A Parameter-Free Density Functional That Works for Noncovalent

LETTER pubs.acs.org/JPCL

A Parameter-Free Density Functional That Works for Noncovalent Interactions Henk Eshuis* and Filipp Furche* Department of Chemistry, University of California, Irvine, 1102 Natural Sciences II, Irvine, California 92697-2025, United States

bS Supporting Information ABSTRACT: The failure of semilocal density functional theory for medium- and longrange noncovalent molecular interactions is a long-standing challenge for computational chemistry. Here, we assess the performance of the random phase approximation (RPA), a parameter-free fifth-rung functional, for reaction energies governed by changes in medium- and long-range noncovalent interactions. Our benchmark data include relative energies of alkane isomers, two sets of isomerization reactions testing intramolecular dispersion, a set of dimers of biological importance, a hierarchy of n-homodesmotic reactions, and the predissociation of a ruthenium-based Grubbs catalyst with bulky ligands. The RPA results are an order of magnitude more accurate than those of popular semilocal functionals such as PBE or B3LYP and more systematic than those of semiempirical functionals parametrized for weak interactions, such as B2PLYP-D or M06-2X. In conclusion, RPA is highly promising for thermochemical applications, particularly if noncovalent interactions are important. SECTION: Molecular Structure, Quantum Chemistry, General Theory

M

edium- and long-range noncovalent interactions play a vital role in many chemical systems and processes, such as protein folding,1 the stacking of base pairs in DNA,2 adsorption of molecules on surfaces, and covalent crystal packing.3 Modern semilocal density functional theory (DFT) fails to capture these interactions.4,5 A striking example for the shortcoming of semilocal DFT is the relative energies of alkane isomers. The energy difference between branched and linear isomers of n-alkanes is mainly due to the change in intramolecular medium-range interactions.6 While the covalent 1,2-bonding interactions between adjacent atoms are very similar for the isomers, the number and range of 1,3-interactions differ, leading to different conformer stabilities.7,8 These elementary stereoelectronic effects in n-alkanes are not accounted for correctly by semilocal DFT;6,9 for example, for the isomerization of octane to 2,2,3,3tetramethylbutane, all common semilocal density functionals yield an incorrect energetic ordering. Similar conclusions were drawn for isodesmic reactions of n-alkanes.10 Even though the mechanism underlying long-range dispersion interactions has been known since the 1930s, incorporating them in mainstream electronic calculations in a systematic and seamless fashion is still difficult. Coupled cluster singles and doubles with perturbative triples [CCSD(T)], touted the “gold standard” for noncovalent interactions,11 is limited to single-reference ground states and small systems due to its steep N7 scaling of computational cost with system size N. The less expensive second-order Møller
Plesset perturbation theory (MP2) includes long-range interactions at the uncoupled monomer level only,12 leading to mixed results for noncovalent bonding.4 This r 2011 American Chemical Society

motivated an empirical long-range correction to second-order perturbation theory in Grimme’s recent B2PLYP-D hybrid functional.13 Empirical damped C6/R6-type corrections have also been proposed in conjunction with semilocal density functionals.14 While these methods, especially the recent -D3 parameter set,15 can perform well on restricted data sets, they tend to suffer from double counting and leave the user with a large and sometimes confusing choice of different parameter sets and functional combinations. The performance of these methods for systems and oxidation states not covered by the training set is unclear. For example, dispersion coefficients can change dramatically upon ionization, which force field type corrections cannot describe. Dispersion coefficients can also be obtained from the exchange-hole dipole moment model or from related schemes.16,17 Some Minnesota functionals18,19 perform well for shorter-ranged noncovalent interactions20 but are based on extensive parametrization and statistical analysis. Similar improvements are reported by Chai and Head-Gordon for long-range corrected (double-) hybrid functionals.21 van der Waals density functionals of the Langreth
Lundqvist type22
25 are a promising alternative26 but have received little testing for intramolecular dispersion and midrange correlation. The performance of these functionals varies strongly with the exchange functional used in conjunction.25 Finally, none of the above methods except for CCSD(T) properly captures the non-pairwise-additive character Received: February 22, 2011 Accepted: April 1, 2011 Published: April 11, 2011 983

