Exploring Avenues Beyond Revised DSD Functionals: II. Random-Phase Approximation and scaled MP3 corrections
1 Exploring Avenues Beyond Revised DSD Functionals: II. Random-Phase Approximation and scaled MP3 corrections
Golokesh Santra, † Emmanouil Semidalas, † and Jan M.L. Martin *,† † Department of Organic Chemistry, Weizmann Institute of Science, 7610001 Reḥovot, Israel. Email: [email protected]
Abstract:
For revDSD double hybrids, the Görling-Levy second-order perturbation theory component is an Achilles’ Heel when applied to systems with significant near-degeneracy (“static”) correlation. We have explored its replacement by the direct random phase approximation (dRPA), inspired by the SCS-dRPA75 functional of Kállay and coworkers. The addition to the final energy of both a D4 empirical dispersion correction, and of a semilocal correlation component lead, to significant improvements, with DSD-PBEdRPA -D4 approaching the performance of revDSD-PBEP86-D4 and the Berkeley ωB97M(2). This form appears to be fairly insensitive to the choice of semilocal functional, but does exhibit stronger basis set sensitivity than the PT2-based double hybrids (due to much larger prefactors for the nonlocal correlation). As an alternative, we explored adding an MP3-like correction term (in a medium-sized basis sets) to a range-separated ωDSD-PBEP86-D4 double hybrid, and found it to have significantly lower WTMAD2 (weighted mean absolute deviation) for the large and chemically diverse GMTKN55 benchmark suite; the added computational cost can be mitigated through density fitting techniques. I. Introduction:
While Kohn-Sham density functional theory (KS-DFT) in principle would be exact if the exact exchange-correlation (XC) functional were known, in practice its accuracy is limited by the quality of the approximate XC functional chosen in electronic structure calculations. Over the past few decades a veritable “zoo” (Perdew’s term ) of such functionals has emerged. Perdew introduced an organizing principle known as the “Jacob’s Ladder”, ascending by degrees from the Hartree “vale of tears” (no exchange, no correlation) to the Heaven of chemical accuracy: on every degree or rung, a new source of information is introduced. LDA (local density approximation) constitutes the 1 st rung, GGAs (generalized gradient approximations) the 2 nd rung, and meta-GGAs (mGGAs, which introduce the density Laplacian or the kinetic energy density) represents the 3 rd rung of the ladder. The 4 th rung introduces dependence on the occupied Kohn-Sham orbitals: hybrid functionals (global, local, and range-separated) are the most important subclass here. Lastly, the fifth rung corresponds to inclusion of virtual orbital information, such as in double hybrids (see Refs. for reviews, and most recently Ref. by the present authors). Building on the earlier work of Görling and Levy who introduced perturbation theory in a basis of Kohn-Sham orbitals, Grimme’s 2006 paper presented the first double hybrid in the current sense of the word. The term refers to the fact that, aside from an admixture of (m)GGA and ‘exact’ Hartree-Fock like exchange, the correlation is treated as a hybrid of (m)GGA correlation and GLPT2 (2 nd -order Görling-Levy perturbation theory). Following a Kohn-Sham calculation with a given semilocal XC functional and given percentage of HF exchange, the total energy is evaluated in the second step as: E !" = E + c &,"( E ),"( + &1 − c &,"( )E ),&* + c *,&* E *,&* + c +,- E +,- + c +.. E +.. + E /0.1 [𝑠 , 𝑠 , 𝑐 , 𝑎 $ , 𝑎 + , 𝑒𝑡𝑐] (1) where E N1e stands for the sum of nuclear repulsion and one-electron energy terms; E
X,HF is the HF-exchange energy and c
X,HF the corresponding coefficient; E
X,XC and E
C,XC are the semilocal exchange and correlation energies respectively, and c
C,XC is the fraction of semi local correlation energy used in the final energy. E and E are the opposite-spin and same-spin MP2-like energies obtained in the basis of the KS orbitals from the first step, and c and c are the linear coefficients for the same. Finally, E disp is a dispersion correction, with its own adjustable parameters. As shown, e.g., in Refs. , modern double hybrids can achieve accuracies for large, chemically diverse validation benchmarks like GMTKN55 (general main-group thermochemistry, kinetics, and noncovalent interactions) that rival those of composite wavefunction theory (cWFT) methods like G4 theory. (See, however, Semidalas and Martin for some ways to improve cWFT at zero to minimal cost. ) One Achilles’ Heel for GLPT2 are molecules with small band gaps (a.k.a absolute near-degeneracy correlation, type A static correlation ), owing to the orbital energy difference in the PT2 denominator becoming very small. One potential remedy would be to replace PT2 by the random phase approximation (RPA) for the nonlocal correlation part. From the viewpoint of wavefunction theory, Scuseria and coworkers have analytically proven the equivalence of RPA and direct ring coupled cluster with all doubles(drCCD). While the coupled-cluster singles and doubles (CCSD) method is not immune to type A static correlation, it is much more resilient compared to PT2. The very first foray in this direction was made by Ahnen et al., who substituted RPA for GLPT2 in the B2PLYP double hybrid. Later, Kállay and coworkers, as well as Grimme and Steinmetz, have explored this possibility in greater depth and came up with their own double hybrids featuring the direct random phase approximation (dRPA, Ref. and references therein). The dRPA75 ‘dual hybrid’ of Kállay and coworkers, which uses orbitals evaluated at the PBE level (with 75% Hartree-Fock exchange and full PBEc correlation), but only includes pure dRPA correlation in the final energy, is closer in spirit to dRPA than to a double hybrid. In contrast, Grimme and Steinmetz’s PWRB95 employs computationally inexpensive mGGA orbitals (specifically, mPW91B95 ) to evaluate a final energy expression consisting of 50% HF exchange, 50% semilocal exchange, 35% dRPA correlation, 71% semilocal