337 lines
19 KiB
TeX
337 lines
19 KiB
TeX
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\section{Quasi-exactly solvable models}
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%****************************************************************
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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}.
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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}.
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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 $k$.
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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}.
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A related system consists of two electrons trapped on the surface of a sphere of radius $R$.
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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}.
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Seidl utilised this system to develop new correlation functionals \cite{Seidl00a, Seidl00b} within the adiabatic connection in DFT \cite{Seidl07b}.
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As mentioned earlier, we will use the term ``spherium'' to describe this two-electron system.
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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 $R$.
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Because the corresponding HF energies are also known exactly, this is now one of the most complete theoretical models for understanding electron correlation effects.
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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 $R$.
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We adopt the convention that a $D$-sphere is the surface of a ($D+1$)-dimensional ball.
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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 $\ree = \abs{\br_1-\br_2}$.
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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}.
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We have also published dedicated studies of the 1D system (that we dubbed ringium) \cite{QR12, Ringium13, NatRing16}.
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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}.
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Finally, the nodal structures of these systems has been investigated in collaboration with Dario Bressanini \cite{Nodes15}.
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%****************************************************************
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\subsection{Singlet ground state}
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%****************************************************************
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The electronic Hamiltonian is
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\begin{equation}
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\HOp = - \frac{\nabla_1^2}{2} - \frac{\nabla_2^2}{2} + \frac{1}{\ree},
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\end{equation}
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and because each electron moves on a $D$-sphere, it is natural to adopt hyperspherical coordinates \cite{Louck60}.
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For $^1S$ states, it can be then shown \cite{TEOAS09} that the wave function $S(\ree)$ satisfies the Schr{\"o}dinger equation
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\begin{equation}
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\label{eq:S-singlet}
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\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).
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\end{equation}
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By introducing the dimensionless variable $x = \ree/2R$, this becomes a Heun equation \cite{Ronveaux} with singular points at $x = -1, 0, +1$.
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Based on our previous work \cite{TEOAS09} and the known solutions of the Heun equation \cite{Polyanin}, we seek wave functions of the form
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\begin{equation}
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\label{eq:S_series}
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S(\ree) = \sum_{k=0}^\infty s_k\,\ree^k,
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\end{equation}
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and substitution into \eqref{eq:S-singlet} yields the recurrence relation
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\begin{equation}
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\label{eq:recurrence-singlet}
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s_{k+2} = \frac{ s_{k+1} + \qty[ k(k+2D-2) \frac{1}{4R^2} - E ] s_k }{(k+2)(k+D)},
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\end{equation}
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with the starting values
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\begin{equation}
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\{s_0,s_1\} = \begin{cases}
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\qty{0,1}, & D = 1,
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\\
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\qty{1,1/(D-1)}, & D \ge 2.
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\end{cases}
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\end{equation}
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Thus, the Kato cusp conditions \cite{Kato57} are
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\begin{align}
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\label{eq:cusp-circle}
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S(0) & = 0,
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&
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\frac{S''(0)}{S'(0)} & = 1,
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\end{align}
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for electrons on a ring ($D=1$), i.e.~ringium, and
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\begin{equation}
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\label{eq:S-cusp}
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\frac{S'(0)}{S(0)} = \frac{1}{D-1},
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\end{equation}
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in higher dimensions.
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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.
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We will return to this point below.
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The wave function \eqref{eq:S_series} reduces to the polynomial
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\begin{equation}
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S_{n,m}(\ree) = \sum_{k=0}^n s_k\,\ree^k,
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\end{equation}
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(where $m$ the number of roots between $0$ and $2R$) if, and only if, $s_{n+1} = s_{n+2} = 0$.
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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
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\begin{equation}
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\label{eq:E_S}
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R_{n,m}^2 E_{n,m} = \frac{n}{2}\qty(\frac{n}{2}+D-1).
