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Figure 2.9 Rotational profile corresponding to eq. 2.10 with V1 = 0.5, V2 = -0.2, V3 = 0.5

encountered rotational profiles can be obtained by combining the three Vi parameters. Figure 2.9 shows an example with one anti and two less stable gauche minima and with a significant cis barrier, corresponding to the combination V1 = 0.5, V2 = -0.2, V3 = 0.5 in eq. (2.10).

As mentioned in Section 2.2.3, the out-of-plane energy may also be described by an "improper" torsional angle. For the example shown in Figure 2.6, a torsional angle ABCD may be defined, even though there is no bond between C and D. The out-of-plane Eoop may then be described by an angle wABCD, for example as a harmonic function (w - W0)2 or eq. (2.10) with a large V2 constant. Note that the definition of such improper torsional angles is not unique, the angle wABDC (for example) is equally good. In practice there is little difference between describing Eoop as in eq. (2.8) or as an improper torsional angle.

For each combination of four atom types, A, B, C and D, there are generally three torsional parameters to be determined, V1ABCD, V2ABCD and V3ABCD.

2.2.5 The van der Waals energy

Evdw is the van der Waals energy describing the repulsion or attraction between atoms that are not directly bonded. Together with the electrostatic term Eel (Section 2.2.6), it describes the non-bonded energy. Evdw may be interpreted as the non-polar part of the interaction not related to electrostatic energy due to (atomic) charges. This may for example be the interaction between two methane molecules, or two methyl groups at different ends of the same molecule.

Evdw is zero at large interatomic distances and becomes very repulsive for short distances. In quantum mechanical terms, the latter is due to the overlap of the electron clouds of the two atoms, as the negatively charged electrons repel each other. At intermediate distances, however, there is a slight attraction between two such electron clouds from induced dipole-dipole interactions, physically due to electron correlation (discussed in Chapter 4). Even if the molecule (or part of a molecule) has no permanent dipole moment, the motion of the electrons will create a slightly uneven distribution at a given time. This dipole moment will induce a charge polarization in the neighbour molecule (or another part of the same molecule), creating an attraction, and it can be derived theoretically that this attraction varies as the inverse sixth power of the distance between the two fragments.

The induced dipole-dipole interaction is only the leading term of such induced multipole interactions: there are also contributions from induced dipole-quadrupole, quadrupole-quadrupole, etc., interactions. These vary as R-8, R-10, etc., and the Rr6 dependence is only the asymptotic behaviour at long distances. The force associated with this potential is often referred to as a "dispersion" or "London" force.9 The van der Waals term is the only interaction between rare gas atoms (and thus the reason why say argon can become a liquid and a solid) and it is the main interaction between non-polar molecules such as alkanes.

Evdw is very positive at small distances, has a minimum that is slightly negative at a distance corresponding to the two atoms just "touching" each other, and approaches zero as the distance becomes large. A general functional form that fits these conditions is given in eq. (2.11).

C AB

It is not possible to derive theoretically the functional form of the repulsive interaction, it is only required that it goes toward zero as R goes to infinity, and it should approach zero faster than the Rr6 term, as the energy should go towards zero from below.

A popular function that obeys these general requirements is the Lennard-Jones (LJ) potential,10 where the repulsive part is given by an Rr12 dependence (C1 and C2 are suitable constants).

The Lennard-Jones potential can also be written as in eq. (2.13).

Here R0 is the minimum energy distance and e the depth of the minimum. There are no theoretical arguments for choosing the exponent in the repulsive part to be 12, this is purely a computational convenience, and there is evidence that an exponent of 9 or 10 gives better results.

The Merck Molecular Force Field (MMFF) uses a generalized Lennard-Jones potential where the exponents and two empirical constants are derived from experimental data for rare gas atoms.11 The resulting buffered 14-7 potential is shown in eq. (2.14).

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