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

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_i. 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 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 ω₀ 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 = e²E/a as well as μ = αE, so that α = e²/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 ≡ ħω₀ 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 I₀ are often taken. This seems to work reasonably well for He (I/I₀ ≅ 1.20) and H₂ (I/I₀ ≅ 1.09). For the noble gases Ne, Ar, Kr and Xe, however, I/I₀ should be taken about 9/4 in order to match more reliable calculations [2]. A similar factor or even higher was noted for N₂, Cl₂, and CH₄. 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 U₀′ 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.