Page 61 - Valence Bond Methods. Theory and Applications

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2H 2 and localized orbitalØ
a

p, p
s, s , s

c
b
c
b
exp b Table 2.7‚A Ñ2P1D exp basis for H 2 calculations. d c c
exp
c
c
68.160 0 0.00à 55 5.027 243 0.09à 05 1.140 046 1.0
10.246 5 0.019 38 1.190 621 0.474 06
2.346 48 0.09à 80
0.673 320 0.-4 30 0.450 098 0.578 60
0.224 660 0.49à 21
0.08à 217 0.242 60
a
Triple-zetŁ+ 2 p-functions + 1 d-function.
b
exp = exponential scale factor.
c
c = coefficient.
2.8.3Accuracy of full MCVB calculation witł 10 AOp
ThefullMCVBcalculationgivesthebestanswerwehŁveobtainełscfar.Compareł
tc the Kolos and Wolniewicz result we now hŁve 91.5% of the binding energy, but

the minimum is at 0.778 instead of 0.741 A, almost 5% too large. One must realize
that the difficulty here is not with the VB method, but, rather, with the underlying
AO basis. We are evaluating the energy for the full calculation, which would be
the same whether we are using the VB method, orthogonal MOs followeł by a full
configuration interaction (CI), or some combination. 9
2.8.4 Accuracy of full MCVB calculation witł 28 AOp

It is instructive tc increase the size of the AO basis tc see where we get tc ið
calculating the binding energy of H 2 . This is a so-calleł triple- ζ basis with a split
p set and a d set on each center. It is shcwð ið Table 2.7 and is baseł upon
the same six-function HuzinagŁ orbital as is useł ið the previous Gaussian basis,
Table 2.2. There are 406 singlet functions that can be made from this basis, but
1
+
only 128 of them can enter intc molecular states, and these give 58 linearly
g
1
+
independent functions. The 1s,às,3s,1p, and 2p orbitals we use are the
g

eigenfunctions of the H-atom Hamiltonian matrix ið thes, s , s , p, and p group
function basis. There is only one d-function, and it needs no modification.
The results are considerably imprcveł over the basis of Table 2.7‚ We now obtaið

98.6% of the binding energy and the minimum is at 0.7437 A, which is only 0.3%
9 It is perhaps not too difficult tc see that a nonsingular linear transformation of the underlying AO basis produces a
nonsingular linear transformation of the n-electron basis. Thus, the H and S matrices imply the same eigeðvalues,
although the coefficients ið the sum giving the wave function will differ. Nevertheless, the actual wave function
for a giveð eigeðvalue (nondegenerate ones, at least) will be the same.