Gelfand tableau

Because of the Pauli principle, no more than two such indices may be equal. The indices occurring in the designation of an individual structure pi,P2,---,fkn, k) may be combined into a single "Gelfand tableau [ni] or [ jp]. The relationship (41) may then be written as... 

The labels for the CSF basis functions can thus be written as a triangular Gelfand tableau... 

Each row of the tableau gives the irrep labels for U n), U n— 1). ..etc. The hierarchical ordering comes from the betweenness conditions . The integers in any row must lie between those of the previous row. For the electronic structure problem the integers rriim can have only three values 2, 1 or 0. Consequently, as Paldus discovered, each row of the Gelfand tableau could be denoted by... 

Then one has the remarkable feature that makes the U n) method so powerful for electronic structure problems. Because of the restriction of the entries in the Gelfand tableau to 2, 1 or 0 there are only four ways of adding a lower row to a tableau and still satisfy the betweenness conditions. This leads to the genealogical representation (labelled by possible steps 0, 1, 2 or 3) of the CSF basis shown (in part) in Eq. (18) for U 4) (four orbitals and four electrons) with a full Cl with triplet spin multiplicity (F = 2 1 0 ). 

---

See also in sourсe #XX -- [ ]

---