Configuration state function

From configurations to configuration state functions

CSFs are derived from configurations. A configuration is just an assignment of electrons to orbitals. For example, ${\displaystyle 1s^2}$ and ${\displaystyle 1{\pi }^2}$ 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. For a configuration the number of electrons is fixed; let's call this ${\displaystyle N}$. When we are creating CSFs from a configuration we have to work with the spin-orbitals associated with the configuration.

For example, given the ${\displaystyle 1s}$ orbital in an atom we know that there are two spin-orbitals associated with this,

${\displaystyle 1s\alpha 1s\beta }$

where

${\displaystyle \alpha ,\beta }$

are the one electron spin-eigenfunctions for spin-up and spin-down respectively. Similarly, for the ${\displaystyle 1\pi }$ orbital in a linear molecule (${\displaystyle C_{\mathrm{\infty }v}}$ point group) we have four spin orbitals:

${\displaystyle 1\pi (+)\alpha ,1\pi (+)\beta ,1\pi (-)\alpha ,1\pi (-)\beta }$.

This is because the ${\displaystyle \pi }$ designation corresponds to z-projection of angular momentum of both ${\displaystyle +1}$ and ${\displaystyle -1}$.

We can think of the set of spin orbitals as a set of boxes each of size one; let's call this ${\displaystyle M}$ boxes. We distribute the ${\displaystyle N}$ electrons among the ${\displaystyle M}$ boxes in all possible ways. Each assignment corresponds to one Slater determinant, ${\displaystyle D_i}$. There can be great number of these, particularly when ${\displaystyle N<<M}$. Another way to look at this is to say we have ${\displaystyle M}$ entities and we wish to select ${\displaystyle N}$ 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 ${\displaystyle c_i}$, derived ultimately from Clebsch–Gordan coefficients. Thus the CSF is a linear combination

${\displaystyle \sum _i{c}_i{D}_i}$.

The Lowdin projection operator formalism may be used to find the coefficients. For any given set of determinants ${\displaystyle D_i}$ 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

At the most fundamental level, a configuration state function can be constructed from a set of ${\displaystyle M}$ orbitals and a number ${\displaystyle N}$ of electrons using the following genealogical algorithm:

1. distribute the ${\displaystyle N}$ electrons over the set of ${\displaystyle M}$ orbitals giving a configuration
2. 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, ${\displaystyle S_z}$ undefined.
3. check that the spatial coupling of all orbitals matches that required for the system wavefunction. For a molecule exhibiting ${\displaystyle C_{\mathrm{\infty }v}}$ or ${\displaystyle D_{\mathrm{\infty }h}}$ this is achieved by a simple linear summation of the coupled ${\displaystyle \lambda }$ value for each orbital; for molecules whose nuclear framework transforms according to ${\displaystyle D_{2h}}$ symmetry, or one of its sub-groups, the group product table has to be used to find the product of the irreducible representation of all ${\displaystyle N}$ orbitals.
4. couple the total spins of the ${\displaystyle N}$ orbitals from left to right; this means we have to choose a fixed ${\displaystyle S_z}$ for each orbital.
5. 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 ${\displaystyle N}$ electrons and ${\displaystyle M}$ 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.

The following table shows the possible couplings for a ${\displaystyle \sigma }$ orbital with one or two electrons.

The situation for orbitals in Abelian point groups mirrors the above table. The next table shows the fifteen possible couplings for a ${\displaystyle \pi }$ orbital. The ${\displaystyle \delta ,\varphi ,\gamma ,\dots }$ 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 ${\displaystyle \dots }$ 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.