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Benchmark Databases for Nonbonded Interactions and Their Use To Test Density Functional Theory

Yan Zhao and Donald G. Truhlar*

Department of Chemistry and Supercomputing Institute, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431

Received December 17, 2004

Abstract:We presentfourbenchmarkdatabasesof bindingenergiesfor nonbondedcomplexes. Four types of nonbondedinteractionsare considered: hydrogenbonding,charge transfer,dipole interactions, and weak interactions. We tested 4 DFT methods and 1 WFT method against the new databases; one of the DFT methods (PBE1KCIS) is new, and all of the other methods are from the literature. Among the tested methods, the PBE, PBE1PBE, B3P86, MPW1K, B97- 1, and BHandHLYP functionals give the best performance for hydrogen bonding. MPWB1K, MP2, MPW1B95, MPW1K, and BHandHLYP give the best performances for charge-transfer interactions,and MPW3LYP, B97-1, PBE1KCIS,B98, and PBE1PBE give the best performance for dipole interactions. Finally, MP2, B97-1, MPWB1K, PBE1KCIS, and MPW1B95 give the best performance for weak interactions. Overall, MPWB1K is the best of all the tested DFT methods, with a relative error (highly averaged) of only 1%, and MPW1K, PBE1PBE, and B98 are the best of the tested DFT methods that do not contain kinetic energy density. Moving up the rungs of Jacob’s ladder for nonempirical DFT, PBE improves significantly over the LSDA, and TPSS improve slightly over PBE (on average) for nonbonded interactions.

1. Introduction

One can classify interatomic interactions as bonded or nonbonded. One can further subdivide bonded interactions into ionic, metallic, covalent, coordinate covalent, and partial bonds (as at transition states), and one can subdivide nonbonded interactions into charge-transfer interactions, hydrogen bonds, dipolar interactions, dispersion (London forces), and so forth. Mixed cases are also possible, such as polar covalent (e.g., an HF bond is about 50% ionic and 50% covalent1) or a much more complicated range of possibilities2 for nonbonded interactions. Nevertheless the distinctionsand the broadly defined categoriesof interactions are useful for understanding chemical phenomena and for testing the abilities of approximate theories and models to interpret chemical phenomena.

Density functional theory (DFT3-87 and wave function theory (WFT)31,34,4,5,87-118 have been widely compared for their abilities to treat bonds and transition states, but comparisons are less complete for nonbonded interactions. There are two reasons for this. First, it has been realized for a long time that DFT, at least with the early functionals, is less accurate for nonbonded interactions than for bonded ones,16-18 and this can be understood in part by the fact that current functionals are not designed to treat dispersion interactions, which are sometimes dominant in nonbonded interactions.Second, no standard databases (analogousto the G3 database105,106,109 or Database/3113 for bond energies, ionization potentials, and electron affinities; the latter also includes partial bond strengths as measured by barrier heights) are available for nonbonded interactions. The purpose of the present article is to remedy the latter problem and to use newly created databases for nonbonded interactions for a systematiccomparisonof DFT and WFT methods.

Considerableinsight into how DFT works can be obtained by detailed analysis of the functionals and the Kohn-Sham electron density. In particular it should be recognized that, for molecules, the separation of exchange-correlation effects into exchange and correlation are different in WFT and* Corresponding author e-mail: truhlar@umn.edu.

10.1021/ct049851d C: $30.25 © 2005 American Chemical Society Published on Web 04/23/2005

simulates orbital relaxation, pair correlation, | ”.46 However |

DFT.10,21,28,46,49In particular,DFT exchangeincludesa certain amount of what is called nondynamical (also called static or internal or, in certain contexts, left-right) correlation in WFT.10,28,43,49Handy and Cohen49 have shown that DFT with an exchange functional (Becke88 or B88X8) but no correlation functional gave lower energies than Hartree-Fock for the multicenter system (for example, molecules), and they concluded that local exchange functionals must introduce nondynamicalcorrelation.Furthermore,He et al.46 found that even for closed-shell systems that are well described without nondynamical correlation (so-called single-reference systems),densitiesobtainedby DFT with an exchangefunctional but no correlation functional look more like those obtained with fourth-order perturbation theory (MP4) than those obtained by uncorrelated Hartree-Fock.46 They concluded that “even though the DFT exchange functional does not include any Coulomb correlation effects by construction, it the resulting electron density is too high in the van der Waals region,and correlationfunctionalscontractthe densitytoward high-density regions (where there is more favorable correlation energy),thus improvingthe descriptionof van der Waals interactions.46 Since correlation functionals make up for deficiencies in exchange functionals, and since the exchange functional gives a much larger contribution to molecular interactions than the correlation one, it is important that the correlation functional be well matched to the exchange functional with which it is used.

