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%****************************************************************
\section{Schr\"odinger equation}
%****************************************************************
In this memoir, we consider atomic and molecular quantum systems (i.e.~systems composed by nuclei and electrons) within the Born-Oppenheimer approximation \cite{Szabo, Helgaker}.
This means that we neglect the kinetic energy of the nuclei and treat the nuclear coordinates as parameters.
We therefore concentrate our attention on the electronic degrees of freedom.
Unless otherwise stated, atomic units are used throughout this memoir.
A chemical system is completely defined at a time $t$ by its electronic wave function $\Psi(\bX,t)$, solution of the time-dependent Schr\"odinger equation
\begin{equation}
\label{eq:time-dependent-schrodinger-equation}
i \pdv{\Psi(\bX,t)}{t} = \HOp \Psi(\bX,t),
\end{equation}
where $\HOp$ is the so-called Hamiltonian and $\bX = (\bx_1,\dots,\bx_n) = (\bs,\bR)$ is a composite coordinate vector gathering the spin coordinates $\bs = (s_1,\ldots,s_n)$ and spatial coordinates $\bR = (\br_1,\ldots,\br_n)$ of the $n$ electrons.
In the case of a stationary system, the time-independent Schr\"odinger equation reads
\begin{equation}
\label{eq:time-independent-schrodinger-equation}
\HOp \Psi(\bX,t) = E\,\Psi(\bX,t),
\end{equation}
where $E$ is the energy of the system and the non-relativistic Hamiltonian is explicitly given by
\begin{equation}
\label{eq:Hamiltonian}
\begin{split}
\HOp & = \TeOp + \VenOp + \VeeOp + \VnnOp
\\
& = - \sum_i^n \frac{\nabla^2_i}{2}
- \sum_i^n \sum_A^\nuc \frac{Z_A}{\abs{\brA - \bri}}
+ \sum_{i < j}^n \frac{1}{\abs{\bri - \brj}}
+ \sum_{A<B}^\nuc \frac{Z_A Z_B}{\abs{\brB - \brA}},
\end{split}
\end{equation}
where $\nabla_i^2$ is the Laplace operator associated with the $i$th electron, $\brA$ and $Z_A$ are the nuclear coordinates and charge of nucleus $A$.
The first term $\TeOp$ is the kinetic energy operator of the electrons, the next term $\VenOp$ corresponds to the Coulombic attraction between electrons and nuclei, while the last two terms corresponds to the interelectronic ($\VeeOp$) and internuclear ($\VnnOp$) Coulombic repulsions, respectively.
Note that the last term $\VnnOp$ is a constant as, within the Born-Oppenheimer approximation, it does not depend on the electronic coordinates, and will be omitted for the sake of clarity.
%****************************************************************
\section{Hartree-Fock approximation}
%****************************************************************
Within the Hartree-Fock (HF) approximation \cite{Szabo}, the electronic wave function $\PsiHF$ is written as a Slater determinant of $n$ spin orbitals
\begin{equation}
\label{eq:Psi-HF}
\PsiHF(\bx_1,\ldots,\bx_n) = \frac{1}{\sqrt{n!}}
\begin{vmatrix}
\sMO{1}(\bx_1) & \sMO{2}(\bx_1) & \ldots & \sMO{n}(\bx_1) \\
\sMO{1}(\bx_2) & \sMO{2}(\bx_2) & \ldots & \sMO{n}(\bx_2) \\
\vdots & \vdots & \ddots & \vdots \\
\sMO{1}(\bx_n) & \sMO{2}(\bx_n) & \ldots & \sMO{n}(\bx_n) \\
\end{vmatrix}.
\end{equation}
Each spin orbital $\sMO{i}(\bx)$ is a product of a spin part $\omega(s)$ and a spatial part $\MO{i}(\br)$ --- also known as a molecular orbital (MO) ---
\begin{equation}
\sMO{i}(\bx) = \omega(s) \MO{i}(\br),
\end{equation}
where
\begin{equation}
\omega(s) =
\begin{cases}
\alpha(s), & \text{for spin-up electrons,}
\\
\beta(s), & \text{for spin-down electrons.}
\end{cases}
\end{equation}
The spin orbitals form an orthonormal set, i.e.
\begin{equation}
\braket{\sMO{i}}{\sMO{j}} = \delta_{ij},
\end{equation}
where
\begin{equation}
\delta_{ij} =
\begin{cases}
1, & \text{if $i = j$},\\
0, & \text{otherwise},
\end{cases}
\end{equation}
is the Kronecker delta \cite{NISTbook}.
The HF energy is defined as
\begin{equation}
\EHF = \mel{\PsiHF}{\HOp}{\PsiHF},
\end{equation}
and yields the following expression:
\begin{equation}
\begin{split}
\EHF & = \sum_i^n \mel{\sMO{i}(\br_1)}{\HcOp}{\sMO{i}(\br_1)}
\\
& + \sum_{i<j}^n \qty[
\mel{\sMO{i}(\br_1)\sMO{j}(\br_2)}{\ree^{-1}}{\sMO{i}(\br_1)\sMO{j}(\br_2)}
- \mel{\sMO{i}(\br_1)\sMO{j}(\br_2)}{\ree^{-1}}{\sMO{j}(\br_1)\sMO{i}(\br_2)} ],
\end{split}
\end{equation}
where the so-called core Hamiltonian (i.e.~the one-electron part of the electronic Hamiltonian) is defined as
\begin{equation}
\HcOp = \TeOp + \VenOp.
\end{equation}
We define the Fock operator as
\begin{gather}
\FOp \sMO{i}(\br_1) = \MOev{i} \sMO{i}(\br_1),
\\
\FOp(\br_1) = \HcOp(\br_1) + \sum_i^n \qty[ \JOp_i(\br_1) - \KOp_i(\br_1) ],
\end{gather}
where $\JOp_i(\br_1)$ and $\KOp_i(\br_1)$ are the Coulomb and exchange operators respectively:
\begin{subequations}
\begin{align}
\JOp_i(\br_1) \sMO{j}(\br_1) & = \sMO{j}(\br_1) \int \sMO{i}(\br_2) \ree^{-1} \sMO{i}(\br_2) d\br_2,
\\
\KOp_i(\br_1) \sMO{j}(\br_1) & = \sMO{i}(\br_1) \int \sMO{i}(\br_2) \ree^{-1} \sMO{j}(\br_2) d\br_2.
\end{align}
\end{subequations}
In the following, we adopt the restricted HF (RHF) formalism which means that we assume that the spatial part of the spin orbital is independent of the spin state of the electron occupying this orbital \cite{Szabo}.
Moreover, unless otherwise stated, the systems treated here are closed-shell systems, i.e.~each MO is doubly occupied by one spin-up and one spin-down electron.
%****************************************************************
\subsection{Roothaan-Hall equations}
%****************************************************************
Within the LCAO approximation, we expand each MO as a linear combination of $N$ atomic orbitals (AOs), such as
\begin{equation}
\label{eq:LCAO}
\MO{i}(\br) = \sum_\mu \cMO{\mu}{i} \AO{\mu}(\br).
\end{equation}
In practice, the AOs $\AO{\mu}(\br)$ are usually chosen as cartesian Gaussian functions due to their computational convenience.
However, other choices (such a Slater functions) are possible depending on the type of systems and the target accuracy.
We will come back to this particular point later in this memoir.
In the AO basis, we have
\begin{align}
\label{eq:FAO}
\FkEl{\mu}{\nu} = \mel{\AO{\mu}}{\FOp}{\AO{\nu}}
\equiv \mel{\mu}{\FOp}{\nu}
& = \HcEl{\mu}{\nu} + \sum_{\lambda \sigma} \PEl{\lambda}{\sigma}
\qty[ \braket{\mu \lambda}{\nu \sigma} - \frac{1}{2} \braket{\mu\lambda}{\sigma \nu} ],
\end{align}
with
\begin{gather}
\HcEl{\mu}{\nu} = \mel{\mu}{\HcOp}{\nu},
\\
\braket{\mu \lambda}{\nu \sigma} = \iint \AO{\mu}(\br_1) \AO{\lambda}(\br_2) \ree^{-1} \AO{\nu}(\br_1) \AO{\sigma}(\br_2) d\br_1 d\br_2,
\end{gather}
and where the density matrix is defined as
\begin{equation}
\PEl{\mu}{\nu} = 2 \sum_i^\occ \cMO{\mu}{i} \cMO{\nu}{i}.
\end{equation}
The HF electronic energy of the system is then given by
\begin{equation}
\label{eq:E-elec}
\EHF = \sum_{\mu \nu} \PEl{\mu}{\nu} \HcEl{\mu}{\nu}
+ \frac{1}{2} \sum_{\mu \nu \lambda \sigma}^N \PEl{\mu}{\nu} \PEl{\lambda}{\sigma} G_{\mu \nu \lambda \sigma}.
\end{equation}
In matrix form, the Fock matrix $\FkMat$ can be decomposed as
\begin{equation}
\FkMat = \HcMat + \GMat,
\end{equation}
where
\begin{align}
\GEl{\mu}{\nu} & = \sum_{\lambda \sigma} \PEl{\lambda}{\sigma} \GEl{\mu\nu}{\lambda\sigma},
&
\GEl{\mu\nu}{\lambda\sigma} & = \braket{\mu \lambda}{\nu \sigma} - \frac{1}{2} \braket{\mu\lambda}{\sigma \nu}.
\end{align}
The stationnarity of the energy with respect to the coefficients $\cMO{\mu}{i}$ yields the Roothaan-Hall equations:
\begin{equation}
\label{eq:Roothaan-Hall}
\sum_\nu \FEl{\mu}{\nu} \cMO{\nu}{i} = \sum_\nu \SEl{\mu}{\nu} \cMO{\nu}{i} \MOev{i},
\end{equation}
or in matrix form
\begin{equation}
\label{eq:Roothaan-Hall-mat}
\FkMat \CMat = \SMat \CMat \MOevMat,
\end{equation}
where the elements of the overlap matrix $\SMat$ are given by
\begin{equation}
\SEl{\mu}{\nu} = \braket{\mu}{\nu}.
