Essentials of Computational Chemistry

14.1 DETERMINANTAL/CONFIGURATIONAL REPRESENTATION OF EXCITED STATES 489
Having dispensed with notational details, let us think more carefully about the chemical
picture implied by Figure 14.1. By picturing an excited state as being a different occupation
of the orbitals of the ground state, we are providing something of a privileged status
to the ground-state orbitals. One must recall that these orbitals, in the HF procedure, are
variationally optimized given an average electrostatic repulsion that depends on the shape
and occupation number of all of the other MOs. When the excited state of Figure 14.1 is
generated, the occupation number of the HOMO is reduced by 1 and the occupation number
of the newly occupied orbital is increased by 1, and thus the HF field acting on every orbital
has changed. This means that none of the occupied orbitals are optimal for the excited state,
and as a result the energy of the excited state, e.g.,
E
1 a
N/2 =
1 a
N/2 |H | 1 a
N/2
(14.5)
evaluated using Eq. (14.3) for the wave function (i.e., the Slater determinant formed from
the optimized orbitals for the ground state), will be unphysically too high.
It is somewhat tempting to impose an incorrect dynamical view on the excited state
when it is generated by ground-state absorption of a photon. One might imagine that the
‘instantaneous’ absorption process generates the wave function of Eq. (14.3), which then
relaxes to the optimal wave function for the excited state by adjustment of all of the orbitals.
This physical picture, however, ignores the common timescale of all electronic motion.
Even as the electron is ‘moving’ from one orbital to the next, the orbitals whose occupation
numbers are not changing are relaxing in response to changing electron–electron interactions.
To digress for a moment, a timescale separation that usually is valid is the Born–
Oppenheimer separation of the nuclear and electronic motions. Thus, as illustrated in
Figure 14.2, when the optimal geometry of the ground state is not the same as that for
the excited state, we may view the absorption of radiation as taking place at the ground-state
geometry, and the energy involved is called the ‘vertical’ excitation energy. On the timescale
of nuclear dynamics, the geometry of the excited state ultimately relaxes to its own optimum.
The energy difference E between the two systems taking each to be at its own optimal
geometry is referred to as the ‘adiabatic’ excitation energy. The same conceptual framework
may be applied to the reverse process, emission. Note that excitation and emission energies
are often expressed not in energy units but instead in terms of the wavelength of radiation
corresponding to that energy according to
E = hc
λ
(14.6)
where h is Planck’s constant, c the speed of light in a vacuum, and λ the radiation wavelength.
AlargerE value is equivalent to a shorter wavelength, so one says that vertical excitations
are blue-shifted (since blue light is on the short wavelength side of the visible spectrum)
relative to adiabatic excitation energies, while vertical emissions with smaller E values
are red-shifted (red being at the opposite end of the visible spectrum) relative to the same
adiabatic standard. The difference between the vertical excitation and emission energies (or
wavelengths) is referred to as the Stokes shift.