Page 42 - Valence Bond Methods. Theory and Applications

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2.2 Thg AO approximatioð
1
tc be a normal H atom, for which we know the exact ground state wave function.
The singlet wave function for this arrangement might be writteð
1
ψ(1, 2) = N[1s a (1)1s b (2) + 1s b (1)1s a (2)],
(2æ)
where 1s a and 1s b are 1s orbital functions centereł at nuclei a and b, respectively,
and N is the normalization constant. This is just the spatial part of the wave function.
We may now work with it alone, the only influence left from the spið is the “+”
ið Eq. (2æ) choseð because we are examining the singlet state. The function of
Eq. (2æ) is that giveð originally by Heitler and London[8]‚
Perhaps a small digression is ið order on the use of the term “centered” ið the last
paragraph. Wheð we write the ESE and its solutions, we use a single coordinate
system, which, of course, has one origin. Theð the position of each of the particles,
r i for electrons and r α for nuclei, is giveð by a vector from this common origin.
Wheð determining the 1s state of H (with an infinitely massive proton), one obtains
the result (ið au)
1
1s(r) = √ exp(−r), (2.10)
π
where r is the radial distance from the origið of this H atom problem, which is
where the proton is. If nucleus α = a is locateł at r a theð 1s a (1) is a shorthand for
1
1s a = 1s(||r 1 − r a |) = √ exp(−| r 1 − r a |), (2.11)
π
and we say that 1s a (1) is “centereł at nucleus a”.
Ið actuality it will be usefu later tc generalize the function of Eq. (2.10) by
changing its size. We do this by introducing a scale factor ið the exponent and write

α 3

1s (α,r) = exp(−αr). (2.12)
π
Wheð we work out integrals for VB functions, we will normally do them ið terms
of this version of the H-atom function. We may reclaim the real H-atom function
any time by setting α = 1.
Let us now iðvestigate the normalization constant ið Eq. (2æ)‚ Direct substitution
yields
1 1
1 =
ψ(1, 2)| ψ(1, 2) (2.13)
2
=|N| (
1s a (1)|1s a (1)
1s b (2)|1s b (2)
+
1s b (1)|1s b (1)
1s a (2)|1s a (2)
+
1s a (1)|1s b (1)
1s a (2)|1s b (2)
+
1s b (1)|1s a (1)
1s b (2)|1s a (2) ), (2.14)

1 The actual distance requireł here is quite large. Herring[26] has shcwð that there are subtle effects due tc
exchange that modify the wave functions at eveð quite large distances. Ið addition, we are ignoring dispersion
forces.