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2-2015

Dispersion Correction Derived from FirstPrinciples for Density Functional Theory andHartree−Fock TheoryEmilie B. GuidezIowa State University, eguidez@ameslab.gov

Mark S. GordonIowa State University, mgordon@iastate.edu

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Dispersion Correction Derived from First Principles for DensityFunctional Theory and Hartree−Fock Theory

AbstractThe modeling of dispersion interactions in density functional theory (DFT) is commonly performed using anenergy correction that involves empirically fitted parameters for all atom pairs of the system investigated. Inthis study, the first-principles-derived dispersion energy from the effective fragment potential (EFP) methodis implemented for the density functional theory (DFT-D(EFP)) and Hartree–Fock (HF-D(EFP)) energies.Overall, DFT-D(EFP) performs similarly to the semiempirical DFT-D corrections for the test casesinvestigated in this work. HF-D(EFP) tends to underestimate binding energies and overestimateintermolecular equilibrium distances, relative to coupled cluster theory, most likely due to incompleteaccounting for electron correlation. Overall, this first-principles dispersion correction yields results that are ingood agreement with coupled-cluster calculations at a low computational cost.

DisciplinesChemistry

CommentsReprinted (adapted) with permission from Journal of Physical Chemistry A 119 (2015): 2161, doi:10.1021/acs.jpca.5b00379. Copyright 2015 American Chemical Society.

This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/chem_pubs/588

Dispersion Correction Derived from First Principles for DensityFunctional Theory and Hartree−Fock TheoryEmilie B. Guidez and Mark S. Gordon*

Department of Chemistry, Iowa State University, Ames, Iowa 50011, United States

ABSTRACT: The modeling of dispersion interactions in density functional theory (DFT)is commonly performed using an energy correction that involves empirically fitted parametersfor all atom pairs of the system investigated. In this study, the first-principles-deriveddispersion energy from the effective fragment potential (EFP) method is implemented forthe density functional theory (DFT-D(EFP)) and Hartree−Fock (HF-D(EFP)) energies.Overall, DFT-D(EFP) performs similarly to the semiempirical DFT-D corrections for thetest cases investigated in this work. HF-D(EFP) tends to underestimate binding energiesand overestimate intermolecular equilibrium distances, relative to coupled cluster theory,most likely due to incomplete accounting for electron correlation. Overall, this first-principles dispersion correction yields results that are in good agreement with coupled-cluster calculations at a low computational cost.

■ INTRODUCTIONIntermolecular dispersion forces arise from the interactionbetween induced multipoles.1 These forces are at the origin ofmany chemical and biological processes such as proteinfolding,2−5 molecular recognition,6 and DNA base pairstacking.7 The modeling of dispersion interactions has beenthe subject of many investigations.8−11 Density functionaltheory (DFT) is frequently used to model molecular systemsthat contain on the order of hundreds of atoms.12,13 However,most commonly used density functionals (as well as Hartree−Fock (HF) theory) cannot account for dispersion interactions.Certain density functionals such as MPWB1K,14 M06-2X,15

M08-HX,16 M08-SO,16 and M1117 developed by Truhlar andco-workers correctly capture attractive noncovalent interactionswhere intermolecular overlap is nonnegligible (at the van derWaals minima). In addition, other functionals such as the vander Waals nonlocal correlation functionals (vdW-DF)18−22 in-clude dispersion. In order to enable popular density functionals(GGAs, hybrids, ...) to account for dispersive interactions,Grimme and co-workers introduced a series of empirical cor-rections,23−25 collectively referred to here as “-D”. In DFT-D,the -D dispersion interaction energy correction is added to theKohn−Sham energy.In general, the dispersion interaction between two molecules

A and B can be expressed as26

= + + +ECR

CR

CR

...disp66

77

88 (1)

R is the intermolecular distance and Cn are coefficients. Theterms with an odd power of R are orientation-dependent,27,28

and average to zero in freely rotating systems. The dispersionenergy expression is often truncated at the R−6 term, whichcorresponds to the interaction between induced dipoles oftwo species A and B. Three empirically parametrized disper-sion corrections, called DFT-Dn (n = 1, 2, 3),23−25 have been

developed by Grimme et al. The expression for the R−6 term inDFT-Dn is given by9

∑ ∑= −=

−

= +

E SCR

f R( )I

N

J I

N IJ

IJIJdisp 6

1

1

1

66 dmp

at at

(2)

