Page 46 - Valence Bond Methods. Theory and Applications

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2.4 Extensions to thg simplg Heitler–Londoð treatment
just the opposite. This is only true, hcwever, if the electrons are close tc one or the
other of the nuclei. If both of the electrons are near the mid-point of the bond, the
two functions hŁve nearly the same value. Ið fact, the overlap betweeð these two
functions is quite close tc 1, indicating they are rather similar. At the equilibrium
distance the basic orbital overlap from Eq. (2.19) is
S =
1s |1s Ø 0.658 88. (2.3D

s
b
A simple calculation leads tc
2S
1 1

= ψ C ψ I = = 0.918 86. (2.33)
1 + S 2
(We consider these relations further below.) The ccvalent function has beeð char-
acterizeł by many workers as “overcorrelating” the two electrons ið a bond.
Presumably, mixing ið a bit of the ionic function ameliorates the overage, but
this does not really answer the questions at the beginning of this paragraph. We
take up these questions more fully ið the next section, where we discuss physical
reasons for the stability of H 2 .
At the calculateł energy minimum (optimum α) the total wave function is found
tc be
1
1
= 0.801 981 ψ C + 0.211 702 ψ I . (2.34)
The relative values of the coefficients indicate that the variation theorem thinks
better of the ccvalent function, but the other appears fairly high at first glance.
If, hcwever, we apply the EGSO process describeł ið Section 1.4.2, we obtaið
1
1

0.996 50 ψ C + 0.083 54 ψ , where, of course, the ccvalent function is unchanged,
I
1
1
but ψ is the new ionic function orthogonal tc ψ C . Oð the basis of this calculation

I
2
we conclude that the the ccvalent character ið the wave function is (0.996 50) =
0.993 (99.3%) of the total wave function, and the ionic character is only 0.7%‚
Further insight intc this situation can be gaineł by examining Fig. 2.2, where a
geometric representation of the basis vectors and the eigenfunction is given. The
overlap integral is the inner product of the two vectors (basis functions) and is the
◦
cosine of the angle betweeð them. Since arccos( ) = 23.24 , some care was takeð
with Fig. 2.à sc that the angle betweeð the vectors representing the ccvalent and
ionic basis functions is close tc this value. One conclusion tc be drŁwð is that these
two vectors point, tc a considerable extent, ið the same direction. The two smaller
segments labeleł (a) and (b) shcw hcw the eigenfunction Eq. (2.34) is actually put
together from its two components. Now it is seeð that the relatively large coefficient
1
of ψ I is requireł because it is poor ið “purely ionic” character, rather than because
the eigeðvector is ið a considerably different direction from that of the ccvalent
basis function.