## lördag 6 augusti 2016

### New Quantum Mechanics 12: H2 Non Bonded

Here are results for two hydrogen atoms forming an H2 molecule at kernel distance R = 1.4 at minimal total energy of -1.17 and a non-bonded molecule for larger distance approaching full separation for R larger than 6-10 at a total energy of -1. The results fit quite well with table data listed below.

The computations were made (on an iPad) in cylindrical coordinates in rotational symmetry around molecule axis on a mesh of 2 x 400 along the axis and 100 in the radial direction. The electrons are separated by a plane perpendicular to the axis through the the molecule center, with a homogeneous Neumann boundary condition for each electron half space Schrödinger equation. The electronic potentials are computed by solving a Poisson equation in full space for each electron.

PS To capture energy approach to -1 as R becomes large, in particular the (delicate) $R^{-6}$ dependence of the van der Waal force, requires a (second order) perturbation analysis, which is beyond the scope of the basic model under study with $R^{-1}$ dependence of kernel and electronic potential energies.

%TABLE II. Born–Oppenheimer total, E
%Relativistic energies of the ground state of the hydrogen molecule
%L. Wolniewicz
%Citation: J. Chem. Phys. 99, 1851 (1993);
for two hydrogen atoms separated by a distance R bohr

R    energy
0.20 2.197803500
0.30 0.619241793
0.40 -0.120230242
0.50 -0.526638671
0.60 -0.769635353
0.80 -1.020056603
0.90 -1.083643180
1.00 -1.124539664
1.10 -1.150057316
1.20 -1.164935195
1.30 -1.172347104
1.35 -1.173963683
1.40 -1.174475671
1.45 -1.174057029
1.50 -1.172855038
1.60 -1.168583333
1.70 -1.162458688
1.80 -1.155068699
2.00 -1.138132919
2.20 -1.120132079
2.40 -1.102422568
2.60 -1.085791199
2.80 -1.070683196
3.00 -1.057326233
3.20 -1.045799627
3.40 -1.036075361
3.60 -1.028046276
3.80 -1.021549766
4.00 -1.016390228
4.20 -1.012359938
4.40 -1.009256497
4.60 -1.006895204
4.80 -1.005115986
5.00 -1.003785643
5.20 -1.002796804
5.40 -1.002065047
5.60 -1.001525243
5.80 -1.001127874
6.00 -1.000835702
6.20 -1.000620961
6.40 -1.000463077
6.60 -1.000346878
6.80 -1.000261213
7.00 -1.000197911
7.20 -1.000150992
7.40 -1.000116086
7.60 -1.000090001
7.80 -1.000070408
8.00 -1.000055603
8.50 -1.000032170
9.00 -1.000019780
9.50 -1.000012855
10.00 -1.000008754
11.00 -1.000004506
12.00 -1.000002546