Water molecule bond length

Classical Mechanics

Newton Equations

$F=ma$ where $F=-\nabla V$.

Integration

The finite differences (Euler method) are

Failed to parse (syntax error): {\displaystyle \begin{align*} v(t) &= \frac{x(t+\Delta t) - x(t)}{\Delta t} \\ a(t) &= \frac{v(t+\Delta t) - v(t)}{\Delta t} \end{align*} } and

$a(t)={\frac {F(x(t))}{m}}$

Velocity Verlet Algorithm

A very good and easy to implement integration method is velocity Verlet:

Failed to parse (syntax error): {\displaystyle \begin{align*} x(t + \Delta t) &= x(t) + v(t) \Delta t + \frac12 a \Delta t^2 \\ v(t + \Delta t) &= v(t) + \frac12\left( a(t) + a(t+\Delta t) \right) \Delta t \end{align*} }

Potential Function

Lennard--Jones potential with parameters for TIPS model:

Failed to parse (syntax error): {\displaystyle \begin{align*} A &= 580.0 \times 10^3 kcal A^{12}/mol \\ B &= 525.0 kcal A^6/mol \end{align*} } where $A$ is Ångströms.

Temperature/ Initial distribution

The initial velocity of the hydrogen atom is chosen randomly from the Maxwell-Boltzmann distribution at given temperature $T$

$p(v)=\left({\frac {m}{2\pi k_{B}T}}\right)^{1/2}\exp \left[-{\frac {1}{2}}{\frac {mv^{2}}{k_{B}T}}\right]$

Issues

1D statement