 1.1 General Introduction
 1.2 What are Dispersion Forces?
 1.3 When are Dispersion Forces Important?
 1.4 Historical Summary of Conceptual Developments
 1.4.1 Late 19th Century: Discovery of vdW Forces
 1.4.2 1900s: Early Microscopic Explanations of vdW Forces
 1.4.3 1920s–1930s: Quantum Mechanical Explanation of vdW Dispersion Forces
 1.4.4 1920s: Model Pair Potential Between Atoms or Molecules
 1.4.5 1930s: Early Pairwise Summation Approaches
 1.4.6 1940s: Electromagnetic Retardation and the Casimir Effect
 1.4.7 1930s–1960s: Theory and Experiments on Colloids
 1.4.8 1950s and 1960s: Macroscopic Lifshitz Theory
 1.4.9 1960s Onwards: Extending Lifshitz to Nonplanar Geometries
 1.4.10 1950s Onwards: is There Life after Lifshitz?
 1.4.11 1960s–Present: Dispersion Forces via the Physics of Many Interacting Electrons
 1.4.12 1960s Onwards: Manyelectron Quantum Perturbation Theory
 1.4.13 1960s Onwards: Electron Density Functional Theory
 1.4.14 2000s Onwards: Explicitly Nonlocal Density Functionals for Dispersion Interactions
 1.4.15 Time Dependent Density Functional Theory and Intermolecular Dispersion Energy
 1.4.16 1990s Onwards: Symmetry Adapted Perturbation Theory
 1.4.17 21st Century: Modern Approaches to Pairwise Additive vdW Analysis
 1.4.18 21st Century: Qualitative Failure of Pairwise vdW Additivity in Polarisable and Lowdimensional Systems
 1.4.19 21st Century: Random Phase Approximation and Related Correlation Calculations of Dispersion Energy
 1.4.20 2010s: “Many Body Dispersion”
 1.5 Direct Experimental Measurement of Dispersion Forces
 1.6 Indirect Measurements of vdW Binding
 1.7 Nomenclature
 References
CHAPTER 1: Introduction

Published:03 Apr 2020

Special Collection: 2020 ebook collection
London Dispersion Forces in Molecules, Solids and Nanostructures: An Introduction to Physical Models and Computational Methods, The Royal Society of Chemistry, 2020, pp. 119.
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We briefly discuss the occurrence and importance of dispersion (van der Waals) interactions. We then give a historical review of the important developments in the understanding and modelling of these interactions, from the 19th century to the present day.
1.1 General Introduction
The four basic forces in nature (strong nuclear, weak nuclear, electromagnetic and gravitational) have very different strengths. Most of the forces that we experience every day, except gravity, are ultimately electromagnetic in origin. Intrinsically, gravity is the weakest force by far: for two protons the ratio of gravitational attraction to electrostatic repulsion is of order 10^{−36} and for two electrons it is of order 10^{−43}.
All living things are made of protons, neutrons and electrons to a first approximation and so are subject to electromagnetic forces yet, despite its relative weakness, gravity is very important to us, as anyone who has fallen over can attest. Indeed it is said that an elephant experiences very serious injuries from a fall of only 0.5 m.
The reason for this apparent contradiction is that electrostatic forces can be repulsive or attractive depending on the sign of the electric charges involved, and since living things are close to being electrically neutral overall, electrostatic forces from external objects very nearly cancel out. Static gravity, on the other hand, is always attractive and so cannot cancel out. The reason gravity is important to us is the enormous amount of matter (the Earth) that lies beneath our feet, together with the noncancelling nature of the force.
Even the large size of the Earth does not always ensure dominance of the gravitational force for objects near its surface. Small creatures can fall without injury because of air friction, but have different problems including becoming trapped in any body of water by surface tension (an ultimately electromagnetic force).
At the nanoscale, it turns out that another aspect of electromagnetism, namely the dispersion (van der Waals, vdW) force, becomes very important. Although the dispersion force is weak compared with covalent and ionic forces, it is primarily attractive and so, like gravity at the macroscale, it can produce strong effects where (relatively) large amounts of nanoscale matter are present.
