Biomedical Engineering Reference
In-Depth Information
individual Slater determinants and classify the resulting wavefunctions at the end of the
calculation.
We write the CI wavefunction as
c
1b
Ψ
1b
+···
where the CI expansion coefficients have to be determined from a variational calculation.
This involves finding the eigenvalues and eigenvectors of a matrix whose elements are
typically
Ψ
CI
=
c
0
Ψ
0
+
c
1a
Ψ
1a
+
Ψ
i
Ψ
j
dτ
The matrix elements can be determined from the Slater-Condon-Shortley rules, giving
typically HF-LCAO orbital energies and various two-electron integrals such as
e
2
4πε
0
1
r
12
ψ
21
(
r
1
) ψ
22
(
r
2
) dτ
1
dτ
2
ψ
21
(
r
1
)ψ
22
(
r
2
)
These have to be calculated from the two-electron integrals over the HF-LCAO basis
functions, at first sight a four-dimensional sum known in the trade as the
four-index
transformation
.
Two of each of the (a), (b), (c) and (d) Slater determinants correspond to spin eigenfunc-
tions having spin quantum number
M
s
=
1 and so
need not be considered because states of different spin symmetry do not mix. In addition,
Brillouin's theorem (Brillouin 1933) tells us that singly excited states constructed using
HF wavefunctions do not mix with the ground state for a closed shell system, so we do not
need to include the ground state Ψ
0
in the variational calculation.
If we take all possible excited Slater determinants and solve the variational problem we
reach a level of theory known as
CI singles (CIS)
. A keynote paper is that by Foresman
et al.
(1992), and their synopsis puts everything into perspective.
0, one to
M
s
=+
1 and one to
M
s
=−
This work reviews the methodological and computational considerations necessary for the
determination of the ab initio energy, wavefunction and gradient of a molecule in an electron-
ically excited state using molecular orbital theory. In particular, this paper re-examines a funda-
mental level of theory which was employed several years ago for the interpretation of the elec-
tronic spectra of simple organic molecules: configuration interaction (CI) among all singly sub-
stituted determinants using a Hartree Fock reference state. This investigation presents several
new enhancements to this general theory. First, it is shown how the CI singles wavefunction can
be used to compute efficiently the analytic first derivative of the energy ...Second, a computer
program is described which allows these computations to be done in a 'direct' fashion.
You should have picked up the words 'direct' and 'gradient'.
To return to my example of the azines; in the earliest and necessarily qualitative
treatments, the first four transitions were identified with the single excitations shown in
Figure 17.7. Benzene can be regarded as a special case of the series and the highest occu-
pied orbitals are doubly degenerate, as are the lowest unoccupied ones. This leads to many
degeneracies in the excited states that simply do not appear (other than by accident) in the
substituted azines. The assignment of bands in the benzene spectrum led to considerable
discussion in the primary literature.
