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First-Principles Models for van der Waals Interactions in Moleculesand Materials: Concepts, Theory, and ApplicationsJan Hermann,† Robert A. DiStasio Jr.,*,‡ and Alexandre Tkatchenko*,†,§

†Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany‡Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, United States§Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg

ABSTRACT: Noncovalent van der Waals (vdW) or dispersion forces are ubiquitous innature and influence the structure, stability, dynamics, and function of molecules andmaterials throughout chemistry, biology, physics, and materials science. These forces arequantum mechanical in origin and arise from electrostatic interactions betweenfluctuations in the electronic charge density. Here, we explore the conceptual andmathematical ingredients required for an exact treatment of vdW interactions, andpresent a systematic and unified framework for classifying the current first-principlesvdW methods based on the adiabatic-connection fluctuation−dissipation (ACFD)theorem (namely the Rutgers−Chalmers vdW-DF, Vydrov−Van Voorhis (VV),exchange-hole dipole moment (XDM), Tkatchenko−Scheffler (TS), many-bodydispersion (MBD), and random-phase approximation (RPA) approaches). Particularattention is paid to the intriguing nature of many-body vdW interactions, whosefundamental relevance has recently been highlighted in several landmark experiments.The performance of these models in predicting binding energetics as well as structural,electronic, and thermodynamic properties is connected with the theoretical concepts and provides a numerical summary of thestate-of-the-art in the field. We conclude with a roadmap of the conceptual, methodological, practical, and numerical challengesthat remain in obtaining a universally applicable and truly predictive vdW method for realistic molecular systems and materials.

CONTENTS

1. Introduction 47152. Conceptual understanding of vdW interactions in

molecules and materials 47173. Experimental evidence for the nonlocal and

nonadditive nature of van der Waals dispersioninteractions 4722

4. Theoretical framework for van der Waals models 47224.1. From point polarizability to nonlocal polar-

izability and back 47234.2. Exact correlation energy from the nonlocal

dynamic polarizability 47244.3. Local effective polarizability from the elec-

tron density 47284.3.1. Anisotropy of αeff(r) 47284.3.2. Harmonic oscillator model of the

dynamic response 47284.4. Classification of approximate van der Waals

methods 47294.4.1. Coarse-graining the ACFD formula and

Dyson equation 47294.4.2. Truncating the many-body expansion of

the long-range correlation energy 47304.5. Methods based on the random-phase

approximation 47314.5.1. Range-separated random-phase ap-

proximation 4731

4.5.2. Approximate exchange-correlation ker-nels 4731

4.5.3. Renormalized perturbation theory tosecond order 4732

4.6. Nonlocal density functionals 47324.6.1. A simple nonlocal functional 47334.6.2. vdW-DF functional 47334.6.3. Variations of vdW-DF 4734

4.7. Fragment-based pairwise methods 47344.7.1. Fragment-based pairwise models from

the ACFD formula 47354.7.2. Models for C6 dispersion coefficients

and other vdW parameters 47354.8. Many-body dispersion framework 4738

4.8.1. Many-body dispersion framework fromthe ACFD formula 4738

4.8.2. Range-separated many-body dispersionmethod 4739

4.8.3. General remarks about MBD and n-body expansions 4739

5. Models for vdW interactions: Applications 47405.1. Benchmark databases 47405.2. Structural and electronic properties 47445.3. Molecular dynamics 4745

Received: July 11, 2016Published: March 8, 2017

Review

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© 2017 American Chemical Society 4714 DOI: 10.1021/acs.chemrev.6b00446Chem. Rev. 2017, 117, 4714−4758

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6. Challenges and outlook 47477. Concluding remarks 4749Associated Content 4749

Special Issue Paper 4749Author Information 4749

Corresponding Authors 4749ORCID 4750Notes 4750Biographies 4750

Acknowledgments 4750References 4750

1. INTRODUCTION

The ongoing challenges associated with modeling increasinglylarge and more complex molecular and condensed-phasematerials have led to the universal realization that van derWaals (vdW) dispersion interactions,1−4 commonly consideredto be only a small contribution to the total energy, are in fact acrucial ingredient of both a qualitatively correct andquantitatively accurate description of the binding propertiesin a majority of molecular systems5−7 and materials.8−10 In fact,it has become increasingly more apparent that the relevance ofvdW interactions extends well beyond binding and cohesiveenergies, encompassing the structural,7,9 mechanical,11,12

spectroscopic,13 kinetic,14 and even electronic15,16 signaturesof condensed matter. With an influence that ranges from drugbinding in proteins and double-helix stability in DNA17 to pedaladhesion in geckos18,19 and cohesion in asteroids,20,21 thesenonbonded forces are quantum mechanical in origin and arisefrom the electrostatic interactions between the constantlyfluctuating electron clouds which constitute molecules andmaterials.22 As such, vdW dispersion interactions are stronglynonlocal, extending to distances exceeding 10 nm in nanoscalematerials,22−25 or even further in mesoscopic materials.26−28

With this context, the importance of understanding andmodeling vdW interactions in realistic systems can hardly beoveremphasized. However, our ability to accurately model theseinteractions from first principles is severely impeded by theprohibitively high computational cost of the high-level wave-function methods of quantum chemistry, as well as the lack ofefficient yet accurate approximations to the many-body electroncorrelation problem for large systems.29 In light of this fact, thesophistication of approximate models for describing vdWinteractions has been steadily increasing, especially within thecontext of electronic structure methods based on density-functional theory (DFT). The majority of these commonlyused vdW-inclusive methods finds a theoretical basis in thequantum-mechanical treatment of two fragments A and B (e.g.,atoms, localized orbitals, or infinitesimal regions of the electrondensity) separated by a distance RAB that places them outside ofdensity or orbital overlap, for which second-order perturbationtheory predicts the vdW interaction energy to be proportionalto RAB

−6 (to leading order). Only recently have more advancedtheoretical models been developed which go beyond thispairwise-additive approximation, resulting in quantitativeimprovements in the predictive capabilities of vdW-inclusiveDFT calculations and novel conceptual insights into the natureof many-body vdW interactions. At this point, the existence ofadvanced theoretical models for describing vdW interactionscoupled with the rapid increase in available computationalresources has allowed us to achieve the accuracy needed formeaningful prediction of binding energetics between small

organic molecules,30 and vdW-inclusive methods are now evenbeing extended toward quantitative predictions of the proper-ties and functions of technologically and biologically relevantmaterials.9,10

Meanwhile, the steep rise in the number of applications ofvdW-inclusive methods to larger and more complex systemshas led to a more comprehensive understanding of theirinherent strengths and weaknesses, as well as a roughdelineation of their respective ranges of applicability. Despitesuch substantial progress in the field of modeling vdWinteractions, many questions still remain unanswered andsignificant further development is required before a universallyapplicable approach emerges. For instance, the aforementionedpairwise-additive interatomic vdW methods are frequently usedto describe adsorption of organic molecules on inorganicsurfaces.31−34 Despite the fact that such an approach effectivelyignores the relatively strong electrodynamic response screeningpresent in bulk materials, the results often have a reasonablelevel of accuracy. Another example comes in the form of use ofthe nonlocal vdW-DF family of functionals35−38 to describebinding between molecules, which often leads to an acceptablelevel of accuracy even though one would not expect theemployed homogeneous dielectric approximation to work wellfor molecules. These examples highlight the need for a deeperunderstanding of the physical reasons why these differentapproaches often yield relatively good results outside theirexpected domains of applicability, as this is a determining factorin the development of more robust approximations.In general, vdW interactions can be remarkably sensitive to

the geometry and electronic structure of a given system.Therefore, these interactions not only affect the structure anddynamics of the system, but can also have a visible effect on theelectronic charge distribution and other properties derived fromthe charge density (e.g., multipole moments, work functions,polarizabilities, electron affinities, etc.),15 in accordance with thetheoretical insight into vdW interactions advocated byFeynman.39 This observation highlights the need for first-principles-based methodsvdW methods that explicitlydepend on the charge density or orbitalsin which the vdWinteractions are treated in a fully self-consistent manner; assuch, these methods are the main focus of this work.In this review, we will use the terms “van der Waals (vdW)

interactions” and “dispersion interactions” interchangeably.Originally, vdW interactions encompassed (especially in thechemistry literature) electrostatic (permanent moment−permanent moment), induction (permanent moment−inducedmoment), and dispersion (induced moment−induced mo-ment) interactions. In the current work, however, this trio isincreasingly being referred to as noncovalent interactions, whilevdW interactions are reserved specifically for the dispersioninteractions, thus aligning the terminology with the physicsliterature. This has perhaps been motivated by the ever-increasing overlap between chemistry and condensed-matterphysics in the field of vdW interactions. Another matter ofterminology that extends well beyond semantics concerns theuse of the phrase “many-body”; here the communitiessignificantly differ in their association of a given physical (ormathematical) entity with the label “body”. For example, a bodywould typically refer to a molecule in the field of molecularmodeling, although it could also refer to an atom, asdemonstrated by the literature on the Axilrod−Teller−Mutothree-body (three-atom) potential.40,41 In the quantumchemistry community, the coupled-cluster hierarchy is a

Chemical Reviews Review

DOI: 10.1021/acs.chemrev.6b00446Chem. Rev. 2017, 117, 4714−4758

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particular class of methods within many-body perturbationtheory, and in this case, the bodies are electrons (electronicexcitations). In this review, we will use the term many-bodywhen referring to interactions between more than two bodies(i.e., beyond pairwise), as the point of this term is to delineatethe degree of coupling in a given model. The meaning of theterm body will therefore change according to the context, andwe hope that our exposition will make it clear why such a fluidicuse of this term is indeed helpful in defining a generalconceptual and mathematical framework for describing vdWinteractions.This review concentrates on the conceptual, methodological,

mathematical, and numerical aspects of various approaches fortreating vdW interactions based on first principles. For theplethora of other aspects associated with vdW interactions, werefer the interested reader to the following set of reviews, whichis by no means an exhaustive list but includes many workswhich are complementary to that presented here.5−7,9,30,42−51

This review begins with a conceptual introduction to thetreatment of vdW interactions in molecules and materials ofincreasing complexity, showcasing the critical ingredientsnecessary for a qualitatively correct and physically sounddescription of these interactions in different classes of systems.We then proceed with a brief discussion of several recent

landmark experiments that clearly highlight the nontrivialnature of vdW interactions at the nanoscale and demonstrate

that our understanding of these quantum-mechanical phenom-ena is only beginning to emerge. This is followed by acomprehensive theoretical basis for the description of vdWinteractions that provides a systematic classification scheme and

derivation for many of the popular first-principles-basedmethods within the framework of the adiabatic-connectionfluctuation−dissipation (ACFD) theorem. A critical assessment

of the performance of these models in predicting bindingenergetics, structural and electronic properties, as well asequilibrium quantities from ab initio molecular dynamics is then

discussed, as we introduce the many challenges that arise whenthese methods are applied to increasingly realistic andexperimentally relevant systems. We conclude this review bypresenting an outlook which not only summarizes the

achievements in the field of modeling vdW interactions inthe past decade, but also highlights the many remainingchallenges that we expect will keep this exciting field quite

active for some time to come.

Figure 1. Structural features, electronic properties, and environmental effects of relevance to noncovalent vdW interactions. Each of these distinctsystem characteristics affects the underlying electronic structure and therefore the associated vdW interactions, leading to complications in theaccurate modeling of these interactions in realistic systems. A seamless and universally applicable vdW method must meet this challenge andaccurately describe these effects across the wide range of systems found in chemistry, biology, and physics.



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2. CONCEPTUAL UNDERSTANDING OF VDWINTERACTIONS IN MOLECULES AND MATERIALS

In this section, we discuss the current conceptual understandingof noncovalent vdW interactions in molecules and materials. Byconsidering a set of fundamental systems, we demonstrate howthe vdW energy depends on local (additive or weaklynonadditive) and nonlocal (nonadditive) effects that originatefrom the underlying electronic structure of the system. Ourfocus here is on identifying and analyzing the ingredients(Figure 1) that are necessary to correctly describe vdWinteractions across a wide array of systems characterized bymarkedly different structural features (e.g., size, dimensionality,geometry/topology), electronic properties (e.g., covalent/ionicbonding, hybridization, oxidation state, charge transfer), andenvironmental effects (e.g., solvation, interfaces, externalpressure).Currently available vdW-inclusive methods are able to

describe how these structural, electronic, and environmentalconsiderations affect the vdW energy only to a certain extentwhich varies from method to method. In general, thesemethods can be classified based on the approximations thatthey use for the two central quantities required for thedescription of vdW interactions, namely the nonlocal micro-scopic frequency-dependent polarizability α(r, r′, u) and theeffective interaction potential v(r, r′) between points r and r′ inspace. This classification scheme is illustrated in Figure 2 andincludes (a) the exact treatment of the electron correlation via

the adiabatic-connection fluctuation−dissipation (ACFD)theorem,52−54 (b) fragment-based (or coarse-grained) many-body vdW methods,55,56 (c) fragment-based (or coarse-grained) pairwise vdW methods,57−61 and (d) nonlocal two-point (fine-grained) density-functional approximations.35−37,62

For a quick overview, we present a condensed description ofthe most widely used vdW methods in Table 1, which includesa brief synopsis of each method as well as a critical assessmentof their performance in treating different aspects of vdWinteractions. Throughout this section we discuss the differentscenarios that arise in the modeling of the polarizability andinteraction potential in systems of increasing complexity. Whilethis section focuses on a conceptual picture of vdWinteractions, Section 4 will address these issues from amathematical point of view. We aim to dispel several mythsabout vdW interactions and note that many of these points willbe revisited and reinforced by the more formal presentation inSection 4.The exact solution of the electronic Schrodinger equation

would automatically incorporate the exact vdW contribution tothe total energy of a many-electron system, as this contributionis strictly contained in the electron correlation energy.However, the explicit solution of the Schrodinger equationfor a system containing more than a few electrons is still acomputationally prohibitive task, even when many approx-imations are used. For this reason, modeling of realisticmaterials often starts with more tractable mean-field models,

Figure 2. Theoretical approaches used by different vdW models based on first-principles. (a) Exact treatment of the correlation energy in theframework of the adiabatic-connection fluctuation−dissipation (ACFD) theorem, which uses the full nonlocal response function χλ(r,r′,u). Thisresponse function can be viewed as describing the propagation of fluctuations throughout the molecule or material. (b) Fragment-based (coarse-grained) many-body methods correlate fluctuations to infinite order by using model response functions in the ACFD formula. (c) Fragment-based(coarse-grained) two-body methods compute the pairwise contribution to the vdW energy by correlating only two dipoles (or multipoles) at a time.(d) Long-range pairwise functionals use an infinitesimal (fine-grained) representation of dipolar fluctuations and correlate only two dipoles at a time.



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such as the Hartree−Fock (HF) approximation, or thealternative suite of density-functional approximations (DFAs),which use the three-dimensional electron charge density, n(r),instead of the exceedingly more complicated many-electronwave function. However, these common approximations areunable to describe the long-range electron correlation energyand therefore fail to describe vdW interactions. This is trueeven for the more advanced semilocal meta-GGA func-tionals.63−65

Even in the simple case of an isolated many-electron atom,an accurate calculation of its (frequency-dependent) polar-izability is a complicated many-body problem. However,substantial progress has been made in this regard with thedevelopment of post-HF quantum-chemical methods and time-dependent DFT (TD-DFT) calculations that can providereliable estimates of the dipole polarizability and dipole−dipolevdW coefficients for all atoms between hydrogen andplutonium (Z = 1 to 102).58,66−69 Calculations that treatmany-electron correlation effects explicitly (either wave-function based or more recently TD-DFT methods) yieldelectronic response properties with an accuracy of 2% to 10%for gas-phase atoms.66−68,70,71 This high level of accuracy makesthe available free-atom polarizabilities and vdW coefficientsuseful as reference data for vdW methods in more complexsystems, as for example in the Tkatchenko−Scheffler (TS)60

and exchange-hole dipole moment (XDM)59 methods.However, since the resulting atoms-in-molecules coefficientsare inevitably approximate, we are faced with the question of

how the errors in asymptotic vdW coefficients affect thebinding energy at equilibrium distances.The binding energy is a result of an interplay between the

Pauli repulsion and vdW attraction, so the vdW energy alone isoften larger (more attractive) than the net binding energy itself.As an illustration, consider noble gas atom dimers, for whichinteraction potentials of spectroscopic accuracy can beconstructed solely from highly accurate vdW coefficients inconjunction with mean-field HF calculations.73,74 One can thensimply calculate that an error of 20% in the asymptotic vdWcoefficients of the xenon atom results in an error ofapproximately 40% in the equilibrium binding energy (Figure3). Already this simple example highlights the importance ofaccuracy in the parameters used for including vdW interactionsin electronic structure calculations. Even errors of 10%, whichare typical, may lead to unacceptable errors of 20% in thecorresponding binding energies. A common way to circumventthese errors is via fitting free parameters of a method toreference binding energies, but this strategy is likely to fail forsystems outside the data set used for the fitting. As one mightexpect, these levels of uncertainty in the asymptotic vdWcoefficients lead to even more serious problems in larger andmore polarizable systems. The vdW contribution to the bindingenergy of a diindenoperylene molecule (C32H16) adsorbed onthe Cu(111) surface is approximately 5.3 eV.75 In this case, anerror of 20% in the vdW coefficient would translate to an errorof 1.0 eV in the equilibrium binding energyexceeding theenergy of thermal fluctuations (kT) at room temperature by a

Table 1. Approximate vdW Methods in Terms of Their General Properties and the Degree to Which They Incorporate SomeEffects Influencing vdW Interactions

MethodaHybridization, coordination and

chemical environment Ions/oxidation statesAccuracy of C6 coefficients of small

organic moleculesPolarization in

materialsNonadditivepolarizability

D1/D2 No No 20% No NoD3 Only coord. effects Requires separate parametrization 5−10% No NoXDM Yes Yes, but strong charge transfer

requires parametrization12% No Short-range

TS Yes Yes, but strong charge transferrequires parametrization

5.5% No Short-range

MBD Yes Yes, but strong charge transferrequires parametrization

6.2% Yes All ranges

vdW-DF1 Yes Yes 20% No Short-rangevdW-DF2 Yes Yes 60% No Short-rangeVV10 Yes Yes 12% No Short-rangeRPA Yes Yes 10% Yes All ranges

MethodAnisotropy in vdW

parameters Many-body vdW energyComputational

costAmount offitting Applicability

D1/D2 No No Very low High Small moleculesD3 No Three-body Axilrod−Teller can

be addedVery low Intermediate Small and midsize molecules

XDM No Three-body Axilrod−Teller canbe added

Low Low Small and midsize molecules

TS No Three-body Axilrod−Teller canbe added

Low Low Small and midsize molecules

MBD Yes Infinite order Low Low Broadly applicable. Take care with metalsvdW-DF1 No No Intermediate None Small molecules and bulk solids, sensitive to

exchange functionalvdW-DF2 No No Intermediate Low Small molecules and bulk solids, sensitive to

exchange functionalVV10 No No Intermediate Low Small and midsize moleculesdRPA Yes Infinite order High None Broadly applicable

aD1,57 D2,107 and D358 are popular vdW methods that do not use electronic structure as input but are presented for comparison. XDM is theexchange-hole dipole moment method.59 TS is the Tkatchenko−Scheffler method.60 MBD is the many-body dispersion method.88 vdW-DF135 andvdW-DF236 are the nonlocal vdW density functionals. VV10 is the vdW functional of Vydrov and van Voorhis.62 dRPA is the direct RPA method;see for example ref 108.



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factor of 35. We note that, in popular pairwise vdW corrections,vdW coefficients of molecules on metal surfaces can beoverestimated by as much as 500%, because these approachesmiss the complex polarization effects in the extended surfacethat are typically responsible for reducing the vdW coefficientsof atoms in metals by 50−500% with respect to the free-atomvalues.76,77

The discrepancies in the vdW energy caused by small errorsin asymptotic coefficients can lead to qualitatively incorrectconclusions about relative stabilities (not absolute energies) aswell, such as when studying molecular crystal polymorphs.78 Insystems of increasing size, the binding arises from an interplayof short- and long-range interactions, so an accurate and reliablevdW method must be able to describe the binding at bothequilibrium and asymptotic separations, as well as everythingin-between, on equal footing. For these reasons, the accuracy ofasymptotic vdW coefficients must be given a significant weightwhen evaluating the performance of vdW-inclusive methods.While accurate vdW parameters can now be obtained for

isolated gas-phase atoms, the situation in real molecules andmaterials is vastly more complicated. Even for small molecules,the molecular polarizability and vdW coefficients cannot beaccurately represented simply as a sum over free-atomparameters. Dobson49 calls this difference between a freeatom and an atom in a molecule type-A nonadditivityit iscaused by the increased kinetic energy of the valence electronsdue to the confining molecular (instead of atomic) externalpotential. In many organic molecules (see Hybridization inFigure 1), the main effect is that orbital hybridization (i.e., arelatively local effect) reduces the effective polarizability andvdW coefficients of the atoms. For example, the C6 coefficientof an isolated carbon atom is 46.6 (a.u.), but this value isreduced to 29.8 for an sp-bonded atom and 22.1 for an sp3-bonded atom.79 However, the hybridization (coordinationnumber) concept alone is insufficient to fully account for theinfluence of the local chemical environment on vdWcoefficients. For example, consider the methane (CH4) andcarbon tetrachloride (CCl4) molecules. In both thesemolecules, the carbon atoms are sp3-hybridized with 4-foldcoordination, and are therefore identical according to these twomeasures of the local chemical environment. Yet the localchemical environments experienced by these atoms are

substantially different, with the C−H bond distance of 1.1 Åin CH4 compared to the C−Cl distance of 1.8 Å in CCl4. Thisresults in increased volume and lowered kinetic energy of thevalence electrons in CCl4 compared to CH4. The TS method isable to capture these local environment effects and yields aC6

C−C coefficient of 43.1 in CCl4 and 24.1 in CH4. As theselocal chemical effects are important in capturing qualitativetrends in vdW parameters, they are now captured by mostvdW-inclusive methods (see Table 1). Indeed, it is quiteremarkable that the C6 coefficients of small molecules can beobtained very accurately (i.e., to within 5.5%) only from theknowledge of the relative volume of an atom in its environmentcompared to the corresponding gas-phase atom.60

The local chemical environment (volume scaling) is only thefirst effect in a series of electronic structure contributions tovdW parameters. Another mostly local effect is caused bycharge transfer and different oxidation states in many-electronsystems (see Oxidation State/Solvation and Charge Transfer/Ionic Compounds in Figure 1). In rock salt (NaCl), being aclassical example of charge transfer, simple volume-basedmodels cannot be used to calculate vdW coefficientsunambiguously, because there are multiple ways to assignatomic densities to the Na and Cl atoms. One limit assumesthat NaCl consists of a superposition of Na and Cl free-atomdensities, yielding a polarizability of 182.7 (a.u.) per unit cell.The other limit assumes Na+ and Cl− ions, yielding asignificantly reduced polarizability of 21.6.80 The largedifference between these two limits is caused by the changein the polarizability from 167.7 for neutral Na atom to 0.9 forNa+. Measurements of dielectric properties81 and high-levelcalculations82 demonstrate that the polarizability of NaCl solidis in fact closer to the sum of Na+ and Cl− ions. Large changesin vdW parameters also occur for different oxidation states ofmetal atoms in molecules and solids.83−85 In contrast to thehybridization effects, charge transfer and oxidation states canonly be treated seamlessly by vdW methods that explicitly useelectronic structure information.35,36,55,59,60,62 Methods thatrely on atomic fragments require sophisticated definitions ofthe atomic fragment densities to yield quantitative accuracy forcharge transfer and oxidation states.86,87

Another manifestation of electronic structure in vdWparameters is their dependence on the contraction or expansionof the crystal lattice (see Lattice Expansion/Contraction inFigure 1). For instance, Zhang et al. demonstrated a substantiallinear dependence of αSi and C6

Si−Si in zinc blende silicon on thelattice constant using TD-DFT calculations.80 This behaviorcannot be explained by the coordination of the silicon, becausethis is unchanged throughout the lattice contraction, but rathercan be related to the change in the electronic band gap.Nonlinear dependence of vdW parameters on the latticeconstant can be observed in semiconductors, such as Ge, wherethe small band gap becomes zero upon the metallic phasetransition induced by the lattice contraction. Such effects arereliably reproduced only by methods that explicitly considerelectronic orbitals and their energies (e.g., RPA and TD-DFTmethods). Consistent treatment of vdW interactions in systemswith widely varying band gaps remains challenging because theelectronic response is difficult to extract from the electrondensity alone. The linear increase of the polarizability withlattice constant in semiconductor solids is reproducedquantitatively by the many-body dispersion (MBD) meth-od,55,88 which models the response of valence electrons byoverlapping quantum harmonic oscillators, qualitatively by

Figure 3. Potential energy curve of the xenon dimer as a function ofthe interatomic distance R. The black curve represents the exactpotential energy curve,72 while the blue (red) curve was obtained byincreasing (decreasing) all (C6, C8, C10) vdW coefficients by 20%. Thischange in the vdW coefficients yields an error of ±40% in the bindingenergy at the equilibrium distance, illustrating the importance of highlyaccurate asymptotic vdW coefficients for equilibrium predictions.



