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  

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