共Received 13 July 2009; accepted 17 September 2009; published online 3 November 2009兲

Long-range corrected double-hybrid density functionals

Jeng-Da Chai

and Martin Head-Gordon

1Department of Chemistry, University of California, Berkeley, Berkeley, California 94720, USA

and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

2Department of Physics, National Taiwan University, Taipei 10617, Taiwan

共Received 13 July 2009; accepted 17 September 2009; published online 3 November 2009兲

We extend the range of applicability of our previous long-range corrected 共LC兲 hybrid functional,

␻ ^B97X 关J.-D. Chai and M. Head-Gordon, J. Chem. Phys. 128, 084106 共2008兲兴, with a nonlocal description of electron correlation, inspired by second-order Møller–Plesset 共many-body兲 perturbation theory. This LC “double-hybrid” density functional, denoted as ␻ B97X-2, is fully optimized both at the complete basis set limit 共using 2-point extrapolation from calculations using triple and quadruple zeta basis sets兲, and also separately using the somewhat less expensive 6-311+ +G共3df ,3pd兲 basis. On independent test calculations 共as well as training set results兲,

␻ B97X-2 yields high accuracy for thermochemistry, kinetics, and noncovalent interactions. In addition, owing to its high fraction of exact Hartree–Fock exchange, ␻ B97X-2 shows significant improvement for the systems where self-interaction errors are severe, such as symmetric homonuclear radical cations. © 2009 American Institute of Physics. 关doi:10.1063/1.3244209兴

I. INTRODUCTION

Over the past two decades, Kohn–Sham density func- tional theory 共KS-DFT兲

has become the most popular elec- tronic structure method for midsize and larger molecular sys- tems due to its satisfactory accuracy and reasonable computational costs.

Although the essential ingredient of KS-DFT, the exchange-correlation density functional E

, has not been exactly known, functionals based on semilocal generalized gradient approximations have performed well in many solid-state applications. Aiming to incorporate some of the nonlocal effects missing in semilocal approximations, hy- brid DFT methods, first proposed by Becke,

have further reduced the remaining errors of semilocal density function- als, which has further expanded the usefulness of DFT for practical calculations.

In common hybrid functionals, a small fraction, c

, of the exact Hartree–Fock 共HF兲 exchange is added to a semilo- cal density functional. However, in certain situations, a large fraction 共even 100%兲 of HF exchange is needed, and such global hybrid functionals can fail qualitatively. These situa- tions mostly occur in the asymptotic regions of molecular systems.

To remedy this, the development of long-range corrected 共LC兲 hybrid functional methods

has recently become an active research direction to obtain improved mod- els for E

. In the LC approach, exact HF exchange is em- ployed for the long-range 共LR兲 part of the interelectron re- pulsion operator, and an approximated exchange density functional is employed for the complementary short-range 共SR兲 part, while the treatment for correlation remains the same 共at least in functional form兲 as for semilocal and com-

mon hybrid functionals. Due to the inclusion of 100% LR exact exchange, this approach has considerably reduced some qualitative failures of semilocal and global hybrid functional methods.

Two issues are important to make an effective LC-DFT.

One is the development of accurate SR exchange density functionals, and the other is to find a suitable sepa- rator for the SR/LR decomposition. In the first LC scheme, an ansatz for the conversion of any E

to E

, was proposed by Iikura et al.,

and has become widely used. However, the resulting LC hybrid generalized gradient approximation 共GGA兲 functionals and variants do not outperform global hybrid GGA functionals for thermochemistry. In 2006, a dif- ferent LC scheme, was proposed by Vydrov et al.,

based on integrating the PBE model of the exchange hole. Their LC- ␻ PBE functional shows improved performance for ther- mochemistry and barrier heights 共BH兲, and is comparable to global hybrid GGA functionals such as B3LYP.

However, further improvements to this approach depend on more ac- curate modeling of the exchange hole, which is challenging.

As an alternative path to more accurate LC functionals, we have recently proposed a simple ansatz to extend any E

to E

, for a SR operator that still retains considerable spatial extent.

First, the SR local spin density exchange energy density is augmented by a flexible basis functional 共we em- ploy Becke’s 1997

form兲 which is to be fully reoptimized to be appropriate for SR exchange 共as is the corresponding correlation functional兲. This yields the LC hybrid ␻ ^B97 functional. Second, recognizing that relatively small optimal

␻ values mean that SR exchange still has finite extent, we also include an adjustable fraction of exact SR exchange to define the ␻ B97X functional. ␻ ^{B97 and} ␻ B97X have shown to be accurate across a diverse set of test data, containing thermochemistry, kinetics, and noncovalent interactions.

We have also considered the problem of optimal parti- tioning, and identified a conserved property 共the fraction of

a兲Electronic mail: [email protected].

b兲Author to whom correspondence should be addressed. Electronic mail:

[email protected].

0021-9606/2009/131共17兲/174105/13/$25.00 131, 174105-1 © 2009 American Institute of Physics

exact exchange retained at midrange—approximately 0.4兲 of the optimal HF operators for the ␻ B97X functional opti- mized at each value of ␻ ^.

We have argued that this is due to the appropriateness of the underlying GGA when mixed with HF according to the distance criterion. Further support for this argument is presented elsewhere.

Similar conclusions have also been made by others with different arguments.

However, other GGA problems associated with the lack of nonlocality of the correlation hole, such as the lack of dispersion interactions 共van der Waals forces are missing兲, are not resolved by the LC hybrid scheme, as the correlation functionals in typical LC hybrids are treated only semilo- cally, which cannot capture LR correlation effects.

In our previous work, we followed the DFT-D scheme

to extend our ␻ B97X with damped atom-atom dispersion corrections, denoted as ␻ ^B97X-D.

␻ B97X-D allows us to obtain dis- persive effects with essentially zero additional computational cost relative to ␻ B97X. However, due to the smaller optimal value of ␻ ⁱⁿ ␻ ^B97X-D, 共 ␻ ^{= 0.2 Bohr}

兲 the self-interaction error 共SIE兲 of ␻ B97X-D is larger than that of ␻ ^B97X 共 ␻

= 0.3 Bohr

兲 and ␻ ^B97 共 ␻ ^{= 0.4 Bohr}

兲. In addition, the use of “-D” corrections introduces a large number of fixed empirical parameters into the functional 共though only a

damping parameter was explicitly optimized in the develop- ment of ␻ B97X-D兲, which is undesirable for first-principles methods.

There are other methods that are under active develop- ment for treating nonlocal correlations. The van der Waals density functional approach pioneered by Rydberg et al.

is one promising direction, though numerical integrations are required, and the level of accuracy is still in the process of being brought to a satisfactory level.

The real-space post-HF models of Becke and Johnson

are a second direc- tion that shows great promise. A third direction attracting considerable interest is the addition of nonlocal correlation based on the random phase approximation.

To date, however, perhaps the most successful approach in practice to including nonlocal correlation effects is pro- vided by the double-hybrid 共DH兲 methods,

which mix both the HF exchange and a nonlocal orbital correlation en- ergy from the second-order perturbation energy expression in wave function theory. There are typically only 1 or 2 empiri- cal parameters for scaling the components of nonlocal corre- lation. Moreover, due to the presence of nonlocal correlation in DH functionals, the corresponding fractions of HF ex- change is noticeably higher than for conventional hybrid functionals, as HF exchange is compatible with nonlocal cor- relation. The sharp increase in HF exchange in DH function- als thus greatly reduces the SIE relative to typical hybrid functionals.

In this work, we demonstrate the usefulness of a com- bined LC and DH scheme. The performance of this LCDH functional is compared with other hybrid, LC hybrid, and DH functionals.

II. THE LCDH SCHEME

Similar to the existing DH scheme,

LC-DFT can be extended to include nonlocal orbital correlation energy from second-order Møller–Plesset perturbation 共MP2兲 theory,

that includes a same-spin 共ss兲 component E

and an opposite-spin 共os兲 component E

. The two parameters, c

and c

, are introduced to adjust the amounts of E

and E

in LCDH.

E

= E

+ c

E

+ c

E

. 共1兲

The first LCDH functional, mixing PT2 correlation with a LC hybrid functional, was proposed by Ángyán et al.

In their LCDH functional, the SR xc energy was treated by density functional and the LR part was treated by MP2 theory. The relevant work in this direction has had promising results for nonbonded complexes.

By contrast, our LCDH in Eq. 共1兲 is a simplified version. The exchange part is treated by the range-separated hybrids, while the correlation part is treated by global hybrids. Our simplified LCDH func- tional thus avoids the need to model the SR correlation,

TABLE I. Basis sets used for ␻^B97X-2共TQZ兲 on the training set.

cc-pV共TQ兲Z denotes the TQ extrapolation to basis set limit used for PT2 correlation.

