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} . The system is 1D, thus gradient will be differential, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F=-{\frac {dV}{dx}}} .

Potential Function

Lennard--Jones potential with parameters for TIPS model:

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

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} A &= 580.0 \times 10^3 kcal A^{12}/mol \\ B &= 525.0 kcal A^6/mol \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} is Ångströms.

Integration

The finite differences (Euler method) are

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} 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

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) = -\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} }

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