S6 is a scaling parameter that depends on the functional.C6IJ, the dispersion coefficient for the atom pair IJ, is a fitted

parameter, Nat is the number of atoms, and RIJ is the distancebetween atoms I and J. The repulsive interaction betweenthe nuclei at very small RIJ is taken into account by using thedamping function fdmp. In the DFT-Dn implementations byGrimme, the damping function also involves a fitted param-eter. While DFT-Dn (n = 1, 2) only include the R−6 term,23,24

DFT-D3 also includes the R−8 term in a recursive manner.25 InDFT-D1 and DFT-D2, dispersion coefficients are predeter-mined using atomic ionization potentials and static polar-izabilitites. There is no dependence on the atomic environment,which can lead to substantial errors. On the other hand,DFT-D3 considers the effective volume of the atoms by takinginto account the number of neighbors in the environment tocalculate the dispersion coefficients. Other methods such as theTkatchenko−Scheffler29−31 and the Becke−Johnson32−34models also consider the chemical environment of the atomsto calculate dispersion coefficients. The former method relieson the Hirshfeld partitioning of the total electron densitybetween the atoms (which is obtained from electronic structurecalculations) to compute the dispersion coefficients.29 Thelatter computes the dipole moment generated by the exchange-correlation hole.34 Both methods are minimally empirical. ThedDsC method, based on the generalized gradient approximationof the Becke−Johnson model, was developed later on.35,36

Received: January 13, 2015Published: February 4, 2015

Article

pubs.acs.org/JPCA

© 2015 American Chemical Society 2161 DOI: 10.1021/acs.jpca.5b00379J. Phys. Chem. A 2015, 119, 2161−2168

One drawback of DFT-D can be the double countingof some of the electron correlation that is already included inthe correlation functional. The parametrization attempts tominimize the effect of double counting. On the other hand, theHartree−Fock method does not include any electron cor-relation except for the Fermi hole.37 Therefore, there is nopossibility of double counting when one adds the -D correctionto HF theory. However, other correlation effects are not in-cluded in HF-D. HF-D3 calculations using the Grimme cor-rections have been performed on several systems.38 Both DFTand HF have similar computational costs, which is one reasonfor the popularity of DFT.The effective fragment potential (EFP) method has been

developed to treat intermolecular interactions.39,40 In the EFPmethod, all interaction energy terms, including the dispersionenergy, are derived from first principles.41 In the EFP method,the interaction energy between the molecules or fragmentsis divided into five contributions: the Coulomb, polarization,exchange−repulsion, charge transfer, and dispersion interactionenergies:40,42,43

= + + + +−E E E E E Ecoul pol ex rep ct disp (3)

The objective of this work is to add the dispersion energyderived from first principles in the EFP method to the DFTKohn−Sham energy (DFT-D(EFP)) and Hartree−Fockenergy (HF-D(EFP)).

■ METHODIn the DFT-D(EFP) and HF-D(EFP) implementations,each molecule represents a fragment within the EFP scheme.The dispersion energy between the molecules (fragments) iscalculated as it would be in an EFP calculation. The disper-sion interaction energy calculated in this manner is then addedto the quantum mechanical (QM) energy of the system(Kohn−Sham or Hartree−Fock). The total energy of thesystem is given by

= +E E Edisp QM (4)

The expression for the dispersion energy derived fromRayleigh−Schrodinger perturbation theory is1,41,44

∫∑ ∑ ∑π

α ω α ω ω= − ℏ

αβγδαβ γδ αγ βδ

∈ ∈

∞E T T

2(i ) (i ) d

i j

x y zij ij i j

dispAB

A B

, ,

0

(5)

In eq 5, i and j label the localized molecular orbitals (LMOs)of molecules A and B, respectively.45 α(iω) represents thedynamic dipole polarizability tensor. The dynamic polarizabilitytensor is calculated by solving the time-dependent Hartree−Fock equations.41,46 T is a second order electrostatic tensorgiven by1,41,44

πε

δ

πε= ∇ ∇ =

−αβ α β

α βαβT

R R

R R R

R1

41 3

4ij

ij ij

ij ij ij

ij0

2

05

(6)

