Prototype RealQM for Helium Atom: Test Yourself!

Real Quantum Mechanics (RealQM) offers a new atom/ion/molecule model in the form of a system of non-overlapping electron densities as a classical continuum mechanics Schrödinger model in 3 space dimensions + time. This model describes a reality/actuality and is readily computable also for many electrons, which is not the case for standard QM (stdQM) as a multi-dimensional statistical model describing possibilities instead of actualities. See previous post for a connection to the 2023 Nobel Prize in Physics depicting electron charge densities, thus supporting RealQM, while real pictures of stdQM are missing. 

A prototype computational realisation of RealQM for the Helium atom with two electrons can be inspected here. The ground state of Helium is computed by minimising the total energy consisting of kinetic energy, attractive kernel potential energy of kernel at $x=0$ and repulsive mutual electron energy:  

• $\frac{1}{2}\int w(x)\vert\nabla u(x)\vert^2 dx -\int\frac{2u^2(x)}{\vert x\vert}dx+\int\int\frac{u^2(x)u^2(y)}{\vert x-y\vert}dxdy$

over a decomposition over a fixed grid of $u(x)=u_1(x)+u_2(x)$ into two electron density functions $u_1(x)$ and $u_2(x)$ with disjoint supports in 3d space given by characteristic functions $w_1(x)$ and $w_2(x)$ with $u_1^2(x)$ the charge density of electron 1 and $u_2^2(x)$ that of electron 2, with the boundary between $w_1$ and $w_2$ acting as a free boundary determined to achieve that u_1 and u_2 agree on the free boundary so that the electrons meet with same density.

The code consists of a couple of lines expressing gradient minimisation involving

1. Explicit relaxation of Hamiltonian in u + charge density normalisation (involves homogeneous Neuman conditions for u_1 and u_2 on free boundary enforced by the presence of $w(x)$ in the kinetic energy).
2. Explicit relaxation of Poisson problem for electron potentials.
3. Explicit level set front tracking of w to reach continuity of u over fixed grid.

which can be seen as a Bernoulli free boundary problem with homogeneous Neumann + continuity as free boundary condition.  

Run the code by clicking the arrow in the p5js code to see electrons initiated away from the kernel approaching the kernel and meeting at a free boundary with continuity (and approx homogeneous Neumann condition because of coarse resolution):   

and experiment further by modifying the code. It is fun and illuminating!

Compare with 1st excited state with 1 electron in 1st shell and 1 electron in 2nd shell (with p5js code):

Here you can test RealQM for the H2 molecule including minimisation over kernel distance:

Here you can test 2-shell atoms/ions starting with Lithium with 2 electrons in 1st shell and 1 electron in 2nd shell, continuing with always 2 electrons in 1st shell and Beryllium/ions with 2 electrons in 2nd shell.

This is a preparation for atoms/ions with more than 2 shells starting with Boron with 2+2+1 in 3 shells, Carbon 2+2+2, Nitrogen 2+2+3, Oxygen 2+2+4, Fluorine 2+3+4 and Neon 2+4+4 filling the first period.

And so on through the whole table... 

The electrons in each shell are captured by one electron density function carrying the total electron charge.

Here a test (with p5js-code to test) for Beryllium with 4 electrons in 2 shells initiated with a gap between of the 2 electrons in 2nd shell which closes under free boundary level set tracking:

See further RealQM results:

• Lithium (2++1)
• Carbon (2+4) 
• more to come

Computations in 3d without assuming spherical symmetry. Compare with earlier Atom Simulator in spherical symmetry.