Exploring Avenues Beyond Revised DSD Functionals: I. range separation, with xDSD as a special case
1 Exploring Avenues Beyond Revised DSD Functionals: I. range separation, with xDSD as a special case
Golokesh Santra, † Minsik Cho, †,§ and Jan M.L. Martin *,† † Department of Organic Chemistry, Weizmann Institute of Science, 7610001 Reḥovot, Israel.
Email: [email protected] § Department of Chemistry,
Brown University, Providence, Rhode Island 02912, USA
Abstract:
We have explored the use of range separation as a possible avenue for further improvement on our revDSD minimally empirical double hybrid functionals. Such ωDSD functionals encompass the XYG3 type of double hybrid (i.e., xDSD) as a special case for ω ->0. As in our previous studies, the large and chemically diverse GMTKN55 benchmark suite was used for evaluation. Especially when using the D4 rather than D3BJ empirical dispersion model, xDSD has a slight performance advantage in WTMAD2. As in previous studies, PBEP86 is the winning combination for the semilocal parts. xDSD n -PBEP86-D4 marginally outperforms the previous ‘best in class’ ω B97M(2) Berkeley double hybrid, but without range separation and using fewer than half the number of empirical parameters. Range separation turns out to offer only marginal further improvements on GMTKN55 itself. While ω B97M(2) still yields better performance for small-molecule thermochemistry, this is outweighed in WTMAD2 by superior performance of the new functionals for conformer equilibria. Results for two external test sets with pronounced static correlation effects may indicate that range-separated double hybrids are more resilient to such effects. I. Introduction:
Kohn-Sham density functional theory (KS-DFT) is presently by far the most widely used family of electronic structure methods. Its combination of reasonable accuracy and comparatively gentle computational cost scaling makes it an appealing choice for medium and large molecules; for small molecules, wavefunction ab initio (WFT) approaches still outperform it. The accuracy of KS-DFT stands or falls with the exchange-correlation (XC) functional. Perdew organized the plethora of available approaches into what he called a “Jacob’s Ladder” arranged by the kinds of information employed in it: LDA (local density approximation) on the first rung, GGAs (generalized gradient approximations) on the second rung, meta-GGAs on the third rung (adding either the density Laplacian or the kinetic energy density), hybrid functionals on the fourth run (adding also occupied orbital information). The fifth rung corresponds to inclusion of virtual orbital information: the most widely used class of such methods are so-called double hybrids (see Refs. – for reviews, and most recently Ref. by the present authors). As shown in Refs. , their accuracy over the very large and diverse GMTKN55 (General Main-Group Thermochemistry, Kinetics, and Noncovalent Interactions, with 55 problem sets) test suite approaches that of WFT methods, yet the CPU cost increase over ordinary hybrid GGAs is actually quite modest if an RI (resolution of the identity ) approximation is applied in the MP2 (2nd-order Møller-Plesset) part. Generally speaking, there are two basic approaches available for double hybrids in literature, which we shall denote gDH (after Grimme ) and xDH (after the XYG3 functional , ) in the paper. In gDH, an iterative Kohn-Sham(KS) calculation is carried out with a fraction(c ’ X,HF ) of Hartree-Fock (HF) exchange and (1 – c ’ X,HF ) of DFA (density functional approximation) exchange, plus DFA correlation scaled by a coefficient c
C,DFA . Next, using the converged orbitals from the KS step, a post-HF GLPT2(second-order G örling−Levy perturbation theory) correlation energy term is evaluated in the basis of the KS orbitals and added in. (As with lower-rung DFT methods, a dispersion correction can optionally be added, though it generally needs a prefactor less than unity as some dispersion is already captured in the GLPT2 term.) In contrast, in xDH, the KS orbitals used for the evaluation of all energy terms following the at the final step are evaluated for a standard hybrid with full DFA correlation (i.e., c C,DFA = 100%), and with c X as appropriate for a typical hybrid functional. It was argued , that such orbitals are more appropriate as a basis for GLPT2 than the damped-correlation orbitals in gDH, though this argument has been refuted on empirical ground by Goerigk and Grimme and by Kesharwani et al. Kozuch and Martin modified the gDH approach into their dispersion-corrected spin-component-scaled double hybrids (DSD), which employ the following energy equation: E DH−DSD = E
N1e + c
X,HF E x,HF + (1 − c X,HF )E X,XC + c
C,XC E C,XC + c E + c E + E disp [𝑠 , 𝑠 , 𝑐 𝐴𝑇𝑀 , 𝑎 , 𝑎 , 𝑒𝑡𝑐] (1) where E N1e stands for the sum of nuclear repulsion and one-electron energy terms; c
X,HF and c
C,XC are the fractions of exact exchange and semilocal correlation; E disp is the dispersion correction term (dependent upon parameters like s ; s ; a ; a ; c ATM … ); and c and c are the two coefficients corresponding to opposite-spin and same-spin GLPT2 correlation. The xDH version thereof, denoted xDSD, was explored in Ref. and found to offer only a minor advantage over the corresponding DSD. It must however be said that both the DSD and the xDSD functionals were originally parametrized and validated using quite modest training sets (for reasons of computational cost): furthermore, the weighting of the subsets is somewhat arbitrary, and experimentation on our part showed considerable dependence of the final parameters on the weights used there. In contrast, the much larger GMTKN55 dataset is not only over 10x larger, but uses a more robust, unambiguously weighted, performance metric in the guise of WTMAD2 (weighted mean absolute deviation, type 2, in which the weights of the subsets are corrected for the different energy scales of the reference data). In Ref. we were able to leverage GMTKN55 to obtain a family of more accurate revDSD functionals, with revDSD-PBEP86-D4 as the winner among them: just for the PBEP86 case, we also considered a single example of the xDH type, and did find xrevDSD-PBEP86-D4 to be slightly more accurate still than revDSD-PBEP86-D4. One objective of the present paper is to explore whether this is true more generally: specifically, we shall investigate xDSD-PBEPBE here we also include xDSD-PBEPW91, xDSD-PBEB95, xDSD-BLYP, and xDSD-SCAN in our arsenal. Two types of dispersion correction, D3(BJ) and the more recent, more flexible and accurate, D4 will be considered, in the latter also with different many-body dispersion terms. (We also will consider xDOD forms, in which same-spin GLPT2 is eliminated: this permits further acceleration for large systems through a Laplace transform algorithm. ) The second objective is to investigate whether revDSD can be improved through introducing range-separated Hartree-Fock exchange (RSH). In the long-distance limit, the exchange potential of global hybrids(GHs) behaves like – c X /r rather than the correct – (r being the interelectronic distance). Hence, Hirao and coworkers proposed a scheme where the interelectronic repulsion operator 1/r is partitioned into a short-range (SR) components to be treated by a (meta)GGA, and a long-range (LR) component to be treated by “exact” exchange, and to ‘crossfade’ from SR to LR using an error function (erf x ) and the complementary error function (erfc x = 1 – erf x ) of r . A more generalized form of this model was later proposed by Handy and coworkers: = 1 − 𝛼 − 𝛽erf(ωr )r ⏟ SR= Short−Range + α + 𝛽erf(ωr )r ⏟ LR= Long−Range (2)
In this equation, ω represents the range separation parameter, which controls the transition between the LR and SR parts, α is the percentage of “exact” HF exchange in the short -range limit, and α + β the corresponding percentage in the long-range limit. (Proper asymptotic behavior can be enforced through α + β =1/ ε , where the dielectric constant ε =1 in vacuo — leading to β =1- α — and ε -> ∞ for a perfect conductor.) ω can be determined empirically using a training set, – or tuned non-empirically minimizing the deviation from the conditions the exact KS functional must obey. Following this approach, several empirical and non-empirical LC-DH functionals have been proposed, such as LC-PBE, LC- ω PBE, M11, CAM-B3LYP, ωΒ , ω B97X, ω B97X-V, ω B97M-V, and many more. We shall denote DSD-type double hybrids with RSH functionals ω DSD, where ω stands for the range separation parameter. Note that for ω =0, ω DSD and ω DOD functionals reduce to the xDSD and xDOD forms, respectively, which ties our two objectives together. The combination of range separation with GLPT2 for the correlation energy was first proposed by Ángyán and coworkers. Chai and Head-Gordon instead obtained orbitals from a RSH calculation and then evaluated the GLPT2 correlation in the basis of these orbitals, the final energy being a mix of GGA exchange, HF exchange, GGA correlation, and GLPT2 correlation. Their most recent elaboration of his concept was the ω B97M(2) functional, which for the GMTKN55 benchmark was found to have the lowest WTMAD2 of all functionals surveyed. (To be fair, however, it has three times the number of empirical parameters of the next best performer, xrevDSD-PBEP86-D4. ) Another effort along these lines was RSX-QIDH by Adamo et. al. , who established a ‘nonempirical’ parametrization combining their ‘ nonempirical ’ quadratic integrated double hybrid(QIDH) model together with Savin’s RSX(range-separated exchange) scheme. Later they introduced another such LC-DH, RSX-0DH. Two very recent empirical
RSDHs, originally developed for an electronic excitation energy benchmark, are ωB2PLYP and ωB2GP -PLYP by Goerigk and coworkers. II.
Computational Methods:
II.A. Reference Data : We can divide parameter space into “linear” parameters such as s , s , c and c , and “ nonlinear ” parameters such as α and ω : every change in the latter requires complete recalculation of the entire GMTKN55 database, which would make a complete survey of ( α , ω ) parameter space for every underlying semilocal functional intractably costly. Fortunately, Gould obtained so- called “diet” versions of GMTKN55, which are statistical reductions to the most representative 50 (diet50), 100 (diet100), or 150 (diet150) reactions. After some experimentation, we settled on diet100 for the prescreening stage: based on this, we will decide which semilocal functionals to retain for in-depth investigation with the full GMTKN55 set. GMTKN55 is the updated and larger form of Grimme group ’s previous GMTKN24 and GMTKN30 database. This dataset consists of 55 types of chemical problems, which can be further categorized into five top-level subsets: thermochemistry of small and medium-sized molecules, barrier heights, large molecule reactions, intermolecular interactions, and conformer energies. One full evaluation of the GMTKN55 needs 2459 single point energy calculations (give or take a few duplicates) to generate 1499 unique energy differences. (Complete details of all 55 subsets and original references can be found in Table S1 in the ESI) Originally proposed by Goerigk et al. , WTMAD2(weighted mean absolute deviation type 2), has been used as the primary metric for this work: WTMAD2 =
1∑ N i55i=1 . ∑ N i55i=1 . kcal mol ⁄|∆E|̅̅̅̅̅̅ i . MAD i (3) where |∆𝐸|̅̅̅̅̅̅ 𝑖 is the mean absolute value of all the reference energies from 𝑖 = 1 to 55, 𝑁 𝑖 is the number of systems in each subset, 𝑀𝐴𝐷 𝑖 is the mean absolute difference between calculated and reference energies for each of the 55 subsets. Note that, from the statistical viewpoint, MAD(mean absolute deviation) is a more ‘robust’ metric than RMSD(root -mean-square deviation), as MAD is more resilient to a small number of large outliers than the RMSD. For a normal distribution without systematic error, RMSD≈5MAD/4. Reference geometries were taken “as is” from
Ref. and not optimized further. II.B. Electronic Structure calculations:
All the calculations were performed using Q-CHEM 5.3 package (except ωB and ωB for which ORCA 4.2.1 has been used), running on the ChemFarm HPC cluster of the Weizmann Institute Faculty of Chemistry. The Weigend−Ahlrichs def2-QZVPP basis set was considered throughout with a few exceptions, like WATER27, RG18, IL16, G21EA, AHB21,BH76 and BH76RC subsets, where the diffuse-function augmented def2-QZVPPD basis set was used instead. However, for the computationally demanding C60ISO and UPU23 subsets, which have small weights in WTMAD2, the more economical def2-TZVPP was employed to curb computational cost. The SG-3 integration grid was used across the board, except for the SCAN (strongly constrained and appropriately normed meta-GGA type) variants, where due to its severe integration grid sensitivity , unpruned (150, 590) grid was employed. In the MP2-like step, the RI (resolution of the identity) approximation was applied in conjunction with the def2-QZVPPD-RI fitting basis set. In this project, while most of the calculations were done using frozen inner-shell orbitals, we made two departures from this recipe to avoid unacceptably small orbital energy gaps between the highest frozen and lowest correlated orbital. First, for the
MB16−43,
HEAVY28, HEAVYSB11, ALK8, CHB6 and ALKBDE10 subsets, we correlated the (n-1)sp subvalence electrons of the metal and metalloid atoms. Second, for HAL59 and HEAVY28, the (n − 1) spd orbitals of the heavy p-block elements were kept unfrozen.
