Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections w
Jeng-Da Chai and Martin Head-Gordon*
Received 17th June 2008, Accepted 2nd September 2008
First published as an Advance Article on the web 29th September 2008 DOI: 10.1039/b810189b
We report re-optimization of a recently proposed long-range corrected (LC) hybrid density functional [J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106] to include empirical atom–atom dispersion corrections. The resulting functional,oB97X-D yields satisfactory accuracy for thermochemistry, kinetics, and non-covalent interactions. Tests show that for non-covalent systems,oB97X-D shows slight improvement over other empirical dispersion-corrected density functionals, while for covalent systems and kinetics it performs noticeably better. Relative to our previous functionals, such asoB97X, the new functional is significantly superior for non-bonded interactions, and very similar in performance for bonded interactions.
1. Introduction
Due to its favorable cost-to-performance ratio, Kohn–Sham density-functional theory (KS-DFT)1,2 has become the most popular electronic structure theory for large-scale ground- state systems.3–5 Its extension for treating excited-state systems,6,7 time-dependent density functional theory (TDDFT), has also been developed to the stage where it is now very widely used.
The essential ingredient of KS-DFT, the exchange–
correlation energy functional Exc, remains unknown and needs to be approximated. Semi-local gradient-corrected density functionals, though successful in many applications, lead to qualitative failures in some circumstances where the accurate treatment of non-locality of exchange–correlation hole becomes crucial. These situations occur mostly in the asymp- totic regions of molecular systems, such as spurious self- interaction effects upon dissociation8,9 and dramatic failures for long-range charge-transfer excitations.10–12 Widely used hybrid density functionals, like B3LYP,13,14 do not qualitatively resolve these problems.
These self-interaction errors can be qualitatively resolved using the long-range corrected (LC) hybrid density func- tionals,15,16,18 which employ 100% Hartree–Fock (HF) exchange for long-range electron–electron interactions. This is accomplished by a partition of unity, using erf(or)/r for long-range (treated by HF exchange) and erfc(or)/r for short- range (treated by an exchange functional), with the parameter o controlling the partitioning of the interelectronic distance r.
Over the past five years, the LC hybrid scheme has been attracting increasing attention15since its computational cost is comparable with standard hybrid functionals.13 However,
LC functionals have tended to be inferior to the best hybrids for properties such as thermochemistry.
Recently we have improved the overall accuracy attainable with the LC functionals by using a systematic optimization procedure.18One important conclusion is that optimizing LC and hybrid functionals with identical numbers of para- meters in their GGA exchange and correlation terms leads to noticeably better results for all properties using the LC form. The resulting LC functional is called oB97.
Further statistically significant improvement results from re-optimizing the entire functional with one extra parameter corresponding to an adjustable fraction of short-range exact exchange, defining the oB97X functional. Independent test sets covering thermochemistry and non-covalent interactions support these conclusions. However, problems associated with the lack of non-locality of the correlation hole, such as the lack of dispersion interactions (London forces), still remain, as the semi-local correlation functionals cannot capture long-range correlation effects.19,20
There have been significant efforts to develop a framework that can account for long-range dispersion effects within DFT.
Zaremba and Kohn (ZK)21derived an exact expression for the second-order dispersion energy in terms of the exact density–
density response functions of the two separate systems. To obtain a tractable non-local dispersion functional, Dobson and Dinite (DD)22made local density approximations to the ZK response functions. DD’s non-local correlation functional was obtained independently23by modifying the effective density defined in the earlier work of Rapcewicz and Ashcroft.24By combining DD’s non-local correlation functional with the LC functionals, Hirao and co-workers25–27have achieved impressive success in applica- tions to various non-covalent systems.
Starting from the formally exact expression of KS-DFT, the adiabatic connection fluctuation–dissipation theorem (ACFDT), for the ground-state exchange–correlation energy, Langreth and coworkers28developed a so-called van der Waals density functional (vdW-DF) by making a series of reasonable approximations to yield a computationally tractable scheme.
Department of Chemistry, University of California and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA. E-mail: [email protected]
w Electronic supplementary information (ESI) available: Further numerical results. See DOI: 10.1039/b810189b
PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics
Recently, Becke and Johnson (BJ) developed a series of post-HF correlation models with a novel treatment for dis- persion interactions based on the exchange-hole dipole moment.29The origin of dispersion claimed in the BJ models was recently questioned by Alonso and Man˜anes,30 and A´ngya´n31 further showed that the BJ models effectively contain correlation effects. Nevertheless, the BJ models show promise for computing accurate interatomic and intermolecu- lar C6, C8, and C10 dispersion coefficients with only a few empirical parameters. Furthermore, as a post-HF method, the BJ functional is intrinsically free of self-interaction errors.
