RealQM Molecule Model 2D

To illustrate the basic feature of the new Schrödinger equation of Real Quantum Mechanics RealQM, let us  reduce from 3d to 2d, to get the following formulation for a molecule with $N$ electrons (like graphene):

Find the wave function $\Psi (x)$ with $x$ a 2d Euclidean coordinate, of the form 

• $\Psi (x) = \Psi_1(x) + \Psi_2 (x) +    + \Psi_N(x)$

where the $\Psi_n(x)$ for $n=1,2,...,N$ are one-electron wave functions with non-overlapping supports meeting a Bernoulli free boundary $\Gamma$, which minimizes the total energy 

• $E(\Psi ) = \frac{1}{2}\sum_{n=1}^N\int \vert\nabla\Psi_n\vert^2dx-\int P(x)\Psi^2dx$  (kinetic + potential energy)

with $P(x)$ a potential with contribution from electron and kernel charges, under the side condition 

• $\int\Psi_n^2 dx = 1$ for $n=1,2,...,N,$

and the Bernoulli free boundary condition:

• $\Psi (x)$ is continuous and 
• the normal derivative of $\Psi_n(x)$ vanishes on $\Gamma$ for $n=1,2,...,N.$  

This minimisation problem is solved with a gradient method realised as an explicit update consisting of three lines for iterative update of (i) wave function, (ii) level set function for $\Gamma$ and (iii) potential $P(x)$ realised in this code.

We consider a molecule consisting of two H atoms in green and one Beryllium atom with valence shell consisting of two electrons as two "half-shells" in red and blue. We start the iteration with the electrons concentrated into disks without overlap:

 

We find the following configuration after energy minimisation using this code to see the formation of a free boundary between electron wave functions:

We see that electron wave functions meet at a free boundary with continuity and zero normal derivative displayed in a horizontal cross-cut in yellow and vertical in magenta. We find a total energy of -12.34.

We compare with the atoms well separated with a total energy of -5.803 after energy minimisation.  

We understand that the reason the energy is lower for electrons in Bernoulli free boundary contact is that the charge density is concentrated between the kernels thus decreasing potential energy without increase of kinetic energy measured by $\nabla\Psi_n(x)$, since RealQM wave functions are not required to tend to zero like the global wave functions of stdQM.