View source for Water molecule bond length

== Introduction ==

== Classical Mechanics ==

=== Newton Equations ===
<math>F = ma</math> where <math>F = - \nabla V</math>. The system is 1D, thus gradient will be differential, and <math>F= - \frac{d V}{dx}</math>.

The mass <math>m</math> should be the [https://en.wikipedia.org/wiki/Reduced_mass reduced mass] of the oxygen--hydrogen system, but we use mass <math>m=1</math> here.

=== Potential Function ===

[https://en.wikipedia.org/wiki/Lennard-Jones_potential Lennard--Jones potential] with dimensionless parameters for [https://en.wikipedia.org/wiki/Water_model TIPS] model: 

<math>
V(x) = 4 \left[ \left( \frac1x\right)^{12} - \left( \frac1 x\right)^6 \right]
</math>

where the distance <math>x</math> is given in Ångstroms.

=== Integration ===

The finite differences (Euler method) are 

<math>
\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}
</math>
and

<math>
a(t) = -\frac{\frac{d}{dx}(V(x(t)))}{m} 
</math>



=== Velocity Verlet Algorithm ===

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

<math>
\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}
</math>

where <math>a(t) = a(x(t)) = - \frac1m \frac{d}{dx}V(x(t))</math> is given at Section . . .

=== Temperature/  Initial distribution ===

The initial velocity of the hydrogen atom is chosen randomly from the [https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution Maxwell-Boltzmann distribution] at given temperature <math>T</math>

<math>
p(v) = \left( \frac{m}{2\pi k_B T} \right)^{1/2}
\exp\left[- \frac12 \frac{mv^2}{k_B T} \right]
</math>

The mean speed (for 3d?) is

<math>
v_\text{mean} = \sqrt{\frac{8k_B T}{\pi m}}
</math>

where <math>k_B</math> is Boltzmann constant, here <math>k_B = 1</math>.

== Results ==

== 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).