The analogy between dispersion and gravitational forces only goes so far, however, as vdW forces can occasionally be repulsive. Furthermore, the floorboards do not shield us from our gravitational attraction to the Earth beneath our feet. A contrasting situation can occur with dispersion forces, where a conducting graphene (carbon) layer one atom thick, inserted between two other nanoobjects, has been shown^{13 } to “screen out” the vdW force between those objects. In keeping with the electromagnetic origin of vdW forces, this screening effect has been shown^{2 } to be analogous to the “Faraday cage” effect whereby electric fields are screened out by a metal cage.
1.2 What are Dispersion Forces?
Dispersion forces are weak forces that occur between any pieces of matter that contain electrons. They are usually but not always attractive. They are longranged in the sense that their decay with separation is via inverse powers of distance, rather than exponential decay as for covalent forces. The ultimate origin of these forces is electromagnetic, but their mechanism is quite subtle, as explained in the following paragraphs.
It is well known that two static electric charges attract or repel each other with a Coulomb force that decays with separation R as 1/R^{2}. For two particles that carry no net charge, but that carry a timeconstant electric dipole moment, the electromagnetic force falls off as 1/R^{4}, has a strong dependence on the angular orientation of the dipoles, and can be attractive or repulsive (see Section 2.1). The interaction between higher static electric multipoles falls off with higher powers of R^{−1}.
For a pair of spherical atoms, such as argon, that have no permanent electric moments,^{1} it is found that there is nevertheless an attractive interatom dispersion force. This clearly cannot arise from any permanent electric moments. The simplest way to understand this is to postulate that quantum mechanical zeropoint motions of the electrons give rise to shortlived dipole moments on each atom, moments that average to zero over time. It is the electromagnetic interaction between these transient dipoles that causes the dispersion interaction. This simple picture is expanded in Section 2.4.
Under some circumstances it is the thermal fluctuations of the electronic moments, rather than the zero point quantal fluctuations, that are predicted to dominate the dispersion force.
There are other ways to understand dispersion forces, for example by directly considering the zeropoint energy of the electromagnetic field in the presence of the pieces of matter concerned, or from second order quantum mechanical perturbation theory of the combined electronic motions. These viewpoints are also discussed at an intuitive level in Sections 2.6 and 2.7 (ii). The rest of the book then develops various methods in full mathematical detail, with guidance for numerical implementation.
1.3 When are Dispersion Forces Important?
Intermolecular interactions and scattering in gas phase chemistry.
Intramolecular energetics and stereochemistry of larger molecules.
Relative stability of hydrocarbon isomers.
Cohesion of raregas and molecular crystals, the latter important in pharmaceuticals.
Cohesion of layered solids such as MoS_{2}, graphite, etc.
Cohesion and selfassembly of nanostructures in general.
Stiction in nanomachines.
Liquid crystals.
Stability and tertiary structure of biopolymers such as proteins and DNA.
Mesoscale biological phenomena such as the sticking of gecko feet.
Colloid properties.
Aerosol properties, toxicity.
Catalysis.
Relative stability of proposed chemical reaction intermediates.
Wetting analysis.
Thin films: adhesion of paints and coatings.
Surface science in general.
Accurate quantification of vdW forces is required for analysis of experiments designed to seek the postulated fifth fundamental force of nature.
1.4 Historical Summary of Conceptual Developments
Good summaries of the history of intermolecular force studies have been given by Rowlinson^{4 } and Stone,^{5 } for example. Here we focus only on dispersion forces.
1.4.1 Late 19th Century: Discovery of vdW Forces
Our discussion begins in the 19th century, but the interested reader can find some material about precursor ideas in the book by Israelachvili.^{6 } By the middle of the 19th century the ideal gas law
was well established to describe the pressure p of a gas in terms of the number of moles, the absolute (Kelvin) temperature T and the universal gas constant R. This law was synthesised by Clapeyron from separate empirical laws for the dependence on volume, temperature and molar content. Eqn (1.1) was explained by Koenig and Clausius from kinetic theory. In this approach the pressure on the containing wall of a gas arises when the gas particles bounce off the wall and transfer momentum to it. If the average kinetic energy of the gas particles is assumed to be proportional to the absolute temperature T, then (eqn (1.1)) follows after averaging over the particle–wall collisions. In this theory no interaction between the particles is either assumed or allowed.