Although the usual DFT functionalsdo not contain dipolar dispersion interactions, there is some debate as to whether DFT methods, with either the usual functionals or new ones, might nevertheless produce useful results for the attractive interaction between rare gas atoms.119-122 Furthermore DFT, even with the usual functionals, does contain the polarizabilities.123 Our goal in the present paper is not, however, to pursue lines of research based on explicit inclusion of dipole polarization but rather to check which of the density functionals in current widespread use disqualify themselves by predicting unrealistic interaction potentials in regimes where the real interaction potentials are dominated by dispersion forces or other nonbonded interactions,and which density functionalsyield reasonableresults in such situations, for whatever reason.

In addition to lacking explicit R-6 terms, DFT (without

Hartree-Fock exchange) predicts no interaction energy for molecules so far apart that they do not overlap (because the density is the same as for infinitely separated molecules). At the equilibrium distance of nonbonded complexes, the lack of explicit R-6 terms need not be a serious issue because the higher terms (R-8, etc.) in the asymptotic expansion are not negligible.124,125 Furthermore, the overlap and exchange forces are also not negligible at the equilibrium internuclear distance of nonbonded complexes.124,126 Thus DFT is not excluded as a potentially useful theory for nonbonded interactions, as is sometimes claimed.

In summary, our goal is to understand the performance of existing density functionals for nonbonded interactions and to compare this performance to that of WFT with the same basis sets. We therefore develop four new databases for such testing:

¥ A hydrogen bond database

¥ A charge-transfer complex database

¥ A dipole complex database

¥ A weak interaction database

Whereas hydrogen bonds are dominated by electrostatic and polarization (also called induction) interactions (with a smaller contribution from charge transfer), charge-transfer complexes derive a considerableportion of their stabilization from electron transfer between the two centers. Dipole complexes involve much smaller amounts of intermolecular charge transfer and have no hydrogen bonds. Weak complexes are defined here as those that are dominated by dispersion interactions.

In the literature, there are many theoretical studies of hydrogen bonds,29,31,39,4,47,54,69,81,84,9,101,110-112,115,127,128 charge-transfer complexes,18,19,27,29,34,70,116 and weak interactions.29,4,54,62,83,92,93,96,114 However there are very few studies31,39,47,68 of dipolar interaction complexes. Several stud- ies39,47,68 treated (HCl)2 dimer as a hydrogen bond complex, but in the present study we will treat (HCl)2 dimer as a dipole interactioncomplexsince there is no classicalhydrogenbond in (HCl)2 dimer. The databases are used to test several types of DFT: (i) the local spin density approximation (LSDA, in which the density functional depends only on density), (i) the generalized gradient approximation (GGA, in which the density functional depends on density and its reduced gradient), (i) meta GGA (in which the functional also depends kinetic energy density), (iv) hybrid GGA (a combination of GGA with Hartree-Fock exchange), and (v) hybrid meta GGA (a combinationof meta GGA with Hartree-Fock exchange). In addition we study one level of WFT: Møller-Plesset second-order perturbation theory87 (MP2).

Section 2 explains the theories, databases, and functionals used in the present work. Section 3 presents results and discussion, and section 4 has concluding remarks.

2. Theory and Databases 2.1. Weizmann 1 (W1) Theory. It is difficult to extract the zero-point-exclusive binding energies De from experiment for nonbonded complexes due to the uncertainties in the experimental ground-state dissociation energy D0 and due to the uncertain effect of anharmonicity on the zero point vibrational energy of these loose complexes. To obtain the best estimates for the binding energies in the new database, we employed the W1 method for most of the nonbonded complexes, and we also took some theoretical and experimental results from the literature.

W1 theory was developed by Martin and Oliveira, and it is a method designed to extrapolate to the complete basis limit of a CCSD(T)89 calculation. Thus W1 theory should be good enough for obtaining best estimates of binding energiesof these nonbondedcomplexes.Boese et al.68,80 have already used W1 and W2 theory to calculate best estimates for some hydrogen bonding dimers, and we will employ W1 theory for several more nonbonded complexes in the present

416 J. Chem. Theory Comput., Vol. 1, No. 3, 2005 Zhao and Truhlar

work. The strengths and limitations of W1 theory have been described elsewhere.97,104,107,108,118 2.2. HB6/04 Database. The hydrogen bond database consists of binding energies of six hydrogen bonding dimers,

(HCOOH)2 are calculated here by the W1 theory. This database is called the HB6/04 database.