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Table 1. Mean Error (ME), Mean Absolute Error (MAE), Absolute Maximum Error (Max) (kcal/mol), and Absolute Maximum Percentage Error (Max%) for n-Butane, n-Pentane, and n-Hexane Relative Energiesa PBEb

TPSSb

B3LYPb

ME

0.61

0.71

MAE Max

0.61 1.27

0.71 1.41

error

Max%

71

78

B2PLYPb

B2PLYP-Db

B2PLYP-D3c

M06-2Xd

RPA

0.96

0.50


0.31

0.07

0.25

0.04

0.96 1.84

0.50 0.91

0.31 0.61

0.07 0.12

0.25 0.50

0.04 0.11

102

51

34

7

28

6

a

Reference values and geometries taken from ref 52. The mean absolute reaction energy is 1.8 kcal/mol, b Taken from ref 53, evaluated using the def2QZVP basis set. c Taken from ref 54, evaluated using the def2-QZVP basis set. d Taken from ref 52, evaluated using pc-2 basis sets.

of dispersion interactions.27 Clearly, a nonempirical, systematic and affordable method that includes weak interactions from the outset is highly desirable. Recently, RPA has received renewed interest as a means to compute the electron correlation energy.28
34 RPA may be viewed as a nonlocal fifth-rung functional in a density functional context35 or as a simplified coupled cluster doubles method in a wave function context.33 RPA can be seen as a seamless extension of symmetry-adapted perturbation theory at the DFT level [SAPT(DFT)]36,37 because it does not require a partitioning of the system. The RPA energy is given by ERPA ¼ EHF þ ECRPA

to large molecules consistently using large basis sets, showing that RPA performs much better for noncovalent interactions than expected from previous work. The first RPA results on a Grubbs-type transition-metal catalyst with over 100 atoms are presented. Medium-range, nonlocal electron correlations are vital for stereoelectronic effects in n-alkanes.6,7 In Table 1, we compare RPA errors in the relative energies of n-butane, n-pentane, and n-hexane conformers to commonly used functionals. The generalized gradient approximation (GGA) functional PBE,48 the meta-GGA TPSS, 49 and the hybrid functional B3LYP50 fail, producing errors comparable to the mean absolute isomer energy difference (1.8 kcal/mol). Results are slightly better for the double-hybrid B2PLYP,51 which combines a hybrid functional with a MP2-based correction. The recent -D3 dispersion correction by Grimme15 to this functional gives excellent results. The meta-GGA M06-2X is surprisingly accurate, but the maximum error still amounts to 28% of the mean energy difference. RPA, which is parameter-free, reduces the error by almost 1 order of magnitude compared to B2PLYP and outperforms all semilocal functionals. The accuracy of RPA is similar to that of B2PLYPD3, and in fact, the RPA error is comparable in magnitude to the much more expensive CCSD(T) method used as a reference.52 To further investigate the ability of RPA to capture mediumrange correlation, we consider a set of six organic reactions (dubbed IDISP) by Goerigk and Grimme53 where intramolecular dispersion effects play a crucial role (Table 2). Semilocal functionals perform poorly for IDISP; the best results are obtained using M06-2X or the double-hybrid B2PLYP with semiempirical dispersion correction. RPA performs remarkably well, yielding the lowest MAE. Importantly, RPA is almost exact for the 2-octane isomerization to iso-octane, which has been proposed as a mandatory benchmark for new density functionals.6 The high accuracy of RPA for this reaction is in line with the excellent performance of RPA for alkane conformers. In addition, the reaction energy for the anthracene dimerization is predicted almost quantitatively with RPA. Among the other functionals, only B2PLYP-D and M06-2X predict the sign of this reaction correctly. For the IDISP set, B2PLYP-D improves considerably upon semilocal functionals (and performs better than B2PLYPD354), but it is empirical and moreover cannot be applied to small gap systems, in contrast to RPA. In our implementation, RPA scales more favorably than B2PLYP, whose cost increases as N5 with the system size N. Similar results are obtained for errors in isomerization energies of 24 large (24
81 atoms) organic molecules (Figure 1).56 Dispersion effects become more important with increasing system size. PBE057 with dispersion correction and the recent double-hybrid B2GP-PLYP-D58 perform well, but RPA outperforms all density functional methods. The errors in RPA are

