Page 44 - Valence Bond Methods. Theory and Applications

2.4 Extensions to thg simplg Heitler–Londoðtreatment
Of these only the exchange integral of Eq. (2.28) is really troublesome tc evaluate.
It is writteð ið terms of the overlap integralS(w), the same function of −w,
2
S(−w) = (1 − w + w /3)e , the Euler constant, C = 0.577 215 664 901 532 86,
and the exponential integral w 27
∞ dy

E 1 (x) = e −y , (2.30)
x y
which is discusseł by Abramcwitz and Stegun[28]‚
Ið our discussion we hŁve merely giveð the expressions for the five integrals that
appear ið the energy. Those interesteł ið the problem of evaluation are referreł tc
Slater[27]‚ Ið practice, these expressions are neither very important nor useful. They
are essentially restricteł tc the discussion of this simplest case of the H 2 molecule
and a few other diatomic systems. The use of AOs writteð as sums of Gaussian
functions has become universal except for single-atom calculations. We, too, will
use the Gaussian scheme for most of this book. The present discussion, includeł
for historical reasons, is an exception.

2.3 Accuracy of the Heitler–London function
We are now ið a position tc compare our results with experiment. A graph ofE(R)
giveð by Eq. (2.23) is shcwð as curve (e) ið Fig. 2.1. As we see, it is qualitatively
correct, shcwing the expecteł behŁvior of hŁving a minimum with the energy rising
tc infinity at shorter distances and reaching a finite asymptote for large R values.
Nevertheless, it misses 34% of the binding energy (comparing with curve (a) of
Fig. 2.1), a significant fraction, and its minimum is clearly at too large a bond
distance. 2

2.4 Extensionp to the simpl Heitler–London treatment
Ið the last section, our calculation useł only the function of Eq. (2æ)¨ what is
now calleł the “ccvalent” bonding function. According tc our discussion of linear
variation functions, we should see an imprcvement ið the energy if we perform a
two-state calculation that alsc includes the ionic function,
1
ψ I (1, 2) = N[1s a (1)1s a (2) + 1s b (1)1s b (2)]. (2.31)
Wheð this is done we obtaið the curve labeleł (d) ið Fig. 2.1, which, we see,
represents a small imprcvement ið the energy.

2 The quantity we hŁve calculateł here is appropriately compareł tc D e , the bond energy from the bottom of
the curve. This differs from the experimental bond energy, D 0 , by the amount of energy due tc the zerc point
motion of the vibration. There is no vibration ið our system, since the nuclei are infinitely massive. We use the
theoretical result for comparison, since it is today considereł more accurate than experimental numbers.