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nonlocal density functionals, in which the polarizabilitydepends on the electron density and its gradient,35,62 and notat all by pairwise fragment-based methods.57−60

All effects discussed so far were mostly local: the localbehavior of the electron density is often sufficient to describecoordination, charge transfer, oxidation states, and variation ofvdW parameters with small changes in structure. However, thefrequency-dependent polarizability is a nonlocal function thatexplicitly depends on two points, r, r′, in space: α(r, r′, u). Themany effects in vdW interactions that require an explicitnonlocal description necessarily have a nonadditive many-bodynature. For instance, the prototypical benzene molecule has astrongly anisotropic response, with the in-plane component ofthe polarizability being approximately twice as large as the out-of-plane component. Already this well-known nonadditivecontribution is neglected by most vdW methods, which relyon an isotropic definition of vdW parameters.35,57−60,62

However, many of these isotropic pairwise methods are ableto quite accurately approximate binding energies between smallmolecules, and this suggests that the effects of anisotropy mightbe partially compensated by the empirical damping functionsused in these methods and the lack of many-body vdW energyterms (Axilrod−Teller and higher order). The importance of

nonadditive effects increases dramatically with system size andlower dimensionality. For example, the in-plane component ofthe polarizability increases in a strongly nonlinear fashion inpolycyclic aromatic hydrocarbons.89,90 This nonlinear increaseof polarizability then translates into vdW coefficients beingstrongly nonadditive.91−93

Dobson calls these effects type-B nonadditivity,49 and theydepend sensitively on the size, structure, and dimensionality ofa given system. To illustrate these effects, we calculated thenonlocal fully interacting polarizability tensor α(r, r′, u = 0) ofthree molecules (Figure 4): adamantane C10H16, naphthaleneC10H14, and decapentaene C10H12. The polarizability matrix isconstructed using a self-consistent coupled dipole ap-proach,55,94,95 which models the response of the valenceelectrons to an internal electric field by a projection to a set ofoverlapping and interacting dipole oscillators. Although thepolarizability matrix in Figure 4 is not the exact α(r, r′, u = 0),as could be obtained by TD-DFT calculations, it provides areliable approximation to the polarization, as has beendemonstrated in numerous studies.55,92,95−97 The polarizabilitytensor is a dense matrix for all three molecules, and a diagonalisotropic approximation of such a tensor, as used in many vdWmethods, is clearly incomplete. The dependence of polarization

Figure 4. Nonlocal polarizability αAB of three hydrocarbon molecules with 10 carbon atoms: bulky, saturated adamantane; planar, aromaticnaphthalene; and linear, conjugated decapentaene. Solution of the dipole screening equation gives the nonlocal polarizability αAB ≡ αAB

ij (A, B labelatoms, i, j Cartesian coordinates). Three different visualizations of this quantity are shown: (a) Molecular structures and Cartesian components ofthe total molecular polarizability ∑ABαAB. x, y, and z components are encoded with green, blue, and red, respectively. All arrows have the same scale.(b) Visualization of αAB

ij for the subset of A and B being carbon atoms. Each 10-by-10 matrix corresponds to the αABij matrix with i and j fixed. The

color encodes the sign (red for positive, blue for negative), and the intensity encodes the magnitude of a given matrix element. (c) Dependence ofthe magnitude of αAB

ij on the distance between carbon atoms A and B in a log plot. The data points are fitted with a single exponential curve.



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on the geometry and bonding is nontrivial, with a much slowerdecay in decapentaene compared to naphthalene andadamantane. The possible effects of such nonlocal behavioron the polarizability can be illustrated on two differentorientations of the benzene dimer: the sandwich and T-shapedconfigurations. The plots in Figure 5 show, for instance, that

the higher symmetry of the sandwich configuration makes thexz component of the polarizability tensor vanish, whereas this isnot the case for the T-shaped structure. The isotropic pairwiseapproximation would be unable to capture the full complexityof these polarization effects.Polarization effects scale nonlinearly, so much stronger

effects can be observed in nanoscale molecules and materialscompared to the illustrative case of the benzene dimer. Thestability and mobility of defects in solids,14 polymorphism inmolecular crystals,7,13,98,99 adsorption of atoms and moleculeson surfaces,100,101 self-assembly of nanostructured materi-als,12,92 and relative stability of secondary structure motifs inpeptides and proteins102 have all been shown to require explicitnonlocal and nonadditive treatment of polarizability and vdW

interactions. In general, the presence of any environment (be itliquid phase, crystal, or presence of edges and surfaces) willstrongly affect the polarizability and vdW interactions withinand between molecules and materials. Last but not least,electromagnetic fields can also change the electron densitydistribution and its response, thereby potentially influencingvdW interactions. Our understanding of these complexpolarization effects is still fairly limited, and much currentwork is dedicated to the study of the behavior of intermolecularinteractions in complex environments.Particularly strong nonadditive and nonlocal effects (called

type-C by Dobson49) occur in nanoscale materials of reduceddimensionality,23,49 such as in fullerenes, nanotubes, nanorib-bons, and multilayered graphene, or in materials without cubic(spherical) symmetry. The coupling between many electronsover extended length scales leads to farsighted quantum-mechanical fluctuations (as opposed to nearsighted correla-tion103) that are responsible for long-range vdW forces.22 Theinherent delocalization of free electrons in metals coupled withlow dimensionality leads to an especially slow decay of the vdWforce between metallic chains and layers as a function of theirseparation,23 a modification of the conventional asymptoticbehavior which dominates at very large distances (e.g., beyond10−20 nm in bilayer graphene).104 In this context, Misquitta etal.105,106 demonstrated that, upon closure of the band gap,semiconducting nanowires may also exhibit unconventionalpower-law behavior at intermediate separations, followed byasymptotic convergence to the pairwise-additive limit.The preceding discussion concentrated on describing many-

body effects in the vdW energy that stem from thenonadditivity of polarizability in complex molecules andmaterials. This is often the dominant many-body effect, butthere are other types of many-body contributions to the vdWenergy, which can be understood using higher-orderperturbation theory on isotropic fragments with fixed polar-izability. This approach was originally used by Axilrod andTeller,40 and independently by Muto,41 resulting in the so-called Axilrod−Teller−Muto (ATM) three-body term, with acharacteristic decay of RABC

−9 , where RABC is the geometric meanof the three mutual distances between fragments A, B, and C.By continuing this perturbative expansion, one can derive anexpression for four-body and higher-order terms that decayfaster with respect to the distance between the objects. Becauseof this short-range nature of higher-order terms, it is commonto argue that many-body terms beyond the three-body ATMterm are negligible for realistic interatomic distances found inmolecules and materials. However, the ATM expansion wasderived assuming fixed isotropic fragments and additivelocalized polarizabilitiesassumptions that are not generalenough, as discussed above. In general, a subtle interplaybetween nonadditive polarization and higher-order vdW energyterms determines the many-body contributions to the vdWenergy, which can in turn be both attractive or repulsive. Thesign will sensitively depend on the topology, size, dimension-ality, and polarizability density distribution. An empirical way toaccount for this interplay is to renormalize the dispersioncoefficients that enter the pairwise and ATM terms. However,such a procedure cannot be transferable, and moresophisticated methods have been developed that include bothnonadditive polarization and higher-order many-body effects byconstruction. This intriguing nature of many-body effects andcoupling between different terms in the many-body expansion

Figure 5. Nonlocal polarizability αAB of two essential configurations ofthe benzene dimer: T-shaped and sandwich geometry. (a) A singlebenzene molecule and the Cartesian components of the totalmolecular polarizability ∑ABαAB. x, y, and z components are encodedwith green, blue, and red, respectively. All arrows have the same scale.(b) Visualization of the full nonlocal polarizability αAB

ij of the T-shapedand sandwich configurations of the benzene dimer with A and Brestricted to carbon atoms. See the caption of Figure 4b for a detaileddescription.



4721



is analyzed in the theoretical part of this review, especially inSection 4.8.3.While the exact ACFD formulation (see Figure 2(a)) is able

to seamlessly include all the described nonadditive contribu-tions to noncovalent vdW interactions, this is not the case formany approximate methods discussed in this section. For quickguidance, Table 1 provides a succinct description of a range ofapproximate methods for modeling vdW interactions, togetherwith their ability to treat some nonadditive aspects of theseforces. We hope that this table together with the discussion inthis and the next two sections will aid understanding of thereasons behind the good performance of some methods forparticular classes of systems, and the failures of varying degreefor other molecules and materials.

3. EXPERIMENTAL EVIDENCE FOR THE NONLOCALAND NONADDITIVE NATURE OF VAN DER WAALSDISPERSION INTERACTIONS

Our understanding of vdW interactions in complex systemsrelies mostly on theoretical concepts and the analysis of theresults of computer modeling. However, several recent seminalexperiments find unexpected collective behavior in vdWinteractions that simple models based on pairwise approx-imations are unable to explain satisfactorily. These experimentalresults are discussed here; taken together, they indicate a needto revise our understanding of vdW interactions based on arigorous, explicit, nonlocal, and many-body description.The group led by Stefan Tautz directly measured the force

gradients caused by the interaction of a single moleculeattached to a tip of an atomic force microscope with clean(111) terraces of a gold surface.109 The high precision of thesemeasurements allowed them to confirm the well-knownLifshitz−Zaremba−Kohn (LZK)110,111 Z−3 power law for theinteraction of a molecule with an underlying surface as afunction of their separation Z. This study also validated theLZK approximation for a range of increasingly large moleculesderived from perylene. The most remarkable finding is that thevdW attraction of perylene derivatives with the gold surfacescales nonlinearly with the number of carbon atoms. Thisfinding was attributed to a superlinear increase of thepolarizability with size, and was explained using calculationsat the RPA level of theory.109

Stronger nonlocality was observed in experiments carried outin the group of Karin Jacobs, who studied the interaction ofnanoscale, mesoscale, and macroscale objects with polarizableheterointerfaces composed of SiO2 thin films grown on a Sisurface.24,25 They studied the force between nanoscale proteins,mesoscale bacteria, and macroscale gecko feet and SiO2/Siinterfaces by varying the thickness of the SiO2 thin film. Thisallows determining the critical SiO2 thickness at which theunderlying Si surface becomes “invisible” for the interactingobject. The conventional pairwise treatment of vdWinteractions yields a critical thickness of approximately 1 nm,after which the vdW interaction should vanish. Remarkably, theexperiments indicate that the measured force converges onlywhen a thickness of 10−20 nm is reached, indicating that vdWinteractions are more nonlocal and farsighted than expectedfrom simple models. In contrast, explicit many-body models forvdW interactions predict much slower convergence of vdWforces for heterointerfaces with complex dielectric profiles.92

Such interfaces are characterized by electric fields thatpropagate far into the vacuum, which explains the observedorder-of-magnitude slower decay of vdW interactions.

Another manifestation of many-body effects is the recentlyobserved complete screening of vdW interactions by a single-layer graphene or MoS2 adsorbed on a SiO2 surface by Tsoi etal.112 This group carried out atomic-force microscopy experi-ments on top of the SiO2 surface and the same surface coveredby a single-layer graphene or MoS2. Whereas for a pristine SiO2surface the usual Lifshitz theory explains the experimentalobservations, covering the surface with graphene or MoS2 leadsto measurements that can only be explained by assuming thatthe tip interacts exclusively with the adsorbed single layer. Apossible explanation for this behavior is that the covering layersare characterized by a strongly anisotropic polarizability tensor,with in-plane fluctuations having a dominant role. Becausethese fluctuations are not significantly modified by adsorptionon the insulating SiO2 substrate, the tip−substrate force isdominated by the interaction with the 2D in-plane fluctuations,making any contribution of the underlying SiO2 surfacenegligible.The nonlocality and nonadditivity of vdW interactions is also

manifested in the nontrivial behavior of nanoclusters adsorbedon extended nanostructures.113,114 A simple additive modelwould lead to a linear dependence of the adsorption density onthe accessible nanotube surface area. Instead, Rance et al.observed quadratic dependence.113 Furthermore, the attractiveinteractions increased nonlinearly for larger, more polarizablegold nanoparticles.While all these experiments provide a strong evidence for the

many-body nature of vdW interactions in complex materials, itis expected that they are just the “tip of the iceberg” in theocean of molecules and materials. The complex interplay ofsize, dimensionality, and polarization response will continue tosurprise us and lead to new insights into vdW interactions, andultimately to better control over the self-assembly of complexpolarizable molecules and materials into useful ensembles.114

4. THEORETICAL FRAMEWORK FOR VAN DER WAALSMODELS

After summarizing the general concepts behind the currentunderstanding of vdW interactions in the previous section, wenow present a mathematical framework for discussing theseconcepts and for describing the different vdW models with asingle terminology. We start with the exact adiabatic-connection fluctuation−dissipation (ACFD) theorem for theelectron correlation energy and formulate a general classi-fication of currently used approximations to this expression. Wewill be concerned only with models that treat the correlationbetween electrons explicitly via an interacting potential and inwhich this potential has a correct asymptotic power-lawbehavior. In this regard, we will not discuss semilocal densityfunctionals that, to some degree, capture the short-range part ofthe correlation energy responsible for vdW interactions, such asthe Minnesota family of functionals,63,64 or the class of explicitvdW models which can only be applied in situations with non-negligible electron density overlap, such as the method of vonLilienfeld et al. based on atomic pseudopotentials.115−119 Wealso do not discuss ab initio methods that are based onsymmetry-adapted perturbation theory (SAPT),120,121 sincesuch approaches use a different definition of the noninteractingreference system that can only be applied to systems withdistinct interacting fragments (although this limitation has beena focus of recent work122).Although the framework presented here is general and could

be used to discuss the modeling of vdW interactions even in



4722


fully empirical methods such as classical force fields, we willfocus only on first-principles approximations, that is, methodswhich explicitly take into account the electronic structure of amolecule or material via the electron density or orbitals. Assuch, we limit the scope of this review to the context ofelectronic structure methods, in particular density-functionaltheory (DFT). In this case, a vdW model can be included in theself-consistent Kohn−Sham (KS) equations via its dependenceon the electron density or orbitals, and hence can predict notonly the effect of the long-range electron correlation on thetotal energy, but also its effect on the electronic structure. Thisobservation is closely related to the well-known Feynmanpicture of vdW interactions,39 which essentially provides aphenomenological connection between these two effects, andwhich does not apply to models that depend only on thegeometry and topology of a molecule or material. (TheHellmann−Feynman theorem applied to the electronicHamiltonian of two well-separated fragments implies that thelong-range electron correlation slightly disturbs the fragmentelectron densities, and in a fully self-consistent method, thesedistortions can be used to alternatively express the vdW forceson atoms as pure electrostatic attraction between the nuclei andthe surrounding electron density. However, an electroncorrelation method is needed to calculate the distortions inthe first place, so the Feynman picture does not practically solvethat problem.)We begin this section with the introduction and discussion of

different variants of the polarizabilitya quantity that is centralto the description of the long-range electron correlation. This isfollowed by a detailed analysis of the ACFD formula andrelated equations, and by the presentation of a theoreticalframework for the development of systematic approximationsto this formula based on range-separation and the quantumharmonic oscillator model. Finally, we introduce a generalclassification scheme for such approximations and apply it tothe currently used vdW models, which are then reviewedindividually in detail in the subsequent sections.

4.1. From point polarizability to nonlocal polarizability andback

The electrical polarizability, α, indicates the willingness of amolecule or a material to respond (become polar) under theinfluence of a perturbation of the electric field. As such, α isonly one of several commonly used response functions fordescribing this behavior, each of which differs by expressing theperturbation (cause) and polarization (effect) as differentquantities. Other typical response functions are the densityresponse function χ and the relative electric permittivity ε.Originally, these quantities of classical electrodynamics wereconsidered as macroscopic, that is, they were “smoothed” overthe rapid variations in the fields and densities that occur in amaterial on the length scale of individual atoms. However, theycan in fact also be used in the microscopic regime, nowphenomenologically describing the behavior of the electronsthat can be fully explained only by quantum mechanics. In itssimplest form, the (point) polarizability αpt determines the(point) dipole p of a molecule or a crystal unit cell that isinduced by an external electric field E,

α=p Ept (1)

We will now generalize the point polarizability to a continuousnonlocal time-dependent tensor quantity, α(r, r′, t, t′), relatethis form to the simple one above and to other response

functions, comment on why this is still not the full picture, andfinally show why this has anything and everything to do withvdW interactions.First, the field and the induced dipole may not have the same

direction in a general anisotropic material. In other words, theinduced dipole in some Cartesian direction, pi, may not dependonly on the component of the perturbing field in that direction,Ei, but also on the other components, Ej. This can be capturedmathematically by considering the polarizability as a tensorquantity α with Cartesian components αij, as was discussed inSection 2 and shown illustratively in Figures 4 and 5. Second,we switch to a continuous time-dependent description, whereboth the external field and the response depend on the positionin space and in time. When a small perturbation in the externalpotential Δϕ(r, t) (or the corresponding field E(r, t) =−∇(Δϕ(r, t))) is applied, assuming fixed nuclear charges, thematerial responds with a change in the electron (charge)density Δn(r′, t′). The time-dependent point dipole p(t) = ∫ drΔn(r, t)r has its continuous analog in the polarization densityP(r, t), which can also be understood as a dipole density, p(t) =∫ dr P(r, t). We note that there is no nontrivial pointcounterpart to Δn, because the total charge in a material cannotbe changed by an applied external field, ∫ dr Δn(r) = 0 (unlessthe field is strong enough to produce emission, which is acompletely different kind of effect). The induced charge densityand polarization density are related by

∇Δ = − ·n t tr P r( , ) ( , ) (2)

The response of the material is in general nonlocal; that is, thepolarization here and now does not depend only on the fieldhere and now. This effect can be captured with α being anonlocal function. In fact, it is this sole ability of thepolarizability to relate the response at a point r to aperturbation at a point r′, which enables it to capture theelectron correlation. Thus, in the most general form, therelationship between the polarization density and the polarizingelectric field is

∫ ∫ ∑ α= ′ ′ ′ ′ ′ ′−∞

P t t t t E tr r r r r( , ) d d ( , , , ) ( , )i

t

jij j

(3)

where the integrals are over all space and time and the sum overCartesian components can be understood as multiplication ofthe matrix α and the vector E. In what follows, we will oftendrop the explicit sum and enumeration over Cartesiancomponents and simply write ∑jαijEj = (αE)i. Equation 3would thus be written as

∫ ∫ α= ′ ′ ′ ′ ′ ′−∞

t t t t tP r r r r E r( , ) d d ( , , , ) ( , )t

(4)

Although α is the dipole polarizability, one can recover all thehigher induced multipoles from its fully nonlocal form, by firstcalculating P, followed by Δn via eq 2, and from that anyinduced multipole moment.Now we follow the path back to the molecular isotropic

polarizability. To do so, one must first assume a static electricfield, and drop the dependence on time, i.e., E(r). (This will bediscussed in more detail later when discussing the vdW energy.)Second, one must assume a spatially homogeneous electricfield, E(r) = E, for which

∫ α α= ′ ′ =( )P r r r r E r E( ) d ( , ) ( )loc (5)



4723


where we introduced a new quantity, αloc(r), termed here as alocal polarizability. Although αloc is now formally a localquantity, it effectively contains the integrated nonlocal responseat point r to the (homogeneous) field at all points r′. This isanalogous to how the local exchange-correlation energy densityin the homogeneous electron gas is formally a local quantity atpoint r, but contains integrated correlation contributions fromall points r′. Integrating over r′ in α(r, r′) reduces the amountof information contained in α; hence, αloc is no longer able tocompletely describe the response of a material to a generalinhomogeneous field. Indeed, using αloc to get a full response ofa material subject to an inhomogeneous perturbation would bean approximation, akin to using the local density approximation(LDA) in DFT when treating inhomogeneous systems such asmolecules.Next, if one requires only the knowledge of the total point

dipole of a molecule or material in a homogeneous field, then

∫ ∫ α α= = =( )p r P r r r E Ed ( ) d ( )loc pt (6)

where αpt is now the anisotropic point polarizability, obtainedby integrating over both r and r′ in α(r, r′), i.e., αpt = ∬ dr dr′α(r, r′). Finally, if one assumes an isotropic material, for whichthe response is the same in all directions, αpt,ij = αptδij, then

α=p Ept (7)

which brings us back to the point polarizability in eq 1.The nonlocal polarizability α(r, r′) is related to other

response functions, such as χ or ε by simple formulas, whichmirrors the fact that all these response functions carryessentially the same information. The density response χrelates a perturbation in the external potential to a response interms of the induced density,

∫ ∫ χ ϕΔ = ′ ′ ′ ′ Δ ′ ′−∞

n t t t t tr r r r r( , ) d d ( , , , ) ( , )t

(8)

By comparing eqs eqs 8 and 3, and using the relationshipsbetween Δϕ, Δn, E, and p, we arrive at

∑

αχ

α

∇ ∇′ ′ = · · ′ ′

= ∂∂

∂∂ ′

′ ′

′t t t t

r rt t

r r r r

r r

( , , , ) ( , , , )

( , , , )ij i j

ij

r r

(9)

That is, the density response is the result of the divergenceoperator applied to both r and r′ of the nonlocal polarizability.We note that whereas α is a tensor, χ is only a scalar. As in thecase of Δn, χ has no nontrivial point analog, because ∬ dr dr′χ(r, r′, t, t′) = 0 by charge conservation. For this reason, we willuse α rather than χ throughout this review. The relationshipbetween the density response χ (and hence the nonlocalpolarizability) and the relative permittivity ε is obtained directlyfrom their macroscopic definitions in the context of theMaxwell equations, and in atomic units (ε0 = 1/4π) reads as