II.C. Optimization of Parameters:
Range separated DSD double hybrids have seven empirical parameters: a. fraction of exact exchange c X,HF or α ; b. fraction of semilocal DFT correlation c C,DFT ; c. fraction of opposite-spin PT2 correlation c ; d. fraction of same-spin PT2 correlation c =c ; e. prefactor s for the D3(BJ) dispersion correction ; f. damping function range parameter a for the D3(BJ).(as recommended in Refs we set a =0 and s =0 ); and g. the range separation parameter, ω . Now, the xDSD family of functionals, being the special case of range separated DSD type (i.e., ω=0 ), have six parameters (a-f) instead of seven. Powell’s BOBYQA(Bound Optimization BY
Quadratic Approximation) derivative-free constrained optimizer and a few scripts and Fortran programs developed in-house, were used for optimizing the parameters. Once a full GMTKN55 evaluation is finished with a fixed set of { c X,HF , c c,DFT , ω } no further electronic structure calculation is necessary to obtain an associated optimal set of (c-f); these can be obtained in what amounts to an inner optimization loop, with c X,HF , c X,DFT , and, where applicable, ω minimized in an outer optimization loop. (We previously found in the revDSD paper that coupling between (a) and (c,d) is too strong to permit placing c C,DFT in the inner loop, as well as that for a fixed value of (a) convergence of DFT correlation parameter up to two decimal places can be achieved within two macro-iterations .) The process is analogous to microiterations vs. macroiterations in CASSCF algorithms (CI coefficients vs. orbitals, see Ref. and references therein), or QM-MM geometry optimizations where geometric parameters in the MM layer are subjected to microiteration for each change of coordinates in the QM layer, and the latter optimized in macroiteration cycles (e.g., Ref. ). The bottleneck during the microiteration steps would normally be the evaluation of all the dispersion corrections for the entire GMTKN55 set for a given combination of parameters. However, this step could be greatly accelerated by parallelization over all CPUs in a 40-core node. A minor I/O access bottleneck was resolved by copying all the required files into a memory file system. In the present case, the optimum value for the range separation parameter ω for a given fixed α is determined manually by interpolation. We repeated this process for six equally spaced α values ranging from 0.57 to 0.72 to construct six different range-separated DSD (i.e., ω DSD) functionals.
III.
Results and Discussion: A. pre-screening of functionals with diet-GMTKN55:
The prescreening experiment was performed using diet100 for DSD n -PBEP86-D3BJ, DSD n -PBEPBE-D3BJ, DSD n -PBEB95-D3BJ, and DSD n -PBEPW91-D3BJ variants, where n stands for the fraction of HF exchange used, i.e. c X,HF or α . cX=0.57 wDSD wDOD cX=0.60 wDSD wDOD Figure 1: Change of WTMAD2(kcal/mol) for DSD X and DOD X -PBEP86-D3BJ with respect to range separation parameter (x-axis) for different cX,HF. [Similar graphs for DSD X -PBEB95-D3BJ, DSD X -PBEPW91-D3BJ, DSD X -PBEPBE-D3BJ and their DOD X versions can be found in Figure S1 in ESI] From Figure 1 and Figure S1 in the ESI it is clear that, for every c
X,HF considered, the DSDn and DODn-PBEP86-D3BJ variants benefited from range separation, whereas for the other exchange-correlation(XC) combinations, only c
X,HF =0.57 and to some extent c
X,HF =0.60 showed some advantage. That being said, we also found, none of the nonlocal XC combination tried above, reached even close to the accuracy of DSD n or DOD n -PBEP86-D3BJ in terms of WTMAD2 (see Table S4 in ESI). So, we decided to proceed further with the range separation experiment only for PBEP86, where we considered simultaneous variation of the short-range HF-exchange(c X ) and the range separation( ω ) parameters — using the full GMTKN55. B. xDSD functionals: In our previous study, for the xrevDSD-PBEP86-D4 functional we did not reoptimize the c X,HF parameter, instead took the earlier reported best value by Martin and coworkers and optimized other linear parameters against GMTKN55. What if we also vary c X,HF (a full evaluation of GMTKN55 is necessary for each cX,HF value considered)? In this current study we look for the optimum c
X,HF for which WTMAD2 is minimum. We call our new functionals, xDSDn-PBEP86-Disp, where ‘ n ’ stands for the percentage of HF exchange used. Following this notation, xDSD -PBEP86-D4 is same as the xrevDSD-PBEP86-D4 functional reported by us in our previous study. xDSDn-PBEP86, we did full GMTKN55 evaluation for eight equally spaced c X,HF points, ranging from 0.57 to 0.78 followed by a parameter optimization, taking both D3BJ and D4 dispersion correction into account. Both with D3BJ and D4, c
X,HF =0.75 offered the lowest WTMAD2 instead of previously reported c X,HF =0.69 for xDSD-PBEP86 — which could be an artifact of optimizing c X,HF against a training set considerably smaller than the GNTKN55. cX=0.63 wDSD wDOD cX=0.66 wDSD wDOD cX=0.69 wDSD wDOD cX=0.72 wDSD wDOD With D3BJ empirical dispersion correction, the calculated WTMAD2 for xDSD -PBEP86-D3BJ is 2.144 kcal/mol — which is essentially identical to ωB97M (2)=2.121 kcal/mol, but with many fewer empirical parameters and (still) no range separation. (The latter is significant, considering that many codes are able to exploit density fitting in global hybrids but not range-separated hybrids.) Increased percentage of HF-exchange in our optimized functional (i.e., going from c X,HF =0.69 for the xrevDSD-PBEP86-D3BJ to c
X,HF =0.75 for the xDSD -PBEP86-D3BJ) mainly benefits small molecule thermochemistry and intermolecular interactions (see Table S5 in ESI). Now, if we apply c =0 constraint (i.e., the xDODn-PBEP86-D3BJ functionals), c X,HF =0.72 offers the lowest WTMAD2 (=2.231 kcal/mol), and small molecule thermochemistry suffers most of the deterioration resulted from applying this constraint.
Next, we repeated the same experiment for five different semilocal XC-combinations, namely, PBEPBE, PBEB95, PBEPW91, SCAN, and BLYP (results are presented in Table 2). Although, as expected, all xDSD n functionals surpass the corresponding revDSD variants, none approached the accuracy of xDSD -PBEP86-D3BJ; the only contender coming close is xDSD -BLYP-D3BJ with 77% HF exchange. For the xDSD n -PBEPBE variants, one finds the “ sweet spot ” at c X,HF =0.72, unlike the previously reported c
X,HF =0.68 in Ref. . Table 1: WTMAD2(kcal/mol) for xDSDn(xDODn)-XC-D3BJ(D4) functionals and final parameters for the D4 variants. Functionals WTMAD2(kcal/mol) Parameters D3BJ a D4 c
X,HF c C,DFT c c s s c ATM a a xDSD PBEP86 PBEB95
BLYP
SCAN
PBEPW91
PBEPBE xDOD PBEP86
PBEB95
BLYP
SCAN
PBEPW91
PBEPBE a All results are with fixed a =5.5, a =0, s =0 For the xDOD functionals (which are particularly important because here we can enable the reduced scaling PT2 algorithms, as well as eliminate one empirical parameter), all except xDOD -SCAN-D3BJ prefers a lesser percentage of exact exchange than the corresponding xDSD variants. The largest penalty for restricting c =0 is paid for xDOD -BLYP-D3BJ — WTMAD2 value drops from 2.254 kcal/mol to 2.564 kcal/mol. Upon further inspection, of all the 55 subsets, W4-11, TAUT15 and BSR36 are the three most affected ones. Similar to our previous observation for DSD-SCAN functionals, xDSD -SCAN-D3BJ also sacrifice almost nothing constraining c to be zero. Instead of using a fixed value 5.5, if we optimize a together with c DFT , c , c and s ; statistics for xDSD variants stay practically unchanged, whereas WTMAD2 for xDOD -PBEB95-D3BJ and xDOD -BLYP-D3BJ reduced by 0.068 and 0.058 kcal/mol respectively. In our prior work, for technical reasons we adopted c ATM =s , where c ATM is the prefactor for the Axilrod-Teller-Muto(ATM) 3-body correction term. If we allow c
ATM as a variable, this somewhat reduced WTMAD2 for xDSDn-PBEPBE-D4, xDSDn-PBEPW91-D4, xDSDn-PBEB95-D4 and their xDOD variants. This leaves us with five adjustable dispersion parameters for the xDSD-D4 functionals. When we optimized all of them along with other parameters using BOBYQA, we noticed s settles on values close to zero, and c ATM on values close to one. Hence if we constrain s =0 and cATM=1, the loss in accuracy is negligible, which permits eliminating two adjustable parameters. (More detailed results are available in Section SI.2 of the ESI.) The only exceptions are SCAN variants, where although s value is not close to zero and c ATM is above two for all cases, restricting c
ATM =1 and s =0 does no harm to the total WTMAD2, e.g, with all parameters varied WTMAD2 for xDSD -SCAN-D4 is 2.351 kcal/mol which increases to 2.379 kcal/mol when we impose those restrictions. Hence, going forward, we decided to freeze s and c ATM throughout and optimize the remaining parameters (see Table 2). With D4 dispersion correction the best performer of xDSD family is again xDSD -PBEP86-D4 (WTMAD2= 2.119 kcal/mol), which now marginally outperforms Mardirossian and Head- Gordon’s ω B97M(2) (2.131 kcal/mol). Among all the 55 subsets, moving from D3BJ to D4 improved performances of BUT14DIOL, HAL59 and MCONF subsets quite a bit for xDSD and xDOD -PBEPBE-D4; BSR36, BUT14DIOL and MCONF got improved for xDSD and xDOD -PBEPW91-D4. Three subsets, BUT14DIOL, PCONF21 and MCONF benefited from relacing D3BJ by D4 dispersion correction for the PBEB95 XC-combination. Lastly, for xDOD -PBEB95-D4, AMINO20X4, BSR36, BUT14DIOL, MCONF and PCONF21 subsets benefited the most. Except for xrevDSD -PBEP86-D4 and xDSD -BLYP-D4, all other functionals prefer a very small fraction of opposite spin MP2-like correction. That is why for those functionals, we sacrifice very little by shifting from xDSD-D4 to xDOD-D4. We should also mention, the accidental similarity of xrevDSD -PBEP86-D4 to Kállay and coworkers’ dRPA75 regarding the preferred percentage of exact exchange. Can the performances of xDSDn-XC-D4 functionals be improved further by replacing the default 3-body Axilrod-Teller-Muto term by the many-body MBD correction of Tkatchenko, and scaling it with a prefactor (now called c MBD rather than c