Rij = Ri − Rj, where Ri and Rj are the coordinates of the LMOcentroids i and j of fragments A and B, respectively. One cansubstitute ∑αβγσ

x,y,z Tαβij Tγσ

ij = 6/Rij6 into eq 5.1 Within the isotropic

approximation, eq 5 further reduces to41

∑ ∑∫

π

α ω α ω ω= − ℏ

∈ ∈

∞

ER

3 (i ) (i ) d

i j

i j

ijdispAB

A B

06

(7)

α represents one-third of the trace of the dynamic polar-izability tensor. The use of the isotropic approximation is justifiedsince anisotropic effects are minimal in this distributed approach.41

In addition, it permits the direct comparison of the C6 coefficientswith experimental and other theoretical data and reduces computa-tional cost.41 The integral in eq 7 is evaluated using a 12-pointGauss−Legendre quadrature formula.47 With a change of variable

ω ω ωω

= +−

=−

+tt

tt

11

, d2 d

(1 )00

2(8)

Equation 7 can be rewritten as

∑ ∑ ∑π

ω α ω α ω= − ℏ−

∈ ∈ =

E Wt R

3 2(1 )

(i ) (i )

i j kk

k

i j

ijdispAB

A B 1

120

2 6(9)

Wk and tk are the Gauss−Legendre weighting factor and abscissa.41

ω0 has an optimal value of 0.3.48 In order to avoid a singularitynear R = 0, each term in eq 9 is multiplied by a damping func-tion.49 An overlap-based damping function is used here:

∑= −− | |

!=

f i j SS

n( , ) 1

( 2 ln )ij

n

ijn

dmp2

0

6 /2

(10)

Sij in eq 10 is the overlap between the localized molecularorbitals i and j. This damping function differs slightly from theone used originally in EFP49 in that it is now equally applicableto even and odd powers of n. An overlap-based damping func-tion is a logical choice since the intermolecular overlap integraldepends on the intermolecular distance, so the dispersioninteractions are diminished as the overlap increases. In addition,unlike other damping functions, the overlap damping functioncontains no empirically fitted parameters.49

The present approach is derived from first principles and doesnot involve any empirically fitted parameters. The Tkatchenko−Scheffler method determines atomic polarizabilities based onthe partitioning of the electron density as well as the effectiveatomic volumes. In contrast, the method described here solvesthe time-dependent Hartree−Fock equations using localized

Figure 1. Systems with a dominant dispersion interaction (first row):(A) benzene (sandwich); (B) methane dimer; (C) hydrogen dimer;(D) π-stacked adenine−thymine. Hydrogen-bonded systems (secondrow): (E) water dimer; (F) ammonia dimer; (G) methanol dimer; (H)adenine−thymine base pair (WC). Mixed systems (third row): (I)ethene−ethyne; (J) T-shape benzene dimer; (K) benzene−waterdimer; (L) benzene−ammonia dimer. Color coding: black = carbon;white = hydrogen; red = oxygen; blue = nitrogen.

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molecular orbitals to calculate molecular polarizabilities. TheDFT-D(EFP) binding energy between two molecules is cal-culated using the formula

∑Δ = −‐E E EdimerDFT D(EFP)

monomerDFT

(11)

An analogous formula is used for HF-D(EFP) calculations:

∑Δ = −‐E E EdimerHF D(EFP)

monomerHF

(12)

In this work, the potential energy surfaces (PESs) of severaltest dimers are generated and compared with the PESs obtainedwith the Grimme DFT-Dn methods and with CCSD(T) and MP2calculations. Note that, for DFT-Dn calculations, intramoleculardispersion forces are included in the energy of the monomer butthey are not included in this new DFT-D(EFP) method.

■ COMPUTATIONAL DETAILSCalculations reported in this work were performed using theGAMESS software package.50,51 The DFT functional B3LYP52,53

was used with the 6-311++G(d,p) basis set. The HF-D(EFP)calculations were performed with the same basis set. For thehydrogen dimer, an additional set of calculations was done withthe 6-311++G(3df,3p) basis set. The dimerization energies(eqs 11 and 12) were calculated at various intermoleculardistances to generate a potential energy surface for each dimer.The internal monomer geometries were held fixed in the dimercalculations. All potential energy surfaces were generated bymoving the molecules along the axis that connects their centersof mass at the equilibrium dimer orientation. The intermo-lecular equilibrium distance R0 and the equilibrium bindingenergy ΔE0, corresponding to the minimum on the potentialenergy surface, are reported for all methods. The equilibriumgeometries of the benzene dimers (sandwich and T-shape)were optimized at the CCSD(T)/aug-cc-pVQZ* level of theorywith frozen monomers.54 Equilibrium geometries of the methane,hydrogen, water, ammonia, and methanol dimers were taken fromref 49. The equilibrium geometries for the Watson−Crick DNApair adenine−thymine (AD-WC) and the adenine−thymine stack

