On the Exact Helium Wave Function Expansion
Abstract
A $^1S$ wave function's
expansion for 2 electron atoms and ions is
obtained and the leading coëfficients are reported.
Introduction
The ground electronic state of Helium (and its isoelectronic ions)
has been studied extensively since the original work
of Hylleraas (E. A. Hylleraas, Uber den Grundzustand des Heliumatoms,
Zeits. f. Physik 48, 469 (1928)).
Absent an exact solution, efforts have been expended to find compact approximate wave functions which give increasingly better energy eigenvalues for the energy
( K. Rodriguez, G. Gasaneo, and D. Mitnik, Accurate
and simple wavefunctions for the Helium isoelectronic sequence with correct cusp conditions, Journal of Physics B:
Atomic, Molecular and Optical Physics 40, 3923 (2007).).
The $^1S$ wave function for Helium's two electrons is not a power series in $r_1, r_2$ and $r_{12}$, as shown by Bartlett, Gibbons and Dunn
( J. H. Bartlett Jr., J. J. Gibbons Jr., and C. G. Dunn,
Phys. Rev. 47, 679 (1935)).
who demonstrated that a unique solution for $\psi(r_1,r_2,r_{12})$ of the form
$$
\psi = \sum_{i,j,k}C_{i,j,k}r_1^ir_2^jr_{12}^k
$$
does not exist.
They showed that the coëfficient of
$\frac{r_2^2}{r_1r_{12}} \hookrightarrow C_{101} = 0$, that the coëfficients of
$\frac{1}{r_1} \hookrightarrow 2C_{100}+C_{010}= 0$ and that the coëfficients of
$\frac{r_{1}}{r_{12}} \hookrightarrow 5C_{101}-\frac{1}{2} C_{100} = 0$.
Fock
(V. A. Fock, Izv. Akad. Nauk SSSR, Ser. Fiz. 18, 161
(1954).,V. A. Fock, On the Schrödinger equation of the Helium atom, Det Koneglige Norske Videnskabers Selskabs
Forhandlinger 31, 138 (1958)., E. Z. Liverts and N. Barnea, Angular fock coefficients: Refinement and further development, Phys. Rev. A92, 42512
(2015).) showed that there must be logarithmic terms in the expansion, and gave the form the expansion had to take.
Defining $R=\sqrt{r_1^2+r_2^2}$, he wrote (in current but mixed notation)
\begin{eqnarray}
\psi = 1 -Z (r_1+r_2) +\frac{1}{2}r_{12}+
\nonumber \\
R
\left (\psi_{2,1} \ell n R
+ \psi _{2,0} \right )
+
\nonumber \\
R^{1/2} \left (\psi_{5/2}R \ell n R
+ R\psi _{5/2,0}
\right )
+
\nonumber \\
R^2 \left (\psi_{3,2}\ell n^2 R +
\psi_{3,1} \ell n R +\psi_{3,0}
\right )
+\dots
\end{eqnarray}
with $Z=2$ for Helium.
$\psi_{2,1}$ was shown by Fock to be a linear combination of
$\cos \alpha$ and $\sin \alpha \cos \vartheta$
where $\cos \alpha \hookrightarrow r_1^2-r_2^2$ and
$\sin \alpha \cos \vartheta \hookrightarrow 1-\frac{r_{12}^2}{r_1^2+r_2^2}$. Here
$R^{1/2}\cos(\alpha/2)$ and $R^{1/2}\sin(\alpha/2)$ are equal to $r_1$ and $r_2$ respectively, while $r_{12}$ is equal to $R^{1/2}\sqrt{(1-\sin\alpha\cos\vartheta)}$.
It is hard to fit the Fock form coherently into the Bartlet et al. formulation of an $ Ansatz$ for the Schrödinger Equation's possible solution.
The idea that the coëfficients of $\psi_{2,1} $ are a linear combination of $\sin \alpha \sin \vartheta$ and $\cos\alpha$ is worrisome, as $\cos{\alpha}$ is an odd function of $r_1$ and $r_2$ and the ground state of 2-electron systems should be even with respect to these two variables. If the coëfficient of $\cos \alpha$ in $\psi_{2,1} $ is indeed zero, then $\psi_{2,1} =\frac{ r_1^2+r_2^2 - r_{12}^2}{r_1^2+r_2^2}$.
The $Ansatz$
The following expansion is proposed:
\begin{eqnarray}
1 + C_{1,0,0}r_1 + C_{0,1,0}r_2 + C_{0,0,1}r_{12} +
\nonumber \\
+C_{1,1,0}r_1r_2 + C_{1,0,1}r_1r_{12} +
C_{0,1,1}r_2r_{12}+
\\ \nonumber
constant*
\left (
r_1r_{12}\ell n r_1r_{12}
+ r_2r_{12}\ell n r_2r_{12}
\right )
+\dots
\label{eqone}
\end{eqnarray}
as the first few terms.
The last term shown in (Eqn. 1) mitigates the Bartlett, Gibbons and Dunn conundrum.