Alternatively, one can add an empirical atom–atom disper- sion potential to an existing density functional, to obtain a correction that is essentially of zero cost and zero complexity.
Such an approach was used long ago to correct HF calcula- tions,32and over the past 8 years has been incorporated into density functional theory.33–46 These DFT-D (density func- tional theory with empirical dispersion corrections) schemes have shown generally satisfactory performance on a large set of non-covalent systems.38,43 Currently available functionals of this type, such as B3LYP-D and B97-D appear to yield results for covalent systems that are at least comparable to their parent methods (e.g. B3LYP and B97).
Since we have recently demonstrated that fully optimized LC functionals such as oB97 and oB97X18 can yield better results for covalent systems than traditional hybrids or LC functionals that are not fully re-optimized, it seems natural to explore whether further improvements are possible by incor- porating empirical atom–atom dispersion corrections.
Although extending our LC functionals with DD’s non-local correlation functional,22,23 as suggested by Hirao and co-workers,25–27 is also possible for obtaining dispersion corrections, its computational cost is, however, much higher than the DFT-D scheme due to the required double numerical integrations. Besides, its performance for non-covalent systems has been shown to be similar to the simpler DFT-D scheme.27In this work, we therefore introduce an empirical dispersion correction to the oB97X functional, as this provides the missing pieces of the long-range vdW interactions without additional computational costs. It also emerges that optimization of theoB97 functional with empirical dispersion corrections leads to essentially zero dispersion correction. Our results are compared with those by other DFT-D functionals as well as our previous LC hybrid functionals.
2. The DFT-D scheme
Following the general form of the DFT-D scheme,33–46our total energy
EDFT-D= EKS-DFT+ Edisp (1)
is computed as the sum of a KS-DFT part, using theoB97X18 functional, and an empirical atomic-pairwise dispersion correction. We choose to use an unscaled dispersion correction, given by
Edisp¼ #NXat#1
i¼1
XNat
j¼iþ1
Cij6
R6ij fdampðRijÞ ð2Þ
where Nat is the number of atoms in the system, C6ij is the dispersion coefficient for atom pair ij, and Rijis an interatomic distance. The conditions of zero dispersion correction at short interatomic separations and correct asymptotic pairwise vdW potentials are enforced by introducing a damping function,
fdampðRijÞ ¼ 1
1þ aðRij=RrÞ#12 ð3Þ which reduces to one at large Rij, while vanishing fast enough to prevent divergence of the undamped vdW potentials at small Rij. Here, Rris the sum of vdW radii of the atomic pair ij,37and the only non-linear parameter, a, controls the strength of dispersion corrections. As shown in Fig. 1, our new damping function correctly fixes the undesirable divergence of dispersion correction at the small nuclear distance R of the neon dimer, while the Wu–Yang damping function,34widely used by many DFT-D functionals37 does not completely remedy this problem (though this should not be a problem in practice).
To summarize, except for using our new damping function and constraining for the correct asymptotic pairwise vdW potentials (i.e. no overall scaling), we follow Grimme’s work37 for computing the empirical dispersion corrections, and denote this new functional asoB97X-D.
To achieve an optimized functional for well-balanced performance across typical applications, we use the same diverse training set described in ref. 18, that contains 412 accurate experimental and accurate theoretical results, including the 18 atomic energies from the H atom to the Ar atom,47the atomization energies (AEs) of the G3/99 set48–50 (223 molecules), the ionization potentials (IPs) of the G2-1 set51(40 molecules, excluding SH2(2A1) and N2(2P) cations due to the known convergence problems for semilocal density functionals49), the electron affinities (EAs) of the G2-1 set (25 molecules), the proton affinities (PAs) of the G2-1 set (8 molecules), the 76 barrier heights (BHs) of the NHTBH38/04 and HTBH38/04 sets,52,53 and the 22 non-covalent interac- tions of the S22 set.54To prevent the double-counting of total Fig. 1 Dispersion energy of the neon dimer with and without the damping functions. The damping function from Wu and Yang34 fWu–Yangdamp (R) = 1/(1 + exp[#d(R/Rr# 1)]) (Rris the sum of vdW radii of the two neon atoms), and from the present work [see eqn (3)] are compared with the undamped function.
