Essentials of Computational Chemistry

496 14 EXCITED ELECTRONIC STATES
looks to be somewhat low, however, again probably reflecting the non-orthogonal nature of
the HF reference for this state compared to the lower energy one.
14.2.2 CI Singles
For many singly excited states, SCF calculations are not an option under any circumstances.
Within the context of using the orbitals of the ground state to describe the excited state, the
simplest way to evaluate the energy of the excited state would be to evaluate the Hamiltonian
for the determinant formed after promotion of the excited electron. Such an approach is rarely
useful, however, and a significant drawback of these singly excited single-configuration
wave functions is that, although each one will be orthogonal to the ground state (because of
Brillouin’s theorem, see Section 7.3.1), they are unlikely to be orthogonal to one another.
However, as long as we limit our consideration to singly excited states, they can be made to
be orthogonal to one another with fairly little computational effort, and in the process better
descriptions of the states, and presumably better energies, may be determined.
This orthogonalization is the essence of the technique known as CI singles (CIS) because
the CI matrix is formed restricting consideration to only the HF reference and all singly
excited configurations (Figure 14.5). The matrix is essentially of size M × N where M is
the number of occupied orbitals from which excitation is allowed, and N is the number of
virtual orbitals into which excitation is considered. If excitation is allowed to occur with
a spin-flip of the excited electron (e.g., permitting generation of triplet excited states from
singlet ground states or vice versa; see, for example, Sears, Sherrill, and Krylov 2003) then
the size increases, although none of the triplet states have matrix elements with any of the
singlet states because of their different spins. Orthogonalization of the CIS matrix takes
place only in the space(s) of the excited states, since they do not mix with the HF reference.
The orthogonalization provides energy eigenvalues each of which has associated with it an
eigenvector detailing the weight of every singly excited determinant in the state. That is, the
CIS wave function for each excited state is written as
Ψ HF
Ψ a
i
Ψ HF
E HF
0
0 dense
k =
occupied
i
virtual
a
diagonalization
ciak a i
Ψ a
Ψ i HF Ψ a
i
Ψ HF
Ψ a
i
E HF
0
E 1
0
E Ω
(14.11)
Figure 14.5 The CIS procedure diagonalizes the CI matrix formed only from the HF reference
and all singly excited configurations. The diagonalization provides energy eigenvalues and associated
eigenvectors that may be used to characterize individual states as linear combinations of single
excitations