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\end{equation}
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$S_{n,m}(\ree)$ is the exact wave function of the $m$-th excited state of $^1S$ symmetry for the radius $R_{n,m}$.
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{table}
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\centering
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\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}
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\centering
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\begin{tabular}{ccccc}
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\hline
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State & $D$ & $2R$ & $E$ & $S(\ree)$ or $T(\ree)$ \\
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\hline
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\mr{4}{*}{$^1S$} & 1 & $\sqrt{6}$ & 2/3 & $\ree(1 + \ree/2)$ \\
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& 2 & $\sqrt{3}$ & 1 & $1 + \ree$ \\
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& 3 & $\sqrt{10}$ & 1/2 & $1 + \ree/2$ \\
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& 4 & $\sqrt{21}$ & 1/3 & $1 + \ree/3$ \\
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\hline
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\mr{4}{*}{$^3P$} & 1 & $\sqrt{6}$ & 1/2 & $1 + \ree/2$ \\
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& 2 & $\sqrt{15}$ & 1/3 & $1 + \ree/3$ \\
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& 3 & $\sqrt{28}$ & 1/4 & $1 + \ree/4$ \\
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& 4 & $\sqrt{45}$ & 1/5 & $1 + \ree/5$ \\
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\hline
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\end{tabular}
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\end{table}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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%****************************************************************
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\subsection{Triplet excited state}
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%****************************************************************
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If we write the $^3P$ state wave function as \cite{Breit30}
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\begin{equation}
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^3\Psi = (\cos \theta_1 - \cos \theta_2)\,T(\ree),
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\end{equation}
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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
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\begin{equation}
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\label{eq:S-triplet}
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\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),
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\end{equation}
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and the antisymmetric part provides an additional kinetic energy contribution $D/(2R^2)$.
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Substituting the power series expansion
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\begin{equation}
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\label{eq:T_series}
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T(\ree) = \sum_{k=0}^\infty t_k\,\ree^k
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\end{equation}
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into \eqref{eq:S-triplet} yields the recurrence relation
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\begin{equation}
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\label{eq:recurrence-triplet}
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t_{k+2} = \frac{ t_{k+1} + \qty[ k(k+2D) \frac{1}{4R^2} - E ] t_k }{(k+2)(k+D+2)},
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\end{equation}
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with the starting values
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\begin{equation}
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\qty{t_0,t_1} = \qty{1, 1/(D+1)},
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\end{equation}
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yielding the cusp condition
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\begin{equation}
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\label{eq:T-cusp}
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\frac{T'(0)}{T(0)} = \frac{1}{D+1}.
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\end{equation}
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The wave function \eqref{eq:T_series} reduces to the polynomial
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\begin{equation}
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T_{n,m}(\ree) = \sum_{k=0}^n t_k\,\ree^k,
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\end{equation}
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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
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\begin{equation}
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\label{eq:E_T}
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R_{n,m}^2 E_{n,m} = \frac{n}{2}\qty(\frac{n}{2}+D).
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\end{equation}
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$T_{n,m}(\ree)$ is the exact wave function of the $m$-th excited state of $^3P$ symmetry for the radius $R_{n,m}$.
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It is illuminating to begin by examining the simplest $^1S$ and $^3P$ polynomial solutions.
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Except in the $D=1$ case, the first $^1S$ solution has
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\begin{align}
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R_{1,0} & = \sqrt{\frac{(2D-1)(2D-2)}{8}}, & E_{1,0} & = \frac{1}{D-1},
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\end{align}
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and the first $^3P$ solution has
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\begin{align}
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R_{1,0} & = \sqrt{\frac{(2D+1)(2D+2)}{8}}, & E_{1,0} & = \frac{1}{D+1}.
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\end{align}
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These are tabulated for $D = 1, 2, 3, 4$, together with the associated wave functions, in Table \ref{tab:lowest}.
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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 ($\sqrt{6}/2 \approx 1.2247$).
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The corresponding wave functions are related by $S_{2,0} = \ree\,T_{1,0}$.