Eqn (1.1) was known to become accurate at high temperatures and low densities, but modifications were clearly required to account for observed deviations such as condensation to a liquid at low temperatures and high densities.
Johannes Diderik van der Waals studied this problem and in 1873 he proposed a modification of eqn (1.1), the van der Waals equation of state, in the form
Here b was in the first instance considered to be the volume occupied by the finitesized particles constituting the gas, so that a reduced volume V − b is available for their motion. Shortranged repulsive forces are thereby tacitly assumed to prevent penetration of one particle into the volume of another particle. The term a/V^{3} reduces the pressure, and was motivated by assuming the existence of a longranged attractive force between the gas particles, a force that reduces the pressure exerted on the walls of the container by those particles that are near the walls.
Eqn (1.2) proved successful in describing gases over a wide range of conditions, even giving qualitatively reasonable results near the critical point. Even quantitative agreement was possible in many regimes if the parameters a and b were allowed to depend on volume and temperature. van der Waals received the physics Nobel prize in 1910 for his work on nonideal gases.
It is interesting to recall that, even in the late 19th century when eqn (1.2) was proposed, many scientists did not accept the reality of the particles (atoms and molecules) that are needed to explain the kinetic pressure of dilute gases from a microscopic point of view. While the reduction of pressure on the walls via intragas attraction could perhaps be obtained in a continuum model without postulating discrete particles in the gas, van der Waals was a staunch supporter of the existence of atoms and molecules (see for example his 1910 Nobel prize address^{7 }). Because of this, his notion of attractive forces within a gas had implications far beyond the macroscopic properties of gases. It implied that all species (atoms or molecules) experience forces attracting them to each other. This attraction does not depend on the species having permanent charges, or permanent dipole or multipole electric moments. This is evidenced by the success of the van der Waals equation of state for (e.g.) the noble gases He, Ar, Kr etc., whose atoms have no such moments.
From a modern point of view, the numbers obtained for the attraction parameter a in eqn (1.2) show that the attractive part of van der Waals' universal force is much weaker than covalent and ionic binding forces, but much stronger than gravity at the relevant length scales. For example, the attractive force between two N_{2} molecules is much weaker than the forces required to break the N– –N bond within a single molecule.
1.4.2 1900s: Early Microscopic Explanations of vdW Forces
Before the advent of quantum mechanics there were several early attempts to understand a longranged attraction between atoms or molecules. These generally assumed that molecules had permanent electric moments, which would lead to an orientationdependent mutual energy, falling off as R^{−3} in the lowestorder (dipolar) case: see Section 2.1. After averaging over random orientations (e.g. due to thermal effects) the dominant residual dipolar energy between two permanent dipoles falls off as R^{−6}. Theories of this type are indeed valid in the appropriate regime for thermally tumbling molecules having permanent dipoles, and this is sometimes known as the Keesom interaction.^{8 }
At around the same time Debye^{9 } proposed that a molecule with a permanent electric dipole could induce a dipole on another molecule that lacked a permanent dipole. The resulting induction energy falls off as R^{−6}, as discussed in Section 2.2.
1.4.3 1920s–1930s: Quantum Mechanical Explanation of vdW Dispersion Forces
It was not until quantum mechanics was established in the 1920s that a successful fully microscopic explanation was given for the attractive vdW force between atoms or molecules in their groundstate without the presence of permanent dipoles or multipoles. Using quantum perturbation theory to second order in the intermolecular Coulomb interaction, Wang,^{10 } followed by London,^{11,12 } obtained an attractive interaction energy falling off as R^{−6} when R→∞, where R is the separation between two atoms in their groundstate. This is the basic “dispersion” or “van der Waals” interaction, although terminology varies widely, as discussed in Section 1.7. See also Section 2.4.
