Molecule Formation by RealQM: Basic Test

To illustrate the potential of RealQM for simulation of the formation of molecules in covalent bonding, let as here consider a 2d model of an X3 molecule formed by three X atoms each represented by a negative electron valence charge outside an inner shell of positive charge with the following initial state:

We see the valence charge in red around an inner shell. We run this code (uniform 100 x 100 mesh) to follow the formation of the molecule as valence charges evolve to meet at a free boundary.

We see cross-cut through lower atoms in green of valence and inner shell charges, together with potentials acting on inner shells in blue. We decrease the inner shell charge from 2 to 1.8 to get using this code:

Here the average potential gradient acting on inner shell charge is displayed in light-blue showing a net attraction signifying negative valence charge accumulation between the atoms overpowering positive charge repulsion. Changing to 2.2 we get with this code:

We see potential gradients change sign with now positive charge repulsion overpowering negative charge attraction. We conclude that equilibrium is reached for C around 2. 

We understand that the secret of molecule formation is the accumulation of negative charge between the atoms without increase of kinetic energy because electron charges meet at a Bernoulli free boundary with non-zero density. This is the essential new physics brought by RealQM, which appears to unlock the secret of covalent bonding.

We here change the inner shell charge rather than the distance between atoms, for display simplicity.

The purpose of this exercise is to show the potential of RealQM for simulation of molecules with many electrons. The computational cost on a given mesh scales linearly with number of valence electrons involved thus with number of atoms.  

Compare with earlier 3d examples under tag Real Quantum Chemistry.vs 

Let us compare RealQM and StdQM vs the key ingredients of (1) kernel potential energy from accumulation of electron density between kernels (-), (2) electron kinetic energy (+) and (3) electron repulsion energy (+), where we indicate sign of contribution to total energy. Here RealQM has an advantage concerning (2) because of the Bernoulli free boundary and (3) because non-overlapping charge densities have smaller electron repulsion energy than overlapping densities. 

Evidence of (3) is given by StdQM energy for Helium of -2.75 with overlapping Hydrogen electron densities, while RealQM gives -2.87. 

We conclude that RealQM has a better chance to capture the secret covalent bonding than StdQM. 

Here is a first version of RealQM code for dynamic molecule formation with kernel geometry determined by electron potentials.

Here is a conversation with chatGPT showing that covalent bonding is not well understood even today 100 years after the advent of quantum mechanics. This is mind boggling…

PS Specifically, we compare

• $R1= \int_{R3}\int_{R3}\frac{\exp(-\vert x\vert)\exp(-\vert y\vert )}{\vert x -y\vert } dxdy$ 
• $R2= 4\int _{R31}\int_{R32}\frac{\exp(-\vert x\vert)\exp(-\vert y\vert )}{\vert x -y\vert }dxdy$ 

where $R3$ is all of 3d space, $R31$ is the half space $x_1<0$ and $R32$ is the half space $y_1>0$ with

$x=(x_1,x_2,x_3)$ and $y=(y_1,y_2,y_3)$, and find that $R2<R1$. Non-overlapping gives smaller electron repulsion energy than overlapping, as expected.