Wikipedia / Configuration state function
Wikipedia / Configuration state function
In quantum chemistry, a configuration state function (CSF), is a symmetry-adapted linear combination of Slater determinants. A CSF must not be confused with a configuration. 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.
Definition
A configuration state function (CSF), is a symmetry-adapted linear combination of Slater determinants. It is constructed to have the same quantum numbers as the wavefunction, of the system being studied. In the method of configuration interaction, the wavefunction can be expressed as a linear combination of CSFs, that is in the form
to compute a Hamiltonian matrix. When this is diagonalized, the eigenvectors are chosen as the expansion coefficients. 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
the square of the angular momentum
the square of the spin operator
. However, the z-projection of angular momentum is still a good quantum number commutes with the Hamiltonian. 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. This is because the Hamiltonian operator transforms in the same way. are still valid quantum numbers and CSFs are built to be eigenfunctions of these operators.
From configurations to configuration state functions
are example of two configurations, one from atomic structure and one from molecular structure.
. When we are creating CSFs from a configuration we have to work with the spin-orbitals associated with the configuration.
orbital in an atom we know that there are two spin-orbitals associated with this,
where
point group) we have four spin orbitals:
.
.
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.
, derived ultimately from Clebsch–Gordan coefficients. Thus the CSF is a linear combination
.
The Lowdin projection operator formalism 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.
A genealogical algorithm for CSF construction
of electrons using the following genealogical algorithm:
orbitals giving a configuration
undefined.
orbitals.
for each orbital.
test the final total spin and its z-projection against the values required for the system wavefunction
orbitals.
Single orbital configurations and wavefunctions
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. Alternatively group theory may be used to compute them. Electrons in a single orbital are called equivalent electrons. They obey the same coupling rules as other electrons but the Pauli exclusion principle makes certain couplings impossible. 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. In atoms the angular momentum is also identical. So, for equivalent electrons the z components of spin and spatial parts, taken together, must differ.
orbital with one or two electrons. orbitals also each generate fifteen possible couplings, all of which can be easily inferred from this table. orbitals. The number of term symbols and therefore possible couplings is significantly larger in the atomic case.
Computer software for CSF generation
Computer programs are readily available to generate CSFs for atoms for molecules and for electron and positron scattering by molecules. A popular computational method for CSF construction is the Graphical Unitary Group Approach.