1.4.4 1920s: Model Pair Potential Between Atoms or Molecules
In 1924 LennardJones^{13 } proposed a simple pair potential (“LJ”) for the vdW interaction between electrically neutral molecules. While this potential predated the quantal derivation of a R^{−6} law, it was perhaps stimulated by the R^{−6} laws derived by Keesom and Debye as described in the previous section. The LJ potential contained an attractive R^{−6} term plus a repulsive R^{−12} term intended to keep molecules from intimate contact, thereby accounting for the volume exclusion phenomenon proposed for gas molecules by van der Waals. The R^{−6} component is justified in detail by the work of London et al. The R^{−12} component has no basis in fundamental theory, however, simply serving to create a fairly hard wall near to contact in a computationally convenient way.
1.4.5 1930s: Early Pairwise Summation Approaches
Once the attraction between different internal parts of matter is accepted as a general phenomenon extrapolated from the vdW imperfect gas analysis, it becomes possible to also predict an overall attraction between two separate extended bodies. The simplest versions are based on an assumed pairwise additivity of the van der Waals atom–atom (or molecule–molecule) interactions. This approach has been carried out in various guises since the work of van der Waals and continues to be useful today under the right conditions. Hamaker^{14 } used this approach to obtain the attractive interaction between two spheres of matter. An early use for this pairwise additive approach was given by Kallmann and Willstaetter,^{15 } who proposed its application to colloidally suspended particles.
1.4.6 1940s: Electromagnetic Retardation and the Casimir Effect
In 1948 Casimir^{16 } predicted an attractive force between noncontacting ideal metal plates, arising from the zero point energy of the normal modes of the electromagnetic field in this geometry. The same approach can also yield repulsions in other geometries (e.g. a hollow metal sphere tends to expand under these Casimir forces^{17 }). It is perhaps not immediately obvious, from the above, that this attractive Casimir force is related to the attractive van der Waals/London force. It is indeed related however, and in 1948 Casimir and Polder^{18 } used quantum electrodynamics to show that the interaction energy between molecules does indeed fall off as R^{−6} at medium separations as predicted by Wang, London and collaborators, but decays as R^{−7} for larger separations, The existence of these two regimes follows from the retardation of the electromagnetic interaction, as discussed in Section 2.10.
1.4.7 1930s–1960s: Theory and Experiments on Colloids
Fine particles of an insoluble substance (much bigger than a molecule, however) can remain suspended in a fluid because of their thermal kinetic energy among other reasons. Such a suspension is also known as a colloidal suspension or colloid. Nevertheless under some circumstances the particles are observed to clump together, a phenomenon known as flocculation. This implies that some attractive force is involved, and gravitation turned out to be far too weak to explain the observed phenomenon. It was found that the attractive van der Waals force fulfils this role, and this was one of the early successes of dispersion force theory. A complete description of colloids is quite complicated, however, and also involves repulsive forces between the particles, forces that arise from an electrical double layer of charge in the fluid just outside each suspended particle, as well as entropic effects related to the motion and orientation of the fluid molecules. In the 1930s Kallman and Willstaetter^{15 } proposed that the attractive force between a pair of colloidal particles could be obtained by summing the dispersion forces between all pairs of atoms, one atom of each pair being located inside one of the particles. There were a number of further analyses of colloidal properties in the 1930s and 1940s from Hamaker, Derjaguin, Verwey and Overbeek, and others. Verwey and Overbeek^{19 } studied the interaction between colloidal particles, and proposed that there was a repulsive interaction due to electric bilayers in the fluid surrounding each particle. They also noted the need for an attractive dispersion force but considered that the (then very new) retarded R^{−7} potential due to Casimir and Polder^{18 } would be needed for typical distances between colloidal particles This is due to the finite propagation speed of the electromagnetic interaction between two bodies, and not essentially due to the presence of the liquid. Liquids can however influence the effective speed of light and hence modify the retardation effect.
Partial quantitative experimental confirmation of this body of theory was given by Schenkel and Kitchener.^{20 }
1.4.8 1950s and 1960s: Macroscopic Lifshitz Theory
In the 1950s and 60s the theory of Lifshitz^{21 } and Dzyaloshinskii et al.^{22 } obtained a (usually attractive) interaction between wellseparated macroscopic objects, using a method that went beyond perturbation theory. This approach allowed for electromagnetic retardation and went a long way to unifying the works of London and Casimir described above. While the specific calculation was limited to parallel thick slabs of matter, the method points the way to more general cases. A simplified discussion of the Lifshitz approach is given in Chapter 3.
