Helium – Two-Electron Ions

The Helium atom as two electron charge densities restricted to half-spaces meeting at plane through the kernel as free boundary under a Bernoulli condition = homogeneous Neumann  + continuity.  The basic “two-valuedness” in stdQM expressed by Pauli’s Exclusion Principle is here replaced by a the two-valuedness of the half-space subdivision, not asking for any ad hoc introduction of the notion of spin.

As first crucial test case beyond Hydrogen, we now consider realQM for Helium and two-electron ions with $N=2$ seeking a minimiser among wave functions $\psi =\psi_1+\psi_2$ arguing that by symmetry we may assume that $\psi_1$ is supported by the half-space $\Omega_1=\{x:x_1\ge 0\, and\, \vert x\vert >\epsilon\}$ and $\psi_2$ by $\Omega_2=\{x:x_1\le 0\, and\, \vert x\vert >\epsilon\}$ with the plane $\Gamma_1=\{x:x_1=0\, and\, \vert x\vert >\epsilon\}$ as free boundary, where $\epsilon > 0$ is the (small) radius of the kernel.  More precisely, we assume by symmetry that $\psi_2(x_1,x_2,x_3)=\psi_1(-x_1,x_2,x_3)$ for $x_1\le 0$, which allows restriction of attention to $\psi_1$.

The ground state $\Psi_1$ then appears as the minimiser over $H^1(\Omega_1)$ of the total energy

• $TE(\psi_1)=\frac{1}{2}\int_{\Omega_1}\vert\nabla \psi_1\vert^2dx+\int_{\Omega_1}W\psi_1^2dx+\int_{\Omega_1}V_2\psi_1^2dx$

under the normalisation

• $\int_{\Omega_1}\psi_1^2(x)dx=1$

where the electron potential $V_2(x)$ for $x\in\Omega_1$ is given by

• $V_2(x)=\int_{\Omega_2}\frac{\psi_1^2(-y_1,y_2,y_3)}{2\vert x-y\vert}dy$.

We observe that $V_2$ is the solution of the equation $-\Delta V_2=2\pi f$ in $\Re^3$ with $f(x)=\psi_1^2(-x_1,x_2,x_3)$ for $x_1<0 \, and \vert x\vert >\epsilon$ and $f(x)=0$ else, which gives an alternative way of computing $V_2$ other then direct integration.

The condition of stationarity with respect to variations of $\Psi_1$ takes the form of the eigenvalue problem

• $H\Psi_1=E_1\Psi_1\quad\mbox{in }\Omega_1,\quad\frac{\partial\Psi_1}{\partial n}=0\quad\mbox{on } \Gamma_1$,

for the Hamiltonian

• $H = -\frac{1}{2}\Delta+W+2V_2$.

Homogeneous Neumann Condition at Free Boundary and Kernel

The wave function  is subject to a homogeneous Neumann condition on the free boundary separating the two electrons and also at a small distance from the kernel thus attributing a small effective radius $\epsilon$ to the kernel.

The homogeneous Neumann condition at the kernel plays an important role to give energies in correspondence with observation, since without this condition energies come out to be too small. For example, for Helium without the homogeneous Neumann condition the energy is about -3.0 which is significantly lower than the observed. With this condition the energy fits the exact energy for a small effective kernel radius, which thus acts as a parameter.

The effective kernel radius corresponds to 2-7 mesh points with the number of mesh points decreasing as the kernel charge increases to a minimum of 2 points required to effectively implement the homogeneous Neumann condition. Increasing the effective radius increases the energy.

Notice that since realQM is a physical model, a positive kernel radius makes more sense than in stdQM which is a non-physical model.

Robin Boundary Condition at Kernel

We compare using a Robin boundary condition at the kernel, which gives slightly smaller kernel radii.

Computational Results

We compute in spherical coordinates with azimuthal symmetry with a decomposition of the wave function into a major polarly symmetric part and a smaller polarly unsymmetrical part with the electronic potential for the symmetrical part computed analytically.

The best results of stdQM for Helium with Hartree-Fock is E = -2.903724376 (Pekeris 1959) to be compared with the experimental  E =  -2.90338 by NIST Atomic Data Base suggesting an accuracy of (modest) 3 decimals.

For an atom with many electrons in a shell configuration, it is natural to homogenise the inner shells and only keep electron individuality in the outermost (valence) shell. This gives a setting with 1 to 8 electrons in an outer shell interacting with an effective kernel of a certain radius, thus connecting to Helium with positive kernel radius.

We give below the computed ground state energies for the first two rows of the periodic table with Z=1,…,10 with the observed energies listed in the paragraph head lines. We note that the the effective radius of the two electrons scales with 1/Z resulting in an effective kernel radius changing from 0.05 for Z=2 to about 0.01 for Z=10. In case Z=1 and N=1 just one electron surrounds the kernel and the effective kernel radius comes out smaller than for Z=2.

The electrons of atoms/ions with more than two electrons arrange in a shell structure with a homogeneous Neumann condition on the free boundary separating electronic shells. In this perspective the kernel with positive radius and a homogeneous Neumann condition appears as core shell fitting into the shell structure.

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