\end{equation}
The coefficient matrix $\bC$ gathers the MO coefficients $\cMO{\mu}{i}$, while the diagonal matrix $\MOevMat$ gathers the MO energies $\MOev{i}$.
We introduce the orthogonalization matrix $\bX$ such as
\begin{equation}
\XMat^\dag \SMat \XMat = \IdMat
\end{equation}
in order to work in an orthogonal AO basis (where $\IdMat$ is the identity matrix).
There are two main orthogonalisation methods, namely the L\"owdin orthogonalisation for which $\XMat = \SMat^{-1/2}$ and the canonical orthogonalisation for which $\XMat = \UMat \sMat^{-1/2}$ (where $\UMat$ and $\sMat$ are the eigenvectors of eigenvalues matrices of $\SMat$, respectively).
Nowadays, the usual procedure consists in performing a singular value decomposition (SVD) of the overlap matrix $\SMat$.
This procedure is efficient, numerically stable and allows to remove the linear dependencies which might be present in the AO basis.
Rotating the Fock matrix $\FMat$ into the orthogonal basis yields
\begin{equation}
\label{eq:Fprime}
\FkMat^{\prime} \CMat^{\prime} = \CMat^{\prime} \MOevMat,
\end{equation}
where
\begin{equation}
\FkMat^{\prime} = \XMat^{\dag} \FMat \XMat.
\end{equation}
The matrices $\CMat^{\prime}$ and $\MOevMat$ can be determined by a straightforward diagonalisation of Eq.~\eqref{eq:Fprime}, and the matrix $\CMat$ is obtained by back-transforming the eigenvectors in the original basis:
\begin{equation}
\CMat = \XMat \CMat^{\prime}.
\end{equation}
%****************************************************************
\subsection{Self-consistent field calculation}
%****************************************************************
In order to obtain the MO coefficients $\CMat$, one must diagonalise the Fock matrix $\FkMat$.
However, this matrix does depend on the MO coefficients itself.
Therefore, one must employ an iterative procedure called self-consistent field (SCF) method.
The SCF algorithm is described below:
\begin{enumerate}
\item Obtain an estimate of the density matrix $\PMat$.
\item Build the Fock matrix: $\FMat = \HcMat + \GMat$.
\item Transform the Fock matrix in the orthogonal matrix: $\FMat^{\prime} = \XMat^{\dag} \FMat \XMat$.
\item Diagonalize $\FMat^{\prime}$ to obtain $\CMat^{\prime}$ and $\MOevMat$.
\item Back-transform the MOs in the original basis: $\CMat = \XMat \CMat^{\prime}$.
\item Compute the new density matrix $\PMat = \CMat \CMat^{\dag}$, as well as the HF energy:
\begin{equation}
\EHF = \frac{1}{2} \Tr{\PMat \qty( \HcMat + \FkMat )}.
\end{equation}
\item Convergence test. If not satisfied, go back to $2.$
\end{enumerate}
Unfortunately, the HF method cannot be used to obtain the exact energy of the system even in the complete basis set (CBS) limit due to the approximate treatment of the electron-electron interaction.
Within the HF method, this interaction is averaged over all the electrons.
In other word, a given electron ``feels'' the averaged repulsion of the $n-1$ remaining electrons (mean-field approach).
We will see in the next section how one can go beyond the HF approximation.
%****************************************************************
\section{Post Hartree-Fock methods}
%****************************************************************
The correlation energy $\Ec$ is defined as the error in the HF approximation, i.e.~the energy difference between the exact energy and the energy calculated within the HF approximation:
\begin{equation}
\label{eq:Ec}
\Ec = E - \EHF.
\end{equation}
Thanks to the variational principle, $\Ec$ is always a negative quantity.
The purpose of post HF methods is to recover some or all of the correlation energy \cite{JensenBook, CramerBook}.
%****************************************************************
\subsection{Configuration interaction methods}
%****************************************************************
One of the most conceptually simple (albeit expensive) approach to recover a large fraction of the correlation energy is the configuration interaction (CI) method.
The general idea is to expand the wave function as a linear combination of ``excited'' determinants.
These excited determinants are built by promoting electrons from occupied to unoccupied (virtual) MOs usually based on the HF orbitals, i.e.
\begin{equation}
\label{eq:Psi-CI}
\PsiCI = \cCI{0}{} \PsiHF
+ \sum_i^\occ \sum_a^\virt \cCI{i}{a} \ExDet{i}{a}
+ \sum_{ij}^\occ \sum_{ab}^\virt \cCI{ij}{ab} \ExDet{ij}{ab}
+ \sum_{ijk}^\occ \sum_{abc}^\virt \cCI{ijk}{abc} \ExDet{ijk}{abc}
+ \ldots,
\end{equation}
where $\ExDet{i}{a}$, $\ExDet{ij}{ab}$ and $\ExDet{ijk}{abc}$ are singly-, doubly- and triply-excited determinants.
$\ExDet{ij}{ab}$ corresponds to the excitations of two electrons from the occupied spinorbitals $i$ and $j$ to virtual spinorbitals $a$ and $b$.
It is easy to show that the CI energy
\begin{equation}
\ECI = \mel{\PsiCI}{\HOp}{\PsiCI}
\end{equation}
is an upper bound to the exact energy of the system.
When all possible excitations are taken into account, the method is called full CI (FCI) and it recovers the entire correlation energy for a given basis set.
Albeit elegant, FCI is very expensive due to the exponential increase of the number of excited determinants.
For example, when only singles and doubles are taken into account, the method is called CISD.
It recovers an important chunk of the correlation.
However, it has the disadvantage to be size-inconsistent.
A method is said to be size-consistent if the correlation energy of two non-interaction systems is egal to twice the correlation energy of the isolated system.
Size-extensivity means that the correlation energy grows linearly with the system size.
%****************************************************************
\subsection{Density-functional theory}
%****************************************************************
Density-functional theory (DFT) is based on two theorems known as the Hohenberg-Kohn (HK) theorems \cite{Hohenberg64}, which states that it exists a non-interacting reference system with an electronic density $\rho(\br)$ equal to the real, interaction system.
The first theorem proves the existence of a one-to-one mapping between the electron density and the external potential, while the second HK theorem guarantees the existence of a variational principle for the ground-state electron density.
%****************************************************************
\subsection{Kohn-Sham equations}
%****************************************************************
Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory \cite{Kohn65}.
In analogy to the HF theory, the electrons are treated as independent particles moving in the average field of all others but now with exchange an correlation included by virtue of an ``exchange-correlation'' functional.
Following the work of Kohn and Sham \cite{Kohn65}, we introduce KS orbitals $\sMO{i}(\br)$, and the energy can be decomposed as
\begin{equation}
\EKS \qty[\rho(\br)] = \Ts \qty[\rho(\br)]
+ \Ene \qty[\rho(\br)]
+ J \qty[\rho(\br)]
+ \Exc \qty[\rho(\br)],
\end{equation}
where
\begin{equation}
\Ts \qty[\rho(\br)] = - \frac{1}{2} \sum_i^\occ \mel{\sMO{i}}{\nabla_i^2}{\sMO{i}}
\end{equation}
is the non-interacting kinetic energy,
\begin{equation}
\Ene \qty[\rho(\br)] = - \sum_A^\nuc \int \frac{Z_A \rho(\br)}{\abs{\brA - \br}} d\br
\end{equation}
is the electron-nucleus attraction energy,
\begin{equation}
J \qty[\rho(\br)] = \frac{1}{2} \iint \frac{\rho(\br_1) \rho(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
\end{equation}
is the classical electronic repulsion, and the one-electron density is
\begin{equation}
\rho(\br) = \sum_i^\occ \abs{\sMO{i}(\br)}^2.
\end{equation}
The exchange-correlation energy
\begin{equation}
\Exc \qty[\rho(\br)] = \qty{ T \qty[\rho(\br)] - \Ts \qty[\rho(\br)] } + \qty{ \Eee \qty[\rho(\br)] - J \qty[\rho(\br)]}
\end{equation}
is the sum of two terms: one coming from the difference between the exact kinetic energy $T\qty[\rho(\br)]$ and the non-interacting kinetic energy $\Ts \qty[\rho(\br)]$, and the other one coming from the difference between the exact interelectronic repulsion $\Eee \qty[\rho(\br)]$ and the classical Coulomb repulsion $J \qty[\rho(\br)]$.
Here, we will only consider the second term as the ``kinetic'' correlation energy is usually much smaller than its ``Coulomb'' counterpart.
Also, as it is usually done, we will split the exchange-correlation energy as a sum of an exchange and correlation components, i.e.
\begin{equation}
\Exc \qty[\rho(\br)] = \Ex \qty[\rho(\br)] + \Ec \qty[\rho(\br)].
\end{equation}
Similarly to the Roothaan-Hall equations, the condition of stationarity of the KS energy with respect to the electron density
\begin{equation}
\frac{\delta E \left[ \rho (\mathbf{r}) \right]}{\delta \rho (\mathbf{r})} = \mu
\end{equation}
(where $\mu$ is the chemical potential) yields the KS equations
\begin{equation}
\qty[ - \frac{\nabla_{\br}^2}{2}
- \sum_A^\nuc \frac{Z_A}{\abs{\br - \brA}}
+ \int \frac{\rho(\br^{\prime})}{\abs{\br - \br^{\prime}}} d\br^{\prime}
+ \fdv{\Exc\qty[\rho(\br)]}{\rho(\br)} ] \sMO{i}(\br)
= \MOev{i} \sMO{i}(\br),
\end{equation}
which can be re-written as
\begin{equation}
\FOp_\text{KS} \sMO{i}(\br) = \MOev{i} \sMO{i}(\br).
