London dispersion force

Interaction energy of anargondimer.The long-range section is due to London dispersion forces.

London dispersion forces(LDF,also known asdispersion forces,London forces,instantaneous dipole–induced dipole forces, fluctuating induced dipole bondsor loosely asvan der Waals forces) are a type ofintermolecular forceacting betweenatomsandmoleculesthat are normally electrically symmetric; that is, the electrons are symmetrically distributed with respect to the nucleus.They are part of thevan der Waals forces.The LDF is named after the German physicistFritz London.They are the weakest intermolecular force.

Introduction

The electron distribution around an atom or molecule undergoes fluctuations in time. These fluctuations create instantaneous electric fields which are felt by other nearby atoms and molecules, which in turn adjust the spatial distribution of their own electrons. The net effect is that the fluctuations in electron positions in one atom induce a corresponding redistribution of electrons in other atoms, such that the electron motions become correlated. While the detailed theory requires a quantum-mechanical explanation(seequantum mechanical theory of dispersion forces),the effect is frequently described as the formation ofinstantaneous dipolesthat (when separated byvacuum) attract each other. The magnitude of the London dispersion force is frequently described in terms of a single parameter called theHamaker constant,typically symbolized$A$.For atoms that are located closer together than thewavelength of light,the interaction is essentially instantaneous and is described in terms of a "non-retarded" Hamaker constant. For entities that are farther apart, the finite time required for the fluctuation at one atom to be felt at a second atom ( "retardation" ) requires use of a "retarded" Hamaker constant.

While the London dispersion force between individual atoms and molecules is quite weak and decreases quickly with separation$R$like${\frac {1}{R^{6}}}$,in condensed matter (liquids and solids), the effect is cumulative over the volume of materials,or within and between organic molecules, such that London dispersion forces can be quite strong in bulk solid and liquids and decay much more slowly with distance. For example, the total force per unit area between two bulk solids decreases by${\frac {1}{R^{3}}}$where$R$is the separation between them. The effects of London dispersion forces are most obvious in systems that are very non-polar (e.g., that lackionic bonds), such ashydrocarbonsand highly symmetric molecules likebromine(Br_2,a liquid at room temperature) oriodine(I_2,a solid at room temperature). In hydrocarbons andwaxes,the dispersion forces are sufficient to causecondensationfrom the gas phase into the liquid or solid phase. Sublimation heats of e.g. hydrocarbon crystals reflect the dispersion interaction. Liquification of oxygen and nitrogen gases into liquid phases is also dominated by attractive London dispersion forces.

When atoms/molecules are separated by a third medium (rather than vacuum), the situation becomes more complex. Inaqueous solutions,the effects of dispersion forces between atoms or molecules are frequently less pronounced due to competition with polarizablesolventmolecules. That is, the instantaneous fluctuations in one atom or molecule are felt both by the solvent (water) and by other molecules.

Larger and heavier atoms and molecules exhibit stronger dispersion forces than smaller and lighter ones.This is due to the increasedpolarizabilityof molecules with larger, more dispersedelectron clouds.The polarizability is a measure of how easily electrons can be redistributed; a large polarizability implies that the electrons are more easily redistributed. This trend is exemplified by thehalogens(from smallest to largest: F₂,Cl₂,Br₂,I₂). The same increase of dispersive attraction occurs within and between organic molecules in the order RF, RCl, RBr, RI (from smallest to largest) or with other more polarizableheteroatoms.Fluorineandchlorinearegasesat room temperature, bromine is a liquid, and iodine is a solid. The London forces are thought to arise from the motion of electrons.

Quantum mechanical theory

The first explanation of the attraction between noble gas atoms was given byFritz Londonin 1930.He used a quantum-mechanical theory based onsecond-order perturbation theory.The perturbation is because of theCoulomb interactionbetween the electrons and nuclei of the two moieties (atoms or molecules). 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 themonomers.Thus, no intermolecular antisymmetrization of the electronic states is included, and thePauli exclusion principleis only partially satisfied.

London wrote aTaylor seriesexpansion of the perturbation in${\frac {1}{R}}$,where$R$is the distance between thenuclear centers of massof the moieties.

This expansion is known as themultipole expansionbecause 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 an expression describing the interaction between instantaneous multipoles (see the qualitative description above). Additionally, an approximation, named afterAlbrecht Unsöld,must be introduced in order to obtain a description of London dispersion in terms ofpolarizability volumes,$\alpha '$,andionization energies,$I$,(ancient term:ionization potentials).

In this manner, the following approximation is obtained for the dispersion interaction$E_{AB}^{\rm {disp}}$between two atoms$A$and$B$.Here$\alpha '_{A}$and$\alpha '_{B}$are the polarizability volumes of the respective atoms. The quantities$I_{A}$and$I_{B}$are the first ionization energies of the atoms, and$R$is the intermolecular distance.

$E_{AB}^{\rm {disp}}\approx -{3 \over 2}{I_{A}I_{B} \over I_{A}+I_{B}}{\alpha '_{A}\alpha '_{B} \over {R^{6}}}$

Note that this final London equation does not contain instantaneous dipoles (seemolecular dipoles). The "explanation" of the dispersion force as the interaction between two such dipoles was invented after London arrived at the proper quantum mechanical theory. The authoritative workcontains a criticism of the instantaneous dipole modeland a modern and thorough exposition of the theory of intermolecular forces.

The London theory has much similarity to the quantum mechanical theory oflight 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 the case of the London dispersion.

Relative magnitude

Dispersion forces are usually dominant over the three van der Waals forces (orientation, induction, dispersion) between atoms and molecules, with the exception of molecules that are small and highly polar, such as water. The following contribution of the dispersion to the total intermolecular interaction energy has been given:

Contribution of the dispersion to the total intermolecular interaction energy

Molecule pair	% of the total energy of interaction
Ne-Ne	100
CH4-CH4	100
HCl-HCl	86
HBr-HBr	96
HI-HI	99
CH3Cl-CH3Cl	68
NH3-NH3	57
H2O-H2O	24
HCl-HI	96
H2O-CH4	87

See also

• Dispersion (chemistry)
• van der Waals force
• van der Waals molecule
• Non-covalent interactions