where we dropped the electron--electron energy and estimated the remaining
terms by the hydrogen ground state energies in the first inequality,
and used the condition on $s_0^\alpha$ in the second inequality.
Assume that $\langle f_1^\alpha |H_0^\alpha | f_1^\alpha\rangle = 0$, then
 $$
\mathcal{E}^{UHF}(f_1^\alpha,\dots,f_{s_0^\alpha}^\alpha)
=\mathcal{E}^{UHF}(0,f_2^\alpha,\dots,f_{s_0^\alpha}^\alpha),
$$
because $\mathcal{E}^{UHF}(f_1^\alpha,\dots,f_{s_0^\alpha}^\alpha)
= \mathcal{E}^{UHF}(0,f_2^\alpha,\dots,f_{s_0^\alpha}^\alpha)
+ \langle f_1^\alpha|H_0^\alpha |f_1^\alpha \rangle$, which contradicts
 \eqref{cor:eqn1} and \eqref{cor:eqn2}.
Therefore, $\langle f_1^\alpha | H_0^\alpha | f_1^\alpha \rangle
= \varepsilon_1^\alpha \| f_1^\alpha \|^2 <0$, which implies $\varepsilon_1^\alpha < 0$
 and thus $\norm{f_1^\alpha}=1$.
\end{proof}

\section{Derivation of the closed shell energy functional}\label{app}

For the reader's convenience we give here a self-contained derivation 
of the restricted Hartree-Fock functional \eqref{rest energy}. 
For this purpose, we begin with a lemma that will be useful for 
the calculation of the electron--electron interaction energy.

Let $P_\ell$ denote the $\ell$-th Legendre polynomial. 
We remark that for $\hat{\mathbf{x}},\hat{\mathbf{y}} \in \mathbb{S}^2$ and
$\ell \in \mathbb{N}_0$ the addition theorem
\begin{equation}\label{LegProp}
\sum_{m=-\ell}^\ell Y_{\ell m}(\hat{\mathbf{x}}) \overline{Y_{\ell m}(\hat{\mathbf{y}})}
= \frac{2\ell +1}{4\pi} P_\ell(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}})
\end{equation}
holds, where $\hat{\mathbf{x}} \cdot \hat{\mathbf{y}}$ is the usual scalar
 product of two vectors in $\mathbb{R}^3$. In addition, we recall the following 
relationship between the Wigner 3j-symbols and the Legendre polynomials:
\begin{equation}
\begin{pmatrix} \ell_1 & \ell_2 & \ell_3 \\ 0 & 0 & 0 \end{pmatrix}^2
 = \frac{1}{2} \int_{-1}^1 P_{\ell_1}(x) P_{\ell_2}(x) P_{\ell_3}(x) \,dx.
\end{equation}

\begin{proposition}
\label{prop:representation formula}
Let $\ell, L \in \mathbb{N}_0$ and $M \in \mathbb{Z}$, $|M| \leq L$. 
Then for all $r,s>0$ and $\hat{\mathbf{x}} \in \mathbb{S}^2$:
\begin{equation}
\label{eqn general}
\frac{1}{4\pi}\int_{\mathbb{S}^2} \frac{P_\ell(\hat{\mathbf{x}} 
\cdot \hat{\mathbf{y}})Y_{LM}(\hat{\mathbf{y}})}{|r\hat{\mathbf{x}}
-s\hat{\mathbf{y}}|} \,d\sigma(\hat{\mathbf{y}}) 
=
Y_{LM}(\hat{\mathbf{x}}) \sum_{n=|L-\ell|}^{L+\ell} 
\begin{pmatrix} L & \ell & n \\ 0 & 0 & 0 \end{pmatrix}^2 
\frac{\min\{r,s\}^n}{\max\{r,s\}^{n+1}}.
\end{equation}
\end{proposition}