1.4.9 1960s Onwards: Extending Lifshitz to Nonplanar Geometries
Back in the 1930s Derjaguin^{23 } had proposed what is now known as the proximity force approximation, PFA, which allowed the energy of interaction between two gently curved surfaces to be approximated using a knowledge of the mutual energy E(d) of two parallel planar surfaces of the same materials at separation d. This work was not restricted to dispersion forces, but it was natural to use it with the Lifshitz approach, which provides the input quantity E(d). This has proved to be extremely fruitful,^{6,24,25 } but it is limited to gently curved cases where the radius of curvature R greatly exceeds the minimum separation d of the interacting objects. More recently the macroscopic quantum electrodynamic approach (essentially the Lifshitz philosophy) has been used to analyse the dispersion interaction between uniform spheroidal or cylindrical objects that can be highly curved.^{26,27 } This was then used to examine the validity of the PFA for more strongly curved systems. Even more recently a systematic expansion has been given for the next correction beyond the PFA.^{28 }
1.4.10 1950s Onwards: is There Life after Lifshitz?
It has sometimes been considered that the major theoretical developments in dispersionforce theory ended with the Lifshitz theory and its extension to nonplanar geometries by Deryaguin, Landau, etc., although of course many details and applications continued to be developed, and direct experimental verification has had ongoing progress. This viewpoint neglects the contributions from developments in manyelectron quantum mechanics, which have made possible the quantitative prediction of dispersion forces in the regime of intimate contact between interacting objects, where chemical binding forces may coexist with dispersion forces. The macroscopic Lifshitz theory cannot make reliable predictions here. This closespaced regime is the subject of much of the present book, and we now continue our historical sketch with a timeline of these developments.
1.4.11 1960s–Present: Dispersion Forces via the Physics of Many Interacting Electrons
Starting from the mid20th century, the study of interacting quantal manyelectron systems experienced a period of enormous expansion, from the viewpoints of both theoretical chemistry and condensed matter physics. Two branches of these developments, manyelectron quantum theory and density functional theory, along with their efficient numerical implementation, culminated in 1998 with the award of the chemistry Nobel prize to John Pople and Walter Kohn. The more sophisticated variants of these computational approaches include the correlations between the motions of electrons at different spatial locations, which in one viewpoint are the cause of the dispersion interaction (see Section 2.7(iii)). These methods will be explained in Chapters 4 and 5 and the Appendices, with emphasis on their application to dispersion forces.
1.4.12 1960s Onwards: Manyelectron Quantum Perturbation Theory
Pople's Nobel prize contributions were in the development, and practical implementation on computers, of various quantum chemical theories based on approximations to the manyelectron wavefunction. He started from the Hartree Fock approach, which cannot describe dispersion interactions. However, he continued with perturbation theory (e.g. the Moeller–Plesset version and resummed perturbation theory (coupled cluster approach)). These latter methods can correctly describe all the binding forces between molecules, including the dispersion force at all separations. This includes the contact regime where dispersion forces may coexist with other types of bonding. This level of detailed understanding of the interaction near contact is out of the reach of macroscopic theories such as the Lifshitz approach. The down side is that the computations are so intensive that initially only relatively small molecules could be treated. Progress continues up to today, however, with theoretical and computational strategies that reduce this computational burden, as well as advances in computer power. See Chapters 4, 5 and 13.