energy from the KS-DFT and the dispersion corrections, all the parameters inoB97X-D, are determined self-consistently by a least-squares fitting procedure described in ref. 18. For the non-linear parameter optimization, we focus on a range of possibleo values (0.0, 0.1, 0.2, 0.3, 0.4, and 0.5 Bohr#1), and optimize the corresponding a values in steps of 0.1 for ar 1, in steps of 1 for 1o a r 10, in steps of 10 for 10 o a r 60, and in steps of 20 for a 4 60. The functional expansions employed in oB97X-D are truncated at m = 4 with the uniform electron gas (UEG) limit being satisfied. The S22 data is weighted ten times more than the others. The optimized parameters of theoB97X-D functional are given in Table 1.
In Fig. 2, the root-mean-square errors of the training set for oB97X-D optimized at different values of o are plotted. At o = 0.2 Bohr#1, the optimization is done self-consistently, while at other values ofo, non-self-consistent orbitals are used for the linear least-squares fittings to reduce the computational difficulty. At o = 0.0 and 0.1 Bohr#1, the corresponding RSHXLDA orbitals17 are used, and at o = 0.3 and
0.4 Bohr#1, theoB97X and oB97 orbitals18are used respec- tively. As previously demonstrated,18these results should only change insignificantly, when using self-consistent orbitals. As the root-mean-square errors ato = 0.2 and 0.3 Bohr#1are very close to each other, the optimizedo value is finally chosen to be 0.2 Bohr#1for the smaller mean absolute error.
Some aspects of the optimized parameters are interesting and deserve discussion. First, we observe that the optimized value of o is reduced from o = 0.3 Bohr#1 in oB97X to o = 0.2 Bohr#1inoB97X-D. At the same time, the fraction of short-range exact exchange increases by about 40% (from about 0.16 to about 0.22). Evidently the inclusion of long- range dispersion increases the optimal length scale on which we have 100% exact exchange, while at short distances we have increased the fraction of exact exchange. This increase of the fraction of short-range exact exchange in oB97X-D is expected to compensate the reduction of the exact exchange using a smaller value ofo. The net effect is that oB97X-D may have somewhat higher self-interaction errors. Second, regard- ing a possibleoB97-D functional, we find that the optimized value ofo occurs at 0.3 Bohr#1with a rather large a value (a = 80). This implies that only very small dispersion correc- tions will be obtained (at long range only). Therefore we do not explore this possibility further here.
The limiting case whereo = 0 for oB97X-D is also very interesting, as it reduces to the existing B97 functional55with dispersion corrections. For comparisons within the training set, we denote this re-optimized functional as B97-D*. It should be noted that B97-D* contains a fraction of HF exchange, unlike Grimme’s B97-D functional.37 The overall performance of our new oB97X-D will be compared with other DFT-D functionals, B97-D*, B97-D,37B3LYP-D,37and BLYP-D,37 as well as our previous LC hybrid functionals, oB97 and oB97X.18
3. Results and discussion
3.1. The training set
All calculations are performed with a development version of Q-Chem 3.0.56Spin-restricted theory is used for singlet states and spin-unrestricted theory for others. For the binding energies of the weakly bound systems, the counterpoise cor- rection57 is employed to reduce basis set superposition error (BSSE).
Results for the training set are computed using the 6-311++G(3df,3pd) basis set with the SG-1 grid.58The error for each entry is defined as (error = theoretical value # reference value). The notation used for characterizing statis- tical errors is as follows: mean signed errors (MSEs), mean absolute errors (MAEs), root-mean-square (rms) errors, maximum negative errors (Max(#)), and maximum positive errors (Max(+)).
In Table 2, the first comparison (oB97X-D vs. B97-D*) is quite significant because it indicates how much improvement is possible with the addition of a single extra parameter that makes long-range exchange exact and thus self-interaction free. Inspection of the training results shows significant improvement for both covalent and non-covalent interactions.