\noindent\textbf{Remark.}
An easy consequence of this proposition is that for all $\ell,\ell' \in \mathbb{N}_0$
\begin{equation}
\label{exchange kernel}
\frac{1}{(4\pi)^2} \int_{(\mathbb{S}^2)^2} 
\frac{P_{\ell}(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}})P_{\ell'}(\hat{\mathbf{x}} 
\cdot \hat{\mathbf{y}})}{|r\hat{\mathbf{x}}-s\hat{\mathbf{y}}|} 
\,d\sigma(\hat{\mathbf{x}},\hat{\mathbf{y}}) 
= \sum_{k=|\ell-\ell'|}^{\ell+\ell'} 
\begin{pmatrix} \ell & \ell' & k \\ 0 & 0 & 0 \end{pmatrix}^2 
\frac{\min\{r,s\}^k}{\max\{r,s\}^{k+1}}.
\end{equation}
This is seen by multiplying \eqref{eqn general} with 
$\overline{Y_{LM}(\hat{\mathbf{x}})}$, integrating over $\mathbb{S}^2$ 
with respect to $\hat{\mathbf{x}}$ and summing over $M=-L,\dots,L$.

\begin{proof}
Assume first that $r\neq s$.
For fixed $\hat{\mathbf{x}} \in \mathbb{S}^2$ the series expansion
$$
\frac{1}{|r\hat{\mathbf{x}}-s\hat{\mathbf{y}}|} 
= \frac{1}{r_>} \sum_{n=0}^\infty \Big( \frac{r_<}{r_>} \Big)^n 
P_n(\hat{\mathbf{x}}\cdot \hat{\mathbf{y}})
$$
converges pointwise for all $\hat{\mathbf{y}}\in \mathbb{S}^2$ and thus 
in $L^2(\mathbb{S}^2)$ because 
$\sum_{n=0}^N \Big( \frac{r_<}{r_>} \Big)^n P_n(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}})$
is bounded uniformly in $N$ and $\hat{\mathbf{y}}$. We get
\begin{align*}
\frac{P_\ell(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}})}{|r\hat{\mathbf{x}}-s\hat{\mathbf{y}}|}
& =  \frac{1}{r_>} \sum_{n=0}^\infty \Big( \frac{r_<}{r_>} \Big)^n 
 P_n(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}}) P_\ell(\hat{\mathbf{x}} 
\cdot \hat{\mathbf{y}}) \\
& =  \frac{1}{r_>} \sum_{n=0}^\infty \Big( \frac{r_<}{r_>} \Big)^n 
 \sum_{k=|\ell-n|}^{\ell+n} (2k+1) 
\begin{pmatrix} k & \ell & n \\ 0 & 0 & 0 \end{pmatrix}^2 
P_k(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}})
\end{align*}
where we used the addition theorem
$$
P_n(z)P_\ell(z) = \sum_{k=|\ell-n|}^{\ell+n} (2k+1) 
\begin{pmatrix} k & \ell & n \\ 0 & 0 & 0 \end{pmatrix}^2 P_k(z).
$$
The addition theorem \eqref{LegProp} allows us to compute
\begin{align*}
&\frac{1}{4\pi}\int_{\mathbb{S}^2} \frac{P_{\ell}(\hat{\mathbf{x}} 
 \cdot \hat{\mathbf{y}})Y_{LM}(\hat{\mathbf{y}})}{|r\hat{\mathbf{x}}
 -s\hat{\mathbf{y}}|} \,d\sigma(\hat{\mathbf{y}}) \\
& =   \frac{1}{r_>} \sum_{n=0}^\infty \left( \frac{r_<}{r_>} \right)^n 
 \sum_{k=|\ell-n|}^{\ell+n} 
 \begin{pmatrix} k & \ell & n \\ 0 & 0 & 0 \end{pmatrix}^2 
 \sum_{m=-k}^k Y_{km}(\hat{\mathbf{x}}) 
 \int_{\mathbb{S}^2} \overline{Y_{km}(\hat{\mathbf{y}})} Y_{LM}(\hat{\mathbf{y}}) 
 \,d\sigma(\hat{\mathbf{y}}) \\
& =   Y_{LM}(\hat{\mathbf{x}}) \sum_{n=0}^{\infty} 
 \begin{pmatrix} L & \ell & n \\ 0 & 0 & 0 \end{pmatrix}^2 
 \frac{\min\{r,s\}^n}{\max\{r,s\}^{n+1}}.
\end{align*}
The desired equation for $r \neq s$ now follows from the fact that the 
Wigner 3j-symbols vanish unless $|L-\ell| \leq n \leq L+\ell$. 
The case $r=s$ can be derived from the above result by choosing a sequence 
$r_n \downarrow s$. Clearly, 
$\frac{1}{|r_n \hat{\mathbf{x}}-s\hat{\mathbf{y}}|} \uparrow 
\frac{1}{|s\hat{\mathbf{x}}-s\hat{\mathbf{y}}|}$ for all 
$\hat{\mathbf{y}} \in \mathbb{S}^2 \setminus \{\hat{\mathbf{x}}\}$ and 
$\frac{1}{|\hat{\mathbf{x}}- \hat{\mathbf{y}}|}$ is integrable with respect 
to $\hat{\mathbf{y}} \in \mathbb{S}^2$. Hence Lebesgue's Dominated Convergence 
Theorem may be used to see that the formula is also true for $r=s$.
\end{proof}