1.4.13 1960s Onwards: Electron Density Functional Theory
Density functional theory (DFT, see Chapter 3) was initiated for manyelectron systems by Kohn and coworkers in the 1960s,^{29,30 } culminating in a share of the chemistry Nobel prize in 1998. DFT builds on ideas from the earlier highly approximate Thomas–Fermi–Slater approaches, but is an in principle exact way to treat interacting manyelectron systems, directly yielding the total energy and groundstate spatial electron density distribution ρ(r⃑). Thus it should be capable of describing the dispersion interaction energy between systems at any separation. In practice, approximations are required. The earliest approximation (local density approximation, LDA^{30 }) used the fictitious uniform electron gas as a starting point, and was immediately useful for the physics of inhomogeneous metals (e.g. metal surfaces) where the electron density varies relatively slowly in space. It was vastly more efficient computationally than the direct quantum chemical approaches, but was not useful for typical chemistry applications because of the highly nonuniform character of the electron density in individual atoms and molecules. From the 1990s the generalized gradient approximation (GGA, e.g. ref. 31 and 32) and metaGGA (e.g. ref. 33) classes of approximation became available, making possible the efficient prediction of many chemical processes to useful accuracy. The tail of the dispersion interaction still resists this approach, however.
1.4.14 2000s Onwards: Explicitly Nonlocal Density Functionals for Dispersion Interactions
The GGA and metaGGA approaches described in Section 1.4.13 do not correctly describe correlations between the motions of electrons in two wellseparated systems, and consequently these approaches lack the asymptotic dispersion interaction (see Section 2.7(iii)). This is the fault of the semilocal approximations made, not of the density functional approach itself. Starting from the late 1990s there were attempts^{34,35 } to obtain a fully nonlocal density functional theory that would include dispersion forces at all separations in a natural way. In 2004 Dion et al. proposed a highly sophisticated nonlocal functional, the “vdWDF”.^{36 } Its correlation energy is intended to be added to the energy from semilocal density functionals. It gives the energy as an explicit though complicated and highly nonlocal functional of the groundstate electron density ρ, taken at two points r⃑ and r⃑′ simultaneously. Despite its complex origins the vdWDF is numerically quite efficient. This functional, when combined with a particular metaGGA^{33 } as a starting point, should be regarded as part of the current state of the art for predicting dispersion forces for systems near their equilibrium separation. It can, however, fail for asymptotic forces under some circumstances. This approach is described in detail in Section 7.12.
1.4.15 Time Dependent Density Functional Theory and Intermolecular Dispersion Energy
The density functional theory described so far is used for groundstate calculations with or without static external fields. Timedependent density functional theory (TDDFT) was formally initiated by Runge and Gross^{37 } although simple versions had been informally introduced earlier.^{38 } TDDFT provides a computationally efficient way to explore the response of manyelectron systems to timevarying electric fields (dynamic response). Since dispersion forces can be related to dynamic response properties (see Chapter 5) this has provided a practical means of predicting dispersion forces between quite large molecules, including those of biological significance.^{39 } Early uses of TDDFT for vdW forces was primarily via the peturbative Generalised Casimir Polder formula (see Chapter 5). Another use of TDDF is in nonperturbative energy calculations using the adiabaticconnectionfluctuation dissipation (ACFD) approach, discussed in detail in Chapter 6.
1.4.16 1990s Onwards: Symmetry Adapted Perturbation Theory
One of the drawbacks of the perturbative methods described in Section 1.4.12 was that one had to treat all the interacting chemical species simultaneously as a giant supermolecule, requiring large computational resources. In the 1990s, Jeziorski and coworkers developed symmetry adapted perturbation theory (SAPT).^{40 } This is a more sophisticated intermolecular perturbation theory, beyond the London type of analysis, and it allows for some degree of overlap between the electronic clouds of the interacting species via a treatment of the exchange antisymmetry of the manyelectron wavefunction. It has the major advantage that, unlike the Moeller–Plesset and coupledcluster perturbation schemes mentioned in Section 1.4.12, it does not treat the whole of the electron–electron Coulomb interaction as a perturbation, only the part of it that couples the interacting molecules. This allows one to use prior knowledge of the properties of each molecule in isolation, including those properties depending on the electron–electron interaction. This approach is one of the important current computational tools for dispersion forces and is described in detail in Chapter 5.
