Essentials of Computational Chemistry

492 14 EXCITED ELECTRONIC STATES
describing excited states that fails to enforce such orthogonality must be viewed with caution.
One might, of course, be tempted to say that the failure of an excited state to be orthogonal to
the ground state would be sufficiently damning to warrant no further use of the excited-state
wave function. However, there is some room for ambiguity, insofar as the excited-state wave
function must be orthogonal to the exact ground-state wave function, but we are almost never
working with that wave function, only some approximation thereto. Thus, an exact excitedstate
wave function may very well fail to be orthogonal to, say, the HF approximation to
the ground-state wave function.
In some cases, orthogonality is ensured by the individual natures of the two states. As
already alluded to above, if the electronic states belong to two different irreps of the molecular
point group, and the product of the two irreps fails to contain the totally symmetric representation,
then the two states are necessarily orthogonal (see Appendix B). Taking again the
phenylnitrene system in Figure 14.3 as an example, the lowest energy singlet is open-shell
and has a single electron occupying each of the two nitrogen p orbitals. By analogy to
Eqs. (14.3), (14.7), and (14.8), this formally two-determinantal wave function is
1
k =
···k b1,kb2,k
(14.9)
where the k subscripts on all orbitals emphasize their possible differences with those optimized
for the wave functions of Eqs. (14.7) and (14.8). The electronic state symmetry of
Eq. (14.9) is 1 A2. Since the product of A2 with A1 in the C2v point group is A2, which is not
the totally symmetric representation, the orthogonality of the A2 wave function of Eq. (14.9)
with the A1 wave functions of Eqs. (14.7) and (14.8) is ensured.
A different guarantee of orthogonality arises if the two states in question have different
spin. Continuing with the phenylnitrene system, the ground state is the triplet version of
Eq. (14.9), i.e.,
3
0 =
···0 b1,0b2,0
(14.10)
Orthogonality of the singlet and triplet spin coordinates ensures that the wave function of
Eq. (14.10) is orthogonal to all of those in Eqs. (14.7)–(14.9).
Methods for generating excited-state wave functions and/or energies may be conveniently
divided into methods typically limited to excited states that are well described as involving
a single excitation, and other more general approaches, some of which carry a dose of
empiricism. The next three sections examine these various methods separately. Subsequently,
the remainder of the chapter focuses on additional spectroscopic aspects of excited-state
calculations in both the gas and condensed phases.
14.2 Singly Excited States
For an average molecule, there are typically one or more low-energy excited states that may
be reasonably well described as valence-MO-to-valence-MO single electronic excitations,
and the language of spectroscopy reflects this point. Thus certain states are referred to as
n → π ∗ , π → π ∗ , etc., indicating the orbital from which the electron is excited on the left