Water molecule bond length

Classical Mechanics

Newton Equations

$F=ma$ where $F=-\nabla V$. The system is 1D, thus gradient will be differential, and $F=-{\frac {dV}{dx}}$.

The mass $m$ should be the reduced mass of the oxygen--hydrogen system, but we use mass $m=1$ here.

Potential Function

Lennard--Jones potential with dimensionless parameters for TIPS model:

$V(x)=4\left[\left({\frac {1}{x}}\right)^{12}-\left({\frac {1}{x}}\right)^{6}\right]$

where the distance $x$ is given in Ångstroms.

Integration

The finite differences (Euler method) are

${\begin{aligned}v(t)&={\frac {x(t+\Delta t)-x(t)}{\Delta t}}\\a(t)&={\frac {v(t+\Delta t)-v(t)}{\Delta t}}\end{aligned}}$ and

$a(t)=-{\frac {{\frac {d}{dx}}(V(x(t)))}{m}}$

Velocity Verlet Algorithm

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

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

where $a(t)=a(x(t))=-{\frac {1}{m}}{\frac {d}{dx}}V(x(t))$ is given at Section . . .

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