In quantum chemistry, a configuration state function (CSF), is a symmetry-adapted linear combination of Slater determinants.
In general, one configuration gives rise to several CSFs; all have the same total quantum numbers for spin and spatial parts but differ in their intermediate couplings.
A configuration state function (CSF), is a symmetry-adapted linear combination of Slater determinants.
In the method of configuration interaction, the wavefunction[1] can be expressed as a linear combination of CSFs, that is in the form
CSFs rather than just Slater determinants can also be used as a basis in multi-configurational self-consistent field computations.
In atomic structure, a CSF is an eigenstate of In linear molecules,
does not commute with the Hamiltonian for the system and therefore CSFs are not eigenstates of
However, the z-projection of angular momentum is still a good quantum number and CSFs are constructed to be eigenstates of
The CSFs are constructed to have the spatial transformation properties of one of the irreducible representations of the point group to which the nuclear framework belongs.
are still valid quantum numbers and CSFs are built to be eigenfunctions of these operators.
CSFs are therefore sometimes also called N-particle symmetry adapted basis functions.
For a configuration the number of electrons is fixed; let's call this
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.
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
Thus the CSF is a linear combination The Lowdin projection operator formalism[3] may be used to find the coefficients.
In fact this simply reflects the different internal couplings of total spin and spatial angular momentum.
At the most fundamental level, a configuration state function can be constructed from a set of
of electrons using the following genealogical algorithm: The above steps will need to be repeated many times to elucidate the total set of CSFs that can be derived from the
Basic quantum mechanics defines the possible single orbital wavefunctions.
In a software implementation, these can be provided either as a table or through a set of logic statements.
The Pauli exclusion principle requires that no two electrons in a system can have all their quantum numbers equal.
For equivalent electrons, by definition the principal quantum number is identical.
So, for equivalent electrons the z components of spin and spatial parts, taken together, must differ.
The situation for orbitals in Abelian point groups mirrors the above table.
orbitals also each generate fifteen possible couplings, all of which can be easily inferred from this table.
Similar tables can be constructed for atomic systems, which transform according to the point group of the sphere, that is for s, p, d, f
The number of term symbols and therefore possible couplings is significantly larger in the atomic case.
[9] A popular computational method for CSF construction is the Graphical Unitary Group Approach.