Molecular Dynamics as Real Physics

This is a clarification of the previous post and this post.

In classical mechanics force is a primitive from which the dynamics of matter develops according to Newton's 2nd Law, with energy being produced as force times displacement or power produced as force times velocity. Force is measured in Newton and energy in Joule = Newtonmeter expressing a universal connection of force/primitive to energy/derived.   

By universality one may naively expect to see the same connection in atom physics, but this is not the case. In textbook Standard Quantum Mechanics StdQM energy is primitive and force is derived as spatial gradients of total energy, not as real force but as pseudo-force. 

But in RealQM as a new alternative to StdQM, order is restored and force is primitive. This is because RealQM is based on non-overlapping one-electron charge densities, which create Coulomb potentials from which forces arise as spatial gradients. 

Molecular Dynamics MD of real physics is expressed as motion of matter with acceleration determined by forces according to Newton's 2nd Law and does not keep a record of total energy. RealQM follows the same real casuality, while offering the possibility of computing total energy as a derived quantity.

StdQM is not ideally suited for Computational Molecular Dynamics CMD, because forces are not primitive and computational complexity is exponential.

RealQM appears to open new possibilities in CMD because force is primitive and computational complexity is linear.  

Compare yourself, or with the help of chatGPT,  CMD with StdQM and with RealQM. Result?

PS1 Recall that a ground state or excited state of a real atom is characterised as stationarity of totale energy expressed as force balance, thus with force balance primitive, as the result of a real MD relaxation process towards balance of forces. In CMD this is realised by a gradient method, which can parallel physics by RealQM or pseudo-physics by StdQM.

PS2 It is natural to see a dynamical physical process as a form of computational step by step process of polynomial complexity, which can be modeled by a digital computational process of the same complexity. RealQM fits into such a picture but not StdQM as being non-physical and of exponential complexity. 

RealQM vs StdQM: Primitives as Force or Energy?

RealQM is a mathematical model of classical continuum mechanics form of atomic physics based on local non-overlapping one-electron charge densities generating Coulomb potentials determining electrostatic forces.

A stationary state of an atom as ground state or excited state, is established in a dynamic process driven by forces towards force balance expressing minimum/stationarity of total energy. Forces include electrostatic forces and forces from presence of kinetic energy.

In classical physics, force is a primitive concept and energy is derived as force x displacement and then measured in Joule = Newtonmeter. Force balance is more primitive than minimum/stationarity of total energy. 

Real physics is governed by forces and does not have any means of measuring total energy to seek minimum/stationarity of total energy. 

Computational physics can compute total energy and through a gradient method proceed towards energy minimum/stationarity. 

RealQM can mimic real physics by computing forces to seek balance of forces, or in computational form use gradient method. 

We compare with textbook Standard Quantum Mechanics StdQM, where electrons do not have local non-overlapping support and so do not express electronic Coulomb potentials and so not electrostatic forces. StdQM is defined by minimum/stationarity of total energy and not by force balance. StdQM only allows determination of pseudo-forces as gradients of total energy. 

Let us compare Molecular Dynamics MD based on RealQM vs StdQM. 

• RealQM is like real physics based on forces as primitives and force balance defining stationary states. 
• StdQM is based on total energy which is not a primitive in real physics.   

It may be that RealQM opens new possibilities in MD. Computational complexity is linear in RealQM and exponential in StdQM.

Molecular Dynamics with RealQM

RealQM is an alternative version of textbook Standard Quantum Mechanics StdQM. RealQM is based on non-overlapping one-electron charge densities in 3d physical space which preserves fundamental principles of classical continuum mechanics of 

• casuality
• determinism
• locality
• separability. 

RealQM can be seen as the model Schrödinger was working towards after formulation his ground-breaking Schrödinger Equation SE capturing the spectrum of the H atom with one electron, as an extension to systems with many electrons. RealQM has shown to capture basic aspects of atoms and molecules.  

But history took another turn with a generalisation breaking with the above classical principles which became StdQM formulated in wave functions defined over $3N$ dimensional unphysical configuration space for a system with $N>1$ electrons. Schrödinger protested but was silenced by Bohr-Born-Heisenberg laying a foundation with is the same today. 

Let us consider the basic problem of formation of the H2 molecule from two neutral H atoms being drawn towards each other to find a stable configuration at a kernel distance of 1.4 atomic units under release of 0.17 Hartree as binding energy. 

RealQM models the dynamic formation of H2 by computing the forces acting on the kernels under Coulomb interaction between electrons and kernels and between kernel and kernel, and finding a balance at the distance 1.4 under release of 0.17 Hartree. The is a casual process driven by physical forces.

StdQM approaches the problem in a different way according to Born-Oppenheimer, by first locking kernel positions and then computing the corresponding electronic energy, which is added to a kernel-kernel potential and then differentiated with respect to kernel positions, followed by updating kernel positions in a process towards energy minimum. This is a non-casual process driven by non-physical forces.

The key difference is:

• RealQM: electron have local support and so build electronic potentials with physical forces.  
• StdQM: electrons have non-local support and so only build electronic energies without physical forces.

It thus appears that RealQM opens a new approach to Molecular Dynamics with possibly new capabilities in protein folding.

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.