System Basis set

G3/99共223兲 cc-pV共TQ兲Z

IP共40兲 cc-pV共TQ兲Z

EA共25兲 aug-cc-pV共TQ兲Z

PA共8兲 cc-pV共TQ兲Z

NHTBH共38兲 aug-cc-pV共TQ兲Z

HTBH共38兲 aug-cc-pV共TQ兲Z

S22共22兲 cc-pV共TQ兲Z

Atoms共10兲 aug-cc-pCV共TQ兲Z

TABLE II. Optimized parameters for the ␻^B97X-2共TQZ兲 and for the

␻^B97X-2共LP兲. Here, the same-spin PT2 coefficient css, and the opposite- spin PT2 coefficient c_os, are defined in Eq.共1兲, and others are defined in Eq.

共28兲 of Ref.10.

␻^B97X-2共TQZ兲 ␻^B97X-2共LP兲

␻ ^{0.3 Bohr−1 0.3 Bohr−1}

cx␴,0 3.15503⫻10⁻¹ 2.51767⫻10⁻¹

cc␴␴,0 9.08460⫻10⁻¹ 1.15698

cc␣␤,0 5.18198⫻10⁻¹ 5.53261⫻10⁻¹

cx␴,1 1.04772 1.57375

cc␴␴,1 ⫺2.80936 ⫺3.31669

cc␣␤,1 −5.85956⫻10⁻¹ ⫺1.16626

cx␴,2 ⫺2.33506 ⫺5.26624

cc␴␴,2 6.02676 6.27265

c_c␣␤,2 4.27080 6.84409

cx␴,3 3.19909 6.74313

c_c␴␴,3 ⫺4.56981 ⫺4.51464

cc␣␤,3 ⫺6.48897 ⫺8.90640

c_x 6.36158⫻10⁻¹ 6.78792⫻10⁻¹

c_ss 5.29319⫻10⁻¹ 5.81569⫻10⁻¹

c_os 4.47105⫻10⁻¹ 4.77992⫻10⁻¹

which is not widely available,

compared to the availability of SR exchange. We adopt our ␻ ^B97X 关see Eq. 共28兲 of Ref.

10 兴 for the LC-DFT part and denote this new LCDH func- tional as ␻ B97X-2, where the “ ⫺2” refers to the post-KS treatment for the nonlocal 共orbital兲 correlation energy taken from second-order perturbation theory.

To achieve an optimized functional for well-balanced performance across typical chemical applications, we use the same diverse training set described in Ref. 10, which con- tains 404 accurate experimental and accurate theoretical re- sults, including the 10 atomic energies from the H atom to the Ne atom,

the atomization energies 共AEs兲 of the G3/99 set

共223 molecules兲, the ionization potentials 共IPs兲 of the G2-1 set

关40 molecules, excluding SH

共

A

兲 and N

共

⌸兲 cations due to the known convergence problems for semilo- cal density functionals

兴, the electron affinities 共EAs兲 of the G2-1 set 共25 molecules兲, the proton affinities 共PAs兲 of the G2-1 set 共8 molecules兲, the 76 BHs of the NHTBH38/04 and HTBH38/04 sets,

and the 22 noncovalent interactions of the S22 set.

To adequately converge the PT2 correlation energy to- ward the complete basis set limit, E

, we adopt the standard two-point extrapolation

for cc-pVXZ 共or aug-cc-pVXZ or aug-cc-pCVXZ兲 to the expression

E

= E共X兲 + cX

, 共2兲

where c is a constant, and X is 3 for T and 4 for Q.

We denote our LCDH functional parameterized with this TQ extrapolation as ␻ B97X-2共TQZ兲. For certain types of calculations, such as EAs and BHs, the use of diffuse basis sets is found to be important, and we adopt the correspond- ing “aug” 共augmented兲 basis sets. For atomic energies, we further include the “core” effects with core-valence 共CV兲 ba- sis set, aug-cc-pCVXZ. For XC grids, we use the extra fine grid, EML 共99,590兲, consisting of 99 Euler–Maclaurin radial grid points

and 590 Lebedev angular grid points

for atomic energies, and EML共75,302兲 for others. In Table I, we summarize the basis sets used to parameterize

␻ B97X-2共TQZ兲. For some calculations, the resolution-of- identity 共RI兲 approximation

is used for the PT2 calculations with suitable auxiliary basis sets.

For practical calculations on large systems, the use of the large cc-pVQZ or aug-cc-pVQZ basis sets becomes compu- tationally prohibitive. To partially circumvent this limitation, we have also parameterized the same LCDH functional with the Large Pople 共LP兲 type basis set, 6-311++G共3df ,3pd兲, which we denote as ␻ B97X-2共LP兲. We use the LP basis set and the SG-1 grid

together with the RI approximation for PT2 calculations, where very large auxiliary basis sets 共i.e., cc-pVQZ or aug-cc-pVQZ兲 are used.

To prevent double-counting of energy contributions from the LC-DF and from the PT2 correlation, all the parameters in ␻ B97X-2, are determined self-consistently by the least- squares fitting procedure described in Ref. 10. For the non-

TABLE III. Statistical errors共in kcal/mol兲 of the training set. The B97-2共LP兲^ⴱfunctional is defined in the text. The results for␻^B97X-2共TQZ兲 are obtained with the basis sets and extrapolation scheme described in TableIand in the text. The results for the␻B97X-D are taken from Ref.11, and the results for the

␻^{B97X and}␻B97 are taken from Ref.10.

System Error ␻B97X-2共TQZ兲 ␻B97X-2共LP兲 B97-2共LP兲^ⴱ ␻B97X-D ␻B97X ␻B97 B2PLYP-D^ⴱ B2PLYP^ⴱ SCS-MP2^ⴱ

G3/99共223兲 MSE 0.02 ⫺0.10 ⫺0.05 ⫺0.10 ⫺0.09 ⫺0.20 ⫺0.56 0.10 0.36

MAE 1.43 1.52 1.46 1.93 2.09 2.56 2.44 3.34 4.19

rms 2.28 2.07 2.02 2.75 2.86 3.51 3.37 4.32 5.55

IP共40兲 MSE ⫺0.66 ⫺0.14 0.19 0.19 ⫺0.15 ⫺0.48 ⫺0.98 ⫺0.91 ⫺2.21

MAE 1.57 1.73 1.71 2.74 2.69 2.65 2.26 2.21 3.82

rms 2.16 2.37 2.38 3.62 3.59 3.58 2.69 2.61 5.47

EA共25兲 MSE 0.06 ⫺0.29 0.03 0.10 ⫺0.43 ⫺1.45 ⫺1.43 ⫺1.11 ⫺2.85

MAE 1.42 1.56 1.47 1.92 2.05 2.67 2.08 1.75 4.75

rms 1.99 2.23 2.42 2.40 2.59 3.10 2.37 2.09 6.40

PA共8兲 MSE ⫺1.06 ⫺0.80 ⫺0.71 1.49 0.60 0.68 ⫺0.89 ⫺1.16 0.57

MAE 1.19 1.09 0.97 1.54 1.22 1.45 1.06 1.19 1.22

rms 1.39 1.32 1.27 2.11 1.72 2.17 1.31 1.43 1.32

NHTBH共38兲 MSE 0.48 0.54 0.21 ⫺0.42 0.56 1.32 ⫺2.38 ⫺2.34 6.47

MAE 1.29 1.67 1.88 1.51 1.75 2.31 2.52 2.48 6.52

rms 1.99 2.53 2.85 2.00 2.08 2.82 3.04 3.07 9.39

HTBH共38兲 MSE ⫺0.04 ⫺0.13 ⫺0.33 ⫺2.52 ⫺1.51 ⫺0.34 ⫺2.60 ⫺2.35 7.32

MAE 0.65 0.74 0.82 2.64 2.24 2.24 2.62 2.43 7.32

rms 0.83 0.94 1.01 3.04 2.58 2.62 2.78 2.63 15.10

S22共22兲 MSE ⫺0.10 ⫺0.08 ⫺0.07 ⫺0.08 0.53 0.16 ⫺0.08 2.23 1.13

MAE 0.26 0.24 0.23 0.22 0.87 0.60 0.16 2.23 1.13

rms 0.29 0.28 0.28 0.27 1.30 0.80 0.22 2.81 1.51

Atoms共10兲 MSE ⫺0.20 ⫺0.08 0.57 0.47 0.17 ⫺0.79 1.89 2.06 37.74

MAE 2.14 2.49 2.84 2.77 1.63 3.53 3.18 3.46 37.74

rms 2.67 2.94 3.30 3.25 1.91 4.26 3.85 4.17 45.26

All共404兲 MSE ⫺0.04 ⫺0.07 ⫺0.02 ⫺0.27 ⫺0.14 ⫺0.15 ⫺0.94 ⫺0.40 2.11

MAE 1.31 1.43 1.42 1.96 2.04 2.42 2.29 2.87 5.31

rms 2.06 2.04 2.08 2.73 2.73 3.26 3.06 3.70 10.15

linear parameter optimization, we focus on a discrete set of possible ␻ ^values 共0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 Bohr

兲.