Let us turn to the derivation of $\mathcal{E}^{RHF}$.
If $f_1,\dots,f_{s_0}$ are in $H^1_0(\mathbb{R}_+)$, then the functions 
$\varphi_{jm\sigma}$ defined by \eqref{restr form} are orthonormal 
in $L^2(\mathbb{R}^3;\mathbb{C}^2)$, and $\varphi_{jm\sigma} \in H^1(\mathbb{R}^3;\mathbb{C}^2)$ by Hardy's inequality
\begin{equation}\label{Hardy}
\int_{\mathbb{R}_+} \frac{|f(r)|^2}{r^2} \,dr \leq 4 \int_{\mathbb{R}_+} |f'(r)|^2 \,dr
\end{equation}
for $f \in H^1_0(\mathbb{R}_+)$.
Using the addition theorem \eqref{LegProp}, the corresponding density 
matrix $\tau$ and electronic density $\rho$ take the form
\begin{gather}
\tau(x,y)  =  \delta_{\mu_x \mu_y} \sum_{j=1}^{s_0} \frac{2\ell_j+1}{4\pi}  
\frac{f_j(|\mathbf{x}|)}{|\mathbf{x}|}
 \frac{\overline{f_j(|\mathbf{y}|)}}{|\mathbf{y}|} 
 P_{\ell_j}(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}}), \label{tau explicit} \\
\rho(x)  =  \sum_{j=1}^{s_0} \frac{2\ell_j+1}{4\pi} 
\frac{|f_j(|\mathbf{x}|)|^2}{|\mathbf{x}|^2}.
\end{gather}
Here we abbreviate $\hat{\mathbf{x}} := \mathbf{x}/ |\mathbf{x} |$ for all 
$0 \neq \mathbf{x} \in \mathbb{R}^3$.
If the general Hartree-Fock functional \eqref{HF functional} is evaluated 
at the functions $\varphi_{jm\sigma}$, the only term which is not trivially
 computed is the exchange term:
\begin{align*}
\iint \frac{|\tau(x,y)|^2}{|x-y|} \,dx\,dy 
& =  2 \sum_{j,k=1}^{s_0} \frac{(2\ell_j+1)(2\ell_k+1)}{(4\pi)^2} 
 \int_{(\mathbb{R}_+)^2} \,dr\,ds \overline{f_j(r)f_k(s)}f_k(r)f_j(s) \\
&\quad\times  \int_{(\mathbb{S}^2)^2} \,d\sigma(\hat{\mathbf{x}}, \hat{\mathbf{y}}) 
 \frac{P_{\ell_j}(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}}) 
 P_{\ell_k}(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}})}{|r\hat{\mathbf{x}} 
 - s \hat{\mathbf{y}}|}.
\end{align*}
Using \eqref{exchange kernel}, the form of \eqref{rest energy} follows 
from the choice
\begin{equation}\label{Uform}
U_{\ell \ell'}(r,s) = \sum_{k=|\ell-\ell'|}^{\ell+\ell'} 
\begin{pmatrix} \ell & \ell' & k \\ 0 & 0 & 0  \end{pmatrix}^2 
\frac{\min\{r,s\}^k}{\max\{r,s\}^{k+1}}.
\end{equation}

\section{Appendix}\label{sec:lm}

The appendix contains two lemmas on some technical properties 
of the functions $U_{\ell\ell'}$ as well as of the restricted Hartree-Fock 
functional and the radial Fock operators.