ð1Þ

where the Hartree
Fock (HF) energy and the RPA correlation energy are both evaluated using Kohn
Sham orbitals.30 In the plasmon picture,38 the correlation energy arises from zero-point vibrational energies of virtual electronic transitions31 ECRPA ¼

1 2

∑n ðΩn
ΩDn Þ

ð2Þ

The Ωn are direct RPA excitation energies that could be obtained from a time-dependent Hartree calculation,33,39,40 and the ΩD n are their first-order equivalents. For weakly interacting monomers, eq 2 generates virtual dipole
dipole, dipole
quadrupole, and higher multipole interactions using monomer transition moments at the coupled RPA level. Thus, RPA is dispersionconsistent,12,41 as opposed to, for example, MP2. Being parameter-free, RPA can be systematically improved by adding beyond-RPA exchange and correlation,42
44 much like the Hartree method is improved by adding ordinary exchange and correlation. RPA is computationally much more involved than semilocal DFT and was long considered too expensive for widespread use in chemistry. However, recent progress in the implementation of RPA using the resolution-of-the-identity (RI) approach and frequency integration has fundamentally changed the scope of RPA,31,45 making applications to systems with over 100 atoms in large basis sets possible. Little is known about the performance of RPA for thermochemistry. For rare gas dimers, RPA captures the long-range part of the interaction correctly,44,46 though for smaller interatomic distances, underbinding can occur. RPA does not improve significantly upon semilocal functionals for atomization energies.47 Total correlation energies are poor,30 but it was speculated that energy differences for isoelectronic reactions might be accurate.28 Recent studies suggested that RPA might be viable for weakly bonded systems of biological importance,44,46 but small basis sets were used. Here, for the first time, we present RPA results for medium- and long-range interactions in medium 984

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Table 2. Errors, Mean Errors (ME), and Mean Absolute Errors (MAE) in Reaction Energies (kcal/mol) for Systems Including Intramolecular Dispersion Interactions (IDISP set)a reaction 2anthracene f anthracene dimer 2p-xylene f [2.2]-paracyclophane þ 2H2 n-octane f iso-octane

PBE

TPSS

B3LYP

B2PLYP

B2PLYP-D

M06-2X

RPA

23.24

29.30

34.59

16.29

4.26

5.33


3.73


10.48 7.38


11.21 8.39


18.24 10.35


8.69 5.12


0.60
0.61


3.16 0.33

0.05 0.49
2.64

n-undecane f 2,2,3,3,4,4-hexamethylpentane

12.74

13.57

18.69

9.24


2.03

0.38

(C14H30)linear f (C14H30)folded


3.17


3.64


4.41


2.48

0.52


0.08

0.17

(C22H46)linear f (C22H46)folded


13.78


16.88


18.53


10.31

2.49


0.74


0.43

ME MAE

2.65

3.25

3.74

1.53

0.67

0.34


1.02

11.80

13.83

17.47

8.69

1.75

1.67

1.25

a

Non-RPA results were taken from refs 53 and 55 (M06-2X). All energies were computed using the def2-QZVP basis set, except M06-2X (cc-pVTZ). Structures and reference values are based on experimental or theoretical data; see ref 53 for details.

