Citethis:Phys. Chem. Chem. Phys.,2011,13 ,2010420107 ...webhome.weizmann.ac.il/home/comartin/OAreprints/238.pdf · SPBE, or DSD-SP86) leads to a very significant deterioration. Perhaps

20104 Phys. Chem. Chem. Phys., 2011, 13, 20104–20107 This journal is c the Owner Societies 2011

Cite this: Phys. Chem. Chem. Phys., 2011, 13, 20104–20107

DSD-PBEP86: in search of the best double-hybrid DFT with

spin-component scaled MP2 and dispersion correctionsw

Sebastian Kozuch*aand Jan M. L. Martin*

ab

Received 11th August 2011, Accepted 27th September 2011

DOI: 10.1039/c1cp22592h

Spin-component scaled double hybrids including dispersion

correction were optimized for many exchange and correlation

functionals. Even DSD-LDA performs surprisingly well.

DSD-PBEP86 emerged as a very accurate and robust method,

approaching the accuracy of composite ab initio methods at a

fraction of their computational cost.

Double hybrid (DH) DFT has shown to be a successful

method for the accurate energy estimation of small and

medium sized molecules.1–6 DH-DFT in its more typical

formulation mixes an exact exchange term (‘‘HF-like’’) with

the DFT exchange functional as simple hybrids, but adds a

perturbational correlation term (‘‘MP2-like’’) to the DFT

correlation in the basis of the Kohn–Sham orbitals.7 The first

DH of this form was the B2-PLYP of Grimme,3 with other

examples being the general purpose B2GP-PLYP2 and the

long-range corrected oB97X-2.5

Some additional flexibility can be provided by setting a

different weight to the same-spin and opposite-spin MP2, in

the spin-component-scaled MP2 method (SCS-MP2).8 Same-

spin MP2 reflects mostly long range correlation, while the

opposite-spin is related to short range interactions.1,8

It is possible (and advisable) to add a dispersion correction

to DFT methods, as typically they do not account for long

range interactions.9 In the case of DH-DFT this is less critical,

as the MP2 term does account for those interactions (at least

partially);2 nevertheless, the accuracy expected for DHs

cannot be attainable for non-bonding interactions without a

dispersion add-on.

When the SCS distinction and a dispersion term are

included, we obtain the DSD-DFT (Dispersion corrected,

Spin-component scaled Double Hybrid).1 The total exchange–

correlation term is then expressed as (see eqn (1)):

EXC = cXEXDFT + (1 � cX)EX

HF + cCECDFT

+ cOEOMP2 + cSES

MP2 + s6ED (1)

where cX is the amount of DFT exchange, cC that of DFT

correlation, cO and cS of opposite and same-spin MP2, and s6of the D2 dispersion correction. In this form (eqn (1)) we have

already presented the DSD-BLYP functional, which had a

remarkable performance.1,4,6,10

Herein we will seek for the best combination of exchange–

correlation functional for the DSD-DFT method. The

different functionals were selected from the ‘‘Popularity Poll of

Density Functionals’’,11 with some choices of our own added.

The parametrization procedure is virtually the same as the

one discussed in our previous DSD-BLYP paper.1 (Further

technical details are relegated to the electronic supporting

information.)

Six training sets were used to optimize the parameters of

eqn (1): W4-08 (atomization energies),12 DBH24 (reaction

kinetics),13 Pd (oxidative additions on a bare Pd atom),14

Grubbs (olefin metathesis with a Ru catalyst),15 Grimme’s

‘‘Mindless benchmark’’ (quasi-randommain group reactions),16

and S22 (for van der Waals forces and H-bonds).17,18 These

training sets cover thermochemistry and kinetics of main

group and transition metals, plus long range interactions.

Fig. 1 presents the overall error statistics over our training

sets for the different functional combinations. Table 1 shows

the parameters of selected combinations (detailed error

statistics over all the subsets are reported in the ESIw).Perhaps the most stunning result to the present investigators

is the surprisingly good performance of the DSD-SVWN5

(that is, DSD-LDA) functional, with an average error of just

1.83 kcal/mol. Even though all double hybrids are, strictly

speaking, on the 5th rung of the ‘‘Jacob’s Ladder’’,19 we can

climb the underlying ‘‘sub-ladder’’ functional. For DSD-PBE

(2nd sub-rung, a GGA) this is improved to 1.75 kcal/mol,

while DSD-TPSS (3rd sub-rung, a meta-GGA) actually does

worse at 2.09 kcal/mol. Essentially all of the improvement for

DSD-PBE derives from a 25% drop in the RMSD for the

‘‘mindless benchmark’’.

