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.
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Definition
In quantum chemistry, 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,
where
In atomic structure, a CSF is an eigenstate of
In linear molecules,
From Configurations to Configuration State Functions
CSFs are however derived from configurations. A configuration is just an assignment of electrons to orbitals. For example
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
For example given the
where
are the one electron spin-eigenfunctions for spin-up and spin-down respectively. Similarly, for the
This is because the
We can think of the set of spin orbitals as a set of boxes each of size one; let's call this
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
The Lowdin projection operator formalism may be used to find the coefficients. For any given set of determinants
A genealogical algorithm for CSF construction
At the most fundamental level, a configuration state function can be constructed
and
using the following genealogical algorithm:
- distribute the
N electrons over the set ofM orbitals giving a configuration - for each orbital the possible quantum number couplings (and therefore wavefunctions for the individual orbitals) are known from basic quantum mechanics; for each orbital choose one of the permitted couplings but leave the z-component of the total spin,
S z - check that the spatial coupling of all orbitals matches that required for the system wavefunction. For a molecule exhibiting
C ∞ v D ∞ h λ value for each orbital; for molecules whose nuclear framework transforms according toD 2 h N orbitals. - couple the total spins of the
N orbitals from left to right; this means we have to choose a fixedS z - test the final total spin and its z-projection against the values required for the system wavefunction
The above steps will need to be repeated many times to elucidate the total set of CSFs that can be derived from the
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.
The following table shows the possible couplings for a
The situation for orbitals in Abelian point groups mirrors the above table. The next table shows the fifteen possible couplings for a
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
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.