The functional expansions employed in ␻ B97X-2 are trun- cated at m = 3, and the S22 data is weighted ten times more than others.

Although the ␻ ^{value of} ␻ B97X-2 was found to be op- timal at 0.2 Bohr

, the rms errors at each ␻ are, however, very insensitive. We thus fix the final ␻ value to be 0.3 Bohr

, based on comparably good performance in the training set as well improved performance in systems with severe SIE issues. The optimized parameters of the

␻ B97X-2 functional parametrized with TQ extrapolation to the basis set limit 共TQZ兲 and with the LP basis sets are given in Table II.

The limiting case where ␻ ^{= 0 for} ␻ B97X-2 is also very interesting, as it reduces to a reoptimized B97 functional

augmented with the scaled PT2 correlation. For comparisons within the training set, we denote this reoptimized functional 共parameterized with the LP basis set兲 as B97-2共LP兲

. We also reoptimize B2PLYP-D, B2PLYP, and SCS-MP2 on the same training set and with the same enhanced weight for the S22 set. We denote these reoptimized functionals as B2PLYP-D

, B2PLYP

, and SCS-MP2

. The overall performance of all these functionals parameterized on the same training set 关in- cluding our previous LC hybrid functionals, ␻ ^B97,

␻ ^B97X,

^and ␻ ^B97X-D 共Ref. 11兲兴 is shown in Table III The overall performance of B2PLYP,

B2PLYP-D,

SCS-MP2,

and MP2,

although not optimized in the train- ing set, is shown in Table IV for comparison with their re-

optimized versions. Note that the SCS-MP2 and MP2 meth- ods are truly self-interaction-free methods with similar cost as all the DH and LCDH functionals.

III. RESULTS AND DISCUSSION A. The training set

All calculations are performed with a development ver- sion of Q-Chem 3.0.

Spin-restricted theory is used for sin- glet states and spin-unrestricted theory for others. For the binding energies of the weakly bound systems, the counter- poise correction

is employed to reduce basis set superposi- tion errors 共BSSE兲.

The error for each entry is defined as error

= theoretical value− reference value. The notation used for characterizing statistical errors is as follows: mean signed errors 共MSEs兲, mean absolute errors 共MAEs兲, rms errors, maximum negative errors 关max共⫺兲兴, and maximum positive errors 关max共+兲兴.

As can been seen in Table III, our new LCDH function- als, ␻ B97X-2共TQZ兲 and ␻ B97X-2共LP兲, achieve very high accuracy. Many comparisons are possible, and we summa- rize a few of the most important conclusions in the following paragraphs.

Let us compare the new LCDH functionals to other DH functionals trained on the same data. It is evident that B97-2共LP兲

performs almost identically to ␻ B97X-2共LP兲, since, as already discussed, the rms errors are very insensi-

TABLE IV. Statistical errors共in kcal/mol兲 of the training set.

System Error B2PLYP-D B2PLYP SCS-MP2 MP2

G3/99共223兲 MSE ⫺1.48 ⫺3.21 ⫺1.58 5.42

MAE 2.63 3.96 4.03 9.16

rms 3.60 5.26 5.34 12.44

IP共40兲 MSE ⫺1.04 ⫺1.04 ⫺2.40 ⫺1.69

MAE 2.29 2.29 3.98 3.59

rms 2.73 2.72 5.58 5.00

EA共25兲 MSE ⫺1.51 ⫺1.50 ⫺3.14 ⫺1.67

MAE 2.17 2.17 5.03 3.95

rms 2.45 2.44 6.57 5.67

PA共8兲 MSE ⫺0.90 ⫺1.08 0.75 ⫺0.93

MAE 1.08 1.17 1.32 0.96

rms 1.33 1.39 1.46 1.20

NHTBH共38兲 MSE ⫺2.29 ⫺2.00 6.61 5.37

MAE 2.44 2.19 6.65 5.48

rms 2.94 2.68 9.49 8.45

HTBH共38兲 MSE ⫺2.49 ⫺2.01 7.55 5.84

MAE 2.52 2.11 7.55 6.54

rms 2.69 2.32 15.20 14.68

S22共22兲 MSE 0.08 2.30 1.30 ⫺0.17

MAE 0.21 2.30 1.30 0.75

rms 0.29 2.93 1.68 1.04

Atoms共10兲 MSE 8.00 8.00 38.00 39.15

MAE 8.00 8.00 38.00 39.15

rms 9.20 9.20 45.75 45.79

All共404兲 MSE ⫺1.28 ⫺2.04 1.06 4.72

MAE 2.51 3.29 5.30 7.81

rms 3.45 4.49 10.19 13.00

tive to ␻ . Therefore there is little benefit to the LCDH scheme relative to a fully reoptimized DH functional on this data set. The principal benefit will be reduced SIE, as will be evaluated later. However, both ␻ B97X-2共LP兲 and B97-2共LP兲

performed significantly better than the reopti- mized B2PLYP-D

and B2PLYP

DH functionals. This in- dicates the importance of flexible GGA forms even in the DH functionals: the GGA is held fixed in the definition of B2PLYP-D

and B2PLYP

.

Comparison between this new LCDH functional and our previous LC hybrids, ␻ ^B97X-D, ␻ ^{B97X, and} ␻ ^{B97, shows} that ␻ B97X-2 consistently improves LC hybrids. This indi- cates the usefulness of augmenting LC hybrids with nonlocal correlation. In particular, relative to ␻ ^B97X-D, ␻ ^B97X-2

performs similarly for the S22 intermolecular interactions, and significantly better for thermochemical calculations.

Finally, the reoptimized SCS-MP2

, though free from SIE, performs most poorly of all approaches shown in Table III, due to the lack of mixing with semilocal density func- tionals. Additionally, for reference the performance of exist- ing methods that are not optimized on the training set, B2PLYP-D, B2PLYP, SCS-MP2, and MP2, are listed in Table IV for comparison. More detailed results for the train- ing set are given in Tables V–VII.

Since the ␻ B97X-2 contains orbital correlation through the PT2 contribution, its energy converges more slowly as the basis set is enlarged than LC hybrids. As can been seen in Table VIII, ␻ B97X-2共TQZ兲 performs best with the TQ ex-

TABLE V. Nonhydrogen transfer BHs共in kcal/mol兲 of the NHTBH38/04 set 共Ref.42兲. The results for the␻B97X-D are taken from Ref.11, and the results for the␻^{B97X and}␻B97 are taken from Ref.10.

