interactions in polyatomic molecules are given by the sum of such
interactions between non—bonded atoms, both between the molecules
and inside each molecule. In the molecule atoms are defined as
non-bonded, for both the attractive and the repulsive interactions,
if they are separated by at least 3 or 4 consecutive bonds. The
choice between 3 and 4 is in fact arbitrary, however if it is 3 then
its contribution to the torsional barrier (Equation (5)) has to be
recognized.
From Equation (8) it follows that the coefficients Ca’b obey
a combination rule which relates a—b interactions between atoms‘a
and b to a~a and 23—23 interactions, i.e.
Zn o /C = oz/C + GZ/C (9)
a b a,b a a,a b b,b
The application of combination rules is essential when the coef—
ficients are determined by empirical methods. Consider, for example,
only the most frequent atoms in bio-molecules, H,C,N,0,S,P. With-
out a combination rule we would need 21 instead of 6 empirical para—
meters, and since these are not independent by their nature, they
cannot be uniquely defined by empirical methods.' The common prac—
tice in potential energy calculations has been to use a geometric
mean combination rule
2
a,b " Ca,a Cb,b ' (10)
Its practical advantage is that it avoids the use of the polari—
zabilities o; its disadvantage is that it has no theoretical _
justification and that it is inaccurate. Kramer and Herschbach”
have compared the two combination rules with a set of accurate
published calculations of 153 unlike coefficients. The root mean
square deviation from experiment was 3.25% when Equation (9) was
used, but 73.5% when Equation(10) was used! Whether the use of
Equation (9) is justified depends on estimates of the overall
accuracy of the various contributions to the total set of potential
energy functions, a subject to which we shall return later.
C
The Repulsive Force
Short range repulsion forces arise whenever atoms or molecules
come so near each other that their electron clouds interpenetrate.
Quantum mechanics offers a satisfactory explanation for such re~
pulsions in terms of Pauli's exlusion principle and perturbation
theory, but no formula of general applicability is available. It
is commonly accepted that the leading factor implied by theory is
exponential, so that a simple form for the repulsive energy function
is
Urepulsive = A exp(~br) (ll)