London Dispersion Force - Quantum Mechanical Theory of Dispersion Forces

Quantum Mechanical Theory of Dispersion Forces

The first explanation of the attraction between noble gas atoms was given by Fritz London in 1930. He used a quantum mechanical theory based on second-order perturbation theory. The perturbation is the Coulomb interaction V between the electrons and nuclei of the two monomers (atoms or molecules) that constitute the dimer. The second-order perturbation expression of the interaction energy contains a sum over states. The states appearing in this sum are simple products of the stimulated electronic states of the monomers. Thus, no intermolecular antisymmetrization of the electronic states is included and the Pauli exclusion principle is only partially satisfied.

London developed the method perturbation V in a Taylor series in, where is the distance between the nuclear centers of mass of the monomers.

This Taylor expansion is known as the multipole expansion of V because the terms in this series can be regarded as energies of two interacting multipoles, one on each monomer. Substitution of the multipole-expanded form of V into the second-order energy yields an expression that resembles somewhat an expression describing the interaction between instantaneous multipoles (see the qualitative description above). Additionally an approximation, named after Albrecht Unsöld, must be introduced in order to obtain a description of London dispersion in terms of dipole polarizabilities and ionization potentials.

In this manner the following approximation is obtained for the dispersion interaction between two atoms and . Here and are the dipole polarizabilities of the respective atoms. The quantities and are the first ionization potentials of the atoms and is the intermolecular distance.

Note that this final London equation does not contain instantaneous dipoles (see molecular dipoles). The "explanation" of the dispersion force as the interaction between two such dipoles was invented after London gave the proper quantum mechanical theory. See the authoritative work for a criticism of the instantaneous dipole model and for a modern and thorough exposition of the theory of intermolecular forces.

The London theory has much similarity to the quantum mechanical theory of light dispersion, which is why London coined the phrase "dispersion effect". In physics the term "dispersion" describes the variation of a quantity with frequency which is the fluctuation of the electrons in case of the London dispersion.

For endohedral fullerene, such as He@C60, the inside atom typically locates at the center of the outer shell. In this case, the intermolecular distance, is zero if the multipole expansion centers are set at the centers of the two subsystems. The London equation would not be applicable. Another formula has been derived in literature

here is a new polarizability.