dimer to DNA base pairs, with a variety of interactions (hydrogen bonds, dispersion interactions, and mixed).62 This set, S22, has become a popular benchmark for the performance of electronic structure methods. Figure 2 shows that RPA performs as well as B2PLYP-D3 and M06-2X. Again, RPA is a vast improvement upon semilocal density functionals. The considerably larger RPA errors for the S22 set previously reported in the literature appear to be due to basis set incompleteness.46 We now assess the performance of RPA for a recently proposed n-homodesmotic hierarchy63 of bond separation reactions satisfying an increasingly strict balance of the chemical bonding environment. This hierarchy ranges from breaking of all bonds (atomization), over changes in covalent bonding (isogyric), to changes in the intramolecular environment involving three (isodesmic), four ((hypo)homodesmotic), and five (hyperhomodesmotic) non-hydrogen-atom fragments.64 The aim of this hierarchy is to predict accurate enthalpies of formation at low levels of theory. Here, we use it to assess and compare the consistency of electronic structure methods. Accurate methods should yield small errors, and more importantly, the errors should decrease with n. We consider three sets of hydrocarbons proposed by Wheeler et al.63 The first set consists of conjugated hydrocarbons, the second consists of nonconjugated hydrocarbons, and the third includes two ring structures. RPA atomization energies (RC0) for all three sets are poor, especially compared to the good performance of B3LYP and M06-2X (see Table 3). A substantial improvement is obtained, though, for the isogyric class (RC1), which includes bond breaking and making. Here, for all three sets, RPA yields smaller errors than B3LYP and M06-2X. This is important because bond breaking and making underly chemical reactions, and a model that captures these successfully should be applicable to a wide range of chemical processes.64 For isodesmic classes (RC2), RPA errors are of similar magnitude for all three sets, indicating that RPA behaves systematically. Errors for B3LYP and M06-2X show much larger scatter. Errors for the homodesmotic classes (RC4-RC6) are mostly small due to cancellation, and the difference between methods is blurred. RPA errors are smaller than 0.75 kcal/mol for all cases, whereas for the other two functionals, errors are larger for the ring structures, presumably due to ring strain. Of all investigated methods, RPA displays the most systematic behavior for the three sets of reactions. Finally, we apply RPA to the tricyclohexylphosphine [(PCy)3] dissociation of a the Grubbs second-generation catalyst, (H2IMes)(PCy3)(Cl)2RudCHPh (1) used for olefin metathesis.67
69 (1) is part of a class of Grubbs catalysts of the general formula

Figure 1. Mean absolute error (MAE) and maximum error (Max) in isomerization energies (kcal/mol) of 24 large organic molecules for several methods. M06-2X results were taken from ref 59. Structures and other non-RPA results were taken from ref 56. RPA and M06-2X results were obtained using the def2-QZVP basis set, and all others using QZV3P.

Figure 2. Mean error (ME), mean absolute error (MAE), and maximum error (Max) in isomerization energies (kcal/mol) of the S22 test set for several methods. M06-2X results are from ref 60 in the 6-311þ G(3df,2p) basis. Structures and other non-RPA results were taken from ref 54. Reference energies were taken from ref 61.

comparable to the estimated errors in the reference energies, which were obtained using spin-component scaled third-order Møller
Plesset perturbation theory with extrapolation to the basis set limit (SCS-MP3/CBS).56 Jurecka et al. constructed a test set of 22 noncovalently bonded dimers of biological importance, ranging in size from the water 985

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Table 3. Mean Absolute Errors (kcal/mol) for the Energetics of Bond Separation Reactionsa of the n-Homodesmotic Reaction Classes reaction classb

B3LYPc

M06-2Xc

RPAd

CCSD(T)d

Conjugated Hydrocarbons (Set A) RC0

7.03

1.96

65.92

RC1

9.87

10.51

7.27

35.10 2.93

RC2

7.06

4.47

1.65

0.31

RC3 RC4

1.75 0.24

0.98 0.22

0.53 0.20

0.55 0.17

RC5

0.11

0.19

0.10

0.14

Figure 3. Errors in reaction enthalpies for PCy3 dissociation (kcal/mol) for various functionals. Non-RPA results are taken from ref 72 (def2TZVPP77 basis sets, with geometries optimized per functional using SVP77 basis sets). For RPA, the PBE-optimized geometry was used, with the def2-TZVPP basis sets on H and C atoms and def2-QZVPP77 basis sets on Ru, P, Cl, and N. All results are corrected for ZPVEs. For RPA, the ZPVE was computed at the TPSS level. The reference value is the experimental collision-induced dissociation threshold energy for the gas phase, 36.8 ( 2.4 kcal/mol.76