\begin{lemma}[Properties of $U_{\ell\ell'}$] \label{lm:U-ell}
Let $\ell,\ell' \in \mathbb{N}_0$, and $r,s >0$. Then the functions $U_{\ell\ell'}$ 
defined by \eqref{Uform} obey:
\begin{itemize}
\item[(U1)] $U_{\ell\ell'}(r,s) = U_{\ell' \ell}(r,s) = U_{\ell \ell'}(s,r)$,
\item[(U2)] $0 \leq U_{\ell \ell'}(r,s) \leq \max\{r,s\}^{-1},$
\item[(U3)] $ U_{\ell\ell}(r,s) \geq \frac{1}{2\ell +1}
\frac{1}{\max\{r,s\}}$,
\item[(U4)] For all $g \in H^1_0(\mathbb{R}_+)$ the integral kernels 
$g(r) U_{\ell\ell'}(r,s) \overline{g(s)}$ define non-neg\-ative 
Hilbert-Schmidt operators on $L^2(\mathbb{R}_+)$.
\end{itemize}
\end{lemma}

\begin{proof}
(U1) and (U3) are obvious from the explicit representation of 
$U_{\ell\ell'}(r,s)$ and
$$
\begin{pmatrix} \ell & \ell & 0 \\ 0 & 0 & 0 \end{pmatrix}^2 
= \frac{1}{2\ell+1}.
$$ 
(U2) The positivity of $U_{\ell \ell'}$ is clear, the upper bound can 
be proved using \eqref{exchange kernel}, \eqref{LegProp} and Cauchy-Schwarz:
\begin{align*}
&U_{\ell \ell'} (r,s)\\
& = \frac{1}{(4\pi)^2} \int_{(\mathbb{S}^2)^2} 
\frac{P_{\ell}(\hat{\mathbf{x}} \cdot \hat{\mathbf{y}})P_{\ell'}(\hat{\mathbf{x}} 
\cdot \hat{\mathbf{y}})}{|r\hat{\mathbf{x}}-s\hat{\mathbf{y}}|} 
\,d\sigma(\hat{\mathbf{x}},\hat{\mathbf{y}}) \\
& \leq  \frac{1}{(2\ell+1)(2\ell' +1)} \sum_{m=-\ell}^{\ell} 
 \sum_{m'=-\ell'}^{\ell'} \Big( \int_{(\mathbb{S}^2)^2} 
 \frac{|Y_{\ell m}(\hat{\mathbf{x}})|^2 |Y_{\ell' m'}
 (\hat{\mathbf{y}})|^2}{|r\hat{\mathbf{x}}-s\hat{\mathbf{y}}|} 
\,d\sigma(\hat{\mathbf{x}},\hat{\mathbf{y}}) \Big)^{1/2}  \\
& \times   \Big( \int_{(\mathbb{S}^2)^2} \frac{|Y_{\ell' m'}(\hat{\mathbf{x}})|^2 
 |Y_{\ell m}(\hat{\mathbf{y}})|^2}{|r\hat{\mathbf{x}}-s\hat{\mathbf{y}}|} 
 \,d\sigma(\hat{\mathbf{x}},\hat{\mathbf{y}}) \Big)^{1/2} \\
& \leq  \frac{1}{(4\pi)^2} \int_{(\mathbb{S}^2)^2} 
\frac{1}{|r\hat{\mathbf{x}}-s\hat{\mathbf{y}}|} 
\,d\sigma(\hat{\mathbf{x}},\hat{\mathbf{y}}) 
= \frac{1}{\max\{r,s\}},
\end{align*}
where we used $2ab \leq a^2 + b^2$ and \eqref{LegProp} 
in the last inequality.

