Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.23 Solution of the hydrogen molecule
To find the approximate solution for the hydrogen molecule, the key is
to be able to find the expectation energy of the approximate wave
functions
.
First, for given , the individual values of and
can be computed from the normalization requirement

(D.11) 
where the value of the overlap integral
was given in derivation
{D.21}.
The inner product
is a sixdimensional integral, but when multiplied out, a lot of it
can be factored into products of threedimensional integrals whose
values were given in derivation {D.21}. Cleaning up the
inner product, and using the normalization condition, you can get:
using the abbreviations
Values for several of the inner products in these expressions are
given in derivation {D.21}. Unfortunately, these involving
the distance between the electrons cannot be
done analytically. And one of the two cannot even be reduced to a
threedimensional integral, and needs to be done in six dimensions.
(It can be reduced to five dimensions, but that introduces a nasty
singularity and sticking to six dimensions seems a better idea.) So,
it gets really elaborate, because you have to ensure numerical
accuracy for singular, highdimensional integrals. Still, it can be
done with some perseverance.
In any case, the basic idea is still to print out expectation
energies, easy to obtain or not, and to examine the printout to see
at what values of and the energy is minimal. That will be
the ground state.
The results are listed in the main text, but here are some more data
that may be of interest. At the 1.62 nuclear spacing of the
ground state, the antisymmetric state 1 has a
positive energy of 7 eV above separate atoms and is therefore
unstable.
The nucleus to electron attraction energies are 82 eV for the
symmetric state, and 83.2 eV for the antisymmetric state, so the
antisymmetric state has the lower potential energy, like in the
hydrogen molecular ion case, and unlike what you read in some books.
The symmetric state has the lower energy because of lower kinetic
energy, not potential energy.
Due to electron cloud merging, for the symmetric state the electron to
electron repulsion energy is 3 eV lower than you would get if the
electrons were point charges located at the nuclei. For the
antisymmetric state, it is 5.8 eV lower.
As a consequence, the antisymmetric state also has less potential
energy with respect to these repulsions. Adding it all together, the
symmetric state has quite a lot less kinetic energy than the
antisymmetric one.