Figure 2. Potential energy curves of a set of test dimer cases with dominant dispersion interactions. (A) Benzene sandwich; (B) methane dimer; (C)hydrogen dimer with a 6-311++G(d,p) basis set; (D) hydrogen dimer with a 6-311++G(3df,3p) basis set; (E) adenine−thymine DNA stack. Thefunctional B3LYP is used for all DFT calculations with a 6-311++G(d,p) basis set except for (D), where a 6-311++G(3df,3p) basis set is used.A 6-311++G(d,p) basis set is also used for HF, MP2 and CCSD(T) calculations except for (D) where a 6-311++G(3df,3p) basis set is used.

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optimized at the MP2/cc-pVTZ level of theory with counter-poise correction (CP) were taken from ref 55. The equilibriumgeometries of the benzene−water and benzene−ammoniacomplexes optimized at the MP2/cc-pVTZ level of theorywith counterpoise correction were also taken from ref 55. Theethene−ethyne complex optimized at the CCSD(T)/cc-pVQZlevel of theory was adapted from ref 55. The CCSD(T)/CBSenergies were taken from refs 56, 54, and 57.

■ RESULTS AND DISCUSSIONThe three classes of systems that were investigated are shown inFigure 1.The first class corresponds to systems with dominant disper-

sion interactions, which include the sandwich configurationof the benzene dimer, the methane dimer, the hydrogen dimer,and the π-stacked adenine−thymine (Figure 1A−D). Thesecond class corresponds to hydrogen-bonded systems, whichinclude the water dimer, the ammonia dimer, the methanoldimer, and the adenine−thymine Watson−Crick DNA basepair (Figure 1E−H). The last class of complexes investigated isthe mixed systems: the ethene−ethyne complex, the T-shapebenzene dimer, and the benzene−water and benzene−ammoniacomplexes (Figure 1I−L). This classification was taken from ref 9.The potential energy surfaces of the first class of systems are

displayed in Figure 2.The intermolecular equilibrium distances R0 and equilibrium

binding energies ΔE0 of all of these systems are shown in Table 1.The benzene dimer is first investigated in a sandwich configura-

tion. Like the semiempirical DFT-Dn methods (n = 1, 2, 3),DFT-D(EFP) and HF-D(EFP) both tend to underestimatethe binding energy of the benzene sandwich in comparison toCCSD(T). MP2 overestimates the binding energy for thissystem. DFT-D(EFP) gives distances and binding energies thatare similar to those of both DFT-D2 and DFT-D3, with anintermolecular equilibrium distance R0 of 3.82 Å and a bindingenergy ΔE0 of −1.11 kcal/mol. HF-D(EFP) overestimates theintermolecular equilibrium distance and underestimates thebinding energy compared to CCSD(T), DFT-D2, DFT-D3,and DFT-D(EFP). In fact, the HF-D(EFP) results are verysimilar to those predicted by DFT-D1. For the methane dimer,all of the DFT-Dn methods underestimate the intermolecularequilibrium distance, while the HF-D(EFP) intermolecular dis-tance is in excellent agreement with the CCSD(T) value. The bind-ing energy at R0 for all of the -D methods is within 0.2 kcal/mol ofthe CCSD(T) value.The CCSD(T) interaction between two hydrogen molecules

is very weak (smaller than 0.1 kcal/mol); it is therefore a chal-lenge to model the PES accurately. As may be seen in Table 1,two basis sets were used for the H2 dimer: 6-311++G(d,p) and6-311++G(3df,3p). The larger basis set increases the MP2 andCCSD(T) binding energies by ∼0.03 kcal/mol. The effect onDFT-Dn is small because the dependence on the basis set ofthe empirically determined -D methods is minimal. This is notthe case for the -D(EFP) method, since it arises from firstprinciples and therefore has an explicit basis set dependence.For the smaller basis set, DFT-D(EFP) does not have aminimum on the H2 dimer potential energy surface, while theHF-D(EFP) binding energy is smaller than that predicted byCCSD(T) by ∼0.03 kcal/mol. The binding energies predictedby both -D(EFP) methods are similar to those predicted byMP2 and CCSD(T).The π-stacking interaction between the DNA bases adenine