Returning to the DSD-LDA result, we note that replace-

ment by a GGA of either the exchange part (DSD-BVWN5,

DSD-PBEVWN5) or the correlation part (DSD-SLYP, DSD-

SPBE, or DSD-SP86) leads to a very significant deterioration.

Perhaps this is not surprising as the performance of LDA

itself, such as it is, would be even worse were it not for an error

compensation between underestimated exchange and excess

correlation.

aDepartment of Organic Chemistry, Weizmann Institute of Science,IL-76100 Rechovot, Israel. E-mail: [email protected]

b Center for Advanced Scientific Computing and Modeling(CASCAM), Department of Chemistry, University of North Texas,Denton, TX 76203-5017, USA. E-mail: [email protected]

w Electronic supplementary information (ESI) available: Completestatistical errors of the training and validation sets, detailed theoreticalmethods, plus guidelines to run DSD-PBEP86 with various softwarepackages. See DOI: 10.1039/c1cp22592h

PCCP Dynamic Article Links

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This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 20104–20107 20105

The DSD-B98, DSD-HCTH407, DSD-tHCTH, and DSD-

BMK results are presented more as curiosa than anything else.

It is quite likely that the high error metrics for these functionals

could be substantially reduced by refitting the coefficients in

their empirical power series expansions simultaneously with the

double-hybrid parameters: however, we doubt that this would

improve on the functional we shall shortly discuss.

Earlier we published1 a DSD-BLYP spin-component scaled

double hybrid: the parameter set given in Table 1 of the

present work is slightly different from that in ref. 1 as we have

used the D2 dispersion correction9 with the standard cutoff

function exponent a = 20.0 throughout the present work,

rather than the almost step-function like a= 60.0 proposed in

ref. 1. At the time we noted that, while its error statistics did

not represent a great improvement over B2GP-PLYP,2 the

functional was more robust to nondynamical correlation as its

error statistics for W4-08MR (the subset of molecules with

severe multireference effects) were substantially improved. In

the present work we find that not only is DSD-BLYP little

better than DSD-LDA, but substantial improvements can be

reached by replacing the correlation functional. DSD-BP86,

with an overall error of just 1.68 kcal/mol, appears to be the

best DSD-DFT combination with B88 exchange.

Among the several GGA correlation functionals considered,

for a given exchange functional the ordering of error metrics

appears to be P86 o PW91 o PBE o LYP. Within the P86

correlation column, DSD-PBEP86 marginally outperforms

DSD-mPW-P86 (by 1.62 vs. 1.63 kcal/mol), followed by

DSD-BP86 (1.86 kcal/mol). The range-separated HSE

hybrid20 achieves slightly better performance statistics with a

given correlation functional than the ordinary GGAs;

however, for the P86 correlation functional, the performance

gain is quite small (1.60 kcal/mol for DSD-HSEP86 vs.

1.62 kcal/mol for DSD-PBEP86). As PBE exchange and P86

correlation are available in all the major quantum chemical

codes, we have decided to focus on the DSD-PBEP86 combi-

nation for more detailed validation. The effect of range

separated DHs5 (including HSE exchange) will be considered

in future work.

As in earlier work on DSD-BLYP,1 we found during

optimization that cS is strongly coupled to s6, with cS + s6varying quite little during the optimization. This reflects the

long-range character of the same-spin contribution. The sum

of both parameters is almost always smaller than one, showing

the overbinding of the MP2 method on dispersion forces.

The HF exchange percentage varies within a narrow band,

as typically does the opposite spin MP2 contribution (cO) and

the DFT correlation. In non-SCS double hybrids the sum of

MP2 and DFT correlation is set to one,2,3 while in the DSD-

DFT version this constraint is lifted. Yet, the sums of the

short-range term coefficients (cO + cC) typically stay very

close to unity.

High exact exchange is typically considered counterproductive

for transition metals in simple hybrid DFT. In contrast, the DHs

considered here perform excellently for the Pd and Grubbs sets.

The counterbalance of the MP2 correlation rectifies the defici-

encies of extreme HF exchange.

