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