Molecule Formation by RealQM

This post gives a summary with more details of recent posts exploring the potential of RealQM for simulation of molecule formation as a collection of atoms find a joint configuration with lower total energy than separate atoms. The main ingredients are: 

1.  Each atom is represented by an outer shell of negative valence electron charge density surrounding an inner shell of positive charge density representing the atom kernel shielded by remaining electrons. Charge densities do not overlap. 
2. Potentials appear as solutions to Poisson equations with charge density input. Each charge density interacts with the potentials from the other densities but not with its own.
3. The total energy $E$ is the sum of kinetic energy as measure of charge density spatial gradients and potential energy as measure of charge densities weighted by potentials.
4. Configuration of valence electrons for given inner shells/kernels are computed by gradient minimization of $E$ over valence charge densities meeting at a Bernoulli free boundary with continuity of charge density and zero normal derivative.
5. Inner shell configurations identified by kernel positions are for given valence charge densities updated from gradients of potentials acting on inner shell charges. Inner shell charges are kept constant over spheres of constant radius (for simplicity).  
6. The computer code consists of essentially three lines for update of (i) charge densities, (ii) potentials and (iii) free boundary followed by update of kernel positions based on gradients of potentials. 

We may compare with StdQM molecule dynamics, where for given kernel positions electron configurations are computed by DFT and then kernel positions are updated from computation of total energy gradients altogether using very complex computer codes.  

You can test RealQM for a 2d model with three atoms like Oxygen O forming an O3 ozon molecule by running this code with this output during iteration to minimum of total energy:

We see three atoms, each atom consisting of an inner shell with atom kernel shielded by electrons surrounded by an outer shell of valence electron charge density (red), which meet at a free boundary between the kernels. We see cross-cut through lower kernels of electron charge density (green), global potentials for lower inner shells (blue) with gradients (light-blue) indicating directions of update of kernel positions. We see kernels slowly moving in the directions given by light-blue gradients which tend to zero in approach to energy minimum.  

This is an exercise to show qualitative performance of RealQM for simulation of formation of a molecule froms atoms coming together in a process of energy minimisation. The computational complexity scales linearly with the total number of valence electrons, and so opens to simulations of formation of large molecules.

PS Here is description in mathematical notation;. The wave function $\Psi (x)$ with $x$ a 3d Euclidean space coordinate for the system/molecule takes the form of a sum of atom wave functions $\psi_i(x)$ and $\psi_{i+N}(x)$ for atom $i=1,...,N$,

• $\Psi (x) = \sum_{i=1}^{2N}\psi_i(x)$ 

where $\psi_1(x),...\psi_N(x)$ represent valence electrons of negative charge, $\psi_{N+1},...\psi_{2N}(x)$ represent inner shells of positive charge and all wave functions have disjoint support with total charge 1: 

• $\int\psi_i^2dx = 1$ for $i=1,...N$ and $\int\psi_i^2dx = -1$ for $i=N+1,...,2N$. 

The total energy $E(x)$ of the valence electrons is given as a sum of kinetic energy and potential energy: 

• $E(x)=\frac{1}{2}\sum_{i=1}^N\int\vert\nabla\psi_i(x)\vert^2dx+\sum_{i=1}^NP_i(x)\psi_i(x)^2dx$

where $P_i(x)$ for $i=1,..,N$ is the total potential acting on valence electron $i$ with contribution from other valence electrons and inner shells through solution to the Poisson problem 

• $-\Delta P_i =\frac{1}{2} \sum_{j=1,j\neq i}^N\psi_j^2 - \sum_{j=N+1}^{2N}\psi_j^2$, 

where the factor $\frac{1}{2}$ balances double count of electrons. 

The formation of the molecule is realised by minimisation of $E(x)$ over valence electrons and kernel positions with valence charge densities meeting at a Bernoulli free boundary. 

To guide motion of kernel positions we compute gradients of the potentials $P_i(x)$ for  $i=N+1,...2N$ acting on inner shells as solutions to  

• $-\Delta\Psi_i =\sum_{j=1}^N\psi_j^2 - \frac{1}{2}\sum_{j=N+1,j\neq i}^{2N}\psi_j^2$.

 

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.