\end{equation}
These equations are solved iteratively, just like the HF equations, by expanding the KS MOs in a AO basis, yielding
\begin{equation}
\FkMat_\text{KS} \CMat = \SMat \CMat \MOevMat.
\end{equation}
%****************************************************************
\subsection{Exchange-correlation functionals}
%****************************************************************
Due to its moderate computational cost and its reasonable accuracy, KS DFT \cite{Hohenberg64, Kohn65} has become the workhorse of electronic structure calculations for atoms, molecules and solids \cite{ParrYang}.
To obtain accurate results within DFT, one only requires the exchange and correlation functionals, which can be classified in various families depending on their physical input quantities \cite{Becke14, Yu16}.
These various types of functionals are classified by the Jacob's ladder of DFT \cite{Perdew01, Pewdew05} (see Fig.~\ref{fig:Jacob-ladder}).
%%% FIGURE 1 %%%
\begin{figure}
\centering
\includegraphics[width=0.25\linewidth]{../Chapter1/fig/fig1}
\caption{
\label{fig:Jacob-ladder}
Jacob's ladder of DFT.
$\rho$, $x$, $\tau$ and $\Ex^\text{HF}$ are the electron density, the reduced gradient, the kinetic energy density, and the HF exchange energy, respectively.}
\end{figure}
%%%
\begin{itemize}
\item The local-density approximation (LDA) sits on the first rung of the Jacob's ladder and only uses as input the electron density $\rho$.
The oldest and probably most famous LDA functional is the Dirac exchange functional (D30) \cite{Dirac30} based on the uniform electron gas (UEG) \cite{WIREs16}.
Based on the work of Ceperley and Alder who used quantum Monte Carlo calculations (see below) to determine the correlation energy of the UEG with respect to the density \cite{Ceperley80},
Vosko, Wilk and Nusair (VWN) proposed a LDA correlation functional by fitting their data \cite{VWN80}.
\item The generalized-gradient approximation (GGA) corresponds to the second rung and adds the gradient of the electron density $\nabla \rho$ as an extra ingredient.
The well-known B88, G96, PW91 and PBE exchange functionals are examples of GGA exchange functionals \cite{B88, G96, PW91, PBE}.
Probably the most famous GGA correlation functional is LYP \cite{LYP}, which gave birth to the GGA exchange-correlation functional BLYP \cite{Becke88b} by combination with B88.
\item The third rung is composed by the so-called meta-GGA (MGGA) functionals \cite{Sala16} which uses, in addition to $\rho$ and $\nabla \rho$, the kinetic energy density
\begin{equation}
\tau = \sum_i^\occ \abs{\nabla\sMO{i} }^2.
\end{equation}
The M06-L functional from Zhao and Truhlar \cite{M06L}, the mBEEF functional from Wellendorff et al.~\cite{mBEEF} and the SCAN \cite{SCAN} and MS \cite{MS0,MS1_MS2} family of functionals from Sun et al.~are examples of widely-used MGGA functionals.
\item The fourth rung (hyper-GGAs or HGGAs) includes the widely-used hybrid functionals, introduced by Becke in 1993 \cite{Becke93}, which add a certain percentage of HF exchange.
Example of such functionals are B3LYP \cite{Becke93}, B3PW91 \cite{PW92, Becke93, Perdew96}, BH\&HLYP \cite{Becke93b} or PBE0 \cite{PBE0}.
Hybrids functionals are known for their accuracy in electronic structure theory.
However, they are more computationally expensive than LDA or GGA functionals due to the calculation of the costly HF exchange.
\item The fifth rung includes double hybrids and RPA-like functionals but we will not be discussing such types of functionals in the present memoir.
\end{itemize}
%****************************************************************
\section{Quantum Monte Carlo methods}
%****************************************************************
%-----------------------------------------------
\subsection{Variational Monte Carlo}
%-----------------------------------------------
In the VMC method, the expectation value of the Hamiltonian with respect to a trial wave function is obtained using a stochastic integration technique.
Within this approach a variational trial wave function $\PsiT(\bR)$ is introduced, and one then calculates its variational energy
\begin{equation}
\EVMC = \frac{\int \PsiT(\bR) \HOp \PsiT(\bR) d\bR}{\int \PsiT(\bR)^2 d\bR},
\end{equation}
using the Metropolis Monte Carlo method of integration \cite{Umrigar99}.
The resulting VMC energy is an upper bound to the exact ground-state energy, within the statistical Monte Carlo error.
Unfortunately, any resulting observables are biased by the form of the trial wave function, and the method is therefore only as good as the chosen $\PsiT$.
%--------------------------------------------
\subsection{Diffusion Monte Carlo}
%--------------------------------------------
DMC is a stochastic projector technique for solving the many-body Schr\"odinger equation \cite{Kalos74, Ceperley79, Reynolds82}.
Its starting point is the time-dependent Schr\"odinger equation in imaginary time
\begin{equation}
\label{eq:DMC}
-\pdv{\Psi(\bR,\tau)}{\tau} = (\HOp - S) \Psi(\bR,\tau).
\end{equation}
For $\tau \to \infty$, the steady-state solution of Eq.~\eqref{eq:DMC} for $S$ close to the ground-state energy is the ground-state $\Psi(\bR)$ \cite{Kolorenc11}.
DMC generates configurations distributed according to the product of the trial and exact ground-state wave functions.
If the trial wave function has the correct nodes, the DMC method yields the exact energy, within a statistical error that can be made arbitrarily small by increasing the number of Monte Carlo steps.
Thus, as in VMC, a high quality trial wave function is essential in order to achieve high accuracy \cite{Umrigar93, Huang97}.
%------------------------------------------
\subsection{Trial wave functions}
%------------------------------------------
Within QMC, trial wave functions are usually defined as \cite{Huang97, Drummond04, LopezRios12}
\begin{equation}
\label{eq:Psi-trial}
\PsiT(\bR) = e^{\Js(\bR)} \sum_I \cCI{I}{} \sdet_I^{\uparrow}(\br^{\uparrow}) \, \sdet_I^{\downarrow}(\br^{\downarrow}),
\end{equation}
where $D_I^{\sigma}$ are determinants of the spin-$\sigma$ electrons.
The fermionic nature of the wave function is imposed by a single- or multi-determinant expansion of Slater determinants made of HF or KS MOs.
$\Js(\bR)$ is called the Jastrow factor and $e^{\Js(\bR)}$ is a nodeless function.
Hence, the nodes of $\PsiT$ are completely determined by the determinantal part of the trial wave function.
%---------------------------------------------------
\subsection{Fixed-node approximation}
%---------------------------------------------------
Considering an antisymmetric (real) electronic wave function $\Psi(\bR)$, the nodal hypersurface (or simply ``nodes'') is a $(n\, D-1)$-dimensional manifold defined by the set of configuration points $\bN$ for which $\Psi(\bN)=0$.
The nodes divide the configuration space into nodal cells or domains which are either positive or negative depending on the sign of the electronic wave function in each of these domains.
In recent years, strong evidence has been gathered showing that, for the lowest state of any given symmetry, there is a single nodal hypersurface (up to all permutations) that divides configuration space into only two nodal domains (one positive and one negative) \cite{Ceperley91, Glauser92, Bressanini01, Bressanini05a, Bajdich05, Bressanini05b, Scott07, Mitas06, Mitas08, Bressanini08, Bressanini12}.
Except in some particular cases, electronic or more generally fermionic nodes are poorly understood due to their high dimensionality and complex topology \cite{Ceperley91, Bajdich05}.
The number of systems for which the exact nodes are known analytically is very limited \cite{Klein76, Bressanini05a, Bajdich05, Nodes15}.
The quality of fermion nodes is of prime importance in QMC calculations due to the fermion sign problem, which continues to preclude the application of in principle exact QMC methods to large systems.
The dependence of the DMC energy on the quality of $\PsiT$ is often significant in practice, and is due to the fixed-node approximation which segregates the walkers in regions defined by $\PsiT$ \cite{Ceperley91}.
The fixed-node error is only proportional to the square of the nodal displacement error, but it is uncontrolled and its accuracy difficult to assess \cite{Kwon98, Luchow07a, Luchow07b}.
The DMC method then finds the best energy \emph{for that chosen nodal surface}, providing an upper bound for the ground-state energy.
The exact ground-state energy is reached only if the nodal surface is exact.
Therefore, one of greatest challenge of QMC methods is to design a well-defined protocol to control the fixed-node error or, equivalently, to be able to build chemical meaningful nodal surfaces for any chemical system.

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\documentclass[aip,jcp,preprint,amsmath,amsfonts,amssymb,showkeys]{standalone}
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By solving the Schr\"odinger equation, one can predict some of the chemistry and most of the physics of a given system.
However, although this statement is true and philosophically important, it was realised many years ago that, for more than \textit{one} electron, it is usually far too difficult from the mathematical point of view to solve this mighty equation.
As Dirac pointed out,
\begin{quote}
\textit{``The aim of science is to make difficult things understandable in a simpler way.''}
\end{quote}
Consequently, it is essential to develop simple approximations that are accurate enough to have chemical and physical usefulness.
To do this, quantum chemists and physicists have developed a variety of simple models that, despite their simplicity, contain the key physics of more complicated and realistic systems.
A few examples are:
\begin{itemize}
\item the Born-Oppenheimer model: the motions of nuclei and electrons are independent;
\item the orbital model: electrons occupy orbitals and move independently of one another;
\item the local density model: the molecular electron density is built as an assembly of uniform electron gas densities.
\end{itemize}
Nowadays, all of these models are routinely applied in theoretical and/or computational studies.
Spherical models are another example.
One of the most popular starting points for modelling complex real life phenomenon by a highly simplified scientific model is the spherical geometry, and the most famous illustration of this is probably the so-called \textit{spherical cow} (Fig.~\ref{fig:cow}).
While appearing completely nonsensical to most people outside the scientific area, these spherical models can be extremely powerful for understanding, explaining and even predicting physical and chemical phenomena in a wide range of disciplines of physics and chemistry.