Our present observation of subpar performance of LYP

appears to conflict with earlier statements2,3 to the effect that

the LYP correlation functional outperforms all the others in

double-hybrid settings. The reason for this apparent contra-

diction is revealed by considering the ratio between optimum

cS and cO coefficients in Table 1. While it is fairly close to unity

for combinations involving LYP correlation, cS/cO is much

lower than one in all other combinations. As both ref. 2,3

imposed cS = cO, a bias in favor of LYP correlation was

naturally introduced.

Why would LYP prefer a ratio cS/cO closer to one, while the

other correlation functionals all prefer cS much smaller than

cO? It has been noted earlier (e.g. by Perdew and coworkers21)

that LYP has a quite imbalanced treatment of same-spin and

opposite-spin correlation: in fact, the LYP correlation

energy of a fully spin-polarized system is zero.22 While this

admittedly guarantees that a single electron has no unphysical

self-correlation in LYP (unlike in other GGAs), a fully spin-

polarized uniform electron gas manifestly does have a correla-

tion energy.

Scaled-opposite spin (SOS) MP2 was introduced by Head-

Gordon et al. as a computationally efficient post-HF

method.23 It was observed that for typical (non-van der Waals

driven) chemical reactions the cS/cO ratio in pure MP2

methods is small (leading to the SCS-MP2 method8). In this

case, it is possible to completely neglect the long-range same-

spin component. The result is the SOS-MP2 method, which

through a Laplace transformation recasts the opposite spin

Fig. 1 Mean error of the DSD-DFT methods (in kcal/mol). They all

present an outstanding accuracy compared to single-hybrid DFT, with

the P86 correlation as a clear winner.

Table 1 Parameters of selected DSD-DFT with different functionalcombinations, including D2 dispersion correction (see eqn (1))

Correlation S-VWN5 B-LYP B-P86 PBE-PBE PBE-P86

cS (Same. Spin MP2) 0.11 0.43 0.24 0.12 0.23cO (Opp. Spin MP2) 0.58 0.46 0.49 0.53 0.51s6 (D2 disp.) 0.28 0.35 0.41 0.42 0.29cC (DFT Corr.) 0.34 0.55 0.49 0.51 0.45cX (DFT Exch.) 0.29 0.29 0.33 0.34 0.32

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20106 Phys. Chem. Chem. Phys., 2011, 13, 20104–20107 This journal is c the Owner Societies 2011

MP2 in a form with O(N4) computational cost scaling, rather

than the O(N5) of full MP2. This approach was already

applied on the B2OS3LYP and PWPB95 double-hybrids.6,24

For the sake of clarity, we will refer as ‘‘DOD-DFT’’ to

DSD-DFT parametrizations with constraint cS = 0. Not

surprisingly, we found a putative DOD-BLYP to have a rather

poor RMSD of 2.82 kcal/mol for the training set. However,

DOD-PBE seems a natural choice, as the optimal cS/cO ratio

is small to begin with. Its RMSD is only 1.82 kcal/mol,

comparable with some of the best double-hybrids (see complete

statistics and parameters in the ESIw).To improve the long-range behaviour of the selected DSD-

PBEP86, we substituted the D2 dispersion correction by the

D3BeckeJohnson version.25 The latter’s parameters were fitted

against the S130 set of weak interaction data employed by

Grimme and coworkers in the original D326 and D3BJ25

papers (it is also a subset of the GMTKN30 benchmark4).

Its 130 datapoints include the S22 set at equilibrium and two

stretched geometries (inter-monomer distances 1.5 and 2 times

equilibrium),27 n-alkane conformer energies28 (which are

particularly sensitive toward proper treatment of dispersion28),

alkane dimers from ethane to n-heptane, small peptide con-

formers, sugar conformers, cysteine conformers, and six rare

gas dimers. The latter were given a weight of 20 as in ref. 25,26.

As the published D3BJ parameters for other functionals25

have zero or insignificant a1 for double-hybrids, a1 was con-

strained to zero. In addition, as the reduction in RMSD from

allowing a nonzero s8 was found to be statistically insignificant

during the fitting, s8 was fixed at zero as well. The resulting

D3BJ parameters were s6 = 0.418 and a2 = 5.65 (a procedure4

based on asymptotic behavior of noble gas dimers yielded a

somewhat higher s6 = 0.50). As the superposition of the DFT

and the dispersion correction parts must be shaped according

to the method, we re-optimized the DH parameters, now with

D3BJ instead of D2. The final and recommended DSD-PBEP86

parameters with the D3BJ dispersion scheme are cS = 0.25,

cO = 0.53, cX = 0.30, cC = 0.43. For this functional, the mean

error for the training set is 1.51. (If D3BJ is not available, the

optimal D2 has s6 = 0.276).