(U4) The integral kernels $K(r,s) := g(r) U_{\ell\ell'}(r,s) \overline{g(s)}$ 
are in $L^2(\mathbb{R}_+^2)$ by (U2) and by Hardy's inequality \eqref{Hardy}, which 
shows that the corresponding integral operators are Hilbert-Schmidt. 
Moreover, let
$$
\varphi_m({\mathbf x},\mu) := \frac{g(|{\mathbf x}|)}{|{\mathbf x}|} Y_{\ell m}
({\mathbf x}) \delta_{\mu,+1}, \quad m=-\ell,\dots,\ell,
$$
and
$$
\tau(x,y)  :=  \sum_{m=-\ell}^\ell \varphi_m(x)\overline{\varphi_m(y)}
 = \delta_{\mu_x,+1} \delta_{\mu_y,+1} \frac{2\ell + 1}{4\pi}  
\frac{g(|\mathbf{x}|)}{|\mathbf{x}|}
\frac{\overline{g(|\mathbf{y}|)}}{|\mathbf{y}|} P_{\ell}(\hat{\mathbf{x}} 
\cdot \hat{\mathbf{y}}).
$$
Given $f \in L^2(\mathbb{R}_+)$, we define
$$
\varphi({\mathbf x},\mu) := \frac{f(|{\mathbf x}|)}{|{\mathbf x}|} Y_{\ell' 0}
({\mathbf x}) \delta_{\mu,+1},
$$
then
$$
\iint \frac{\overline{\varphi(x)}\tau(x,y)\varphi(y)}{|x-y|} \,dx\,dy 
= (2\ell + 1)\iint \overline{f(r)} K(r,s) f(s) \,dr\,ds.
$$
The last equality can be computed using \eqref{eqn general} and \eqref{Uform}.
Hence, the non-negativity of the integral operator corresponding to $K$ 
follows from the non-negativity of the term on the left-hand side.
\end{proof}

\begin{lemma}\label{lm:technical}
(i) For all $f \in H^1_0(\mathbb{R}_+)$ and $\varepsilon >0$: 
$ \sprod{f}{\frac{1}{r}f} \leq \varepsilon \norm{f'}^2 + \frac{1}{\varepsilon} \norm{f}^2$.

(ii) Let $s_0 \in \mathbb{N}$, $\ell_1,\dots,\ell_{s_0} \in \mathbb{N}_0$, $Z>0$, 
$f_1,\dots,f_{s_0} \in H^1_0(\mathbb{R}_+)$, and $\varepsilon>0$. Then
$$
\mathcal{E}^{RHF}(f_1,\dots,f_{s_0}) \geq 2 \sum_{j=1}^{s_0} (2\ell_j+1) 
\big[ (1-Z\varepsilon) \| f_j' \|^2 - \frac{Z}{\varepsilon} \norm{f_j}^2 \big].
$$

(iii) Let $s_0 \in\mathbb{N}$, $\ell_1,\dots, \ell_{s_0} \in \mathbb{N}_0$, and 
$f_1,\dots,f_{s_0} \in H_0^1(\mathbb{R}_+)$. Then for all $\ell \in \mathbb{N}_0$:
$$
0 \leq K_{\ell} \leq U \leq \sum_{k=1}^{s_0} (2\ell_k+1) (\|f_k'\|^2+\|f_k\|^2).
$$

(iv) Let $\ell,\ell' \in \mathbb{N}_0$. Then the following maps are weakly sequentially 
continuous on $H_0^1(\mathbb{R}_+)$ resp.~$H_0^1(\mathbb{R}_+) \times H_0^1(\mathbb{R}_+)$:
\begin{gather*}
f  \mapsto  \sprod{f}{\frac{1}{r}f}, \\
(f,g)  \mapsto  \form{f \otimes g}{\max\{r,s\}^{-1}}{f \otimes g}, \\
(f,g)  \mapsto  \form{f \otimes g}{U_{\ell \ell'}}{g \otimes f}.
\end{gather*}

(v) The functional $\mathcal{E}^{RHF}$ is weakly sequentially lower semicontinuous 
on the set $\times_{i=1}^{N} H_0^1(\mathbb{R}_+)$.
\end{lemma}