Besides, they offer unparalleled mathematical simplicity, while retaining much of the key physics.
An explicit example is the spherical model introduced by Haldane \cite{Haldane83} to explain the fractional quantum Hall effect (FQHE), for which Laughlin, St\"ormer and Tsui received the Nobel prize in physics.
This geometry has been instrumental in establishing the validity of the FQHE theory, and provides the cleanest proof for many properties.
In this chapter, we will show that the spherical geometry can be also useful to better understand the structure of the exact electronic wave function.
Almost ten years ago, following this idea, we undertook a comprehensive study of two electrons on the surface of a sphere of radius $R$ \cite{TEOAS09, Concentric10, Hook10}.
We used quantum chemistry electronic structure models ranging from HF to state-of-the-art explicitly correlated treatments, the last of which leads to near-exact wave functions and energies.
This helped us to understand not only the complicated relative motion of electrons, but also the errors inherent to each method.
It eventually led to the important discovery that the system composed of two electrons restricted to the surface of a $D$-sphere (where $D$ is the dimensionality of the surface of the sphere) is exactly solvable for a countable infinite set of values of $R$ \cite{QuasiExact09, ExSpherium10, QR12, ConcentricExact14}.
In other words, it means that the \textit{exact} solution of the Schr\"odinger equation can be obtained for certain ``magic'' values of the radius of the sphere \cite{QuasiExact09}.
This discovery propelled the two-electrons-on-a-sphere model (subsequently named spherium), into the exclusive family of exactly solvable two-electron models.
Moreover, after an exhaustive study of the ground state of two electrons confined by various external potentials \cite{EcLimit09, Ballium10}, we noticed that the correlation energy is weakly dependent on the external potential, and we conjectured that the behaviour of the two-electron correlation energy, in the limit of large dimension, is \textit{universal}!
The rigorous proof of this conjecture has been published in Ref.~\cite{EcProof10}.
In particular, we showed that the limiting correlation energy at high-density in helium and spherium are amazingly similar \cite{Frontiers10}.
However, while the closed-form expression of the limiting correlation energy has never been found for helium, the value for spherium is quite simple to obtain.
This shows the superiority of the spherical geometry approach and that it can be used in quantum chemistry to provide robust and trustworthy models for understanding, studying and explaining ``real world'' chemical systems.
In this chapter, we will summarise some of our key discoveries.
%%% FIG 1 %%%
\begin{figure}
\centering
\includegraphics[width=0.2\textwidth]{../Chapter2/fig/sphericalcow}
\caption{\label{fig:cow} A spherical cow.}
\end{figure}
%%% %%%

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%****************************************************************
\section{Quasi-exactly solvable models}
%****************************************************************
Quantum mechanical models for which it is possible to solve explicitly for a finite portion of the energy spectrum are said to be quasi-exactly solvable \cite{Ushveridze}.
They have ongoing value and are useful both for illuminating more complicated systems and for testing and developing theoretical approaches, such as DFT \cite{Hohenberg64, Kohn65, ParrYang} and explicitly-correlated methods \cite{Kutzelnigg85, Kutzelnigg91, Henderson04, Bokhan08}.
One of the most famous quasi-solvable model is the Hooke's law atom which consists of a pair of electrons, repelling Coulombically but trapped in a harmonic external potential with force constant $k$.
This system was first considered nearly 50 years ago by Kestner and Sinanoglu \cite{Kestner62}, solved analytically in 1989 for one particular $k$ value \cite{Kais89}, and later for a countably infinite set of $k$ values \cite{Taut93}.
A related system consists of two electrons trapped on the surface of a sphere of radius $R$.
This has been used by Berry and collaborators \cite{Ezra82, Ezra83, Ojha87, Hinde90} to understand both weakly and strongly correlated systems and to suggest an ``alternating'' version of Hund's rule \cite{Warner85}.
Seidl utilised this system to develop new correlation functionals \cite{Seidl00a, Seidl00b} within the adiabatic connection in DFT \cite{Seidl07b}.
As mentioned earlier, we will use the term ``spherium'' to describe this two-electron system.
In Ref.~\cite{TEOAS09}, we examined various schemes and described a method for obtaining near-exact estimates of the $^1S$ ground state energy of spherium for any given $R$.
Because the corresponding HF energies are also known exactly, this is now one of the most complete theoretical models for understanding electron correlation effects.
In this section, we consider the $D$-dimensional generalisation of this system in which the two electrons are trapped on a $D$-sphere of radius $R$.
We adopt the convention that a $D$-sphere is the surface of a ($D+1$)-dimensional ball.
Here, we show that the Schr\"odinger equation for the $^1S$ and the $^3P$ states can be solved exactly for a countably infinite set of $R$ values and that the resulting wave functions are polynomials in the interelectronic distance $\ree = \abs{\br_1-\br_2}$.
Other spin and angular momentum states can be addressed in the same way using the ansatz derived by Breit \cite{Breit30} and we will discuss these excited states later in this section \cite{ExSpherium10}.
We have also published dedicated studies of the 1D system (that we dubbed ringium) \cite{QR12, Ringium13, NatRing16}.
The case of two concentric spheres has also been considered in two separate publications \cite{Concentric10, ConcentricExact14}, as well as the extension to excitonic wave functions \cite{Exciton12}.
Finally, the nodal structures of these systems has been investigated in collaboration with Dario Bressanini \cite{Nodes15}.
%****************************************************************
\subsection{Singlet ground state}
%****************************************************************
The electronic Hamiltonian is
\begin{equation}
\HOp = - \frac{\nabla_1^2}{2} - \frac{\nabla_2^2}{2} + \frac{1}{\ree},
\end{equation}
and because each electron moves on a $D$-sphere, it is natural to adopt hyperspherical coordinates \cite{Louck60}.
For $^1S$ states, it can be then shown \cite{TEOAS09} that the wave function $S(\ree)$ satisfies the Schr{\"o}dinger equation
\begin{equation}
\label{eq:S-singlet}
\qty[ \frac{\ree^2}{4R^2} - 1 ] \dv[2]{S(\ree)}{\ree} + \qty[ \frac{(2D-1)\ree}{4R^2} - \frac{D-1}{\ree} ] \dv{S(\ree)}{\ree} + \frac{S(\ree)}{\ree} = E\,S(\ree).
\end{equation}
By introducing the dimensionless variable $x = \ree/2R$, this becomes a Heun equation \cite{Ronveaux} with singular points at $x = -1, 0, +1$.
Based on our previous work \cite{TEOAS09} and the known solutions of the Heun equation \cite{Polyanin}, we seek wave functions of the form
\begin{equation}
\label{eq:S_series}
S(\ree) = \sum_{k=0}^\infty s_k\,\ree^k,
\end{equation}
and substitution into \eqref{eq:S-singlet} yields the recurrence relation
\begin{equation}
\label{eq:recurrence-singlet}
s_{k+2} = \frac{ s_{k+1} + \qty[ k(k+2D-2) \frac{1}{4R^2} - E ] s_k }{(k+2)(k+D)},
\end{equation}
with the starting values
\begin{equation}
\{s_0,s_1\} = \begin{cases}
\qty{0,1}, & D = 1,
\\
\qty{1,1/(D-1)}, & D \ge 2.
\end{cases}
\end{equation}
Thus, the Kato cusp conditions \cite{Kato57} are
\begin{align}
\label{eq:cusp-circle}
S(0) & = 0,
&
\frac{S''(0)}{S'(0)} & = 1,
\end{align}
for electrons on a ring ($D=1$), i.e.~ringium, and
\begin{equation}
\label{eq:S-cusp}
\frac{S'(0)}{S(0)} = \frac{1}{D-1},
\end{equation}
in higher dimensions.
We note that the ``normal'' Kato value of 1/2 arises for $D=3$ --- a system we called glomium as the name of a 3-sphere is a glome --- suggesting that this may the most appropriate model for atomic or molecular systems.
We will return to this point below.
The wave function \eqref{eq:S_series} reduces to the polynomial
\begin{equation}
S_{n,m}(\ree) = \sum_{k=0}^n s_k\,\ree^k,
\end{equation}
(where $m$ the number of roots between $0$ and $2R$) if, and only if, $s_{n+1} = s_{n+2} = 0$.
Thus, the energy $E_{n,m}$ is a root of the polynomial equation $s_{n+1} = 0$ (where $\deg s_{n+1} = \lfloor (n+1)/2 \rfloor$) and the corresponding radius $R_{n,m}$ is found from \eqref{eq:recurrence-singlet} which yields
\begin{equation}
\label{eq:E_S}
R_{n,m}^2 E_{n,m} = \frac{n}{2}\qty(\frac{n}{2}+D-1).
\end{equation}
$S_{n,m}(\ree)$ is the exact wave function of the $m$-th excited state of $^1S$ symmetry for the radius $R_{n,m}$.