correlation, and 65% nonlocal dispersion correction — making it an obvious double hybrid. One major issue with the dRPA75 was its poor performance for total atomization energies (TAEs, the computational cognates of heats of formation). The authors later remedied that by spin-component scaling: although dRPA is a spin-free method and thus such scaling would have no effect for closed-shell systems, it will affect open-shell cases (most relevantly for TAEs, atoms), particularly as dRPA has an spurious self-correlation energy for unpaired electrons. The so called SCS-dRPA75 functional employs c X =0.75, c o-s =1.5 and c s-s =(2-c o-s )=0.5 — addressing the issue for atoms and other open-shell species while being equivalent to dRPA75 for closed-shell species. In their revision of the S66x8 noncovalent interactions dataset, Brauer et al. found that the ostensibly good performance of dRPA75/aug-cc-pVTZ resulted from a spurious error compensation between basis set superposition error and the absence of a dispersion correction. They also observed, as expected, that the basis set convergence behavior of dRPA is similar to that of CCSD. A D3BJ dispersion correction was parametrized for use with dRPA75 and its parameters found to be very similar to those optimized on top of CCSD (coupled cluster with all singles and doubles ); from a symmetry-adapted perturbation theory perspective, the most important dispersion term not included in dRPA and CCSD is the 4 th -order connected triple excitations term. In addition, as already mentioned, the dRPA75 and SCS-dRPA75 forms do not include any semilocal correlation contribution in their final energy expressions. The first research question to be answered in this paper is (see subsection III.A) whether (SCS)dRPA75 can be further improved by not only admitting modern dispersion corrections and semilocal correlation, but also reparametrizing against a large and chemically diverse database. The functional form is denoted DSD- XC dRPAn-Disp, where XC stands for the nonlocal exchange-correlation combination used for both the orbital generation in the first step and energy calculation in the second step; n is the percentage of HF-exchange used for both the steps. The final energy for DSD-XCdRPAn-Disp has the form: E !" = E + c &,"( E &,"( + &1 − c &,"( )E &,&* + c *,&* E *,&* + c E *9*98/:;<78. + c .8. E *9*98/:;<.8. + E /0.1 [s , s , c <=* , a $ , a + , etc] (2) where, c o-s and c s-s stands for opposite spin and same spin dRPAc coefficient, respectively. All other terms are same as Eq (1). In this notation, the SCS-dRPA75 dual hybrid is a special case where c X,HF =0.75, c
C,XC =0 and s =s =0. As we will show later on, the answer to our research question is affirmative, and the resulting functionals approach the accuracy of the best PT2-based double hybrids known thus far – Mardirossian and Head-Gordon’s ωB97M(2), and our own revDSD-PBEP86-D4. The second research question (to be answered in III.B) is: would taking GLPT2 beyond second-order improve the performances of revDSD functionals further? Radom and coworkers considered MP3, MP4, and CCSD instead of MP2 and found no significant improvement over regular double hybrids. However, this may simply have been an artifact of the modest basis sets and relatively small training set used in Ref. Such considerations have been examined in Ref. , where it was also found that the benefits of including an MP3 “middle step” in a 3-tier cWFT can be realized also with a medium-sized basis set for this costly term. In the chapters below, we shall consider its addition to global double hybrid revDSD and range-separated ωDSD type double hybrids using the GMTKN55 dataset for training/calibration. Newly developed functionals will be denoted DSD3 for global DHs, and ωDSD3 for range-separated DHs. The final energy expression of a DSD3 functional has the following form: 𝐸 >?>@ = E + c &,"( E &,"( + &1 − c &,"( )E &,&* + c *,&* E *,&* + c +,- E +,- + c +.. E +.. + 𝑐 @ 𝐸 +𝐸 BCDE [𝑠 , 𝑠 , 𝑐 , 𝑎 $ , 𝑎 + , 𝑒𝑡𝑐] (3) where 𝐸 !" stands for the MP3 energy component calculated in a basis of HF orbitals, and c is a corresponding scaling parameter. All other parameters and energy components is same as regular DSD functionals in Eq(1). For ωDSD3, the range separation of the HF exchange introduces one additional parameter, the range-separation exponent ω. II.
Computational Methods:
A. Reference Data : The primary parametrization and validation set used in this work is the GMTKN55 (General Main-group Thermochemistry, Kinetics, and Noncovalent interactions) benchmark by Grimme, Goerigk, and coworkers. This database is an updated and expanded version of its predecessors GMTKN24 and GMTKN30. GMTKN55 comprises 55 types of chemical model problems, which can be further classified into five major (top-level) subcategories: thermochemistry of small and medium-sized molecules, barrier heights, large-molecule reactions, intermolecular interactions, and conformer energies (or intra-molecular interactions). One full evaluation of the GMTKN55 requires a total of 2459 single point energy calculations, leading to 1499 unique energy differences. (Complete details of all 55 subsets and original references can be found in Table S1 in the ESI.) The WTMAD2 (weighted mean absolute deviation, type 2) as defined in the GMTKN55 paper has been used as the primary metric of choice throughout the current work: WTMAD2 = $∑ !""! . ∑ N . I2.3L kcal mol ⁄|∆Q|RRRRRR ! . MAD (4) where |∆𝐸|%%%%%% ’ is the mean absolute value of all the reference energies from 𝑖 = 1 to 55, 𝑁 ’ is the number of systems in each subset, 𝑀𝐴𝐷 ’ is the mean absolute difference between calculated and reference energies for each of the 55 subsets. MAD is a more ‘robust’ metric than RMSD, in the statistical sense of the word that it is more resilient to a small number of large outliers than the RMSD (root-mean-square difference). For a normal distribution without systematic error, RMSD≈5MAD/4. Reference geometries were downloaded from the ESI of Ref. and used without further geometry optimization.
