Essentials of Computational Chemistry

14.1 DETERMINANTAL/CONFIGURATIONAL REPRESENTATION OF EXCITED STATES 491
out-of-plane p orbital belonging
to b 2 irreducible representation
3 A2 (T0)
N
N
in-plane p orbital belonging
to b 1 irreducible representation
1 A2 (S1)
1 1 A 1 (S2) 2 1 A 1 (S3)
Figure 14.3 Electronic configurations of phenylnitrene, differing in occupation of the nitrogen p
orbitals, labeled according to their spin and spatial symmetries. Relative energy orderings for the four
configurations are indicated in parentheses; T0 is the triplet ground state, and Sn represents the nth
lowest singlet excited state
differing in whether the occupied nitrogen p orbital is in the plane of the aromatic ring (S2)
or perpendicular to it and conjugated with the π system (S3). As both states are closedshell,
both belong to the totally symmetric irreducible representation (e.g., the two states
are 1A1 since phenylnitrene belongs to the C2v point group), but the different symmetries
of the alternative lone pair orbitals (one being b1 and the other b2) makes it relatively
straightforward to converge variationally optimized wave functions that may be written as
1
i = ···i b 2 1
(14.7)
and
1
j = ···j b 2 2
N
N
(14.8)
(for HF example, see Kim, Hamilton, and Schaefer 1992; for corresponding DFT example,
see Smith and Cramer 1996; Johnson and Cramer 2001). Note that in this case the ellipsis in
each wave function carries a subscript to emphasize that while the orbitals thereby implied
in Eqs. (14.7) and (14.8) are qualitatively similar, they are not identical (since each set was
variationally optimized subject to different HOMOs). This point is not merely technical, but
represents a critical problem associated with the approach, namely that there is no guarantee
that the two wave functions of Eqs. (14.7) and (14.8) are orthogonal, as they should be.
The issue of orthogonality is an important one. Every excited state must be orthogonal
to the ground state (as well as to all of the other excited states), and any technology for