and thymine (A−T) for the methods considered here are Table

1.Interm

olecular

Equ

ilibrium

Distances

R0in

ÅandBinding

EnergiesΔE0in

kcal/m

olof

System

swithDom

inantDispersionCon

tributionsa

dimer

DFT

-D1

DFT

-D2

DFT

-D3

DFT

-D(EFP

)HF-D(EFP

)MP2

CCSD

(T)

CCSD

(T)/CBSb

benzene(sandw

ich)

4.02

(−0.55)

3.82

(−1.21)

3.86

(−1.66)

3.82

(−1.11)

4.24

(−0.60)

3.68

(−4.55)

3.88

(−3.08)

3.9c

(−1.70)

(CH

4)2

3.90

(−0.21)

3.76

(−0.38)

3.96

(−0.38)

3.88

(−0.24)

4.10

(−0.28)

4.12

(−0.34)

4.08

(−0.38)

3.71

(−0.53)

(H2)

2(6-311++

G(d,p))

3.16

(−0.049)

3.22

(−0.035)

3.30

(−0.068)

N/A

3.92

(−0.022)

3.72

(−0.042)

3.68

(−0.052)

3.36d(−

0.11)

(H2)

2(6-311++

G(3df,3p))

3.16

(−0.055)

3.20

(−0.040)

3.30

(−0.074)

3.30

(−0.027)

3.60

(−0.060)

3.50

(−0.073)

3.44

(−0.087)

3.36d(−

0.11)

A−T(stack)

3.28

(−10.45)

3.10

(−13.97)

3.22

(−12.51)

3.08

(−12.27)

3.26

(−10.98)

3.10

(−18.21)

−3.17

(−11.66)

aThe

bindingenergy

isthevalueindicatedinparentheses.bEn

ergy

values

wereobtained

attheCCSD

(T)/CBS(Δa(DT)Z)leveloftheorywith

theoptim

ized

MP2

/cc-pV

TZCPgeom

etry,exceptforthe

methane

dimer

which

was

optim

ized

attheCCSD

(T)/cc-pVTZlevelof

theory.56

c These

values

areobtained

attheCCSD

(T)/aug-cc-pVQZ*levelof

theory

from

ref54.dThese

values

arefrom

the

CCSD

(T)/CBSpotentialenergy

surfaceof

theT-shape

hydrogen

dimer

byJohnsonandDiep.57

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summarized in Table 1. CCSD(T) calculations were notperformed for this system, but the value of R0 optimized at theMP2/cc-pVTZ CP level of theory55 and the value of ΔE0obtained at the CCSD(T)/CBS level56 are reported in Table 1.MP2 greatly overestimates the binding energy of A−T:18.2 kcal/mol vs 11.66 kcal/mol for CCSD(T)/CBS. In con-trast, all of the DFT-D methods predict binding energies thatare within 2 kcal/mol of the CCSD(T)/CBS value. Specifically,DFT-D(EFP) predicts a binding energy that is 0.61 kcal/molhigher than the CCSD(T)/CBS value. The HF-D(EFP) energyis just 0.68 kcal/mol smaller than the CCSD(T)/CBS value. Allof the -D methods predict intermolecular distances that are ingood agreement with the MP2/cc-pVTZ CP optimized value.The second class of systems is the hydrogen-bonded com-

plexes. The PESs of these systems are shown in Figure 3, andthe values of R0 and ΔE0 are reported in Table 2.For the water dimer, all DFT-D methods overestimate the

binding energy in comparison to CCSD(T) by 0.5−0.7 kcal/mol,while the HF-D(EFP) binding energy is only 0.35 kcal/mol toolow. The intermolecular equilibrium distances obtained withthe DFT-D methods are between 2.84 and 2.90 Å, which is

