Biomedical Engineering Reference
In-Depth Information
where the two
G
matrices contain Coulomb and exchange contributions similar to those
defined by Equation (16.12) for the closed shell case. We then allow
R
1
and
R
2
to vary,
subject to orthonormality, just as in the closed shell case. Once again, Roothaan (1960)
showed how to write a Hamiltonian matrix whose eigenvectors give the columns
U
1
and
U
2
above.
16.7 Unrestricted Hartree-Fock (UHF) Model
The methyl radical CH
3
can be easily produced by breaking the C-C bond in ethane:
C
2
H
6
→
2CH
3
It has nine electrons and there are two ways we can progress, as shown in Figure 16.3.
ROHF
UHF
Figure 16.3
Methyl radical
First we can insist that the lowest energy molecular orbitals are doubly occupied, with the
extra electron occupying the highest energy orbital (ROHF). On the other hand we can let
the α and β spin electrons occupy different spatial orbitals (the
unrestricted Hartree-Fock
(UHF) model).
The treatment will be familiar to you by now. We write a single Slater determinant but
allow the
p
α and the
q
β spin electrons to have different spatial orbitals. Application of the
Slater-Condon-Shortley rules gives an energy expression in terms of the HF orbitals as