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\centering
\caption{\label{tab:lowest} Radius $R$, energy $E$ and wave function $S(\ree)$ or $T(\ree)$ of the first $^1S$ and $^3P$ polynomial solutions for two electrons on a $D$-sphere}
\centering
\begin{tabular}{ccccc}
\hline
State & $D$ & $2R$ & $E$ & $S(\ree)$ or $T(\ree)$ \\
\hline
\mr{4}{*}{$^1S$} & 1 & $\sqrt{6}$ & 2/3 & $\ree(1 + \ree/2)$ \\
& 2 & $\sqrt{3}$ & 1 & $1 + \ree$ \\
& 3 & $\sqrt{10}$ & 1/2 & $1 + \ree/2$ \\
& 4 & $\sqrt{21}$ & 1/3 & $1 + \ree/3$ \\
\hline
\mr{4}{*}{$^3P$} & 1 & $\sqrt{6}$ & 1/2 & $1 + \ree/2$ \\
& 2 & $\sqrt{15}$ & 1/3 & $1 + \ree/3$ \\
& 3 & $\sqrt{28}$ & 1/4 & $1 + \ree/4$ \\
& 4 & $\sqrt{45}$ & 1/5 & $1 + \ree/5$ \\
\hline
\end{tabular}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%
%****************************************************************
\subsection{Triplet excited state}
%****************************************************************
If we write the $^3P$ state wave function as \cite{Breit30}
\begin{equation}
^3\Psi = (\cos \theta_1 - \cos \theta_2)\,T(\ree),
\end{equation}
where $\theta_1$ and $\theta_2$ are the $D$-th hyperspherical angles of the two electrons \cite{Louck60}, the symmetric part satisfies the Schr{\"o}dinger equation
\begin{equation}
\label{eq:S-triplet}
\qty[ \frac{\ree^2}{4R^2} - 1 ] \dv[2]{T(\ree)}{\ree} + \qty[ \frac{(2D+1)\ree}{4R^2} - \frac{D+1}{\ree} ] \dv{T(\ree)}{\ree} + \frac{T(\ree)}{\ree} = E\,T(\ree),
\end{equation}
and the antisymmetric part provides an additional kinetic energy contribution $D/(2R^2)$.
Substituting the power series expansion
\begin{equation}
\label{eq:T_series}
T(\ree) = \sum_{k=0}^\infty t_k\,\ree^k
\end{equation}
into \eqref{eq:S-triplet} yields the recurrence relation
\begin{equation}
\label{eq:recurrence-triplet}
t_{k+2} = \frac{ t_{k+1} + \qty[ k(k+2D) \frac{1}{4R^2} - E ] t_k }{(k+2)(k+D+2)},
\end{equation}
with the starting values
\begin{equation}
\qty{t_0,t_1} = \qty{1, 1/(D+1)},
\end{equation}
yielding the cusp condition
\begin{equation}
\label{eq:T-cusp}
\frac{T'(0)}{T(0)} = \frac{1}{D+1}.
\end{equation}
The wave function \eqref{eq:T_series} reduces to the polynomial
\begin{equation}
T_{n,m}(\ree) = \sum_{k=0}^n t_k\,\ree^k,
\end{equation}
when the energy $E_{n,m}$ is a root of $t_{n+1} = 0$ and the corresponding radius $R_{n,m}$ is found from \eqref{eq:recurrence-triplet} which yields
\begin{equation}
\label{eq:E_T}
R_{n,m}^2 E_{n,m} = \frac{n}{2}\qty(\frac{n}{2}+D).
\end{equation}
$T_{n,m}(\ree)$ is the exact wave function of the $m$-th excited state of $^3P$ symmetry for the radius $R_{n,m}$.
It is illuminating to begin by examining the simplest $^1S$ and $^3P$ polynomial solutions.
Except in the $D=1$ case, the first $^1S$ solution has
\begin{align}
R_{1,0} & = \sqrt{\frac{(2D-1)(2D-2)}{8}}, & E_{1,0} & = \frac{1}{D-1},
\end{align}
and the first $^3P$ solution has
\begin{align}
R_{1,0} & = \sqrt{\frac{(2D+1)(2D+2)}{8}}, & E_{1,0} & = \frac{1}{D+1}.
\end{align}
These are tabulated for $D = 1, 2, 3, 4$, together with the associated wave functions, in Table \ref{tab:lowest}.
In the ringium ($D=1$) case (\textit{i.e.}~two electrons on a ring), the first singlet and triplet solutions have $E_{2,0} = 2/3$ and $E_{1,0} = 1/2$, respectively, for the same value of the radius ($\sqrt{6}/2 \approx 1.2247$).
The corresponding wave functions are related by $S_{2,0} = \ree\,T_{1,0}$.
Unlike $T_{1,0}$, the singlet wavefunction $S_{2,0}$ vanishes at $\ree = 0$, and exhibits a second-order cusp condition, as shown in \eqref{eq:cusp-circle}.
For spherium ($D=2$ case), we know from our previous work \cite{TEOAS09} that the HF energy of the lowest $^1S$ state is $\EHF = 1/R$.
It follows that the exact correlation energy for $R = \sqrt{3}/2$ is $\Ec = 1-2/\sqrt{3} \approx -0.1547$ which is much larger than the limiting correlation energies of the helium-like ions ($-0.0467$) \cite{Baker90} or Hooke's law atoms ($-0.0497$) \cite{Gill05}.
This confirms our view that electron correlation on the surface of a sphere is qualitatively different from that in three-dimensional physical space.
For glomium ($D=3$ case), in contrast, possesses the same singlet and triplet cusp conditions --- Eqs.~\eqref{eq:S-cusp} and \eqref{eq:T-cusp} --- as those for electrons moving in three-dimensional physical space.
Indeed, the wave functions in Table \ref{tab:lowest}
\begin{align}
S_{1,0}(\ree) & = 1 + \ree/2, & (R & = \sqrt{5/2}),
\\
T_{1,0}(\ree) & = 1 + \ree/4, & (R & = \sqrt{7}),
\end{align}
have precisely the form of the ansatz used in Kutzelnigg's increasingly popular R12 methods \cite{Kutzelnigg85, Kutzelnigg91}.
Moreover, it can be shown \cite{EcLimit09} that, as $R \to 0$, the correlation energy $\Ec$ approaches $-0.0476$, which nestles nicely between the corresponding values for the helium-like ions ($-0.0467$) \cite{Baker90} and the Hooke's law atom ($-0.0497$) \cite{Gill05}.
Again, this suggests that the $D=3$ model (``electrons on a glome'') bears more similarity to common physical systems than the $D=2$ model (``electrons on a sphere'').
We will investigate this observation further in the next section.
%For fixed $D$, the radii increase with $n$ but decrease with $m$, and the energies behave in exactly the opposite way.
%As $R$ (or equivalently $n$) increases, the electrons tend to localize on opposite sides of the sphere, a phenomenon known as Wigner crystallization \cite{Wigner34} which has also been observed in other systems \cite{Thompson04a, Thompson04b, Taut93}.
%As a result, for large $R$, the ground state energies of both the singlet and triplet state approach $1/(2R)$.
%Analogous behavior is observed when $D \to \infty$ \cite{Yaffe82, Goodson87}.
%****************************************************************
\subsection{Other electronic states}
%****************************************************************
As shown in Ref.~\cite{ExSpherium10}, one can determine exact wave functions for other electronic states, but not all of them.
These states are inter-connected by subtile interdimensional degeneracies (see Table \ref{tab:summary}) using the transformation $(D,L) \rightarrow (D+2,L-1)$, where $L$ is the total angular momentum of the state.
We refer the interested readers to Refs.~\cite{Herrick75a, Herrick75b, ExSpherium10, eee15, Nodes15} for more details.
The energies of the $S$, $P$ and $D$ states ($m=0$) for glomium are plotted in Fig.~\ref{fig:ES} (the quasi-exact solutions are indicated by markers), while density plots of spherium ($n=1$ and $m=0$) are represented on Fig. \ref{fig:ES-on-sphere}.
% FIGURE 1
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{../Chapter2/fig/ES}
\caption{
\label{fig:ES}
Energy of the $S$, $P$ and $D$ states of glomium
(${}^1S^{\rm e} < {}^3P^{\rm o} \leq {}^1P^{\rm o} < {}^3P^{\rm e} < {}^3D^{\rm e} < {}^1D^{\rm o} \leq {}^3D^{\rm o}$).
The quasi-exact solutions are shown by the markers.}
\end{figure}
% FIGURE 2
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{../Chapter2/fig/ES-on-sphere}
\caption{
\label{fig:ES-on-sphere}
Density plots of the $S$, $P$ and $D$ states of spherium.
The squares of the wave functions when one electron is fixed at the north pole are represented.
The radii are $\sqrt{3}/2$, $\sqrt{15}/2$, $\sqrt{5}/2$, $\sqrt{21}/2$, $\sqrt{21}/2$, $3\sqrt{5}/2$ and $3\sqrt{3}/2$ for the ${}^1S^{\rm e}$, ${}^3P^{\rm o}$, ${}^1P^{\rm o}$, ${}^3P^{\rm e}$, ${}^3D^{\rm e}$, ${}^1D^{\rm o}$ and ${}^3D^{\rm o}$ states, respectively.}
\end{figure}
%****************************************************************
\subsection{Natural/unnatural parity}
%****************************************************************
In attempting to explain Hund's rules \cite{Hund25} and the ``alternating'' rule \cite{Russel27, Condon} (see also Refs.~\cite{Boyd84, Warner85}), Morgan and Kutzelnigg \cite{Kutzelnigg92, Morgan93, Kutzelnigg96} have proposed that the two-electron atomic states be classified as:
\begin{quote}
\textit{A two-electron state, composed of one-electron spatial orbitals with individual parities $(-1)^{\ell_1}$ and $(-1)^{\ell_1}$ and hence with overall parities $(-1)^{\ell_1+\ell_2}$, is said to have natural parity if its parity is $(-1)^L$. [\ldots] If the parity of the two-electron state is $-(-1)^{L}$, the state is said to be of unnatural parity.}
\end{quote}
After introducing spin, three classes emerge.
In a three-dimensional space, the states with a cusp value of $1/2$ are known as the {\em natural parity singlet states} \cite{Kato51,Kato57}, those with a cusp value of $1/4$ are the {\em natural and unnatural parity triplet states} \cite{Pack66}, and those with a cusp value of $1/6$, are the {\em unnatural parity singlet states} \cite{Kutzelnigg92}.
%In the previous section, we have observed that the $^1S^{\rm e}$ ground state and the first excited $^3P^{\rm o}$ state of glomium possess the same singlet ($1/2$) and triplet ($1/4$) cusp conditions as those for electrons moving in three-dimensional physical space and we have therefore argued that glomium may be the most appropriate model for studying ``real'' atomic or molecular systems.
%As mentioned previously, this is supported by the similarity of the correlation energy of glomium to that in other two-electron systems.