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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}.
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For spherium ($D=2$ case), we know from our previous work \cite{TEOAS09} that the HF energy of the lowest $^1S$ state is $\EHF = 1/R$.
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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}.
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This confirms our view that electron correlation on the surface of a sphere is qualitatively different from that in three-dimensional physical space.
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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.
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Indeed, the wave functions in Table \ref{tab:lowest}
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\begin{align}
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S_{1,0}(\ree) & = 1 + \ree/2, & (R & = \sqrt{5/2}),
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\\
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T_{1,0}(\ree) & = 1 + \ree/4, & (R & = \sqrt{7}),
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\end{align}
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have precisely the form of the ansatz used in Kutzelnigg's increasingly popular R12 methods \cite{Kutzelnigg85, Kutzelnigg91}.
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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}.
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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'').
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We will investigate this observation further in the next section.
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%For fixed $D$, the radii increase with $n$ but decrease with $m$, and the energies behave in exactly the opposite way.
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%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}.
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%As a result, for large $R$, the ground state energies of both the singlet and triplet state approach $1/(2R)$.
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%Analogous behavior is observed when $D \to \infty$ \cite{Yaffe82, Goodson87}.
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%****************************************************************
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\subsection{Other electronic states}
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%****************************************************************
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As shown in Ref.~\cite{ExSpherium10}, one can determine exact wave functions for other electronic states, but not all of them.
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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.
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We refer the interested readers to Refs.~\cite{Herrick75a, Herrick75b, ExSpherium10, eee15, Nodes15} for more details.
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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}.
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% FIGURE 1
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\begin{figure}
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\centering
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\includegraphics[width=0.6\textwidth]{../Chapter2/fig/ES}
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\caption{
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\label{fig:ES}
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Energy of the $S$, $P$ and $D$ states of glomium
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(${}^1S^{\rm e} < {}^3P^{\rm o} \leq {}^1P^{\rm o} < {}^3P^{\rm e} < {}^3D^{\rm e} < {}^1D^{\rm o} \leq {}^3D^{\rm o}$).
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The quasi-exact solutions are shown by the markers.}
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\end{figure}
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% FIGURE 2
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\begin{figure}
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\centering
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\includegraphics[width=0.6\textwidth]{../Chapter2/fig/ES-on-sphere}
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\caption{
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\label{fig:ES-on-sphere}
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Density plots of the $S$, $P$ and $D$ states of spherium.
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The squares of the wave functions when one electron is fixed at the north pole are represented.
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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.}
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\end{figure}
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%****************************************************************
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\subsection{Natural/unnatural parity}
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%****************************************************************
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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:
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\begin{quote}
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\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.}
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\end{quote}
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After introducing spin, three classes emerge.
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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}.
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%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.
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%As mentioned previously, this is supported by the similarity of the correlation energy of glomium to that in other two-electron systems.
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Most of the higher angular momentum states of glomium, possess the ``normal'' cusp values of $1/2$ and $1/4$.
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However, the unnatural $^1D^{\rm o}$ and $^1F^{\rm e}$ states have the cusp value of $1/6$.
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%****************************************************************
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\subsection{First-order cusp condition}
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%****************************************************************
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The wave function, radius and energy of the lowest states are given by
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\begin{align}
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\Psi_{1,0} (\ree) & = 1 + \gamma\,\ree,
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&
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R_{1,0}^2 & = \frac{\delta}{4\gamma},
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&
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E_{1,0} & = \gamma,
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\end{align}
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which are closely related to the Kato cusp condition \cite{Kato57}
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\begin{equation}
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\label{eq:Kato}
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\frac{\Psi^{\prime}(0)}{\Psi(0)} = \gamma.
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\end{equation}
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We now generalise the Morgan-Kutzelnigg classification \cite{Morgan93} to a $D$-dimensional space.
