%**************************************************************** \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.