Essentials of Computational Chemistry

14
Excited Electronic States
14.1 Determinantal/Configurational Representation
of Excited States
An excited electronic state is one in which at least one electron is not in as low energy
an orbital as it could be given the molecular geometry. Such a state may be generated by
various processes, e.g., absorption of radiation by the ground state or as the product of a
chemical reaction. Although unstable relative to collapse to the lower energy ground state,
some excited states may have significant lifetimes owing to inefficient coupling with the
ground state, as described in more detail below.
Given M doubly occupied molecular orbitals and N empty virtual orbitals (and possibly
some number of singly occupied orbitals), the number of possible excited states that can be
generated is enormous. For the moment, we will restrict our discussion to the case where
the ground state is closed-shell and in the excited state only a single excited electron exists
(implying it must have come from the HOMO). Extending the discussion to more general
cases is intuitively straightforward, but notationally somewhat tedious.
The usual manner in which chemists think about an excited state is to take the ground state
0 as context. Thus, as shown in Figure 14.1, one considers our excited state to be generated
by the removal of one electron from the HOMO of the ground state and its placement into
some higher-energy orbital. Since the wave function for the ground state can be represented
as a single Slater determinant, e.g.
1 0 =
ψ 2 1 ψ 2 2 ψ 2 3 ···ψ 2
N/2
(14.1)
(the Slater determinant appropriate for the RHF wave function of a singlet having N electrons),
then the excited state might in general be written as
a N/2 = ψ 2 1 ψ 2 2 ψ 2 3 ···ψN/2ψ
a
(14.2)
where we use the compact notation that if an orbital is not doubly occupied, it is multiplied
by an α spin function unless there is a bar over it, in which case it is multiplied by a β
Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer
© 2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)