only slightly smaller than the CCSD(T) value of 2.92 Å. Theintermolecular equilibrium distance of 2.94 Å obtained withHF-D(EFP) is very close to the CCSD(T) and MP2 values.Similar trends are observed for the ammonia and methanoldimers, for which the DFT-D(EFP) method underestimates theintermolecular equilibrium distance and overestimates the bind-ing energy. HF-D(EFP) overestimates R0 by only 0.02 Å forthese two systems in comparison to CCSD(T) but under-estimates the binding energy by about 0.55 kcal/mol. Theadenine−thymine Watson−Crick pair contains two hydrogenbonds. The PESs obtained with all DFT-D methods for this com-plex are similar to the ones obtained with MP2. The values of R0obtained with DFT-Dn (n = 1, 2, 3), and with DFT-D(EFP) andHF-D(EFP) are in good agreement with the MP2/cc-pVTZ CPoptimized value. The DFT-Dn and HF-D(EFP) binding energiesare all within 2 kcal/mol of the CCSD(T)/CBS value, whilethe DFT-D(EFP) method overestimates the binding energy bynearly 3.4 kcal/mol.The last class of systems investigated is the mixed systems.

The PESs are shown in Figure 4.The values of R0 and ΔE0 are displayed in Table 3.

Figure 3. Potential energy curves of a set of hydrogen-bonded complexes. (A) Water dimer; (B) ammonia dimer; (C) methanol dimer; (D)adenine−thymine Watson−Crick pair. The functional B3LYP is used for all DFT calculations with a 6-311++G(d,p) basis set. A 6-311++G(d,p)basis set is also used for HF, MP2 and CCSD(T) calculations.

Table 2. Intermolecular Equilibrium Distances R0 in Å and Binding Energies ΔE0 in kcal/mol of Hydrogen-Bonded Systemsa

dimer DFT-D1 DFT-D2 DFT-D3 DFT-D(EFP) HF-D(EFP) MP2 CCSD(T) CCSD(T)/CBSb

(H2O)2 2.90 (−6.25) 2.88 (−6.48) 2.90 (−6.41) 2.84 (−6.52) 2.94 (−5.50) 2.92 (−5.93) 2.92 (−5.85) 2.91 (−5.02)(NH3)2 3.28 (−3.89) 3.28 (−4.19) 3.28 (−4.06) 3.20 (−4.24) 3.36 (−3.13) 3.32 (−3.75) 3.34 (−3.67) 3.21 (−3.17)(CH3OH)2 3.58 (−6.44) 3.54 (−6.88) 3.56 (−6.82) 3.50 (−6.94) 3.60 (−5.90) 3.58 (−6.47) 3.58 (−6.46) −A···T (WC) 5.96 (−16.55) 5.94 (−18.17) 5.96 (−17.90) 5.84 (−20.11) 5.92 (−17.07) 5.98 (−16.03) − 5.97 (−16.74)

aThe binding energy is the value indicated in parentheses. bEnergy values were obtained at the CCSD(T)/CBS(Δa(DT)Z) level of theory with theoptimized CCSD(T)/cc-pVQZ geometry, except for the adenine−thymine DNA pair which was optimized with MP2/cc-pVTZ CP.56

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For the ethene−ethyne complex, the R0 values predictedby all of the DFT-D methods are smaller than those predictedby CCSD(T). DFT-D(EFP) and HF-D(EFP) slightly under-estimate the binding energy (by 0.08 and 0.3 kcal/mol respec-tively), while the DFT-D2 and DFT-D3 methods overestimatethe binding energy by 0.3 kcal/mol. For this system, the DFT-D1method is in very good agreement with CCSD(T).For the T-shape benzene dimer, all DFT-Dn methods as

well as HF-D(EFP) and DFT-D(EFP) underestimate thebinding energy in comparison to CCSD(T). MP2 on the otherhand overestimates the interaction between the monomers.DFT-D(EFP) predicts an equilibrium intermolecular distancethat is identical to that of CCSD(T) and a binding energy thatis nearly 2 kcal/mol too small. HF-D(EFP) overestimatesR0 and also underestimates the equilibrium binding energy

by ∼2 kcal/mol. The DFT-D2 and DFT-D3 methodsunderestimate the binding energy by ∼1 kcal/mol, while theDFT-D1 method is very similar to HF-D(EFP).For the water−benzene and ammonia−benzene complexes,

DFT-D(EFP) slightly underestimates the intermolecular equi-librium distances and underestimates the equilibrium bind-ing energies. On the other hand, HF-D(EFP) overestimatesR0 for both the benzene−ammonia complex and the water−benzene complex. Overall, DFT-D(EFP) and HF-D(EFP)yield PESs that are similar to those of DFT-D1, while theDFT-D3 method is in better agreement with CCSD(T).In general, for the mixed species, the -D(EFP) method tendsto underestimate binding energies slightly more than theDFT-D3 method, but the differences are typically 1 kcal/molor less.

