A Tour of the Foundations of
Computational Chemistry and Material Science
Lecture 1: ab initio Methods



Transistion Amplitudes in Single Configuration and Multi-configuration Cases

In the single configuration case, the ground state initial state is denoted by tex2html_wrap_inline386. The excited final state is denoted by tex2html_wrap_inline388, where the configuration state function tex2html_wrap_inline390, defining the channel and for the case of an open channel, containing the continuum orbital. The transistion amplitude for channel j is,

equation214

where, tex2html_wrap_inline392 represents the state containing the outgoing wave in channel (j) only and incoming waves in all channels, i.e.,

equation219

For the multi-configuration case, the ground state is denoted by tex2html_wrap_inline394 and the excited state is tex2html_wrap_inline396. Let tex2html_wrap_inline398 contain normalized outgoing waves, i.e., tex2html_wrap_inline400. Then,

equation230

Note, tex2html_wrap_inline402 is normalized to satisfy normalization condition above.

For low energy photoionization (dipole approximation), the differential cross section is,

equation240

with the cross section given by,

equation249

where the amplitude tex2html_wrap_inline404 for channel tex2html_wrap_inline406 is written as,

equation258

tex2html_wrap_inline408 denotes whether the multipolarity J is magnetic tex2html_wrap_inline410 or electric tex2html_wrap_inline412. For low energy photoionization, tex2html_wrap_inline414. (dipole electric case) The matrix element in the above expression for tex2html_wrap_inline404 is a reduced matrix element between the ground state and the ionization (excited) state in the channel tex2html_wrap_inline418. 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 tex2html_wrap_inline404 can be expressed in terms of reduced many particle matrix elements between ground state and excited states for one channel only.

equation268

Now the reduced many particle matrix element can be expressed in terms of reduced single particle matrix elements,

equation274

where tex2html_wrap_inline422 is the tensorial tex2html_wrap_inline424 coefficient. The reduced single particle matrix element can be factorized into angular coefficients tex2html_wrap_inline426 and a radial integral tex2html_wrap_inline428

equation279

So that,

equation281




Author: Dr. Warren Perger and Ken Flurchick