Strangely, the original DSD-BLYP1 with D3BJ dispersion6

actually worsens in the training set compared to D2 (mean

error = 2.34 vs. 1.80). The reason for this deterioration lies in

the parameters for this functional being optimized for D2

dispersion, and not for D3. This shows that the dispersion

must be optimized jointly with the DFT part, taking into

account not only dispersion forces but also the short-range

interactions (as we did for the final DSD-PBEP86 parameters).

Several validation sets were studied to test the validity of the

method. Some of them were already considered in our previous

DSD-BLYP paper,1 such as the NHTBH38 (Non Hydrogen

Transfer Barrier Heights) and HTBH38 (Hydrogen Transfer

Barrier Heights),29 H-bonded dimers,30 van der Waals

complexes,31 pericyclic reactions,32,33 alkanes thermochemistry

and isomerization,34 and harmonic frequencies.1 Selected

results are presented in Table 2, while the rest is relegated to

the ESI.w However, it must be noted that all the DSD-DFT

methods have an outstanding performance compared to

lower rungs on Perdew’s ladder.19 To these validation sets,

we add here the anthracene dimerization35 and W4-11

(an expanded version of the W4-08 small molecule thermo-

chemistry benchmark).36

For the W4-11 set, restricting ourselves to the subset of 124

molecules not dominated by nondynamical correlation effects

leads to an RMSD of 1.9 kcal/mol, on par with the G4

composite ab initio method. The latter is still more resilient

for strongly multireference species, although much of the

DSD-PBEP86 RMSD there comes from a few pathological

systems such as trans-HOOO.

Improvements over the earlier DSD-BLYP and B2GP-

PLYP functionals are seen especially for the Houk pericyclic

reaction set and, to a lesser extent, for hydrogen-bonded

dimers. Unlike DSD-BLYP, DSD-PBEP86 does quite well

for alkane TAEs.

The [4 + 4] cycloadditive photodimerization of anthracene

to a D2h ‘‘sandwich’’ is a challenging test for DFT methods35

as it involves both covalent bonding between the central rings

and dispersion forces between the outer rings. (The mechanism

has been explored in detail by Bendikov and coworkers37).

At B3LYP-D/Def2-TZVP reference geometries, we obtain

11.15 kcal/mol at the CCSD(T)/cc-pVTZ(no f on H) level:

lower level results with this basis set are: MP2 20.59, SCS-MP2

13.13, and CCSD 9.85 kcal/mol. At the MP2/cc-pVQZ level,

we obtained 20.21 kcal/mol (SCS-MP2 12.55 kcal/mol): using

these latter two results (which stretched our hardware to the

limit) for estimated basis set expansion effects beyond

cc-pVTZ(no d on H), we obtain 10.77 and 10.57 kcal/mol,

respectively, leading us to propose a best estimate of

10.7� 1 kcal/mol where the error bar is somewhat conservative.

The earlier best estimate35 of 9 � 3 kcal/mol was obtained from

MP2/TZV(2d,2p) geometries at a combination of QCISD(T)/

cc-pVDZ and MP2/cc-pVQZ levels.

As noted earlier,35,38 most conventional DFT methods get

the sign wrong even with dispersion corrections. We do note

that M06-2X38,39 yields a semiquantitative answer of about

6 kcal/mol. At the B2GP-PLYP/TZVPP level, we obtain

–0.90 kcal/mol raw and +7.39 kcal/mol with a D2 dispersion

correction. DSD-BLYP yields 3.04 (raw) and 10.50 (D2 or

D3BJ) kcal/mol.