Nonconjugated Hydrocarbons (Set B) RC0

10.62

3.18

63.48

37.00

RC1

13.02

8.06

6.20

2.85

RC2

2.67

1.30

2.42

0.16

RC4e

0.38

0.18

0.39

0.19

RC5

0.14

0.21

0.05

0.19

Cyclopentadiene and 1,3-Cyclohexadiene (Set C) RC0 RC1

10.72 13.51

2.17 10.69

54.61 7.61

34.97 3.71

RC2

0.14

0.58

2.23

0.59

RC3

0.37

1.40

0.75

0.13

RC4

1.45

1.82

0.67

0.15

RC5

1.15

2.16

0.26

0.35

RPA comes close to the experimental dissociation energy, and improves considerably upon popular semilocal functionals, whose errors are more than half of the experimental enthalpy of 36.8 kcal/mol.76 Application of RPA to the smaller model systems yields similar errors (see Supporting Information), and thus, RPA behaves systematically for both the model and real catalyst. As bond breaking is involved in this process, short-range correlation is important, which might explain the relatively large RPA error compared to M06. This points to the need for “beyond-RPA” methods.

a

Non-RPA results from ref 63. Errors are relative to focal point analysis benchmarks. All energies were evaluated at B3LYP/6-31G(d) optimized geometries. b Reaction classes: RC0, atomization; RC1, isogyric; RC2, isodesmic; RC3, hypohomodesmotic; RC4, homodesmotic; RC5, hyperhomodesmotic. c Using the 6-31G(d)65 basis set. d Using the cc-pVTZ66 basis set. e For set B, RC3 is equal to RC4.

[L(PR3)X2RudCHR0 ]; their high activity and functional group tolerance have made these catalysts increasingly popular in organic synthesis.70 The mechanism of olefin metathesis involving Grubbs catalysts has been the subject of many computational studies.70 However, because both transition metal to ligand and carbon
carbon bonds are broken, olefin metathesis is challenging for theory. Here, we focus on the well-studied first part of the reaction mechanism, namely, the phosphine dissociation of (1) ðH2 IMesÞðPCy 3 ÞðClÞ2 RudCHPh This example also demonstrates the computational efficiency of RPA. With our present preliminary implementation, the computation of the RPA correlation energy for (1), which contains 117 atoms, takes approximately 8 CPU hours on 16 processors of a single 2.2 GHz AMD Opteron 6174 node using multithreaded BLAS routines and 10 Gb of memory. A CCSD(T) calculation for (1) is presently not feasible even on supercomputers. There are two fundamentally different approaches to improving upon semilocal density functionals. One is to add simple corrections and introduce parameters fitted to empirical data. This approach can be very useful on restricted data sets and yield excellent results at very low computational cost. The downsides include a lack of transparency, outliers that statistics do not account for, artifacts from overfitting, and too much user choice. The other approach is to use a parameter-free model that captures the essential physics missing in semilocal functionals. This strategy requires much longer development cycles and produces methods

f ðH2 IMesÞðClÞ2 RudCHPh þ PCy 3 which leads to a catalytically active 14-electron Ru complex.71 Because (1) contains a transition-metal atom and has a relatively small gap, perturbation theory is not the method of choice here. Several authors have benchmarked DFT for model Grubbs catalysts against high-level theoretical methods.70,72,73 In these models, the large ligands are substituted by smaller side groups. An important conclusion is that trends obtained for the smaller models cannot be readily transferred to the real system (1) because of the noncovalent mid- and long-range interactions of the bulky ligands with the reaction center.71,74,75 For example, PBE0 was found to describe the phosphine dissociation for the model systems accurately, but it fails for the real catalyst.71 The M06 functional was found to perform best for both scenarios.71 To gauge the performance of RPA, errors in reaction enthalpies (kcal/mol) for the dissociation of (1) are presented in Figure 3. 986

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whose up-front costs are higher. However, parameter-free methods are more robust and have more predictive power than semiempirical models. To describe mid- and long-range noncovalent interactions correctly for a large range of systems and properties, a model must include long-range correlations between induced density fluctuations, which is the fundamental mechanism underlying these interactions. Of all density functionals tested here, RPA is the only one satisfying this condition. This is why RPA achieves uniformly high accuracy for noncovalent interactions.