Most of the higher angular momentum states of glomium, possess the ``normal'' cusp values of $1/2$ and $1/4$.
However, the unnatural $^1D^{\rm o}$ and $^1F^{\rm e}$ states have the cusp value of $1/6$.
%****************************************************************
\subsection{First-order cusp condition}
%****************************************************************
The wave function, radius and energy of the lowest states are given by
\begin{align}
\Psi_{1,0} (\ree) & = 1 + \gamma\,\ree,
&
R_{1,0}^2 & = \frac{\delta}{4\gamma},
&
E_{1,0} & = \gamma,
\end{align}
which are closely related to the Kato cusp condition \cite{Kato57}
\begin{equation}
\label{eq:Kato}
\frac{\Psi^{\prime}(0)}{\Psi(0)} = \gamma.
\end{equation}
We now generalise the Morgan-Kutzelnigg classification \cite{Morgan93} to a $D$-dimensional space.
Writing the interparticle wave function as
\begin{equation}
\label{eq:Psi-1}
\Psi(\ree) = 1 + \frac{\ree}{2\kappa+D-1} + O(\ree^2),
\end{equation}
we have
\begin{equation}
\label{eq:classification}
\begin{split}
\kappa = 0, & \text{ for natural parity singlet states,}
\\
\kappa = 1, & \text{ for triplet states,}
\\
\kappa = 2, & \text{ for unnatural parity singlet states.}
\end{split}
\end{equation}
The labels for states of two electrons on a $D$-sphere are given in Table \ref{tab:summary}.
% TABLE 1
\begin{table*}
\centering
\caption{
\label{tab:summary}
Ground state and excited states of two electrons on a $D$-sphere}
\begin{tabular}{ccccccc}
\hline
State & Configuration & $\delta$ & $\gamma^{-1}$ & $\Lambda$ & $\kappa$ & Degeneracy \\
\hline
$^1S^{\rm e}$ & $s^2$ & $2D-1$ & $D-1$ & 0 & 0 & $^3P^{\rm e}$ \\
\hline
$^3P^{\rm o}$ & $sp$ & $2D+1$ & $D+1$ & $D/2$ & 1 & $^1D^{\rm o}$ \\
$^1P^{\rm o}$ & $sp$ & $2D+1$ & $D-1$ & $D/2$ & 0 & $^3D^{\rm o}$ \\
$^3P^{\rm e}$ & $p^2$ & $2D+3$ & $D+1$ & $D$ & 1 & \\
\hline
$^3D^{\rm e}$ & $sd$ & $2D+3$ & $D+1$ & $D+1$ & 1 & $^1F^{\rm e}$ \\
$^1D^{\rm o}$ & $pd$ & $2D+5$ & $D+3$ & $3D/2+1$ & 2 & \\
$^3D^{\rm o}$ & $pd$ & $2D+5$ & $D+1$ & $3D/2+1$ & 1 & \\
\hline
$^1F^{\rm e}$ & $pf$ & $2D+7$ & $D+3$ & $2D+3$ & 2 & \\
\hline
\end{tabular}
\end{table*}
%****************************************************************
\subsection{Second-order cusp condition}
%****************************************************************
The second solution is associated with
\begin{gather}
\Psi_{2,0} (\ree) = \Psi_{1,0} (\ree)
+ \frac{\gamma ^2 (\delta +2)}{2 \gamma (\delta +2)+4 \delta +6} \ree^2,
\\
R_{2,0}^2 = \frac{(\gamma+2)(\delta+2)-1}{2\gamma},
\\
E_{2,0} = \frac{\gamma(\delta+1)}{(\gamma+2)(\delta+2)-1}.
\end{gather}
For two electrons on a $D$-sphere, the second-order cusp condition is
\begin{equation}
\label{eq:Psi2}
\frac{\Psi^{\prime\prime}(0)}{\Psi(0)}
= \frac{1}{2D} \left( \frac{1}{D-1} - E \right).
\end{equation}
Following \eqref{eq:Psi2}, the classification \eqref{eq:classification} can be extended to the second-order coalescence condition, where the wave function (correct up to second-order in $u$) is
\begin{equation}
\Psi(\ree)
= 1 + \frac{\ree}{2\kappa+D-1}
+ \frac{\ree^2}{2(2\kappa+D)} \left( \frac{1}{2\kappa+D-1} - E \right) + O(\ree^3).
\end{equation}
Thus, we have, for $D = 3$,
\begin{equation}
\frac{\Psi^{\prime\prime}(0)}{\Psi(0)} =
\begin{cases}
\frac{1}{6} \qty( \frac{1}{2} - E ), & \text{ for } \kappa = 0,
\\
\frac{1}{10} \qty( \frac{1}{4} - E ), & \text{ for } \kappa = 1,
\\
\frac{1}{14} \qty( \frac{1}{6} - E ), & \text{ for } \kappa = 2.
\end{cases}
\end{equation}
For the natural parity singlet states ($\kappa=0$), the second-order cusp condition of glomium is precisely the second-order coalescence condition derived by Tew \cite{Tew08}, reiterating that glomium is an appropriate model for normal physical systems.

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%****************************************************************
\section{Universality of correlation effects}
%****************************************************************
Understanding and calculating the electronic correlation energy is one of the most important and difficult problems in theoretical chemistry.
In this pursuit, the study of high-density correlation energy using perturbation theory has been particularly profitable, shedding light on the physically relevant density regime and providing {\em exact} results for key systems, such as the uniform electron gas \cite{GellMann57} and two-electron systems \cite{BetheSalpeter}.
The former is the cornerstone of the most popular density functional paradigm (the local-density approximation) in solid-state physics \cite{ParrYang}; the latter provide important test cases in the development of new explicitly-correlated methods \cite{Kutzelnigg85,Nakashima07} for electronic structure calculations \cite{Helgaker}.
%****************************************************************
\subsection{High-density correlation energy}
%****************************************************************
The high-density correlation energy of the helium-like ions is obtained by expanding both the exact \cite{Hylleraas30} and HF \cite{Linderberg61} energies as series in $1/Z$, yielding
\begin{subequations}
\begin{align}
\label{eq:Eex}
E(Z,D,V)
& = E^{(0)}(D,V) Z^2
+ E^{(1)}(D,V) Z
+ E^{(2)}(D,V)
+ \frac{E^{(3)}(D,V)}{Z}
+ \ldots,
\\
\label{eq:EHF}
E_{\rm HF}(Z,D,V)
& = E^{(0)}(D,V) Z^2
+ E^{(1)}(D,V) Z
+ E_{\rm HF}^{(2)}(D,V)
+ \frac{E_{\rm HF}^{(3)}(D,V)}{Z}
+ \ldots,
\end{align}
\end{subequations}
where $Z$ is the nuclear charge, $D$ is the dimension of the space and $V$ is the external Coulomb potential.
Equations \eqref{eq:Eex} and \eqref{eq:EHF} share the same zeroth- and first-order energies because the exact and the HF treatment have the same zeroth-order Hamiltonian.
Thus, in the high-density (large-$Z$) limit, the correlation energy is
\begin{equation}
\begin{split}
\Ec^{(2)}(D,V)
& = \lim_{Z\to\infty} \Ec(Z,D,V)
\\
& = \lim_{Z\to\infty} \qty[ E(Z,D,V)-E_{\rm HF}(Z,D,V) ]
\\
& = E^{(2)}(D,V) - E_{\rm HF}^{(2)}(D,V).
\end{split}
\end{equation}
Despite intensive study \cite{Schwartz62, Baker90}, the coefficient $E^{(2)}(D,V)$ has not yet been reported in closed form.
However, the accurate numerical estimate
\begin{equation}
\label{eq:E2-He-3D}
E^{(2)} = -0.157\;666\;429\;469\;14
\end{equation}
has been determined for the important $D=3$ case \cite{Baker90}.
Combining \eqref{eq:E2-He-3D} with the exact result \cite{Linderberg61}
\begin{equation}
\label{eq:E2HF-He-3D}
E_{\rm HF}^{(2)} = \frac{9}{32} \ln \frac{3}{4} - \frac{13}{432}
\end{equation}
yields a value of
\begin{equation}
\Ec^{(2)} = -0.046\;663\;253\;999\;48
\end{equation}
for the helium-like ions in a three-dimensional space.
In the large-$D$ limit, the quantum world reduces to a simpler semi-classical one \cite{Yaffe82} and problems that defy solution in $D=3$ sometimes become exactly solvable.
In favorable cases, such solutions provide useful insight into the $D=3$ case and this strategy has been successfully applied in many fields of physics \cite{Witten80, Yaffe83}.
Indeed, just as one learns something about interacting systems by studying non-interacting ones and introducing the interaction perturbatively, one learns something about $D = 3$ by studying the large-$D$ case and introducing dimension-reduction perturbatively.
Singularity analysis \cite{Doren87} reveals that the energies of two-electron atoms possess first- and second-order poles at $D=1$, and that the Kato cusp \cite{Kato57, Morgan93} is directly responsible for the second-order pole.
In our previous work \cite{EcLimit09, Ballium10}, we have expanded the correlation energy as a series in $1/(D-1)$ but, although this is formally correct if summed to infinite order, such expansions falsely imply higher-order poles at $D=1$.
For this reason, we now follow Herschbach and Goodson \cite{Herschbach86, Goodson87}, and expand both the exact and HF energies as series in $1/D$.