Reactions ⌬Eref ␻B97X-2共LP兲 ␻B97X-D ␻B97X ␻B97 B2PLYP-D B2PLYP SCS-MP2 MP2

H + N₂O→OH+N2 V^f 18.14 21.14 17.45 19.22 20.67 16.50 16.67 36.81 36.05

V^r 83.22 84.13 77.73 80.57 81.93 76.42 76.62 97.54 88.32

H + FH→HF+H V^f 42.18 42.07 40.54 43.10 44.78 36.53 36.66 48.61 46.57

V^r 42.18 42.07 40.54 43.10 44.78 36.53 36.66 48.61 46.57

H + ClH→HCl+H V^f 18.00 19.17 18.24 20.73 23.17 15.67 15.70 23.50 22.64

V^r 18.00 19.17 18.24 20.73 23.17 15.67 15.70 23.50 22.64

H + FCH₃→HF+CH3 V^f 30.38 30.61 30.10 32.14 33.46 26.18 26.55 38.00 36.14

V^r 57.02 56.38 54.56 55.41 55.83 52.27 52.70 62.85 60.23

H + F₂→HF+F V^f 2.27 9.71 ⫺0.64 0.86 1.96 0.81 0.86 29.13 28.32

V^r 106.18 115.83 103.97 104.27 103.66 106.35 106.41 135.50 133.73

CH₃+ FCl→CH3F + Cl V^f 7.43 7.31 2.84 3.93 4.62 1.85 2.53 20.79 18.72

V^r 60.17 61.74 56.60 58.52 59.96 56.16 56.77 77.76 75.15

F⁻+ CH₃F→FCH3+ F⁻ V^f ⫺0.34 ⫺3.52 ⫺1.36 ⫺2.27 ⫺2.60 ⫺3.69 ⫺3.20 1.40 0.37

V^r ⫺0.34 ⫺3.52 ⫺1.36 ⫺2.27 ⫺2.60 ⫺3.69 ⫺3.20 1.40 0.37

F⁻¯CH3F→FCH3¯F⁻ V^f 13.38 12.13 12.91 13.28 13.32 11.45 11.29 15.39 14.75

V^r 13.38 12.13 12.91 13.28 13.32 11.45 11.29 15.39 14.75

Cl⁻+ CH₃Cl→ClCH3+ Cl⁻ V^f 3.10 2.38 3.71 4.71 6.21 0.22 1.01 6.40 5.50

V^r 3.10 2.38 3.71 4.71 6.21 0.22 1.01 6.40 5.50

Cl⁻¯CH3Cl→ClCH3¯Cl⁻ V^f 13.61 13.90 14.37 16.09 17.74 11.33 11.41 16.89 16.54

V^r 13.61 13.90 14.37 16.09 17.74 11.33 11.41 16.89 16.54

F⁻+ CH₃Cl→FCH3+ Cl⁻ V^f ⫺12.54 ⫺13.92 ⫺13.47 ⫺13.11 ⫺11.72 ⫺15.67 ⫺15.07 ⫺9.84 ⫺10.28

V^r 20.11 18.08 21.36 20.83 20.15 17.98 18.62 22.51 20.70

F⁻¯CH3Cl→FCH3¯Cl⁻ V^f 2.89 3.39 3.16 4.23 5.39 1.78 1.72 5.63 5.66

V^r 29.62 28.62 30.70 31.19 30.95 27.75 27.73 32.01 30.64

OH⁻+ CH₃F→HOCH3+ F⁻ V^f ⫺2.78 ⫺5.28 ⫺3.32 ⫺3.70 ⫺4.05 ⫺5.88 ⫺5.06 ⫺0.35 ⫺1.71

V^r 17.33 14.90 18.05 17.64 17.86 14.02 14.61 18.98 17.96

OH⁻¯CH3F→HOCH3¯F⁻ V^f 10.96 10.09 10.50 11.47 11.52 8.74 8.99 13.44 12.58

V^r 47.20 47.05 49.18 49.33 49.13 45.91 46.16 49.71 50.20

H + N₂→HN2 V^f 14.69 16.56 12.32 13.99 15.47 12.26 12.33 28.17 27.64

V^r 10.72 11.56 13.42 14.32 15.06 10.65 10.71 9.86 8.65

H + CO→HCO V^f 3.17 3.52 3.37 4.55 5.65 1.68 1.82 6.37 5.93

V^r 22.68 24.33 26.16 26.72 27.07 24.00 24.09 22.87 22.70

H + C₂H₄→CH3CH₂ V^f 1.72 3.44 2.99 4.07 4.94 1.31 1.81 10.00 9.35

V^r 41.75 45.42 45.49 47.07 48.49 42.30 42.52 48.05 46.58

CH₃+ C₂H₄→CH3CH₂CH₂ V^f 6.85 7.61 4.57 5.04 4.81 5.66 7.15 14.99 12.87

V^r 32.97 35.86 33.59 35.21 36.59 32.02 32.36 41.45 41.52

HCN→HNC V^f 48.16 49.63 46.43 46.29 45.89 48.76 48.68 52.29 52.21

V^r 33.11 33.59 33.22 33.12 32.80 33.38 33.32 35.41 34.80

MSE 0.54 ⫺0.42 0.56 1.32 ⫺2.29 ⫺2.00 6.61 5.37

MAE 1.67 1.51 1.75 2.31 2.44 2.19 6.65 5.48

rms 2.53 2.00 2.08 2.82 2.94 2.68 9.49 8.45

Max共⫺兲 ⫺3.18 ⫺5.49 ⫺3.50 ⫺2.81 ⫺6.80 ⫺6.60 ⫺0.86 ⫺2.07

Max共+兲 9.65 3.74 5.32 6.74 1.32 1.41 29.32 27.55

trapolation procedure 共TQZ兲, while it performs less satisfac- torily with smaller basis sets. This data shows the importance of using ␻ B97X-2 with the basis sets for which it is optimized—either the approximation to the complete basis set limit, ␻ B97X-2共TQZ兲, or the alternative development for the LP basis, ␻ B97X-2共LP兲.

B. The test sets

To test the performance of ␻ B97X-2 outside its training set, we also evaluate its performance on various test sets involving the additional 48 AEs in the G3/05 test set

共other than the 223 AEs in the G3/99 test set

兲, 30 chemical

reaction energies taken from the NHTBH38/04 and HTBH38/04 databases,

29 noncovalent interactions,

and 4 dissociation curves of symmetric radical cations. There are a total of 111 pieces of data in the test sets. Due to the large sizes of molecules in the test sets, we have only tested the performance of ␻ ^B97X-2 共LP兲, which was parameterized with the LP basis set. More detailed information about the test sets as well as the basis sets, and numerical grids used is given in Ref. 10. We use either full PT2 correlation 共without RI approximation兲, or, for efficiency, we evaluate PT2 corre- lation with the RI approximation and large auxiliary basis sets. There is no chemically significant difference.

The additional 48 AEs in the G3/05 test set

are com-

TABLE VI. Hydrogen transfer BHs共in kcal/mol兲 of the HTBH38/04 set 共Refs.41and42兲. The results for the␻B97X-D are taken from Ref.11, and the results for the␻^{B97X and}␻B97 are taken from Ref.10.

