- Configuration state function
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, Ψ, 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
Ψ = ∑ ckψk k
where ψk denotes the set of CSFs. The coefficients, ck, are found by using the expansion of Ψ 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 operator, .
- the z-projection of angular momentum
- the square of the spin operator .
- the z-projection of the spin operator
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 and . In non-linear (which implies polyatomic) molecules, neither nor commute 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. and are still valid quantum numbers and CSFs are built to be eigenfunctions of these operators.
From Configurations to Configuration State Functions
CSFs are however derived from configurations. A configuration is just an assignment of electrons to orbitals. For example 1s2 and 1π2 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 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 1s orbital in an atom we know that there are two spin-orbitals associated with this,
are the one electron spin-eigenfunctions for spin-up and spin-down respectively. Similarly, for the 1π orbital in a linear molecule ( point group) we have four spin orbitals:
This is because the π designation corresponds to z-projection of angular momentum of both + 1 and − 1.
We can think of the set of spin orbitals as a set of boxes each of size one; let's call this M boxes. We distribute the N electrons among the M boxes in all possible ways. Each assignment corresponds to one Slater determinant, Di. There can be great number of these, particularly when N < M. Another way to look at this is to say we have M entities and we wish to select 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 ci, 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 Di 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 M orbitals
- a number N of electrons
using the following genealogical algorithm:
- distribute the N over the set of M 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, Sz undefined.
- check that the spatial coupling of all orbitals matches that required for the system wavefunction. For a molecule exhibiting or this is achieved by a simple linear summation of the coupled
λ value for each orbital; for molecules whose nuclear framework transforms according to D2h 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 N orbitals.
- couple the total spins of the N orbitals from left to right; this means we have to choose a fixed Sz for each orbital.
- 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 N electrons and M orbitals.
Single Orbital configurations and wavefunctions
Basic quantum mechanics defines the possible single orbital wavefunctions. In any software implementation, these will be provided either as a table or thourgh a set of logic statements.
The following table shows the possible couplings for a σ orbital with one or two electrons.
Orbital Configuration Term symbol Sz projection σ1 2Σ + σ1 2Σ + σ2 1Σ + 0
The next table shows the fifteen possible couplings for a π orbital. The orbitals also each generate fiftenn possible couplings, all of which can be easily inferred from this table.
Orbital Configuration Term symbol Lambda coupling Sz projection π1 2Π + 1 π1 2Π + 1 π1 2Π − 1 π1 2Π − 1 π2 3Σ − 0 + 1 π2 3Σ − 0 0 π2 3Σ − 0 − 1 π2 1Δ + 2 0 π2 1Δ − 2 0 π2 1Σ + 0 0 π3 2Π + 1 π3 2Π + 1 π3 2Π − 1 π3 2Π − 1 π4 1Σ + 0 0
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 orbitals. The number of term sysmbols 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.
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