∫ε χ′ ′ = + ″ ″ ″ ′ ′−t t v t tr r r r r r r( , , , ) 1 d ( , ) ( , , , )1(10)

where v(r, r′) = |r − r′|−1 is the Coulomb potential.Although α, χ, and ε are fully nonlocal time-dependent

response functions, they still do not describe the response of amaterial in its full complexity, but only within the linear regime.In general, the response of a material is a nonlinear functionalof the perturbation, and the response functions presented

above are the corresponding first functional derivatives of thesequantities,

α δδ

χ δδϕ

′ ′ =′ ′

′ ′ =′ ′

t ttt

t tn t

tr r

P rE r

r rr

r( , , , )

( , )( , )

, ( , , , )( , )

( , )(11)

This additional complexity is however not significant for adiscussion of the electron correlation energy, because theelectron pair density, n2(r, r′), from which the correlationenergy can be calculated directly, can be fully recovered fromthe linear response functions only. The fundamental reason forthis is that the Coulomb potential 1/|r − r′| acts between twopoints in space, whereas the higher functional derivatives of thenonlinear response depend on three or more positions in space.Thus, these quantities essentially carry more information aboutthe electronic system than is needed for the calculation of thecorrelation energy.Before proceeding further, the final step is to transform these

linear response functions from the time domain to thefrequency domain. All approaches discussed in this review areconcerned with systems in equilibrium (stationary states inquantum mechanics). Then, the response depends only on thetime difference t − t′. This allows one to Fourier transform thepolarizability α(r, r′, t − t′) to the frequency domain, α(r, r′,u), with the frequency denoted as u; namely:

α α α′ ′ = ′ − ′ → ′t t t t ur r r r r r( , , , ) ( , , , 0) ( , , )FT

(12)

The frequency-dependent polarizability α(r, r′, u) is oftencalled the dynamic polarizability. This transformation intro-duces a new level of complication by making the polarizability acomplex quantity with real and imaginary components.However, these two components have a clear physicalinterpretation: the real part describes a non-lossy response ofa material and the imaginary part represents dissipation.4.2. Exact correlation energy from the nonlocal dynamicpolarizability

So far, we have not discussed the role that the nonlocaldynamic polarizability plays in the description of vdWinteractions. This connection is formulated as an exactexpression for the electron correlation energy that is usuallyreferred to as the ACFD theorem. We will not repeat itselaborate derivation here, as it can be found elsewhere,54,123 butinstead cover the physical ideas embodied in the derivation.The connection between the correlation energy and thepolarizability starts with the realization that the response of amedium to an electromagnetic field, which is captured by theimaginary part of the dynamic polarizability, leads to dissipationof the field in the medium. The fluctuation−dissipationtheorem then relates the behavior of this dissipation to thecharge density fluctuations that are present in every quantum-mechanical system even at equilibrium. In essence, this theoremclaims that the relaxation of a system from a perturbed statebehaves according to the same laws both in the case when theperturbation was caused by an external field (such as whenprobing a molecule spectroscopically) or when it is an internalspontaneous fluctuation.124 This enables one to describe thedensity fluctuations in terms of dynamic response functions,such as the polarizability, α(r, r′, u). Once we have thedescription of these fluctuations, the electron correlation energythen follows directly from their Coulomb interaction. In thisway, one obtains the full electron correlation energy, of whichthe long-range part is responsible for vdW interactions. The



4724


reason why the ACFD framework is especially relevant for the

problem of vdW interactions is that it is possible to construct

good approximations to both α(r, r′, u) and the ACFD

expression itself when one is concerned only with the long-

range part of the correlation energy.In the literature, the ACFD is typically written in terms of the

density response function χ(r, r′, u) (see eqs 8 and 12),

∫ ∫ ∬πλ

χ χ

= − ′

× ′ − ′ ′λ

∞E

u

u u v

r r

r r r r r r

d2

d d d

( ( , , i ) ( , , i )) ( , )

c0 0

1

0 (13)

where r, r′ are spatial coordinates, u is the frequency, v(r, r′) =1/|r − r′| is the Coulomb potential, λ is a coupling strength that

connects the Kohn−Sham (KS) electronic system (λ = 0) to

the real system (λ = 1), χλ is the density response function of

the system at coupling strength λ, and χ0 is simply χλ for λ = 0,

that is, the density response function of the noninteracting KS

system. The integral over λ from 0 to 1 is the adiabatic-

connection that gives the other half of the name to the ACFD

formula and originates from the Hellmann−Feynman theorem

applied to the full electronic Hamiltonian with the electronic

repulsion term scaled by λ. The integral of χ over imaginary

frequencies is simply a mathematical trick, and is in fact

equivalent to the integral over real frequencies of the imaginary

part of χ,125

∫ ∫χ χ′ = ′∞ ∞

u u u ur r r rd ( , , i ) d Im ( , , )0 0 (14)

It is the form on the right-hand side which comes directly from

the fluctuation−dissipation theorem, because the imaginary

part of the response function, Im χ, accounts for dissipation.The KS response, χ0, is directly given by the Adler−Wiser

sum over the KS orbitals ϕμ(r),126,127

∑χϕ ϕ ϕ ϕ

′ = −* ′ * ′

ϵ − ϵ +μνμ ν

μ μ ν ν

μ νu f f

ur r

r r r r( , , i ) ( )

( ) ( ) ( ) ( )

i0

(15)

where ϵμ are the orbital energies and fμ are the occupation

numbers.For the purposes of this review, we will express the ACFD

formula in an equivalent128 form in terms of the polarizability

α(r, r′, u) and the dipole potential T(r, r′),

∇∇′ = − ′

= − ′ − ′ − | − ′|| − ′|

′vT r r r r

r r r r r r Ir r

( , ) ( , )

3( )( )

r r

2

2

Using the relationship between α and χ in eq 9, the divergence

theorem,129 and the limit χ(r, r′)v(r, r′) → 0 for |r − r′| → ∞,

one can transform

∬ ∬

∬

∬

∬

∬

∑

∑

∑

∑

α

χ α

α

α

α

′ = ′ ∂∂

∂∂ ′

= − ′ ∂∂

∂∂ ′

= ′ ∂∂

∂∂ ′

= − ′

= − ′

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

vr r

v

r rv

r rv

T

r r r r

r r

r r

r r

r r T

d d d d

d d

d d

d d

d d Tr[ ]

ij i jij

ij iij

j

ijij

i j

ijij ij

(16)

where the trace is over Cartesian components of αT. TheACFD formula can then be rewritten as

∫ ∫ ∬α απ

λ= ′

× ′ − ′ ′λ

∞E

u

u u

r r

r r r r T r r

d2

d d d

Tr[( ( , , i ) ( , , i )) ( , )]

c0 0

1

0 (17)

Both the interacting αλ and noninteracting α0 are in generalnonlocal functions, but only the former contains informationabout the electron correlation. To explain this point, we notethat the statement above, that α0 is the response of the KSsystem of noninteracting electrons, does not mean that it is theresponse one would get from a KS calculation in the presenceof an electric field. Indeed, in such a calculation the density andhence the mean field for the electrons would change under theinfluence of the perturbing electric field, so that all correlationeffects described by the exchange-correlation potential wouldbe propagated into the obtained response. In fact, with theexact exchange-correlation functional, one would obtain theexact α1 from such a calculation. On the other hand, theresponse described by α0 assumes that the mean field for theelectrons (comprising the Hartree and exchange-correlationpotential) is not changed by the perturbing field. That is, α0describes a process where each electron responds individuallyto the perturbing field while moving within the mean field ofthe unperturbed electrons. This is a crude approximation thatneglects the essential mechanism in which the response of anelectron is influenced by the response of all the other electrons.In the aforementioned KS calculation in the presence of anelectric field, this mechanism is actually ensured by the self-consistent adaptation of the mean field, whereas, in the Dysonapproach introduced below, it is achieved by direct coupling ofthe electrons. In both cases, the response of an electron takesinto account all the other electrons as it must. An indirectconsequence of the approximations made in α0 is that its spatialextent and nonlocal character are determined only by single-electron behavior. As a result, α0 is exponentially localized inmaterials with a finite gap;130 that is, α0(r, r′) goesexponentially quickly to zero as |r − r′| increases. We notethat this may not be the case for α1 even in such materials,because the long-range correlation between electrons caninduce long-ranged character in α1. It is this difference betweenαλ and α0 that is exploited by the adiabatic-connection formulain eq 17 to obtain the correlation energy.The interacting polarizability αλ(r, r′,u) can be expressed in

terms of α0 by means of the self-consistent Dyson equation,



4725


∬

∑

α α

α α

α α α

α α α

α α α

α α

λ

λ

λ

λ

λ

′ = ′

− ″ ‴ ″ ″ ‴ ‴ ′

≡ ′ − ⟩ ′

= ′ − ⟨ ⟩ ′

+ ⟨ ⟩ ′ −

= ⟨ − ⟩ ′

λ

λ λ

λ λ

λ

λ λ

λ=

∞

r r r r

r r r r T r r r r

r r T r r

r r T r r

T T r r

T r r

( , ) ( , )

d d ( , ) ( , ) ( , )

( , ) ( , )

( , ) ( , )

( , ) ...

( ) ( , )n

n

0

0 xc,

0 0 xc,

0 0 xc, 0

20 xc, 0 xc, 0

00 xc, 0

(18)

where Txc,λ(r, r′, u) = T(r, r′) − λ−1∇r∇r′ f xc,λ(r, r′, u) is thedipole potential accompanied by the exchange-correlationkernel f xc,λ(r, r′, u) = δvxc,λ(r, u)/δn(r′, u), ⟨·⟩ denotes theuse of the shorthand implicit notation for the spatial integrals,and the dependence on the frequency has been omitted forclarity. The polarizability of the real system is expressed here asan infinite sum over progressively more coupled terms builtfrom the polarizability of the noninteracting Kohn−Shamsystem. Straightforward use of the ACFD formula together withthe Dyson equation to calculate the correlation energy from α0is impossible due to the lack of knowledge of the exactexchange-correlation kernel f xc. This quantity is notoriouslymore difficult to approximate than the exchange-correlationfunctional itself.The ACFD formula in eq 17 and the Dyson equation in eq

18 are the two basic building blocks for the first-principlesdescription of vdW interactions. The latter is essentially theexact expression for the response of a material, while the formeruses this response to give the exact correlation energy. So far,we have discussed an exact theoretical framework without anyapproximations. Now, we connect the ACFD framework toDFT by introducing new effective quantities. In a typical DFTcalculation, the short-range part of the electron correlation isobtained from a semilocal or hybrid density functional.131,132

However, the ACFD formula gives the total correlation energy,including the short-range part, and if it was used in its full form,the short-range part would be double-counted. To solve thisissue, the ACFD expression can be range-separated; that is, thedouble integral over all space in eq 17 can be split into twoparts,

∬ ∬∬

′ = ′ − | − ′|

+ ′ | − ′|

f

f

r r r r r r

r r r r

d d d d (1 ( ))

d d ( )(19)

where f(R) is some range-separating function for which f(0) = 0and f(R) → 1 as R → ∞. The first and second terms on theright-hand side correspond to the short- and long-range parts,respectively. Such a range separation usually leads toapproximate approaches as described below; however, despitebeing arbitrary, it is in principle exact if the treatment of the twoparts is done consistently. For a more rigorous introduction ofthe range separation, see refs 133 and 134. The partition splitsthe electron correlation energy into two parts,

= +E E Ec c,sr c,lr (20)

The short-range part, Ec,sr, can then be approximated with a KScalculation, while the long-range correlation energy, Ec,lr, isgiven by the long-range part of the ACFD formula, that is, byeq 17 with ∬ dr dr′ replaced with ∬ dr dr′ f(|r − r|).

We note at this point that we do not identically equate thelong-range correlation energy with vdW dispersion interactions.Whereas the former is uniquely defined by specifying aparticular range separation, the latter has no formal non-perturbational definition. With that said, however, the overlapof the two phenomena is large and it is safe to say that the long-range electron correlation provides the microscopic explanationof vdW interactions. It is in this context that we present vdWmethods as methods for calculating the long-range correlationenergy. We also stress that, at large enough distances, thecontribution of long-range electron correlation to the electronicenergy can be understood fully in terms of vdW interactions (orcorresponding retarted Casimir interactions). In particular,although static (left−right, on-site) electron correlation ismanifest at large separations and is only partially captured bysemilocal density functionals, it is nevertheless caused by theshort-range part of the Coulomb potential. In other words,issues with static correlation would persist in a hypotheticalworld in which the long-ranged Coulomb potential would bereplaced with a short-ranged Yukawa potential.The range-separating function f must depend on the

underlying density functional that is used for the short-rangecorrelation; more precisely, f must depend on the range towhich the functional is able to capture the electron correlation.In addition, the range separation must certainly also be system-dependent, as can be seen from the following examples. First,consider the case of the homogeneous electron gas, for whichall LDA-based semilocal functionals give the exact energy; thatis, range separation is unnecessary in this case. In contrast, inthe case of nonoverlapping fragments, no such functional cancapture the electron correlation between these entities.However, neither the functional nor the system dependenceare completely known or understood theoretically, and theparticular form of the range-separating function is thus a matterof a long and unsettled debate in the literature. We will discussthe various approaches to this range separation in the respectivesections below.In analogy to the ACFD formula, the Dyson screening eq 18

can also be split into short- and long-range parts, albeit in a lessstraightforward manner. First, the coupling potential, Txc,λ, canbe split as follows:

′

≈ − | − ′| ′ + | − ′| ′

≡ ′ + ′

λ

λ

λ

f f

T r r

r r T r r r r T r r

T r r T r r

( , )

(1 ( )) ( , ) ( ) ( , )

( , ) ( , )

xc,

xc,

sr, lr (21)

where f is again some (in general different) range-separatingfunction, and we replaced the screened dipole potential in thelong-range part with the bare dipole potential, since it isassumed that the range of the exchange-correlation kernel isshorter than that of the range-separating function f. Thisseparation splits the n-th term in the Dyson equation into 2n

terms, each of which is formed by some particular combinationof Tsr and Tlr.Now we contract all the short-range segments ...α0Tsrα0...

contained in these terms and introduce an effective short-rangescreened polarizability αsr(r, r′) such that the Dyson equationbecomes

∑ α αα λ′ = ⟨ − ⟩ ′λ=

∞

r r T r r( , ) ( ) ( , )n

n

0sr lr sr

(22)



4726


Here, only the long-range coupling is considered explicitly viaTlr, whereas the short-range coupling enters implicitly in theeffective short-range screened polarizability αsr.The range separation of both the energy in the ACFD

expression and α in the Dyson equation enables us to combinethese two equations, after performing one final simplification.The functions used for range separating both of these quantitiesshould in principle be different. In the case of the energy, theseparation should be governed by the underlying semilocal orhybrid density functional. In the case of the polarizability, theseparation should be constructed based on the model for αsr(r,r′). However, we will invoke a final approximation here, doneby all vdW approaches known to us, by equating these tworange-separating functions. In principle, one can alwaysconstruct a model for αsr(r, r′) such that these two functionsmatch. Inserting eq 22 into the long-range part of eq 17 andintegrating over λ, we arrive at

∫ ∬∑

α

π= − − ′

× ⟨ ⟩ ′=

∞ ∞E

nu

u

r r

T r r

( 1) d2

d d

Tr[ ( ) ( , , i )]

n

n

n

c,lr2 0

sr lr (23)

Starting at second order, eq 23 expresses the long-rangeelectron correlation energy as an infinite sum over all orders ofthe long-range coupling of the short-range screened polar-izability. To clarify the various shorthands, the lowest ordercontribution Ec,lr

(2), written out in full form, is

∫ ∫ ∫ ∫ ∫∑

π

α α

= − ′ ″ ‴

× ′ ′ ″ ″ ‴ ‴

∞E

u

u T u T

r r r r

r r r r r r r r

12

d2

d d d d

( , , i ) ( , ) ( , , i ) ( , )ijkl

ij jk kl li

c,lr(2)

0

sr lr sr lr

(24)

The structure of the individual terms in eq 23 is illustrated inFigure 6b for the case of the benzene dimer. These terms canbe viewed as closed chains (loops) of quantum-mechanical

Figure 6. Schematic representation of the different types of interaction terms present in exact and approximate models for van der Waals interactionsas shown for the benzene dimer. (a) The legend for the individual quantities that are present in the different interaction terms. In the left columnfrom top, the long-range dipole potential Tlr(r,r′) (eq 21), its square, and the coarse-grained long-range dipole potential Tlr,AB (Section 4.4.1). In theright from top, the short-range screened nonlocal polarizability αsr(r, r′) (eq 22), the local effective polarizability αeff(r) (eq 25), and the coarse-grained local effective polarizability αeff,A (Section 4.7.2). The clouds around αeff hint at the effective spatial extent of the local polarizability. (b)Examples of terms in the formally exact many-body expansion of the long-range correlation energy in eq 23. Points in space interact via chainedshort-range polarizabilities and long-range dipolar interactions. The red and orange colors distinguish examples of the second- and third-order terms,respectively. αsr(r, r′) is largest when spatially localized on atoms and decays exponentially (in nonzero gap materials) with distance (see Figure 4c).On the other hand, αlr is damped at short distances, but decays only polynomially with distance. The terms in the figure illustrate contributions notonly from atoms, but also from bonds and even the tails of atomic densities (Section 4.3). (c) Examples of long-range correlation energy termscaptured by nonlocal density functionals (Section 4.6). In this case, only second-order terms are accounted for and the short-range nonlocalpolarizability is approximated by a density-derived effective local polarizability αeff(r) (Section 4.3). (d) Examples of terms in a coarse-grained versionof the full many-body expansion of the long-range correlation energy (Section 4.8). In this case, the second- (red) and third-order (orange) terms aredepicted. Here, the interactions terms occur only between whole atoms.



4727



events, in which a spontaneous fluctuation at some pointgenerates a perturbation at another, which produces a responseat yet another, which again generates a perturbation, etc., untilsome fluctuation interacts with the initial one. The closed formof these terms corresponds to the fact that these events are nottriggered by an external perturbing field, but are spontaneousinternal fluctuations.Equation 23 represents a general framework in which one

can discuss all major vdW-inclusive methods in terms of thevarious realizations and approximations of this expression.When assessing a given vdW model, one can start with adiscussion of how αsr and Tlr are constructed, followed by adiscussion of how are they coupled. The fundamentalmotivation for the separation of the whole problem intoshort- and long-range components is that the explicit couplingT is relatively easy to handle at long range and the properties ofa material captured by α can be reasonably modeled from theelectron density at short range, as we show in the next section.

4.3. Local effective polarizability from the electron density

In this section, we argue that it is reasonable to approximateαsr(r, r′) by its local counterpart αeff(r). We already introduceda local polarizability, αloc(r) = ∫ dr′ α(r, r′), which captures thelocal response to a homogeneous perturbation at all points. Themain difference between αloc(r) and αeff(r) is that the former isobtained from the full α(r, r′), whereas the latter comes fromαsr(r, r′), which is α0(r, r′) that was screened at short rangeonly. As argued above, α0 has mostly local character in finite-gap systems, which implies that αsr(r, r′) is also mostly short-ranged. That is, αsr(r, r′) describes only the response at points rthat are near the perturbing field at point r′. It is thenreasonable to assume that the electric field is approximatelyhomogeneous in this local neighborhood M(r) surrounding r,and hence the response can be well described with a localeffective polarizability given by

∫α α= ′ ′′∈

r r r r( ) d ( , )Mr r

eff( )

sr(25)

This integration can be limited to the local neighborhoodbecause αsr(r, r′) goes quickly to zero for large |r − r′|. Sinceαeff only depends on the local properties of a material, it turnsout that reasonably good approximations for this quantity canbe constructed using the electron density, its derivatives, orother density-derived quantities, which essentially was themotivation for range-separating the Dyson equation in the firstplace. In addition, the effective local polarizability alsosignificantly simplifies the Dyson equation and hence ourcentral correlation energy expression, eq 23. Here, αeff(r) mustenter these equations as δ(r − r′)αeff(r′), where δ is the Diracdelta function. For example, the second-order contribution ineq 24 reduces in the following way after the substitution ofαeff(r):

∫ ∬α α

π= − ′

× ′ ′ ′

∞E

u

u u

r r

r T r r r T r r

12

d2

d d

Tr[ ( , i ) ( , ) ( , i ) ( , )]

c0

eff lr eff lr (26)

Several distinct regions of the electron density can beconsidered when constructing a density model for αeff(r),which may influence different types of materials as discussed inSection 2. The inner-shell electrons are relatively unresponsive,yet they cannot be safely neglected in all circumstances,especially for heavier atoms. Most of the response comes from

the valence electrons, and hence, a good model must capturethe changes in the response associated with the formation ofcovalent bonds. In this regard, the tails of the electron densitycontain indirect information about the unoccupied electronicstates, and hence, they are useful in models of αeff. Ionic(charge-transfer) states are reflected in the depletion andaccumulation of the electron density and its derivatives withrespect to neutral atoms. In all these examples, the response canbe in principle captured well by a semilocal model for αeff, andthe difficulties lie mostly in constructing a model that is generalenough to describe all the cases as well as the continuoustransitions between them on the same footing.

4.3.1. Anisotropy of αeff(r). The (point) polarizability of amolecule is in general an anisotropic quantity. A well-knownexample is the benzene molecule, in which the in-planecomponents of the polarizability are approximately two timeslarger than the out-of-plane component (see Figure 5a).Because this quantity is coupled with the highly anisotropicdipole potential Tlr, a proper account of the anisotropy in thepolarizability is crucial for an accurate estimate of the long-range correlation energy Ec,lr. In the context of eq 23, thecontribution from the anisotropy emerges in two distinct ways.First, αsr(r, r′) (or αeff(r)) introduces the anisotropy directly

into the expression for Ec,lr. This aspect of the anisotropy isneglected in most vdW models. This approximation enablesone to write the polarizability as a scalar, αeff(r) ≈ αeff(r), whichsimplifies the equations by making αeff and Tlr commute. Forexample, the second-order contribution to Ec,lr in eq 24 reducesto the familiar 1/R6 dependence on the distance R=|r − r′|:

∫ ∬

∬∫

∬

πα α

πα α

= − ′

× ′ ′ ′

= − ′

× ′

′

= − ′ ′| − ′|

| − ′|

∞

∞⎜ ⎟⎛⎝

⎞⎠

⎡⎣⎢

⎤⎦⎥

Eu

u u

u u u

Cf

r r

r T r r r T r r

r r

r r

T r r

r r r rr r

r r

12

d2

d d

Tr[ ( , i ) ( , ) ( , i ) ( , )]

12

d d

3d ( , i ) ( , i )

Tr16

( , )

12

d d ( , )( )

c,lr(2)

0

eff lr eff lr

0eff eff

lr2

6

2

6

(27)

where we introduced ∫ α α′ = ′π

∞C u u ur r r r( , ) d ( , i ) ( , i )6

30

,

which is a generalization of the Casimir−Polder integral, and fis the damping function used for range-separating Tlr = f(R)T(see eq 21). C6(r, r′) defined in this way is just a more generalform of the familiar atomic C6,AB coefficients.Second, the anisotropy is also accounted for automatically by

the long-range coupling in eq 23. This follows from the Dysonscreening (eq 18), which was incorporated into the ACFDTformula (eq 17). In this regard, αλ obtained from the Dysonequation can be strongly anisotropic even if α0 was isotropic.

