8.6 Molecular systems 233stretchtorsionbendFigure 8.4. Different types of motion of atoms within a molecule.bond lengthl:
Vstretch(l)=^12 αS(l−l 0 )^2 (8.103)
wherel 0 is the equilibrium bond length.
Now consider three atoms bonded in a chain-like configuration A–B–C. This
chain is characterised by a bending orvalence angleφwhich varies around an
equilibrium valueφ 0 and the potential is described in terms of a cosine:
Vvalence(φ)=−αB[cos(φ−φ 0 )+cos(φ+φ 0 )] (8.104)where the equivalence of the anglesφ 0 and−φ 0 is taken into account. Often, a
harmonic approximation cos(φ)≈ 1 −φ^2 /2, valid for small angles, is used.
Finally there is an interaction associated with chain configurations of four atoms
A–B–C–D. The plane through the first three atoms A, B, C does not coincide in
general with that through B, C and D. Thetorsioninteraction is similar to the bend
interaction, but the angle (calleddihedralangle), denoted byθ, is now that between
these two planes:
Vtorsion(θ)=−αT[cos(θ−θ 0 )+cos(θ+θ 0 )]. (8.105)This interaction is also often replaced by its harmonic approximation. Other inter-
actions and more complicated forms of these potentials can be used – we have only
listed the simplest ones.
Characteristic vibrations associated with the different degrees of freedom dis-
tinguished here can be derived from the harmonic interactions – the frequencies