Essentials of Computational Chemistry

14.2 SINGLY EXCITED STATES 497
where the coefficients c are the components of the eigenvector for state k. With large enough
basis sets, even the CIS matrix may grow cumbersomely large with which to work, and
iterative methods designed to locate only lower energy roots are employed, just as in CI
treatments considering higher excitations. Analytic gradients are available for CIS wave
functions, so it is possible to optimize the geometry of a particular state, making CIS
a useful method for obtaining either vertical excitation energies, or adiabatic excitation
energies.
Note the difference in objectives between a ground-state CI calculation and a CIS calculation.
In the former, the goal is to improve the description of the ground state, and excitations
must be included at least through doubles (since singles do not mix with the ground state).
In the CIS calculation, the ground state is important only to the extent it determines the
orbitals, and the CI is carried out to orthogonalize the singly excited states.
Insofar as the latter process does not involve any orbital reoptimization for any particular
state, it provides a wave function that is roughly equivalent in quality only to an HF wave
function for the ground state. Of course, this may still be useful for a number of purposes.
CIS results for six excited states of benzene are included in Table 14.2, as are results from
other levels of theory that will be discussed later. The CIS results are qualitatively useful,
insofar as the states are correctly ordered, and the error is fairly systematic – all states are
predicted to be too high in energy by an average of 0.7 eV. The worst prediction is for the
lowest excited state, which is known to have significant dynamical electron correlation, and
is therefore challenging for the CIS method.
To improve CIS results beyond their roughly HF quality, various options may be considered.
Particularly for spectroscopic predictions, semiempirical parameterization of the CIS
matrix elements may be preferred over their direct evaluation in an ab initio sense using
Eq. (7.12), and the most complete realization of this formalism is the INDO/S parameterization
of Zerner and co-workers. A few examples of the excellent performance of this
highly efficient model for the computation of excited-state energies have already been
discussed (Table 5.1). Of additional interest, Hutchison, Ratner, and Marks (2002) found
that CIS/INDO/S provided the highest accuracy of several methods (including ab initio CIS,
RPA, and TDDFT; the last two are discussed later in this chapter) for predictions of first
excited-state energies in 60 oligomers of various aromatic heterocycles. With increasing
Table 14.2 Energies (eV) for singlet excited states of benzene relative to the 1 A1g,
ground state as predicted by various methods a
Excited state CIS RPA TD-BPW91 TD-B3LYP Expt.
1B2u 6.15 5.96 5.19 5.40 4.9
1B1u 6.31 6.01 5.93 6.06 6.2
1E1g 7.13 7.12 6.34 6.34 6.33
1A2u 7.45 7.43 6.87 6.84 6.93
1E2u 7.75 7.74 6.85 6.88 6.95
1E1u 7.94 7.52 6.84 6.96 7.0
Mean abs. error: 0.7 0.6 0.1 0.1
a From Stratmann, Scuseria, and Frisch (1998). All calculations employed the 6-31+G(d) basis set.