4.3.2. Harmonic oscillator model of the dynamicresponse. The exact dependence of the polarizability on thefrequency is in general unknown. Here, we briefly introduce themodel system of a quantum harmonic oscillator (QHO), whichis used to a different degree by many vdW approaches inmodeling the frequency dependence of the local effectivepolarizability. Because of the harmonic force and associatedquadratic potential, the QHO is the simplest model for



4728


studying small perturbations of a system in equilibrium. In ourcase, the equilibrium system is the electronic ground state, orthe ground-state electron density, and the small perturbationsare the quantum-mechanical charge fluctuations. The QHOalso happens to be a quantum system with equidistant energylevels, which makes it a perfect tool for “counting” electronicexcitations, which in turn describe all response properties ofmolecules and materials. It is for both these intertwined reasonsthat QHOs are such a useful model for describing vdWinteractions, among others.Many properties of a QHO can be derived analytically. In

particular, the dynamic (point) polarizability of a QHO is

αω δ

=− −

uq

m u u( )

( i )QHO

2

2 2(28)

where q is the charge of the oscillator, m is its mass, ω is itsexcitation (characteristic) frequency, and δ is an infinitesimallysmall number. The oscillator may be considered alsoinfinitesimal, in which case m and q would be infinitesimalsdm and dq, respectively. The infinitesimal imaginary part in thedenominator is necessary to satisfy the Kramers−Kronig causalidentities for response functions, and its physical meaning issimply that the absorption spectrum is a Dirac delta functionlocated at ω (in the imaginary part of αQHO). In other words,the QHO only absorbs electromagnetic radiation correspond-ing to its own excitation frequency. Furthermore, thisinfinitesimal imaginary part disappears in the integral over theimaginary frequency iu, as can be seen for example in the formof eq 14 for αQHO

∫ ∫αω

πω

=+

=∞ ∞

u u uq

m uq

md (i ) d

( ) 20

QHO

0

2

2 2

2

(29)

∫ ∫α π δ ω πω

= − =∞ ∞

u u uqmu

uq

md Im ( ) d

2( )

20

QHO

0

2 2

(30)

The QHO model, with its single-frequency absorptionspectrum, is insufficient to describe the full spectrum of a realatom. However, a frequency integral of such a complicatedspectrum (imaginary part of α) is a single number, and as suchcan be effectively captured by an integral of a simple spectrumcharacterized by a single frequency. The choice of such aneffective frequency necessarily depends on the particular formof the integrand, and the use of a single effective frequency forall cases constitutes the so-called Unsold approximation.135 Forinstance, consider the generalization of the Casimir−Polderintegral for the C6 coefficient in eq 27. This expressiondemonstrates that the second-order correlation energy betweentwo atoms (with complicated absorption spectra) can becaptured exactly by the interaction between two QHOs (withsingle-frequency spectra) as long as the pairs of atoms andQHOs have the same C6 coefficient. The C6 coefficient for twoidentical QHOs depends only on the static dipole polarizabilityand the characteristic excitation frequency:

∫πα α α ω= =

∞C u u u

3d (i ) (i )

34

( (0))6QHO

0

QHO QHO QHO 2

(31)

A useful combination rule holds for the C6 coefficient of twoQHOs A and B (with different sets of parameters q, ω, and m):

∫πα α=

=+α

ααα

∞C u u u

C C

C C

3d (i ) (i )

2

AB A B

AA BB

AA BB

6,0

6, 6,

6,(0)(0) 6,

(0)(0)

B

A

A

B (32)

where C6,AA are the C6 coefficients between two identicaloscillators A from eq 31 and αA(0) are the corresponding staticdipole polarizabilities. However, using eq 32 for combining theC6 coefficients of real atoms is precisely an example of theUnsold approximation, as the effective frequencies needed toexactly capture the integrals in eqs 31 and 32 would in generalbe different. On the other hand, it has been shown that thisapproximation leads to a mean absolute relative error of only3% in the C6 coefficients for heteronuclear atomic pairs.

60

4.4. Classification of approximate van der Waals methods

Here, we introduce a general classification of vdW methodsbased on two independent approximations to the generalexpression for Ec,lr in eq 23. The first one involves coarse-graining spatial quantities such as α(r, r′), αeff(r), or T(r, r′).The second approximation involves truncating the infiniteseries, or the many-body expansion, in eq 23 to some finiteorder. Methods that do not use either of these twoapproximations (but use others instead) comprise the class ofmodels based on the random-phase approximation (see Section4.5). Nonlocal density functionals, which will be discussed inSection 4.6, constitute a class of truncated, but non−coarse-grained methods. The many-body dispersion (MBD) frame-work of coupled harmonic oscillators represents a class ofcoarse-grained, but nontruncated models (see Section 4.8).Finally, both the coarse-graining and truncation approximationsare used in the popular class of pairwise methods (see Section4.7).

4.4.1. Coarse-graining the ACFD formula and Dysonequation. Each of the spatial integrals (over r and r′) in theACFD expression in eq 17 can be partitioned into N parts withthe use of a set of functions, wA(r), which satisfy ∑A = 1

N wA(r) =1 at each point r. When these functions are spatially localized,they naturally define a set of N fragments. Using such apartitioning scheme breaks the ACFD formula into N2 terms,

∫ ∫ ∬∑

α α

πλ= ′

× ′ − ′ ′ ′λ

∞E

u

u u w w

r r

r r r r T r r r r

d2

d d d

Tr[( ( , , i ) ( , , i )) ( , )] ( ) ( )

AB

A B

c0 0

1

0(33)

One may now coarse-grain the two-point dipole potential,T(r,r′), via

′ ′ ≈ ′w w w wr r T r r r r T R R( ) ( ) ( , ) ( ) ( ) ( , )A B A B A B (34)

in which T(RA, RB) is the dipole potential between the centersRA and RB of fragments A and B. Such an approximation thenenables one to reduce the spatial integrals in the ACFDformula,



4729


∫ ∫ ∬

∫ ∫

∑

∑

α

α

α α

πλ

πλ

= ′ ′

− ′ ′

≡ −

λ

λ

∞

∞

⎡⎣⎢⎢

⎤⎦⎥⎥

Eu

u

u w w

uu u

r r r r

r r r r T R R

T

d2

d Tr [ d d ( ( , , i )

( , , i )) ( ) ( )] ( , )

d2

d Tr[( (i ) (i )) ]

AB

A B A B

ABAB AB

c0 0

1

0

0 0

1

, 0,AB

(35)

where we defined αAB = ∬ dr dr′ α(r, r′) wA(r) wB(r′) and TAB= T(RA, RB), which are now coarse-grained versions of α(r, r′)and T(r, r′), respectively. A simple, yet illustrative example ofsuch a coarse-graining procedure is any numerical, real-spacegrid implementation of the ACFD formula, in which RA aresimply the grid points. In this case, the error associated withapproximating T(r, r′) with TAB would be considered anumerical error arising from the use of a grid that is too coarse.In the case of vdW models, however, one is only concernedwith the long-range part of T, Tlr, and hence the fragments Acan be much larger than grid points, while still maintaining areasonable level of accuracy. Here, the fragments are typicallychosen to be atoms, whereby an effective atoms-in-moleculespartitioning scheme wA(r) is used; however, other fragmenta-tion schemes can also be used with good results. Using ananalogy from electrostatics, we now connect the coarse-grainingto the standard multipole expansion procedure. The electro-static interaction energy of two charge distributions, nA(r), isgiven by ∬= ′ ′ ′−E n v nr r r r r rd d ( ) ( , ) ( )n n

12 1 2 . The same

quant ity for two dipole distr ibut ions , PA(r), is

∬= ′ ′ ′−E r r P r T r r P rd d ( ) ( , ) ( )P P12 1 2 . En−n can be expanded

into the multipole series q−q, q−p, p−p, p−Q, etc., where q, p,and Q denote monopole (charge), dipole, and quadrupolemoments, respectively. Likewise, the dipole distributioninteraction energy EP−P can also be expanded into an analogousseries starting with the p−p term. Here, we consider onlycharge distributions nA(r) that arise from dipole distributionsPA(r) via nA(r) = −∇·PA(r) and are therefore characterized byzero monopole (charge) moments. It then follows that theenergies En−n and EP−P, as well as their multipole expansions,would in fact be identical. Within this context, the coarse-graining of eq 34 simply corresponds to considering only thep−p term in the multipole expansion of EP−P.The feasibility of such an approximation is determined by the

spatial extent of each wA(r) around RA relative to the distancesbetween the fragments. When the fragments are atoms, theissue of considering only the dipole−dipole term is a well-known and well-analyzed approximation. Some modelsimprove upon this approximation by including higher-orderterms from the multipole expansion, such as the dipole−quadrupole or quadrupole−quadrupole terms. However, themultipole expansion of Tlr(r, r′) = f(|r − r′|)T(r, r′) is notsimply the multipole expansion of T(r, r′) multiplied by thedamping function f(|r − r′|). Therefore, the coupling of thehigher-order moments is not separable from the dampingfunction itself; rather, it can be partially and effectively capturedin f, as this function often contains some empiricism anyway.An alternative path to mitigate this error would simply involvethe use of a finer graining for α(r, r′). This returns to the factthat the fully nonlocal dipole polarizability is sufficient for anexact description of the long-range electron correlation.

As was done for the ACFD formula, one can also coarse-grain the Dyson equation in eq 18:

∑α α α α

α α α

λ

λ

= −

≡ − ⟨ ⟩

λ λ λ

λ λ

T

T

AB ABCD

AC CD DB

AB AB

, 0, 0, xc, , ,

0, 0 xc, (36)

where the meaning of ⟨·⟩ has been extended to denote alsoimplicit summation over fragments in addition to implicitintegration over space. As a result, the many-body expansion ofEc,lr in eq 23 can also be coarse-grained,

∫ ∫

∫

∑ ∑

∑ ∑

α α

α

πλ λ

π

= ⟨ − ⟩

= − − ⟨ ⟩

∞

=

∞

=

∞ ∞

⎡⎣⎢⎢

⎤⎦⎥⎥E

u

nu

u

T T

T

d2

d Tr ( )

( 1) d2

Tr[ ( ) (i )]

AB n

nAB AB

n

n

AB

nAB

c,lr0 0

1

1sr lr sr lr,

2 0sr lr

(37)

The coarse-grained versions of these equations will serve as abasis for discussing the pairwise and many-body dispersionvdW methods. We note that the only approximation made sofar is the replacement of the full multipole expansion with thedipole−dipole term. As long as all short-range correlationeffects are properly accounted for in αsr,AB, there is no reductionin generality in this regard.

4.4.2. Truncating the many-body expansion of thelong-range correlation energy. In eq 23, Ec,lr is expanded inthe number of interactions of αsr via the coupling potential Tlr.The second-order (i.e., the lowest-order) contribution corre-sponds to two interactions, the third-order contribution tothree interactions, etc. These interactions occur between bodies(atoms, molecules, points in space), which motivates theterminology of two-, three-, and four-body contributions to theinteraction energy. We point out that the correspondencebetween the m-body and n-th order contributions is notstraightforward. Starting at n = 4, some terms at n-th orderdescribe interactions between m < n bodies and hencecontribute to the m-body interaction energy. For instance, theterm at fourth order contains four Tlr between two bodies, andso contributes to the 2-body interaction energy (at fourthorder). On the other hand, the lowest-order contribution to agiven n-body interaction energy is typically the largest, whichjustifies the colloquial equivalence of these two distinctexpansions.The lowest-order contribution to Ec,lr corresponds to two Tlr

(e.g., red lines in Figure 6b). The higher-order contributions ineq 23 tend to cancel out in weakly polarizable systems or forsymmetrical geometries, which is the basis for pairwise schemesand nonlocal density functionals. In general, the quality of thissecond-order approximation is difficult to assess; however,three universal factors can be distinguished. First, the increasingnumber of the interactions Tlr decreases the magnitude of thehigher-order terms. This effect dominates at long range, whereTlr is small, and indeed the asymptotic interaction betweenfinite bodies corresponds to the second-order truncation.However, this requires that the correlation is treated to infiniteorder within the interacting bodies to obtain their respectivenonlocal polarizabilities. Second, the increasing number ofterms within each order may increase the total magnitude ofthe higher-order contributions. This effect is important in largermolecular systems and materials. (The number of terms at n-thorder in a coarse-grained model with N fragments grows as



4730


O(Nn), and hence the relative number of terms in subsequentorders grows as O(N).) The third universal factor is thesymmetry and dimensionality of the system. Whereas thesecond-order terms always lower the total energy of the system(i.e., stabilize), the signs of the individual higher-order termshave an angular dependence. In high-symmetry crystals withsimple unit cells and lattices, the higher-order terms tend tocancel out, leading to higher-order contributions that may becollectively negligible. In contrast, this is not the case in low-dimensional materials, in which nontrivial behavior in Ec,lr andresponse properties can be observed.In the present context, the often used Axilrod−Teller−Muto

(ATM) three-body potential constitutes a third-order con-tribution to Ec,lr. The previous discussion makes it clear thatwhile such a correction may extend the applicability of asecond-order model, it does not provide a universal descriptionof vdW interactions. A more detailed discussion of these termscan be found in Section 4.8, which deals with the many-bodydispersion framework.

4.5. Methods based on the random-phase approximation

Here, we discuss electronic structure methods within theACFD formalism that do not coarse-grain nor truncate theexpression for the correlation energy, but work with α0 in themolecular orbital basis as obtained directly from a KScalculation via eq 15. These methods construct a model forthe nonlocal polarizability in a way that is also applicable for theshort-range correlation energy, which distinguishes them fromthe explicit models for Ec,lr discussed in the next three sections.For the purpose of this review, we discuss these methods onlyfrom the point of view of long-range electron correlation andrefer the reader elsewhere for a more general exposition.136,137

A starting point for the methods of this section is therandom-phase approximation (RPA), in which the exchange-correlation kernel f xc(r, r′, u) = δvxc(r, u)/δn(r′, u) in theDyson equation (eq 18) is set to zero. Its name comes from theoriginal derivation of the approximation, in which thefluctuations of electrons are assumed to be influenced onlyby in-phase Coulomb contributions, whereas out-of-phaseterms are considered to be random and hence cancel out.This was later shown to be equivalent to the f xc = 0 definition inDFT. The application of RPA to the Dyson equation leads to aconceptually simple method for the electron correlation energy,which amounts to evaluating the expression eq 23 with αsr andTlr replaced by α0 and T, respectively, that is

∫ ∬∑

α

π= − − ′

× ⟨ ⟩ ′=

∞ ∞E

nu

u

r r

T r r

( 1) d2

d d

Tr[ ( ) ( , , i )]

n

n

n

c2 0

0 (38)

Using the explicit Adler−Wiser expression in eq 15 for the bareresponse in the χ-representation, this expression can beevaluated numerically in a straightforward manner, yieldingthe total electron correlation energy.108 In DFT, α0 is obtainedfrom a KS calculation carried out with some approximatedensity functional, and hence is only an approximation to thetrue α0 that would be obtained with the exact functional. Theremainder of the total electronic energy is usually taken as thesum of the kinetic, nuclear-electron attraction, Hartree, andexact-exchange terms, which are themselves approximate due tothe fact that they were obtained from KS orbitals based on anapproximate density functional.

RPA treats not only the long-range but also the short-rangeelectron correlation explicitly via the coupling potential, bothon the level of the energy, Esr, in the ACFD formula, and thelevel of the response, αsr, in the Dyson equation. This raisestwo issues. First, the short-range correlation energy may beinadequate, leading to an incorrect description of vdW systemsin equilibrium geometries, where this quantity plays asignificant role. Second, the fully coupled αλ may lackcontributions from f xc at short range, leading to inaccuratemolecular polarizabilities and hence incorrect long-range vdWcoefficients. Below, we discuss three different approaches tomitigate these issues.

4.5.1. Range-separated random-phase approxima-tion. The first attempt at correcting the behavior of RPA atshort range was given by Kurth and Perdew.138 They suggesteda correction to RPA, termed RPA+, given by

= − ++E E E EcRPA

cRPA

cGGA@RPA

cGGA

(39)

where EcRPA is the RPA correlation energy, Ec

GGA@RPA is thecorrelation energy calculated with a special GGA designed forcapturing Ec

RPA, and EcGGA is the correlation energy from a

standard GGA. The idea of the RPA+ approach is to removethe incorrect short-range part of the correlation energy fromRPA with the use of a special GGA, and then reintroduce abetter approximation to it using a standard GGA, such asPBE.139 The special GGA designed for this purpose wasconstructed by replacing the LDA part of the PBE correlationfunctional with an LDA parametrized to reproduce the RPAcorrelation energy of the homogeneous electron gas. This waslater refined by specially constructing also the density-gradientdependent part using additional properties of the short-rangebehavior of RPA.140 Here, the short-range correlation energy,Ec,sr, comes from a GGA density functional, while the RPA isused only for the long-range part, Ec,lr. However, the rangeseparation is accomplished only approximately using Ec

GGA@RPA

and depends on how well this GGA functional describes theshort-range part of RPA. The issue of explicit range separationof RPA was later attacked by Toulouse et al.134 In their work, arange-separated version of the KS scheme was used, in whichonly the short-range part of the electron correlation wasincluded in the exchange-correlation potential for the electrons,while the long-range part was treated explicitly via the long-range version of the ACFD formula. In such a scheme, the longrange Dyson equation in eq 22 is essentially solved with α0instead of αsr; that is, the long range is treated as in RPA, but atshort-range, it is assumed that v(r, r′) and f xc(r, r′) partiallycancel each other.

4.5.2. Approximate exchange-correlation kernels. In adifferent approach, the ACFD framework is used in full withoutrange separation, but a suitable approximation to f xc isconstructed. A straightforward nontrivial approximation canbe obtained from any density functional Exc[n] via

δδδ

′ ≈ − ′f uE nn

r r r rr

( , , ) ( )[ ]

( )xc

2xc

2(40)

where the kernel is approximated as local and independent offrequency. This so-called adiabatic local density approximation(ALDA) violates several known limits of f xc, which make itunsuitable for use in the ACFD formula. Olsen and Thygesenanalyzed the ALDA in the homogeneous electron gas inreciprocal space and noticed that whereas ALDA provides agood description of the small-q (large-wavelength) behavior of



4731


the electron correlation hole, it decays too slowly to zero forlarge q compared to the exact behavior. This observation leadthem to introduce the renormalized ALDA (rALDA) bymodifying the dependence of f xc in ALDA on large q.141−143 Bytransforming back to real space and generalizing toinhomogeneous densities, this defines a universal approxima-tion to f xc and hence a method for obtaining the electroncorrelation energy via the ACFD formula. Compared to RPA,the short-range part of the coupling in the Dyson equation isdescribed more realistically here, which propagates into a betterdescription of both the short-range correlation energy and theshort-range effects in the polarizability.A more straightforward, but computationally more demand-

ing approach is to approximate f xc with the exact exchangekernel, f x, leading to the so-called tdEXX method.144,145 Thedevelopment of more accurate and sophisticated kernels is thedomain of TD-DFT and the associated study of responseproperties, which is beyond the scope of this review. For a wayto make these approaches faster and hence more easilyapplicable to studying vdW interactions, as well as for anoverview of this rich field, see, for example, ref 146 and the refstherein.4.5.3. Renormalized perturbation theory to second

order. Yet a different path to fixing the issues of RPA comesfrom the standard many-body perturbation theory for electrons.The Møller−Plesset correlation energy at second order (MP2)consists of the Coulomb and exchange terms, of which only theformer is significant at all ranges (decays polynomially withdistance) whereas the latter is significant only at short range(decays exponentially). In this context, RPA can be described asan infinite-order sum of the so-called ring excitations, of whichthe Coulomb term of MP2 is the first one (the higher ordersare combinations of the single Coulomb term). Thisrenormalization of the Coulomb term resolves many of thedeficiencies of MP2, such as its inapplicability to zero-gapsystems and its tendency to overestimate Ec,lr. In the samebreath, RPA neglects exchange effects, which are partiallycaptured at the MP2 level, which can lead to issues at shortrange, where these effects become important. The second-orderscreened exchange (SOSEX) is constructed in a similar way asRPA by renormalizing the exchange part of MP2, that is, bysumming the corresponding type of the perturbation term toinfinite order. Together, RPA and SOSEX form a method thatextends well beyond MP2 via renormalization of both theCoulomb and exchange terms, resulting, for instance, in adescription of one-electron systems free of self-correlation.A renormalized form of the full second-order perturbation

theory (rPT2) can be obtained by adding the first-orderperturbation term, the so-called single-excitation (SE)correction.147 This term is trivially zero in the standard MPnscheme, because Brillouin’s theorem guarantees that singleexcitations of the HF orbitals do not contribute to thecorrelation energy. In a KS-based calculation, however, thisterm is required because the KS orbitals are not stationary withrespect to the RPA and SOSEX energy expression. In addition,the SE correction was found to approximately compensate forthe difference between the exact exchange energies evaluatedwith HF and KS orbitals. In analogy to RPA and SOSEX, theSE term can be summed to infinite order, leading to therenormalized SE (rSE) correction. The resulting rPT2 methodcombines the rSE, RPA, and SOSEX schemes.148 Klimes et al.derived an alternative formulation of rSE, in which the singlesexcitation energies are taken from the GW approximation.149 A

detailed discussion of many different flavors of RPA-likeapproximations can be found in other publications.150−152

4.6. Nonlocal density functionals

Nonlocal density functionals derive their name from their two-point dependence on the electron density via

∬= ′ ′ Φ ′E n n nr r r r r r12

d d ( ) ( ) [ ]( , )c (41)

where Φ is a so-called nonlocal kernel, constructed to capturethe long-range part of the correlation energy. The first universalfunctional of this type, called vdW-DF, was derived by Dion etal.35 as a culmination of the body of work established byLangreth and Lundqvist throughout the course of three decades(for a detailed discussion of the vdW-DF class of nonlocalfunctionals, see refs 38, 153, and 154. We start this section witha review of the general approximations to the ACFD formulaleading to nonlocal functionals and continue with a detaileddescription of particular functionals, including the vdW-DFfamily. The section concludes with a discussion of therelationships between nonlocal functionals and other vdWmethods.Nonlocal density functionals represent the class of non-

coarse-grained and second-order truncated approximations tothe ACFD formula. In addition, they use an isotropic density-dependent model for the effective polarizability αeff which wasdefined in Section 4.3.1. We start the discussion by rewritingthe second-order contribution to Ec,lr from eq 27 in a differentform:

∬ ∫π

α α

= − ′

× ′| − ′|

| − ′|

∞E u

n u n uf n u

r r

r rr r

r r

12

d d3

d

[ ]( , i ) [ ]( , i )[ ]( , )

c(2)

0

eff eff

2

6 (42)

where the functional dependence on the electron density isexplicitly denoted in all relevant quantities and the range-separating function f is generalized to depend on the frequency.Examples of the interaction terms under the spatial integral signare depicted in Figure 6c. Here, pairs of infinitesimalfluctuations with properties described by αeff are correlatedvia the coupling potential f 2/R6.The use of αeff is the first of two local approximations used by

all nonlocal functionals. As described in Section 4.3, αeff(r) isthe local counterpart of the short-range nonlocal polarizabilityαsr(r, r′) and in general can be constructed as an integral overthe local neighborhood surrounding r as in eq 25. We note thateq 42 does not specify the explicit functional form for thedependence of αeff on the electron density. In this sense, thesecond local approximation commonly enters via the modelsused for αeff(r), which depend only on the electron density andits gradient,

