Dispersion Interaction - Basic Energetics: Intermolecular Interactions - Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

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 ei and mass mi. 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

c3-math-5013

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).

c3-fig-0004

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

c3-math-5014

where n denotes the quantum number n = 0, 1, 2, 3, … and ω0 the circular frequency of the oscillator. The latter is given by

c3-math-5015

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

c3-math-5016

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

c3-math-5017

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) c3-math-0013

so that the resulting total energy U is

c3-math-5018

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

c3-math-5019

c3-math-5020

Hence, the solution becomes

c3-math-5021

with the new frequencies

c3-math-5022

and force constants

c3-math-5023

The new ground-state energy becomes

c3-math-5024

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

(3.14) c3-math-0014

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) c3-math-0015

The dispersion energy is thus proportional to the polarizability α squared, a characteristic energy Iħω0 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) c3-math-0016

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

(3.17) c3-math-0017

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 Vint for the in-line configuration and the parallel orientation using the Drude model and the expressions for the dipole interactions is given by

c3-math-5025

Problem 3.8

Verify for interacting oscillators the expression for U0′ and Win-line.

Problem 3.9*

Verify for interacting oscillators the expression for Wparallel.

Problem 3.10*

Derive the London dispersion interaction between dissimilar molecules.