Essentials of Computational Chemistry

14.2 SINGLY EXCITED STATES 495
Table 14.1 Energies (kcal mol −1 ), where available, for lowest
singlet excited states of phenylnitrene relative to the 3 A2 ground
state a
Source 1 A2 1 1 A1 2 1 A1
HF/6-31G(d) 64.6 80.1
BLYP/6-31G(d) 14.5 b , 22.8 c 31.4 48.7
BPW91/cc-pVDZ (SCF) d 14.3 b 33.9 43.0
BLYP/cc-pVTZ (SCF) e 29.5 41.0
CCSD(T)/cc-pVDZ e 35.2 47.2
CAS(8,8)/cc-pVDZ d 17.8 42.1 76.2
CASPT2/cc-pVDZ d 19.3 37.4 57.8
CASPT2/cc-pVTZ e 19.3 34.8 54.5
MRCISD/DZP f 21.0 39.8 (52.0)
Experiment g 18. 30. n.a.
a Zero-point vibrational energies are not included in the theoretical energies,
but ZPVE differences between alternative electronic states are predicted to
be small at the few levels where they have been evaluated.
b Determined from sum method.
c Determined from spin projection.
d Johnson and Cramer (2001).
e Smith and Cramer (1996).
f Hrovat, Waali, and Borden (1992); Kim, Hamilton, and Schaefer (1992).
g Travers et al. (1992); Ellison, G. B., unpublished results.
levels of theory that would be expected to be reasonably accurate suggests that the DFT
predictions are substantially too low. The DFT wave function for the 2 1 A1 state is that of
Eq. (14.8) and, as discussed above, is found by fortuitous convergence of the SCF equations
for this occupation scheme where variational collapse to the 1 1 A1 state of Eq. (14.7) would
otherwise be expected. The apparently rather poor accuracy of the energy for the higher state
suggests that this orthogonality issue cannot be ignored here, and the SCF procedure must
be regarded as unreliable.
As for the HF level, the SCF approach for the closed-shell singlet states is identical to
that in the DFT case (in this instance, the two-determinantal nature of the lower energy openshell
singlet requires an MCSCF description, so HF values are not reported for this state).
However, both of the closed-shell singlets are subject to large non-dynamical correlation
effects (as a consequence, in part, of being so close in energy to one another). Since HF
theory is much more sensitive to such correlation than DFT, the energies of these two states
are predicted to be much too high. This error is in some sense even worse than it appears,
because severe spin contamination in the triplet, which exhibits an expectation value for S 2
in excess of 2.7, probably causes it too to be poorly represented at the HF level.
Of course, with HF wave functions in hand, it is possible to carry out post-HF calculations
to partially correct for electron correlation effects. The poor quality of the HF wave functions,
however, militate against any treatment much less sophisticated than coupled-cluster. At the
CCSD(T)/cc-pVDZ level, the predicted energy of the lowest closed-shell singlet is in fair
agreement with experiment (other data in the table suggest that use of a triple-ζ basis set
would improve the CCSD(T) estimate). The energy of the second closed-shell singlet state