Each frame of this simulation showcases the outcomes of Sequential Quadratic Programming (SQP) minimization applied to a Lennard-Jones system. The system’s energy is defined by the following expression:
![\begin{equation*} E_{LJ}=\sum_{\{ A, B \} } \frac{1}{2}\sum_{a \in A} \sum_{b \in B} 4\epsilon\left[\left(\frac{\sigma}{r_{ab}}\right)^{12} - \left(\frac{\sigma}{r_{ab}}\right)^{6}\right] \end{equation*}](https://milotorda.net/wp-content/ql-cache/quicklatex.com-e667e732de97bd4a1f260d50e3bd9bbb_l3.png "Rendered by QuickLaTeX.com")
where the parameters are set as 
and 
. Here, 
represents the ordered pair of molecules 
and 
. The simulation involves a system of 
molecules, each comprising 
atoms. At the start, a uniform random configuration was generated, with each molecule represented by two center of mass coordinates and one rotational coordinate.
As the simulation progresses, the boundaries of the box are incrementally reduced following each SQP minimization. This effectively simulates an increase in pressure on the system. Notice the hexagonal tiling patches as the simulation advances. These are depicted as blue regular hexagons, each scaled by a factor of 1.93185 and rotated by an angle of 0.17862 relative to the molecules.
It’s important to note that the empty spaces observed in the simulation are a result of the rate at which pressure is increased. A sufficiently slow increase in pressure would eventually lead to a 
regular tiling.
and ![\begin{equation*} E_{LJ}=\frac{1}{2}\sum_{a \in A} \sum_{b \in B} 4\epsilon\left[\left(\frac{\sigma}{r_{ab}}\right)^{12} - \left(\frac{\sigma}{r_{ab}}\right)^{6}\right] \end{equation*}](https://milotorda.net/wp-content/ql-cache/quicklatex.com-22d9342dd4cf52e04dee80ca78afe812_l3.png "Rendered by QuickLaTeX.com")
(
below.
and 
with the unit cell given by ![\begin{equation*} U=\left[\begin{array}{ccc} a\sin(\beta)\sqrt{1-\left(\frac{\cos(\alpha)cos(\beta) - cos(\gamma)}{\sin(\alpha)\sin(\beta)}\right)^2} & 0 & 0 \\ -a\frac{\cos(\alpha)cos(\beta) - cos(\gamma)}{\sin(\alpha)} & b\sin(\alpha) & 0 \\ a\cos(\beta) & b\cos(\alpha) & c \end{array}\right] \end{equation*}](https://milotorda.net/wp-content/ql-cache/quicklatex.com-46346f911c2c7e97ee0e38a4da1bc482_l3.png "Rendered by QuickLaTeX.com")
equals to 
and combined with the volume of the unit sphere 
gives the optimal packing density of spheres 
. See a visualization of this configuration below.
, 
and 
, then the determinant of the unit cell equals to 
and again we get the optimal packing density of spheres 
. See the visualization of this configuration below.
, 
, 
and 
and again we get the optimal packing density of spheres 
. See the visualization of this configuration below.
