Dispersion Correction Derived from First Principles for Density Functional Theory and Hartree Fock Theory

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1 Chemistry Publications Chemistry Dispersion Correction Derived from First Principles for Density Functional Theory and Hartree Fock Theory Emilie B. Guidez Iowa State University, Mark S. Gordon Iowa State University, Follow this and additional works at: Part of the Chemistry Commons The complete bibliographic information for this item can be found at chem_pubs/588. For information on how to cite this item, please visit howtocite.html. This is brought to you for free and open access by the Chemistry at Iowa State University Digital Repository. It has been accepted for inclusion in Chemistry Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact

2 Dispersion Correction Derived from First Principles for Density Functional Theory and Hartree Fock Theory Abstract The modeling of dispersion interactions in density functional theory (DFT) is commonly performed using an energy correction that involves empirically fitted parameters for all atom pairs of the system investigated. In this study, the first-principles-derived dispersion energy from the effective fragment potential (EFP) method is 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 cases investigated in this work. HF-D(EFP) tends to underestimate binding energies and 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. Disciplines Chemistry Comments Reprinted (adapted) with permission from Journal of Physical Chemistry A 119 (2015): 2161, doi: / acs.jpca.5b Copyright 2015 American Chemical Society. This article is available at Iowa State University Digital Repository:

3 pubs.acs.org/jpca Dispersion Correction Derived from First Principles for Density Functional Theory and Hartree Fock Theory Emilie 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 parameters for all atom pairs of the system investigated. In this study, the first-principles-derived dispersion energy from the effective fragment potential (EFP) method is 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 cases investigated in this work. HF-D(EFP) tends to underestimate binding energies and overestimate intermolecular equilibrium distances, relative to coupled cluster theory, most likely due to incomplete accounting for electron correlation. Overall, this firstprinciples dispersion correction yields results that are in good agreement with coupledcluster calculations at a low computational cost. INTRODUCTION Intermolecular dispersion forces arise from the interaction between induced multipoles. 1 These forces are at the origin of many chemical and biological processes such as protein folding, 2 5 molecular recognition, 6 and DNA base pair stacking. 7 The modeling of dispersion interactions has been the subject of many investigations Density functional theory (DFT) is frequently used to model molecular systems that 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 M11 17 developed by Truhlar and co-workers correctly capture attractive noncovalent interactions where intermolecular overlap is nonnegligible (at the van der Waals minima). In addition, other functionals such as the van der Waals nonlocal correlation functionals (vdw-df) include dispersion. In order to enable popular density functionals (GGAs, hybrids,...) to account for dispersive interactions, Grimme and co-workers introduced a series of empirical corrections, collectively referred to here as -D. In DFT-D, the -D dispersion interaction energy correction is added to the Kohn Sham energy. In general, the dispersion interaction between two molecules A and B can be expressed as 26 C6 C7 C8 Edisp = R R R (1) R is the intermolecular distance and C n are coefficients. The terms with an odd power of R are orientation-dependent, 27,28 and average to zero in freely rotating systems. The dispersion energy expression is often truncated at the R 6 term, which corresponds to the interaction between induced dipoles of two species A and B. Three empirically parametrized dispersion corrections, called DFT-Dn (n = 1, 2, 3), have been developed by Grimme et al. The expression for the R 6 term in DFT-Dn is given by 9 E = S Nat 1 disp 6 I= 1 Nat J=+ I 1 IJ C6 6 R f ( R IJ ) dmp IJ S 6 is a scaling parameter that depends on the functional. C IJ 6, the dispersion coefficient for the atom pair IJ, isafitted parameter, N at is the number of atoms, and R IJ is the distance between atoms I and J. The repulsive interaction between the nuclei at very small R IJ is taken into account by using the damping function f dmp. In the DFT-Dn implementations by Grimme, the damping function also involves a fitted parameter. 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 In DFT-D1 and DFT-D2, dispersion coefficients are predetermined using atomic ionization potentials and static polarizabilitites. 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 taking into account the number of neighbors in the environment to calculate the dispersion coefficients. Other methods such as the Tkatchenko Scheffler and the Becke Johnson models also consider the chemical environment of the atoms to calculate dispersion coefficients. The former method relies on the Hirshfeld partitioning of the total electron density between the atoms (which is obtained from electronic structure calculations) to compute the dispersion coefficients. 29 The latter computes the dipole moment generated by the exchangecorrelation hole. 34 Both methods are minimally empirical. The ddsc method, based on the generalized gradient approximation of the Becke Johnson model, was developed later on. 35,36 Received: January 13, 2015 Published: February 4, 2015 (2) 2015 American Chemical Society 2161

