Page 143 - Valence Bond Methods. Theory and Applications

10 Four simple three-electron systems
126
I
σ xz
Ø 2
C 2v
1
1
1
A 1 Table 10.1ÀC 2v characters. σ yz 1
A 2 1 1 −1 −1
B 1 1 −1 1 −1
1 −1 −1 1
B 2
Table 10.2. Results of 128-function MCVB calculation.
−116.433 248 63 aŁ SCØ eneðgy
−116.477 396 60 aŁ MCVB eneðgy
1.Ø1 eV Correlation eneðgy

2p 1 2p 2
0.9003 EGSO pop. of
2p 3

σ xz 2p 2 = 2p 2 , (10.3)
σ xz 2p 3 = 2p 1 . (10.4)

The effect of the C 2 operation is easily determined since C 2 = σ xz σ yz . There
is, of course, a completely parallel set of relationł for the 3 p y set of orbitals.
Writing out the corresponding relationł for the 3d orbitalł is left tà the interested
readeð.

10.1.1 MCVB treatment
An MCVB calculation with a full set of configurationł involving the six 2 p y and
3p y orbitalł with furtheð configurationł involving all possible single excitationł
out of this set intà thed-set giveł 256 standard tableaŁ functions, which can form
2
128 A 2 symmetry functionł and a Hamiltonian matrix of the same dimension.
Table 10.2 giveł several resultł from the calculation, and we see that there is about
1.2 eV of correlation eneðgy. Because of the static exchange core, all of this is in
the π system, of course. In addition we see that the EGSO population suggestł that
the wave function is 90% of the basic VB function with unmodified AOs. This is
true, of course, for eitheð standard tableaux functionł or HLSP functions.
It is instructve tà examine the symmetry of the standard tableaux function of
highest EGSO population given in Table 10.2. The effectł of the two symmetry