ATM )? We found that, except for xDSD -PBEB95-D4MBD (WTMAD2 drops from 2.403 to 2.288 kcal/mol), no other xDSDn functionals showed any improvement that can be considered statistically significant for the present dataset. (We have also looked at the influence of considering c ATM as one separate parameter for the Axilrod-Teller-Muto(ATM) term and MBD correction beyond the three body term for our previously reported revDSD functionals, WTMAD2 and updated parameters for those revDSD-D4 functional can be found in Table S3 in ESI.) In a recent study, Semidalas and Martin have reported significant improvement for their composite methods, by switching from frozen core to core-valence correlation and used complete basis set extrapolation from aug-ccpwCVTZ(-PP), and aug-cc-pwCVQZ(-PP) level calculations. Hence, we also checked whether further improvement of WTMAD2 statistics is possible by using sufficiently large basis set and including subvalence correlation in the MP2-like part? Extrapolating from the core-valence aug-ccpwCVTZ(-PP) and aug-cc-pwCVQZ(-PP) energies for xDSD -PBEP86-D3BJ, using the L – formula for opposite spin and L -5 for same spin MP2-like correlation, proposed by Halkier et al., we found no change in WTMAD2 to three decimal places (0.00014 kcal/mol). We have therefore not explored this further for the other double hybrids. C. Range Separation:
We revisit the range separation experiment now using the full GMTKN55. With D3BJ dispersion correction, we found the lowest WTMAD2(2.108 kcal/mol) for ωDSD -PBEP86-D3BJ ( ω=0.13 ), which is very close to what ωDSD -PBEP86-D3BJ( ω=0.16 ) exhibited (2.112 kcal/mol). In general, reduction of c X,HF entails an increase in ω in comparison. Similar to the previous section, here also we checked, how much performance we sacrificed by switching from ωDSD to ωDOD . With WTMAD=2.204 kcal/mol, the ωD OD -PBEP86-D3BJ( ω=0.1
0) appeared to be the best performer in this category. By and large, we gave up about 0.1 kcal/mol accuracy by constraining c =0; small molecule thermochemistry is consistently the category most affected by this restriction. Upon further inspection over all the 55 subsets, we found, BSR36 and TAUT15 are mainly the reason behind this degradation. Both the ωDSD -PBEP86-D3BJ( ω=0.13 ) and ωDOD -PBEP86-D3BJ( ω=0.1
0) only marginally outperforms the corresponding ω =0 variants, xDSD -PBEP86-D3BJ and xDOD -PBEP86-D3BJ respectively. Next, instead of freezing a2 at 5.5 when we optimized it together with other parameters, we found almost no change in WTMAD2 statistics either for ωDSD or ωD OD. Aiming for further improvement, we considered replacing D3BJ term by D4 energy components. Like earlier in this paper, we found that imposing s =0 and c ATM =1 caused only an insignificant increase of WTMAD2, although here optimal c
ATM were somewhat further from unity. The lowest WTMAD2 we can get by shifting from D3BJ to D4 is 2.081 kcal/mol for ωD SD -PBEP86-D4 — at the cost of eight adjustable parameters. Table 2: WTMAD2 (kcal/mol) and final recommended parameters for the ωDSD n( ωDOD n)-PBEP86-D4 functionals Functional WTMA2(kacl/mol) Parameters D3BJ D4 ω c X,HF c C,DFT c c s s c ATM a a ω DSD-PBEP86 ω DOD-PBEP86
Now, we shift our focus from ωD SD-D4 to ωD OD-D4 functionals. The ωD OD -PBEP86-D4( ω =0.10) is the best performer here with WTMAD2=2.175 kcal/mol. At the expense of one less parameter (seven instead of eight) we sacrificed only 0.09 kcal/mol accuracy and small molecule thermochemistry is the reason behind this loss (see Table S5). Similar to xDSD cases, here also, including core valence correlation for the MP2-like term or considering MBD term beyond three-body ATM correction did not help either. We also considered eliminating the empirical dispersion correction altogether. Similar to the global DH cases, this approach significantly degrades accuracy here too. The general trend shows, improvement of performance with the increase of HF exchange and requirement of higher ω value with respect to their ωDSD and ωDOD counterparts. D. “External” benchmarks : Our new functionals are tested further against two different test suits, which are not the subsets of GMTKN55. These two data sets are: MOBH35 originally proposed by Iron and Janes, and POLYPYR21. (a)MOBH35: This database comprises 35 reactions, including both early and late and 3d, 4d, and 5d transition metals. We extracted the best reported reference energies from the erratum to the original MOBH35 paper. The def2-QZVPP basis set was used with grids and auxiliary basis sets as described above in the Methods section.
Figure 2: MAD (kcal/mol) for our new xDSD and ω DSD functionals tested against modified MOBH35
Using a variety of multireference diagnostics, our group has recently found [E. Semidalas, M. A. Iron, and J. M. L. Martin, to be published] that reaction 9 exhibits severe static correlation in all three structures, that gets progressively worse from reactant via transition state to product as the HOMO-LUMO gap narrows. Under those circumstances, as previously found for polypyrroles, a large gap opens between canonical CCSD(T) and DLPNO-CCSD(T) — yet for the product, diagnostics are so large that one can legitimately question whether CCSD(T) itself is adequate for the problem. Hence, omitting that reaction from the MOBH35 dataset, we have recalculated MAD values for the remaining 34 reactions (see Figure 2). In general, xDOD and ω DOD functionals perform better than their DSD counterparts, irrespective of the choice of dispersion correction. Shifting from D3BJ to D4, benefits across the board by 0.2-0.3 kcal/mol. ω DOD -PBEP86-D4( ω =18) achieves the lowest MAD=1.0 kcal/mol, closely followed by the other ω DOD variants. Among the DOD variants, ω =0 corresponds to the special case of ω DOD functionals. Here, xDOD72-PBEP86-D4 has the lowest MAD=1.1 kcal/mol, again close to what was found for the ω DOD -PBEP86-D4( ω =18). That being said, the other empirical range separated double hybrids ωB2PLYP , ωB2 GP-PLYP and ωB ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) x D S D - P B E P - D B J x D S D - S C A N - D B J x D S D - P B E B - D B J x D S D - P B E P W - D B J x D S D - B L Y P - D B J x D S D - P B E P B E - D B J x D O D - P B E P - D B J x D O D - S C A N - D B J x D O D - P B E B - D B J x D O D - P B E P W - D B J x D O D - B L Y P - D B J x D O D - P B E P B E - D B J ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) x D S D - P B E P - D x D S D - P B E B - D x D S D - P B E P W - D x D S D - B L Y P - D x D S D - S C A N - D x D S D - P B E P B E - D x D O D - P B E P - D x D O D - P B E B - D x D O D - P B E P W - D x D O D - B L Y P - D x D O D - S C A N - D x D O D - P B E P B E - D ω B G P - P L Y P ω B P L Y P ω B M ( ) ω B M - V ω B X - V MAD (kcal/mol) : MOBH35 without reaction 9 that these reaction were omitted from Dohm et al.’s recent revision of MOBH35.] So, if we also drop reactions 17-20 together with reaction 9 and recalculate MADs for the remaining 30 reactions, MAD drops across the board. Also, while MAD for ω B2PLYP and ω B2GP-PLYP remains elevated, that for ω B97M(2) now is in the same cohort as our best functionals (see Figure 3).
Figure 3: MAD (kcal/mol) statistics for xDSD and wDSD functionals evaluated against modified MOBH35 (b) POLYPYR21:
This database contains 21 unique structures of penta-, hexa- and heptaphyrins, which are [4n] π -electron expanded porphyrins that have generated considerable interest recently because of their potential application as molecular switches (see the introduction to Ref. for a brief review). The structures are H ü ckel, M ö bius, and figure-eight minima as well as the various transition states between them. Among them, the most troublesome are the M ö bius rings, which exhibits pronounced multireference character (for more details see Refs. , ). CCSD(T)/CBS level reference energies were extracted from Ref . We have used the def2-TZVP basis for all calculations here. With D3BJ dispersion correction, it appears that xDOD functionals perform noticeably better than their xDSD counterparts. In Ref. for the problem at hand, as well as in Iron and Janes for MOBH35, this was observed for DOD vs DSD functionals, and ascribed to the greater resilience of spin-opposite-scaled GLPT2 to static correlation. However, if here we replace D3BJ by D4, most of that difference goes away. xDOD -BLYP-D3BJ still offers the lowest RMSD (1.64 kcal/mol). ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) ω D S D - P B E P - D B J ( ω = . ) ω D O D - P B E P - D B J ( ω = . ) x D S D - P B E P - D B J x D S D - S C A N - D B J x D S D - P B E B - D B J x D S D - P B E P W - D B J x D S D - B L Y P - D B J x D S D - P B E P B E - D B J x D O D - P B E P - D B J x D O D - S C A N - D B J x D O D - P B E B - D B J x D O D - P B E P W - D B J x D O D - B L Y P - D B J x D O D - P B E P B E - D B J ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) ω D S D - P B E P - D ( ω = . ) ω D O D - P B E P - D ( ω = . ) x D S D - P B E P - D x D S D - P B E B - D x D S D - P B E P W - D x D S D - B L Y P - D x D S D - S C A N - D x D S D - P B E P B E - D x D O D - P B E P - D x D O D - P B E B - D x D O D - P B E P W - D x D O D - B L Y P - D x D O D - S C A N - D x D O D - P B E P B E - D ω B G P - P L Y P ω B P L Y P ω B M ( ) ω B M - V ω B X - V MAD (kcal/mol): MOBH35 without reaction 9, 17-20 Table 3: MAD (kcal/mol) and RMSD (kcal/mol) error statistics for the new xDSD(DOD) and ωDSD(DOD) functionals evaluated against POLYPYR21.