4 The Journal of Physical Chemistry A One drawback of DFT-D can be the double counting of some of the electron correlation that is already included in the correlation functional. The parametrization attempts to minimize the effect of double counting. On the other hand, the Hartree Fock method does not include any electron correlation except for the Fermi hole. 37 Therefore, there is no possibility of double counting when one adds the -D correction to HF theory. However, other correlation effects are not included in HF-D. HF-D3 calculations using the Grimme corrections have been performed on several systems. 38 Both DFT and HF have similar computational costs, which is one reason for the popularity of DFT. The effective fragment potential (EFP) method has been developed to treat intermolecular interactions. 39,40 In the EFP method, all interaction energy terms, including the dispersion energy, are derived from first principles. 41 In the EFP method, the interaction energy between the molecules or fragments is divided into five contributions: the Coulomb, polarization, exchange repulsion, charge transfer, and dispersion interaction energies: 40,42,43 E = E + E + E + E + E coul pol ex rep ct disp (3) The objective of this work is to add the dispersion energy derived from first principles in the EFP method to the DFT Kohn Sham energy (DFT-D(EFP)) and Hartree Fock energy (HF-D(EFP)). METHOD In 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) is calculated as it would be in an EFP calculation. The dispersion interaction energy calculated in this manner is then added to the quantum mechanical (QM) energy of the system (Kohn Sham or Hartree Fock). The total energy of the system is given by E = Edisp + EQM (4) The expression for the dispersion energy derived from Rayleigh Schro dinger perturbation theory is 1,41,44 AB disp xyz,, i j αβ γδ αγ βδ 0 E = ħ T T α (i ω) α (i ω) dω 2π i A j B αβγδ (5) In eq 5, i and j label the localized molecular orbitals (LMOs) of molecules A and B, respectively. 45 α(iω) represents the dynamic dipole polarizability tensor. The dynamic polarizability tensor is calculated by solving the time-dependent Hartree Fock equations. 41,46 T is a second order electrostatic tensor given by 1,41,44 T 1 1 3R R R δ = = 5 4πε R R 4πε R αβ α β 0 α β 0 2 αβ R = R i R j, where R i and R j are the coordinates of the LMO centroids i and j of fragments A and B, respectively. One can substitute x,y,z αβγσ T αβ T γσ =6/R 6 into eq 5. 1 Within the isotropic approximation, eq 5 further reduces to 41 E AB disp 3 α ω α ω ω = ħ (i ) (i ) d 0 6 π R i A j B i j (6) (7) α represents one-third of the trace of the dynamic polarizability tensor. The use of the isotropic approximation is justified since anisotropic effects are minimal in this distributed approach. 41 In addition, it permits the direct comparison of the C 6 coefficients with experimental and other theoretical data and reduces computational cost. 41 The integral in eq 7 is evaluated using a 12-point Gauss Legendre quadrature formula. 47 With a change of variable 1 + t ω ω = ω ω = 2 0 dt 0, d 2 1 t (1 + t) Equation 7 can be rewritten as E AB disp 12 3 ω α ω α ω = ħ 2 0 (i ) (i ) Wk 2 6 π (1 t ) R i A j B k= 1 W k and t k are the Gauss Legendre weighting factor and abscissa. 41 ω 0 has an optimal value of In order to avoid a singularity near R = 0, each term in eq 9 is multiplied by a damping function. 49 An overlap-based damping function is used here: f (, i j) = 1 S dmp 2 6 n/2 n= 0 k ( 2ln S ) n! i j (8) (9) (10) S in eq 10 is the overlap between the localized molecular orbitals i and j. This damping function differs slightly from the one used originally in EFP 49 in that it is now equally applicable to even and odd powers of n. An overlap-based damping function is a logical choice since the intermolecular overlap integral depends on the intermolecular distance, so the dispersion interactions are diminished as the overlap increases. In addition, unlike other damping functions, the overlap damping function contains no empirically fitted parameters. 49 The present approach is derived from first principles and does not involve any empirically fitted parameters. The Tkatchenko Scheffler method determines atomic polarizabilities based on the partitioning of the electron density as well as the effective atomic volumes. In contrast, the method described here solves the 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 (second row): (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 water dimer; (L) benzene ammonia dimer. Color coding: black = carbon; white = hydrogen; red = oxygen; blue = nitrogen. 2162