Reactions ⌬Eref ␻B97X-2共LP兲 ␻B97X-D ␻B97X ␻B97 B2PLYP-D B2PLYP SCS-MP2 MP2

H + HCl→H2+ Cl V^f 5.7 5.25 4.22 5.33 6.68 2.78 2.84 9.08 10.68

V^r 8.7 6.62 4.51 5.24 6.49 5.38 5.44 10.21 7.45

OH+ H₂→H+H2O V^f 5.1 5.12 2.24 2.56 3.27 2.71 2.92 10.05 7.27

V^r 21.2 21.77 18.76 19.50 20.39 18.35 18.55 29.14 31.87

CH₃+ H₂→H+CH4 V^f 12.1 11.62 9.14 9.63 10.29 9.69 10.23 14.56 12.76

V^r 15.3 14.99 13.67 15.13 16.35 12.30 12.78 19.11 19.76

OH+ CH₄→CH3+ H₂O V^f 6.7 5.60 3.19 3.97 4.53 3.30 3.88 10.49 7.19

V^r 19.6 18.87 15.17 15.41 15.59 16.33 16.96 25.04 24.79

H + H₂→H2+ H V^f 9.6 9.86 9.22 10.74 12.38 7.03 7.12 13.49 13.24

V^r 9.6 9.86 9.22 10.74 12.38 7.03 7.12 13.49 13.24

OH+ NH₃→H2O + NH₂ V^f 3.2 3.16 0.01 1.62 2.83 ⫺0.46 0.12 10.10 6.40

V^r 12.7 13.42 9.68 10.97 12.10 9.80 10.41 21.10 18.62

HCl+ CH₃→Cl+CH4 V^f 1.7 0.50 ⫺1.67 ⫺1.07 ⫺0.47 ⫺0.90 ⫺0.18 64.92 64.14

V^r 7.9 5.24 3.15 4.34 5.41 4.31 4.97 70.60 67.91

OH+ C₂H₆→H2O + C₂H₅ V^f 3.4 3.06 0.33 1.26 1.85 0.46 1.11 8.17 4.94

V^r 19.9 19.83 16.61 17.22 17.31 17.23 18.19 25.46 25.50

F + H₂→HF+H V^f 1.8 0.95 ⫺3.70 ⫺3.89 ⫺3.76 ⫺1.79 ⫺1.67 5.92 4.37

V^r 33.4 33.88 28.98 29.17 29.80 30.28 30.40 41.17 46.07

O + CH₄→OH+CH3 V^f 13.7 12.70 9.36 9.94 10.45 10.05 10.47 18.12 15.88

V^r 8.1 8.59 4.69 4.85 4.82 6.27 6.76 13.00 11.49

H + PH₃→PH2+ H₂ V^f 3.1 3.10 3.22 4.62 5.88 1.08 1.31 5.81 5.50

V^r 23.2 23.83 23.41 24.11 25.25 22.93 23.48 26.31 24.04

H + HO→H2+ O V^f 10.7 11.59 8.78 9.79 10.68 8.12 8.18 17.30 17.76

V^r 13.1 12.32 8.92 9.38 10.26 9.28 9.33 17.87 15.15

H + H₂S→H2+ HS V^f 3.5 4.02 4.07 5.54 6.94 1.78 1.95 6.94 6.66

V^r 17.3 16.89 16.27 17.16 18.42 15.96 16.25 19.59 16.11

O + HCl→OH+Cl V^f 9.8 10.91 5.63 7.07 14.21 6.53 6.57 17.99 15.25

V^r 10.4 11.55 5.77 7.38 14.45 7.97 8.01 18.54 14.63

NH₂+ CH₃→CH4+ NH V^f 8.0 8.91 5.67 6.50 6.95 6.93 7.74 12.61 10.61

V^r 22.4 21.52 18.54 19.42 20.07 18.82 19.57 25.43 23.70

NH₂+ C₂H₅→C2H₆+ NH V^f 7.5 9.83 7.38 8.48 8.82 8.04 9.13 13.05 10.93

V^r 18.3 18.93 15.95 16.87 17.54 16.18 16.97 23.13 21.06

C₂H₆+ NH₂→NH3+ C₂H₅ V^f 10.4 10.89 8.81 10.16 11.01 8.83 9.83 14.57 11.19

V^r 17.4 17.41 15.42 16.77 17.21 15.35 16.62 20.85 19.54

NH₂+ CH₄→CH3+ NH₃ V^f 14.5 13.51 11.39 12.64 13.49 11.57 12.46 16.99 13.84

V^r 17.8 16.52 13.70 14.72 15.29 14.34 15.25 20.53 19.23

s-trans cis-C₅H₈→s-trans cis-C5H₈ V^f 38.4 37.99 39.06 41.44 42.84 37.55 38.07 39.84 33.30

V^r 38.4 37.99 39.06 41.44 42.84 37.55 38.07 39.84 33.30

MSE ⫺0.13 ⫺2.52 ⫺1.51 ⫺0.34 ⫺2.49 ⫺2.01 7.55 5.84

MAE 0.74 2.64 2.24 2.24 2.52 2.11 7.55 6.54

rms 0.94 3.04 2.58 2.62 2.69 2.32 15.20 14.68

Max共⫺兲 ⫺2.66 ⫺5.50 ⫺5.69 ⫺5.56 ⫺3.82 ⫺3.77 1.44 ⫺5.10

Max共+兲 2.33 0.66 3.04 4.44 0.54 1.63 63.22 62.44

puted by various density functionals. This test set can be regarded as a very stringent test, as it contains third-row elements 共none is in our training set兲, and the accuracy of density functionals for AEs is usually very sensitive to their functional forms. As can been seen in Table IX, all of our

␻ B97 types of functionals, B2PLYP-D and B2PLYP perform noticeably better than SCS-MP2 and MP2. This shows the importance of hybrid methods. Relative to B2PLYP-D,

␻ ^{B97X and} ␻ B97X-D, which are the best performing exist- ing functionals, the performance of ␻ B97X-2 is generally similar, but not superior.

We also test their performance on 30 reaction energies taken from NHTBH38 and HTBH38 with unequal forward and reverse barriers. These involve lighter elements than are in the G3/05 test set. As can be seen in Table X, all the functionals perform very well, with far smaller errors than MP2. For these molecules, the new LCDH functional,

TABLE VII. Interaction energies共in kcal/mol兲 for the S22 set 共Ref.43兲. The counterpoise corrections are used to reduce the basis set superposition errors.

Monomer deformation energies are not included. The results for the␻B97X-D are taken from Ref.11, and the results for the␻^{B97X and}␻B97 are taken from Ref.10.

Complex关symmetry兴 ⌬Eref ␻B97X-2共LP兲 ␻B97X-D ␻B97X ␻B97 B2PLYP-D B2PLYP SCS-MP2 MP2 Hydrogen bonded complexes

共NH3兲2关C2h兴 ⫺3.17 ⫺3.35 ⫺3.07 ⫺3.58 ⫺3.64 ⫺3.32 ⫺2.56 ⫺2.50 ⫺2.92

共H2O兲2关Cs兴 ⫺5.02 ⫺5.24 ⫺4.97 ⫺5.59 ⫺5.64 ⫺5.08 ⫺4.66 ⫺4.14 ⫺4.61

Formic acid dimer关C2h兴 ⫺18.61 ⫺18.76 ⫺19.30 ⫺19.96 ⫺20.13 ⫺18.90 ⫺17.52 ⫺15.47 ⫺17.16

Formamide dimer关C2h兴 ⫺15.96 ⫺15.80 ⫺16.16 ⫺16.65 ⫺16.78 ⫺15.95 ⫺14.49 ⫺13.27 ⫺14.68

Uracil dimer关C2h兴 ⫺20.65 ⫺20.13 ⫺20.44 ⫺20.30 ⫺20.31 ⫺20.51 ⫺18.67 ⫺17.29 ⫺19.22

2-pyridoxine· 2-aminopyridine关C1兴 ⫺16.71 ⫺16.70 ⫺17.06 ⫺16.37 ⫺16.40 ⫺17.11 ⫺14.93 ⫺14.23 ⫺16.29

Adenine· thymine WC关C1兴 ⫺16.37 ⫺16.11 ⫺16.45 ⫺15.91 ⫺16.05 ⫺16.48 ⫺14.15 ⫺13.54 ⫺15.47

MSE 0.06 ⫺0.14 ⫺0.27 ⫺0.35 ⫺0.12 1.36 2.29 0.88

MAE 0.22 0.24 0.60 0.63 0.17 1.36 2.29 0.88

Dispersion complexes

共CH4兲2关D3d兴 ⫺0.53 ⫺0.70 ⫺0.57 ⫺0.57 ⫺0.44 ⫺0.40 0.05 ⫺0.25 ⫺0.42

共C2H₄兲2关D2d兴 ⫺1.51 ⫺1.76 ⫺1.78 ⫺1.77 ⫺1.92 ⫺1.45 ⫺0.27 ⫺0.82 ⫺1.35

Benzene· CH₄关C3兴 ⫺1.50 ⫺1.69 ⫺1.68 ⫺1.41 ⫺1.55 ⫺1.33 ⫺0.13 ⫺0.99 ⫺1.64

Benzene dimer关C2h兴 ⫺2.73 ⫺3.25 ⫺3.19 ⫺1.57 ⫺2.33 ⫺2.32 0.83 ⫺2.50 ⫺4.58

Pyrazine dimer关Cs兴 ⫺4.42 ⫺4.82 ⫺4.25 ⫺2.86 ⫺3.68 ⫺4.12 ⫺0.70 ⫺4.14 ⫺6.38

Uracil dimer关C2兴 ⫺10.12 ⫺9.72 ⫺9.79 ⫺7.84 ⫺8.90 ⫺9.68 ⫺4.72 ⫺7.59 ⫺10.32

Indole· benzene关C1兴 ⫺5.22 ⫺5.23 ⫺5.05 ⫺2.39 ⫺3.58 ⫺4.36 0.33 ⫺4.42 ⫺7.56

Adenine· thymine stack关C1兴 ⫺12.23 ⫺11.81 ⫺11.81 ⫺8.40 ⫺10.26 ⫺11.85 ⫺4.58 ⫺9.73 ⫺13.86

MSE ⫺0.09 0.02 1.43 0.70 0.34 3.63 0.98 ⫺0.98

MAE 0.30 0.25 1.51 0.82 0.34 3.63 0.98 1.05

Mixed complexes

Ethene· ethine关C2v兴 ⫺1.53 ⫺1.72 ⫺1.64 ⫺1.67 ⫺1.63 ⫺1.54 ⫺0.99 ⫺1.18 ⫺1.52

Benzene· H₂O关Cs兴 ⫺3.28 ⫺3.51 ⫺3.50 ⫺3.39 ⫺3.56 ⫺3.33 ⫺2.05 ⫺2.62 ⫺3.25

Benzene· NH₃关Cs兴 ⫺2.35 ⫺2.55 ⫺2.54 ⫺2.31 ⫺2.46 ⫺2.26 ⫺1.01 ⫺1.80 ⫺2.43

Benzene· HCN关Cs兴 ⫺4.46 ⫺4.88 ⫺4.79 ⫺4.61 ⫺4.89 ⫺4.77 ⫺3.07 ⫺3.87 ⫺4.78

Benzene dimer关C2v兴 ⫺2.74 ⫺2.99 ⫺2.89 ⫺2.11 ⫺2.38 ⫺2.63 ⫺0.63 ⫺2.21 ⫺3.36

Indole· benzene T-shape关C1兴 ⫺5.73 ⫺5.82 ⫺5.63 ⫺4.44 ⫺4.82 ⫺5.71 ⫺2.79 ⫺4.86 ⫺6.53