B. Electronic Structure calculations:
The MRCC2020 program package was used for all calculations involving dRPA correlation. The Weigend−Ahlrichs def2-QZVPP basis set was used for all of the subsets except WATER27, RG18, IL16, G21EA, BH76, BH76RC and AHB21 – where the diffuse-function augmented def2-QZVPPD was employed – and the C60ISO and UPU23 subsets, where we settled for the def2-TZVPP basis set to reduce computational cost. The LD0110-LD0590 angular integration grid was used for all the DFT calculations; this is a pruned Lebedev-type integration grid similar to Grid=UltraFine in Gaussian, or SG-3 in Q-Chem. In their original GMTKN55 paper, Goerigk et al. correlated all electrons in the post-KS steps. However, in a previous study by our group, we have shown that core-valence correlation is best omitted while using the def2-QZVPP basis set (which has no core-valence functions), while in a more recent study on composite wavefunction methods indicated that even with correlation consistent core-valence sets, the effect of subvalence electrons on WTMAD2 of GMTKN55 is quite small — benefits gained there are mostly from the added valence flexibility of the basis sets. Exceptions were made for MB16−43, HEAVY28, HEAVYSB11, ALK8, CHB6 and ALKBDE10 subsets — where the orbital energy gaps between halogen and chalcogen valence and metal subvalence shells can drop below 1eV, such that subvalence electrons of metal and metalloid atoms must be unfrozen — as well as for the HAL59 and HEAVY28 subsets, where (n − 1)spd orbitals on heavy p-block elements were kept unfrozen. For the DSD3 and ωDSD3 functionals, QCHEM In order to reduce the computational cost, all the MP3 calculations were done using the def2-TZVPP basis set; all other energy components were evaluated using the same basis set combination mentioned above. For technical reasons, HF reference orbitals had to be used for the MP3 steps. All the calculations were performed on the ChemFarm HPC cluster in the Faculty of Chemistry at the Weizmann Institute of Science. C. Optimization of Parameters:
A fully-optimized dRPA-based double hybrid will have six empirical parameters: the fraction of global ("exact”, HF-like) exchange, c
X,HF (c X,DFT = 1- c
X,HF ); the fraction of semilocal DFT correlation, c
C,DFT ; that of opposite-spin dRPA correlation, c o-s ; of same-spin dRPA correlation c s-s ; a prefactor s for the D3(BJ) dispersion correction and parameter a for the D3(BJ) damping function (like in refs we constrain a =0 and s =0). However, DSD3-type functionals (see below) introduce one additional parameter ( c ) for the MP3 correlation term. For the ωDSD3 family, yet another parameter ω needs to be considered for range-separation, which brings the total number of empirical parameters to eight — still only half the number involved in the current “best in class” double hybrid ωB97M(2), which has sixteen empirical parameters. We employed Powell’s BOBYQA (Bound Optimization BY Quadratic Approximation) derivative-free constrained optimizer, together with scripts and Fortran programs developed in-house, for the optimization of all parameters. Once a full set of GMTKN55 calculations is done for one set of fixed nonlinear parameters c X,HF and c
C,DFT ( for ωDSD3 also ω ), the associated optimal values of the remaining parameters {c , c , (c ), s , a } can be obtained in a “microiteration” process. This entire process corresponds to one step in the “macroiterations” in which we minimize WTMAD2 with respect to {c X,HF , c
C,DFT } and, where applicable, (ω). The process is somewhat akin to microiterations in CASSCF algorithms w.r.t. 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 (e.g., Ref. ). III. Results and Discussion: A.I. The GMTKN55 Suite:
In our previous study, we found that refitting of the original DSD functionals to the large and chemically diverse GMTKN55 dataset led to greatly improved performance, particularly for noncovalent interaction and large-molecule reaction energy. Motivated by this prior finding, we attempted first to reoptimize the spin-component-scaling factors in SCS-dRPA75 and obtained WTMAD2=4.71 kcal/mol — just a marginal improvement over the original dual hybrid (WTMAD2=4.79 kcal/mol). In the S66x8 noncovalent interactions benchmark paper, dRPA75-D3BJ with basis set extrapolation was found to be the best performer of all DFT functionals. Inspired by this observation, we added a D3BJ correction on top of the Kállay SCS-dRPA75 dual hybrid and found that WTMAD2 dropped from 4.79 to 2.89 kcal/mol. (For perspective, it should be pointed out that the lowest WTMAD2 thus far found for a rung-four functional is 3.2 kcal/mol for ωB97M-V. ) By additionally relaxing the opposite spin and same spin (SS-OS) balance of dRPA correlation in the optimization, WTMAD2 can be further reduced to 2.76 kcal/mol (see Table 1). As expected, the majority of the improvement comes from the noncovalent interaction and large molecule reaction subsets (Figure 1). Considering that the energy expression for optSCS-dRPA75-D3BJ contains full dRPA correlation — unlike revDSD double hybrids, where the GLPT2 correlation is scaled down by ~50% — one can reasonably expect basis set sensitivity. Would improving the basis set beyond def2-QZVPP reduce WTMAD2 further? Extrapolating from def2-TZVPP and def2-QZVPP using the familiar L –3 formula of Halkier et al., we found a reduction by only 0.03 kcal/mol – while using a compromise extrapolation exponent between the L –3 opposite-spin and L –5 for same-spin correlation, α=3.727 from solving [((4/3) -1) –1 + ((4/3) -1) –1 ]/2 = ((4/3) α -1) –1 reduced WTMAD2 further to 2.70 kcal/mol. What if we “upgrade” D3BJ to the recently-published D4 dispersion term? Aside from the usual four adjustable two-body D4 parameters s , s , a , and a , the prefactor of the 3-body Axilrod-Teller-Muto term, c ATM , cannot simply be fixed at c