1.4.17 21st Century: Modern Approaches to Pairwise Additive vdW Analysis
While the physics and mathematics community had been using pairwise additive descriptions of vdW forces for decades (e.g. for graphite^{41 } and carbon nanotubes^{42 }), it was not until the 21st century that theoretical chemists began to take the pairwise approach seriously for vdW energetics within and between molecules. Indeed the inner parts of the early empirical pairwise atom–atom potentials, such as the R^{−12} term in the LJ potential, were completely unrealistic for chemical binding situations. Wu and Yang^{43 } and Elstner et al.^{44 } were among the first to start from a chemically realistic density functional description able to deal with chemical bonding. They then added a compatible correction for dispersion forces via a sum of contributions from atom pairs. This was taken up by many workers and has now reached quite a sophisticated level^{45,46 } with many successes.
1.4.18 21st Century: Qualitative Failure of Pairwise vdW Additivity in Polarisable and Lowdimensional Systems
Since at least the 1970s examples have been discovered where the pairwise summation approach (see Section 1.4.17) is not just quantitatively inaccurate for dispersion forces, but qualitatively wrong.^{4752 } One of the outcomes of these investigations was that the pairwise additive approach can predict the wrong power law for the asymptotic decay of the dispersion interaction with separation. Dobson et al.^{49 } showed that this phenomenon is strongest in lowdimensional systems with a small or zero electronic energy gap. Further insight was given in ref. 53 where three types of nonpairwise additivity were noted. Other incorrect predictions of pairwise theory relate to the dependence of the vdW interaction on the number of atoms in large polarisable systems.^{54,55 }
1.4.19 21st Century: Random Phase Approximation and Related Correlation Calculations of Dispersion Energy
In one interpretation, dispersion energies arise from correlations between the motions of electrons. The random phase approximation (RPA) was introduced in the 1960s^{56 } as an approximation to the correlation energy of a fictitious metallic system, the homogeneous electron gas. Subsequent developments^{57,58 } showed how to apply this method to general inhomogeneous electronic systems. Its use for dispersion energy problems in general was advocated in the intervening years,^{59 } but it is only quite recently that computational developments have allowed this method to be applied in practice to predict correlation energies of real molecules, nanostructures and solids.^{6064 } This method, when applied starting from GGA groundstate calculations, includes all the physical nonadditive processes (types A, B, C) identified in ref. 53 (see Section 1.4.18 and also Sections 8.7 and 8.8). The RPA also has the major advantage that it is seamless – it gives results similar to the Lifshitz theory for large separations, but unlike that theory it remains reliable right down to intimate contact where atomic detail is important and chemical as well as dispersion forces may occur. The RPA and its extensions are part of the current state of the art for dispersion interactions. The RPA also has some deficiencies discussed in Chapter 6, but the evidence is that these do not invalidate the method for applications where energy differences between similar electronic states are concerned. This covers, in particular, the energetics of layered compounds.
1.4.20 2010s: “Many Body Dispersion”
The pairwise approach in its modern forms, as described above in Section 1.4.17, is often very successful. Clearly, however, it needs revision in order to deal with the separation dependence of asymptotic vdW forces in lowdimensional systems, and the dependence of the number of atoms in large highly polarisable systems, as outlined in Section 1.4.18. In order to deal with manyatom vdW interactions, Tkatchenko et al.^{65 } introduced a harmonicoscillator model of each atom, and considered the collective oscillations of these model systems coupled by their mutual Coulomb interaction. This “many body dispersion” (MBD) approach was applied to the longranged part of the Coulomb interaction that gives rise to nonadditive vdW interactions. The resulting formalism is available in standard software packages and is part of the current state of the art for vdW interactions of large molecular and nanoscale systems. It correctly describes two out of the three nonadditive processes identified in ref. 53. In practice it gives a good account of near–contact interactions in many systems. The situation is less clear away from the equilibrium geometry, and MBD does not correctly treat, for example, the asymptotic vdW interactions in lowdimensional metallic systems.
1.5 Direct Experimental Measurement of Dispersion Forces
Attempts to measure the dispersion force directly have quite a long history. Here we will only touch on a few examples.
A problem with direct confirmation of the Lifshitz predictions for the force between thick parallel plates is that it is hard to ensure complete flatness and parallel orientation, and to know precisely their separation when it is of the order of microns or less. There are also competing forces from stray electric charges, especially for insulating samples. These difficulties were largely surmounted in the late 1990s following the use of metallised nanospheres near to a flat substrate. The issue of parallelism does not arise with a sphere, and sophisticated means were used to account for stray electrostatic forces.