This suggests the following $Ansatz$:
\begin{eqnarray}
\psi = \sum_{i,j,k,\ell}C_{i,j,k,\ell}r_1^ir_2^jr_{12}^k
\left [
\left (r_1r_{12}
\ell n r_1r_{12}
\right)^\ell
+
\left (r_1r_{12}
\ell n r_2r_{12}
\right)^\ell
\right ]
\end{eqnarray}
(with restrictions on $i,j,k$ and $\ell$), but it fails the Bartlett Gibbons and Dunn criterion.
Instead, the following is proposed:
\begin{eqnarray}
\psi = \sum_{i,j,k=0}C_{i,j,k,0}r_1^ir_2^jr_{12}^k
\nonumber \\
+
\sum_{\{i,j,k\}=0,\ell=1}C_{i,j,k,\ell}r_1^ir_2^jr_{12}^k
\left (
r_1r_{12}\left (
\ell n r_1r_{12}
\right)^\ell
+
r_2r_{12} \left (
\ell n r_2r_{12}
\right)^\ell
\right )
\label{ansatz}
\end{eqnarray}
with $C_{0,0,0,0}=1$.
The Hamiltonian and Leading Terms in the Expansion
The Hamiltonian Operator used in the best calculation to date
(
H. Nakashima and H. Nakatsujja, J. Chem. Phys. 127,
224104 (2007).)
applicable to the ground state of the Helium-like atom's/ion's two electrons is:
\begin{eqnarray}
-\frac{1}{2 }
\left (
\frac{1}{r_1^2}
\frac{\partial \left (r_1^2\frac{\partial}{\partial r_1}\right ) }{\partial r_1} +
\frac{1}{r_2^2}
\frac{\partial\left ( r_2^2\frac{\partial}{\partial r_2}\right ) }{\partial r_2}
\right )-
\frac{1}{r_{12}^2}
\frac{\partial\left ( r_{12}^2\frac{\partial}{\partial r_{12}}\right ) }{\partial r_{12}}
\nonumber \\
-
\frac{r_1^2-r_2^2+r_{12}^2}{2r_1r_{12}}
\frac{\partial^2}{\partial r_{1}\partial r_{12}}
-
\frac{r_2^2-r_1^2+r_{12}^2}{2r_2r_{12}}
\frac{\partial^2}{\partial r_{2}\partial r_{12}}
\nonumber \\
-\frac{Z}{r_1}
-\frac{Z}{r_2}
+\frac{\lambda}{r_{12}}
\label{ham}
\end{eqnarray}
where $\lambda = 1$ and $Z$ is the atomic number of the nucleus.
Substituting our $Ansatz$ into the Hamiltonian of the Schrödinger Equation
$H\psi=E\psi$ and setting coëfficients to zero sequentially,
we obtain $C_{1,0,0,0}=C_{0,1,0,0} = -Z$,
and $C_{0,0,1,0}= \lambda /2$ as expected.
Results and Discussion
Table 1, Sequential coëfficients
| Term |
Coëfficient |
| $\frac{r_1^2}{r_2r_{12}}$ |
$ C_{1,0,1,0}= -2*(C_{0,0,0,1}+C_{0,0,0,2})$ |
| $\frac{r_2^2}{r_1r_{12}}$ |
$ C_{0,1,1,0}= -2*(C_{0,0,0,1}+C_{0,0,0,2})$ |
| $\frac{1}{r_1}$, $\frac{1}{r_2}$
|
$C_{1,1,0,0}=Z^2$ |
| $\frac{r_2^2\ell n(r_1r_{12})}{r_1r_{12}}$,
$\frac{r_1^2\ell n(r_2r_{12})}{r_2r_{12}}$ |
$C_{0,0,0,1}=-(4C_{0,0,0,2}+6C_{0,0,0,3}) $ |
| $\frac{r_2r_{12}}{r_1}$,
$\frac{r_1r_{12}}{r_2}$ |
$C_{1,1,1,0}=-(12ZC_{0,0,0,2}+24ZC_{0,0,0,3})/3$ |
| $\frac{r_{12}}{r_1}$,$\frac{r_{12}}{r_2}$ |
$ C_{0,0,0,2}=2(Z\lambda/2+3C_{0,0,0,3})$ |
Table 1 indicates the beginning sequence of coëfficients determined sequentially substituting terms of the expansion of the wave function $Ansatz$
(Eqn. \ref{ansatz})
into the Hamiltonian
(Eqn. \ref{ham})
of the Schrödinger equation.
Demurral
The continuation of developments with respect to coëfficient recursion relationships, variational calculations based on wave function choices shaped by the results shown here, and the ultimate finding of simple functional forms for which this series is the expansion, exceeds the ability and timeline of the author.
It is not apparent what sequence one needs to solve the succeeding equations.
However, approximate calculations which include logarithmic terms such as those indicated above, should allow smaller expansions with increased accuracy and a well defined path for improvement.
It is worth noting that literature statements that equate more accurate energy values (to more digits) to more accurate wave functions is not, in fact, true.
The contrary is expected to be true however, i.e., the more accurate the wave function the better the eigenenergy.
Acknowledgements
The mathematics employed herein was done in part using SageMath/CoCalc.
SageMath/CoCalc code
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