Introduction

Overview

Teaching: 30 min min
Exercises: 0 min

Questions

• How can we simulate the motion of molecules subject to realistic intramolecular forces?

Objectives

• We can solve Newton’s equations of motion for relative motion of atoms subject to intramolecular forces.

Molecular Dynamics simulations provide a powerful tool for studying the motion of atomic, molecular, and nanoscale systems (including biomolecular systems). In molecular dynamics simulation, the motion of the atoms are treated with classical mechanics; this means that the positions and velocities/momenta of the atoms are precisely specified throughout the simulation, and these quantities are updated using Newton’s laws of motion (or equivalent formulations).

A brief overview of different applications of molecular dynamics simulations, as well as an introduction to the key working equations can be found here, with some additional details here, as well as to references cited therein. An even more concise summary of this excercise can be found in these slides.

The motion of the atoms over the duration of a molecular dynamics simulation is known as a trajectory. The analysis of molecular dynamics trajectories can provide information about thermodynamic and kinetic quantities relevant for your system, and can also yield valuable insights into the mechanism of processes which occur over time. Thus molecular dynamics simulations are employed to study an enormous range of chemical, material, and biological systems.

One of the central quantities in a molecular dynamics simulation comes from the inter-particle forces. From Newton’s second law, we know that the acceleration a particle feels is related to the force it experiences;

The acceleration of each particle in turn determines how the position and momentum of each particle changes. Therefore, the trajectories one observes in a molecular dynamics simulation is governed by the forces experienced by the particles being simulated.

In this exercise, we will compute the forces experienced by our particles using the tools of quantum chemistry. In particular, we will simulate the vibrational motion of the diatomic molecule HF where the relevant force is a one-dimensional force acting along the interparticle separation. We will derive this force from the potential curve of the HF molecule, which depends on the HF separation or bond-length as follows,

where $V(r)$ denotes the potential as a function of the bondlength that can be readily computed from ab initio calculations. The resulting molecular dynamics simulation is often referred to as an ab initio molecular dynamics simulation, since the underlying forces are determined from first principles, i.e. through quantum mechanics. To the extent that our ab initio method is accurate, we will obtain accurate forces to recover realistic dynamics of the system we are studying.

Key Points

• We can use quantum chemistry to obtain a potential energy surface, from which realistic intramolecular forces can be derived. We can then perform molecular dynamics simulations to solve Newton’s equation of motion for the relative motion of atoms subject to realistic intramolecular forces.