Functionals MAD (kcal/mol) RMSD (kcal/mol) Total Möbius Structures H ü ckel & figure-eight structures xDSD -PBEPBE-D3BJ 2.31 3.33 5.36 0.90 xDOD -PBEPBE-D3BJ 1.69 2.37 3.68 0.76 xDSD -PBEB95-D3BJ 2.50 3.59 5.73 0.95 xDOD -PBEB95-D3BJ 1.30 1.75 2.34 0.81 xDSD -PBEPW91-D3BJ 2.38 3.44 5.53 0.91 xDOD -PBEPW91-D3BJ 1.63 2.27 3.47 0.75 xDSD -SCAN-D3BJ 1.96 2.73 4.29 0.86 xDOD -SCAN-D3BJ 1.63 2.24 3.42 0.80 xDSD -BLYP-D3BJ 3.08 4.51 7.28 1.14 xDOD -BLYP-D3BJ 1.19 1.64 2.13 0.75 xDSD -PBEP86-D3BJ 2.46 3.58 5.74 0.90 xDOD -PBEP86-D3BJ 1.42 1.96 2.88 0.70 xDSD -PBEPBE-D4 1.82 2.49 3.27 1.02 xDOD -PBEPBE-D4 1.72 2.39 3.55 0.79 xDSD -PBEB95-D4 1.58 2.18 2.60 1.01 xDOD -PBEB95-D4 1.54 2.13 2.54 1.05 xDSD -PBEPW91-D4 1.74 2.48 3.58 0.79 xDOD -PBEPW91-D4 1.68 2.35 3.43 0.80 xDSD -SCAN-D4 1.76 2.42 3.25 0.97 xDOD -SCAN-D4 1.63 2.28 3.27 0.82 xDSD -BLYP-D4 1.56 2.20 2.49 1.01 xDOD -BLYP-D4 1.31 1.87 2.23 0.86 xDSD -PBEP86-D4 2.08 3.02 4.52 0.80 xDOD -PBEP86-D4 1.42 1.96 2.88 0.70 ω DSD -PBEP86-D3BJ( ω =0.13) 1.61 2.34 3.77 0.72 ω DOD -PBEP86-D3BJ( ω =0.08) 1.28 1.85 3.09 0.63 ω DSD -PBEP86-D3BJ( ω =0.16) 0.92 1.34 2.12 0.54 ω DOD -PBEP86-D3BJ( ω =0.10) 0.70 1.00 1.56 0.49 ω DSD -PBEP86-D3BJ( ω =0.18) 0.53 0.72 1.09 0.44 ω DOD -PBEP86-D3BJ( ω =0.15) 0.38 0.49 0.69 0.38 ω DSD -PBEP86-D3BJ( ω =0.20) 0.97 1.33 1.86 0.60 ω DOD -PBEP86-D3BJ( ω =0.16) 0.78 1.06 1.41 0.55 ω DSD -PBEP86-D3BJ( ω =0.22) 0.38 0.49 0.45 0.44 ω DOD -PBEP86-D3BJ( ω =0.18) ω DSD -PBEP86-D3BJ( ω =0.22) ω DOD -PBEP86-D3BJ( ω =0.20) 0.55 0.75 1.26 0.36 ω DSD -PBEP86-D4( ω =0.13) 1.49 2.16 3.32 0.59 ω DOD -PBEP86-D4( ω =0.08) 1.08 1.58 2.36 0.50 ω DSD -PBEP86-D4( ω =0.16) 0.84 1.25 1.80 0.46 ω DOD -PBEP86-D4( ω =0.10) 0.56 0.84 1.08 0.43 ω DSD -PBEP86-D4( ω =0.18) 0.43 0.59 0.67 0.40 ω DOD -PBEP86-D4( ω =0.15) 0.42 0.55 0.64 0.42 ω DSD -PBEP86-D4( ω =0.20) 0.99 1.39 1.70 0.64 ω DOD -PBEP86-D4( ω =0.16) 0.82 1.16 1.34 0.59 ω DSD -PBEP86-D4( ω =0.22) 0.45 0.65 0.41 0.50 ω DOD -PBEP86-D4( ω =0.18) 0.43 0.59 0.47 0.48 ω DSD -PBEP86-D4( ω =0.22) 0.55 0.55 0.99 0.48 ω DOD -PBEP86-D4( ω =0.20) 0.63 0.83 1.32 0.48 ω B97M(2) ω B2PLYP ω B2GP-PLYP The story largely repeats itself for the range separated ω DOD vs. ω DSD functionals. In general, switching from D3BJ to D4 dispersion deteriorates performance for the ω DOD variants. However, for the ω DSD functionals D4 is a better choice than D3BJ. Among all the xDSD and ω DSD functionals, ωDOD -PBEP86- D3BJ(ω=0.16) has the lowest RMSD=0.45 kcal/mol, which in fact, slightly outperforms the previously reported top performer ωB97M(2) (0.63 kcal/mol). However, in light of remaining uncertainties in the reference values, that difference should not be considered significant: inspection of the Möbius structure data in isolation does reveal, across the board, that range-separated DHs cope with them much more than global double hybrids (see Table 3). Once again, using MBD instead of ATM does not have any significant effect. IV.
Conclusions:
Aiming to improve our previous revDSD family functionals further, we have considered both range separation, ω DSD, and xDSD — the latter, while analogues of the XYG3 family of functionals, are also recovered as the ω =0 limit of range separated double hybrids. Concerning our first research objective, from an extensive survey we can conclude the following: a) xDSD-D3BJ functionals have a slight advantage over our prior revDSD family functionals, which can be further improved upon by replacing D3BJ with D4. b) For D4, allowing c
ATM , the prefactor for the three-body Axilrod-Teller-Muto term, to take on values different from 1 does not reduce WTMAD2 by a statistically significant amount that would justify the introduction of the extra adjustable parameter. Replacing ATM by the many-body dispersion model of Tkatchenko and coworkers achieves no significant benefit, although the systems in GMTKN55 may not be large enough to rule this out. c)
For the xDSD n -PBEP86-D4 variants, c X,HF =0.75 offers the lowest WTMAD2, unlike the previously reported c
X,HF =0.69 for DSD-PBEP86-D4. However, when we imposed the c =0 constraint, WTMAD2 reaches a minimum at c
X,HF =0.72. d)
In terms of WTMAD2, xDSD -PBEP86-D4 marginally outperforms ω B97M(2), hitherto the ‘record holder’ for lowest WTMAD2, but without its range separation and using less than half the number of empirical parameters. Concerning the second research objective: applying range separation over the HF-exchange part, we found lowest WTMAD2 for c X,HF =0.72 and ω =0.13. With D3BJ, WTMAD2 is 2.108 kcal/mol, which can be lowered slightly further by substituting D4 (2.083 kcal/mol). So, range separation helped to improve performance slightly beyond xDSD -PBEP86-D3BJ(D4), and in turn a little further beyond ω B97M(2) – again, using only half the number of adjustable parameters present in ω B97M(2). Although ω B97M(2) outperforms all the new ω DSD and ω DSD functionals for small-molecule thermochemistry, this is outweighed in WTMAD2 by superior performance of the new functionals for conformer equilibria. Finally, external test sets of metal-organic barrier heights (MOBH35) and of isomer equilibria and interconversion barriers in polypyrroles (POLYPYR21) put the functionals’ performance in the presence of static correlation to the test. The largest improvement from range separation is seen here for POLYPYR21, and specifically for the Möbius structures (which have the largest amount of static correlation). Acknowledgments:
We would like to acknowledge helpful discussions with Mr. Emmanouil Semidalas (WIS), Dr. Mark A. Iron (WIS), and Prof. Mercedes Alonso Giner (Free University of Brussels).
We also thank Mr. Nitai Sylvetsky for critical comments on the draft manuscript. GS acknowledges a doctoral fellowship from the Feinberg Graduate School (WIS). MC was a Karyn Kupcynet-Getz International Summer School Fellow at WIS in 2019.
Funding Sources:
This research was funded by the Israel Science Foundation (grant 1969/20) and by the Minerva Foundation (grant 20/05) . Supporting Information:
The Supporting Information (in PDF format) is available free of charge at https://doi.org/10.1021/xxxxxxx. Abridged details of all 55 subsets of GMTKN55 with proper references. Improvement of
WTMAD2 for all five rungs of Jacob’s ladder by considering def2 -QZVPPD for BH76 and BH76RC subset. The benefit of using c
ATM and c
MBD for the revDSD functionals. Prescreening for wDSDn-PBEB95-D3BJ, wDSDn-PBEPW91-D3BJ, wDSDn-PBEPBE-D3BJ and wDODn variants using diet100. Breakdown of total WTMAD2 into top five subsets for all the xDSD and range separated DSD functionals. QCHEM sample inputs for ω DSD-D3BJ and wDOD-D3BJ functionals.
References: (1) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects.
Phys. Rev. , (4A), A1133 – A1138. https://doi.org/10.1103/PhysRev.140.A1133. (2) Semidalas, E.; Martin, J. M. L. Canonical and DLPNO-Based G4(MP2)XK-Inspired Composite Wave Function Methods Parametrized against Large and Chemically Diverse Training Sets: Are They More Accurate and/or Robust than Double-Hybrid DFT?
J. Chem. Theory Comput. , (7), 4238 – – Valence Correlation, and F12 Alternatives. J. Chem. Theory Comput. , (12), 7507 – ’s Ladder of Density Functional Approximations for the Exchange-Correlation Energy.
AIP Conf. Proc. , (1), 1 –
20. https://doi.org/10.1063/1.1390175. (5) Zhang, I. Y.; Xu, X. Doubly Hybrid Density Functional for Accurate Description of Thermochemistry, Thermochemical Kinetics and Nonbonded Interactions.
Int. Rev. Phys. Chem. , (1), 115 – Wiley Interdiscip. Rev. Comput. Mol. Sci. , (6), 576 – Chemical Modelling: Vol. 13 ; Springborg, M., Joswig, J.-O., Eds.; Royal Society of Chemistry: Cambridge, UK, 2017; pp 191 – Martin, J. M. L.; Santra, G. Empirical Double‐Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT.
Isr. J. Chem. , (8 – – J. Phys. Chem. A , (24), 5129 – Phys. Chem. Chem. Phys. , (48), 32184 – Theor. Chem. Acc. , (1 – – Theor. Chem. Acc. , (1 – – J. Chem. Phys. , (3), 034108. https://doi.org/10.1063/1.2148954. (14) Zhang, Y.; Xu, X.; Goddard, W. A. Doubly Hybrid Density Functional for Accurate Descriptions of Nonbond Interactions, Thermochemistry, and Thermochemical Kinetics. Proc. Natl. Acad. Sci. , (13), 4963 – J. Chem. Phys. , (17), 174103. https://doi.org/10.1063/1.3703893. (16) Görling, A.; Levy, M. Exact Kohn-Sham Scheme Based on Perturbation Theory. Phys. Rev. A , (1), 196 – Phys. Chem. Chem. Phys. , , 6670 – J. Phys. Chem. C , (48), 20801 – (19) Kozuch, S.; Martin, J. M. L. DSD-PBEP86: In Search of the Best Double-Hybrid DFT with Spin-Component Scaled MP2 and Dispersion Corrections. Phys. Chem. Chem. Phys. , (45), 20104. https://doi.org/10.1039/c1cp22592h. (20) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. , (15), 154104. https://doi.org/10.1063/1.3382344. (21) Grimme, S.; Ehrlich, S.; Goerigk, L. Effect of the Damping Function in Dispersion Corrected Density Functional Theory. J. Comput. Chem. , (7), 1456 – J. Chem. Phys. , (16), 161708. https://doi.org/10.1063/1.4991798. (23) Caldeweyher, E.; Ehlert, S.; Hansen, A.; Neugebauer, H.; Spicher, S.; Bannwarth, C.; Grimme, S. A Generally Applicable Atomic-Charge Dependent London Dispersion Correction. J. Chem. Phys. , (15), 154122. https://doi.org/10.1063/1.5090222. (24) Almlöf, J. Elimination of Energy Denominators in Møller-Plesset Perturbation Theory by a Laplace Transform Approach. Chem. Phys. Lett. , (4), 319 – – Plesset Perturbation Theory.
J. Chem. Phys. , (1), 489 – Theor. Chim. Acta , (1 – – Theor. Chim. Acta , (1 – –
62. https://doi.org/10.1007/BF02329241. (28) Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid Exchange – Correlation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP).
Chem. Phys. Lett. , (1 – –
57. https://doi.org/10.1016/j.cplett.2004.06.011. (29) Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. A Long-Range-Corrected Time-Dependent Density Functional Theory.
J. Chem. Phys. , (18), 8425 – Phys. Rev. B , (8), 081204. https://doi.org/10.1103/PhysRevB.88.081204. (31) Refaely-Abramson, S.; Jain, M.; Sharifzadeh, S.; Neaton, J. B.; Kronik, L. Solid-State Optical Absorption from Optimally Tuned Time-Dependent Range-Separated Hybrid Density Functional Theory. Phys. Rev. B - Condens. Matter Mater. Phys. , (8), 1 –
6. https://doi.org/10.1103/PhysRevB.92.081204. (32) Rohrdanz, M. A.; Martins, K. M.; Herbert, J. M. A Long-Range-Corrected Density Functional That Performs Well for Both Ground-State Properties and Time-Dependent Density Functional Theory Excitation Energies, Including Charge-Transfer Excited States.
