MCTD approximation for excited states is the movement of excited particles
in the 'frozen field' of other MCHF/MCDF ground state orbitals the
field.
The multi-channel MCTD excited state function can be written as,
where,
and
and
is the configuration state function (CSF) corresponding to transistions
(channels) to excited orbitals, discrete and continuous ones too.
is the CSF corresponding to transistions among valence orbitals
.
The coefficients
are equal to the coefficients of the corresponding parent ground state
configuration. The equation for the coefficients
where
is,
One can derive the MCTD orbital equations, in the same way the
MCDF/MCHF equations are derived, from the variational principle by
taking variations only with respect to the excited orbitals
.
The MCTD orbital equations could then be deduced from
the MCDF/MCHF equations by the following replacements: (which can be
easily seen from the MCDF equations)
and,
Note, the configuration averaged angular coefficients, which are the
sums of the coefficients in the MCDF case. Reduced to only one term
because there is only one excited state configuration with the given
excited orbital
.