Phenol dimer关C1兴 ⫺7.05 ⫺7.18 ⫺6.98 ⫺6.49 ⫺6.93 ⫺6.95 ⫺4.71 ⫺5.79 ⫺7.22

MSE ⫺0.22 ⫺0.12 0.30 0.07 ⫺0.01 1.70 0.69 ⫺0.28

MAE 0.22 0.17 0.42 0.33 0.10 1.70 0.69 0.29

MSE ⫺0.08 ⫺0.08 0.53 0.16 0.08 2.30 1.30 ⫺0.17

MAE 0.24 0.22 0.87 0.60 0.21 2.30 1.30 0.75

rms 0.28 0.27 1.30 0.80 0.29 2.93 1.68 1.04

Max共⫺兲 ⫺0.52 ⫺0.69 ⫺1.35 ⫺1.52 ⫺0.40 0.36 0.23 ⫺2.34

Max共+兲 0.52 0.42 3.83 1.97 0.86 7.65 3.36 1.45

TABLE VIII. Statistical errors共in kcal/mol兲 of the training set. MAE 共in kcal/mol兲 of␻^B97X-2共TQZ兲 on the training set, using four different basis sets, are listed. TQZ denotes the TQ extrapolation scheme with the basis sets described in TableI, while QZ and TZ are the corresponding basis sets for the extrapolation. LP is the 6-311+ +G共3df ,3pd兲 basis set.

System TQZ QZ TZ LP

G3/99共223兲 1.43 2.41 6.29 4.99

IP共40兲 1.57 2.04 2.95 2.19

EA共25兲 1.42 1.46 1.65 2.39

PA共8兲 1.19 0.99 1.13 1.12

NHTBH共38兲 1.29 1.33 1.53 1.42

HTBH共38兲 0.65 0.63 0.70 0.71

S22共22兲 0.26 0.29 0.56 0.37

All共394兲 1.29 1.89 4.23 3.44

␻ B97X-2, and the existing DH functionals appear to perform somewhat better than the LC-DF methods. These results con- trast with the what we observe for the G3/05 test set. Con- sidering that PT2 correlation for noble gas atoms is known to change from underestimation to overestimation on going down the periodic table,

it appears that ␻ B97X-2 may per- form slightly better for the lighter elements on which it was trained. A similar comment may be made for B2PLYP and B2PLYP-D.

The performance of various functionals are examined on several sets of noncovalent interactions.

As can been seen in Table XI, ␻ ^{B97X-2 and} ␻ B97X-D perform similarly to the best method here, MP2. B2PLYP and SCS-MP2 per- form less satisfactorily in this application. B2PLYP-D also performs very well here, though its performance is slightly inferior to both ␻ ^{B97X-D and} ␻ B97X-2. Considering that B2PLYP-D contains both empirical atom-atom dispersion in- teractions, and nonlocal PT2 correlation, it is encouraging

TABLE IX. Statistical errors of the additional 48 AEs共in kcal/mol兲 in the G3/05 set 共Ref.52兲. The results for the␻B97X-D are taken from Ref.11, and the results for the␻^{B97X and}␻B97 are taken from Ref.10.

Error ␻B97X-2共LP兲 ␻B97X-D ␻B97X ␻B97 B2PLYP-D B2PLYP SCS-MP2 MP2

MSE ⫺0.36 0.24 0.76 1.28 ⫺1.91 ⫺2.87 ⫺3.13 5.88

MAE 3.25 3.01 3.60 4.25 3.66 4.29 6.13 12.45

rms 4.57 3.95 4.52 5.41 4.60 5.88 8.32 18.37

Max共⫺兲 ⫺10.31 ⫺6.28 ⫺5.96 ⫺6.50 ⫺11.51 ⫺18.56 ⫺29.66 ⫺20.39

Max共+兲 20.64 13.06 14.88 18.14 10.84 10.15 15.97 61.56

TABLE X. Comparison of errors of different functionals for the reaction energies共in kcal/mol兲 of the 30 chemical reactions in the NHTBH38/04 and HTBH38/04 database共Refs.41and42兲. The results for the␻B97X-D are taken from Ref.11, and the results for the␻^{B97X and}␻B97 are taken from Ref.

10.

Reactions ⌬Eref ␻B97X-2共LP兲 ␻B97X-D ␻B97X ␻B97 B2PLYP-D B2PLYP SCS-MP2 MP2

H + N₂O→OH+N2 ⫺65.08 2.09 4.80 3.72 3.82 5.16 5.13 4.35 12.81

H + FCH₃→HF+CH3 ⫺26.64 0.86 2.18 3.37 4.27 0.56 0.49 1.79 2.55

H + F₂→HF+F ⫺103.91 ⫺2.21 ⫺0.70 0.50 2.21 ⫺1.64 ⫺1.64 ⫺2.47 ⫺1.50

CH₃+ FCl→CH3F + Cl ⫺52.74 ⫺1.69 ⫺1.02 ⫺1.85 ⫺2.60 ⫺1.56 ⫺1.50 ⫺4.23 ⫺3.70

F⁻+ CH₃Cl→FCH3+ Cl⁻ ⫺32.65 0.65 ⫺2.18 ⫺1.29 0.78 ⫺0.99 ⫺1.04 0.30 1.67

F⁻¯CH3Cl→FCH3¯Cl⁻ ⫺26.73 1.49 ⫺0.80 ⫺0.23 1.17 0.76 0.73 0.35 1.74

OH⁻+ CH₃F→HOCH3+ F⁻ ⫺20.11 ⫺0.07 ⫺1.25 ⫺1.22 ⫺1.80 0.21 0.44 0.78 0.43 OH⁻¯CH3F→HOCH3¯F⁻ ⫺36.24 ⫺0.72 ⫺2.44 ⫺1.62 ⫺1.36 ⫺0.94 ⫺0.93 ⫺0.03 ⫺1.38

