## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 3. Basic Energetics: Intermolecular Interactions

### 3.4. Dispersion Interaction

Although the motion of electrons in a molecule is rather fast, at any moment the charge distribution of a molecule will have a (nonpermanent) dipole moment. The field associated with this dipole moment induces a (also nonpermanent) dipole moment in a neighboring molecule. Both induced dipoles interact, and this leads to the so-called (*London*) *dispersion interaction*, which is relatively weak but omnipresent and (in vacuum) always attractive. The exact derivation is outside the scope of these notes. However, a relatively simple model, proposed by London [19]__ ^{7)}__ and using the Drude model for atoms, yields good insight in the nature of the interaction. Moreover, it yields the correct expression apart from the exact numerical pre-factor.

In the *Drude model*, one assumes that an atom or molecule can be considered as a set of particles (electrons) with charge *e _{i}* and mass

*m*. Each of these particles is harmonically and isotropically bound to its equilibrium position. The configuration and notation as used for a single electron is sketched in

_{i}__Figure 3.4__. By considering the in-line configuration first, the dynamics of this electron are thus given by those of a harmonic oscillator for which we have

** Figure 3.4** Schematic of the momentary interaction between the electrons (o) bonded to the nucleus (•) of two Drude atoms for an in-line orientation (a) and a parallel orientation (b).

From quantum mechanics (see Section 2.2) we know that the energy expression becomes

where *n* denotes the quantum number *n* = 0, 1, 2, 3, … and *ω*_{0} the circular frequency of the oscillator. The latter is given by

where *a* is the force constant with which the electron is bound to the nucleus and *m* the electron mass. Without any interaction the total energy is given by

In the ground state, *n* = 0, the energy is

At any moment electron 1 of atom 1 exerts a force on electron 2 of atom 2 and for a distance *r*, using __Eq. (3.8)__, the potential is given by

(3.13)

so that the resulting total energy *U* is

This equation can be reduced to a sum of two independent harmonic oscillators, but with different frequencies. This is accomplished by the transformation

Hence, the solution becomes

with the new frequencies

and force constants

The new ground-state energy becomes

Expanding the square roots by the binomial theorem up to second order results in

(3.14)

Finally, we link the force constant *a* to the polarizability *α* using the force *eE* of the electric field *E* as compensated by the restoring force *az*, that is, *z* = *eE*/*a*. The induced dipole moment is given by *μ* = *ez* = *e*^{2}*E*/*a* as well as *μ* = *αE*, so that *α* = *e*^{2}/*a*. The final expression for the in-line interaction energy becomes

(3.15)

The dispersion energy is thus proportional to the polarizability *α* squared, a characteristic energy *I* ≡ *ħω*_{0} and the reciprocal sixth power of the distance *r*.

For the two parallel vibrations, perpendicular to the joining axis, of which the solutions are degenerated, a similar calculation leads to one-quarter of the above mentioned expression for each orientation, so that the total interaction becomes

(3.16)

For dissimilar molecules, a similar but more complex calculation leads to

(3.17)

This result is thus exact for the Drude model of a molecule with only one characteristic frequency, apart from the “binomial approximation”. For real molecules, the energies *I* should be chosen in accordance with the strongest absorption frequencies. In the absence of this information for *I* the ionization energies *I*_{0} are often taken. This seems to work reasonably well for He (*I*/*I*_{0} ≅ 1.20) and H_{2} (*I*/*I*_{0} ≅ 1.09). For the noble gases Ne, Ar, Kr and Xe, however, *I*/*I*_{0} should be taken about 9/4 in order to match more reliable calculations [2]. A similar factor or even higher was noted for N_{2}, Cl_{2}, and CH_{4}. In general, although estimates for *α* are fairly reliable (see __Chapter 10__), estimates for *I* are less trustworthy.

Problem 3.7

Show that *V*_{int} for the in-line configuration and the parallel orientation using the Drude model and the expressions for the dipole interactions is given by

Problem 3.8

Verify for interacting oscillators the expression for *U*_{0}′ and *W*_{in-line}.

Problem 3.9*

Verify for interacting oscillators the expression for *W*_{parallel}.

Problem 3.10*

Derive the London dispersion interaction between dissimilar molecules.