Figure 4. Potential energy curves of a set of mixed complexes. (A) Ethene−ethyne complex; (B) T-shape benzene dimer; (C) water−benzene com-plex; (D) ammonia−benzene complex. The functional B3LYP is used for all DFT calculations with a 6-311++G(d,p) basis set. A 6-311++G(d,p)basis set is also used for HF, MP2 and CCSD(T) calculations.

Table 3. Intermolecular Equilibrium Distances R0 in Å and Binding Energies ΔE0 in kcal/mol of Mixed Systemsa

dimer DFT-D1 DFT-D2 DFT-D3 DFT-D(EFP) HF-D(EFP) MP2 CCSD(T) CCSD(T)/CBSb

ethene−ethyne 4.44 (−1.45) 4.30 (−1.85) 4.40 (−1.81) 4.36 (−1.45) 4.60 (−1.23) 4.44 (−1.68) 4.48 (−1.53) 4.42 (−1.51)benzene (T-shape) 5.06 (−2.10) 4.86 (−3.25) 4.98 (−3.14) 4.88 (−2.51) 5.10 (−2.16) 4.80 (−5.35) 4.88 (−4.38) 5.0c (−2.61)benzene−H2O 3.42 (−3.31) 3.28 (−4.43) 3.36 (−4.21) 3.32 (−3.26) 3.52 (−2.93) 3.32 (−4.34) 3.36 (−4.04) 3.38 (−3.29)benzene−NH3 3.62 (−1.98) 3.42 (−2.96) 3.54 (−2.81) 3.50 (−2.04) 3.74 (−1.80) 3.48 (−3.42) 3.52 (−3.09) 3.56 (−2.32)

aThe binding energy is the value indicated in parentheses. bEnergy values were obtained at the CCSD(T)/CBS(Δa(DT)Z) level of theory with theoptimized MP2/cc-pVTZ geometry, except for the ethene−ethyne complex which was optimized at the CCSD(T)/cc-pVQZ level of theory.56cThese values are obtained at the CCSD(T)/aug-cc-pVQZ* level of theory from ref 54.

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■ CONCLUSIONS

The main conclusion of the present work is that theDFT-D(EFP) and HF-D(EFP) methods predict potentialenergy surfaces for a variety of types of intermolecular com-plexes with an accuracy that is comparable to that of the -Dmethods of Grimme and co-workers. The cost of computingthe dispersion correction with this new method is slightlyhigher than the empirical -D methods but still small in compar-ison to the DFT part of the calculation. The importance of the-D(EFP) approach is that the dispersion correction is derivedfrom first principles, without the need for empirically fittedparameters. In principle, this means that the EFP-based -Dmethod is more broadly applicable. In the -D(EFP) methods,the intermolecular dispersion energy is calculated using theapproach of the effective fragment potential method. TheDFT-D(EFP) and HF-D(EFP) methods were applied tomultiple test sets: complexes with dominant dispersion interac-tions, hydrogen-bonded complexes, and mixed complexes. Forall of these systems, DFT-D(EFP) performs similarly to thesemiempirical DFT-Dn (n = 1, 2, 3) corrections by Grimme.The HF-D(EFP) method also performs surprisingly well.Unlike the DFT methods, HF contains no other source ofelectron correlation (no correlation functional). Therefore, onecannot expect the HF-D method to perform consistentlyas well as DFT-D. Nonetheless, the HF-D method is appealingas a low-cost approximate alternative to MP2. The dispersioncorrection in DFT-D(EFP) and HF-D(EFP) only contains theR−6 term. Results could potentially be improved by includinghigher order terms.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: mark@si.msg.chem.iastate.edu. Tel.: 515-294-0452.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported by a U.S. National Science Founda-tion Software Infrastructure (SI2) grant, ACI-1047772.

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