Essentials of Computational Chemistry

500 14 EXCITED ELECTRONIC STATES
E
State A
State B
∗
∗
orbitals optimized
for state A
q
∗
orbitals optimized
for state B
Figure 14.6 Two electronic states of an arbitrary system having a conical intersection. The inset
region illustrates the effect on each curve of optimizing the orbitals for either State A or State B. At
the coordinate position marked by an asterisk, the relative energies of the two states depend on which
is chosen for orbital optimization, which can lead to root switching problems in an MCSCF calculation.
Additionally, geometry optimization can cause root switching as well, if optimization passes through
the conical intersection
To finesse this problem, it is possible to carry out a so-called ‘state-averaged’ MCSCF.
In a state-averaged calculation, the orbitals are variationally optimized not for any one state
energy, but rather for the average of the two (or more than two, if a larger number of states
are of interest). A drawback to such a calculation is that the quality of any one state’s wave
function is lower than it would be were it to be the only state under consideration. On the
other hand, a virtue of a state-averaged calculation is that all states are expressed using
the same MOs, thereby ensuring orthogonality, which is critical if, say, transition dipoles
between states are to be computed.
Nevertheless, root switching may still be problematic for geometrical reasons in the vicinity
of conical intersections. Thus, for instance, any optimization of State B in Figure 14.6 that
begins to the left of the asterisk in coordinate q will ultimately proceed to the right until
State B falls below State A in energy, at which point it is the first root for chemical reasons,
not technical reasons. The only remedy in this situation is careful analysis in the construction
of state PESs.
MCSCF results for phenylnitrene using a complete active space formed from the six
phenyl π orbitals and the two nitrogen p orbitals and the eight electrons contained therein
are presented in Table 14.1. Note that, because of symmetry and spin restrictions, only the
2 1 A1 state must be determined as the second root of the MCSCF. The CAS results are quite