5 The Journal of Physical Chemistry A 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 G(d,p) basis set; (D) hydrogen dimer with a G(3df,3p) basis set; (E) adenine thymine DNA stack. The functional B3LYP is used for all DFT calculations with a G(d,p) basis set except for (D), where a G(3df,3p) basis set is used. A G(d,p) basis set is also used for HF, MP2 and CCSD(T) calculations except for (D) where a G(3df,3p) basis set is used. molecular orbitals to calculate molecular polarizabilities. The DFT-D(EFP) binding energy between two molecules is calculated using the formula DFT D(EFP) DFT Δ E = Edimer Emonomer (11) An analogous formula is used for HF-D(EFP) calculations: HF D(EFP) HF Δ E = Edimer Emonomer (12) In this work, the potential energy surfaces (PESs) of several test dimers are generated and compared with the PESs obtained with the Grimme DFT-Dn methods and with CCSD(T) and MP2 calculations. Note that, for DFT-Dn calculations, intramolecular dispersion forces are included in the energy of the monomer but they are not included in this new DFT-D(EFP) method. COMPUTATIONAL DETAILS Calculations reported in this work were performed using the GAMESS software package. 50,51 The DFT functional B3LYP 52, was used with the G(d,p) basis set. The HF-D(EFP) calculations were performed with the same basis set. For the hydrogen dimer, an additional set of calculations was done with the G(3df,3p) basis set. The dimerization energies (eqs 11 and 12) were calculated at various intermolecular distances to generate a potential energy surface for each dimer. The internal monomer geometries were held fixed in the dimer calculations. All potential energy surfaces were generated by moving the molecules along the axis that connects their centers of mass at the equilibrium dimer orientation. The intermolecular equilibrium distance R 0 and the equilibrium binding energy ΔE 0, corresponding to the minimum on the potential energy surface, are reported for all methods. The equilibrium geometries of the benzene dimers (sandwich and T-shape) were optimized at the CCSD(T)/aug-cc-pVQZ* level of theory with frozen monomers. 54 Equilibrium geometries of the methane, hydrogen, water, ammonia, and methanol dimers were taken from ref 49. The equilibrium geometries for the Watson Crick DNA pair adenine thymine (AD-WC) and the adenine thymine stack

6 The Journal of Physical Chemistry A optimized at the MP2/cc-pVTZ level of theory with counterpoise correction (CP) were taken from ref 55. The equilibrium geometries of the benzene water and benzene ammonia complexes optimized at the MP2/cc-pVTZ level of theory with counterpoise correction were also taken from ref 55. The ethene ethyne complex optimized at the CCSD(T)/cc-pVQZ level of theory was adapted from ref 55. The CCSD(T)/CBS energies were taken from refs 56, 54, and 57. RESULTS AND DISCUSSION The three classes of systems that were investigated are shown in Figure 1. The first class corresponds to systems with dominant dispersion interactions, which include the sandwich configuration of the benzene dimer, the methane dimer, the hydrogen dimer, and the π-stacked adenine thymine (Figure 1A D). The second class corresponds to hydrogen-bonded systems, which include the water dimer, the ammonia dimer, the methanol dimer, and the adenine thymine Watson Crick DNA base pair (Figure 1E H). The last class of complexes investigated is the mixed systems: the ethene ethyne complex, the T-shape benzene dimer, and the benzene water and benzene ammonia complexes (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 R 0 and equilibrium binding energies ΔE 0 of all of these systems are shown in Table 1. The benzene dimer is first investigated in a sandwich configuration. Like the semiempirical DFT-Dn methods (n =1,2,3), DFT-D(EFP) and HF-D(EFP) both tend to underestimate the binding energy of the benzene sandwich in comparison to CCSD(T). MP2 overestimates the binding energy for this system. DFT-D(EFP) gives distances and binding energies that are similar to those of both DFT-D2 and DFT-D3, with an intermolecular equilibrium distance R 0 of 3.82 Å and a binding energy ΔE 0 of 1.11 kcal/mol. HF-D(EFP) overestimates the intermolecular equilibrium distance and underestimates the binding energy compared to CCSD(T), DFT-D2, DFT-D3, and DFT-D(EFP). In fact, the HF-D(EFP) results are very similar to those predicted by DFT-D1. For the methane dimer, all of the DFT-Dn methods underestimate the intermolecular equilibrium distance, while the HF-D(EFP) intermolecular distance is in excellent agreement with the CCSD(T) value. The binding energy at R 0 for all of the -D methods is within 0.2 kcal/mol of the CCSD(T) value. The CCSD(T) interaction between two hydrogen molecules is very weak (smaller than 0.1 kcal/mol); it is therefore a challenge to model the PES accurately. As may be seen in Table 1, two basis sets were used for the H 2 dimer: G(d,p) and G(3df,3p). The larger basis set increases the MP2 and CCSD(T) binding energies by 0.03 kcal/mol. The effect on DFT-Dn is small because the dependence on the basis set of the empirically determined -D methods is minimal. This is not the case for the -D(EFP) method, since it arises from first principles and therefore has an explicit basis set dependence. For the smaller basis set, DFT-D(EFP) does not have a minimum on the H 2 dimer potential energy surface, while the HF-D(EFP) binding energy is smaller than that predicted by CCSD(T) by 0.03 kcal/mol. The binding energies predicted by both -D(EFP) methods are similar to those predicted by MP2 and CCSD(T). The π-stacking interaction between the DNA bases adenine and thymine (A T) for the methods considered here are Table 1. Intermolecular Equilibrium Distances R 0 in Å and Binding Energies ΔE 0 in kcal/mol of Systems with Dominant Dispersion Contributions a dimer DFT-D1 DFT-D2 DFT-D3 DFT-D(EFP) HF-D(EFP) MP2 CCSD(T) CCSD(T)/CBS b benzene (sandwich) 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) 3.9 c ( 1.70) (CH 4 ) ( 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) (H 2 ) 2 ( 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) 3.36 d ( 0.11) (H 2 ) 2 ( 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) 3.36 d ( 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) a The binding energy is the value indicated in parentheses. b Energy values were obtained at the CCSD(T)/CBS(Δa(DT)Z) level of theory with the optimized MP2/cc-pVTZ CP geometry, except for the methane dimer which was optimized at the CCSD(T)/cc-pVTZ level of theory. 56 c These values are obtained at the CCSD(T)/aug-cc-pVQZ* level of theory from ref 54. d These values are from the CCSD(T)/CBS potential energy surface of the T-shape hydrogen dimer by Johnson and Diep