α α ∇≈n n nr r r[ ]( ) ( ( ), ( ))eff eff (43)

Because the polarizability in these methods is alsoapproximated as isotropic, the directional information in thedensity gradient is not used; this leads to the furthersimplification that only the norm of the density gradient entersthe formulas: αeff[n](r) ≈ αeff(n(r),|∇n(r)|). This leavespotential room for further improvement, as the density gradientin fact does contain information about the directionality ofbonds, for example, and hence the differential response of the



4732


electron density parallel and perpendicular to them (see Figure5 in Section 2 and the discussion there).The density gradient is the key ingredient for a successful

density-functional model of the local effective polarizability.Although early nonlocal functionals did not depend explicitlyon the density gradient, even these approaches used thisquantity to determine hard cutoffs for the spatial integrals in eq42 when treating nonoverlapping fragments.155 The usefulnessof the gradient is apparent from the fact that the ratio of thedensity gradient to the density in the density tail region of anatom or a molecule determines their ionization potential. Theionization potential is approximately proportional to theaverage excitation energy in the Unsold approximation, whichis in turn inversely proportional to the polarizability, thusestablishing the connection. The dependence on the densitygradient enables αeff[n] to capture electron correlation effects inthe neighborhood of a given point, similar to how GGAfunctionals achieve the same compared to LDA. From thisanalogy, a natural idea is to extend the current functionals forαeff to depend on the kinetic energy density in the spirit ofmeta-GGA density functionals; such functionals have not beensuggested so far.The dependence of αeff(r, iu) on the frequency in all nonlocal

functionals is approximated with that of an infinitesimalharmonic oscillator:

αω ω

ωα

=+

=+

⎯ →⎯⎯⎯⎯+

=→

uq

m uq

u

nn u

ur rr

r r

(i )( )

( ) d[ ]( )

( , i ) dq q

QHO2

2 2 2 2

d2 2 eff

(44)

where dq = ndr and it is assumed that the ratio q/mcorresponds to that of an electron, that is, it is equal to 1 (inatomic units). Here, the characteristic frequency ω correspondsto the plasma frequency, the frequency of the macroscopicoscillations in a homogeneous electron gas of the same density.Using this form for αeff[n], the correlation energy expression ineq 42 can be recast into the standard form for a nonlocalfunctional in eq 41 with the nonlocal kernel

∫π

ω ω

Φ ′ = −

×+ ′ +

| − ′|| − ′|

∞u

n u n uf n u

r r

r rr r

r r

( , )3

d

1[ ]( )

1[ ]( )

[ ]( , )

0

2 2 2 2

2

6

(45)

where the differences between distinct nonlocal functionalsreside in the specific functional forms of ω[n] and f [n].4.6.1. A simple nonlocal functional. Vydrov and van

Voorhis designed a simple model for the local effectivefrequency ω[n] and the range-separating function f [n], whichtogether form the VV10 nonlocal functional. Here, we providean alternative derivation of the VV10 nonlocal functionalequivalent to the original one. In VV10, ω[n](r) isapproximated with156

ω π ∇= +n n Cn

nr[ ]( )

43

24

(46)

where C is an empirical parameter fitted to a reference set of C6coefficients. There are three distinct ideas used in thisexpression. First, the 4πn term is motivated by thehomogeneous electron gas. Second, the factor 1/3 comes

from the Clausius−Mossotti equation and makes the expressionexact for the limiting case of a jellium sphere in a vacuum.Third, the gradient term is constructed as a local effectivecorrection to the characteristic frequency due to the nonzerogap between occupied and unoccupied orbitals. In the densitytails of atoms, the gradient term goes to a constant whichsmoothly damps the polarizability expression and avoids theneed for hard cut-offs.The specification of ω[n] already fully describes the

asymptotic behavior of the functional, leaving out only theshort-range behavior. The construction of the range-separatingfunction f in VV10 is motivated by considering the overlap ofthe two oscillators at short distances, which effectivelyattenuates their mutual interaction. This is modeled with asecond pair of oscillators in which ω depends on the distancebetween them and increases in such a way that they in factnever overlap. The damping mechanism can be then written asa range-separating function of the form

α αα α

′ = ′

′f u

u R u Ru u

r rr r

r r( , , )

( , i , ) ( , i , )( , i ) ( , i )

2 eff eff

eff eff (47)

where αeff(r, iu, R) is the polarizability of an oscillator modifiedby the presence of the other oscillator at distance R. The R-dependent term in ω contains the second empirical parameterof VV10, b, which effectively controls the onset of the rangeseparation. This parameter is fitted to reference interactionenergies and is specific for each semilocal functional, with whichthe VV10 functional is combined.157 The two empiricalparameters, C and b, control two distinct properties of VV10:C controls the local effective response, and b controls the rangeseparation. The VV10 nonlocal kernel was later slightlymodified by Sabatini et al. to enable efficient evaluation in aplane-wave basis.158 Two reparametrizations of VV10 targetedat binding in layered materials were presented by Bjork-man.159,160

4.6.2. vdW-DF functional. We start the discussion of thevdW-DF functional with the model it uses for ω[n]:

ω π π ∇= ϵ ≈ ϵ +⎛⎝⎜⎜

⎞⎠⎟⎟n n n Z

nn

r[ ]( ) 4 [ ] 4 ( )2 2xc

4 2xcLDA

7/6

2 4

(48)

where ϵxc is the exchange-correlation energy density and Z is aconstant.161 The first equality is obtained by using ω[n] for acalculation of the total exchange-correlation energy of a slowlyvarying electron gas. In vdW-DF, ϵxc is then approximated witha GGA functional. The GGA functional is used here only forthe parametrization of ω[n], and its choice is completelyindependent from the functional used for calculating the short-range exchange-correlation energy. Despite ω[n] in vdW-DFhaving a completely different form from that of VV10, bothfunctionals feature a single parameter which controls theessential density gradient term. In vdW-DF, however, the valueof this parameter is determined from first-principles arguments.We find it remarkable that the functional forms of ω[n] invdW-DF and VV10 differ even in powers of n and |∇n|, yetboth functionals provide relatively accurate polarizabilities. Thisagreement might deteriorate for heavier elements, where theratio of the density and the density gradient changessignificantly compared to the lighter elements.The range-separation mechanism in vdW-DF is more

elaborate than in VV10, partly because it is constructed in



4733


reciprocal space. There are two distinct mechanisms thatsaturate the vdW-DF functional at short range. The first one isanalogous to that in VV10: the local frequency ω is made toincrease as q2 for large q (short wavelengths), akin to ωincreasing with R−2 in VV10. This leads to a contraction of theinfinitesimal oscillators and a decrease of the local effectivepolarizability αeff. The second mechanism comes from therequirement that the vdW-DF functional evaluates to zero forthe homogeneous electron gas. This requirement forces thenonlocal kernel to positive values at short distances R whentransformed back to real space. In VV10, this condition istrivially satisfied by an a posteriori addition of a constant energyterm for a given number of electrons, wheras, in vdW-DF, thisrequirement is built directly into the nonlocal kernel. As aresult, the range-separating function f [n](|r − r′|,u) in vdW-DFhas a complicated form and we do not present it here; however,this function still serves merely to damp the nonlocal functionalat short range and goes to unity at long range.The original vdW-DF functional used a particular value of

the parameter Z that was derived earlier in the context ofsemilocal functionals. This choice provides relatively accuratepolarizabilities when used in conjunction with the GGAfunctional for ω[n] in eq 48. By construction, Z controls notonly ω[n] and hence the long-range asymptotic behavior, butalso the onset of the short-range damping mechanism. Thisseemingly elegant property introduces complications when onewants to couple Ec,lr from vdW-DF with the short-range partEc,sr from some semilocal density functional. Essentially, there isno way to adapt the range separation of vdW-DF withoutmodifying its asymptotic behavior at the same time. Hence,instead of adapting a single parameter for a given semilocalfunctional as is done in VV10, vdW-DF (with Z fixed byenforcing the correct long-range behavior) requires some apriori unknown semilocal functional that would complement itin the short-range. The original work suggests that the revisedPBE (revPBE) functional162 is such a complement, but otheralternatives were found later, as discussed in the next section. Inany case, vdW-DF cannot be easily paired with a given densityfunctional that might be the best choice for the short-range partof the electron correlation energy. Attempts at changing thevalue of Z were also made, but this in general leads to incorrectlong-range behavior.36

The vdW-DF functional was implemented self-consis-tently,163−165 which enables calculations of the electron densitypolarization stemming from vdW interactions. Roman-Perezand Soler used the fast Fourier transformation to efficientlyevaluate the nonlocal functional in periodic systems,166

avoiding the otherwise costly evaluation of the double integralin eq 41.4.6.3. Variations of vdW-DF. There are two popular ways

to modify nonlocal density functionals: first, by changing thenonlocal functional itself, and second, by using a differentsemilocal or hybrid functional for the short-range part of theelectron correlation energy. A simple modification of vdW-DFcan be done by changing its single parameter Z, which controlsthe density gradient contribution to the local frequency ω[n].The value used in the original formulation (vdW-DF) predictsgood C6 coefficients and was combined with the semilocalrevPBE functional. In the second version, vdW-DF2, Z is morethan twice as large, leading to larger values of ω[n] and hencesmaller polarizabilities and C6 coefficients, which turn out to beseverely underestimated. This is compensated at short rangewith a change of the underlying semilocal functional: here,

PW86167 was suggested as a good candidate. The combinationof Z and the GGA functional used in vdW-DF2 gives moreaccurate results for the binding energies of small vdWcomplexes, where the short-range behavior dominates overthe long range.More significant modifications of vdW-DF were presented by

Vydrov and van Voorhis. In vdW-DF-09, a different form forω[n] was used as well as a slightly different range-separationmechanism.168 This modification makes the damping in-dependent of the magnitude of the gradient term in ω[n], incontrast to the original vdW-DF; this allows one to determinethe short- and long-range behavior of the functional separatelyby fitting to reference binding energies and C6 coefficients,respectively. This makes the nonlocal functional usable inconjunction with any semilocal or hybrid functional. The ideaof having two distinct parameters in the nonlocal functionalthat control the short- and long-range behavior turned out tobe decisive for achieving better accuracy. The follow-up tovdW-DF-09, called VV09, can be considered a transitionbetween vdW-DF and VV10.37,169,170 Whereas the dampingmechanism was still constructed in reciprocal space as in vdW-DF, the functional form for ω[n] in VV09 was already the sameas that used later in VV10.Several attempts were made to find semilocal or hybrid

density functionals that would be more compatible with vdW-DF than the original suggestion of revPBE.171 Cooper designeda new GGA exchange functional, termed C09, whose gradientbehavior interpolates between that of a simple gradientexpansion of LDA to that of revPBE.172 Pernal et al. deviseda “dispersion-less” GGA functional by using the symmetry-adapted perturbation theory without the dispersion componentas a reference.173 A different approach to the same problem wassuggested by Rajchel et al., in which electronic motion is treatedby full DFT within interacting fragments, but the interfragmentdegrees of freedom are described by Coulomb and exchangeterms, resulting in an effective dispersionless description readyto be supplemented by a vdW model.174 Klimes et al.reparametrized the PBE and B88175 functionals (termedoptPBE and optB88) by fitting their parameters to referencebinding energies of a set of small vdW complexes.176,177 Thesereparametrizations break some of the exact constraints used inthe construction of PBE and B88 with unclear consequences.Wellendorf et al. used statistical methods, several reference sets,and a relatively general form of a GGA exchange functional tofind an optimal short-range complement to vdW-DF2, calledBEEF.178 They found that the optimal exchange functionalsignificantly depends on the particular system under study. Asthe original vdW-DF functional has better long-range behaviorthan vdW-DF2, better results might have been obtained ifBEEF used vdW-DF. Hamada reparametrized B86b179 to bettersuit vdW-DF2,180 while Berland and Hyldgaard devised a newexchange functional that interpolates between the Langreth−Vosko functional and revised PW86, and combined it withvdW-DF.181 Several authors identified that vdW-DF is notcompatible with exact exchange and the use of hybridfunctionals leads to considerable overbinding when coupledwith the vdW-DF family.

4.7. Fragment-based pairwise methods

The simplest pairwise models of vdW interactions areconstructed using the following formula,



4734


∑= −| − |

| − |E C

g R R

R R12

( )

ABAB

A B

A Bc 6, 6

(49)

where the sum is over pairs of fragments A and B (e.g., atoms),RA is the position of fragment A, g is some short-range dampingfunction, and the C6 coefficient was defined via the Casimir−Polder integral in eq 27. We use the term “fragment” whenreferring to A and B to cover all vdW methods; the fragmentsare atoms in most cases, but not in all.The R−6 potential, which describes the asymptotic attraction

between two fragments at distance R, was derived for the firsttime from quantum mechanics by Wang in 1927182 for the caseof two hydrogen atoms and later generalized by London.183−185

This potential served as a basis for vdW corrections to theoriesranging from the HF mean-field approximation,186 to theLennard−Jones potential, to semiempirical quantum-chemicalmethods, long before having been used in DFT.79,187,188. In2004, Grimme popularized this approach in DFT by illustratingthat even a simple empirical scheme of this kind improves theperformance of standard semilocal exchange-correlation func-tionals for vdW-bound systems.57 This was then followed bythe development of several models for C6 coefficients thatexplicitly take into account the electronic structure of a givenmolecule or material, which are discussed here.Equation 49 is a nontrivial generalization of the textbook R−6

potential between a single pair of fragments to a pairwise sum.In this section, we show which approximations to theexpression for Ec,lr in eq 23 lead to this pairwise sum andthen present an overview of several particular pairwise models.4.7.1. Fragment-based pairwise models from the

ACFD formula. Like the nonlocal functionals, the pairwisemethods use a second-order truncation of Ec,lr and an isotropiclocal effective model for the short-range polarizability. Unlikethe nonlocal functionals, however, these methods also usecoarse-graining of the spatial quantities: αeff, Tlr, and thedamping function g. It then follows that the pairwise formula ineq 49 is obtained from the second-order long-range correlationenergy in eq 27 by using the coarse-graining presented inSection 4.4.1:

∬

∬∑

∑

= − ′ ′| − ′|

| − ′|

= − ′ ′| − ′|

| − ′|′

≈ −| − |

| − |

E Cf

Cf

w w

Cf

r r r rr r

r r

r r r rr r

r rr r

R RR R

12

d d ( , )( )

12

d d ( , )( )

( ) ( )

12

( )

ABA B

ABAB

A B

A B

c(2)

6

2

6

6

2

6

6,

2

6

(50)

This formula is identical to eq 49 after identifying f 2 ≡ g andC6,AB = ∬ dr dr′ C6(r, r′)wA(r)wB(r′). Formally, this expressioncould be obtained by replacing the double spatial integral in thenonlocal functional formula (in eq 41) with a double sum overpairs of fragments. In this view, the role of the model for thelocal frequency ω in nonlocal functionals is played by themodels for the C6 coefficients in pairwise methods, and thesquare of the range-separating function f corresponds to thedamping function g.As discussed in Section 4.4.1, the coarse-graining is

equivalent to considering only the (lowest) dipole−dipoleterm from the multipole expansion of the polarizability αeff(r)wA(r) and the coupling potential T(r, r′). In pairwise schemes,

including the higher multipole terms is straightforward andleads to the well-known series starting with C8/R

8 and C10/R10.

We have also argued that these higher-order contributions canbe at least partially captured in the empiricism of the dampingfunction. This is further supported by the existence of simpleapproximate recursive formulas for the higher-order Cncoefficients based on the C6 coefficients and various integralmoments of the fragment densities.189,190

The accuracy and reliability of a given pairwise method isinfluenced both by the particular model for the C6 coefficientsand the choice of the damping function g. The quality of thesetwo components can be evaluated separately, because only theC6 coefficients determine the asymptotic behavior. Such adistinction is useful, because the C6 coefficients and dampingfunction g influence the performance of a method in differentways. For one, the relative importance of C6 and g for theresulting vdW energies changes with system size. The C6coefficients become more important in larger molecular orcrystal systems, whereas the damping function plays asubstantial role in small vdW complexes.Another difference comes from the fact that the models for

C6 coefficients discussed here are typically based on someunderlying first-principles arguments and assumptions, whereasthe damping functions are mostly empirical. For the most part,this is caused by the lack of general knowledge about thebehavior of the short-range part of the electron correlationenergy as described by the underlying exchange-correlationdensity functionals in the range of typical vdW equilibriumdistances. Furthermore, all other noncovalent interactions(electrostatics, induction, Pauli repulsion) also in generalcontribute to the binding energies in this regime. Whendamping functions are fitted to reference binding energies,which is a common technique, any errors in the non-vdW partof the energies propagate into the fitted parameters, making theresulting vdW model less transferable. This complicates thediscussion of the vdW methods separate from the underlyingfunctionals. We note that this issue is in fact common to allmethods that use a density-functional approximation for theshort-range correlation energy, including the range-separatedRPA methods, the nonlocal functionals, as well as the many-body dispersion framework that will be discussed in the nextsection.

4.7.2. Models for C6 dispersion coefficients and othervdW parameters. Although we set out to cover strictly vdWmethods that explicitly take into account the electronicstructure of a system, we open this section with a particularfamily of methods that do not leverage any information about amolecule or a solid besides its geometry. The DFT-D1,57 DFT-D2,107 and DFT-D358 methods largely popularized theapproach of correcting DFT calculations for vdW interactions,providing a context for further development of less empiricalmodels that use electronic structure as an input. As discussed inSection 2, the spectrum of behaviors of vdW interactions in realmaterials is vast, reflecting the intricacies of electronic structuretheory, and an effort to capture all this complexity purely fromgeometry is a formidable task, akin to classical force fieldsattempting to calculate the electronic energy without explicitlyconsidering electrons. This is reflected in more elaborateconstructions of the DFT-Dn methods with increasing n, asthey tried to cover a wider array of systems.DFT-D1 used fixed atomic C6 coefficients that did not

depend on any molecular environment. The heteronuclearcoefficients were calculated using an expression that is



4735


equivalent to eq 32 with polarizabilities of all atoms consideredequal. The damping function was constructed using vdW radiiin such a way that it did not go to unity for R → ∞, leading toincorrect asymptotic behavior. DFT-D2 followed the sameprotocol (with some numerical reparametrizations) except forthe C6 combination rule, which used an expression equivalentto eq 32 with α α = C C/ /A B AA BB0, 0, 6, 6, , arguably a more

reasonable model than in DFT-D1. DFT-D3 introduced severalchanges. First, the damping was modified to satisfy the expectedasymptotic behavior ( f(R) → 1 for R → ∞). Second, fixed C6coefficients were estimated from reference hydride dimers byapproximate decomposition of their long-range interaction intocontributions from the corresponding heavy atoms and thehydrogen atoms. Third, the vdW radii were replaced withatomic radii calculated as a distance at which an interactionenergy between the corresponding atoms calculated in a certainspecial way58 equals an arbitrary threshold. Fourth, anelementary dependence of the C6 coefficients on the environ-ment was included via empirical geometrical factors thatestimate a coordination number of the atoms. The reference C6coefficients are then interpolated for a given estimatedcoordination number in a complicated but arbitrary waymotivated by numerical results. However, the true dependenceof the electronic structure on geometry is still much morecomplex than what this comparably simple geometricalinterpolation captures, leading to unphysical bumps in thepotential energy surface of many molecular systems besidessimple organic compounds. Apart from these changes, C8coefficients were added to the model as a fixed part, andthree-atom C9 coefficients and the corresponding 3-atompotential as an optional part, which seems to improve theaccuracy of DFT-D3 for large molecular systems, but worsens itfor smaller systems, without a clear consensus of the crossoverregime. In general, it is the absence of the dependence of C6coefficients on the environment in DFT-D1/2 and the difficultywith which it was modeled based purely on the geometry inDFT-D3 which motivated development of methods based onelectronic-structure information, described in the following.Becke and Johnson devised a model for treating vdW

interactions based on the dipole moment of the exchange hole,termed XDM. In a series of papers, they used this quantity toconstruct a heuristic model for the C6 coefficients of isolatedatoms and molecules.191 Later, the model was generalized tothe case of atoms in molecules with the use of a partitioningscheme devised by Hirshfeld.192 In this scheme, a reference setof free-atom densities is used to construct the followingstockholder model for the partition functions wA,

=| − |

∑ | − |w

nn

rr R

r R( )

( )( )A

A A

B B B

Hirshfree

free(51)

where nAfree(r) is the (radially averaged) density of atom A in

vacuo (i.e., a free atom) at a distance r from the nucleus and 0 <wAHirsh(r) ≤ 1 at each point in space by construction. (The

Hirshfeld-partitioned density of an atom A is then given bynA(r) = n(r)wA

Hirsh(r).) The partitioned atomic C6 coefficientswere then combined with a damping function to formulate afull vdW model in ref 193. The approach was also extended toinclude higher-order Cn coefficients and modified so that thedependence on the static molecular polarizability as an externalinput was removed.194 Finally, all of these incrementalimprovements were combined into a single general pairwisemethod for vdW interactions in ref 195.

The dipole moment of the exchange hole used by the XDMmodel is calculated from occupied KS orbitals.196−198 Theexchange hole represents the electron density that is pushedaway from an electron due to the Pauli exclusion principle. Thecombined charge of the exchange hole and the referenceelectron is always zero, but it possesses a nonzero dipolepolarization density, Px(r), which was argued to correspond toinstantaneous quantum-mechanical charge fluctuations. The C6coefficients are thus obtained from the square of the dipolemoment of the exchange-hole polarization as

∫α= | |C r P r12

(0) d ( )6 pt x2

(52)

where αpt(0) is an isotropic static point polarizability of thegiven atom or molecule.(For an alternative density-functional explanation for vdW

interactions based on the correlation hole, see ref 199.) Toobtain the atomic C6 coefficients, the integral above can becoarse-grained with the Hirshfeld partitioning,

∫α= | |C wr P r r12

(0) d ( ) ( )AA A A6, eff, x2 Hirsh

(53)

where the static atomic polarizabilities αeff,A(0) are obtained byscaling the reference free-atom polarizabilities66 via

∫∫

α α

α

=| − |

| − |

≡

n w

n

V nV

r r r r R

r r r R(0) (0)

d ( ) ( )

d ( )

(0)[ ]

A AA A

A A

A

eff,free

3

free 3

free eff

free (54)

The integrated quantities Veff and Vfree in the numerator anddenominator are measures of the “volume” of the atom in amolecule and the free atom, respectivelyquantities thatcorrelate with atomic polarizabilities.200 To get C6 coefficientsbetween different atoms, XDM uses the combination rulederived in eq 32 for two harmonic oscillators. Similarexpressions can be used to calculate higher-order Cn coefficientsfrom the higher multipole moments of the exchange-holepolarization density, Px. The XDM model can also be usedtogether with density-functional approximations for theexchange hole,201,202 which removes the dependence on theoccupied KS orbitals.203 The triple-dipole ATM-like term canbe obtained from the exchange-hole dipole model as well.204

Two distinct damping functions were suggested forXDM.195,203 The first one uses an approach based oncorrelation energy ratios,

= ++

−⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟⎞⎠⎟⎟g R c

C R

E E( ) 1

/AB

AB

A B

XDM1 6,6

c,free

c,free

1

(55)

where Ec,A is the correlation energy of a free atom and c is afitted empirical parameter. The second suggested dampingfunction follows a more common approach, which is based on aratio of length scales,

= ++ +

−⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟g R c

R R cR

( ) 1ABA BXDM2

1

vdW vdW2

6 1

(56)

where RAvdW is a vdW radius for atom A and c1 and c2 are

empirical constants.