In 1997 Lamoreaux made a torsion pendulum using a metalcoated spherical lens and a metalcoated quartz plate. This was used to deduce the dispersion force between them. Agreement with Lifshitz/Casimir/Deryaguin theory was achieved within about 5%. In 1998 Mohideen and Roy^{66,67 } used atomic force microscopy in direct measurements of the force between an aluminised polystyrene sphere and a sapphire plate. The results were compared with dispersion force theory including surface roughness and finite temperature. Good agreement was obtained.
Atomic force microscopy (AFM) also permits direct measurement of forces of molecularsized objects. For example, Wagner et al.^{68 } used an AFM to pull chain molecules off a surface, measuring the changes in force as each monomer detaches. By looking at the results for different chain lengths, they obtained evidence that the dispersion force is nonadditive (see also Chapter 11).
Capasso et al.^{69 } gave experimental confirmation of the theoretical prediction, going back to the work of Lifshitz et al., that dispersion forces can be repulsive when media with three different dielectric functions are involved.
There has been interest in theoretical predictions of thermal effects on dispersion force. Sushkov et al.^{70 } observed the thermal component of the dispersion force in their experiment. No such component was found by CastilloGarza^{71 } but their experiment was able to confirm quantitatively the predictions of theory at liquid nitrogen temperature.
Anisotropic media can experience a dispersiondriven mutual torque. Somers et al. confirmed this experimentally^{72 }
1.6 Indirect Measurements of vdW Binding
In 1975 deflection experiments^{73 } were performed on beams of heavy alkali metal atoms passing near a gold surface. For this geometry theory gives an atom–surface vdW interaction energy of form −C_{3}R^{−3} where R is the atom–surface distance (see Section 12.1). The experiment confirmed the R^{−3} dependence but the experimental coefficient disagreed with the theory employed by 60% or more.
The contact binding energy of molecules to a surface can be measured indirectly via the kinetics of thermal desorption. In 2004 Zacharia et al.^{74 } measured desorption kinetics of polycyclic aromatic hydrocarbons (PAHs) from a graphite surface. Because PAHs are essentially small flakes of graphene terminated by hydrogens, these authors extrapolated to the limit of largesized coronenes and used this to estimate the layer binding energy of graphite, which is substantially determined by dispersion forces. They obtained 52 mH per carbon, in good agreement with predictions of highlevel microscopic theory.^{63,75 } Later work by Thrower et al.^{76 } obtained a somewhat larger value 65 meV per atom, which they supported with lowerlevel B99vdW theory.
1.7 Nomenclature
The nomenclature arising from the discoveries outlined above can be confusing. The terms “van der Waals force”, “dispersion force”, “London dispersion force”, “Casimir force” and “Casimir effect” are all in common use.
“Dispersion forces”^{24,77,78 } are generally understood to be that part of the noncovalent interaction between pieces of matter that cannot be attributed to any permanent electric charges (monopoles or multipoles). In the chemistry community, the whole of this nonionic, nonchemically bonded interaction, at all separations including the repulsive regime near contact, is often termed the “van der Waals” (vdW) interaction. In the physics community, however, this term is usually reserved for the more distant parts of the force versus separation curve, outside the repulsive contact region. A useful categorisation of the many components of the total force between molecules is given in ref. 40 from a perturbation theory standpoint. The noncontact parts of the vdW interaction are also frequently termed “London dispersion forces” after the work of London^{11,12 } who, soon after Wang,^{10 } obtained an interaction of form −C_{6}R^{−6} via intersystem perturbation theory. The term “Casimir force” is most often used to describe the dispersion forces between macroscopic objects at larger separations, where the retardation of the electromagnetic interaction between the species is important. The term “Casimir effect” is also used in a more general way. It refers to the geometry dependence of the quantal zeropoint and/or thermal energy of wave motions in any medium. In the case of the Casimir/dispersion force, these waves are electromagnetic waves and/or electronic plasma oscillations.
The moments used here are reduced moments: e.g. the quadrupole moment