ATM =1 since unlike GLPT2, dRPA does contain n-body dispersion. Note that when optimized together with the other variables, s systematically settled on values near zero; hence, we have constrained s =0 throughout, leaving essentially four dispersion parameters. D4 has thus slightly improved WTMAD2 for SCS-dRPA75 from 2.89 (using D3BJ) to 2.83 kcal/mol. For optSCS-dRPA75, however, it dropped from 2.76 to 2.70 kcal/mol (see table 1). Among all 55 subsets, BSR36, MCONF, and to some extent WATER27 and PNICO23 benefitted by considering D4. Figure 1: Breakdown of total WTMAD2 into five top-level subsets for the dRPA based dual hybrids (left) and PT2 based vs dRPA based DSD double hybrids (right). (
THERMO=Small Molecule Thermochemistry; BARRIER=barrier heights; LARGE=reaction energies for large systems; CONF=conformer/intramolecular interactions; and INTER=intermolecular interactions)
Thus far, we have only considered dRPA correlation for the nonlocal correlation part of the dual hybrids. Can further improvement be achieved by also mixing some semilocal correlation component into the final energy (i.e., by transforming Kállay’s dual hybrid into the true DHDF form)? By doing so, we obtained the DSD-PBEdRPA -D3BJ functional, for which WTMAD2 is reduced by an additional 0.38 kcal/mol (see Table 1) at the expense of introducing one additional parameter (c C,DFT ). The intermolecular interactions subset is the only one that does not show a net improvement. The individual datasets that do benefit most are SIE4x4, AMINO20X4, ISOL24, PCONF21, BH76 and PNICO23. (For S66 and BSR36, performance deteriorates.) Indeed, this DSD-PBEdRPA -D3BJ (WTMAD2=2.36 kcal/mol) compares favorably to its GLPT2-based counterpart, revDSD-PBE-D3BJ (WTMAD2=2.67 kcal/mol): a detailed inspection suggests significant improvements for BUT14DIOL, AMINO20x4, TAUT15, HAL59, G21EA and BHPERI, and degradations for SIE4x4 and RG18. If we additionally relax a from its fixed value (while keeping a =s =0 fixed) WTMAD2 drops slightly further to 2.33 kcal/mol. Supplanting D3BJ with the D4 correction leads to a further drop in WTMAD2 to 2.32 kcal/mol — slightly better than its PT2 based counterpart revDSD-PBEPBE-D4 (WTMAD2=2.39 kcal/mol). Comparing these two for the five top-level subsets, we found that the dRPA-based double hybrid performs worse for the intermolecular interaction (the lion’s share of that due to RG18), comparably for conformer energies, and better for the remaining THERMO BARRIERSLARGECONFINTERMOLoptSCS-dRPA75-D3BJ optSCS-dRPA75SCS-dRPA75 SCS-dRPA75-D3BJ
THERMO BARRIERSLARGECONFINTERMOL revDSD-PBEPBE-D4 DSD-PBEdRPA75-D4revDSD-PBEP86-D4 DSD-PBEP86dRPA75-D4 three (see Figure 1), despite an exception of SIE4x4 due to increased self-interaction error. TAUT15 and G21EA are the two subsets which benefit the most, whereas the two subsets that deteriorate most are SIE4x4 and RG18. Table 1: Total WTMAD2(kcal/mol) and final parameters for dRPA based dual hybrids and their PT2 based counterparts. (constant parameters are in square brackets)
Functionals WTMAD2 (kcal/mol) c X,HF c X,DFT c C,DFT c O-s c S-S s s c ATM a a SCS-dRPA75 4.79 0.75 0.25 N/A 1.5 0.5 — — — — — optSCS-dRPA75 4.71 0.75 0.25 N/A 1.35 0.65 — — — — — SCS-dRPA75-D3BJ 2.89 0.75 0.25 N/A 1.50 0.50 0.2528 [0] N/A [0] 4.505 optSCS-dRPA75-D3BJ 2.76 0.75 0.25 N/A 1.3111 0.6889 0.2546 [0] N/A [0] 4.505 DSD-PBEdRPA -D3BJ 2.38 0.75 0.25 0.1151 1.2072 0.5250 0.3223 [0] N/A [0] 4.505 DSD-PBEP86dRPA -D3BJ 2.36 0.75 0.25 0.1092 1.1936 0.5268 0.3012 [0] N/A [0] 4.505 SCS-dRPA75-D4 2.83 0.75 0.25 N/A 1.50 0.50 0.3692 [0] 0.6180 -0.014 5.388 optSCS-dRPA75-D4 2.70 0.75 0.25 N/A 1.31 0.69 0.3376 [0] 0.4276 -0.049 5.198 DSD-PBEP86dRPA -D4 2.35 0.75 0.25 0.1219 1.1890 0.5281 0.3818 [0] 0.4571 -0.251 6.772 DSD-PBEdRPA -D4 2.32 0.75 0.25 0.1339 1.1967 0.5371 0.4257 [0] 0.6342 -0.145 6.398 The poor performance of DSD-PBEdRPA -D4 for SIE4x4 can be mitigated by applying the constraints c s-s =0 and c o-s =2: MAD for SIE4x4 drops from 9.0 to 4.7 kcal/mol, at the expense of spoiling thermochemical performance. In a previous study, we found that including subvalence electron correlation in the GLPT2 step marginally improved WTMAD2 further. This is not the case here: in fact, correlating subvalence electrons with the given basis sets (which do not contain core-valence correlation functions) actually does more harm than good. Therefore, we have not pursued this avenue further. (For a detailed discussion and review on basis set convergence for core-valence correlation energies, see Ref. ) Figure 2: Trend of WTMAD2 and top five sub-categories with respect to the fraction of HF exchange (c
X,HF ) in DSD-PBEdRPAn-D4(left) and DSD-PBEP86dRPAn-D4(right)
Thus far, we have kept c
X,HF fixed at 0.75. What if we include it too in the optimization process? For each value of c
X,HF , a complete evaluation of the entire GMTKN55 dataset is required. We performed such evaluations for five fixed c
X,HF points (c
X,HF =0.0, 0.25, 0.50, 0.75 and 0.90), where the same fraction of HF-exchange was used for both the orbital generation and the final energy calculation steps. Interpolation to the aforementioned data points ⎼ suggests a minimum in WTMAD2 near c X,HF =0.68 (see Figure S1 in ESI); however, upon actual GMTKN55 evaluation at that point, we found that the corresponding WTMAD2 value (2.34 kcal/mol) is very close to the minimum WTMAD2 calculated, 2.32 kcal/mol for c
X,HF =0.75. It thus appears that the WTMAD2 hypersurface in that region is rather flat with respect to variations in c
X,HF . Performance of the barrier heights subset deteriorates sharply beyond c
X,HF =0.75; for all other subsets, however, trends are not as straightforward. Error statistics for conformer energies remain more-or-less unchanged beyond 50% HF exchange. For c
X,HF < 0.5, a high WTMAD2 value is obtained due to poor performance for small-molecule thermochemistry (see left side of Figure 2). For each c
