In the single configuration case, the ground state initial state is
denoted by
.
The excited final state is denoted by
,
where the configuration state function
,
defining the channel and for the case of an open channel, containing
the continuum orbital. The transistion amplitude for channel j is,
where,
represents the state containing the outgoing wave in channel (j)
only and incoming waves in all channels, i.e.,
For the multi-configuration case, the ground state is denoted by
and the excited state is
.
Let
contain normalized outgoing waves, i.e.,
.
Then,
Note,
is normalized to satisfy normalization condition above.
For low energy photoionization (dipole approximation), the differential cross section is,
with the cross section given by,
where the amplitude
for channel
is written as,
denotes whether the multipolarity J is magnetic
or electric
.
For low energy photoionization,
.
(dipole electric case) The matrix element in the above expression for
is a reduced matrix element between the ground state and the ionization
(excited) state in the channel
.
Because of a coupling between different channels, the final state
contains admixtures of all channels, but has an outgoing wave only
in the considered channel while having incoming waves on all other
channels.
The amplitude
can be expressed in terms of reduced many particle matrix elements
between ground state and excited states for one channel only.
Now the reduced many particle matrix element can be expressed in terms of reduced single particle matrix elements,
where
is the tensorial
coefficient. The reduced single particle matrix element can be
factorized into angular coefficients
and a radial integral
So that,