Determinants or configuration state functions (CSFs)?


Determinants or configuration state functions (CSFs) may be used to form the many electron basis set. A determinant is a simple object: a product of spin orbitals with a given Sz quantum number, that is, the number of alpha spins and number of beta spins are a constant.  Matrix elements involving determinants are correspondingly simple, but unfortunately determinants are not necessarily eigenfunctions of the S**2 operator.

    To expand on this point, consider the four familiar 2e-functions which satisfy the Pauli principle.  Here u, v are space orbitals, and a, b are the alpha and beta spin functions.  As you know, the singlet and triplets are:

       S1 = (uv + vu)/sqrt(2) * (ab - ba)/sqrt(2)
       T1 = (uv - vu)/sqrt(2) *  aa
       T2 = (uv - vu)/sqrt(2) * (ab + ba)/sqrt(2)
       T3 = (uv - vu)/sqrt(2) *  bb

It is a simple matter to multiply out S1 and T2, and to expand the two determinants which have Sz=0,

       D1 = |ua vb|          D2 = |va ub|

This reveals that

       S1 = (D1+D2)/sqrt(2)   or   D1 = (S1 + T2)/sqrt(2)
       T2 = (D1-D2)/sqrt(2)        D2 = (S1 - T2)/sqrt(2)

Thus, one must take a linear combination of determinants in order to have a wavefunction with the desired total spin. There are two important points to note:
  a) A two by two Hamiltonian matrix over D1 and D2 has eigenfunctions with -different- spins, S=0 and S=1.
  b) use of all determinants with Sz=0 does allow for the construction of spin adapted states.  D1+D2, or D1-D2, are -not- spin contaminated.

By itself, a determinant such as D1 is said to be "spin contaminated", being a fifty-fifty admixture of singlet and riplet.  (It is curious that calculations with just one such determinant are often called "singlet UHF", when this is half triplet!).  Of course, some determinants are spin adapted all by themselves, for example the spin adapted functions T1 and T3 above are single determinants, as are the closed shells

       S2 = (uu) * (ab - ba)/sqrt(2).
       S3 = (vv) * (ab - ba)/sqrt(2).

It is possible to perform a triplet calculation, with no singlet states present, by choosing determinants with Sz=1 such as T1, since then no state with Sz=0 exists in the determinant basis set (as is required when S=0).  To summarize, the eigenfunctions of a Hamiltonian formed by determinants with any particular Sz will be spin states with S=Sz, S=Sz+1, S=Sz+2, ... but will not contain any S values smaller than Sz.

    CSFs are an antisymmetrized combination of a space orbital product, and a spin adapted linear combination of simple alpha-beta products.  Namely, the following CSF

       C1 = A (uv) * (ab-ba)/sqrt(2)

which has a singlet spin function is identical to S1 above if you write out what the antisymmetrizer A does, and the CSFs

       C2 = A (uv) * aa
       C3 = A (uv - vu)/sqrt(2) * (ab + ba)/sqrt(2)
       C4 = A (uv) * bb

equal T1-T3.  Since the three triplet CSFs have the same energy, GAMESS works with the simpler form C2.  Singlet and triplet computations using CSFs are done in separate runs, because when spin-orbit coupling is not considered, the Hamiltonian is block diagonal in a CSF basis.  Technical information about the CSFs is that they use Yamanouchi- Kotani spin couplings, and matrix elements are obtained using a GUGA, or graphical unitary group approach.

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