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Writing the interparticle wave function as
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\begin{equation}
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\label{eq:Psi-1}
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\Psi(\ree) = 1 + \frac{\ree}{2\kappa+D-1} + O(\ree^2),
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\end{equation}
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we have
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\begin{equation}
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\label{eq:classification}
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\begin{split}
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\kappa = 0, & \text{ for natural parity singlet states,}
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\\
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\kappa = 1, & \text{ for triplet states,}
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\\
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\kappa = 2, & \text{ for unnatural parity singlet states.}
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\end{split}
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\end{equation}
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The labels for states of two electrons on a $D$-sphere are given in Table \ref{tab:summary}.
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% TABLE 1
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\begin{table*}
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\centering
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\caption{
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\label{tab:summary}
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Ground state and excited states of two electrons on a $D$-sphere}
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\begin{tabular}{ccccccc}
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\hline
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State & Configuration & $\delta$ & $\gamma^{-1}$ & $\Lambda$ & $\kappa$ & Degeneracy \\
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\hline
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$^1S^{\rm e}$ & $s^2$ & $2D-1$ & $D-1$ & 0 & 0 & $^3P^{\rm e}$ \\
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\hline
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$^3P^{\rm o}$ & $sp$ & $2D+1$ & $D+1$ & $D/2$ & 1 & $^1D^{\rm o}$ \\
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$^1P^{\rm o}$ & $sp$ & $2D+1$ & $D-1$ & $D/2$ & 0 & $^3D^{\rm o}$ \\
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$^3P^{\rm e}$ & $p^2$ & $2D+3$ & $D+1$ & $D$ & 1 & \\
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\hline
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$^3D^{\rm e}$ & $sd$ & $2D+3$ & $D+1$ & $D+1$ & 1 & $^1F^{\rm e}$ \\
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$^1D^{\rm o}$ & $pd$ & $2D+5$ & $D+3$ & $3D/2+1$ & 2 & \\
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$^3D^{\rm o}$ & $pd$ & $2D+5$ & $D+1$ & $3D/2+1$ & 1 & \\
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\hline
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$^1F^{\rm e}$ & $pf$ & $2D+7$ & $D+3$ & $2D+3$ & 2 & \\
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\hline
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\end{tabular}
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\end{table*}
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%****************************************************************
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\subsection{Second-order cusp condition}
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%****************************************************************
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The second solution is associated with
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\begin{gather}
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\Psi_{2,0} (\ree) = \Psi_{1,0} (\ree)
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+ \frac{\gamma ^2 (\delta +2)}{2 \gamma (\delta +2)+4 \delta +6} \ree^2,
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\\
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R_{2,0}^2 = \frac{(\gamma+2)(\delta+2)-1}{2\gamma},
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\\
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E_{2,0} = \frac{\gamma(\delta+1)}{(\gamma+2)(\delta+2)-1}.
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\end{gather}
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For two electrons on a $D$-sphere, the second-order cusp condition is
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\begin{equation}
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\label{eq:Psi2}
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\frac{\Psi^{\prime\prime}(0)}{\Psi(0)}
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= \frac{1}{2D} \left( \frac{1}{D-1} - E \right).
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\end{equation}
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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
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\begin{equation}
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\Psi(\ree)
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= 1 + \frac{\ree}{2\kappa+D-1}
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+ \frac{\ree^2}{2(2\kappa+D)} \left( \frac{1}{2\kappa+D-1} - E \right) + O(\ree^3).
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\end{equation}
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Thus, we have, for $D = 3$,
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\begin{equation}
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\frac{\Psi^{\prime\prime}(0)}{\Psi(0)} =
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\begin{cases}
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\frac{1}{6} \qty( \frac{1}{2} - E ), & \text{ for } \kappa = 0,
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\\
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\frac{1}{10} \qty( \frac{1}{4} - E ), & \text{ for } \kappa = 1,
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\\
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\frac{1}{14} \qty( \frac{1}{6} - E ), & \text{ for } \kappa = 2.
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\end{cases}
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\end{equation}
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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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