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  • 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*a and 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 = cXEX DFT + (1 � cX)EXHF + cCECDFT

    + cOEO MP2 + 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 s6 of 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: sebastian.kozuch@weizmann.ac.il

    b Center for Advanced Scientific Computing and Modeling (CASCAM), Department of Chemistry, University of North Texas, Denton, TX 76203-5017, USA. E-mail: gershom@unt.edu

    w Electronic supplementary information (ESI) available: Complete statistical errors of the training and validation sets, detailed theoretical methods, plus guidelines to run DSD-PBEP86 with various software packages. See DOI: 10.1039/c1cp22592h

    PCCP Dynamic Article Links

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    H View Online / Journal Homepage / Table of Contents for this issue

    http://dx.doi.org/10.1039/c1cp22592h http://dx.doi.org/10.1039/c1cp22592h http://dx.doi.org/10.1039/c1cp22592h http://pubs.rsc.org/en/journals/journal/CP http://pubs.rsc.org/en/journals/journal/CP?issueid=CP013045

  • 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 + s6 varying 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

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