Although various possibilities exist for this dimensional expansion \cite{Doren86, Doren87, Goodson92, Goodson93}, it is convenient to write
\begin{subequations}
\begin{align}
E^{(2)}(D,V)
& = \frac{E^{(2,0)}(V)}{D^2}
+ \frac{E^{(2,1)}(V)}{D^3}
+ \ldots,
\label{eq:E2DV}
\\
E_{\rm HF}^{(2)}(D,V)
& = \frac{E_{\rm HF}^{(2,0)}(V)}{D^2}
+ \frac{E_{\rm HF}^{(2,1)}(V)}{D^3}
+ \ldots,
\label{eq:EHF2DV}
\\
\Ec^{(2)}(D,V)
& = \frac{\Ec^{(2,0)}(V)}{D^2}
+ \frac{\Ec^{(2,1)}(V)}{D^3}
+ \ldots,
\label{eq:Ec2DV}
\end{align}
\end{subequations}
where
\begin{subequations}
\begin{align}
\Ec^{(2,0)}(V)
& = E^{(2,0)}(V) - E_{\rm HF}^{(2,0)}(V),
\\
\Ec^{(2,1)}(V)
& = E^{(2,1)}(V) - E_{\rm HF}^{(2,1)}(V).
\end{align}
\end{subequations}
Such double expansions of the correlation energy were originally introduced for the helium-like ions, and have lead to accurate estimations of correlation \cite{Loeser87a, Loeser87b} and atomic energies \cite{Loeser87c, Kais93} {\em via} interpolation and renormalisation techniques.
Equations \eqref{eq:E2DV}, \eqref{eq:EHF2DV} and \eqref{eq:Ec2DV} apply equally to the $^1S$ ground state of any two-electron system confined by a spherical potential $V(r)$.
%****************************************************************
\subsection{The conjecture}
%****************************************************************
For the helium-like ions, it is known \cite{Mlodinow81, Herschbach86, Goodson87} that
\begin{align}
\Ec^{(2,0)}(V) & = - \frac{1}{8},
&
\Ec^{(2,1)}(V) & = - \frac{163}{384},
\end{align}
and we have recently found \cite{EcLimit09} that $\Ec^{(2,0)}(V)$ takes the same value in hookium (two electrons in a parabolic well \cite{Kestner62, White70, Kais89, Taut93}), spherium (two electrons on a sphere \cite{Ezra82, Seidl07b, TEOAS09, QuasiExact09}) and ballium (two electrons in a ball \cite{Thompson02, Thompson05, Ballium10}).
In contrast, we found that $\Ec^{(2,1)}(V)$ is $V$-dependent.
The fact that the term $\Ec^{(2,0)}$ is invariant, while $\Ec^{(2,1)}$ varies with the confinement potential allowed us to explain why the high-density correlation energy of the previous two-electron systems are similar, but not identical, for $D=3$ \cite{EcLimit09, Ballium10}.
On this basis, we conjectured \cite{EcLimit09} that
\begin{equation}
\label{eq:conjecture}
\Ec^{(2)}(D,V) \sim - \frac{1}{8D^2} - \frac{C(V)}{D^3}
\end{equation}
holds for \emph{any} spherical confining potential, where the coefficient $C(V)$ varies slowly with $V(r)$.
% BEGIN TABLE 1
\begin{table}
\centering
\caption{
\label{tab:Ec}
$E^{(2,0)}$, $E_{\rm HF}^{(2,0)}$, $\Ec^{(2,0)}$ and $\Ec^{(2,1)}$ coefficients for various systems and $v(r) = 1$.}
\begin{tabular}{lrcccc}
\hline
System & $m$ & $-E^{(2,0)}$ & $-E_{\rm HF}^{(2,0)}$ & $-\Ec^{(2,0)}$ & $-\Ec^{(2,1)}$ \\
\hline
Helium & $-1$ & $5/8$ & $1/2$ & $1/8$ & $0.424479$ \\
Airium & $1$ & $7/24$ & $1/6$ & $1/8$ & $0.412767$ \\
Hookium & $2$ & $1/4$ & $1/8$ & $1/8$ & $0.433594$ \\
Quartium & $4$ & $5/24$ & $1/12$ & $1/8$ & $0.465028$ \\
Sextium & $6$ & $3/16$ & $1/16$ & $1/8$ & $0.486771$ \\
Ballium & $\infty$ & $1/8$ & $0$ & $1/8$ & $0.664063$ \\
\hline
\end{tabular}
\end{table}
% END TABLE 1
%****************************************************************
\subsection{The proof}
%****************************************************************
Here, we will summarise our proof of the conjecture \eqref{eq:conjecture}.
More details can be found in Ref.~\cite{EcProof10}.
We prove that $\Ec^{(2,0)}$ is universal, and that, for large $D$, the high-density correlation energy of the $^1S$ ground state of two electrons is given by \eqref{eq:conjecture} for any confining potential of the form
\begin{equation}
\label{eq:V-proof}
V(r) = \text{sgn}(m) r^m v(r),
\end{equation}
where $v(r)$ possesses a Maclaurin series expansion
\begin{equation}
v(r) = v_0 + v_1 r + v_2 \frac{r^2}{2} + \ldots.
\end{equation}
After transforming both the dependent and independent variables \cite{EcProof10}, the Schr\"odinger equation can be brought to the simple form
\begin{equation}
\label{eq:Hersch-trans}
\qty( \frac{1}{\Lambda} \Hat{\mathcal{T}}
+ \Hat{\mathcal{U}}
+ \Hat{\mathcal{V}}
+ \frac{1}{Z} \Hat{\mathcal{W}} ) \Phi_D
= \mathcal{E}_D \Phi_D,
\end{equation}
in which, for $S$ states, the kinetic, centrifugal, external potential and Coulomb operators are, respectively,
\begin{gather}
-2 \Hat{\mathcal{T}} =
\qty( \frac{\partial^2}{\partial r_1^2} + \frac{\partial^2}{\partial r_2^2} )
+ \qty( \frac{1}{r_1^2} + \frac{1}{r_1^2} )
\qty( \frac{\partial^2}{\partial \theta^2} + \frac{1}{4} ),
\\
\Hat{\mathcal{U}} =
\frac{1}{2 \sin^2 \theta}
\qty( \frac{1}{r_1^2} + \frac{1}{r_1^2} ),
\\
\Hat{\mathcal{V}} =
V(r_1) + V(r_2),
\\
\Hat{\mathcal{W}} =
\frac{1}{\sqrt{r_1^2 + r_2^2 -2 r_1 r_2 \cos \theta}},
\end{gather}
and the dimensional perturbation parameter is
\begin{equation}
\Lambda = \frac{(D-2)(D-4)}{4}.
\end{equation}
In this form, double perturbation theory can be used to expand the energy in terms of both $1/Z$ and $1/\Lambda$.
For $D=\infty$, the kinetic term vanishes and the electrons settle into a fixed (``Lewis'') structure \cite{Herschbach86} that minimises the effective potential
\begin{equation}
\label{eq:X}
\Hat{\mathcal{X}}
= \Hat{\mathcal{U}} + \Hat{\mathcal{V}}
+ \frac{1}{Z} \Hat{\mathcal{W}}.
\end{equation}
The minimization conditions are
\begin{gather}
\frac{\partial \Hat{\mathcal{X}}(r_1,r_2,\theta)}{\partial r_1} =
\frac{\partial \Hat{\mathcal{X}}(r_1,r_2,\theta)}{\partial r_2} = 0,
\label{eq:dW-r}
\\
\frac{\partial \Hat{\mathcal{X}}(r_1,r_2,\theta)}{\partial \theta} = 0,
\label{eq:dW-theta}
\end{gather}
and the stability condition implies $m > -2$. Assuming that the two electrons are equivalent, the resulting exact energy is
\begin{equation}
\label{eq:Einf}
\mathcal{E}_{\infty}
= \Hat{\mathcal{X}} (r_{\infty},r_{\infty},\theta_{\infty}).
\end{equation}
It is easy to show that
\begin{gather}
r_{\infty} = \alpha + \frac{\alpha^2}{m+2}
\qty(\frac{1}{2\sqrt{2}} - \Lambda \frac{m+1}{m} \frac{v_1}{v_0} ) \frac{1}{Z}
+ \ldots,
\label{eq:r-eq}
\\
\cos \theta_{\infty} = - \frac{\alpha}{4\sqrt{2}} \frac{1}{Z}
+ \ldots,
\label{eq:tetha-eq}
\end{gather}
where $\alpha^{-(m+2)} = \text{sgn}(m) m v_0$.
For the HF treatment, we have $\theta_{\infty}^{\rm HF} = \pi/2$.
Indeed, the HF wave function itself is independent of $\theta$, and the only $\theta$ dependence comes from the $D$-dimensional Jacobian, which becomes a Dirac delta function centred at $\pi/2$ as $D\to\infty$.
Solving \eqref{eq:dW-r}, one finds that $r_{\infty}^{\rm HF}$ and $r_{\infty}$ are equal to second-order in $1/Z$.
Thus, in the large-$D$ limit, the HF energy is
\begin{equation}
\label{eq:EinfHF}
\mathcal{E}_{\infty}^{\rm HF}
= \Hat{\mathcal{X}} \qty(r_{\infty}^{\rm HF},r_{\infty}^{\rm HF},\frac{\pi}{2}),
\end{equation}
and correlation effects originate entirely from the fact that $\theta_\infty$ is slightly greater than $\pi/2$ for finite $Z$.
Expanding \eqref{eq:Einf} and \eqref{eq:EinfHF} in terms of $Z$ and $D$ yields
\begin{align}
E^{(2,0)}(V)
& = - \frac{1}{8} - \frac{1}{2(m+2)},
\\
E_{\rm HF}^{(2,0)}(V)
& = - \frac{1}{2(m+2)},
\label{eqEHF20}
\end{align}
thus showing that both $E^{(2,0)}$ and $E_{\rm HF}^{(2,0)}$ depend on the leading power $m$ of the external potential but not on $v(r)$.
Subtracting these energies yields
\begin{equation}
\label{eq:Ec00}
\Ec^{(2,0)}(V) = - \frac{1}{8},
\end{equation}
and this completes the proof that, in the high-density limit, the leading coefficient $\Ec^{(2,0)}$ of the large-$D$ expansion of the correlation energy is universal, {\em i.e.} it does not depend on the external potential $V(r)$.