DSD-PBEP86 yields 6.32 (raw) 12.04 (D2), and 12.07

(D3BJ) kcal/mol. The dispersion-corrected results are in rather

pleasing agreement with the best estimate, but even the

raw answer is still semiquantitative. This is not the case for

Table 2 Selected RMSD errors for validation sets1 (in kcal/mol)

DSD-BLYPa

DSD-PBEP86b

B2GP-PLYPa,c B3LYPa,c M06c

NHTBH38 2.08 2.18 1.61 5.95 2.83HTBH38 1.14 1.12 1.33 5.78 2.41H-Bonds 0.45 0.38 0.47 0.95 0.42VdW 0.43 0.47 0.40 0.82 0.49Pericyclicd 3.15 2.05 2.65 2.56 2.08W4-11 non MR 2.2 1.9 1.9 4.2 3.9W4-11 all 2.6 2.5 3.0 5.0 4.4Alkanes:TAEs 3.48 1.07 1.00 0.61 3.67Isomeriz. 0.40 0.16 0.19 0.43 0.25

a With D2 dispersion correction; b With D3BJ dispersion correction.c As appeared in ref. 1; d Using W1 data as reference.

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B2GP-PLYP (-0.90 raw, 7.39 with D2) or even DSD-BLYP

(3.04 raw, 10.50 with D2). SCS-CCSD40 overshoots

(15.69 kcal/mol) and SCS(MI)-MP241 severely undershoots

(4.69 kcal/mol) despite being ostensibly optimized for inter-

molecular interactions.

Performance of double hybrids for harmonic frequencies

was considered in ref. 1 for the HFREQ27 set of harmonic

frequencies of 27 small molecules. The optimum double hybrid

for frequencies was found to be 43% exchange and 18% MP2

(not useful parameters for thermochemistry nor, especially,

kinetics), with an RMS deviation of 18.6 cm�1. B2GP-PLYP

and DSD-BLYP yields much worse RMSD of 29.8 and

30.9 cm�1, respectively, compared to 33.9 for B3LYP and

44.0 cm�1 for MP2. Using scaling factors (given in parentheses),

this can be reduced to 17.7 (0.989) for B2GP-PLYP and

22.4 (0.990) for DSD-BLYP.

For DSD-PBEP86, even the unscaled frequencies have an

RMSD = 19.7 cm�1, which drops to 17.7 cm�1 when F2 is

removed as an outlier. A fitted scaling factor of 0.995 yields a

just marginally better RMSD = 16.6 cm�1. We wish to stress

that no parametrization whatsoever was made for this property:

therefore, this strongly suggests that DSD-PBEP86 is a more

‘universal’ functional than either B2GP-PLYP or DSD-BLYP.

For the sake of completeness, DSD-BP86 and DSD-mPWP86

yield unscaled RMSDs of 19.2 and 18.8 cm�1, respectively.

Conclusions

Following up on our earlier DSD-BLYP functional,1 we have

optimized spin-component scaled, dispersion-corrected9,25,26

double hybrids using many different exchange and correlation

functional forms. The resulting functionals have been validated

against a variety of thermodynamic and kinetic benchmarks.

Somewhat surprisingly, the quality of the results is only mildly

sensitive to the choice of the underlying DFT exchange and

correlation components, and even DSD-LDA yields respectable

performance. Simple, nonempirical GGAs appear to work best:

meta-GGAs offer no advantage. P86 correlation is a clear

winner, followed by PW91C and PBEC. The choice of the

exchange DFT part is less critical, and we selected the PBE

exchange for its accuracy and wide availability. Our earlier

assertion that LYP correlation is essential for double hybrids

is found to be an artifact of constraining the same-spin and

opposite-spin coefficients to be equal (although other non-LYP

based double hybrids were found to be efficient5,6). Our best

functional, DSD-PBEP86 (with D3BJ dispersion25), emerges as a

very accurate and robust quantum mechanical method, also

performing well for properties not included in its parametriza-

tion (such as harmonic frequencies). It approaches the accuracy

of ab initio composite methods at a fraction of their computa-

tional cost. Lastly, it can be run using unmodified versions of

several common quantum chemistry programs (see ESIw).

Notes and references

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2 A. Karton, A. Tarnopolsky, J.-F. Lamere, G. C. Schatz andJ. M. L. Martin, J. Phys. Chem. A, 2008, 112, 12868–12886.

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13 J. Zheng, Y. Zhao and D. G. Truhlar, J. Chem. Theory Comput.,2009, 5, 808–821.

14 M. M. Quintal, A. Karton, M. A. Iron, A. D. Boese and J. M. L.Martin, J. Phys. Chem. A, 2006, 110, 709–716.

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