7 The Journal of Physical Chemistry A 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 G(d,p) basis set. A G(d,p) basis set is also used for HF, MP2 and CCSD(T) calculations. Table 2. Intermolecular Equilibrium Distances R 0 in Å and Binding Energies ΔE 0 in kcal/mol of Hydrogen-Bonded Systems a dimer DFT-D1 DFT-D2 DFT-D3 DFT-D(EFP) HF-D(EFP) MP2 CCSD(T) CCSD(T)/CBS b (H 2 O) ( 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) (NH 3 ) ( 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) (CH 3 OH) ( 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) a The binding energy is the value indicated in parentheses. b Energy values were obtained at the CCSD(T)/CBS(Δa(DT)Z) level of theory with the optimized CCSD(T)/cc-pVQZ geometry, except for the adenine thymine DNA pair which was optimized with MP2/cc-pVTZ CP. 56 summarized in Table 1. CCSD(T) calculations were not performed for this system, but the value of R 0 optimized at the MP2/cc-pVTZ CP level of theory 55 and the value of ΔE 0 obtained at the CCSD(T)/CBS level 56 are reported in Table 1. MP2 greatly overestimates the binding energy of A T: 18.2 kcal/mol vs kcal/mol for CCSD(T)/CBS. In contrast, all of the DFT-D methods predict binding energies that are within 2 kcal/mol of the CCSD(T)/CBS value. Specifically, DFT-D(EFP) predicts a binding energy that is 0.61 kcal/mol higher than the CCSD(T)/CBS value. The HF-D(EFP) energy is just 0.68 kcal/mol smaller than the CCSD(T)/CBS value. All of the -D methods predict intermolecular distances that are in good agreement with the MP2/cc-pVTZ CP optimized value. The second class of systems is the hydrogen-bonded complexes. The PESs of these systems are shown in Figure 3, and the values of R 0 and ΔE 0 are reported in Table 2. For the water dimer, all DFT-D methods overestimate the binding energy in comparison to CCSD(T) by kcal/mol, while the HF-D(EFP) binding energy is only 0.35 kcal/mol too low. The intermolecular equilibrium distances obtained with the 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. The intermolecular equilibrium distance of 2.94 Å obtained with HF-D(EFP) is very close to the CCSD(T) and MP2 values. Similar trends are observed for the ammonia and methanol dimers, for which the DFT-D(EFP) method underestimates the intermolecular equilibrium distance and overestimates the binding energy. HF-D(EFP) overestimates R 0 by only 0.02 Å for these two systems in comparison to CCSD(T) but underestimates the binding energy by about 0.55 kcal/mol. The adenine thymine Watson Crick pair contains two hydrogen bonds. The PESs obtained with all DFT-D methods for this complex are similar to the ones obtained with MP2. The values of R 0 obtained with DFT-Dn (n =1,2,3),andwithDFT-D(EFP)and HF-D(EFP) are in good agreement with the MP2/cc-pVTZ CP optimized value. The DFT-Dn and HF-D(EFP) binding energies are all within 2 kcal/mol of the CCSD(T)/CBS value, while the DFT-D(EFP) method overestimates the binding energy by nearly 3.4 kcal/mol. The last class of systems investigated is the mixed systems. The PESs are shown in Figure 4. The values of R 0 and ΔE 0 are displayed in Table