The result \eqref{eq:Ec00} is related to the cusp condition \cite{Kato57, Morgan93, Pan03}
\begin{equation}
\label{eq:cusp}
\left. \pdv{\Psi_D}{\ree}\right|_{\ree=0}
= \frac{1}{D-1} \Psi_D(\ree=0),
\end{equation}
which arises from the cancellation of the Coulomb operator singularity by the $D$-dependent angular part of the kinetic operator \cite{Helgaker}.
The $E^{(2,1)}$ and $\EHF^{(2,1)}$ coefficients can be found by considering the Langmuir vibrations of the electrons around their equilibrium positions \cite{Herschbach86, Goodson87}.
The general expressions depend on $v_0$ and $v_1$, but are not reported here.
However, for $v(r)=1$, which includes many of the most common external potentials, we find
\begin{equation}
\Ec^{(2,1)}(V) = - \frac{85}{128} - \frac{9/32}{(m+2)^{3/2}}
+ \frac{1/2}{(m+2)^{1/2}} + \frac{1/16}{(m+2)^{1/2}+2},
\end{equation}
showing that $\Ec^{(2,1)}$, unlike $\Ec^{(2,0)}$, is potential-dependent.
Numerical values of $\Ec^{(2,1)}$ are reported in Table \ref{tab:Ec} for various systems.

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%****************************************************************
\section{Summary}
%****************************************************************
In the first section of this chapter, we have reported exact solutions of a Coulomb correlation problem, consisting of two electrons on a $D$-dimensional sphere.
The Coulomb problem can be solved exactly for an infinite set of values of the radius $R$ for both the ground and excited states, on both the singlet and triplet manifolds.
The corresponding exact solutions are polynomials in the interelectronic distance $\ree$.
The cusp conditions, which are related to the behaviour of the wave function at the electron-electron coalescence point, have been analysed and classified according to the natural or unnatural parity of the state considered.
In the second section, we proved that the leading term in the large-$D$ expansion of the high-density correlation energy of an electron pair is invariant to the nature of the confining potential.
For any such system, the correlation energy is given by $\Ec \sim -\gamma^2/8$, where $\gamma = 1/(D-1)$ is the Kato cusp factor in a $D$-dimensional space.

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The final decades of the twentieth century witnessed a major revolution in solid-state and molecular physics, as the introduction of sophisticated exchange-correlation models \cite{ParrYang} propelled DFT from qualitative to quantitative usefulness.
The apotheosis of this development was probably the award of the 1998 Nobel Prize for Chemistry to Walter Kohn and John Pople but its origins can be traced to the prescient efforts by Thomas \cite{Thomas27}, Fermi \cite{Fermi26} and Dirac \cite{Dirac30}, more than 70 years earlier, to understand the behaviour of ensembles of electrons without explicitly constructing their full wave functions.
These days, DFT so dominates the popular perception of molecular orbital calculations that many non-specialists now regard the two as synonymous.
In principle, the cornerstone of modern DFT is the HK theorem \cite{Hohenberg64} but, in practice, it rests largely on the presumed similarity between the electronic behaviour in a real system and that in the hypothetical ``infinite'' uniform electron gas (IUEG) or jellium \cite{Fermi26, Thomas27, Dirac30, Wigner34, Macke50, GellMann57, Onsager66, Stern73, Rajagopal77, Isihara80, Hoffman92, Seidl04, Sun10, 2DEG11, 3DEG11, WIREs16, Vignale}.
In 1965, Kohn and Sham \cite{Kohn65} showed that the knowledge of an analytical parametrisation of the IUEG correlation energy allows one to perform approximate calculations for atoms, molecules and solids.
The idea --- the local-density approximation (LDA) --- is attractively simple: if we know the properties of jellium, we can understand the electron cloud in a molecule by dividing it into tiny chunks of density and treating each as a piece of jellium.
The good news is that the properties of jellium are known from DMC calculations \cite{Ceperley80, Tanatar89, Kwon93, Ortiz94, Rapisarda96, Kwon98, Ortiz99, Attaccalite02, Zong02, Drummond09a, Drummond09b}.
Such calculations are possible because jellium is characterised by just a \emph{single} parameter $\rho$, the electron density.
This spurred the development of a wide variety of spin-density correlation functionals (VWN \cite{VWN80}, PZ \cite{PZ81}, PW92 \cite{PW92}, etc), each of which requires information on the high- and low-density regimes of the spin-polarised IUEG, and are parametrised using numerical results from QMC calculations \cite{Ceperley78, Ceperley80}, together with analytic perturbative results.
The bad news is that jellium has an infinite number of electrons in an infinite volume and this unboundedness renders it, in some respects, a poor model for the electrons in molecules.
Indeed, the simple LDA described above predicts bond energies that are much too large and this led many chemists in the 70's to dismiss DFT as a quantitatively worthless theory.
Most of the progress since these days has resulted from concocting ingenious corrections for jellium's deficiencies (GGAs, MGGAs, HGGAs, etc).
However, notwithstanding the impressive progress since the 70's, modern DFT approximations still exhibit fundamental deficiencies in large systems \cite{Curtiss00}, conjugated molecules \cite{Woodcock02}, charge-transfer excited states \cite{Dreuw04}, dispersion-stabilised systems \cite{Wodrich06}, systems with fractional spin or charge \cite{Yang08}, isodesmic reactions \cite{Brittain09}, and elsewhere.
Because DFT is in principle an exact theory, many of these problems can be traced ultimately to the use of jellium as a reference system and the \textit{ad hoc} corrections that its use subsequently necessitates.

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%****************************************************************
\section{Uniform electron gases}
%****************************************************************
In the previous chapter, we considered the behaviour of electrons that are confined to the surface of a sphere.
This work yielded a number of unexpected discoveries \cite{TEOAS09, EcLimit09, QuasiExact09, Concentric10, Hook10, EcProof10, ExSpherium10, Frontiers10, Glomium11} but the one of relevance here is that such systems provide a beautiful new family of UEGs.
These finite UEGs (FUEGs) have been thoroughly studied in Ref.~\cite{Glomium11}.
Here, we only report their main characteristics \cite{UEGs12}.
\begin{table}
\centering
\caption{
\label{tab:Ylm}
The lowest free-particle orbitals on a 2-sphere}
\begin{tabular}{lccc}
\hline
Name & $l$ & $m$ & $\sqrt{4\pi} \,Y_{lm}(\theta,\phi)$ \\
\hline
$s$ & $0$ & $0$ & $1$ \\
\hline
$p_0$ & $1$ & $0$ & $3^{1/2} \cos\theta$ \\
$p_{+1}$ & $1$ & $+1$ & $(3/2)^{1/2} \sin\theta \exp(+i\phi)$ \\
$p_{-1}$ & $1$ & $-1$ & $(3/2)^{1/2} \sin\theta \exp(-i\phi)$ \\
\hline
$d_0$ & $2$ & $0$ & $(5/4)^{1/2} (3\cos^2\theta-1)$ \\
$d_{+1}$ & $2$ & $+1$ & $(15/2)^{1/2} \sin\theta \cos\theta \exp(+i\phi)$ \\
$d_{-1}$ & $2$ & $-1$ & $(15/2)^{1/2} \sin\theta \cos\theta \exp(-i\phi)$ \\
$d_{+2}$ & $2$ & $+2$ & $(15/8)^{1/2} \sin^2\theta \exp(+2i\phi)$ \\
$d_{-2}$ & $2$ & $-2$ & $(15/8)^{1/2} \sin^2\theta \exp(-2i\phi)$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Number of electrons in $L$-spherium and $L$-glomium atoms}
\label{tab:fullshell}
\begin{tabular}{ccccccccc}
\hline
$L$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
$L$-spherium & 2 & 8 & 18 & 32 & 50 & 72 & 98 & 128 \\
$L$-glomium & 2 & 10 & 28 & 60 & 110 & 182 & 280 & 408 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{The lowest free-particle orbitals on a glome (\textit{i.e.}~a 3-sphere)}
\label{tab:Ylmn}
\begin{tabular}{lcccc}
\hline
Name & $l$ & $m$ & $n$ & $\pi\, Y_{lmn}(\chi,\theta,\phi)$ \\
\hline
$1s$ & $0$ & $0$ & $0$ & $2^{-1/2}$ \\
\hline
$2s$ & $1$ & $0$ & $0$ & $2^{1/2} \cos\chi$ \\
$2p_0$ & $1$ & $1$ & $0$ & $2^{1/2} \sin\chi \cos\theta$ \\
$2p_{+1}$ & $1$ & $1$ & $+1$ & $\sin\chi \sin\theta \exp(+i\phi)$ \\
$2p_{-1}$ & $1$ & $1$ & $-1$ & $\sin\chi \sin\theta \exp(-i\phi)$ \\
\hline
$3s$ & $2$ & $0$ & $0$ & $2^{-1/2} (4\cos^2\chi-1)$ \\
$3p_0$ & $2$ & $1$ & $0$ & $12^{1/2}\sin\chi \cos\chi \cos\theta$ \\
$3p_{+1}$ & $2$ & $1$ & $+1$ & $6^{1/2}\sin\chi \cos\chi \sin\theta \exp(+i\phi)$ \\
$3p_{-1}$ & $2$ & $1$ & $-1$ & $6^{1/2}\sin\chi \cos\chi \sin\theta \exp(-i\phi)$ \\
$3d_0$ & $2$ & $2$ & $0$ & $\sin^2\chi \ (3\cos^2\theta-1)$ \\
$3d_{+1}$ & $2$ & $2$ & $+1$ & $6^{1/2}\sin^2\chi \sin\theta \cos\theta \exp(+i\phi)$ \\
$3d_{-1}$ & $2$ & $2$ & $-1$ & $6^{1/2}\sin^2\chi \sin\theta \cos\theta \exp(-i\phi)$ \\
$3d_{+2}$ & $2$ & $2$ & $+2$ & $(3/2)^{1/2}\sin^2\chi \sin^2\theta \exp(+2i\phi)$ \\