Water molecule bond length

Classical Mechanics

Newton Equations

$F=ma$ where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F=-\nabla V} .

Integration

The finite differences (Euler method) are

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

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

Velocity Verlet Algorithm

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

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

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