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

We define some arrays:

Electrical charges on the clouds and nuclei:

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 (Bohr units):

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

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

Minimization of the total energy

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

3D Plot of the computed structure

Projection of Kimball spheres into diagonal plane through the C atom

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.

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

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.

Compute kinetic energy terms, bonding clouds, core clouds:

Total kinetic energy except for π - clouds and lone pairs

Connectivity matrix:

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.

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.

Bonding clouds : Atom pair, radius of cloud:

Plot of PolyPro and its partial constituents

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

Compute energy components:

Politzerratio

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