X,HF , the optimized parameters, the WTMAD2, and its breakdown into five top-level subset components can be found in Table S3 in the ESI. We also noticed that, with increasing %HF for our functionals, the fraction of DFT correlation in the final energy expression decreases almost linearly and approaches zero near c
X,HF =0.85. For the GLPT2-based double hybrids, we found that in both the original and revised parametrizations, the P86c semilocal correlation functional yielded superior performance to PBEc (and indeed all other options considered), while we earlier found that pretty much any good semilocal exchange functional will perform equally well. Presently, however, we found that DSD-PBEP86dRPA n alternatives yield only negligible improvements over their DSD-PBEP86dRPA n counterparts — presumably because the coefficient for the semilocal correlation is so much smaller here. That being said, our own DSD-PBEdRPA75-D4 and DSD-PBEP86dRPA75-D4 are still inferior to Mardirossian and Head-Gordon’s combinatorially-optimized range separated double hybrid, ωB97M(2) (WTMAD2=2.13 kcal/mol) (see table S2 in ESI). A.II. “External” benchmarks:
Next, we tested our new dRPA based double hybrids against two separate datasets very different from GMTKN55: the metal-organic barrier heights (MOBH35) database by Iron and Janes (see also erratum ) and the polypyrrols (extended porphyrins) dataset POLYPYR21. Both datasets are known to exhibit moderately strong static correlation (a.k.a., near-degeneracy correlations) effects. a) MOBH35:
This database comprises 35 reactions ranging from σ-bond metathesis over oxidative addition to ligand dissociations. We extracted the reported ‘best reference energies’ from the erratum to the original Ref. . The def2-QZVPP basis set was used for all of our calculations reported here. Note that these are all closed-shell systems, hence dRPA75, SCS-dRPA75 and optSCS-dRPA75 are equivalent for this problem. Unless semilocal correlation is introduced into the final energy expression, adding a D3BJ or D4 dispersion correction appears to do more harm than good. However, if the association reactions 17-20 are removed from the statistics, the difference goes away — strongly pointing toward basis set superposition error as the culprit. (Omitting dispersion corrections would lead to an error cancellation. ) Among all the functionals tested, DSD-PBEP86dRPA74-D4 and DSD-PBEdRPA74-D4 are the two best performers, with MAD=0.9 and 1.0 kcal/mol respectively. Both with D3BJ and D4 correction, DSD-pBEdRPA75 and DSD-PBEP86dRPA75 are better performers compared to their GLPT2-based revDSD counterparts (the violet columns in Figure 3). Semidalas et al. have recently investigated MOBH35 using a variety of diagnostics for static correlation, as well as recalculated some of the reference energies using canonical CCSD(T) rather than the DLPNO-CCSD(T) approximation. They found that severe type A static correlation in all three structures for reaction 9 (but especially the product) led to a catastrophic breakdown of DLPNO-CCSD(T), to the extent that it can legitimately be asked if even canonical CCSD(T) is adequate. So, omitting that particular reaction and recalculating MADs using the remaining 34 reactions (the orange bars in Figure 2) causes all MADs for the revDSD double hybrids to drop significantly. In contrast, performances for dRPA based double hybrids remains more or less unchanged. Here too, DSD-PBEdRPA-D4 and DSD-PBEP86dRPA75-D4 are the two best performers. If, in addition to reaction 9, we also leave out the bimolecular reactions 17-20 and calculate mean absolute deviations (MADs) for the remaining 30 reactions (the green bars in Figure 3), the MAD values are seen to drop across the board. However, unlike the full MOBH35, here all the dual hybrids perform similarly, whether we include any dispersion correction or not. Same is true for all the double hybrids. From Figure 3, it is clear that, for DSD-PBEP86dRPA75-D4 and DSD-PBEdRPA75-D4, the MAD values drop slightly compared to the MADs calculated against the original MOBH35.
Figure 3: MAD (kcal/mol) statistics for the complete and two modified versions of MOBH35 b) POLYPYR21:
This data set contains 21 structures with Hückel, Möbius, and figure-eight topologies for representative [4n] π-electron expanded porphyrins, as well as the various transition states between them.
Among these 21 unique structures, Möbius structures and transition states resembling them, exhibit pronounced multireference character (for more details see Ref ). We have used def2-TZVP basis set throughout; CCSD(T)/CBS reference energies have been extracted from Ref. . S C S - d R P A t - S C S - d R P A S C S - d R P A - D B J S C S - d R P A - D t S C S - d R P A - D B J op t S C S - d R P A - D D S D - P B E d R P A - D B J D S D - P B E d R P A - D D S D - P B E P R P A - D B J D S D - P B E P R P A - D r e v D S D - P B E P - D B J r e v D S D - P B E P - D r e v D S D - P B E - D B J r e v D S D - P B E - D MAD (kcal/mol) for full MOBH35 and its two modified versions
Full MOBH35 without reaction 9 without 9 & 17-20 As these are all closed-shell systems, changing the OS-SS balance has no effect on the RMSD value, hence dRPA75, SCS-dRPA75, and optSCS-dRPA75 offer identical error statistics. Adding either D3BJ or D4 dispersion correction on top of that does more harm than good. Next, similar to what we found for GMTKN55, adding semilocal correlation (i.e., DSD-XCdRPAn-Disp), helps quite a bit. Considering the D3BJ dispersion correction, both the dRPA based double hybrids outperform their PT2 based revDSD counterparts. On the contrary, with D4 dispersion correction, revDSD-D4 functionals have a slight edge over the dRPA based double hybrids. As expected, the performance variation mainly come from the Möbius structures, whereas RMSD statistics for the Hückel and twisted-Hückel topologies stay more or less the same for all DSD-DHs (3 rd and 4 th columns of Table 2). Table 2: Mean Absolute Deviations (kcal/mol) and Root Mean Squared Deviations (kcal/mol) for new dRPA based DSD-DHs and original PT2 based revDSD functionals on the POLYPYR21 dataset
