Essentials of Computational Chemistry

18 2 MOLECULAR MECHANICS
Energy
0
r eq
Figure 2.1 The first seven vibrational energy levels for a lighter (solid horizontal lines) and heavier
(horizontal dashed lines) isotopomer of diatomic AB. Allowed vibrational transitions are indicated by
solid vertical arrows, forbidden transitions are indicated by dashed vertical arrows
of the potential energy at an arbitrary point by taking a Taylor expansion about req
U(r) = U(req) + dU
dr (r − req) +
r=req
1
2!
d2U dr2
(r − req)
r=req
2
+ 1 d
3!
3U dr3
(r − req) 3 +··· (2.1)
r=req
Note that the first two terms on the r.h.s. of Eq. (2.1) are zero, the first by arbitrary choice,
the second by virtue of req being the minimum. If we truncate after the first non-zero term,
we have the simplest possible expression for the vibrational potential energy
r AB
U(rAB) = 1
2 kAB(rAB − rAB,eq) 2
(2.2)
where we have replaced the second derivative of U by the symbol k. Equation (2.2) is
Hooke’s law for a spring, where k is the ‘force constant’ for the spring; the same term
is used for k in spectroscopy and molecular mechanics. Subscripts have been added to
emphasize that force constants and equilibrium bond lengths may vary from one pair of
atoms to another.
Indeed, one might expect that force constants and equilibrium lengths might vary substantially
even when A and B remain constant, but the bond itself is embedded in different
molecular frameworks (i.e., surroundings). However, as more and more spectroscopic data
became available in the early 20th century, particularly in the area of organic chemistry,
where hundreds or thousands of molecules having similar bonds (e.g., C–C single bonds)