4736


Tkatchenko and Scheffler (TS) devised a pairwise methodfor vdW interactions in which the free-atom scaling approach isextended to all relevant vdW parameters, that is, the atomicpolarizabilities, C6 coefficients, and vdW radii,60 hence makingthem all functionals of the electron density

α α= =

=

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

V nV

C CV n

V

R RV n

V

(0) (0)[ ]

,[ ]

[ ]

A A AA AA

A A

eff,free eff

free6, 6,

free eff

free

2

vdW, vdW,free eff

free

1/3

(57)

The TS approach effectively describes a larger portion of theshort-range correlation effects based on accurate free-atomvalues, whereas the XDM model tries to recover them from theproperties of the exchange hole. The strategy of using free-atomreference values could also be viewed as constructing thepolarizability model in such a way that it satisfies the exactlyknown (free-atom) limits, an approach that has beensuccessfully used for exchange-correlation density functionals,where it is known as satisfying exact norms. In the case ofdensity functionals for polarizability, this normalization was sofar used only in the atomic models (αeff,A)properly normedpoint functionals, αeff(r), are yet to be developed. Finally, theTS model for the C6 coefficients is combined with a logisticdamping function,

= + −+

−

−⎛⎝⎜⎜

⎡⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟⎤⎦⎥⎥⎞⎠⎟⎟g R c c

RR R

( ) 1 exp 1ABA B

TS1 2 vdW vdW

1

(58)

The TS method can be straightforwardly combined with awide range of semilocal and hybrid functionals by adjusting asingle range-separation parameter.205 The value of the range-separation parameter is close to 1 when TS is coupled with thenonempirical PBE, PBE0, or HSE functional. Values slightlylarger are obtained for meta-GGA functionals,63,64,206 whichconfirms that these functionals can effectively describe electroncorrelation at intermediate interatomic distances.In larger molecules and solids, many-body and screening

effects become important, which motivated several improve-ments to the original TS approach. Bucko et al. appliedelectrodynamic screening from eq 18 to the atomic C6coefficients, leading to improvements in the description oflayered materials.207 Zhang et al. and Ruiz et al.76,80 used thedielectric function of bulk materials to define atomic polar-izabilities in solids as an alternative to the free-atom referencedata used in eq 57. Such renormalized bulk referenceparameters can be used to investigate cohesion in solids80,208

and adsorption of molecules on surfaces.76,77,101,209 Further-more, the TS approach was also extended from DFT60 to thedensity-functional tight-binding (DFTB) method,210 as well asclassical force fields.211,212

Sato and Nakai developed an atomic pairwise method thatcan be considered a direct bridge to the class of nonlocalfunctionals, by directly expanding an ALL-like density func-tional into a multipole series around the atomic centers.213,214

They used the general formula for the dynamic polarizability ofan infinitesimal harmonic oscillator in eq 44,215 combined withthe characteristic excitation frequency from the vdW-DF-09nonlocal functional of Vydrov and van Voorhis,168

ω π λπ

∇= +⎛⎝⎜

⎞⎠⎟n n

nn

r[ ]( )19

(3 ) 11

4(3 )2 2 4/3

2 2/3 4/3

2 4

(59)

where λ is an empirical parameter to be fitted to a set of C6coefficients. This formula shows yet another density functionalfor the effective local frequency which was the main topic in thesection about nonlocal functionals. A molecular systemdescribed by this local effective polarizability is then partitionedinto atomic fragments by means of the Becke scheme,216 whichis otherwise often used in DFT calculations to construct real-space integration grids, but serves here as an alternative to theHirshfeld partitioning used by the XDM and TS methods. Byassuming that the contribution to the polarizability from areasof significant atomic density overlaps is small, one obtains acomplete representation of αeff(r) in terms of the effectiveatomic dipole polarizabilities αeff,A and higher moments. Thisthen leads directly to C6, C8, and higher-order coefficients viathe Casimir−Polder integral (the series is truncated after C10 inpractice). This method, named local response dispersion(LRD), is then combined with the following damping function(shown here only for the C6 coefficients)

α α= −

+ +

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟g R

Rc c

( ) exp( )AB

A B

LRD

1 eff, eff, 2

6

3 3(60)

where c1 and c2 are parameters to be determined for eachunderlying density functional by fitting to a set of accurateinteraction energies, and the atomic polarizabilities serve hereas measures of atomic radii. We note that essentially anynonlocal density functional for vdW interactions could beturned into a fragment-based pairwise method by the samemultipole-expansion technique used in LRD. A relatedapproach to LRD was presented by Heßelmann,217 in whichthe polarizability model is replaced by a more rigorous formobtained by directly approximating the expression for anuncoupled response function.Silvestrelli formulated a pairwise method, named vdW-

WF,218 where the interacting fragments are not atoms, butmaximally localized Wannier functions (MLWFs).219 MLWFsare essentially a generalization of localized Boys molecularorbitals to periodic systems; as such, they are the counterpartsto the periodically symmetrized Bloch functions, which are bydefinition fully delocalized. The vdW-WF model is constructedby approximating each occupied MLWF with a single Slater-type s-function of the same spread S (spatial extent). Here, theassumption of an exponential decay of the MLWF isappropriate in systems with a nonzero gap, while the omissionof any angular structure is in general an approximation. Givenan approximate s-orbital, its dynamic polarizability is thencalculated using the ALL polarizability functional,155 where thedensity gradient (of the approximate s-orbital) enters via thedefinition of a MLWF-specific integration domain ΩA for thefollowing spatial integral,

∫απ

=+∈Ω

un

n ur

rr

(i ) d( )

4 ( )AA

Areff, 2

A (61)

with

∇Ω = | | <n knr r r{ : ( ) ( ) }A A A7/6

(62)



4737


where nA(r) is the electron density of the MLWF A asapproximated by the square of the corresponding s-functionand k is a nonempirical constant. The C6 coefficients are thendirectly obtained via the Casimir−Polder integral and thedamping function used in vdW-WF is the same as in the TSmodel with the following modification. The atomic vdW radiiin gTS in eq 58 are replaced by distances from the centers of theMLWFs at which nA(r) falls below a predetermined threshold;the choice for this distance was the subject of a latermodification.220 The MLWF partitioning used in vdW-WFsignificantly differs from the atomic partitioning used in theXDM and TS models. Given that the ALL polarizabilityfunctional is nonlinear in the input density (αALL[n1 + n2] ≠αALL[n1] + αALL[n2]), using it with the individual fragmentdensities as the input constitutes a different scheme thanevaluating the functional integrand with the total density andthen partitioning the integral according to the MLWFfragments. Furthermore, the use of the approximate s-functionsfor the MLWFs results in a partitioning of the total density thatis not exact, that is, ∑AnA(r) ≠ n(r).Tao et al. developed a density functional for the point

dynamic polarizability (and hence the C6 coefficient), whichuses the static polarizability as an input,221

∫α

α

=+

Ω = | | <

π∈Ωu a

na n u

a

r

r r

(i ) d ,

{ : (0)}

rpt

343

4 2

3(63)

where Ω is an integration domain, and a is a constantdetermined by satisfying the exact high-frequency limit ofαpt(iu) and the limiting case of a classical metallic sphere(consisting of a uniform electron density). The choice of theintegration domain in this functional guarantees that the inputstatic polarizability α(0) is recovered for u = 0. This model waslater extended to describe higher-multipole coefficients aswell.222 In the context of general vdW methods, this model isnot fully complete, as the C6 coefficient model is not combinedwith a damping function, and still requires a prescription for theinput static polarizability. However, such a generalization couldbe obtained in a straightforward manner, for example with theapproach used in the XDM or TS methods, and hence wechose to present this model to illustrate the flexibility withwhich density-functional models for the local effective polar-izability can be constructed.Besides developing new functionals for the C6 coefficients,

novel pairwise methods can be constructed by using alternativepartitioning schemes. For instance, Steinmann and Cormin-boeuf adapted the XDM model by using the self-consistentHirshfeld partitioning scheme,61,223 which provides a morerobust description of ionic systems than the original Hirshfeldpartitioning.224 Similarly, Bucko et al. introduced this schemeinto the TS model,86,87 and demonstrated that it substantiallyimproves the performance of DFA+TS for systems with stronglocal charge transfer. In these modified Hirshfeld schemes, theoriginal free-atom reference densities nA

free are modified intogeneralized reference densities n0,A, which themselves dependon the partitioned fragment densities nA. This makes eq 51 aself-consistent equation that needs to be solved in an iterativefashion. Several suggestions were made for the construction ofn0,A in terms of the fragment densities nA.

225 The most commonapproach, used in the modified XDM and TS methods, is toconsider n0,A as a linear combination of the free-atom and free-ion reference densities for a given species, where the

coefficients are determined in such a way that the charges ofthe Hirshfeld density nA and the reference density n0,A areequivalent.

4.8. Many-body dispersion framework

So far, we have covered three of the four classes of methodswhich arise from the independent use of the second-ordertruncating and coarse-graining approximations. In this section,we conclude this overview by discussing the recent many-bodydispersion (MBD) framework, which represents a class ofnontruncated (akin to the RPA-based methods) and coarse-grained (akin to the fragment-based methods) approximations.The central idea of the MBD model is the approximation of thedynamic response of a molecule or a material by that of a fullycoupled system of QHOs, where each oscillator corresponds tosome fragment of the system of interest. Apart from theresponse properties of the QHO model discussed above, themotivation for this approach comes from the fact that thecorrelation energy of such a system can be evaluated exactly bydiagonalizing the corresponding Hamiltonian, which is given by

∑ ∑

∑ ξ ξ

ω ξ

ω ω α α

∇= − +

+

ξH

T

12

12

12

(0) (0)

A AA A

ABA B A B A AB B

MBD2 2 2

A

(64)

where ξ = −m r R( )A A A A is the mass-weighted displacementof the oscillator A from its equilibrium position RA (i.e., thefragment center), ωA is the characteristic frequency of theQHO, and αA(0) is its static polarizability. The first two termsin this Hamiltonian represent the single-oscillator kinetic andpotential energy terms, respectively, while the third termcorresponds to the interoscillator coupling. In the context ofvdW interactions, this model Hamiltonian was first introducedby Bade226,227 in 1957 and used later by Mahan,228 Lucas,229

Renne and Nijboer,230 and Donchev231 in the investigation ofvarious qualitative aspects of long-range interactions betweenparticles. In recent years, a renewed interest in this approachwas initiated by the introduction of a general scheme forparametrizing and range-separating this Hamiltonian to obtaina universal and quantitative model for treating the energetic55,88

and structural232 effects of vdW interactions in molecules andmaterials. In this section, we briefly review how the MBDHamiltonian emerges from the long-range correlation energyexpression in eq 23, discuss several properties of this approachin the context of other vdW methods, and finally describe therange-separated MBD method in particular. We note that theterm “MBD” may refer to the general framework (Section4.8.1) as well as a specific method within this framework(Section 4.8.2).

4.8.1. Many-body dispersion framework from theACFD formula. The starting point for obtaining the MBDHamiltonian as a model for electron correlation is the coarse-grained version of the exact expression for the long-rangecorrelation energy in eq 37. In addition to the coarse-graining,MBD also approximates the short-range polarizability in thatformula by a local effective polarizability, which leads to thefollowing expression:

∫∑ ∑ απ

= − − ⟨ ⟩=

∞ ∞E

nu

uT( 1) d

2Tr[ ( ) (i )]

n

n

AB

nABc,lr

MBD

2 0eff lr

(65)



4738


Although the local approximation to αsr(r, r′) in the MBDframework is shared with both the nonlocal functionals andpairwise methods, it is in general not required here that αeff(r)be isotropic. We note that, here, the long-range correlationenergy is expanded into an infinite series, and such anexpansion could be converted to the class of pairwise methodsdiscussed in the previous section if truncated at second order.To proceed further, the dynamic polarizability αeff(iu) isapproximated by the corresponding expression for a QHO ineq 28. This set of approximations then enables one to performthe integration in the frequency domain in eq 65 analytically,233

leading to the so-called plasmon-pole formula for thecorrelation energy,

∑ ω ω= −=

E12

( )p

N

p pc1

3

(66)

where N is the number of fragments (depending on the systemand the choice of partitioning), the sum is over all 3Ncharacteristic frequencies of the system, ωp are the frequenciesof the fully interacting system (corresponding to α1), and ωpare the frequencies of the noninteracting system (correspond-ing to α0).The plasmon-pole formulation of the correlation energy is a

general approach that can be used for many nontruncatedapproximations to the ACFD formula, in particular the class ofRPA-based methods discussed in Section 4.5. In thisexpression, the analytical integration in the frequency domainis essentially performed by transforming the integral overimaginary frequencies iu to the imaginary part of the integralover real-valued frequencies u. In the latter form of the integral,αeff,A(u) directly describes the absorption spectrum of a systemwhich is characterized by delta functions located at theexcitation frequencies for both the interacting and non-interacting cases. Therefore, the integral over real-valuedfrequencies is reduced to a sum of the excitation frequencydifferences in eq 66. It can also be shown that the plasmon-poleexpression can be obtained as a difference between the groundstate energy of a system of coupled QHOs and the ground stateenergy of a system of uncoupled QHOs, thereby establishingthe equivalence between the plasmon-pole formulation and theHamiltonian formulation in eq 64.233 In other words, whereasthe plasmon-pole expression is only an approximation to theexact correlation energy of the full electronic system, it is in factexact for the correlation energy of a system of QHOs coupledwithin the dipole approximation.The Hamiltonian formulation of the long-range correlation

energy gives a useful insight into the nature of vdWinteractions. In the noninteracting limit (λ = 0), the chargefluctuations are fully localized on the individual fragments; as λis increased toward the interacting limit (λ = 1), these localizedfluctuations couple to form plasmon-like delocalized chargeoscillations. These delocalized fluctuations have lower charac-teristic frequencies on average than their localized counterparts,and hence the coupling decreases (stabilizes) the total energy ofthe system.4.8.2. Range-separated many-body dispersion meth-

od. At first, Tkatchenko et al. developed a MBD method basedon fully screened atomic polarizabilities,55 which lead to adouble counting of the long-range correlation effects in somecases. Later, this was avoided by using a range-separatingapproach, in which the atomic polarizabilities are screened bythe surrounding atoms only at short range.88 Here, we describe

only the latter approach, termed MBD@rsSCS, where “rsSCS”stands for range-separated self-consistent screening. To define aparticular method within the MBD framework, one needs tospecify the partitioning scheme, the models for the staticpolarizabilities αeff,A(0) and the noninteracting characteristicfrequencies ωA, and the range-separating function f(R) whichdefines Tlr. The MBD@rsSCS method partitions the electronicsystem into atomic fragments by means of the Hirshfeldscheme, which is also used in the XDM and TS pairwisemethods. The effective atomic polarizabilities αeff,A(0) are thenobtained in a two-step manner. First, the volume-scalingapproach of the TS method is applied to the reference free-atom polarizabilities, while the characteristic frequencies arekept unchanged. Second, the Dyson screening in eq 18 withonly the short-range part of the dipole potential is applied tothe atomic dynamic polarizabilities, which produces the short-range nonlocal polarizability αsr,AB(iu). The range separationused in this second step is the same that is later used to defineTlr, which is used to obtain the long-range correlation energy.In this regard, the short-range response and the long-rangecoupling are constructed consistently in the [email protected] obtain the local effective dynamic polarizabilities of the

atomic fragments, αsr,AB(iu) is then summed along onefragment coordinate, αeff,A = ∑Bαsr,AB, which is equivalent toeq 25 for the coarse-grained case when the neighborhood istaken to be the entire system. Finally, the fragment polar-izabilities are spherically averaged, and in this regard, theMBD@rsSCS scheme does not use the full flexibility of theMBD framework, which in general supports anisotropic localeffective polarizabilities. To fully specify MBD@rsSCS, therange-separating function f(R) is of the same type as thedamping function in eq 58 used in the TS method, and isparametrized on a reference set of small- and medium-sizedvdW complexes at various binding distances. The range-separating functions f(R) used to define Tlr are in general muchsmoother than the damping functions g(R) of the pairwisemethods, as the latter are in fact the square of thecorresponding f(R). With all relevant quantities determined,the MBD@rsSCS long-range correlation energy is thenobtained by diagonalizing the general MBD Hamiltonian ineq 64 and evaluating the plasmon-pole formula in eq 66. Buckoet al. derived and implemented a formulation of MBD@rsSCSfor periodic boundary conditions in reciprocal space, whichresults in faster and more easily converged calculations.234

4.8.3. General remarks about MBD and n-bodyexpansions. As briefly discussed in Section 4.4.2, the infiniten-th order expansion of the long-range correlation energy in theMBD framework is not equivalent to the general (and finite) n-body expansion, where the latter is often used to discussbeyond-pairwise corrections to the pairwise methods. Here, wediscuss the correspondence between various higher-order termsof the gn(R)Cn/R

n type and the n-th order terms in MBD inmore detail.To go beyond the pairwise approximation, the three-body

(triple-dipole) ATM term of the form g9(R)C9/R9 is sometimes

used in the class of pairwise models. For the n = 3 case, the n-body and n-th order expansions are equivalent, and hence, thesetriple-dipole terms are contained in the third order of MBD.However, the origins of the corresponding terms differsignificantly in MBD and in the beyond-pairwise models andso does the effective contribution of these terms to the long-range correlation energy. In particular, the three-body term in



4739


the beyond-pairwise methods is constructed independently ofthe pairwise term; that is, the C9 coefficients are not in exactcorrespondence with the C6 coefficients (via the generalCasimir−Polder integrals) and the damping function g9contains parameters that are independent of the parametersin g6. Therefore, the contributions of all the higher n-bodyterms (n ≥ 3) in such a scheme influence the parametrizationof the three-body term. As such, the resulting three-bodyenergetic contribution effectively captures the effects of thesehigher-order terms, thus raising the question of transferability.In contrast, the three-body triple-dipole term arises automati-cally in the MBD framework and is constructed with the sameatomic polarizabilities and long-range potential as the second-order dipole−dipole term. In other words, the range-separatingfunction f and the polarizability αeff,A are the same for thesecond, third, and higher orders. Furthermore, the higher-ordercontributions in MBD have typically an alternating sign, asgiven by eq 65, and hence their sum, that is, the differencebetween the total long-range correlation energy and thesecond-order approximation to it, can be significantly smallerthan the third-order contribution. Finally, the triple-dipole termis an example of an n-th order term acting between n bodies, yetthere is also the other class of n-th order terms acting betweenm < n bodies. For m = 2, these terms correspond to the higher-order contributions to the two-body energy and these are notaccounted for explicitly in pairwise approximations; forexample, the lowest 4-th order two-body term wouldcorrespond to the C12/R

12 term in the language of dispersioncoefficients.The other class of higher-order Cn/R

n terms that come fromhigher-order multipoles of the polarizability (dipole−quadru-pole, quadrupole−quadrupole) can be in principle obtained inthe MBD framework in three ways. First, the higher multipolescan be partially captured in the parametrization of the range-separating function, as was the case with pairwise methods.Second, additional oscillators and coupling potentials describ-ing quadrupole oscillations could be added. The contributionsof such oscillators would reduce to the higher-order Cn/R

n

terms when evaluated at second order. Third, the highermultipole terms could be evaluated as a perturbation to thedipole-coupled Hamiltonian due to the full Coulomb couplingby means of the perturbation theory. We note that none ofthese approaches have been yet introduced in a workingmethod.

5. MODELS FOR VDW INTERACTIONS: APPLICATIONSIn this section, we continue with an assessment of thenumerical performance of vdW models in treating systems forwhich vdW interactions play a key role. First, we discussbenchmark databases that consist of small noncovalently bounddimers, molecular crystals, and supramolecular complexes. Wethen follow with a brief survey of other important results forselect systems, focusing on structural and electronic propertiesbeyond binding and cohesive energies. This section is thenconcluded with a concise review of recent ab initio moleculardynamics simulations that use approaches based on firstprinciples for describing vdW interactions.Critical evaluation of exchange-correlation functionals

through benchmarking and extensive testing plays a centralrole in DFT. Such an approach is particularly crucial for densityfunctionals with a higher degree of empiricism, but served alsoto identify the robustness of many first-principles functionals,for which theoretical reasons were often found only later. A

similar situation can be found in the field of vdW methods,where critical assessment is essential for the more empiricalapproaches, but is of great value even for the first-principles-based methods covered in this review. There are several reasonswhy a quantitative assessment is needed, as opposed to aqualitative evaluation. First, it provides evidence for the level ofuncertainty in predicted quantities that one may expect whenusing a particular method for a particular class of systems. Thisbecomes ever more important as computational chemistry andmaterials physics play an increasingly predictive, rather thansupportive, role to experiment. Second, quantitative evaluationcan identify both specific and systematic issues of a givenmethod and hence provide guidance for further improvement.This is particularly important as quantitative differences in therelative energies of molecular systems and materialstheprimary results of electronic structure calculationsoftentranslate into qualitative differences in derived quantities suchas structural, thermodynamic, and response properties. In thisregard, the systematic verification of a given vdW method isusually achieved through comparison against the results ofhigher-level (more costly and more accurate) theoreticalmethods or to experimental results with sufficient resolution.The presented vdW methods consist of many ingredients

(models for response, partitioning schemes, range separationapproaches, treatments of higher-order contributions) which allcontribute to their overall performance. As a result of thiscomposition, evaluations of vdW methods are often interpretedin terms of these individual ingredients. However, it is notalways clear to what degree the individual choices for theingredients contribute to the overall difference in performancebetween different methods. Detailed analyses of the perform-ance in this regard are rare and remain to be performed formany popular vdW methods, even on simple model systems.This presents one clear path forward for the development ofmore robust and transferable vdW models, in which suchcomparative analyses can be used to identify the bestingredients from individual models, which can then becombined to form new methods with better performance.

