130_notes.dvi

By settingd〈dyH〉= 0, we can get the distance between atoms and the energy.

Distance Energy

Calculated 1.3 ̊A -1.76 eV

Actual 1.06 ̊A -2.8 eV

Its clear we would need to introduce some wfn. parameters to get good precision.

27.2 The H 2 Molecule

The H 2 molecule consists of four particles bound together: e 1 ,e 2 , protonA, and protonB. The
Hamiltonian can be written in terms of the H+ 2 Hamiltonian, the repulsion between electrons, plus
a correction term for double counting the repulsion between protons.

H=H 1 +H 2 +

e^2

r 12

−

e^2

RAB

H 1 =

p^21

2 m

−

e^2

rA 1

−

e^2

rB 1

+

e^2

RAB

We wish to compute variational upper bound onRABand the energy.

We will again use symmetric electron wavefunctions,

ψ(r 1 ,r 2 ) =

1

2[1 +S(RAB)]

[ψA(~r 1 ) +ψB(~r 1 )] [ψA(~r 2 ) +ψB(~r 2 )]χs