2.3. CT7/04 Database. The charge transfer (CT) database consists of binding energies of seven charge-transfer com- plexes, in particular C2H4âââF2,N H3âââF2,C 2H2âââClF, HCNâ ââClF, NH3âââCl2,H 2OâââClF, and NH3âââClF. The best estimates of De for all complexes in the charge-transfer database are calculated here by the W1 model. This database is called the CT7/04 database.

2.4. DI6/04Database.The dipoleinteraction(DI) database consists of binding energies of six dipole inteaction com- plexes: (H2S)2, (HCl)2, HClâââH2S, CH3ClâââHCl, CH3SHâ ââHCN, and CH3SHâââHCl. The binding energy of (HCl)2 is taken from Boese and Martin’s80 W2 calculation. The best estimates of De for the other complexes in the dipole interaction database are calculated here by the W1 theory.

This database is called the DI6/04 database.

2.5. WI9/04 Database. The weak interaction database consists of binding energies of weak interaction complexes, namely HeNe, HeAr, Ne2, NeAr, CH4âââNe, C6H6âââNe, (CH4)2,( C2H2)2, and (C2H4)2. The binding energies of HeNe, HeAr,Ne2, and NeArare takenfromOgilvieand Wang’s129,130 analysis. The binding energy of C6H6âââNe is taken from Capplelleti et al.’s131 experimental study. The best estimates of De for CH4âââNe, (CH4)2,( C2H2)2, and (C2H4)2 are calculated by W1 theory. This database is called the WI9/

04 database.

2.6. AE6 Benchmark Database. We parametrized one new hybrid meta GGA method, namely PBE1KCIS (see Table 3). It has one parameter, the fraction X of Hartree- Fock exchange, and this was optimized against the AE6132 benchmark database of atomization energies for six covalently bonded nonmetallic molecules. We have previously used this database as a training set to optimize the MPW1B95,78 TPSS1KCIS,85 and MPW1KCIS86 methods.75 The AE6 database is listed for reference in the Supporting Information.

To parametrize the PBE1KCIS model, we optimize the fraction of Hartree-Fock exchange, X, to minimize the rootmean-square error (RMSE) for the six data in the AE6 database. The optimized X parameter for the PBE1KCIS method is given in Table 3.

2.7. Theoretical Methods Tested. We tested a number of DFT-type methods against the new four-part (HB6, CT7, DI6, WI9) nonbonded-interactiondatabase. In particular, we assessed three LSDAs: SVWN3,5,133 SVWN5,5,133 and SPWL.12,133 We tested twelve GGAs: BP86,7,8 BLYP,8,9 BPW91,8,1 BPBE,8,2 mPWPBE,2,134 G96LYP,9,20 HCTH,3 mPWLYP,9,29 mPWPW91,29 OLYP,9,49 PBE,2 and XLYP.73 We testedsevenmetaGGA methods: BB95,21 mPWB95,21,29

Table 1. Components of W1 Calculations for Binding Energies De (kcal/mol) complex

SCFalimit

CCSDblimit

(T)climit core corr &relativistic final

Hydrogen Bonding

Charge Transfer

Dipole Interaction

Weak Interaction a Hartree-Fock. b Coupled clusters theory with single and double excitations. c Quasiperturbative triple excitations.

Table 2. Benchmark Databases of Binding Energies De (kcal/mol) for Hydrogen Bonding (HB), Charge Transfer (CT), Dipole Interaction (DI), and Weak Interaction (WI)

HB6/04 CT7/04 DI6/04 WI9/04 complex De ref complex De ref complex De ref complex De ref

NH3âââClF 10.62 this work (CH4)2 0.51 this work

Benchmark Databases for Nonbonded Interactions J. Chem. Theory Comput., Vol. 1, No. 3, 2005 417

Table 3. Summary of the DFT Methods Tested (in Chronological Order) method Xa year type ex functionalb corr functionalc refs

SVWN3d 0 1981 LSDA Slater’s local ex 5, 133

VWN no.3

(Parte **1** de 8)