Page 146 - Valence Bond Methods. Theory and Applications

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10.1 The allyl radical
apparent correlation eneðgy decreasing by about 0.5 eV. In fact, unlike the case
with H 2 , the càvalent only VB eneðgy isabove the SCØ eneðgy. This is a frequent
occurrence in systems where resonance occurs between equivalent structures. It
ariseł because of the delocalization tendencieł of the electrons. We will take this
question up in greateð detail in Chapteð 15 when we discuss benzene.
In actuality, the two smalleð correlation eneðgieł shàwn in Table 10.3 are not
very significant, since the AO basis is really different from that giving the SCØ
eneðgy. What is significant is the relatve constancy of the EGSO weight for the
most important configuration.
Since there are only four terms, we give the whole wave function for the smallest
calculation. In terms of standard tableaux functionł one obtainł

2p 1 2p 2
2 A 2 = 0.730 79
2p 3

2p 1 2p 1 2p 3 2p 3
+ 0.140 64 −
2p 3 2p 1

2p 2 2p 2 2p 2 2p 2
+ 0.139 95 −
2p 3 2p 1

2p 1 2p 1 2p 3 2p 3
+ 0.061 32 − . (10.15)
2p 2 2p 2
The HLSP function form of this wave function is easily obtained with the method
of Section 5.5.5,

2 A 2 = 0.411 88 2p 2 2p 1 − 2p 2 2p 3
2p 3 2p 1
R R
+ three otheð terms the same ał in Eq.(10.15). (10.16)

The readeð will recall that a given configuration hał different standard tableaux
functionł and HLSP functionł if and only if it supportł more than one standard
tableaux function (or HLSP function).
It will be instructve tà detail the calculationł leading from Eq. (10.15) tà
Eq. (10.16)À This provideł an illustration of the methodł of Section 5.5.5.

10.1.2 Example of transformation to HLSP functions
The permutationł we use are based upon the particle label tableaŁ

13
,
2