5.1. Benchmark databases

In this section, we briefly discuss four benchmark databases fornoncovalent interactions that were used to test many of themethods reviewed here. The tradition of such high-qualitynoncovalent databases started with the S22 test set of Jurecka etal.,251 which contains 22 binary complexes with a mediannumber of atoms of 19, molar mass of 131 g/mol, and bindingenergy of 4.7 kcal/mol. The S22 set constitutes a representativesample of hydrogen-bonded, vdW bound, and mixedcomplexes. A similarly designed but more extensive set of 66dimers (the S66 test set), with a median number of atoms of20, molar mass of 126 g/mol, and binding energy of 4.1 kcal/mol, was published several years later.252 The referenceenergies in both these databases were obtained using high-level coupled-cluster calculations with single, double, andperturbative triple excitations (CCSD(T)) extrapolated to thecomplete basis set (CBS) limit, which represents the gold-standard electronic structure method in quantum chemistry.The benchmark C21 database240 extends the analysis tocondensed-phase systems via 21 molecular crystals withreference binding energies obtained from experimentalsublimation enthalpies. This work was later extended andrefined to cover 23 molecular crystals in the X23 database,253

with a median number of atoms per molecule of 10, molar mass



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of 89 g/mol, and binding energy of 7.3 kcal/mol. The finaldatabase considered in this section, the S12L set, deals withlarge molecules and consists of 12 supramolecular complexeswith a median number of atoms of 134, molar mass of1060 g/mol, and binding energy of 28 kcal/mol, with referencebinding energies obtained from experimental Gibbs freeenergies of association, or alternatively from quantum MonteCarlo calculations.254,255 Although the available data in theliterature are far from complete (i.e., binding energies of everymethod discussed in this review on every one of these databaseshave not been calculated to date), Figure 7 still illustrates thefollowing general trend: the differences in the predictionsamong the individual vdW methods increase as the size andcomplexity of the systems in question grow. In other words, thereliability of the vdW methods seems to decrease withincreasing system size and complexity. However, this trend isalso accompanied by a corresponding decrease in the accuracyand reliability of the underlying benchmark energies. In thefollowing, we elaborate on various reasons for this observation,accompanied by an illustrative analysis of the differencesbetween particular methods on select systems from thesedatabases.The S22 and S66 databases are relatively similar in design

and will be discussed together. Figure 7 shows that the errorranges corresponding to the S22 and S66 databases arenarrowest among the four benchmark sets. (We note that theerror distribution is narrower for the S66 set due to itsincreased size with respect to the S22 seta larger “sample”leads to a smaller variance between methods.) The consistentperformance observed for these data sets illustrates that, afterintense development throughout the past decade, the majorityof popular vdW methods have managed to capture this first-rung level of complexity of nonbonded interaction energies. Inthe same breath, however, this does not mean that these simplesystems cannot provide further insight into the performance ofa given vdW model. On the contrary, a detailed analysis of aparticular system can reveal valuable information about therelative contributions of different physical phenomena to thetotal binding energy as well as the degree to which those

phenomena are accounted for in a given model. We will focuson two systems from these data sets, namely the benzene dimerand the adenine−thymine complex, each of which will serve toillustrate different potential complications that may occur whendescribing vdW interactions.The benzene dimer has been the primary model system for

noncovalent interactions over the past 40 years.256−263 Inparticular, the fine balance between the Pauli repulsion,electrostatic (quadrupole−quadrupole) interactions, and vdWdispersion results in an intricate potential energy surface (PES)with many local minima.264 The nearly degenerate T-shapedand parallel-displaced (PD) conformations of the benzenedimer are contained in both S22 and S66 sets, but whether aparticular method reproduces this degeneracy is naturally notreflected in the corresponding statistical errors. Another reasonwhy these two geometries are particularly interesting isprecisely the fact that they are two probes of the same PESand hence the number of potential sources of error in theirdescription is limited. In this system, a physically sounddescription of the anisotropy in the vdW interactions becomescrucial for quantitative prediction of the finer details of the PES.As discussed in Section 2, the PD conformation exhibitsalignment among the distinct in-plane charge fluctuations,whereas such fluctuations are orthogonal in the T-shapedconformation (see Figure 5). To capture these effects, a givenvdW model has to correctly describe both the monomerscreening and polarizability as well as the intermolecularcoupling of these anisotropic fluctuations (see the short-rangeαsr(r,r′) and its long-range coupling in eq 22). This difficulttask may be one of the underlying reasons why many second-order vdW methods, which neglect such effects, and even RPA-based methods fail to predict this degeneracyboth versions ofvdW-DF either predict a difference of 0.6 kcal/mol (20%) atthe true equilibrium geometries36 or overestimate the bindingdistances by as much as 0.5 Å;265 the error of the pairwise TSmethod on the PD conformation is 0.8 kcal/mol,50 and allRPA-derived methods reviewed in Section 4.5 overbind the PDconformation by ∼0.7 kcal/mol.148 In contrast, the MBDmethod, which explicitly accounts for the intra- and

Figure 7. Overall performance of vdW methods on four benchmark data sets. The selection includes the methods XDM, MBD, TS, VV10, RPA,rPT2, vdW-DF, and vdW-DF2. When a vdW method specifies a particular exchange-correlation functional, results with that functional were taken;otherwise, the PBE functional was considered. The blue and red data points reflect the mean and range of mean absolute percentage errors (MAPE)and mean absolute errors (MAE), respectively, on a given data set. The dotted line denotes 1 kcal/mol. The displayed structures are illustrativeexamples of systems of a given size and complexity. Data are collected from refs 36, 62, 88, 98, 142, 148, 157, 204, 205, and 235−250.



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intermolecular many-body effects, reduces the error of the TSdescription to only 0.2 kcal/mol. Exceptions to this cleardistinction are some parametrizations of the XDM model238

and the VV10 functional,62 which predict the degeneracy withina 10% margin, while being only second-order approximations.However, the XDM predictions substantially depend on aparticular choice of the exchange-hole model, which suggeststhat other effects may effectively cancel out with the higher-order contributions in this particular system.The adenine−thymine complex can be considered as a

prototypical biological system and is represented twice in theS22 set: first, as a vdW bound stacked conformation; second, asa classical Watson−Crick nucleobase pair. While the first case issimilar in character to the PD benzene dimer, the latter casecontains strong directional hydrogen bonds in which severaltypes of noncovalent binding contribute (electrostatic inter-actions, induction, and dispersion), posing a difficultchallenge.266−268 Of all these interactions, only dispersion isdescribed by vdW models whereas the other components aretreated at the level of the underlying exchange-correlationfunctional. In this regard, an accurate description of electro-static interactions relies predominantly on the correct assign-ment of permanent moments to the individual interactingmolecules. For the case of induction, this must also beaccompanied by a consistent treatment of the molecularpolarizabilities. In DFT, all these quantities are predicted onlyapproximately, yet these inaccuracies are rarely decoupled oreven discussed separately from the errors introduced by thevdW models. A similar conclusion can also be made for thePauli repulsion, the magnitude of which can be relatively largecompared to the total binding energy, yet its description viadifferent exchange functionals can vary greatly. The decom-position of the total binding energy of hydrogen-bondedcomplexes into the individual components varies greatly withdistance,269 which may potentially lead to decreased trans-ferability between different system sizes if some of the errorsmentioned above are effectively incorporated into the short-range behavior or mutually compensate at a certain size range.Indeed, while the Watson−Crick pair is described relativelywell by most vdW methods, with differences within 1.5 kcal/mol (10%), the relative errors across the range of hydrogen-bonded complexes of different sizes vary significantly.270 Theseobservations suggest that at least a part of the remaining errorsin the predicted binding energies can not only be attributed tothe vdW methods themselves, but rather also to the underlyingdensity functionals.The fact that certain pairwise dispersion corrections are

seemingly able to achieve good accuracy for molecular crystalsstems from their more empirical nature and parametrization onlarge molecular systems. Essentially all pairwise methods (D3,XDM, TS) can be made accurate enough for the X23 databaseby adjusting two empirical parameters.253 However, suchadjustments just serve the purpose of mimicking some of thedielectric screening effects and can only be successful forrelatively symmetric systems. The true nature of dielectricscreening in arbitrary systems and geometries can only becaptured with explicit many-body approaches.One large class of complex molecular environments consists

of molecular crystals. The X23 database contains 23 suchsystems with a similar profile of the local intermolecularbonding as in the S22 and S66 sets (ranging from hydrogen-bonded to vdW bound and mixed complexes). While this mightlead one to expect a comparable range of errors for the different

vdW methods, Figure 7 reveals that there is in fact a significantdecrease in the reliability among the methods when treatingmolecular crystals as opposed to gas-phase molecular dimers.As a model case for discussion, consider the benzene crystal,which is a natural extension of the benzene dimer from themolecular databases, yet it is significantly more compli-cated.271−274 For instance, the second-order, pairwise TSmethod overestimates the lattice energy by 3 kcal/mol, whiletreatment of the electron correlation to all orders within theMBD method results in a prediction within 1 kcal/mol (thougha slightly better result is obtained with PBE0 compared to PBE,suggesting small error cancellation in play). Similarly,considering only individual pairs of benzene molecules in thecrystal within SAPT(DFT) or only double excitations withinthe coupled-cluster method leads to overestimation of theenergy by 2 kcal/mol, while increasing the degree of correlationdecreases the error under 1 kcal/mol. In contrast, the DFT-D3method achieves 1 kcal/mol without considering any many-body effects, and their addition worsens the results byapproximately 1 kcal/mol.275 There are several possible reasonsfor the increased complexity and less predictable behavior of acrystal compared to the molecular dimer. First, the benzenecrystal contains individual pairs of benzene molecules in manydifferent orientations, each of which needs to be described withthe same level of accuracy; as such, the molecular crystal servesas another indirect probe into the quality of the dimer PES.Second, the solid state introduces stronger intermolecularpolarization in addition to the intramolecular polarization thatwas already present in the isolated molecular complex; thiscoupling of short- and long-range many-body effects furtherincreases the polarizability of a molecule within the crystal.Third, the presence of the three-dimensional crystal latticeeffectively screens the long-range dipole correlations, resultingin an overall decrease in the lattice energies. The totalcontribution of the latter two competing long-range effects canrange from negative to positive in the prediction of latticeenergies and, hence, adds to the mixed conclusions regardingthe applicability of pairwise schemes and three-body correctionsfor the description of molecular crystals.To put these numerical differences into perspective, consider

that a correct ordering of crystal polymorphs necessary forsuccessful crystal structure prediction requires an accuracy ofonly fractions of a kcal/mol.7,78,276,277 More than half of thecompounds in a pharmaceutically relevant data set showpolymorphism,278 and the energy differences between poly-morphs are usually less than 1 kcal/mol and often even lessthan 1 kJ/mol (0.24 kcal/mol). The X23 database contains twopolymorphs of oxalic acid, for which the experimental energydifference between the α and β forms amounts to only 0.05kcal/mol.240,253,279 Yet, current vdW-inclusive first-principleapproaches yield energy differences between about −1 and 1kcal/mol. PBE+MBD has an absolute error in the difference ofonly 0.12 kcal/mol but predicts the wrong ordering,7 whereasPBE0+MBD correctly predicts the ordering with an absoluteerror of 0.22 kcal/mol, illustrating how subtle the differences inthe two polymorphs are. The three polymorphs of glycine havealso been studied with a variety of methods, of which onlyPBE0+MBD was able to correctly predict the relative stabilityordering as observed in experiment.78 Molecular crystals areusually studied experimentally at finite temperature andpressure, requiring that free energies are considered insteadof bare lattice energies. For instance, form I of the aspirincrystal is much more abundant in nature than form II, yet both



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forms have almost degenerate bare lattice energies. The higherstability of form I could only be explained by consideringharmonic free energies at room temperature.280 Thermal andpressure effects are not only crucial for the stability underthermodynamic conditions, but are also essential for the unit-cell structure itself and a variety of response properties, such asvibrational spectra and elastic constants.281 The most recentstructure prediction blind test for organic crystals282 featured apractically relevant and highly polymorphic system involving aquite flexible molecule. This flexibility leads to a much largernumber of local minima within a narrow energy windowcompared to crystals containing only rigid molecules. Whilemost polymorphs have been found by at least one participantwithin the top 100 predicted structures, the relative ordering ofthe polymorphs is still challenging. That said, inclusion ofmany-body effects in the description of vdW interactions andaccounting for free energies clearly improves the relativestability prediction.A related class of solid-state systems, which shares similar

features to the vdW binding motifs in molecular crystals,consists of layered materials. To date, there is no systematicbenchmark database for such systems; however, we can stilllearn from particular representative systems. We will considergraphite, one of the prototypical layered materials for whichthere is an abundant number of both experimental andtheoretical results,283−288 regarding both its energetic andstructural properties.289 Graphite features the same set ofmany-body effects as found in molecular crystals, but theseeffects are amplified by the extreme difference in responseproperties between the in-plane and out-of-plane directions inthe graphene layers. The correlation of the long-rangefluctuating dipole waves within each layer decays very slowlywith distance, as demonstrated by the slow convergence of thebinding energy of multilayer graphene.9,22,92 On the otherhand, these correlations are nontrivially screened by the innerlayers, resulting in a competition between these two effects ascaptured by the various correlation orders in eq 23. Althoughapproximate, RPA describes these effects at least on asemiquantitative basis, and so can serve as a first-orderreference with the predicted binding energy of 1.11 kcal/molper carbon atom. Many vdW methods, including all nonlocaldensity functionals and most atom-based pairwise methods, fallshort from this result, with predictions ranging from 0.46 to1.86 kcal/mol per carbon atom. When structural data such aslattice constants and bulk moduli are considered as well, onlymethods that treat the dipole correlations to all orders arecapable of matching the experimental reference results. Forinstance, while both the XDM and MBD methods providegood estimates of the interlayer lattice parameter and thebinding energy, XDM underestimates the curvature of theenergy minimum more than 2-fold, whereas MBD predicts avalue within 15% of experiment.12,289

As a final benchmark data set, we consider the S12L (S30L)set of 12 (30) supramolecular complexes.246,290 In thesesystems, the total binding energies are mostly dominated byvdW interactions and hence the choice of a particularunderlying density functional is of less importance than inthe previous data sets, which makes this database an idealplayground for analyzing vdW methods. As a prototypicalexample, we will discuss the so-called buckyball catchercomplex, as it was studied most extensively both theoreticallyand experimentally. A prominent feature of this system is thestrong anisotropy in the polarizability of the pincers of the

catcher, which results in nonlocal dipole fluctuations spanningthe whole complex. This situation is partially analogous to themechanisms occurring in graphite and graphene, but thescreening of the dipole correlations is much weaker in this case(in contrast to graphite) and there is no cancellation of thehigher-order effects due to symmetry (in contrast to graphene).This results in the nonlocal fluctuations having a largedestabilizing contribution to the binding energy, which canreduce this quantity by as much as 10 kcal/mol with respect toa pairwise-additive decomposition of the long-range correlationenergy in the TS, VV10, and bare XDM model, resulting in 2-to 4-fold reduction in the mean relative error.88,246 Byaccounting for the effect of such fluctuations (either inherentlyas in MBD or by higher-order corrections in the XDM, VV10,or DFT-D3 models), the reference binding energies can bepredicted within 2 kcal/mol (5%), leading to many-body effectsplaying a crucial role in the binding of supramolecularcomplexes.Apart from the established data sets discussed above, we

briefly list others that have not seen broader adoption. All datasets mentioned below were benchmarked using the CCSD(T)/CBS method. The X40 set comprises complexes of smallhalogenated organic molecules and provides a good test of howwell a method handles strongly ionic and charge-transfercompounds.291 The L7 set consists of larger organic complexesthan those included in the S22 and S66 sets, but smaller thanthe supramolecular complexes in the S12L set.292 The 3B-69data set is of particular interest to the discussion here, as itbenchmarks molecular three-body contributions to interactionsenergies.293 An interesting approach was used by Faver et al.,294

who dissected a protein−ligand complex 1HSG into 21 smallerfragment dimers bound by noncovalent interactions.A final point in the discussion of the benchmark data sets

concerns the reliability of the underlying reference data. For theS22 and S66 sets, well-converged state-of-the-art electronicstructure theory methods were used by several groups to obtainthe benchmark data. These efforts leave very little room forpotential improvement, with the possible exception of almostdegenerate structures such as the aforementioned benzenedimer conformations, where an accuracy of less than 0.1 kcal/mol is needed to decisively determine the most stablegeometry. The X23 set presents a more difficult challengenot only for the approximate vdW methods, but also for thecomputational techniques used for obtaining the benchmarkenergetics. The common method of deriving these energiesfrom experimental sublimation enthalpies involves severalapproximations, the particular choice of which can lead todifferences as large as 1 kcal/mol.240,253 In contrast, Yang et al.recently calculated the lattice energy of a benzene crystal at theCCSD(T) level with an estimated uncertainty of only ∼0.1kcal/mol,295 which may mark a future path toward obtainingsuitably accurate benchmark energetics for molecular crystals.Finally, the S12L data set is a difficult problem due to the sheersize of the complexes in question and the fact that they havenot been isolated experimentally in the gas phase. The originalreference binding energies were obtained by subtracting thevibrational and solvation contributions from the experimentalGibbs free energies of association.254 The largest uncertainty inthis approach comes from the solvation contributions that areobtained by using semiempirical models with uncertainties thatcan exceed ±5 kcal/mol. Indeed, the differences between theexperimentally derived binding energies and those obtained bythe diffusion quantum Monte−Carlo (DQMC) method



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(statistically converged to 1 kcal/mol) are similar in magnitudeto the uncertainties in the solvation contributions.Besides systematic benchmark databases, much work has also

been done on select systems bound by vdW interactions, whichare in some ways representative of a wider class of molecules ormaterials. The vdW-WF method was tested on small dimersand adsorption of small molecules on various surfaces in severalstudies.296−299 Sorescu et al. performed an extensive study ofmany different methods on a set of 26 nitrogen containingmolecular crystals.300 With the benzene dimer being such a wellstudied complex, its natural extension is the adsorption of asingle benzene molecule on a surface. Such a system has anappeal both from an experimental point of view as a model forheterogeneous catalysis, as well as from a theoretical stance, asit combines contributions coming from electrostatic inter-actions (benzene quadrupole moment) and long-range polar-ization induced in the surface. Typical choices for the surfaceare a single graphene sheet301,302 and metallic surfa-ces.100,208,303−305 In the latter case, the surface chargefluctuations of the metallic electrons have a large effect onthe polarizability of the adsorbed molecule and substantiallychange its response compared to the gas-phase case, thusinfluencing the resulting binding energy. However, the metallicresponse has an intricate many-body character, and a generalfirst-principles approximate method for the description of vdWinteractions in hybrid interfaces is yet to be seen. A further stepin complexity can be taken by studying the interaction of asingle graphene sheet with a metallic surface.143,306−310 Here,the long-range response of a covalent two-dimensional zero-gapsemiconductor is coupled with the response of surfaceplasmons of the metal, presenting a severe test for vdWmodels. Maurer et al. summarized the performance of differentmethods for graphene/Ag(111) binding energy, with resultsranging from 0.77 to 1.67 kcal/mol per carbon atom.311 Thiswork demonstrates the sheer complexity of describing vdWinteractions in complex nanostructured materials.

To fully assess the complicated nature of realistic systemswhen testing approximate models, they need to be probed inmany different ways, either by using different configurationsand conformations, or by investigating other properties besidesbinding energies, such as polarizabilities, stress tensors, andwork functions or dynamical properties. Such studies are thetopic of the following sections.5.2. Structural and electronic properties

Up to this point, we have been concerned primarily with thebinding energy effects of long-range electron correlation, asthose are typically large in relative magnitude. In this section,we discuss the effects of vdW interactions on structural andelectronic properties. While the energy ordering of differentconformers of the benzene dimer poses a significant challengeto all vdW models, an even more sensitive probe into therobustness of a given model and a gateway to obtainingequilibrium structures as well as dynamical properties ofsystems at nonzero temperatures is provided by derivatives ofthe energy with respect to the nuclear coordinates, −dE/dRA.Furthermore, the vdW models discussed here all directlydepend on the electron density and, therefore, via thefunctional derivative of the energy, δE/δn, and the Hohen-berg−Kohn variational principle, necessarily influence theelectron density and hence all electronic properties. Whereasthe structural effects have been recognized since the beginningof the empirical treatment of vdW interactions, the electroniceffects have been largely neglected and it is only in recent yearsthat they have become the topic of investigation.Although searching for local minima on a PES forms the

basis of most computational chemistry studies, it is used tosystematically evaluate the accuracy of vdW methods to a muchsmaller degree than the corresponding binding energies.Recently, several researchers attempted to close this gap.Witte et al. investigated the performance of may electronicstructure methods in predicting the equilibrium geometries ofsmall to large molecular complexes.312 They found that the

Figure 8. Structural changes induced by vdW interactions as predicted by several methods. D3 is the semiempirical DFT-D3 model of Grimme etal.58 (a) Errors in monomer distances of ten locally stable conformations of the benzene dimer with respect to reference DFT/CCSD(T) results.264

(b) Predicted structures of 76 conformers of 5 isolated small peptides using different methods are aligned with reference MP2 structures byminimizing the root-mean-square deviation (RMSD) of the atomic coordinates. The minimized RMSDs are represented using their mean, quartiles,and whiskers covering data within the 1.5-fold quartile range. The outliers are plotted as individual points. (c) The equilibrium geometries of thebuckycatcher−C60 complex as predicted by PBE+MBD (blue) and PBE+D3 (red). [Adapted with permission from Blood-Forsythe et al.232].



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nonlocal density functionals vdW-DF2 and VV10 performreasonably well on small systems with respect to CCSD(T);however, the relative errors associated with vdW-DF2 increasewith system size, whereas the accuracy associated with VV10was more transferable to larger systems. This issue can bepossibly traced back to the way that the vdW-DF2 functional isconstructed, as explained in Section 4.6. Compared to theoriginal vdW-DF functional, the second version uses a largerdensity gradient contribution, leading to more rapid cutoff ofthe tail densities in the polarizability density functional andhence decreased C6 coefficients. When coupled with a moreattractive semilocal functional than the original revPBE, thisapproach may reach better accuracy than vdW-DF on smallsystems, but as the system size grows, the part of the bindingthat comes from the short-range diminishes, while theunderestimation of the long-range contribution begins tomanifest itself.Blood-Forsythe et al. investigated the performance of the

MBD and TS methods together with the DFT-D3 model in thedescription of the PES of the benzene dimer, a set of smallpeptides, and a large supramolecular complex.232 For thebenzene dimer, they found an almost perfect agreementbetween MBD and the reference structures of CCSD(T)quality (RMS error of 0.01 Å), whereas both the pairwise TSand D3 methods yield errors that are several times larger inmagnitude (Figure 8a). Comparing the differences betweenMBD and TS for the individual conformations, it can be seenthat the pairwise approximation shortens the binding distancesin stacked conformations, whereas they are increased in the T-shaped structures. This can be explained by the anisotropy inthe polarizability of the benzene molecule, which is neglected inthe pairwise approximation. Because the destabilizing many-body effects are pronounced in cases where the largercomponents of the polarizability are aligned, the bindingdistances are shortened when these effects are neglected,consistent with the observed behavior. Similar trends were alsoobserved in the prediction of the secondary structures of selectsmall peptides (Figure 8b), governed by subtle intramolecularnonbonded interactions. The fragment separation in thesupramolecular buckycatcher complex was found to decreaseby 0.2 Å with the MBD model (Figure 8c), demonstrating adistinct effect of many-body vdW interactions on the structureof larger and more realistic molecular systems of interest.After discussing the nuclear degrees of freedom of the total

molecular Hamiltonian, we turn to the electronic degrees offreedom. The long-range electron correlation energy is typicallyaround 5 orders of magnitude smaller than the total electronicenergy, and accordingly the electron density changes associatedwith vdW interactions are of a similar relative magnitude withrespect to the total electron density. This is the reason whyvdW methods can be applied in a perturbative fashion as an aposteriori correction to the total energy, without affecting theelectron density, which is obtained a priori from the self-consistent Kohn−Sham equations. On the other hand, for anymethod that is formulated as a functional of the density, Ec[n],one can derive the corresponding contribution to the exchange-correlation potential, δEc/δn, and include it in the Kohn−Shamequations, resulting in subtle changes in the electron density. Infact, the Hellman−Feynman theorem shows that it is thischarge polarization induced by the long-range correlation ofquantum electron fluctuations that acts as the origin of the vdWforces on the nuclei.39,313 Self-consistent implementations havebeen recently provided for several vdW methods discussed

throughout Section 4, including RPA,314 nonlocal densityfunctionals,163−165 as well as the TS15 and XDM315 pairwisemethods. In many of these works, the effects of self-consistencyon the binding energies are reported as negligible and, ingeneral, the electron density is polarized in such a way that itshifts slightly away from the atoms toward the intermolecularregions. However, these conclusions were reached on small,typically organic gas-phase molecular dimers. In contrast,Figure 9 shows that the charge polarization induced by vdW

interactions on a metallic surface depends significantly on theparticular metal: whereas the charge density is accumulatedabove the surface in the case of copper and silver, it is depletedin the case of gold. The effects of this charge redistribution canbe directly compared to experiment via the surface workfunction. This quantity is the energy associated with extractingan electron from the surface and is directly influenced by thedipole density of the surface, which is in turn connected to thecharge distribution along the direction normal to the surface. Asa result of the charge polarization shown in Figure 9, the workfunctions of the coinage metals can be changed by as much as0.3 eV due to long-range electron correlation.15 These workfunctions can then be additionally tuned by noncovalentadsorption of organic molecules on the metallic surface inhybrid organic−inorganic interfaces.15,303,316

5.3. Molecular dynamics

After considering the effects of vdW interactions on micro-scopic structural and electronic properties, we now considerhow these forces influence the macroscopic properties ofmaterials. As argued previously, a first-principles description ofvdW interactions is required to accurately treat the variety ofstructural, electronic, and environmental features that influencethe properties of systems found throughout biology, chemistry,physics, and materials science (see Figure 1). Ab initiomolecular dynamics (AIMD), in which PES is generated “onthe fly” from the electronic ground state without the need forempirical input, provides such a framework. The AIMDtechnique has been applied to many problems317−319 andallows for a quantum-mechanical treatment of the structure anddynamics of a given molecular system as well as its electronicand dielectric properties, and potential chemical reactions. Inthis section, we consider several examples of studies whichdemonstrate that the predictive power of AIMD simulations

Figure 9. Charge density polarization of metallic surfaces due to long-range electron correlation. Δn is the integrated difference between thetotal electron densities obtained by the PBE functional and the self-consistent PBE+vdWsurf method. [Reprinted with permission from ref15. Copyright 2015 American Physical Society.]



