Kimball Model of Methane

We will write a simple Mathematica version 10 code to study methane “ab initio”.
Our objective is to compute the ground state 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 a parametrization run whose results are used below.

Definitions

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We define some arrays:

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Electrical charges on the clouds and nuclei:

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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 MethaneRetro_new1_4.gif (Bohr units):

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

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

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Minimization of the total energy

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

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3D Plot of the computed structure

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Projection of Kimball spheres into diagonal plane through the C atom

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Derive Methane from its structure

We use the result of the optimization, above, to check the program template for deriving the energy components from a given structure.
All computations are transparent. The run lasts about 1 sec on a i7-2600 CPU. (ES 16 June 2013/27 October 2014).

Input and Definitions

The structure has been produced by the Kimball optimizer using G3/6-311+G(d) parameters  (cnofcpp.pas), (cnofcpp.ex_ to be renamed into runnable cnofcpp.exe after download). The corresponding standalone notebook is Methane_new.nb.
The coordinates are read in atomic units, the universally applied system of theoretical chemistry and (micro) physics, see NIST. Length data are in Bohr : 1 a0 = 0.52917721 Å = 52.917721 pm; electric charges in ± electron charges, and energies in Hartrees : 1 Eh = 2 Rydberg = 627.5095 kcal/mol = 2625.50 kJ/mol.

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Show nuclear charges, number of non-hydrogen atoms and hydrogens

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Compute Kimball radii from distance matrix, show core radii derived from CH4, NH3, H2O gauge molecules (cnofhydb.pas), (cnofhydb.ex_ to be renamed into runnable cnofhydb.exe after download), H excentricities, and number of σ bonds.

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Compute kinetic energy terms, bonding clouds, core clouds:

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Total kinetic energy except for π - clouds and lone pairs

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Connectivity matrix:

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Localize double bonds and positions of π-clouds (PItrans.m)

Transform the triangle of every target atom with two of its neighbors into the xy-plane and attach π-clouds above and below the plane to the target. Then back transform the π-clouds into the molecular coordinate array.

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Localize lone pairs, compute size and orientation: Subroutines: XOtrans.m, XOYtrans.m, CNCtrans.m, LpyrNtrans.m

Transform the triangle of every target atom with two of its neighbors into the xy-plane and attach lone pair(s). Then back transform the lone pair(s) into the molecular coordinate array. See one of the subroutines. LpyrNtrans puts the base atoms of a pyramid into the xy plane and attaches LP’s as needed, the moves these back into the molecule frame.

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Bonding clouds : Atom pair, radius of cloud:

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Graphics:Statistics of cloud radii

Plot of PolyPro and its partial constituents

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Graphics:Methane

Graphics:Core skeleton

Preparation of interaction matrices (w-w, w-n, n-n, size arrays, charges):

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Compute energy components:

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Politzerratio

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Results (energies in [Eh] Hartree), compare to first part of notebook

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