1. Introduction

1.  Characteristic length and times scales for molecular dynamics

Types	Time scale	Time scale (s)
Bond vibration	1 fs; Femtosecond	\(10^{-15}\) s
Collective vibration	1 ps; Picosecond	\(10^{-12}\) s
Conformational transition	ps or longer	\(>10^{-12}\)
Enzyme catalysis	\(\mu\)s to ms	\(10^{-6}\sim 10^{-3}\) s
Ligand binding	\(\mu\)s to ms	\(10^{-6}\sim 10^{-3}\)
Protein folding	ms to s	\(10^{-3}\sim 10^0\) s

For an atomistic simulation, it typically use an iteration time step of 1 fs (to capture bond vibrations)

And the accessible time scale for us with atomistic simulation MD is \(100\text{ns} = 10^5 \text{ ps}=10^8 \text{ fs}=10^{-7} \text{s }\)

\[ N_\text{steps}=\frac{T_\text{total}}{t_\text{step}} \]

e.g. for an materials with 1000 atoms, to simulate the 100 ns of real-world motion, we need to calculate the forces on every single atom and update their positions \(10^8\) times.

2.  Empirical potential energy

2.1.  Lennard Jones potential

\[\begin{split} \begin{aligned} V(r)&= \frac{B}{r^{12}}-\frac{A}{r^6}\\ V(r)&=4\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^6] \end{aligned} \end{split}\]

• \(\sigma\) is the unit of length scale

• \(\epsilon\) is the unit of energy scale

2.2.  Morse potential

$$ \begin{aligned} V(r) &= D[1-e^{(-\alpha \cdot (r - r_0))}]^2 - D \\ &= D[\exp(-2\alpha \cdot (r - r_0)) - 2\exp(-\alpha \cdot (r - r_0))] \end{aligned} $$

• D is the unit energy scale, also often refers to bond dissociation energy

• \(\alpha\) is the elastic properties, or force constant (the spring constant) of the bond

• \(r_0\) is the equilibrium distance.

2.3.  Buckingham potential

\[\begin{split} V(r)=Ae^{-\frac{x-x_0}{\rho}}-\frac{C}{(x-x_0)^6}\\ =A\exp[-\frac{r}{\rho}]-\frac{C}{r^6}-\frac{D}{r^8} \end{split}\]

\(\frac{D}{r^8}\): sometimes a \(2^{nd}\) order term is added to satisfy van der Waals perturbation theory (\(r^{-8}\))

Fundamental issues of pair potentials

[!NOTE]

• Pair potentials can not “count bonds”, and do not care about the organization of atoms (angles, etc.)

• In pair potential models the cohesive energy on an atom is largely determined by how many bonding partners are around the atom