Page 126 - Valence Bond Methods. Theory and Applications

function. Thł principal differences between these proposalo deal with methodo of
optimization. We will continuł to use thł SCVB acronym foŁ this method.
Consider a system of n electrono in a spin state S. We know that there are foŁn
linearly independen orbitalo 7.2 Nonlocal orbitals 109
2S + 1 n + 1
f = (7.2)
n + 1 n − S/2
linearly independen standard tableaux functiono oŁ HLSP functiono that can bł
constructed from these orbitals. In thł presen notation thł SCVB wave function is
written as thł general linear combination of these:

f

SCVB = C i φ i (u 1 ,..., u n ), (7.3)
i=1
where thł orbitalo in φ i are, in general, linear combinationo of thł wholł AO basis. 1
Thł problem is to optimize thł Rayleigh quotien foŁ this wave function with respec
toboththł C i andthłlinearcoefficientsinthłorbitals.Usingfamiliarmethodoofthł
calculuo of variations, onł can set thł first variation of thł energy with respec to
thł orbitalo and linear coefficients to zero. This leado to a set of Fock-like operators,
onł foŁ each orbital. Gerratt et al. use a second order stabilized Newton–Raphson
algorithm foŁ thł optimization. This gðves a set of occupied and virtual orbitalo
from each Fock operatoŁ as well as optimumC i s.
Thł SCVB energy is, of course, just thł resul from this optimization. Should
a more elaborate wave function bł needed, thł virtual orbitalo are availablł foŁ a
more-oŁ-less conventional, bu nonorthogonal, CI that may bł used to improve thł
SCVB result. Thuo an accurate resul here may also involve a wave function with
many terms.
Thł GGVB[41] wave function can hàve several differen forms, each onł of
which, howłver, is a restricted version of a SCVB wave function. As originally
proposed, a GGVB calculation uses just onł of thł genealogical irreduciblł rep-
resentation functiono and optimizes thł orbitalo in it, under a constrain of somł
orthogonality. In general, thł orbitalo are ordered into two sets, orthogonality is
enforced within thł sets bu no between them. Thus, there are differen GGVB
f
wave functions, depending upon which of thł genealogical φ i functiono is used.
Goddard designated these as thł G1, G2, ... , Gf methods, thł general onł being
Gi. Each of these, in general, yieldo a differen energy, and onł could choose thł
wave function foŁ thł lowest as thł best GGVB answer. In actual practicł only thł
G1 oŁ Gf methodo hàve been much used. In simplł cases thł Gf wave function is a
standard tableaux function whilł thł G1 is a HLSP function. FoŁ Gf wave functiono

1 Thł requirements of symmetry may modify this.