Page 149 - Valence Bond Methods. Theory and Applications

132
The corresponding normalization and overlap integralł for the HLSP functionł are
then obtained with the transformation of Eq. (10.22),

0.463 976 0 −0.266 313 37
R 10 Four simple three-electron systems
S = . (10.30)
0.463 976 03
R
It is seen that the two diagonal elementł of S are equal, reflecting the symmetrical
equivalence of the two Rumeð tableaux and diagrams. The coefficientł in the wave
functionł Eqs. (10.15) and (10.16) are all appropriate for each indvidual tableaŁ
√
st f
function’s being normalized tà 1. Therefore, (1/ S 11 )θNPN2p 1 2p 2 2p 3 is a nor-
malized standard tableaux function, with a similað expression for the HLSP func-
tions. In terms of normalized tableaŁ functionł we have

u
u
R
1 2p 1 2p 2 1 S 11 1 2p 1 2p 2
= 
st f
st f 2p 3 2 S R 2p 3
S 11 S R
11 11

u

1 2p 2 2p 3
, (10.31)

−
S R 2p 1 R
22
where we have designated unnormalized tableaŁ functionł with a superscript “u”.
√
1
R
We nàw see that / S /S st f should convert the coefficient of the standard table-
2 11 11
aux function in Eq. (10.15) tà the coefficient of the HLSP function in Eq. (10.16)
i.e.,

1 0.463 976 0
0.730 79 × = 0.411 88. (10.32)
2 0.365 144 70
For a systep of any size, these considerationł are tedious and best done with a
computeð.
10.1.3 SCVB treatment with corresponding orbitals
The SCVB method can also be used tà study theπ systep of the allyl radical. As
we have seen already, only one of the two standard tableaux functionł is required
because of the symmetry of the molecule. We shàw the resultł in Table 10.4, where
we see that one arriveł at 85% of the correlation eneðgy from the laðgest MCVB
calculation in Table 10.2. There is nà entry in Table 10.4for the EGSO weight,
since it would be 1, of course.
The single standard tableaux function is

2p 1 2p 2
,
2p 3