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depends crucially upon the accuracy of the underlying potentialand, in particular, the inclusion of and choice for the respectivevdW description.As a first example, we consider the effects of vdW

interactions on the structure and properties of liquidwater.323 Many AIMD studies demonstrated that vdWinteractions significantly influence the microscopic structureand macroscopic equilibrium density324−326 of ambient liquidwater. However, the influence of vdW forces on its propertieslargely depends on the particular vdW approach. For instance,the semiempirical pairwise DFT-D approach of Grimme57

tends to reduce the intensity of the first maximum in theoxygen−oxygen radial distribution function (gOO(r)) of liquidwater over a wide range depending on the underlyingfunctional.324,327,328 With the use of the nonlocal densityfunctionals from the vdW-DF family, one can obtain mixedresults in reproducing the second coordination shell in ambientliquid water depending on the choice of the underlyingexchange-correlation functional.329−331 On the other hand,AIMD simulations using a self-consistent (SC) exchange-correlation potential that accounts for both exact exchange (viathe PBE0 functional) and vdW interactions (via the pairwise TSmodel) are able to quantitatively reproduce the experimentallyderived gOO(r) of liquid water322 (see Figure 10a). In this case,the PBE0 hybrid functional reduces the amount of self-interaction error in the underlying exchange-correlation

potential and thereby weakens the hydrogen-bond strength inliquid water relative to a GGA such as PBE; in doing so,hydrogen-bonded molecules in the first shell move outward,resulting in a marked increase in the population of moleculeslocated in the interstitial region, i. e., the region between thefirst and second coordination shells (see Figure 10b). On theother hand, vdW forces, as described by the TS method, arenondirectional and strengthen the interactions between a givenwater molecule and the molecules in its first and secondcoordination shells (and stabilize broken hydrogen-bondconfigurations in general), which causes the molecules in thesecond shell to move inward and populate the interstitial region(see Figure 10b). In terms of dynamical properties, one alsoobserves a significant increase in the predicted diffusioncoefficient of liquid water when going from PBE (D =0.02 Å 2·ps−1) to PBE0+TS-vdW(SC) (D = 0.15 Å2·ps−1),which makes it much closer to the experimental value of0.23 Å2·ps−1.332 As such, both these effects work collectively toyield a microscopic description of liquid water that is lessstructured, more fluidic, and significantly closer to experiment.As a second example, we consider the use of vdW-inclusive

AIMD (via the optB88-vdW nonlocal functional) to map themolecular structure and energetics of liquid water films onlayered materials, such as graphene and hexagonal boron nitride(BN), to the complex transport of water at the nanoscale.333

Here, AIMD simulations reveal that the structures of the liquidwater films, substantially influenced by vdW interactions, are infact very similar for both graphene and BN; however, slightdifferences in the contact layer (that are not related to thewetting properties of these interfaces) give rise to a remarkablydifferent water slippage of these sheets (see Figure 11). This 3-

fold increase in the friction on BN with respect to graphene isnot captured by force-field-based molecular dynamics simu-lations, thus stressing the importance of a first-principles-baseddescription in predicting transport properties at complexinterfaces.As a third and final case, we consider the role played by vdW

interactions in stabilizing key helical domains in the secondarystructure of gas-phase polypeptides.102,335−338 In a recentAIMD study,334 it was shown that the inclusion of vdW forcescan explain the notable stability of polyalanine (Ac-Alan-LysH

+)

Figure 10. Individual and collective effects of vdW interactions andexact exchange on the structure of liquid water. (a) The oxygen−oxygen radial distribution functions, gOO(r), of liquid water obtainedfrom theory (via DFT-based AIMD simulations) and variousscattering experiments.320,321 (b) Contributions from the first andsecond shells of neighboring water molecules to the gOO(r) of liquidwater. The arrows indicate systematic shifts in the main peak positionsand intensities of the computed distributions with the addition of self-consistent vdW interactions and exact exchange in the underlyingexchange-correlation functional. [Adapted with permission from ref322. Copyright 2014 AIP Publishing LLC.]

Figure 11. Similar structure but different friction of water on grapheneand hexagonal boron nitride surfaces. (a) Average density profile of aliquid water film computed at the optB88-vdW level as a function ofthe height from a graphene (GRA) and hexagonal boron nitride (BN)sheet. (b) Comparison between the Green−Kubo estimate of thefriction coefficient of liquid water on GRA and BN (with shaded areasrepresenting the uncertainties in the coefficients). [Reprinted withpermission from ref 333. Copyright 2014 American Chemical Society.]



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helices up to a temperature of 725 K.339 Figure 12 illustrates thefact that AIMD simulations at the PBE and PBE+TS levels oftheory paint very different pictures of the dynamical helixstructure over a wide range of temperatures. For instance,AIMD simulations at 700 K with PBE+TS give a structure thatis comprised of both α and 310-helical motifs with an overallhelical structure that is preserved after 65 ps of simulation,whereas PBE predicts that the α-helical motifs quicklydisappear within 5−7 ps, in contradiction to the experimentalevidence.

6. CHALLENGES AND OUTLOOK

Recent developments of sophisticated methods for modelingvdW interactions have brought unprecedented insights into thenature of noncovalent interactions in complex molecules andmaterials. The development of vdW methods based on first-principles vastly extended the applicability of density-functionalcalculations in chemistry, biology, and materials science. Thecontinuing efforts to increase the accuracy and reliability ofthese methods will further improve our understanding andcapacity to model the noncovalent interactions in largemolecular systems.Despite all of these advances, a unified method that could

accurately model noncovalent interactions in a wide range ofmolecular and extended systems has yet to become available.All existing vdW-inclusive methods have their advantages andlimitations. Recognizing the limitations is perhaps the mostimportant area of current research, which will eventually enablesystematic developments toward a universally applicablemethod. Aiming toward this goal, we present here a set of

challenges which we deem as particularly important to addressto enable progress in the field of modeling vdW interactions:

• Is it possible to develop an efficient density-basedmethod for vdW interactions which would allow reliabledescription of the wide range of phenomena and systemsillustrated in Figure 1? Could such a unified method treatneutral small and large molecules, salts, ions, metals, andinterfaces between these systems, on equal footing?Besides developing improved methods, this goal alsorequires creating systematic data sets of validatedstructures, binding and cohesive energies, and responseproperties for a wide range of noncovalently boundsystems.

• It has been convincingly demonstrated that the many-body effects in the vdW energy are crucial for bothqualitatively correct and quantitatively accurate modelingof large molecular systems. Can simple rules of thumb bedeveloped that would allow design of molecular systemsthat exhibit desirable properties stemming from collectivemany-body effects? While explicitly correlated quantum-chemical methods allow achieving quantitative accuracy,insights are more easily obtained from coarse-grainedmodels.22

• As the molecular size grows, eventually quantumelectrodynamic or Casimir-like effects, not consideredby existing vdW-inclusive methods, become important:retardation due to the finite speed of light or thermalfield fluctuations, among others. Can we developmethods that seamlessly cross from the vdW into theCasimir regime? Can we devise rules based on system

Figure 12. Importance of vdW interactions in determining the thermodynamic stability of polypeptide helices. (a) Overall number of hydrogenbonds recorded throughout AIMD simulations at the PBE+TS-vdW and PBE levels of theory, with the α-helical bonds (red) and 310-helical bonds(blue) counted separately. (b) Snapshots of the helical conformations from the corresponding AIMD simulations. [Reprinted in part with permissionfrom ref 334. Copyright 2011 American Physical Society.]



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sizes and topologies for when such a crossover becomesnecessary?

• Can we develop efficient density-based methods formodeling vdW interactions in systems subject to bothhomogeneous and inhomogeneous electric fields,between excited-state (and ground-state) molecules, orin general molecules embedded in complex static orfluctuating environments? How can we properly includethe effect of a generalized frequency-dependent bath onvdW interactions in complex molecular systems?

• How do nuclear (classical or quantum) fluctuations atfinite temperature affect vdW interactions betweenmolecules? Do they compensate or accentuate many-body effects? Are there novel phenomena resulting fromthe coupling between quantum nuclear fluctuations andlong-range electronic correlations?

• The crucial role of vdW interactions in the structure andstability of molecular systems is convincingly establishedby now. What about the role of vdW interactions invibrational, mechanical, electronic, optical, and otherresponse properties of molecular systems?

• What is the role of solvents on intermolecularinteractions? This question combines the contributionof both thermal effects and electrodynamic fluctua-tions.340

We now proceed by raising methodological questionsrelevant for the development of improved methods for vdWinteractions and for better understanding of long-rangeelectronic correlation in general:

• Is there a systematic way to combine semilocal densityfunctionals with long-range models for vdW interactions?As density functionals become more farsighted and vdWmodels become more nearsighted, what is the optimalcrossover between semilocal and nonlocal approaches toelectron correlation? How can balance between many-body electronic effects in the semilocal functional and thenonlocal electron correlation be achieved?341

• Is there a systematic way to combine classical force fieldsthat approximately model bonding and electrostatics withmethods for long-range vdW interactions? For example,recent work by Martyna et al. lays the foundation forquantum-mechanical force fields based on quantumDrude oscillators.342−344 How much quantum mechanicscan (should) an empirical force field capture?

• Some of the vdW methods discussed in this reviewprovide a link between DFT, semiempirical density-functional approaches, and classical force fields. Forexample, the TS method was extended from DFT60 tothe tightbinding method210 and force fields.211,212 Cansuch flexibility be transferred to other vdW methods?

• What is the nature of nonlocal fluctuations contained inχ0(r, r′) and α0(r, r′)? How well can they be representedlocally? Could the freedom in the gauge of the α vectorfield be used to some advantage?

• Can narrow-gap or zero-gap materials be described witha local α0? Is the atom-dipole description, which is anefficient approximation for finite-gap materials, efficientalso for metals?

• Is there a clear-cut theoretical distinction between dipoleand charge fluctuations? Can charge fluctuations bemodeled efficiently with semilocal dipole fluctuationseven in metallic systems?

• How good is the single-oscillator Unsold approximationin general? Can we systematically generalize this modelto a many-body approach with multiple frequencies persite?

• How much can be achieved with finer coarse-graining ofthe response? Apart from mitigating the dipole-approximation error, can more complex electronic effectsbe encoded in finely grained local response functions?

• Is the equivalence of electronic χ and α transferable tothe coarse-grained (oscillator) descriptions?

• How can the anisotropic local polarizability beconstructed in both the atomic case and density-functional case? Can we use the directionality of thedensity gradient?

• Can we have truly nonlocal functionals for αeff(r) =∫ r′∈M(r) dr′ F(r, r′)?

• Can we have more exact constraints and limits on α[n]?Can we go to meta-GGA functionals for α[n]? Can wehave point functionals that are exact for isolated atoms?

• When information from the density is used to constructthe local effective α[n], what is the “range” of suchmodels?

Finally, we mention a range of possible applications thatshould enable better understanding of the strengths andweaknesses of existing vdW-inclusive methods and guide thedevelopment of the next generation of unified methods:

• Databases of high-quality binding or cohesive energies ofmolecular systems at equilibrium geometries (S22/S66,S12, C21/X23) have substantially contributed to thedevelopment of better vdW-inclusive methods during thepast decade. Some of these methods are now able toconsistently achieve a relative accuracy of 5−10% and anabsolute accuracy of 1 kcal/mol on these data sets.However, the performance seems to deteriorate once themethods are applied outside the domain of thosedatabases. Substantially more reference data shouldbecome available in the next few years for increasinglyheterogeneous and complex molecular systems, includingnonequilibrium geometries and transition states. Itremains to be seen how well existing vdW-inclusivemethods perform for such more complex benchmarks,representing molecules in more realistic nonequilibriumsituations.

• The performance of the methods strongly varies with thecomplexity and size of a system (Figure 7). For smallmolecular dimers in the S22/S66 data sets, all methodsperform well, but this changes as the systems becomeincreasingly nontrivial. Can we better understand thescaling of vdW interactions with size and complexity?When do many-body or higher-multipole effects becomemore important? Answering such questions wouldrequire development of an efficient method that includesboth multipolar and many-body effects to infinite orderin perturbation theory.

• How well do current vdW-inclusive methods modeldifferent phases of solids? The nature of bondingstrongly depends on the solid structure, and differentphases of the same building block (atom, molecule, orcluster) can exhibit very different polarization responses.Substantial understanding can be gained by analyzing theperformance of vdW-inclusive methods for cohesive andelastic properties of different solid phases. This applies to



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both hard solids (semiconductors, ionic solids, met-als)80,83 and molecular crystals when studying poly-morphism.7,78,99

• Is the performance of vdW-inclusive methods uniformacross the periodic table? There is some evidence thatperformance deteriorates for heavier elements andmaterials with nontrivial electronic states, such asnarrow-gap semiconductors, metals, polarizable 2Dmaterials, and topological insulators. Many more bench-marks are required to gain better understanding of theperformance for different elements in the periodic table.

• Hybrid interfaces between organic and inorganic systemsplay an important role in fundamental science andtechnological applications. The structure and binding ofsuch hybrid interfaces is often determined by vdWinteractions.209 Considerable experimental data is avail-able for structures and energetics of such interfaces.Continuous assessment of vdW methods for suchinterface properties is necessary to enable predictivesimulations of hybrid systems.

• Do existing vdW-inclusive methods perform consistentlyfor dimers, clusters, and solids of polarizable elements,such as semiconductors and metals? When going from adimer through larger clusters to a solid, the polarizabilityscales nonlinearly, and this can have a strong influenceon vdW interactions.55,91,92 Do methods exhibit correctscaling of vdW interactions with system size from dimersto large clusters and extended solids?

• Biological systems provide a rich playground for theassessment of vdW-inclusive methods. Much work hasbeen done evaluating different vdW methods on smallpeptides,337 protein secondary-structure elements,334

DNA,17 crystals of biological molecules,13,17,78,345 andions interacting with biological molecules,346,347 amongother systems.

• Can we control intermolecular vdW interactions by thepresence of nanostructures? Experiments clearly demon-strate that it is possible to dramatically enhance24,25 orattenuate112 vdW interactions by the presence ofheterostructured surfaces with complex dielectric profiles.Hence, it seems that it should be possible to designmolecular interaction profiles by carefully chosennanostructures with specific geometries and dielectricproperties.

• What is the role of microsolvation to full solvation onvdW interactions between solutes? Similar to theprevious point, this field is highly controversial340 anddemands systematic experimental and theoretical inves-tigations of a wide range of solutes in solvents of varyingpolarity and polarizability.

7. CONCLUDING REMARKSDuring the past decade, the atomistic modeling community hasstarted to understand the broad implications of vdW dispersioninteractions for a wide variety of molecules and materials. Thedevelopment of sophisticated first-principles approximations tomodel vdW interactions lead to an exponentially growing“black-box” use of these approaches in combination with semi-local density functionals. However, as is the case with semi-localdensity-functional approximations, all existing vdW-inclusivemethods have their advantages and limitations, and care mustbe taken when applying these methods to systems outside oftheir “comfort zone”. The contributions of many-body effects,

polarization screening, polarizability anisotropy, multipolarexpansion, nontrivial electronic states (metallic or topological),coarse-graining of the electronic response functionall thesequestionsremain open to debate in the modeling of vdWinteractions in realistic materials.In this review, we presented a concise discussion of the

ingredients necessary for an accurate description of vdWinteractions in a wide range of materials (Section 2). Thevarious existing vdW-inclusive methods have been classifiedaccording to their treatment of the microscopic polarizabilityand the interaction potentialthe two fundamental quantitiesrequired for the modeling of vdW interactions. We discussedsystems for which the polarizability can be reasonablyapproximated with local models, and where such localapproximations fail. We emphasized the high accuracy in thepolarizabilities (and vdW coefficients) of 5−6% that is requiredto obtain reliable results for the long-range vdW energy;deviations of just 10% in these quantities can lead tounacceptably large errors.Based on recent landmark experiments probing the nontrivial

nature of vdW interactions at the nanoscale, we argued that theubiquitous many-body effects are crucial for both a qualitativeunderstanding and a quantitative description of the vdWenergy. Furthermore, these experiments demonstrate that ourunderstanding of quantum-mechanical vdW effects is still onlyemerging (see Section 3).We followed by introducing the theoretical framework for

calculating vdW interactions based on the adiabatic-connectionfluctuation−dissipation (ACFD) theorem in Section 4. Thispart also discussed the crucial role of different responsefunctions and the relation between them as used by differentapproximate vdW methods. We emphasize here that the exacttreatment of vdW interactions in arbitrary materials can beformulated exclusively using the microscopic dipolar polar-izability of the system.Finally, we reviewed the performance of widely used vdW-

inclusive methods on a variety of molecular binding or cohesiveenergy databases in Section 5. Although the performance isgenerally very good for small molecular dimers, only someavailable methods perform well for larger more complexsupramolecular systems and extended molecular crystals.Current efforts directed at obtaining reliable benchmarks formore realistic systems and applying vdW methods toincreasingly large molecules will provide significant insightsinto the successes and limitations of different methods.In summary, painstaking and systematic work on the

concepts, theory, and applications of vdW-inclusive methodsshould eventually enable us to develop an efficient method forvdW interactions with broad applicability in chemistry, biology,physics, and materials science. Such a unified method should befirmly based in the ACFD formalism and combine the bestfeatures of interatomic approaches, nonlocal density func-tionals, and many-body perturbation theory.

ASSOCIATED CONTENTSpecial Issue Paper

This paper is an additional review for Chem. Rev. 2016, 116,issue 9, “Noncovalent Interactions”.

AUTHOR INFORMATIONCorresponding Authors

*E-mail: [email protected].



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mailto:[email protected]


*E-mail: [email protected]

Alexandre Tkatchenko: 0000-0002-1012-4854Notes

The authors declare no competing financial interest.

Biographies

Jan Hermann received a double B.Sc. in Chemistry and Physicsfollowed by a M.Sc. in Material Modeling from Charles University inPrague, while working under Ota Bludsky and Petr Nachtigall onintermolecular interactions in zeolites. Now a Ph.D. candidate in theTkatchenko group at the Fritz Haber Institute of the Max PlanckSociety in Berlin, Hermann focuses on the development of generalmodels for van der Waals interactions applicable to realistic materials.

Robert A. DiStasio Jr. is a Professor of Chemistry and ChemicalBiology at Cornell University. He earned a B.S. summa cum laude inBiology and Chemistry from Portland State University while workingwith Carl C. Wamser and the late George S. Hammond. He received aPh.D. in Theoretical Chemistry from the University of California,Berkeley for his work on local and canonical approximations inMøller−Plesset perturbation theory with applications to dispersioninteractions with Martin Head-Gordon. This was followed byPostdoctoral Research at Princeton University, where he workedwith Roberto Car, Salvatore Torquato, and Frank H. Stillinger, as wellas Alexandre Tkatchenko and Matthias Scheffler (at the Fritz HaberInstitute of the Max Planck Society in Berlin). DiStasio is an author ofand active contributor to both the Q-Chem and Quantum ESPRESSOsoftware packages, has given more than 75 seminars and colloquiaworldwide, and published more than 40 articles in peer-reviewedacademic journals.

Alexandre Tkatchenko is a Professor of Theoretical Chemical Physicsat the University of Luxembourg. He obtained his bachelor degree inComputer Science and a Ph.D. in Physical Chemistry at theUniversidad Autonoma Metropolitana in Mexico City. In 2008−2010, he was an Alexander von Humboldt Fellow at the Fritz HaberInstitute (FHI) of the Max Planck Society in Berlin. Between 2011 and2016, he led an independent research group at FHI. Tkatchenko hasgiven more than 70 invited talks plus 70 seminars and colloquiaworldwide, published more than 100 articles in peer-reviewedacademic journals, and serves on the editorial board of ScienceAdvancesan open-access journal in the Science family. He hasreceived a number of awards, including the Gerhard Ertl YoungInvestigator Award of the German Physical Society in 2011, and twoimportant grants from the European Research Council: a StartingGrant in 2011 and a Consolidator Grant in 2016.

ACKNOWLEDGMENTSA.T. acknowledges financial support by the European ResearchCouncil (ERC-StG VDWCMAT) and the DFG SPP-1807Research Network. R.D. acknowledges partial support fromCornell University through startup funding and the CornellCenter for Materials Research through funding from the NSFMRSEC program (DMR-1120296). This research usedresources of the Argonne Leadership Computing Facility atArgonne National Laboratory, which is supported by the Officeof Science of the U.S. Department of Energy under ContractNo. DE-AC02-06CH11357. This research also used resourcesof the National Energy Research Scientific Computing Center,which is supported by the Office of Science of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231.

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component, ∭ ∬∇· = ·∂

r F S Fd dV V

A useful form, which resembles

integration by parts, is obtained by substituting = gF G (where g andG a r e s c a l a r a n d v e c t o r fi e l d s , r e s p e c t i v e l y ) ,

∭ ∬ ∭∇ ∇· = · − ·∂

g g gr G G r Gd dS d .V V V

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