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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(t) = a(x(t)) = - \frac1m \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2} \exp\left[- \frac12 \frac{mv^2}{k_B T} \right] }

Issues

1D statement