(4) Sherrill, C. D. In Rev. Comp. Chem.; John Wiley & Sons, Inc.: Hoboken, NJ, 2008; Vol. 26, pp 1
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erny , J.; Hobza, P. Non-covalent Interactions in Biomacromo(5) C lecules. Phys. Chem. Chem. Phys. 2007, 9, 5291. (6) Grimme, S. Seemingly Simple Stereoelectronic Effects in Alkane Isomers and the Implications for Kohn
Sham Density Functional Theory. Angew. Chem., Int. Ed. 2006, 45, 4460–4464. (7) Wodrich, M. D.; Jana, D. F.; Schleyer, P. V. R.; Corminboeuf, C. Empirical Corrections to Density Functional Theory Highlight the Importance of Nonbonded Intramolecular Interactions in Alkanes. J. Phys. Chem. A 2008, 112, 11495–11500. (8) Kemnitz, C. R.; Mackey, J. L.; Loewen, M. J.; Hargrove, J. L.; Lewis, J. L.; Hawkins, W. E.; Nielsen, A. F. Origin of Stability in Branched Alkanes. Chem.—Eur. J. 2010, 16, 6942. (9) Wodrich, M. D.; Corminboeuf, C.; Schleyer, P. V. R. Systematic Errors in Computed Alkane Energies Using B3LYP and Other Popular DFT Functionals. Org. Lett. 2006, 8, 3631–3634. (10) Grimme, S. n-Alkane Isodesmic Reaction Energy Errors in Density Functional Theory Are Due to Electron Correlation Effects. Org. Lett. 2010, 12, 4670–4673. (11) Cramer, C. J. Essentials of Computational Chemistry: Theories and Models, 2nd ed.; Wiley: New York, 2004; p 226. (12) Szabo, A.; Ostlund, N. S. The Correlation Energy in the Random Phase Approximation: Intermolecular Forces Between Closed-Shell Systems. J. Chem. Phys. 1977, 67, 4351–4360. (13) Schwabe, T.; Grimme, S. Double-Hybrid Density Functionals with Long-Range Dispersion Corrections: Higher Accuracy and Extended Applicability. Phys. Chem. Chem. Phys. 2007, 9, 3397–3406. (14) Grimme, S. Accurate Description of Van der Waals Complexes by Density Functional Theory Including Empirical Corrections. J. Comput. Chem. 2004, 25, 1463–1473. (15) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H
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’ COMPUTATIONAL DETAILS All calculations were performed using TURBOMOLE.78 Input orbitals were obtained using the TPSS functional49 with the RIDFT module of TURBOMOLE. Using PBE input orbitals instead of TPSS orbitals for the alkane conformers resulted in very similar reaction energies (see Supporting Information). Tight integration grids were used throughout (m5).79 RPA correlation energies were obtained with a modified version of the RIMP2 module,80 using the RI approximation and frequency integration.45 The frequency integration was performed using 50 grid points. The def2-QZVP77 basis sets and corresponding auxiliary basis sets were used, unless otherwise stated. Basis set extrapolation47 using quadruple and quintuple zeta valence sets showed little improvement for n-alkane isomerization energies (see Supporting Information). Core electrons were kept frozen. ’ ASSOCIATED CONTENT

bS

Supporting Information. Basis set convergence and extrapolation results for the alkane conformer set. RPA results for the individual processes in the alkane conformer set, the 24 large organic molecules set, the S22 set, and the three sets used in the n-homodesmotic hierarchy. RPA results for a model Grubbs II catalyst, compared to other density functionals. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (H.E.); Filipp.Furche@ uci.edu (F.F.).

’ ACKNOWLEDGMENT The authors would like to thank R. Ahlrichs for helpful comments. This work was supported by the National Science Foundation under Grant No. CHE-0911266. ’ REFERENCES (1) Bashford, D.; Chothia, C.; Lesk, A. M. Determinants of a Protein Fold: Unique Features of the Globin Amino Acid Sequences. J. Mol. Biol. 1987, 196, 199–216. (2) Da-bkowska, I.; Gonzalez, H. V.; Jurec
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