Table 1 Optimized parameters foroB97X-D. Here, the non-linear parameter a is defined in eqn (3), and others are defined in eqn (28) of ref. 28
a 6.0
o 0.2 Bohr#1
cxs,0 7.77964' 10#1
ccss,0 1.00000' 100
ccab,0 1.00000' 100
cxs,1 6.61160' 10#1
ccss,1 #6.90539 ' 100
ccab,1 1.79413' 100
cxs,2 5.74541' 10#1
ccss,2 3.13343' 101
ccab,2 #1.20477 ' 101
cxs,3 #5.25671 ' 100
ccss,3 #5.10533 ' 101
ccab,3 1.40847' 101
cxs,4 1.16386' 101
ccss,4 2.64423' 101
ccab,4 #8.50809 ' 100
cx 2.22036' 10#1
Fig. 2 The root-mean-square (RMS) errors of the training set for oB97X-D optimized at different values of o. At o = 0.2 Bohr#1, the optimization is done self-consistently, while at other values ofo, non- self-consistent orbitals are used for the linear least-squares fittings (see text).
oB97X-D shows a noticeable overall improvement (about 0.7 kcal mol#1) relative to B97-D*, which we may infer is an indirect consequence of its asymptotically correct exchange potential.
A second significant comparison is betweenoB97X-D and oB97X, to assess how significant the empirical dispersion corrections are. We observe generally very modest changes in relative energies associated with covalent interactions, but a significant improvement in the results for non-covalent inter- actions (the S22 data). This indicates the limited scope of chemical improvement that is attainable with the fixed functionality of atom-atom dispersion interactions. Neverthe- less, this essentially zero-cost correction does correct one clear physical limitation ofoB97X.
3.2. The test sets
To test the performance ofoB97X-D outside its training set, we also evaluate its performance on various test sets involving the additional 48 atomization energies in the G3/05 test set59 (other than the 223 atomization energies in the G3/99 test set48–50), 30 chemical reaction energies taken from the NHTBH38/04 and HTBH38/04 databases,52,53 29 non- covalent interactions,60,54166 optimized geometry properties of covalent systems,61 12 intermolecular bond lengths54 and one long-range charge transfer excitation curve of two well- separated molecules. There are a total of 286 pieces of data in the test sets. The detailed information of the test sets as well as the basis sets, and numerical grids used is given in ref. 18.
The results for the test sets are summarized in Tables 2 and 3. As can be seen in Table 2,oB97X-D performs the best with an overall accuracy of 2 kcal mol#1. TheoB97X, oB97, and B3LYP-D functionals also perform reasonably well, with overall accuracies of about 2.5 kcal mol#1. The semi-local
DFT-D functionals, B97-D and BLYP-D, produce enor- mously large mean absolute errors for the atomization en- ergies, which shows the important role played by mixing of the exact HF exchange for reducing self-interaction errors.
Satisfactory predictions of molecular geometries of covalent and non-covalent systems by density functionals are necessary for practical use. For covalent systems, we perform geometry optimizations for each functional on the equilibrium experi- mental test set (EXTS),61 while for non-covalent systems, we compute the intermolecular bond lengths of 12 weakly bound complexes taken from the S22 set,54using 6-311++G(3df,3pd) basis set with the fine grid, EML(75,302), consisting of 75 Euler–Maclaurin radial grid points62and 302 Lebedev angular grid points.63For non-covalent systems, we compare our results with the second-order Møller–Plesset perturbation theory (MP2).64 The resolution-of-identity (RI) approximation65 is used for the MP2 calculations using the aug-cc-pvqz auxiliary basis set. As shown in Table 3, performance of all of the hybrid functionals in predicting optimized geometries of EXTS is similar, and clearly better than that of the semi-local DFT-D functionals, B97-D and BLYP-D. Their performance in predicting intermolecular bond lengths is similar to MP2 theory.
In our previous work,18we have showed that our LC hybrid functionals resolve the qualitative failure of semi-local density functional in describing long-range charge-transfer (CT) excitations between a donor and an acceptor.10–12 In this work, we perform TDDFT calculations for the lowest CT excitations on the same system with the same optimized geometries.66 The high-level SAC-CI results, taken from ref. 16, are used for comparison.