Functionals MAD (kcal/mol) RMSD (kcal/mol) Total Möbius structures Hückel & figure-eight structures
SCS-dRPA75 2.82 4.10 6.94 0.98 optSCS-dRPA75 2.82 4.10 6.94 0.98 SCS-dRPA75-D3BJ 2.88 4.18 7.09 0.96 optSCS-dRPA75-D3BJ 2.88 4.18 7.09 0.96 DSD-PBEdRPA75-D3BJ 2.06 2.92 4.88 0.83 DSD-PBEP86dRPA75-D3BJ 1.96 2.78 4.64 0.79 revDSD-PBEPBE-D3BJ 2.14 3.07 5.16 0.86 revDSD-PBEP86-D3BJ 2.07 2.94 4.94 0.80 SCS-dRPA75-D4 2.87 4.20 7.11 0.93 optSCS-dRPA75-D4 2.89 4.23 7.18 0.92 DSD-PBEdRPA75-D4 2.05 2.95 4.90 0.83 DSD-PBEP86dRPA75-D4 1.95 2.80 4.64 0.81 revDSD-PBEP86-D4 1.93 2.87 4.78 0.82 revDSD-PBEPBE-D4 1.90 2.81 4.66 0.84
B. DSD3 and ωDSD3 family functionals: Introducing scaled third-order correlation
As mentioned in the introduction, Radom and coworkers tried to improve on double hybrids by introducing MP3, MP4, and CCSD correlation. Unfortunately, using fairly modest basis sets and fitting correlation energy coefficients to the smallish and chemically one G2/97 database of atomization energies, they failed to discern any significant improvement beyond regular double hybrids. From our previous experience, we know that the use of small, idiosyncratic training sets for empirical functionals may lead to its poor performance. So, here we are instead employing GMTKN55, which is more than an order of magnitude larger and covers many other types of energetic properties. All the “ micro-iteration ” (i.e., linear) parameters were refitted (i.e., c DFT , c , c , c ; s for D3BJ subject to s =a =0, a = 5.5 fixed; s , a , and a for D4 subject to s =0, c ATM =1). Two functionals, DSD-PBEP86 and ωDSD -PBEP86 (ω=0.16) are considered as the representatives of global and range-separated DHs for the present study. It was previously found, in a cWFT context, that the MP3 term does not change greatly beyond the def2-TZVPP basis set, hence we restrict ourselves to it in an attempt to control computational cost. Total WTMAD2 and optimized parameters for all the DSD3, ωDSD3 and corresponding revDSD functionals are presented in Table 3. Analyzing the results, we can conclude the following. • Considering PT2 and MP3 correlation together and scaling MP3 term by an extra parameter(c ) does improve performance for both the DSD3 and ωDSD3 functionals, at the expense of the extra computational cost entailed by the MP3/def-TZVPP calculations. • For DSD3 with D4 dispersion correction, the improvement is 0.17 kcal/mol compared to revDSD-PBEP86-D4. Among all 55 individual subsets, the RSE43 subset benefited the most and performance for BHPERI and TAUT15 also improved to some extent. However, for wDSD3 the performance gain is more pronounced, 0.29 kcal/mol (see Table 3). Inspection of all 55 individual subsets reveals that the RSE43 and TAUT15 subsets showed significant gain in accuracy and AMINO20x4, RG18, ADIM6 and S66 only marginally improved. • For neither DSD3 nor ωDSD3 can the dispersion correction term be neglected, even if we consider correlation terms beyond PT2.
Table 3: WTMAD2(kcal/mol) and all the optimized parameters for the global and range-separated DHs with PT2c(revDSD and ωDSD) and PT2c+MP3c (DSD3 and ωDSD3 functionals). [a] (Parameters which are kept constant in the optimization cycle are in third bracket) Functionals
WTMAD2 ω c
X,HF c DFT c c c s s c ATM a a DSD3-PBEP86-D4 2.03 N/A 0.69 0.3784 0.6136 0.2069 0.2443 0.6301 [0] 1 0.3201 4.76901 DSD3-PBEP86-D3BJ 2.12 N/A 0.69 0.3782 0.6085 0.2174 0.2525 0.4582 [0] N/A [0] [5.5] revDSD-PBEP86-D4 2.20 N/A 0.69 0.4210 0.5930 0.0608 [0] 0.5884 [0] 1 0.3710 4.2014 revDSD-PBEP86-D3BJ 2.33 N/A 0.69 0.4316 0.5746 0.0852 [0] 0.4295 [0] N/A [0] [5.5] DSD3-PBEP86 3.34 N/A 0.69 0.3726 0.5402 0.5311 0.2410 N/A N/A N/A N/A N/A ωDSD3-PBEP86-D4 1.76 0.16 0.69 0.3048 0.6717 0.3526 0.3057 0.5299 [0] 1 0.0659 6.0732 ωDSD3-PBEP86-D3BJ 1.78 0.16 0.69 0.3063 0.6693 0.3363 0.2842 0.3871 [0] N/A [0] [5.5] ωDSD-PBEP86-D4 2.05 0.16 0.69 0.3595 0.6610 0.1228 [0] 0.5080 [0] 1 0.1545 5.1749 ωDSD-PBEP86-D3BJ 2.08 0.16 0.69 0.3673 0.6441 0.1490 [0] 0.3870 [0] N/A [0] [5.5] ωDSD3-PBEP86 2.86 0.16 0.69 0.2749 0.6417 0.6648 0.3620 N/A N/A N/A N/A N/A [a]
50 systems out of 1499 are omitted: UPU23, C60, ten largest ISOL24, three INV24 and one IDISP
N/A= not applicable
Using the same GMTKN55 test suite, Semidalas and Martin achieved WTMAD2=1.93 kcal/mol for their G4(MP3|KS)-D-v5 cWFT method, which employs the following energy expression, E = E ()/+,- + c . E /01|3-,5-/67819:;<00= + c E /01|3-,--/67819:;<00= + c E [/0 + s A [E(D3BJ)] It differs from the present work in that the semilocal starting energy is 100% Hartree-Fock without semilocal correlation, rather than a hybrid GGA as here. Clearly the latter offers an advantage. Although both the G4(MP3|KS)-D-v5 and DSD3 method use spin component scaled PT2 correlation and scaled MP3 correlation, the key differences between these two are: no DFT correlation component is present in the final G4(MP3|KS)-D-v5 energy expression, while DSD3 has both scaled HF and DFT exchange, unlike 100% E HF for G4(MP3|KS)-D-v5. Unlike presently, Semidalas and Martin reported that the coefficient for dispersion term is very small and can be neglected without compromising any significant accuracy (G4(MP3|KS)-D-v6). With a D3BJ dispersion correction ωDSD3-PBEP86 surpasses the accuracy of G4(MP3|KS)-D-v5 method by 0.15 kcal/mol — which can be slightly improved further by considering D4. However, it should be pointed out that DSD3-PBEP86-D3BJ has six adjustable parameters (compared to only four for G4(MP3|KS)-D-v5, and three for G4(MP3|KS)-D-v6), while ωDSD3-PBEP86-D4 has as many as nine. IV. Conclusions:
Analyzing the results presented above for the dRPA based double hybrids; original and reparametrized form of SCS-dRPA75 dual hybrid; and DSD3 and wDSD3 type double hybrid functionals (all evaluated against GMTN55) we are able to state the following conclusions. Concerning the first research question: a)