J. Chem. Phys. , (5), 054112. https://doi.org/10.1063/1.3073302. (33) Henderson, T. M.; Izmaylov, A. F.; Scalmani, G.; Scuseria, G. E. Can Short-Range Hybrids Describe Long-Range-Dependent Properties? J. Chem. Phys. , (4), J. Chem. Phys. , (8), 3540 – J. Phys. Chem. Lett. , (21), 2810 – J. Chem. Phys. , (8), 84106. https://doi.org/10.1063/1.2834918. (37) Manna, A. K.; Refaely-Abramson, S.; Reilly, A. M.; Tkatchenko, A.; Neaton, J. B.; Kronik, L. Quantitative Prediction of Optical Absorption in Molecular Solids from an Optimally Tuned Screened Range-Separated Hybrid Functional. J. Chem. Theory Comput. , (6), 2919 – Annu. Rev. Phys. Chem. , (1), 85 – J. Chem. Phys. , (7), 074106. https://doi.org/10.1063/1.2244560. (40) Mardirossian, N.; Head- Gordon, M. ΩB97X -V: A 10-Parameter, Range-Separated Hybrid, Generalized Gradient Approximation Density Functional with Nonlocal Correlation, Designed by a Survival-of-the-Fittest Strategy.
Phys. Chem. Chem. Phys. , (21), 9904 – Gordon, M. ΩB97M -V: A Combinatorially Optimized, Range-Separated Hybrid, Meta-GGA Density Functional with VV10 Nonlocal Correlation.
J. Chem. Phys. , (21), 214110. https://doi.org/10.1063/1.4952647. (42) Ángyán, J. G.; Gerber, I. C.; Savin, A.; Toulouse, J. Van Der Waals Forces in Density Functional Theory: Perturbational Long-Range Electron-Interaction Corrections. Phys. Rev. A , (1), 012510. https://doi.org/10.1103/PhysRevA.72.012510. (43) Chai, J.-D.; Head-Gordon, M. Long-Range Corrected Double-Hybrid Density Functionals. J. Chem. Phys. , (17), 174105. https://doi.org/10.1063/1.3244209. (44) Mardirossian, N.; Head-Gordon, M. Survival of the Most Transferable at the Top of Jacob’s Ladder: Defining and Testing the ΩB97M(2) Double Hybrid Density Functional.
J. Chem. Phys. , (24), 241736. https://doi.org/10.1063/1.5025226. (45) Brémond, É.; Savarese, M.; Pérez-Jiménez, Á. J.; Sancho-García, J. C.; Adamo, C. Range-Separated Double-Hybrid Functional from Nonempirical Constraints. J. Chem. Theory Comput. , (8), 4052 – J. Chem. Phys. , (3), 031101. https://doi.org/10.1063/1.4890314. (47) Toulouse, J.; Colonna, F.; Savin, A. Long-Range – Short-Range Separation of the Electron-Electron Interaction in Density-Functional Theory.
Phys. Rev. A , (6), 062505. https://doi.org/10.1103/PhysRevA.70.062505. (48) Brémond, É.; Pérez-Jiménez, Á. J.; Sancho-García, J. C.; Adamo, C. Range-Separated Hybrid Density Functionals Made Simple.
J. Chem. Phys. , (20), 201102. https://doi.org/10.1063/1.5097164. (49) Casanova- Páez, M.; Dardis, M. B.; Goerigk, L. ΩB2PLYP and ΩB2GPPLYP: The First Two
Double-Hybrid Density Functionals with Long-Range Correction Optimized for Excitation Energies.
J. Chem. Theory Comput. , (9), 4735 – Gould, T. ‘Diet GMTKN55’ Offers Accelerated Benchmarking through a Representative
Subset Approach.
Phys. Chem. Chem. Phys. , (44), 27735 – Kinetics, and Noncovalent Interactions − Assessment of Common and
Reparameterized ( Meta -)GGA Density Functionals.
J. Chem. Theory Comput. , (1), 107 – Robust Statistics ; Wiley Series in Probability and Statistics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2009. https://doi.org/10.1002/9780470434697. (53) Geary, R. C. The Ratio of the Mean Deviation to the Standard Deviation as a Test of Normality.
Biometrika , (3 – – Mol. Phys. , (2), 184 – Wiley Interdiscip. Rev. Comput. Mol. Sci. , (1), 4 –
9. https://doi.org/10.1002/wcms.1327. (56) Weigend, F.; Ahlrichs, R. Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy.
Phys. Chem. Chem. Phys. , (18), 3297 – J. Chem. Phys. , (13), 134105. https://doi.org/10.1063/1.3484283. (58) Dasgupta, S.; Herbert, J. M. Standard Grids for High-Precision Integration of Modern Density Functionals: SG-2 and SG-3. J. Comput. Chem. , (12), 869 – Phys. Rev. Lett. , (3), 036402. https://doi.org/10.1103/PhysRevLett.115.036402. (60) Brandenburg, J. G.; Bates, J. E.; Sun, J.; Perdew, J. P. Benchmark Tests of a Strongly Constrained Semilocal Functional with a Long-Range Dispersion Correction. Phys. Rev. B - Condens. Matter Mater. Phys. , (11), 17 –
19. https://doi.org/10.1103/PhysRevB.94.115144. (61) Hättig, C. Optimization of Auxiliary Basis Sets for RI-MP2 and RI-CC2 Calculations: Core-Valence and Quintuple-Zeta Basis Sets for H to Ar and QZVPP Basis Sets for Li to Kr.
Phys. Chem. Chem. Phys. , (1), 59 –
66. https://doi.org/10.1039/b415208e. (62) Hellweg, A.; Rappoport, D. Development of New Auxiliary Basis Functions of the Karlsruhe Segmented Contracted Basis Sets Including Diffuse Basis Functions (Def2- SVPD, Def2-TZVPPD, and Def2-QVPPD) for RI-MP2 and RI-CC Calculations.
Phys. Chem. Chem. Phys. , (2), 1010 – Non-Covalent Interactions in Quantum Chemistry and Physics ; Elsevier, 2017; pp 195 – J. Comput. Chem. , (27), 2327 – J. Chem. Phys. , (19), 194106. https://doi.org/10.1063/1.5094644. (67) Melaccio, F.; Olivucci, M.; Lindh, R.; Ferré, N. Unique QM/MM Potential Energy Surface Exploration Using Microiterations. Int. J. Quantum Chem. , (13), 3339 – Kesharwani, M. K.; Kozuch, S.; Martin, J. M. L. Comment on “Doubly
Hybrid Density Functional XDH-PBE0 from a Parameter-
Free Global Hybrid Model PBE0” [J. Chem.
Phys. 136, 174103 (2012)].
J. Chem. Phys. , (18), 187101. https://doi.org/10.1063/1.4934819. (69) Mezei, P. D.; Csonka, G. I.; Ruzsinszky, A.; Kállay, M. Construction and Application of a New Dual-Hybrid Random Phase Approximation. J. Chem. Theory Comput. , (10), 4615 – Phys. Rev. Lett. , (23), 1 –
5. https://doi.org/10.1103/PhysRevLett.108.236402. (71) Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Basis-Set Convergence in Correlated Calculations on Ne, N2, and H2O.
Chem. Phys. Lett. , (3 – – Iron, M. A.; Janes, T. Correction to “Evaluating Transition Metal Barrier Heights with the Latest Density Functional Theory Exchange – Correlation Functionals: The MOBH35
Benchmark Database.”
J. Phys. Chem. A , (29), 6379 – –Möbius Interconversions in Extended π -Systems. J. Phys. Chem. A , (12), 2380 – – Möbius Interconversions in Expanded Porphyrins.
J. Chem. Theory Comput. , (6), 3641 – J. Chem. Theory Comput. , (3), 2002 – Table of contents: x D S D - P B E P - D D S D - P B E P - D D O D - P B E P - D ω D S D - P B E P - D ω D S D - P B E P - D ω D O D - P B E P - D ω B M ( ) WTMAD2 (kcal/mol)
ThermochemistryBarrier heightsReaction energies of largespeciesConformers/IntramolecularIntermolecular interactions Electronic Supporting Information (ESI)
Exploring Avenues Beyond Revised DSD Functionals: I. range separation, with xDSD as a special case
Golokesh Santra, † Minsik Cho, †,§ and Jan M.L. Martin *,† † Department of Organic Chemistry, Weizmann Institute of Science, 7610001 Reḥovot, Israel. Email: [email protected] § Department of Chemistry,
Brown University, Providence, Rhode Island 02912, USA 2
Contents:
SI.1
Details of all 55 subset of GMTKN55 and corresponding references……. 3
SI.2
Details of BH76/def2-QZVPPD patch……………………...………………………… 9
SI.3
Is Inclusion of many body dispersion correction beyond three body ATM term beneficial over revDSD functionals? ………………………………………………..…. 4
SI.4
Diet-GMTKN55 prescreening for w DSD n -PBEB95-D3BJ, w DSD n -PBEPW91-D3BJ, w DSD n -PBEPBE-D3BJ and w DOD n version of those……………………………… 6 SI.5
Division of WTMAD2 into five major subcategories…………………………… XX
SI.6
QCHEM Sample Inputs……………………………………………………………………….. 10
SI.7
References…………………………………………………………………………………..…… 12 SI.1 Details of all 55 subsets of GMTKN55:
Below are the abbreviations used and concise description of all fifty-five subsets of GMTKN55. Table S1: Abbreviations used and their descriptions
Abbreviation
Description
ACONF Relative energies of alkane conformers
ADIM6 Interaction energies of n-alkane dimers
AHB21 Interaction energies in anion–neutral dimers
AL2X6 Dimerisation energies of AlX3 compounds
ALK8 Dissociation and other reactions of alkaline compounds
ALKBDE10 Dissociation energies in group-1 and -2 diatomics
AMINO20X4 Relative energies in amino acid conformers
BH76RC
30 reaction energies of the BH76 set
BH76
Barrier heights of hydrogen transfer, heavy atom transfer, nucleophilic substitution, unimolecular and association reactions
BHDIV10 Diverse reaction barrier heights
BHPERI , , Barrier heights of pericyclic reactions
BHROT27 Barrier heights for rotation around single bonds
BSR36 , Bond-separation reactions of saturated hydrocarbons
BUT14DIOL Relative energies in butane-1,4-diol conformers
C60ISO Relative energies between C isomers CARBHB12 Hydrogen-bonded complexes between carbene analogues and H2O, NH3, or HCl
CDIE20 Double-bond isomerisation energies in cyclic systems
CHB6 Interaction energies in cation–neutral dimers
DARC , Reaction energies of Diels–Alder reactions
DC13 , , ,
13 difficult cases for DFT methods
DIPCS10 Double-ionisation potentials of closed-shell systems
FH51
Reaction energies in various (in-)organic systems
G21EA
Adiabatic electron affinities
G21IP
Adiabatic ionisation potentials
G2RC Reaction energies of selected G2/97 systems
HAL59
Binding energies in halogenated dimers (incl. halogen bonds)
HEAVY28 Noncovalent interaction energies between heavy element hydrides
HEAVYSB11 Dissociation energies in heavy-element compounds
ICONF Relative energies in conformers of inorganic systems
IDISP
Intramolecular dispersion interactions
IL16 Interaction energies in anion–cation dimers
INV24 Inversion/racemisation barrier heights
ISO34 Isomerisation energies of small and medium-sized organic molecules
ISOL24 Isomerisation energies of large organic molecules
MB16-43 Decomposition energies of artificial molecules
MCONF Relative energies in melatonin conformers
NBPRC
Oligomerisations and H2 fragmentations of NH3/BH3 systems; H2 activation reactions with PH3/BH3 systems
PA26 Adiabatic proton affinities (incl. of amino acids)
PArel Relative energies in protonated isomers
PCONF21
Relative energies in tri- and tetrapeptide conformers PNICO23 Interaction energies in pnicogen-containing dimers
PX13 Proton-exchange barriers in H2O, NH3, and HF clusters
RC21 Fragmentations and rearrangements in radical cations
RG18 Interaction energies in rare-gas complexes
RSE43 Radical-stabilisation energies
S22 Binding energies of noncovalently bound dimers
S66 Binding energies of noncovalently bound dimers
SCONF
Relative energies of sugar conformers
SIE4X4 Self-interaction-error related problems
TAUT15 Relative energies in tautomers
UPU23 Relative energies between RNA-backbone conformers
W4-11 Total atomisation energies
WATER27 Binding energies in (H2O)n, H+(H2O)n and OH−(H2O)n
WCPT18 Proton-transfer barriers in uncatalysed and water-catalysed reactions
YBDE18 Bond-dissociation energies in ylides
SI.2 Details of the BH76/def2-QZVPPD patch:
The BH76 set is a combination of the HTBH38 and NHTBH38 databases by Truhlar and co-workers. Among the 19 reactions of NHTBH38 there are 4 bimolecular nucleophilic substitution (S N
2) reactions and 4 unimolecular S N should be employed instead of def2-QZVPP (we used in ref ), although the latter one is suitable enough for the remaining systems of BH76. As all the systems BH76 is comprised of is small enough, we can use the larger basis def2-QZVPPD throughout. Now, for revDSD functionals, upon further optimization of micro-iteration parameters using BOBYQA, we noticed little to no change of the coefficients, compared to what we reported earlier. However, total WTMAD2 decreases at the range of 0.4-0.6 kcal/mol across the board for double hybrids as well as other methods tested from all the five rungs of Jacob’s ladder (see Table S2).
