Molecular mechanics concern variety parts of physics and mathematics: classical mechanics and kinetics, thermodynamics,
statistical physics, vector and tensor algebra, linear algebra and geometry, the theory of differential equations,
and finally, the theory of numerical methods. This description is an invitation to follow the same path that the author did,
to perceive the theoretical material necessary for molecular dynamics simulation.
The program includes algorithms that can be grouped as follows.
The physical modeling algorithms: numerical integration of Newton's and Euler's equations, an algorithm
for calculating the principal values and principal axes of the inertia tensor, an algorithm for calculating physical
average values, an algorithm for calculating the volume given by an irregular set of points, a cluster search algorithm.
The visualizing of 3D world algorithms: an algorithm for constructing a central projection on an arbitrary plane,
an algorithm for triangulating a surface, a ray tracing algorithm for a sphere and a plane element, an algorithm for a
video file creating.
The superstructure algorithms: thermostat algorithm, barostat algorithm, a series of calculation supervisor, a
three-dimensional molecular editor.
Some results
First step, full energy conservation test (with music ☺)
Molecular fantasy.
The molecular geometry shown on the picture, one atom has positive charge σ,
the other has negative charge σ/2. σ=1.0, l=0.3, α=25° Atom-atom potential parameters:
ε=0.046, α=5.0, rm=0.56.
Molecules: 50;
Video stream: 0.5 nS/frame;
Cube length: 8 nm;
Cooling: 10 nS mixing; 50 nS multiplicate all velocities 0.9999 per 1.0 nS; 240 nS constant energy.
If the system is heated at a constant rate,
then it can be assumed that the internal energy and temperature of the system
will increase linearly \(E(t) = E_0 + K_E t\) and \(T(t) = T_0 + K_T t\).
Then, we obtain an expression for the heat capacity
\(C_V = \left( \dfrac{\partial E}{\partial T} \right)_V =
\dfrac{\partial E}{\partial t} \left. \right/ \dfrac{\partial T}{\partial t} = \dfrac{K_E}{K_T} \)
recalculating to [Jxmol-1xK-1] we get 12,581 vs standard value 12,555 for Helium.
Heat capacity.
A more standard way to calculate the heat capacity is to measure
the mean square of energy fluctuations \({\left\langle {\delta {E_k^2}} \right\rangle _{NVE}} = {k_b}{T^2}{C_V}\),
recalculating the average value to the NVT ensemble gives the value of 12.586 [Jxmol-1xK-1]
for Helium.
Harmonic confinement.
The harmonic confinement to impose the particles
to gather into a single cluster, the volume of cluster calculated by Convex Hull algorithm.
But the system like this is non-homogenous, so the pressure or other thermodynamic values
calculation are open for discussion question.
Obtained result for 150 Xenon atoms give surprisingly good agreement for condensation isotherms.
The solid lines are the experimental data, the dots are the result of calculations by the molecular
dynamic system at harmonic confinement condition.
MolRun.pdf
MolRun.zip