Following the recommendation of Martin and coworkers, adding a dispersion correction on top of the original SCS-dRPA75 significantly improved the WTMAD2 statistics, D4 slightly more so than D3BJ. b) By additionally admitting a semilocal correlation component into the final energy expression, we were able to obtain DSD-PBEdRPA -D3BJ and DSD-PBEdRPA75-D4 functionals that actually slightly outperform their PT2-based counterparts, revDSD-PBE-D3BJ and revDSD-PBE-D4. c) We considered different percentages of HF exchange, but found the WTMAD2 curve flat enough in the relevant region, for both the DSD-PBEdRPAn-D4 and DSD-PBEP86dRPAn-D4 variants, that c
X,HF =0.75 is a reasonable choice. d)
Judging from SIE4x4 subset, we found that the refitted SS-OS balance in dRPAc apparently causes significant self-interaction error. This issue can be eliminated by applying the constraint, c s-s =0, c o-s =2 — at the expense of spoiling small-molecule thermochemistry. Concerning the second research question, we considered a different post-MP2 alternative, namely the addition of a scaled MP3 correlation term (evaluated in a smaller basis set, and using HF orbitals, for technical reasons). Particularly when using range-separated hybrid GGA orbitals, we achieved a significant improvement in WTMAD2. A possible way to make such calculations amenable to larger molecules would be using density-fitted MP3 (e.g.,Ref. ) instead of conventional MP3. Head-Gordon and coworkers have very recently shown that the use of DFT orbitals for regular MP3 level calculation results significantly improved performance for thermochemistry, barrier heights, noncovalent interactions, and dipole moments compared to the conventional HF-based MP3. Unlike what Semidalas and Martin observed for their G4(MP3|KS)-D-v5 method, we have found dispersion correction term cannot be neglected for DSD3 or wDSD3 functionals. Acknowledgments:
We would like to acknowledge helpful discussions with Dr. Mark A. Iron (WIS), and Prof. Mercedes Alonso Giner (Free University of Brussels). Mr. Minsik Cho (Brown University) is thanked for parallelizing the parameter optimization process while he was a Kupcynet-Getz International Summer School intern in our group. We also thank Mr. Nitai Sylvetsky for critical comments on the draft manuscript. GS acknowledges a doctoral fellowship from the Feinberg Graduate School (WIS).
Funding Sources: This research was funded by the Israel Science Foundation (grant 1969/20) and by the Minerva Foundation (grant 20/05) . The work of E.S. on this scientific paper was supported by the Onassis Foundation – Scholarship ID: F ZP 052-1/2019-2020.
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. Breakdown of total WTMAD2 into five major subcategories for the original and refitted SCS-DRPA75 variants, DSD-PBEdRPA , DSD-PBEP86dRPA and corresponding revDSD functionals with dispersion correction. Total WTMAD2, optimized parameters and breakdown of total WTMAD2 into five top-level subsets for DSD-PBEdRPAn and DSD-PBEP86dRPAn. Figure showing mean absolute deviation (MAD) for only reaction 17-20 of MOBH35. References: (1) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects.
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Table of contents: S C S - d R P A o p t S C S - d R P A o p t S C S - d R P A - D D S D - P B E d R P A - D D S D - P B E P d R P A - D r e v D S D - P B E - D r e v D S D - P B E P - D ThermochemistryBarrier heightsReaction energies of large speciesConformers/IntramolecularIntermolecular interactions Electronic Supporting Information (ESI)
Exploring Avenues Beyond Revised DSD Functionals: II. Random-Phase Approximation and scaled MP3 corrections
Golokesh Santra, † Emmanouil Semidalas, † and Jan M.L. Martin *,† † Department of Organic Chemistry, Weizmann Institute of Science, 7610001 Reḥovot, Israel. Email: [email protected] 2
Abbreviations and descriptions used for the GMTKN55 database:
The abbreviations and a concise description of all fifty-five subsets of the GMTKN55 database proposed by Goerigk, Grimme and coworkers are listed in Table S1 below. For a more detailed description of all of them and individual reactions refer to refs. . Table S1: Abbreviations used and their descriptions for the 55 datasets in GMTKN55. 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 AlX 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 H O, NH , 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 H fragmentations of NH3/BH3 systems; H activation reactions with PH /BH 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 H O, NH , 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 (H O) n , H+(H O) n and OH−(H O) n WCPT18 Proton-transfer barriers in uncatalysed and water-catalysed reactions
YBDE18 Bond-dissociation energies in ylides
Table S2: WTMAD2 (kcal/mol) and its breakdown into five major subcategories for original and refitted SCS-DRPA75, DSD-PBEdRPA75, DSD-PBEP86dRPA75 and corresponding revDSD functionals with D3BJ and D4 dispersion correction . Functionals WTMAD2 (kcal/mol) THERMO BARRIERS LARGE CONF INTERMOL dRPA75 5.072 1.221 0.295 0.931 1.074 1.552 SCS-dRPA75 4.791 0.939 0.310 0.917 1.074 1.552 optSCS-dRPA75 4.712 0.867 0.300 0.919 1.074 1.552 SCS-dRPA75-D3BJ 2.894 0.863 0.299 0.610 0.550 0.572 SCS-dRPA75-D4 2.826 0.848 0.310 0.556 0.533 0.579 optSCS-dRPA75-D3BJ 2.758 0.739 0.287 0.608 0.553 0.571 optSCS-dRPA75-D4 2.700 0.724 0.312 0.534 0.523 0.608 DSD-PBEdRPA -D3BJ 2.377 0.614 0.228 0.513 0.432 0.590 DSD-PBEdRPA -D4 2.321 0.596 0.236 0.477 0.434 0.577 DSD-PBEP86dRPA -D3BJ 2.359 0.613 0.216 0.517 0.433 0.579 DSD-PBEP86dRPA -D4 2.349 0.597 0.226 0.466 0.457 0.603 revDSD-PBE-D3BJ 2.668 0.640 0.299 0.532 0.610 0.586 revDSD-PBE-D4 2.393 0.643 0.290 0.555 0.413 0.491 revDSD-PBEP86-D3BJ 2.367 0.521 0.268 0.552 0.457 0.569 revDSD-PBEP86-D4 2.248 0.545 0.260 0.573 0.406 0.463 ωB97M(2) 2.131 0.430 0.214 0.418 0.577 0.492 Table S3: WTMAD2(kcal/mol), optimized parameters and division of total WTMAD2 into five top-level subsets for DSD-PBEdRPAn and DSD-PBEP86dRPAn with both D3BJ and D4 dispersion correction.