Table S2: WTMAD2 for GMTKN55 with and without def2-QZVPPD patch for BH76 for few of the representative functionals of
Jacob’s Ladder
Functionals WTMAD2(kcal/mol) with BH76/def2-QZVPP † with BH76/def2-QZVPPD ωB97M(2) 2.19 2.13 revDSD-PBEP86-D4 B2NC-PLYP-D3BJ 2.96 2.91 noDispSD-SCAN † These values are taken from our review paper on empirical double hybrids. SI.3. Is inclusion of many body dispersion correction beyond three body ATM term beneficial for revDSD functionals? and later popularized by Tkatchenko et al. , to calculate the many-body term of dispersion(MBD) correction—which is basically based on coupling of atomic dipole polarizabilities taking quantum harmonic oscillator(QHO) approximation into consideration. The beauty of this simple yet robust model is that the coupled QHOs can be used as an approximation of the density-density response function and this particular choice reduces the degrees of freedom considerably and hence the computational cost. The energy expression for E disp part can be written as follows, 𝐸 !" = 𝐸 !" + (𝐸 !" − 𝐸 !" ) where the first term represents the two-body terms only (dipole-dipole and dipole-quadrupole terms) and the second term represents all the remaining dipole-dipole interactions up to infinite order. In our prior study, for revDSD-D4 functionals, we used c ATM =s due to technical reason, where c ATM is the parameter for the three body Axilrod-Teller-Muto(ATM) correction term. Here 6 we show, how scaling only three body term separately improve performances further for revDSD functionals (see Table S3). For DSD and DOD-SCAN-D4 this improvement is most prominent, whereas for other cases it is not so outspoken. However, the optimum s for all other functionals seems be settled near zero and c ATM near one. So, if we impose s =0 and c ATM =1 constraints into the micro iteration cycle, we practically loose no accuracy (0.02 kcal/mol or even less). Table S3 has the optimized parameters for s =0, and c ATM =1. The next question we ask is, whether inclusion of the MBD energy term beyond ATM is still helpful? The WTMAD2 results in Table S3 suggests that, considering MBD term does more harm than good for revDSD-BLYP-D4, whereas for DSD and DOD-PBEB95-D4 it is helps quiet a bit. For other cases, the improvement is just marginal.
Table S3: WTMAD2 values and optimized parameters for different functionals with D4 dispersion correction. Functionals WTMAD2(kcal/mol) Final Parameters revD4 old ‡ revD4 revD4 s =0 revD4 s =0 c ATM =1 c
X,HF c C,DFT c c s s c ATM (c MBD ) a a with ATM DSD-SCAN 2.577 2.458 2.474 2.477 0.66 0.4715 0.6406 0.0155 0.4012 [0] [1.0] 0.2548 4.5058 DSD-BLYP 2.535 2.428 2.428 2.428 0.71 0.5231 0.5578 0.1891 0.6808 [0] [1.0] 0.1565 4.8781 DSD-PBEPW91 — 2.312 2.311 2.311 0.67 0.4622 0.5861 0.0357 0.7219 [0] [1.0] 0.3679 3.9422 DSD-PBEB95 2.632 2.559 2.568 2.568 0.66 0.4686 0.5197 0.0649 0.5298 [0] [1.0] 0.3345 3.7291 DSD-PBEPBE 2.435 2.393 2.396 2.393 0.68 0.4376 0.6108 0.0240 0.7296 [0] [1.0] 0.5457 2.9900 DSD-PBEP86 2.274 2.246 2.246 2.2474 0.69 0.4224 0.5935 0.0566 0.5917 [0] [1.0] 0.3710 4.2014 with MBD DSD-SCAN 2.423 0.66 0.4707 0.6462 0.0149 0.1908 0.8382 2.7886 0.5739 3.8328 DSD-BLYP 2.507 0.71 0.5370 0.5457 0.2032 0.4978 0.2221 0.6780 0.2847 4.4000 DSD-PBEPW91 2.301 0.67 0.4706 0.5881 0.0078 0.5912 0.4418 1.7278 0.5476 3.2814 DSD-PBEB95 2.442 0.66 0.4737 0.5346 0.0343 0.3370 0.5567 2.3138 0.6015 2.7941 DSD-PBEPBE 2.368 0.68 0.4463 0.6122 0.0033 0.4585 0.8390 2.1183 0.7397 2.4901 DSD-PBEP86 2.276 0.69 0.4252 0.5926 0.0551 0.5343 0.0401 0.7362 0.3604 4.2306 with ATM DOD-SCAN 2.578 2.460 2.468 2.479 0.66 0.4780 0.6433 [0] 0.4088 [0] [1.0] 0.2571 4.5012 DOD-BLYP [0] [1.0]
DOD-PBEPW91 — 2.315 2.315 2.316 0.67 0.4669 0.5967 [0] 0.7436 [0] [1.0] 0.3688 3.8960 DOD-PBEB95 2.645 2.583 2.583 2.587 0.66 0.4799 0.5472 [0] 0.5548 [0] [1.0] 0.3319 3.6735 DOD-PBEPBE 2.439 2.398 2.396 2.396 0.68 0.4449 0.6161 [0] 0.7438 [0] [1.0] 0.5376 3.0254 DOD-PBEP86 2.305 2.266 2.267 2.271 0.69 0.4301 0.6131 [0] 0.6158 [0] [1.0] 0.3440 4.2427 with MBD DOD-SCAN 2.432 0.66 0.4740 0.6477 [0] 0.2509 0.4330 1.7060 0.4740 3.8328 DOD-BLYP 2.688 0.71 0.5623 0.6244 [0] 0.5644 0.1069 0.5561 0.1603 4.4327 DOD-PBEPW91 2.298 0.67 0.4691 0.5961 [0] 0.5114 0.6384 1.8187 0.5945 3.1249 DOD-PBEB95 2.443 0.66 0.4775 0.5495 [0] 0.3083 0.6255 2.2835 0.6128 2.6867 DOD-PBEPBE 2.367 0.68 0.4477 0.6142 [0] 0.4328 0.9006 2.1241 0.7444 2.4918 DOD-PBEP86 2.296 0.69 0.4324 0.6097 [0] 0.5226 0.0799 0.7861 0.3833 3.9385 ‡ Parameters taken from ref SI.4. Diet-GMTKN55 prescreening:
Table S4: WTMAD2 value for different w and cX,HF(=n) for w DSD n and w DOD n -XC-D3BJ [XC=PBEP86, PBEB95, PBEPBE, PBEPW91] cX w WTMAD2(kcal/mol) w DSD-PBEP86-D3BJ w DOD-PBEP86-D3BJ 0.57 0.00 2.582 2.574 0.20 1.996 1.994 0.22 1.977 1.977 0.24 1.976 1.993 0.26 2.015 2.042 0.60 0.00 2.438 2.425 0.15 2.004 1.991 0.18 1.940 1.937 0.20 1.922 1.917 0.22 1.924 1.929 0.25 1.959 1.997 0.30 2.080 2.166 0.35 2.196 2.355 0.63 0.00 2.292 2.230 0.10 2.023 1.988 0.12 1.968 1.943 0.14 1.901 1.901 0.16 1.880 1.880 0.20 1.879 1.880 0.22 1.883 1.908 0.66 0.00 2.062 2.062 0.10 1.875 1.865 0.12 1.844 1.842 0.14 1.836 1.836 0.16 1.844 1.833 0.18 1.839 1.839 0.20 1.852 1.863 0.22 1.866 1.915 0.24 1.910 1.986 0.69 0.00 1.973 1.936 0.10 1.804 1.799 0.12 1.801 1.795 0.14 1.800 1.795 0.16 1.800 1.806 0.18 1.818 1.834 0.20 1.841 1.893 0.22 1.877 1.961 0.30 2.092 2.252 0.72 0.00 1.861 1.861 0.10 1.778 1.768 0.12 1.776 1.777 Table S4: (Continued) cX w WTMAD2(kcal/mol) wDSD-PBEPW91-D3BJ wDSD-PBEB95-D3BJ wDSD- PBEPBE-D3BJ wDOD-PBEPW91-D3BJ wDOD-PBEB95-D3BJ wDOD-PBEPBE-D3BJ 0.57 0.00 2.579 3.107 2.643 2.541 3.106 2.640 0.12 2.394 3.060 2.466 2.394 3.054 2.465 0.16 2.324 3.025 2.412 2.326 3.046 2.412 0.18 2.298 3.022 2.393 2.297 3.062 2.388 0.22 2.335 3.022 2.426 2.330 3.125 2.415 0.60 0.00 2.414 2.977 2.476 2.414 2.965 2.476 0.12 2.300 2.935 2.353 2.301 2.922 2.354 0.15 2.241 2.907 2.353 2.240 2.931 2.354 0.18 2.218 2.913 2.305 2.218 2.973 2.303 0.20 2.247 2.915 2.331 2.244 3.012 2.324 0.63 0.00 2.264 2.844 2.330 2.264 2.827 2.336 0.10 2.189 2.803 2.412 2.189 2.810 2.415 0.12 2.174 2.804 2.251 2.174 2.825 2.249 0.14 2.160 2.802 2.234 2.160 2.837 2.237 0.16 2.148 2.803 2.221 2.148 2.865 2.222 0.20 2.214 2.819 2.293 2.211 2.946 2.284 0.66 0.00 2.131 2.740 2.195 2.131 2.705 2.193 0.10 2.098 2.705 2.169 2.098 2.732 2.169 0.12 2.092 2.707 2.160 2.092 2.748 2.164 0.14 2.084 2.707 2.158 2.085 2.766 2.158 0.16 2.101 2.714 2.180 2.101 2.799 2.183 0.18 2.143 2.728 2.221 2.150 2.838 2.222 0.69 0.00 2.023 2.619 2.093 2.023 2.628 2.094 0.10 2.028 2.622 2.108 2.028 2.657 2.092 0.12 2.033 2.624 2.101 2.032 2.680 2.108 0.14 2.055 2.628 2.131 2.055 2.699 2.131 0.16 2.102 2.646 2.183 2.106 2.736 2.184 0.18 2.204 2.670 2.230 2.205 2.777 2.236 0.72 0.00 1.964 2.529 2.027 1.964 2.559 2.028 0.10 2.016 2.545 2.083 2.015 2.601 2.074 0.12 2.039 2.562 2.105 2.041 2.622 2.104 0.14 2.073 2.576 2.146 2.084 2.654 2.153 0.16 2.129 2.608 2.206 2.148 2.694 2.206 Figure S1: Change of WTMAD2(kcal/mol) for w DSD X and w DOD X -XC-D3BJ [XC=PBEB95, PBEPBE, PBEPW91] with respect to range separation parameter w (x axis) for different cX,HF. SI.5. Division of WTMAD2 into five major subcategories:
