# Physics

## Molecular Dynamics

Molecular Dynamic (MD) is a type of molecular simulation method, which aims to study the dynamic evolution of physical systems through computer simulations of atoms and molecules. Based on MD simulations and statistical mechanics, many macroscopic thermodynamic properties, for instance, free energy or density, can be evaluated. Typically, trajectories of atoms in a simulation are generated by solving Newton’s laws of motion, where the potential energy function \(V\) comes either from force fields or Quantum Mechanic (QM) ab-initio calculations:

Depending on the smallest indivisible unit during a simulation, MD simulations can be roughly divided into two major categories: All-atom Molecular Dynamics and Coarse-grained Molecular Dynamics (CGMD):

**All-atom Molecular Dynamics**: each individual atom is treated as the smallest indivisible unit for motion and force calculations**Coarse-grained Molecular Dynamics**: a set of adjacent atoms (such as an amino acid residue, a water molecule) is treated as a coarse grained unit, usually referred as a “bead”. Only interactions between beads are considered, while all intra-bead interactions are neglected during a CGMD. This treatment makes CGMD capable of performing simulations on a larger time scale and for larger physical systems with reduced cost of computation and increased loss of accuracy.

Depending on the accuracy of potential energy functions used during a simulation, MD simulations can be divided into three categories: Classical Molecular Dynamics (Classical MD, cMD), Ab-initio Molecular Dynamics (AIMD) and Machine Learning Molecular Dynamics (MLMD):

**Classical Molecular Dynamics**: potential energy functions of the physical system come from a force field;**Ab-initio Molecular Dynamics**: potential energy functions of the physical system come from ab-initio calculations;**Machine Learning Molecular Dynamics**: potential energy functions of the physical system come from a machine learning force field.

## Potential Energy Function

Potential Energy Function, usually shortened as “Potential”, refers to the function that is used to describe the energy of interaction within a physical system. In an all-atom MD simulation the potential is a function of the atom types and atomic coordinates within the given physical system, and it could be given by quantum mechanics (QM), molecular mechanics (MM) force fields, or machine learning (ML) force fields.

## Force Field

Force Field, conventionally called Molecular Mechanics (MM) Force Field, refers to a collection of empirical functions with fixed mathematical formats to describe the potential energy of the physical system. Parameters for these empirical functions are determined by fitting against experimental data or QM-derived data. Compared to ab-initio methods, MM force fields are less accurate but much faster (usually several magnitudes).

## Hamiltonian

Under the context of classical mechanics, the concept of the Hamiltonian refers to the total energy of a physical system, which is the sum of the potential energy and the kinetic energy of all particles within the given system.

In quantum mechanics, the Hamiltonian should be considered as an Hamiltonian operator.

## Statistical Mechanics

In physics, statistical mechanics is a sub-discipline which applies statistical methods and probability theory to describe large assemblies of microscopic particles so that macroscopic behavior of the physical system (for instance, temperature, pressure) can be related to the behavior of microscopic particles.

## State Function

State Function is a physical property to describe the macroscopic property of a physical system. State functions have fixed values for a physical system under certain thermodynamic equilibria and depend only on the current equilibrium state of the system, rather than the path on which the system reaches equilibrium. Examples of State Functions include internal energy, enthalpy, entropy, free energy, etc.

## Ensemble

Ensemble is a concept in statistical mechanics, which refers to a collection of a large number of independent systems with identical properties and structures in various motion states under certain macroscopic conditions.

## Free Energy

The thermodynamic free energy refers to the energy of a thermodynamic system that can be used to do external work. It can be used as a criterion for whether a thermodynamic process can proceed spontaneously. Under given constraints, the system always tends to transition to a state with low free energy. For example, the process of protein folding is the spontaneous transition from an unfolded state with higher free energy to a folded state with lower free energy. According to the different qualifications, it can be divided into Helmholtz free energy (common notation \(F\)) and Gibbs free energy (common notation \(G\)). Note: free energy is different from potential energy although many people may confuse them.

