Configuration State Function - From Configurations To Configuration State Functions

From Configurations To Configuration State Functions

CSFs are however derived from configurations. A configuration is just an assignment of electrons to orbitals. For example and are example of two configurations, one from atomic structure and one from molecular structure.

From any given configuration we can, in general, create several CSFs. CSFs are therefore sometimes also called N-particle symmetry adapted basis functions. It is important to realize that for a configuration the number of electrons is fixed; let's call this . When we are creating CSFs from a configuration we have to work with the spin-orbitals associated with the configuration.

For example given the orbital in an atom we know that there are two spin-orbitals associated with this,

where

are the one electron spin-eigenfunctions for spin-up and spin-down respectively. Similarly, for the orbital in a linear molecule ( point group) we have four spin orbitals:

.

This is because the designation corresponds to z-projection of angular momentum of both and .

We can think of the set of spin orbitals as a set of boxes each of size one; let's call this boxes. We distribute the electrons among the boxes in all possible ways. Each assignment corresponds to one Slater determinant, . There can be great number of these, particularly when . Another way to look at this is to say we have entities and we wish to select of them, known as a combination. We need to find all possible combinations. Order of the selection is not significant because we are working with determinants and can interchange rows as required.

If we then specify the overall coupling that we wish to achieve for the configuration, we can now select only those Slater determinants that have the required quantum numbers. In order to achieve the required total spin angular momentum (and in the case of atoms the total orbital angular momentum as well), each Slater determinant has to be premultiplied by a coupling coefficient, derived ultimately from Clebsch-Gordan coefficients. Thus the CSF is a linear combination

.

The Lowdin projection operator formalism may be used to find the coefficients. For any given set of determinants it may be possible to find several different sets of coefficients. Each set corresponds to one CSF. In fact this simply reflects the different internal couplings of total spin and spatial angular momentum.