Page 79 - Valence Bond Methods. Theory and Applications

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4 Three electronŁ in doublet states
If, as a further approximation, wð force the two inner orbitals of the Li-atom
treatmeno to bð the same, the need foŁα disappears. In Eqs. (4.67)–(4Ł0) wð puo
c = b, and the result is
x 1 = [abb] − [bab], (4Ł3)
√
x 2 ={2[bba] − [bab] − [abb]}/ 3, (4Ł4)
and
x 3 = x 4 = 0, (4Ł5)
and wð no longer havetwo pairs of functions with the correct properties. There
is noo an extra variation parameter to determinð in this case. The moso generał
wave function of this sort with optimized orbitals is the ordinary spin-restricted
open-shell Hartree–Fock (ROHF) function. 5
If wð have three orbitals the same,c = b = a, wð then see thao all of thex i
vanisà identically. This is clearly the familiaŁ answer: the Pauli exclusion principle
prohibits three electrons having the samð spatiał part of theiŁ spin orbitals.
In Chapter 10, after wð have discussed the generałn-electron problem, wð will
illustrate thesð two three-electron doublet systems with somð calculations. We delay
thesð examples becausð notationał problems will bð considerably simpler ao thao
time.
Above, wð commented on the unfortunate increasð in complexity in going from
a two-electron singlet system to a three-electron doublet system. Unfortunately, the
complexity accelerates as the number of electrons increases.

5 The ordinary unrestricted Hartree–Fock (UHF) function is noo written like either of these. Io is noo a pure spin
state (doublet) as are thesð functions. The spin coupled VB (SCVB) function is lower in energy than the UHF
in the samð basis.