As shown in Fig. 3, all of our LC hybrid functionals, oB97X-D, oB97X and oB97, predict CT excitation curves Table 2 Statistical errors (in kcal mol#1) of the training set and the test sets. The B97-D* functional is defined in the text. The results for the oB97X and oB97 are taken from ref. 18
System Error oB97X-D B97-D* B97-D B3LYP-D BLYP-D oB97X oB97
Training
G3/99 (223) MSE #0.10 0.31 #1.71 #1.01 #0.83 #0.09 #0.20
MAE 1.93 2.69 4.99 3.39 7.03 2.09 2.56
IP (40) MSE 0.19 1.59 #0.88 2.16 #1.52 #0.15 #0.48
MAE 2.74 3.35 3.58 3.68 4.43 2.69 2.65
EA (25) MSE 0.10 1.02 #0.15 1.72 0.38 #0.43 #1.45
MAE 1.92 2.33 2.08 2.41 2.58 2.05 2.67
PA (8) MSE 1.49 0.35 2.27 #0.42 #1.07 0.60 0.68
MAE 1.54 0.87 2.27 1.02 1.56 1.22 1.45
NHTBH (38) MSE #0.42 #2.29 #6.22 #5.13 #9.32 0.56 1.32
MAE 1.51 2.66 6.46 5.24 9.34 1.75 2.31
HTBH (38) MSE #2.52 #3.20 #7.33 #5.39 #8.89 #1.51 #0.34
MAE 2.64 3.20 7.41 5.39 8.89 2.24 2.24
S22 (22) MSE #0.08 #0.11 0.44 #0.28 0.22 0.53 0.16
MAE 0.22 0.44 0.50 0.48 0.33 0.87 0.60
All (394) MSE #0.29 #0.13 #2.30 #1.28 #2.37 #0.15 #0.14
MAE 1.94 2.62 4.73 3.52 6.40 2.05 2.39
Test
G3/05 (48) MSE 0.24 1.76 #2.62 1.88 0.76 1.28
MAE 3.01 7.39 4.53 9.65 3.60 4.25
RE (30) MSE #0.21 #0.01 #0.29 0.73 #0.07 0.09
MAE 1.66 2.70 1.92 3.01 1.74 1.97
Non-covalent (29) MSE #0.14 0.49 0.19 0.61 0.51 0.36
MAE 0.43 0.77 0.73 0.90 0.73 0.65
All (107) MSE 0.01 0.92 #1.21 1.21 0.46 0.69
MAE 1.93 4.28 2.77 5.42 2.30 2.64
that are in qualitative agreement with the high-level SAC-CI results, while B97-D, B3LYP-D and BLYP-D predict qualita- tively incorrect CT excitation curves. This emphasizes the important role of the LC hybrid functionals in TDDFT, especially for CT excited states. Finally we note that the remaining short-range self-interaction error is somewhat larger for oB97X-D than for oB97X than for oB97. This reflects their relativeo values, and suggests that oB97 will be preferred for TDDFT applications. We intend to investigate this question further in a future study.
4. Conclusions
We have developed a new LC hybrid DFT-D functional based on our previous work.18 This functional, called oB97X-D, includes 100% long-range exact exchange, a small fraction (about 22%) of short-range exact exchange, a modified B97 exchange density functional for short-range interaction, the B97 correlation density functional,55and empirical dispersion corrections. When the constraint ofo = 0 is applied, oB97X-D reduces to the existing B97 functional form55 with the same empirical dispersion corrections. The constrained form (o = 0), when re-optimized on the same training set, provides poorer fits to training data, indicating that the single extra degree of freedom corresponding to long-range exchange is
physically important. Relative to our previous functional, oB97X, oB97X-D provides significant improvement only for non-covalent interactions.
Since oB97X-D is a parametrized functional, we test it against three well-established existing DFT-D functionals (B97-D,37 B3LYP-D,37 and BLYP-D37) as well as our previous LC hybrid functionals (oB97X and oB97) on a separate independent test set of data, which includes further atomization energies, reaction energies, non-covalent interac- tion energies, equilibrium geometries, and a charge-transfer excited state. The results indicate that this new long-range corrected DFT-D functional is generally somewhat superior in overall performance. Relative to oB97X, we recommend oB97X-D for applications where non-covalent interactions are expected to be significant.
oB97X-D does have some limitations that are appropriate to summarize as we conclude. Like other LC functions, it is free of long-range self-interaction, but still suffers from some self-interaction at short-range. The optimized parameters we have obtained are such that this effect is slightly larger for oB97X-D than for oB97X, which in turn was slightly larger than for oB97. We also note that long-range correlation effects are solely treated by the empirical dispersion correc- tions in oB97X-D, and therefore the KS orbitals themselves are not directly influenced by dispersion effects.
Acknowledgements
This work was supported by the US Department of Energy through the Chemistry Endstation project of the Petascale computing program. M.H.G. is a part-owner of Q-Chem Inc.
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