Notes on Molecular Modelling

Here, Molecular modelling is a set of all methods/technique used to study particles (atoms, molecules, electrons, …) dynamics and behaviour.

Objects: particles (sphere(hard, soft), atoms, molecules, electrons, …) Method: molecular modelling

• simulation methods:
  • molecular dynamics (MD)
  • ab-initio MD (uses QM for forces, e.g. DFT)
    • electronic structure methods (engine for ab-initio MD)
    • DFT, HF
  • classical MD (uses force fields)
    • modeling strategies:
    • coarse-grained methods
    • all-atom models
  • Monte Carlo (MC)

Simulation in Principle

The goal of this section is to introduce the fundamentals of molecular modelling. The first part of the section are an introduction on the underlying concepts of how, why, and what makes molecular dynamic simulation works. The next part will be a more advance level, that is getting a graduate level understanding of some undergraduate topic.

Bird’s Eye View of Molecular Modelling

In essence, molecular modelling simulation concerns with modeling the dynamics of a set of atomic and molecular system under a thermodynamic condition. The goal is to study its behaviour in the most detailed but in a computationally efficient manner. Different methods have been introduced to reduce the representation into a manageable system. For instance, you have a hard-sphere, a soft-sphere, a coarse-graind, a rigid molecules, a flexible molecule, and a dissociative or reactive molecules (see Particle Model). Each system has increasing level of details that adds an additional overhead to the simulation.

The interactions within an system of particles are governed by an interatomic potential, also known as a force field. Starting with the time-dependent Schrodinger equation, you can derive an approximate equation for its dynamics (see MD simulation) or a more precise one(see ab-initio MD), though at a higher computational cost. Many methods have been developed to approximate these force fields, but they are often applicable only under specific conditions and must be used with caution. For computational efficiency, interactions are typically calculated only within a certain distance, or cut-off. To properly account for interactions beyond this range (like long-range electrostatic forces), correction terms must also be included. To achieve higher chemical accuracy, several combinatorial machine learning methods have been developed to generate surrogate models for solving the Schrödinger equation (or its equivalent e.g. DFT) more quickly.

Once a simulation generates a trajectory (a sequence of particle positions and velocities) statistical mechanics provides the theoretical framework to extract meaningful macroscopic properties. It serves as the crucial bridge between the microscopic dynamics of the simulation and the macroscopic properties (e.g., pressure, temperature) of the real-world system. Different statistical ensembles (such as NVE, NVT, or NPT) can be chosen to study different sets of these properties.

The significant computational demands of molecular modelling have made parallel programming a key driver of its development. Consequently, a strong foundation in high-performance computing (HPC) is essential in this field. Proficiency in programming languages like Fortran, C++, and Python is also crucial for developing and running efficient simulations that solve the system’s equations of motion.

Particle Model

• molecular dynamics is an example of an application of particle model. This is based on the Born-Oppenheimer approximation where we can think of the electron instantaneously changing its position as we change the nuclei. The motion of the nuclei is under the influence of the electronic solution of the Schrodinger equation.
• the trajectory of the particle is basically the trajectory of the nuclei

Foundations

• In MD, the basic idea is the use of Newton’s equation and a predefined potential for a system to get the dynamics of a system
  • use of Hamiltonian and Lagrangian formulation to show that conventional MD is NVE
  • the concept of erogidicity is exploited in the field of MD, the idea of ensemble average is equal to the time average measurement

Molecular Dynamics Simulation

Uses an approximate potential where the nuclei can move, requires a parameter which has to be adjusted using fitting method to reproduce thermodynamic properties.

Ab-initio molecular dynamics

It’s goal is to “reduce the amount of fitting and guesswork” [3].

Chemical Reactions Modeling with Force Fields

• characterized by bond-breaking and bond-forming processes
  • mixed QM/MM
  • RMD (Reactive Molecular Dynamics)
  • EVB (Empirical Valence Bond)
  • AVB (Approximate Valence Bond)
  • MCMM (multiconfiguration molecular mechanics)
  • the use of geometric formulas to switch on and off interactions individually (e.g. ReaxFF)
  • adiabatic reative MD (ARMD)
  • multi-surface variant (MS-ARMD)
  • molecular mechanics with proton transfer (MMPT)

Computer Simulation

• MD is like an experiment - adapt a mentality of an experimenter not a theorist
  • instead of merely performing a calculation, the computer becomes the virtual laboratory in which a system is studied - a numerical experiment.
• variety of modeling techniques developed over the years
  • molecular mechanics
  • classical Monte Carlo, biased Monte Carlo
  • quantum based techniques involving path-integral and MCmethods
  • MD + electron density function theory
  • cellular automata
  • Lattice-Boltzmann method
  • hybrid QM/MM

Application

• drug discovery and design
• materials science
• biochemistry and structural biology
• chemistry/physics

References

1. Rapaport, D. C. (2004). The art of molecular dynamics simulation. Cambridge university press.
2. Allen, M. P., & Tildesley, D. J. (2017). Computer simulation of liquids. Oxford university press.
3. Frenkel, D., & Smit, B. (2023). Understanding molecular simulation: from algorithms to applications. Elsevier.
4. Allen, M. P. (2004). Introduction to molecular dynamics simulation. Computational soft matter: from synthetic polymers to proteins, 23(1), 1-28.