H + N₂→HN2 3.97 1.03 ⫺5.06 ⫺4.30 ⫺3.57 ⫺2.36 ⫺2.35 14.34 15.02

H + CO→HCO ⫺19.51 ⫺1.30 ⫺3.28 ⫺2.66 ⫺1.91 ⫺2.81 ⫺2.77 3.01 2.74

H + C₂H₄→CH3CH₂ ⫺40.03 ⫺1.95 ⫺2.47 ⫺2.97 ⫺3.52 ⫺0.95 ⫺0.68 1.98 2.80

CH₃+ C₂H₄→CH3CH₂CH₂ ⫺26.12 ⫺2.13 ⫺2.90 ⫺4.05 ⫺5.65 ⫺0.24 0.91 ⫺0.33 ⫺2.53

HCN→HNC 15.05 0.99 ⫺1.85 ⫺1.87 ⫺1.97 0.34 0.31 1.83 2.35

H + HCl→H2+ Cl ⫺3.0 1.63 2.71 3.09 3.18 0.40 0.40 1.87 6.23

OH+ H₂→H+H2O ⫺16.1 ⫺0.54 ⫺0.41 ⫺0.84 ⫺1.02 0.46 0.47 ⫺2.99 ⫺8.50

CH₃+ H₂→H+CH4 ⫺3.2 ⫺0.17 ⫺1.33 ⫺2.30 ⫺2.86 0.58 0.65 ⫺1.35 ⫺3.80

OH+ CH₄→CH3+ H₂O ⫺12.9 ⫺0.37 0.92 1.46 1.84 ⫺0.12 ⫺0.18 ⫺1.65 ⫺4.70

OH+ NH₃→H2O + NH₂ ⫺9.5 ⫺0.76 ⫺0.17 0.15 0.23 ⫺0.75 ⫺0.79 ⫺1.51 ⫺2.71

HCl+ CH₃→Cl+CH4 ⫺6.2 1.45 1.38 0.79 0.33 0.98 1.05 0.52 2.43

OH+ C₂H₆→H2O + C₂H₅ ⫺16.5 ⫺0.28 0.22 0.54 1.04 ⫺0.27 ⫺0.58 ⫺0.79 ⫺4.06

F + H₂→HF+H ⫺31.6 ⫺1.33 ⫺1.09 ⫺1.46 ⫺1.96 ⫺0.47 ⫺0.47 ⫺3.65 ⫺10.11

O + CH₄→OH+CH3 5.6 ⫺1.50 ⫺0.92 ⫺0.50 0.03 ⫺1.82 ⫺1.89 ⫺0.48 ⫺1.21

H + PH₃→PH2+ H₂ ⫺20.1 ⫺0.62 ⫺0.09 0.61 0.74 ⫺1.75 ⫺2.07 ⫺0.40 1.56

H + HO→H2+ O ⫺2.4 1.67 2.26 2.80 2.82 1.24 1.24 1.82 5.01

H + H₂S→H2+ HS ⫺13.8 0.93 1.60 2.17 2.32 ⫺0.39 ⫺0.50 1.15 4.35

O + HCl→OH+Cl ⫺0.6 ⫺0.04 0.46 0.29 0.36 ⫺0.84 ⫺0.84 0.05 1.22

NH₂+ CH₃→CH4+ NH ⫺14.4 1.79 1.53 1.49 1.28 2.52 2.56 1.58 1.31

NH₂+ C₂H₅→C2H₆+ NH ⫺10.8 1.70 2.23 2.41 2.08 2.66 2.97 0.72 0.67

C₂H₆+ NH₂→NH3+ C₂H₅ ⫺7.0 0.48 0.39 0.39 0.81 0.48 0.21 0.72 ⫺1.35

NH₂+ CH₄→CH3+ NH₃ ⫺3.3 0.29 0.99 1.21 1.50 0.53 0.51 ⫺0.24 ⫺2.09

MSE 0.05 ⫺0.21 ⫺0.07 0.09 ⫺0.03 ⫺0.01 0.57 0.58

MAE 1.09 1.66 1.74 1.97 1.16 1.21 1.91 3.75

rms 1.27 2.06 2.10 2.36 1.57 1.61 3.22 5.11

Max共⫺兲 ⫺2.21 ⫺5.06 ⫺4.30 ⫺5.65 ⫺2.81 ⫺2.77 ⫺4.23 ⫺10.11

Max共+兲 2.09 4.80 3.72 4.27 5.16 5.13 14.34 15.02

TABLE XI. Binding energies共in kcal/mol兲 of several sets of noncovalent interactions. The first three sets are taken from Ref.53with monomer deformation energies taken into considerations. The last three sets are taken from Ref. 43 without considering monomer deformation energies. The counter-point corrections are applied for all the cases. The results for the␻B97X-D are taken from Ref.11, and the results for the␻^{B97X and}␻B97 are taken from Ref.

10.

Complex ⌬Eref ␻B97X-2共LP兲 ␻B97X-D ␻B97X ␻B97 B2PLYP-D B2PLYP SCS-MP2 MP2

Charge-transfer complexes

C₂H₄¯F2 1.06 1.37 0.78 1.03 1.09 1.54 1.03 0.81 1.19

NH₃¯F2 1.81 2.05 1.50 1.93 1.98 2.35 2.02 1.26 1.60

C₂H₂¯ClF 3.81 4.32 3.66 4.43 4.50 3.95 3.45 2.59 3.73

HCN¯ClF 4.86 5.56 4.21 5.32 5.42 4.89 4.66 4.09 4.92

NH₃¯Cl2 4.88 5.25 4.81 5.18 4.89 5.36 4.93 3.59 4.50

H₂O¯ClF 5.36 5.96 5.18 6.16 6.21 5.61 5.27 4.21 4.95

NH₃¯ClF 10.62 11.53 11.12 11.10 10.49 12.29 11.69 8.45 10.33

MSE 0.52 ⫺0.16 0.39 0.31 0.51 0.09 ⫺1.06 ⫺0.17

MAE 0.52 0.31 0.40 0.35 0.51 0.29 1.06 0.22

Dipole-dipole interaction complexes

H₂S¯H2S 1.66 1.83 1.54 1.99 1.99 1.48 1.05 1.06 1.52

HCl¯HCl 2.01 2.06 1.69 2.30 2.33 1.76 1.34 1.21 1.68

H₂S¯HCl 3.35 3.56 3.38 3.90 3.93 3.30 2.84 2.42 3.15

CH₃Cl¯HCl 3.55 3.54 3.17 3.82 3.97 3.26 2.43 2.25 3.07

HCN¯CH3SH 3.59 3.87 3.72 3.99 4.05 3.67 2.89 2.83 3.46

CH₃SH¯HCl 4.16 5.01 4.87 5.28 5.38 4.85 3.94 3.48 4.61

MSE 0.26 0.01 0.50 0.56 0.00 ⫺0.64 ⫺0.85 ⫺0.14

MAE 0.26 0.28 0.50 0.56 0.26 0.64 0.85 0.29

Weak interaction complexes

He¯Ne 0.04 0.06 0.00 0.01 ⫺0.05 0.03 ⫺0.01 0.00 0.01

He¯Ar 0.06 0.10 0.00 0.05 ⫺0.03 0.01 ⫺0.03 0.00 0.02

Ne¯Ne 0.08 0.08 ⫺0.02 ⫺0.02 ⫺0.07 0.10 0.01 ⫺0.01 0.00

Ne¯Ar 0.13 0.15 ⫺0.01 0.05 ⫺0.04 0.12 ⫺0.01 0.00 0.04

CH₄¯Ne 0.22 0.22 0.13 0.10 0.00 0.18 0.00 0.04 0.08

C₆H₆¯Ne 0.47 0.51 0.24 0.30 0.32 0.46 ⫺0.04 0.07 0.26

CH₄¯CH4 0.51 0.73 0.60 0.64 0.55 0.42 ⫺0.13 0.20 0.40

MSE 0.05 ⫺0.08 ⫺0.05 ⫺0.12 ⫺0.03 ⫺0.24 ⫺0.17 ⫺0.10

MAE 0.05 0.11 0.09 0.13 0.03 0.24 0.17 0.10

Hydrogen-bonded DNA base pairs

G¯A HB ⫺11.30 ⫺12.94 ⫺13.48 ⫺12.29 ⫺12.44 ⫺13.43 ⫺10.75 ⫺10.68 ⫺12.85

C¯G WC ⫺30.70 ⫺31.49 ⫺32.45 ⫺31.92 ⫺32.12 ⫺32.10 ⫺29.11 ⫺27.62 ⫺30.37

G¯C WC ⫺31.40 ⫺31.44 ⫺32.28 ⫺31.85 ⫺32.07 ⫺31.96 ⫺29.03 ⫺27.62 ⫺30.32

MSE ⫺0.82 ⫺1.60 ⫺0.89 ⫺1.08 ⫺1.36 1.50 2.49 ⫺0.05

MAE 0.82 1.60 0.89 1.08 1.36 1.50 2.49 0.99

Interstrand base pairs

G¯G IS ⫺5.20 ⫺5.31 ⫺5.43 ⫺4.83 ⫺4.70 ⫺5.15 ⫺4.24 ⫺4.74 ⫺5.17

G¯G IS 0.80 0.67 1.15 2.21 2.13 1.14 3.13 1.21 0.08

C¯C IS 3.10 3.21 3.29 3.62 3.69 3.18 3.35 3.06 2.92

MSE ⫺0.05 0.10 0.77 0.80 0.16 1.18 0.28 ⫺0.29

MAE 0.12 0.25 0.77 0.80 0.16 1.18 0.30 0.31

Stacked base pairs

A¯G S ⫺6.50 ⫺6.78 ⫺6.66 ⫺3.46 ⫺4.62 ⫺6.35 ⫺0.63 ⫺5.29 ⫺8.61

C¯G S ⫺12.40 ⫺11.10 ⫺10.83 ⫺8.57 ⫺9.61 ⫺10.38 ⫺5.55 ⫺9.37 ⫺12.07

G¯C S ⫺11.60 ⫺10.93 ⫺11.06 ⫺8.77 ⫺9.69 ⫺10.40 ⫺5.77 ⫺9.19 ⫺11.70

MSE 0.56 0.65 3.23 2.19 1.12 6.19 2.22 ⫺0.63

MAE 0.16 0.76 3.23 2.19 1.12 6.19 2.22 0.85

MSE 0.16 ⫺0.14 0.51 0.36 0.11 0.75 0.04 ⫺0.19

MAE 0.37 0.43 0.73 0.65 0.46 1.18 0.99 0.36

rms 0.55 0.68 1.18 0.93 0.76 2.16 1.39 0.59

that ␻ ^B97X-2 共which does not contain the atom-atom inter- actions 兲 achieves slightly better results on intermolecular in- teractions.

Common semilocal functionals are generally accurate for systems near equilibrium. However, due to considerable SIEs in semilocal functionals, spurious fractional charge dis- sociation occurs.

This situation becomes amplified for symmetric charged radicals X

, such as H

, He

, Ne

, and Ar

. To test the extent to which LCDH methods improve upon the SIE problem, we performed unrestricted calcula- tions with the aug-cc-pVQZ basis set and a high-quality EML共250,590兲 grid. The DFT results are compared with re- sults from HF theory, MP2 theory, and the very accurate CCSD共T兲 theory.