Table S5: Division of total WTMAD2(kcal/mol) into five major subsets: Small Molecule Thermochemistry (THERMO), Intermolecular Interactions (INTERMOL), Conformers/Intramolecular Interactions (CONF), Barrier Heights (BARRIERS), Small Molecule Thermochemistry (THERMO) and Large-species Reaction Energies (LARGE) cX=0.57 wDSD-PBEPW91 wDSD-PBEB95 wDSD-PBEPBEwDOD-PBEPW91 wDOD-PBEB95 wDOD-PBEPBE2.202.402.602.803.003.20 0 0.05 0.1 0.15 0.2 cX=0.63 wDSD-PBEPW91 wDSD-PBEB95 wDSD-PBEPBEwDOD-PBEPW91 wDOD-PBEB95 wDOD-PBEPBE 2.002.202.402.602.803.003.20 0 0.05 0.1 0.15 0.2 cX=0.63 wDSD-PBEPW91 wDSD-PBEB95 wDSD-PBEPBEwDOD-PBEPW91 wDOD-PBEB95 wDOD-PBEPBE2.002.202.402.602.803.00 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 cX=0.66 wDSD-PBEPW91 wDSD-PBEB95 wDSD-PBEPBEwDOD-PBEPW91 wDOD-PBEB95 wDOD-PBEPBE1.902.102.302.502.702.903.10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 cX=0.69 wDSD-PBEPW91 wDSD-PBEB95 wDSD-PBEPBEwDOD-PBEPW91 wDOD-PBEB95 wDOD-PBEPBE 1.802.002.202.402.602.80 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 cX=0.72 wDSD-PBEPW91 wDSD-PBEB95 wDSD-PBEPBEwDOD-PBEPW91 wDOD-PBEB95 wDOD-PBEPBE Functionals THERMO BARRIERS LARGE CONF INTERMOL WTMAD2 xDSD -PBEP86-D4 0.508 0.252 0.493 0.409 0.457 2.119 xDSD -PBEP86-D4 0.550 0.277 0.502 0.425 0.532 2.285 xDSD -PBEB95-D4 0.569 0.246 0.392 0.706 0.490 2.403 xDSD -BLYP-D4 0.495 0.291 0.479 0.452 0.530 2.242 xDSD -SCAN-D4 0.570 0.314 0.541 0.420 0.534 2.378 xDSD -PBEPW91-D4 0.580 0.270 0.471 0.404 0.478 2.203 xDSD -PBEPBE-D4 0.594 0.271 0.475 0.413 0.486 2.238 xDOD -PBEP86-D4 0.572 0.234 0.512 0.408 0.470 2.196 xDOD -PBEB95-D4 0.608 0.215 0.377 0.774 0.518 2.491 xDOD -BLYP-D4 0.676 0.291 0.520 0.480 0.577 2.543 xDOD -SCAN-D4 0.582 0.308 0.542 0.422 0.532 2.385 xDOD -PBEPW91-D4 0.604 0.244 0.473 0.414 0.484 2.219 xDOD -PBEPBE-D4 0.620 0.249 0.465 0.421 0.488 2.243 ωDSD -PBEP86-D4(ω=0.13) 0.536 0.244 0.415 0.419 0.470 2.083 ωDSD -PBEP86-D4(ω=0.16) 0.552 0.238 0.391 0.427 0.481 2.089 ωDSD -PBEP86-D4(ω=0.18) 0.573 0.241 0.377 0.438 0.488 2.116 ωDSD -PBEP86-D4(ω=0.20) 0.596 0.242 0.376 0.446 0.494 2.154 ωDSD -PBEP86-D4(ω=0.22) 0.614 0.245 0.384 0.453 0.506 2.202 ωDSD -PBEP86-D4(ω=0.22) 0.649 0.259 0.381 0.466 0.503 2.258 ωDOD -PBEP86-D4(ω=0.08) 0.580 0.237 0.475 0.415 0.478 2.185 ωDOD -PBEP86-D4(ω=0.10) 0.587 0.227 0.459 0.423 0.480 2.175 ωDOD -PBEP86-D4(ω=0.15) 0.600 0.228 0.414 0.437 0.498 2.176 ωDOD -PBEP86-D4(ω=0.16) 0.615 0.232 0.413 0.451 0.488 2.199 ωDOD -PBEP86-D4(ω=0.18) 0.636 0.241 0.397 0.461 0.507 2.241 ωDOD -PBEP86-D4(ω=0.20) 0.651 0.253 0.399 0.466 0.532 2.302 ωB97M(2) 0.430 0.214 0.418 0.577 0.492 2.131 SI.6. QCHEM sample inputs: (a) w DSD -PBEP86-D3BJ( w =0.13): $comment wDSD72-PBEP86-D3BJ with w=0.13 $end $rem SYMMETRY false BASIS def2-QZVPP XC_GRID 3 ECP def2-ECP N_FROZEN_CORE FC AUX_BASIS_CORR rimp2-def2-QZVPPD MAX_SCF_CYCLES 1000 SCF_CONVERGENCE 7 MEM_STATIC 2000 SET_ITER 100 MOLDEN_FORMAT false PRINT_ORBITALS 10 SCF_FINAL_PRINT 1 THRESH 12 LRC_DFT TRUE OMEGA 130 combine_K TRUE exchange gen correlation rimp2 DH TRUE SSS_FACTOR 153700 SOS_FACTOR 673830 DFT_D D3_BJ DFT_D3_S6 34760 DFT_D3_S8 0 DFT_D3_A2 550000 DFT_D3_A1 000000 $end $xc_functional X HF 0.72 X wPBE 0.28 C P86 1.00 $end $molecule 0 1 C 0.000000000000 0.000000000000 0.000000000000 O 0.000000000000 0.000000000000 1.131400000000 $end (b) w DOD -PBEP86-D3BJ( w =0.10): $comment wDOD69-PBEP86-D3BJ with w=0.10 $end $rem SYMMETRY false BASIS def2-QZVPP XC_GRID 3 ECP def2-ECP N_FROZEN_CORE FC AUX_BASIS_CORR rimp2-def2-QZVPPD MAX_SCF_CYCLES 1000 SCF_CONVERGENCE 7 MEM_STATIC 2000 SET_ITER 100 MOLDEN_FORMAT false PRINT_ORBITALS 10 SCF_FINAL_PRINT 1 THRESH 12 LRC_DFT TRUE OMEGA 100 combine_K TRUE exchange gen correlation rimp2 DH TRUE SSS_FACTOR 000000 SOS_FACTOR 702490 DFT_D D3_BJ DFT_D3_S6 42353 DFT_D3_S8 0 DFT_D3_A2 550000 DFT_D3_A1 000000 $end $xc_functional X HF 0.72 X wPBE 0.28 C P86 1.00 $end $molecule 0 1 C 0.000000000000 0.000000000000 0.000000000000 O 0.000000000000 0.000000000000 1.131400000000 $end
SI.7 References: (1) Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A Look at the Density Functional Theory Zoo with the Advanced GMTKN55 Database for General Main Group Thermochemistry, Kinetics and Noncovalent Interactions.
Phys. Chem. Chem. Phys. , , 32184–32215. (2) Gruzman, D.; Karton, A.; Martin, J. M. L. Performance of Ab Initio and Density Functional Methods for Conformational Equilibria of C n H 2 n +2 Alkane Isomers ( n = 4−8) †. J. Phys. Chem. A , , 11974–11983. (3) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. , , 154104. 13 (4) Lao, K. U.; Schäffer, R.; Jansen, G.; Herbert, J. M. Accurate Description of Intermolecular Interactions Involving Ions Using Symmetry-Adapted Perturbation Theory. J. Chem. Theory Comput. , , 2473–2486. (5) Yu, H.; Truhlar, D. G. Components of the Bond Energy in Polar Diatomic Molecules, Radicals, and Ions Formed by Group-1 and Group-2 Metal Atoms. J. Chem. Theory Comput. , , 2968–2983. (6) Kesharwani, M. K.; Karton, A.; Martin, J. M. L. Benchmark Ab Initio Conformational Energies for the Proteinogenic Amino Acids through Explicitly Correlated Methods. Assessment of Density Functional Methods. J. Chem. Theory Comput. , , 444–454. (7) Goerigk, L.; Grimme, S. A General Database for Main Group Thermochemistry, Kinetics, and Noncovalent Interactions − Assessment of Common and Reparameterized ( Meta -)GGA Density Functionals. J. Chem. Theory Comput. , , 107–126. (8) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Multi-Coefficient Extrapolated Density Functional Theory for Thermochemistry and Thermochemical Kinetics. Phys. Chem. Chem. Phys. , , 43. (9) Zhao, Y.; González-Garda, N.; Truhlar, D. G. Benchmark Database of Barrier Heights for Heavy Atom Transfer, Nucleophilic Substitution, Association, and Unimolecular Reactions and Its Use to Test Theoretical Methods. J. Phys. Chem. A , , 2012–2018. (10) Goerigk, L.; Grimme, S.; Chemie, T. O. A General Database for Main Group Thermochemistry , Kinetics , and Non-Covalent Interactions – Assessment of Common and Reparameterized ( Meta- ) GGA Density Functionals Supporting Information. , 1–32. (11) Guner, V.; Khuong, K. S.; Leach, A. G.; Lee, P. S.; Bartberger, M. D.; Houk, K. N. A Standard Set of Pericyclic Reactions of Hydrocarbons for the Benchmarking of Computational Methods : The Performance of Ab Initio , Density Functional , CASSCF , CASPT2 , and CBS-QB3 Methods for the Prediction of Activation Barriers , Reaction Energetic. , 11445–11459. (12) Ess, D. H.; Houk, K. N. Activation Energies of Pericyclic Reactions : Performance of DFT , MP2 , and CBS-QB3 Methods for the Prediction of Activation Barriers and Reaction Energetics of 1 , 3-Dipolar Cycloadditions , and Revised Activation Enthalpies for a Standard Set of Hydroc. , 9542–9553. (13) Dinadayalane, T. C.; Vijaya, R.; Smitha, A.; Sastry, G. N. Diels−Alder Reactivity of Butadiene and Cyclic Five-Membered Dienes ((CH) 4 X, X = CH 2 , SiH 2 , O, NH, PH, and S) with Ethylene: A Benchmark Study. J. Phys. Chem. A , , 1627–1633. (14) Steinmann, S. N.; Csonka, G.; Corminboeuf, C. Unified Inter- and Intramolecular Dispersion Correction Functional Theory. J. Chem. Theory Comput. , , 2950–2958. (15) Krieg, H.; Grimme, S. Thermochemical Benchmarking of Hydrocarbon Bond Separation Reaction Energies: Jacob’s Ladder Is Not Reversed! Mol. Phys. , , 2655–2666. (16) Kozuch, S.; Bachrach, S. M.; Martin, J. M. L. Conformational Equilibria in Butane-1,4-Diol: A Benchmark of a Prototypical System with Strong Intramolecular H - Bonds. . (17) Sure, R.; Hansen, A.; Schwerdtfeger, P.; Grimme, S. Comprehensive Theoretical Study of All 1812 C60isomers.