Functional
Dispersion
WTMAD2 (kcal/mol) Parameters Five top-level subsets c
X,HF c X,DFT c C,DFT co-s cs-s s6 s8 cATM a1 a2 THERMO BARRIERS LARGE CONF INTERMOL
DSD-PBEdRPAn
D3BJ 8.81 0.00 1.00 0.5877 0.2078 -0.6099 1.4768 0 N/A 0 4.505 2.712 1.214 1.729 1.423 1.733 D4 7.22 0.00 1.00 0.7861 0.0996 -0.6646 1.5924 0 -0.3616 0.213 4.390 2.643 0.917 1.507 0.969 1.187 D3BJ 6.09 0.25 0.75 0.5128 0.1971 0.0011 1.0310 0 N/A 0 4.505 1.679 1.038 1.199 0.979 1.195 D4 4.98 0.25 0.75 0.6942 -0.0781 -0.0098 1.4014 0 0.3157 0.206 4.731 1.776 0.621 1.045 0.725 0.812 D3BJ 3.37 0.50 0.50 0.3332 0.9030 -0.2004 0.8127 0 N/A 0 4.505 0.927 0.371 0.724 0.634 0.718 D4 2.85 0.50 0.50 0.4372 0.8435 -0.2262 1.0004 0 0.4909 0.114 5.061 0.775 0.315 0.633 0.518 0.607 D3BJ 2.41 0.68 0.32 0.1901 1.1052 0.3138 0.4563 0 N/A 0 4.505 0.675 0.240 0.517 0.428 0.547 D4 2.34 0.68 0.32 0.2293 1.0797 0.3264 0.6050 0 0.7034 0.028 5.633 0.632 0.231 0.494 0.443 0.540 D3BJ 2.38 0.75 0.25 0.1151 1.2072 0.5250 0.3223 0 N/A 0 4.505 0.614 0.228 0.513 0.432 0.590 D3BJ (a2 opt) 2.33 0.75 0.25 0.1273 1.1991 0.5368 0.3383 0 N/A 0 4.904 0.608 0.228 0.507 0.456 0.531 D4 2.32 0.75 0.25 0.1339 1.1967 0.5371 0.4257 0 0.6342 -0.145 6.398 0.596 0.236 0.477 0.434 0.577 D3BJ 3.75 0.90 0.10 -0.0524 1.4564 0.9315 0.0694 0 N/A 0 4.505 0.670 1.148 0.634 0.597 0.702 D4 3.57 0.90 0.10 -0.0790 1.4760 0.9732 0.1396 0 0.6544 -0.187 6.188 0.666 1.123 0.604 0.486 0.687
DSD-PBEP86dRPAn
D3BJ 8.55 0.00 1.00 0.5793 0.1823 -0.7157 1.4229 0 N/A 0 4.505 2.322 1.077 1.837 1.457 1.856 D4 7.07 0.00 1.00 0.7332 0.0865 -0.7483 1.2576 0 -0.8011 0.132 4.427 2.169 0.886 1.706 1.053 1.259 D3BJ 3.48 0.50 0.50 0.3326 0.8751 -0.2262 0.7488 0 N/A 0 4.505 0.908 0.379 0.806 0.615 0.768 D4 3.08 0.50 0.50 0.3990 0.8241 -0.2193 0.7759 0 -0.0008 -0.007 5.333 0.810 0.344 0.715 0.549 0.657 D3BJ 2.43 0.69 0.31 0.1826 1.0984 0.3239 0.4036 0 N/A 0 4.505 0.658 0.235 0.538 0.410 0.590 D4 2.40 0.69 0.31 0.2023 1.0799 0.3444 0.4709 0 0.3168 -0.144 6.194 0.630 0.230 0.492 0.460 0.590 D3BJ 2.36 0.73 0.27 0.1400 1.1525 0.4591 0.3321 0 N/A 0 4.505 0.625 0.219 0.519 0.417 0.578 D4 2.35 0.73 0.27 0.1482 1.1490 0.4785 0.4043 0 0.3983 -0.234 6.677 0.606 0.221 0.470 0.458 0.596 D3BJ 2.36 0.75 0.25 0.1092 1.1936 0.5268 0.3012 0 N/A 0 4.505 0.613 0.216 0.517 0.433 0.579 D3BJ (a2 opt) 2.35 0.75 0.25 0.1204 1.1978 0.5046 0.3156 0 N/A 0 4.752 0.611 0.222 0.512 0.439 0.562 D4 2.35 0.75 0.25 0.1219 1.1890 0.5281 0.3818 0 0.4571 -0.251 6.772 0.597 0.226 0.466 0.457 0.603 D3BJ 3.01 0.90 0.10 -0.0553 1.4644 0.9419 0.0744 0 N/A 0 4.505 0.641 0.430 0.621 0.594 0.727 D4 2.85 0.90 0.10 -0.0744 1.4857 0.9565 0.1606 0 0.5931 -0.258 6.465 0.627 0.427 0.584 0.493 0.715 Figure S1: Fitting of WTMAD2 vs cX,HF curve for DSD-PBEdRPAn-D4 and DSD-PBEP86dRPAn-D4.
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