Lower limit for atomic number in case of imaginary ion with Z < 1
In my programming project I calculate the minimal energy of an atom with 2 electrons in the $L=0, S=0$ state, using a Hylleraas wave function.
The values I find for $Z=2$ (He) and $Z=1$ (H$^-$) are in good correspondence with what can be found in literature (Pekeris, 1962):
* $E_0({\rm He}) \approx -2.903~{\rm a.u.}$
* $E_0({\rm H^-}) \approx -0.528~{\rm a.u.}$
Here a.u. are the atomic units in which the energy of the hydrogen atom is $-0.5$. However, the next part of the project is to find a lower limit for $Z$, e.i. imagine a theoretical ion with $0<Z<1$ $(Z\in \mathbb{R})$, what is the minimal value for $Z$ in order for the two-electron system to remain bound? This is equivalent with asking: for which value of $Z$ is $E_0=0$?
I cannot find any references about this theoretical lower limit online, but I don't think my results are correct: the $E(Z)$ relation, also shown in the graphs below, looks like a power law. 474
This means that $E_0\to0$ for $Z\to0$, but in this case you just have two electrons and you would expect a very positive energy as the two will strongly repel each other.
I do not expect you to help me with the programming project, but maybe someone could provide some useful thoughts or a reference about this theoretical lower limit? I keep finding it strange that the results for $Z=1,2$ are correct with errors of only 0.01% and everything seems to go wrong for small $Z$.
Note: how are these energies exactly calculated? We start with a Hylleraas wavefunction subjected to a coordinate rescaling factor $\alpha$: $$ \langle\vec{r}1\vec{r}_2|\Psi\alpha\rangle = \sum_{STU}C_{STU}N_{STU}{\rm e}^{-\alpha s/2}(\alpha s)^S(\alpha t)^T(\alpha u)^U $$ with $s = r_1 + r_2, t = r_1-r_2, u=r_{12}=|\vec{r}1-\vec{r}_2|$, $S,U\in\mathbb{N}$, $T\in2\mathbb{N}$ and $\alpha\in\mathbb{R}^+$. Three matrices can be calculated: the overlap matrix $[M]$, the kinetic energy $[T]$ and potential energy $[V].$ They scale like: $$\langle\Psi\alpha|\Psi_\alpha\rangle = \langle\Psi|\Psi\rangle/\alpha^6 $$ $$\langle\Psi_\alpha|T|\Psi_\alpha\rangle = \langle\Psi|T|\Psi\rangle/\alpha^4 $$ $$\langle\Psi_\alpha|V|\Psi_\alpha\rangle = \langle\Psi|V|\Psi\rangle/\alpha^5 $$ Variation to the expansion coefficients in $\langle\Psi_\alpha|(T+V)|\Psi_\alpha\rangle/\langle\Psi_\alpha|\Psi_\alpha\rangle$ leads to a generalized eigenvalue problem: $$ \left(\alpha^2[T]+\alpha[V]\right) C_\alpha = E_\alpha[M]C_\alpha $$ The lowest energy eigenvalue $E_\alpha^0$ gives a function of $\alpha$ of which the minimum needs to be determined. This minimum is the best variational approximation of the ground state energy.
UPDATE: It is interesting to plot the evolution of the coordinate rescaling factor $\alpha$ as a function of $Z$. Apparently, there is a discontinuity around $Z\approx 0.89841$, which is close to the critical value of $Z\approx 0.91$ found in literature. I would suppose this point has a significant meaning, but I have failed so far to give it a physical interpretation (I hope to find an interpretation which has to do with the ionization of at least 1 electron). 394