Phys. Chem. Chem. Phys. , , 14296–14305. (18) Yu, L.; Karton, A. Assessment of Theoretical Procedures for a Diverse Set of Isomerization Reactions Involving Double-Bond Migration in Conjugated Dienes. Chem. Phys. , 14 , 166–177. (19) Johnson, E. R.; Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Delocalization Errors in Density Functionals and Implications for Main-Group Thermochemistry.
J. Chem. Phys. , , 204112. (20) Zhao, Y.; Truhlar, D. G. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Function. Theor. Chem. Acc. , , 215–241. (21) Grimme, S. Semiempirical Hybrid Density Functional with Perturbative Second-Order Correlation. J. Chem. Phys. , , 0–16. (22) Grimme, S.; Mück-Lichtenfeld, C.; Würthwein, E. U.; Ehlers, A. W.; Goumans, T. P. M.; Lammertsma, K. Consistent Theoretical Description of 1,3-Dipolar Cycloaddition Reactions. J. Phys. Chem. A , , 2583–2586. (23) Piacenza, M.; Grimme, S. Systematic Quantum Chemical Study of DNA-Base Tautomers. J. Comput. Chem. , , 83–98. (24) Woodcock, H. L.; Schaefer, H. F.; Schreiner, P. R. Problematic Energy Differences between Cumulenes and Poly-Ynes: Does This Point to a Systematic Improvement of Density Functional Theory? J. Phys. Chem. A , , 11923–11931. (25) Schreiner, P. R.; Fokin, A. A.; Pascal, R. A.; Meijere, A. De. Many Density Functional Theory Approaches Fail To Give Reliable Large Hydrocarbon Isomer Energy Differences. , 10–13. (26) Lepetit, C.; Chermette, H.; Heully, J.; Lyon, D.; Uni, V.; Umr, C. Description of Carbo -Oxocarbons and Assessment of Exchange-Correlation Functionals for the DFT Description of Carbo -Mers. , 136–149. (27) Lee, J. S. Accurate Ab Initio Binding Energies of Alkaline Earth Metal Clusters. J. Phys. Chem. A , , 11927–11932. (28) Karton, A.; Martin, J. M. L. Explicitly Correlated Benchmark Calculations on C 8 H 8 Isomer Energy Separations : How Accurate Are DFT , Double-Hybrid , and Composite Ab Initio Procedures ? , . (29) Zhao, Y.; Tishchenko, O.; Gour, J. R.; Li, W.; Lutz, J. J.; Piecuch, P.; Truhlar, D. G. Thermochemical Kinetics for Multireference Systems: Addition Reactions of Ozone. J. Phys. Chem. A , , 5786–5799. (30) Manna, D.; Martin, J. M. L. What Are the Ground State Structures of C 20 and C 24 ? An Explicitly Correlated Ab Initio Approach. . (31) Friedrich, J.; Hänchen, J. Incremental CCSD(T)(F12*)|MP2: A Black Box Method to Obtain Highly Accurate Reaction Energies. J. Chem. Theory Comput. , , 5381–5394. (32) Friedrich, J. Efficient Calculation of Accurate Reaction Energies—Assessment of Different Models in Electronic Structure Theory. J. Chem. Theory Comput. , , 3596–3609. (33) You, A.; Be, M. A. Y.; In, I. Gaussian-2 Theory for Molecular Energies of First- and Second-Row Compounds Compounds. , . (34) Curtiss, L. A.; Redfern, P. C. Assessment of Gaussian-2 and Density Functional Theories for the Computation of Enthalpies of Formation for the Computation of Enthalpies of Formation. , . (35) Kozuch, S.; Martin, J. M. L. Halogen Bonds: Benchmarks and Theoretical Analysis. J. Chem. Theory Comput. , , 1918–1931. (36) Řezáč, J.; Riley, K. E.; Hobza, P. Benchmark Calculations of Noncovalent Interactions of Halogenated Molecules. J. Chem. Theory Comput. , , 4285–4292. (37) Schwabe, T.; Grimme, S. Double-Hybrid Density Functionals with Long-Range Dispersion Corrections: Higher Accuracy and Extended Applicability. Phys. Chem. Chem. Phys. , , 3397–3406. (38) Grimme, S. Seemingly Simple Stereoelectronic Effects in Alkane Isomers and the Implications for Kohn–Sham Density Functional Theory. Angew. Chemie Int. Ed. , , 4460–4464. (39) Goerigk, L.; Grimme, S. Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals—Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. , , 291–309. (40) Grimme, S.; Steinmetz, M.; Korth, M. How to Compute Isomerization Energies of Organic Molecules with Quantum Chemical Methods. J. Org. Chem. , , 2118–2126. (41) Goerigk, L.; Sharma, R. The INV24 Test Set: How Well Do Quantum-Chemical Methods Describe Inversion and Racemization Barriers? Can. J. Chem. , , 1133–1143. (42) Huenerbein, R.; Schirmer, B.; Moellmann, J.; Grimme, S. Effects of London Dispersion on the Isomerization Reactions of Large Organic Molecules: A Density Functional Benchmark Study. Phys. Chem. Chem. Phys. , , 6940–6948. (43) Fogueri, U. R.; Kozuch, S.; Karton, A.; Martin, J. M. L. The Melatonin Conformer Space: Benchmark and Assessment of Wave Function and DFT Methods for a Paradigmatic Biological and Pharmacological Molecule. J. Phys. Chem. A , , 2269–2277. (44) Grimme, S.; Kruse, H.; Goerigk, L.; Erker, G. The Mechanism of Dihydrogen Activation by Frustrated Lewis Pairs Revisited. Angew. Chemie Int. Ed. , , 1402–1405. (45) Setiawan, D.; Kraka, E.; Cremer, D. Strength of the Pnicogen Bond in Complexes Involving Group Va Elements N, P, and As. . (46) Karton, A.; O’Reilly, R. J.; Chan, B.; Radom, L. Determination of Barrier Heights for Proton Exchange in Small Water, Ammonia, and Hydrogen Fluoride Clusters with G4(MP2)-Type, MPn, and SCS-MPn Procedures-a Caveat. J. Chem. Theory Comput. , , 3128–3136. (47) Neese, F.; Schwabe, T.; Kossmann, S.; Schirmer, B.; Grimme, S. Assessment of Orbital-Optimized , Spin-Component Scaled Second-Order Many-Body Perturbation Theory for Thermochemistry and Kinetics. , 3060–3073. (48) Jurečka, P.; Šponer, J.; Černý, J.; Hobza, P. Benchmark Database of Accurate (MP2 and CCSD(T) Complete Basis Set Limit) Interaction Energies of Small Model Complexes, DNA Base Pairs, and Amino Acid Pairs. Phys. Chem. Chem. Phys. , , 1985–1993. (49) Řezáč, J.; Riley, K. E.; Hobza, P. S66: A Well-Balanced Database of Benchmark Interaction Energies Relevant to Biomolecular Structures. J. Chem. Theory Comput. , , 2427–2438. (50) French, A. D.; Johnson, G. P.; Stortz, C. A. Evaluation of Density Functionals and Basis Sets for Ga. , 679–692. (51) Karton, A.; Rabinovich, E.; Martin, J. M. L.; Ruscic, B. W4 Theory for Computational Thermochemistry: In Pursuit of Confident Sub-KJ/Mol Predictions. J. Chem. Phys. , , 144108. 16 (52) Kruse, H.; Mladek, A.; Gkionis, K.; Hansen, A.; Grimme, S.; Sponer, J. Quantum Chemical Benchmark Study on 46 RNA Backbone Families Using a Dinucleotide Unit. J. Chem. Theory Comput. , , 4972–4991. (53) Karton, A.; Daon, S.; Martin, J. M. L. W4-11: A High-Confidence Benchmark Dataset for Computational Thermochemistry Derived from First-Principles W4 Data. Chem. Phys. Lett. , , 165–178. (54) Iii, W. A. G.; Park, U. V; Pennsyl, V. Evaluation of B3LYP , X3LYP , and M06-Class Density Functionals for Predicting the Binding Energies of Neutral , Protonated , and Deprotonated Water Clusters. , 1016–1026. (55) Karton, A.; O’Reilly, R. J.; Radom, L. Assessment of Theoretical Procedures for Calculating Barrier Heights for a Diverse Set of Water-Catalyzed Proton-Transfer Reactions. J. Phys. Chem. A , , 4211–4221. (56) Zhao, Y.; Ng, H. T.; Peverati, R.; Truhlar, D. G. Benchmark Database for Ylidic Bond Dissociation Energies and Its Use for Assessments of Electronic Structure Methods. J. Chem. Theory Comput. , , 2824–2834. (57) Weigend, F.; Ahlrichs, R. Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy. Phys. Chem. Chem. Phys. , , 3297–3305. (58) Santra, G.; Sylvetsky, N.; Martin, J. M. L. Minimally Empirical Double-Hybrid Functionals Trained against the GMTKN55 Database: RevDSD-PBEP86-D4, RevDOD-PBE-D4, and DOD-SCAN-D4. J. Phys. Chem. A , , 5129–5143. (59) Sancho-García, J. C.; Brémond; Savarese, M.; Pérez-Jiménez, A. J.; Adamo, C. Partnering Dispersion Corrections with Modern Parameter-Free Double-Hybrid Density Functionals. Phys. Chem. Chem. Phys. , , 13481–13487. (60) Hui, K.; Chai, J. Da. SCAN-Based Hybrid and Double-Hybrid Density Functionals from Models without Fitted Parameters. J. Chem. Phys. , , 044114. (61) Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid Exchange–Correlation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. , , 51–57. (62) Verma, P.; Wang, Y.; Ghosh, S.; He, X.; Truhlar, D. G. Revised M11 Exchange-Correlation Functional for Electronic Excitation Energies and Ground-State Properties. J. Phys. Chem. A , , 2966–2990. (63) Santra, G.; Martin, J. M. L. Some Observations on the Performance of the Most Recent Exchange-Correlation Functionals for the Large and Chemically Diverse GMTKN55 Benchmark. In AIP Conference Proceedings ; 2019; p 030004. (64) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0 Model.
J. Chem. Phys. , , 6158–6170. (65) Zhao, Y.; Truhlar, D. G. Design of Density Functionals That Are Broadly Accurate for Thermochemistry, Thermochemical Kinetics, and Nonbonded Interactions. J. Phys. Chem. A , , 5656–5667. (66) Becke, A. D. A New Mixing of Hartree-Fock and Local Density-Functional Theories. J. Chem. Phys. , , 1372–1377. (67) Mardirossian, N.; Head-Gordon, M. Mapping the Genome of Meta-Generalized Gradient Approximation Density Functionals: The Search for B97M-V. J. Chem. Phys. , , 074111. 17 (68) Sun, J.; Ruzsinszky, A.; Perdew, J. Strongly Constrained and Appropriately Normed Semilocal Density Functional. Phys. Rev. Lett. , , 036402. (69) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Constantin, L. A.; Sun, J. Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry. Phys. Rev. Lett. , , 026403. (70) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. , , 3865–3868. (71) Perdew, J. P.; Ernzerhof, M.; Burke, K. [ERRATA] Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. , , 3865–3868. (72) Slater, J. C. A Simplification of the Hartree-Fock Method. Phys. Rev. , , 385–390. (73) Perdew, J. P.; Wang, Y. Accurate and Simple Analytic Representation of the Electron-Gas Correlation Energy. Phys. Rev. B , , 13244–13249. (74) Martin, J. M. L.; Santra, G. Empirical Double-Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT. Isr. J. Chem. , , 787–804. (75) Cao, J.; Berne, B. J. Many-body Dispersion Forces of Polarizable Clusters and Liquids. J. Chem. Phys. , , 8628–8636. (76) Hermann, J.; DiStasio, R. A.; Tkatchenko, A. First-Principles Models for van Der Waals Interactions in Molecules and Materials: Concepts, Theory, and Applications. Chem. Rev. ,117