Molecular Dynamics with RealQM

Molecular dynamics describes the internal motion of a single molecule (or set of molecules) as a collection of atomic kernels surrounded by electrons determined by Newtonian mechanics from a potential $V(R)$ depending on the geometric configuration of the kernels represented by $R$, from which kernel forces are determined as the gradient $\nabla_RV(R)$ with respect to $R$.

The potential $V(R)$ for a specific configuration $R$ is determined as the corresponding quantum mechanical electronic ground state energy $E(R)$, assuming that electrons quickly adjust to a new configuration, so that kernels move on a slower time scale than electrons.  

In particular, a stationary ground state of a molecule is determined as a configuration $R$ with minimal $E(R)$ or $\nabla_RV(R)=0$, the search of which only requires at path of $R$ over configurations.

In stdQM the cost of computing the potential $V(R)$ for many configurations is prohibitive, because already the cost for a single configuration scales with $100^{3N}$ where $N$ is the number of electrons, thus beyond any thinkable computer for $N>10$. Ab initio computation of $V(R)$ is thus unthinkable in stdQM and instead various reduced models have been tried such as Carr-Parrinello.  

Here RealQM appears to open entirely new possibilities because the cost of ab initio computation of  $V(R)$ for a single configuration instead scales with $N\times 100^3$, allowing computation of $V(R)$ over a wide range of $R$ with readily available computer power, and so directly $\nabla_RV(R)$ as difference quotient. 

As an example, which you can test yourself running this code and changing the parameter D, is the hydrogen molecule H2 as 2 +1 kernels each surrounded by 1 electron, which computes the following potential $V(R)$ depending on the distance $R$ between the kernels (in atomic units):  

• $V(1.0) = -1.040$
• $V(1.2) = -1.158$
• $V(1.4) = -1.170$
• $V(1.6) = -1.170$
• $V(1.8) = -1.157$
• $V(2)  = -1.145$
• $V(2.2) = -1.127$
• $V(3) = -1.106$
• $V(4) = -1.102$
• $V(5) = -1.013$

We see a minimum of $-1.170$ for $R=1.4-6$ in agreement with observations. Each 

computation is 3d and takes seconds on an iPad and so RealQM delivers the full potential function $V(R)$ for H2 in a minute. Similarly the potential function for other molecules covered in previous posts can be computed. 

It may be that RealQM can open a new window to ab initio molecular dynamics, simply because RealQM is computable while stdQM is not.