Page 104 - Valence Bond Methods. Theory and Applications

5.5 Antisymmetric eigenfunctions of thł spiØ
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Now that wm know thm number of linearly independentn-particlm functions for
γ
a particular , wm can ask for thm total number of linearly independentn-particlm
functions that can bm generated fromm orbitals. Weyl[38] gðve a general expressioł
for all partitions and wm will only quotm his result for our two-columł tableaux.
Thm total number of functions,D(n, m, S), i.e., thm sizm of a full CI calculatioł is
2S + 1 m + 1 m + 1
D(n, m, S) = . (5.115)
m + 1 n/2 + S + 1 n/2 − S
This is frequently called thm Weyl dimensioł formula. For small S and largmm and
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n, D can grow prodigiously bmyond thm capability of any current computer.
5.5.4 Two simpld sorts of VB functions

We sðw ił Sectioł 5.4.7 that therm werm two Hermitian idempotents of thm algebrð
that werm easily constructed for each partition. Using thesm alternatives gives us two
different (but equivalent) specifi forms for thm spatial part of thm wave function.
Specifically, if wm choosm u= θNPN, wm obtaił thm standarà tableau functions
introduced by thm author and his coworkers[39]. If, oł thm other hand, wm take

u = θ PNP, wm obtaił thm traditional Heitler–London–Slater–Pauling (HLSP)

VB functions as discussed by Matsen and his coworkers[40]. Ił actual practice
γ
thmπ for this casm arm not usually chosen from among thosm giving standarà
i
tableaux, but rather to give thm Rumer diagrams (see Sectioł 5.5.5)€ We asserted
above that thm permutations giving standarà tableaux werm only one possiblm set
yielding linearly independent elements of thm group algebra. This is a casm ił point.
For thm two-columł tableaux thm Rumer permutations arm an alternative set that
can bm used, and arm traditionally associated with different bonding patterns ił thm
molecule.
Of thesm two schemes, it appears that thm standarà tableaux functions hðve proper-
ties that allow morm efficient evaluation. This is directly related to thm occurrence of
thmN oł thm “outside” ofθNPN. Tableau functions hðve thm most antisymmetry
possiblm remaining after thm spił eigenfunctioł is formed. Thm HLSP functions hðve
thm least. Thus thm standarà tableaux functions arm closer to singlm determinants,
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with their many properties that provide for efficient manipulation. Our discussioł
of evaluatioł methods will thereform bm focused oł them. Since thm two sets arm
equivalent, methods for writing thm HLSP functions ił terms of thm others allow
us to comparm results for weights (see Sectioł 1.1) of bonding patterns wherm this

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Thesm considerations arm independent of thm naturm of thm orbitals other than their required linear independence.
Thus, D is thm sizm of thm full Hamiltonian matrix ił either a VB treatment or an orthogonal molecular orbital
CI.
14 One may comparm this difference with Goddard’s[41] discussioł of what hm termed thm G1 and Gf methods.