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The Excited State Gradient
2.4 The Excited State Gradient
In this section the expression for the geometrical gradient of the singlet excited state is derived, to
illustrate how expressions for properties can straightforwardly be derived in the AO response
framework.
As for the derivations in Section 2.2 we assume that the wave function of the ground state is
optimized at the point of the potential surface, x 0 , where the excited state gradient is evaluated. The
variational condition is thus fulfilled at that point
FDS − SDF = 0, (2.75)
and the ground-state energy at x 0 is further obtained as
E
0
= 2TrhD + TrDG ( D ) + h , (2.76)
nuc
where h is the one-electron Hamiltonian matrix in the AO basis, h nuc is the nuclear-nuclear
repulsion, G holds the two-electron AO integrals and the Fock matrix F is given by h + G(D).
As mentioned previously, the excitation energy corresponding to the excitation from the ground
state 0 to the excited state f can be found from the poles of the linear response function for the
optimized ground state, 62 i.e. as the eigenvalue of the linear response generalized eigenvalue
equation as Eq. (2.45)
where ω f is the electronic excitation energy
and b f is the normalized eigenvector. 61,62
( ω f )
[2] [2] f
0
The excitation energy can then be obtained from Eq. (2.77) as
E − S b = , (2.77)
f
0
ω f = E − E
(2.78)
f
f † [2]
assuming that the eigenvectors b f satisfy the normalization condition
f
ω = b E b , (2.79)
f † [2] f
b S b = 1. (2.80)
Since we are interested in the molecular gradient for the excited state, f , the energy of the excited
state should be defined at arbitrary points on the potential surface.
2.4.1 Construction of the Lagrangian
The analytic expression for the excited state gradient is found using the Lagrangian technique 65 . We
construct the Lagrangian for the excited state energy E f = E 0 + ω f , using a matrix-vector notation,
( 1) ( )
f 0 f † [2] f f † [2] f
†
L = E + b E b −ω
b S b − −X FDS−SDF . (2.81)
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