As shown in Fig. 1, the predicted H

binding energy of

␻ B97X-2共LP兲 is very close to the HF 共exact兲 curve, and all the LC hybrids predict no spurious barriers on the dissocia- tion curves. It appears that the SIE associated with

␻ B97X-2共LP兲 is more than two times smaller than the next best DFT method shown. From Fig. 2, ␻ B97X-2共LP兲 greatly improves upon the LC hybrids and B2PLYP for the He

binding energy curve. Finally, for larger cations, such as Ne

and Ar

, ␻ B97X-2共LP兲 can dissociate them correctly 共see

Figs. 3 and 4 兲. This is a very encouraging result, which shows the value of the LCDH approach. The errors of cation binding energy curves at dissociation are summarized in Table XII, and quantify the significant improvements seen with ␻ B97X-2共LP兲.

However, all is not entirely well with the ␻ ^B97X-2 共LP兲 LCDH potential curves. Inspection of Figs. 3 and 4 shows that there is a discontinuity 共also see Figs. 5 and 6 兲 in the derivative of ␻ ^B97X-2 共LP兲 binding energy curves for Ne

and Ar

. There is a similar problem for MP2. This problem arises because the post-KS 共or post-HF兲 PT2 correction causes the Hellman–Feynman theorem to no longer hold.

The energy derivative therefore depends on the derivative of the orbitals, which can change discontinuously when the or- bital Hessian exhibits a zero eigenvalue, such as at a spin- unrestriction point, or, as in this case, at the point where left-right symmetry-breaking occurs. These issues, including a related violation of N-representability have been discussed in detail elsewhere.

This undesirable property can, how- ever, be removed by orbital-optimized 共OO兲 methods.

It thus appears desirable to pursue OO-DH and OO-LCDH ap- proaches in the future, although they will necessarily be somewhat more costly.

-80 -60 -40 -20 0 20

1 2 3 4 5 6 7 8 9 10

E( k ca l/mo l)

R (angstrom)

ωB97X-2(LP)HF ωB97X-D ωB97X ωB97 B2PLYP B3LYP

FIG. 1. Dissociation curve of H₂⁺curve. Zero level is set to E共H兲+E共H⁺兲 for each method.

-80 -60 -40 -20 0 20 40

1 2 3 4 5 6 7 8 9 10

E (k ca l/ mo l)

R (angstrom)

CCSD(T) ωB97X-2(LP)HF ωB97X-D ωB97X ωB97 B2PLYP B3LYP

FIG. 2. Dissociation curve of He₂⁺ curve. Zero level is set to E共He兲 + E共He⁺兲 for each method.

-60 -40 -20 0 20

2 3 4 5 6 7 8 9 10

E( k ca l/mo l)

R (angstrom)

CCSD(T) ωB97X-2(LP)HF ωB97X-D ωB97X B2PLYPωB97 B3LYP

FIG. 3. Dissociation curve of Ne₂⁺ curve. Zero level is set to E共Ne兲 + E共Ne⁺兲 for each method.

-40 -30 -20 -10 0 10 20 30

2 3 4 5 6 7 8 9 10

E (k ca l/ mo l)

R (angstrom)

CCSD(T) ωB97X-2(LP)HF ωB97X-D ωB97X ωB97 B2PLYP B3LYP

FIG. 4. Dissociation curve of Ar₂⁺ curve. Zero level is set to E共Ar兲 + E共Ar⁺兲 for each method.

IV. CONCLUSIONS

We have developed a new LCDH functional based on our previous work.

This functional, called ␻ B97X-2, in- cludes 100% LR exact exchange, a large fraction 共about 65 percent兲 of SR exact exchange, a modified B97 exchange density functional for SR interaction, the B97 correlation density functional,

and spin-component scaled PT2 corre- lations. There are a total of 16 parameters that must be speci- fied, and we have done this by fitting to a large training set containing more than 400 pieces of data.

Since ␻ B97X-2 is a parameterized functional, we also test it against MP2,

three well-established existing DH functionals 关B2PLYP,

B2PLYP-D,

and SCS-MP2 共Ref.

49兲兴 as well as our previous LC hybrid functionals 共 ␻ ^B97,

␻ ^{B97X, and} ␻ B97X-D兲 on a separate independent test set of data, which includes further AEs, reaction energies, nonco- valent interaction energies, and 4 symmetrical radical cat- ions. The results indicate that this new LCDH functional is generally comparable or superior in performance for de- manding cases such as AEs and base-stacking interactions.

All the LC hybrid functionals are dramatically superior for radical cation problems that are sensitive to self-interaction errors, and ␻ B97X-2 significantly improves upon the best of them.

As with all approximate density functionals, some limi- tations remain, and should be clearly laid out. First, while

␻ B97X-2 is free of LR self-interaction, it still suffers from some self-interaction at SR, which means that its perfor- mance for the demanding problems of radical cations still

shows errors relative to truly self-interaction free methods.

Second, because the fraction of exact exchange is signifi- cantly increased relative to semilocal functionals, the perfor- mance of ␻ B97X-2 for systems with small gaps and thus potentially strong “static” correlation effects may be poorer than existing functionals. In general ␻ B97X-2 is likely to be most suitable for applications to lighter elements. Third, LR correlation effects are solely treated by a post-KS treatment of the nonlocal correlation effects in ␻ B97X-2, meaning that the KS orbitals are not affected by such corrections, which can cause problems 共as seen in the Ne

and Ar

potential curve discontinuities兲. Orbital optimization would resolve these issues. Fourth, due to the use of wave function PT2 correlation, ␻ B97X-2 is more sensitive to the choices of ba- sis set than normal density functionals, which increases the computational cost for high-quality calculations. This prob- lem can be reduced by using dual-basis methods

or R12 methods.

ACKNOWLEDGMENTS

This work was supported by the Director, Office of En- ergy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No. DE-AC0376SF00098. J.D.C. is grateful to the Start-up Funds 共Grant No. 98R0034-44 and 98R0654兲 from National Taiwan University and is grateful to the Computer and Information Networking Center, National Taiwan Uni- versity for the partial support of high-performance comput- ing facilities. M.H.G. is a part-owner of Q-Chem Inc.

1P. Hohenberg and W. Kohn,Phys. Rev. 136, B864共1964兲.

2W. Kohn and L. J. Sham,ibid. 140, A1133共1965兲.

3R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Mol- ecules共Oxford University, New York, 1989兲.

4R. M. Dreizler and E. K. U. Gross, Density Functional Theory: An Ap- proach to the Quantum Many Body Problem 共Springer-Verlag, Berlin, 1990兲.

5W. Kohn, A. D. Becke, and R. G. Parr, J. Phys. Chem. 100, 12974 共1996兲.

6A. D. Becke,J. Chem. Phys. 98, 5648共1993兲.

7T. Bally and G. N. Sastry,J. Phys. Chem. A101, 7923共1997兲; B. Braïda, P. C. Hiberty, and A. Savin,ibid.102, 7872共1998兲; P. Mori-Sánchez, A.

TABLE XII. Binding energies of symmetric radical cations at bond length R = 100 共Å兲, Eb= E关X2+, R = 100 共Å兲兴−E共X兲−E共X⁺兲 共in kcal/mol兲. The re- sults for the␻^{B97X and}␻B97 are taken from Ref.10.

Molecule ␻B97X-2共LP兲 ␻B97X-D ␻B97X ␻B97 B2PLYP B3LYP

H₂⁺ ⫺6.7 ⫺28.3 ⫺21.3 ⫺17.9 ⫺31.3 ⫺53.5

He₂⁺ ⫺13.8 ⫺47.5 ⫺38.4 ⫺34.5 ⫺44.0 ⫺78.2

Ne₂⁺ ⫺0.0 ⫺42.0 ⫺34.7 ⫺33.6 ⫺41.0 ⫺70.5

Ar₂⁺ ⫺0.0 ⫺21.5 ⫺13.8 ⫺9.5 ⫺27.6 ⫺47.3

-40 -30 -20 -10 0 10 20 30

1.5 2 2.5 3 3.5 4

E (k ca l/ mo l)

R (angstrom)

CCSD(T) HF ωB97X-2(LP)MP2 ωB97X-2(LP) [SCF]

FIG. 5. Same as Fig.3, but with a focus on the unrestricted region.

-40 -30 -20 -10 0 10 20

2 2.5 3 3.5 4 4.5 5

E (k ca l/ mo l)

R (angstrom)

CCSD(T) HF ωB97X-2(LP)MP2 ωB97X-2(LP) [SCF]

FIG. 6. Same as Fig.4, but with a focus on the unrestricted region.