\begin{proof}
(i) and (iii) follow easily from the Cauchy-Schwarz and the Hardy 
inequalities \eqref{Hardy}, (U1), (U2), and (U4).
To prove (ii) fix $j,k \in \{1,\dots,s_0\}$. Using Cauchy-Schwarz, 
(U1) and (U2) we obtain
\begin{align*}
& \Big| \iint \overline{f_j(r) f_k(s)} U_{\ell_j \ell_k}(r,s) 
f_k(r) f_j(s) \,dr\,ds \Big| \\
& \leq  \Big( \iint |f_j(r)|^2 |f_k(s)|^2 U_{\ell_j \ell_k}(r,s) \,dr \,ds 
\Big)^{1/2} \\
&\quad\times \Big( \iint |f_k(r)|^2 |f_j(s)|^2 U_{\ell_j \ell_k}(r,s) \,dr \,ds 
\Big)^{1/2} \\
& =  \iint |f_j(r)|^2 |f_k(s)|^2 U_{\ell_j \ell_k}(r,s) \,dr \,ds \\
&\leq \iint \frac{|f_j(r)|^2 |f_k(s)|^2}{\max\{r,s\}}  \,dr \,ds.
\end{align*}
Therefore,
$$
\mathcal{E}^{RHF}(f_1,\dots,f_{s_0}) \geq 2 \sum_{j=1}^{s_0} (2\ell_j +1) 
\Big( \|f_j' \|^2 - Z \sprod{f_j}{\frac{1}{r} f_j} \Big).
$$
The claim now follows immediately from (i).

(iv) Let $f_n \rightharpoonup f$ weakly in $H^1_0(\mathbb{R}_+)$. 
Due to the Rellich-Kondrashov theorem, $f_n$ converges to $f$ uniformly 
in $\mathbb{R}_+$.
To prove the weak continuity of the Coulomb potential we first use
$$
| \sprod{f_n}{\frac{1}{r}f_n} - \sprod{f}{\frac{1}{r}f} | 
\leq | \sprod{f_n-f}{\frac{1}{r}f_n} | 
+| \sprod{\frac{1}{r}f}{f_n-f}| = (*) + (**).
$$
For $R>0$ we obtain using Cauchy--Schwarz and Hardy's inequality \eqref{Hardy}
\begin{align*}
(*) & \leq  \int_0^R \frac{|f_n(r)-f(r)| |f_n(r)|}{r} \,dr 
 + \frac{1}{R} \int_R^\infty |f_n(r)-f(r)| |f_n(r)| \,dr \\
& \leq  \Big( \int_0^R |f_n(r)-f(r)|^2 \,dr \Big)^{1/2} 
\Big( \int_0^\infty \frac{|f_n(r)|^2}{r^2} \,dr \Big)^{1/2} 
 + \frac{1}{R}\norm{f_n-f}\norm{f_n} \\
& \leq  2 \sqrt{R} \sup_{r \in (0,R)} \{|f_n(r)-f(r)|\} \norm{f_n'} 
+ \frac{1}{R} \big( \norm{f_n} + \norm{f} \big) \norm{f_n}.
\end{align*}
Since $\| f_n' \|$, $\| f \|,$ and $\| f_n\|$ are uniformly bounded in $n$, 
we can first choose $R$ large to make the second term small, then choose $n$ 
large to make the first term small. $(**)$ can be estimated analogously.
The weak continuity of the other maps can be seen with a similar decomposition 
argument as shown above for the Coulomb potential.

(v) Let $f_j^{(n)} \rightharpoonup f_j$ weakly in $H_0^1(\mathbb{R}_+)$ for $j=1,\dots,N$. 
Clearly,
$$
\form{f_j}{-\partial_r^2 + \frac{\ell_j(\ell_j+1)}{r^2}}{f_j} 
\leq \liminf_{n \to \infty} \form{f_j^{(n)}}{-\partial_r^2 
+ \frac{\ell_j(\ell_j+1)}{r^2}}{f_j^{(n)}},
$$
since $f_j^{(n)} \rightharpoonup f_j$ in $H^1_0(\mathbb{R}_+)$ implies 
$\partial_r f_j^{(n)} \rightharpoonup \partial_r f_j$ in $L^2(\mathbb{R}_+)$ 
for the first term, and using the lemma of Fatou for the second term. 
The remaining terms of $\mathcal{E}^{RHF}$ are weakly sequentially continuous 
as shown in (iv).
\end{proof}


\subsection*{Acknowledgments} 
The author wants to thank M.~Griesemer
for drawing his attention to the problem and for helpful discussions.
The author was supported by the Studienstiftung des Deutschen Volkes.


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\end{document}