## Boltzmann Distribution

In statistical mechanics, the Boltzmann distribution describes the In statistical mechanics, the Boltzmann distribution describes the probability distribution of particles in a system in possible microscopic quantum states, and has the following form:

where \(E\) is the quantum state energy, \(k\) is the Boltzmann constant \(T\) is the temperature, \(p_i\) is the probability that the particle is in the \(i\) quantum state, and ε\(_i\) is the energy of the \(i\) quantum state.

## Collective Variables (Reaction Coordinates)

The representative parameters that can quantitatively describe the change process of the system are called Collective Variables (CV) or Reaction Coordinates (RC). For example, in the chemical reaction shown in the figure below, the distance between O and C \(d(\mathrm{C-O})\) can be regarded as the reaction coordinate, and the distance between C and Br \(d(\mathrm{Br-C})\) can also be regarded as the reaction coordinate.

Given that the reaction coordinates are well defined, methods such as umbrella sampling can be used to estimate the free energy difference between different reaction coordinates through molecular simulation, and then the free energy change along with the reaction coordinates during the transforming process can be described, which is the basis of kinetic and thermodynamic research.

## Slow Degrees of Freedom

In the process of dynamic simulation, some degrees of freedom change rapidly with time (such as bond length, bond angle, etc., usually on the order of fs or ps). And some degrees of freedom change slowly with time (such as the dihedral angle, usually on the order of ns, \(\mu\) s , or even ms).

## Enhanced Sampling

Enhanced sampling refers to accelerating the sampling of slow degrees of freedom in the simulation process by some technical means, which are classified as collective variable-based (e.g. umbrella sampling), and collective variable-free (e.g. replica exchange).

## Quantum Mechanics

Quantum Mechanics is a branch of physics that studies microscopic systems. By describing the motion and interaction of microscopic particles (such as electrons, protons, etc.), quantum mechanics can explain many experimental phenomena that cannot be explained under the framework of classical mechanics, including blackbody radiation and the spectrum of the hydrogen atom.

## Operator

Generally, an operator acts on the state space of a physical system, making the physical system transform from one state to another. Within the context of quantum mechanics, the state of a system can be described by a state vector. Physical observables (such as position, momentum, Hamiltonian, etc.) all correspond to a (Hermitian) operator.

## Schrödinger Equation

In quantum mechanics, the Schrödinger equation is a partial differential equation that describes the time evolution of the quantum state of a physical system and is the fundamental equation of quantum mechanics. The Schrödinger equation can be divided into two types: the “time-dependent Schrödinger equation”

and the “time-independent Schrödinger equation” (also known as the steady-state Schrödinger equation)

The time-dependent Schrödinger equation describes how the wave function of a quantum system evolves over time, while the time-independent Schrödinger equation describes the physical properties of a stationary quantum system.

## First Principle

First Principle, also called ab initio, refers to derivation and calculation based on the basic laws of physics without additional assumptions and empirical fitting. For example, the of use the Schrodinger equation to solve electronic structure.

## Wave Function

In quantum mechanics, the state of a quantum system can be described by a wave function. The wave function Ψ(**r**,t) is a complex-valued function. According to Bonn’s statistical interpretation, \(|Ψ|^2\) is the probability density of finding a particle at position **r**, time \(t\).

## Born-Oppenheimer Approximation

The Born-Oppenheimer approximation refers to the approximate variable separation of the nuclear coordinates and the electron coordinates when solving quantum mechanical equations containing the nucleus and electrons, to decompose the wave function of the whole system into separately solving the nuclear wave function and the electron wave function, which are two relatively simple processes. The basis of this approximation is that the mass of the nucleus is 3 to 4 orders of magnitude larger than that of the electron, and the speed of the nucleus is much smaller than that of the electron, so the electron can be regarded as being in the potential field formed by the stationary nucleus, and the nucleus won’t be affected by the specific position of the electron, only the average force of electrons counts.

## Density Functional Theory

Density functional theory (DFT) is a quantum mechanical method to study the electronic structure of multi-electron systems, and it is one of the most commonly used methods in the fields of condensed matter physics and computational chemistry. Since the classical method of electronic structure theory needs to solve the multi-electron wave function with a higher dimension (\(3N\) for a system containing \(N\) electrons), the basic idea of the density function is to use the electron density instead of the wave function as the basic amount of research, thereby reducing the computational complexity. The most common application of density functional theory is implemented with the Kohn-Sham method.