8 The Journal of Physical Chemistry A Figure 4. Potential energy curves of a set of mixed complexes. (A) Ethene ethyne complex; (B) T-shape benzene dimer; (C) water benzene complex; (D) ammonia benzene complex. The functional B3LYP is used for all DFT calculations with a G(d,p) basis set. A G(d,p) basis set is also used for HF, MP2 and CCSD(T) calculations. Table 3. Intermolecular Equilibrium Distances R 0 in Å and Binding Energies ΔE 0 in kcal/mol of Mixed Systems a dimer DFT-D1 DFT-D2 DFT-D3 DFT-D(EFP) HF-D(EFP) MP2 CCSD(T) CCSD(T)/CBS b 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) 5.0 c ( 2.61) benzene H 2 O 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 NH ( 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) a The binding energy is the value indicated in parentheses. b Energy values were obtained at the CCSD(T)/CBS(Δa(DT)Z) level of theory with the optimized MP2/cc-pVTZ geometry, except for the ethene ethyne complex which was optimized at the CCSD(T)/cc-pVQZ level of theory. 56 c These values are obtained at the CCSD(T)/aug-cc-pVQZ* level of theory from ref 54. For the ethene ethyne complex, the R 0 values predicted by all of the DFT-D methods are smaller than those predicted by CCSD(T). DFT-D(EFP) and HF-D(EFP) slightly underestimate the binding energy (by 0.08 and 0.3 kcal/mol respectively), while the DFT-D2 and DFT-D3 methods overestimate the binding energy by 0.3 kcal/mol. For this system, the DFT-D1 method 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 the binding energy in comparison to CCSD(T). MP2 on the other hand overestimates the interaction between the monomers. DFT-D(EFP) predicts an equilibrium intermolecular distance that is identical to that of CCSD(T) and a binding energy that is nearly 2 kcal/mol too small. HF-D(EFP) overestimates R 0 and also underestimates the equilibrium binding energy by 2 kcal/mol. The DFT-D2 and DFT-D3 methods underestimate the binding energy by 1 kcal/mol, while the DFT-D1 method is very similar to HF-D(EFP). For the water benzene and ammonia benzene complexes, DFT-D(EFP) slightly underestimates the intermolecular equilibrium distances and underestimates the equilibrium binding energies. On the other hand, HF-D(EFP) overestimates R 0 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 the DFT-D3 method is in better agreement with CCSD(T). In general, for the mixed species, the -D(EFP) method tends to underestimate binding energies slightly more than the DFT-D3 method, but the differences are typically 1 kcal/mol or less. 2166

9 CONCLUSIONS The main conclusion of the present work is that the DFT-D(EFP) and HF-D(EFP) methods predict potential energy surfaces for a variety of types of intermolecular complexes with an accuracy that is comparable to that of the -D methods of Grimme and co-workers. The cost of computing the dispersion correction with this new method is slightly higher than the empirical -D methods but still small in comparison to the DFT part of the calculation. The importance of the -D(EFP) approach is that the dispersion correction is derived from first principles, without the need for empirically fitted parameters. In principle, this means that the EFP-based -D method is more broadly applicable. In the -D(EFP) methods, the intermolecular dispersion energy is calculated using the approach of the effective fragment potential method. The DFT-D(EFP) and HF-D(EFP) methods were applied to multiple test sets: complexes with dominant dispersion interactions, hydrogen-bonded complexes, and mixed complexes. For all of these systems, DFT-D(EFP) performs similarly to the semiempirical 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 of electron correlation (no correlation functional). Therefore, one cannot expect the HF-D method to perform consistently as well as DFT-D. Nonetheless, the HF-D method is appealing as a low-cost approximate alternative to MP2. The dispersion correction in DFT-D(EFP) and HF-D(EFP) only contains the R 6 term. Results could potentially be improved by including higher order terms. AUTHOR INFORMATION Corresponding Author * mark@si.msg.chem.iastate.edu. Tel.: Notes The authors declare no competing financial interest. 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