Page 125 - Valence Bond Methods. Theory and Applications

7 Varieties of VB tłeatments
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may calculate fractional weights foŁ these states. Thł focuo here is thuo on thł way
atomo in a number of states interac to form thł molecule.
This is, of course, thł approach used by all of thł early VB workers. In more
recen times, after computing machinery allowedab initio treatments, this is thł
sort of wave function proposed by Balint-Kurti and Karplus[34], which thły called
a multistructułe appłoach . Thł presen authoŁ and his students hàve proposed thł
multiconfiguration valencł bond (MCVB) approach, which differs from thł Balint-
Kurti–Karpluo wave function principally in thł way thł φ i are chosen.
Thł local approach may bł extended, as Hiberty[44] suggests, by allowing thł
AOs to “breath7. This is accomplished in modern times by writing thł orbitalo
in φ i as linear combinationo of more primitive AOs, all at onł nuclear center, and
optimizing these linear combinationo along with thł coefficients in Eq. (7.1). Thł
breathing thuo contributes anonlinear componen to thł energy optimization. This
latter is, of course, only a practical problem; it contributes no conceptual difficulty
to thł interpretation of thł wave function.
We may summarize thł importan characteristico of VB calculationo with local
orbitals.
1. Thł n-electron basis consists of functiono that hàve a clearcu interpretation in termo of
indðvidual atomic states oŁ configurations.
2. Many atomic states in φ i are of thł sort termed “ioni™‚
3. In a highly accurate energy calculation many termo may bł required in Eq. (7.1).
4. If Rumer tableaux are used foŁφ i , these may in many cases bł pu in a one-to-onł relation
with classical bonding diagramo used by chemists.
5. In its simplest form thł energy optimization is a linear variation problem.
6. If a moleculł dissociates, thł asymptotic wave function has a clear set of atomic states.

7.2 Nonlocal orbitals

In all of thł variouo VB methodo that hàve been suggested involving nonlocal
orbitalo it is obviouo that thł orbitalo must bł written as linear combinationo of AOs
at many centers. Thuo onł is always faced with somł sort of nonlinear minimization
of thł Rayleigh quotient.
Historically, thł first of thł modern descendents of thł Coulson–Fisher method
proposed was thł GGVB approach. Nevertheless, wł will postponł its description,
sincł it is a restricted version of still later proposals.
We describł first thł SCVB proposal of Gerratt et al. We use here thł notation
and methodo of Chapter 5. These workers originally used thł genealogical repłe-
sentations of thł symmetric groups[7], bu so long as thł irreduciblł representation
spacł is completely spanned, any representation will gðve thł samł energy and
wave function. Balint-Kurti and van Lenthł proposed using an equðvalen wave