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:

${\begin{aligned}x(t+\Delta t)&=x(t)+v(t)\Delta t+{\frac {1}{2}}a\Delta t^{2}\\v(t+\Delta t)&=v(t)+{\frac {1}{2}}\left(a(t)+a(t+\Delta t)\right)\Delta t\end{aligned}}$

Potential Function

Lennard--Jones potential with parameters for TIPS model:

${\begin{aligned}A&=580.0\times 10^{3}kcalA^{12}/mol\\B&=525.0kcalA^{6}/mol\end{aligned}}$ 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