Kimball Model of Methane

We will write a simple Mathematica version 10 code to study methane.
Our objective is to compute the groundstate energy of a tetrahedral structure.
We use the kinetic energy and screening constant parameters from the optimization with G3//6-311+G(d)
ab initio results. Here is the output of an optimization run whose results are used below.

Definitions

Methane_n_1.gif

We define some arrays:

Methane_n_2.gif

Electrical charges on the clouds and nuclei:

Methane_n_3.gif

r Radii of clouds, exc excentricities of protons in CH-clouds, d1 distances of C,H nuclei, d2 distances of C-H clouds from C, n x,y,z coordinates of protons and w of clouds on the corners of cubes, defining tetrahedron; all lengths in Methane_n_4.gif (Bohr units):

Methane_n_5.gif

Preparation of the interaction matrices and summation of terms within one cloud:

Methane_n_6.gif

Kinetic energy of electrons, components of the potential energy, using matrix operations:

Methane_n_7.gif

Minimization of the total energy

Methane_n_8.gif

Methane_n_9.gif

Results, extracted from the solution, above; all energies in [Eh] (Hartree)

Methane_n_10.gif

Methane_n_11.gif

Methane_n_12.gif

Methane_n_13.gif

Methane_n_14.gif

Methane_n_15.gif

Methane_n_16.gif

Methane_n_17.gif

Methane_n_18.gif

Methane_n_19.gif

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3D Plot of the computed structure

Methane_n_22.gif

Methane_n_23.gif

Projection of Kimball spheres into diagonal plane through the C atom

Methane_n_24.gif

Methane_n_25.gif

Created with the Wolfram Language