Water molecule bond length

Classical Mechanics

Newton Equations

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

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

Potential Function

Lennard--Jones potential with dimensionless 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 \left[ \left( \frac1x\right)^{12} - \left( \frac1 x\right)^6 \right] }

where the distance 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 x} is given in Ångstroms.

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 (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}}}

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

References

D.T.W. Lin and C.-K. Chen: A molecular dynamics simulation of TIP4P and Lennard-Jones water in nanochannel, acta Mechanica 173, 181.194 (2004).