Efficient yet Accurate Dispersion-Corrected Semilocal Exchange-Correlation Functionals For Non-Covalent Interactions
Abhilash Patra, Subrata Jana, Lucian A. Constantin, Prasanjit Samal
aa r X i v : . [ phy s i c s . a t m - c l u s ] A p r Efficient yet Accurate Dispersion-Corrected Semilocal Exchange-CorrelationFunctionals For Non-Covalent Interactions
Abhilash Patra, a) Subrata Jana, b) Lucian A. Constantin, c) and Prasanjit Samal d) School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050,India Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti,I-73010 Arnesano, Italy (Dated: 28 April 2020)
Due to several attractive features, the meta-generalized-gradient approximations (meta-GGAs) are consideredto be the most advanced and potentially accurate semilocal exchange-correlation functionals in the rungs ofthe Jacob’s ladder of Density Functional Theory. So far, several meta-GGA are proposed by fitting to the testsets or/and satisfying as many as known exact constraints. Although the density overlap is treated by modernmeta-GGA functionals efficiently, for non-covalent interactions, a long-range dispersion correction is essential.In this work, we assess the benchmark performance of different variants of the Tao-Mo semilocal functional(i.e. TM of Phys. Rev. Lett. , 073001 (2016) and revTM of J. Phys. Chem. A , 6356 (2019))with Grimme’s D3 correction for the several non-covalent interactions, including dispersion and hydrogenbonded systems. We consider the zero, Becke-Johnson(BJ), and optimized power (OP) damping functionswithin the D3 method, with both TM and revTM functionals. It is observed that the overall performanceof the functionals gradually improved from zero to BJ and to OP damping. However, the constructed “OP”corrected (rev)TM+D3(OP) functionals perform considerably better compared to other well-known dispersioncorrected functionals. Based on the accuracy of the proposed functionals, the future applicability of thesemethods is also discussed.
I. INTRODUCTION
Semilocal exchange-correlation (XC) density function-als are the most preferred choice of doing electronic struc-ture calculations within the Kohn-Sham (KS) DensityFunctional Theory (DFT) . Starting from the localdensity approximation (LDA) to the higher rungs ofthe Jacob’s ladder classification of XC functionals , thesemilocal approximations are characterized as the gen-eralized gradient approximations (GGAs) and meta-GGAs . Higher rungs than meta-GGA use non-localinformation from KS orbitals and eigenvalues, and arerecognized from the point of view of their sophistica-tion, as the so-called rung 3.5 , hybrids and hyper-GGA functionals , double hybrids and DFT coupled-cluster based methods , adiabatic-connection meth-ods and generalizations of the random phase approxima-tion (RPA) .Meta-GGA XC functionals improve the overall perfor-mance of GGAs, and the hybrid methods do the sameover their bare semilocal counterparts. But none of thesefunctionals able to incorporate the long-range correla-tion, which is essential for systems dominated by weakbonds. For the last couple of decades, the formulation ofmeta-GGA functionals has been made very physical in-sightful through the inclusion of short- and intermediate-range behavior of the weakly bonded systems . How- a) Electronic mail: [email protected] b) Electronic mail: [email protected], [email protected] c) Electronic mail: [email protected] d) Electronic mail: [email protected] ever, studies show that semilocal approximations donot incorporate short- and intermediate-range disper-sion . Designing density functionals, irrespective ofthe short- and intermediate-range dispersion or van derWaals (vdW) interactions as well as to retain their ac-curacy for the density overlap region, a long-range vdWcorrection is always necessary to describe the functionalperformance correctly for the binding energies of weaklybonded systems .The long-range vdW interaction can be captured via E vdW = − X i
25, and α ≈ . . This is known as “optimize-power” damping with the following analytic form f D OP ) dmp,n ( r ij ) = r β n ij r β n ij + ( α r ,ij + α ) β n . (6)The similarities between D3(BJ) and D3(OP) are notice-able. Most importantly, the parameter β controls therate of dispersion interaction. Here, β = β + 2, andthe same dispersion coefficients and vdW radii are usedin D3(OP) damping. Also, similar to the D3(BJ), theparameters α and α control the distance where thedamping function corresponding to the dispersion cor-rection will be switched on or off. It was also shownthat the D3(OP) improves the descriptions of weaklybonded molecular systems when coupled with any den-sity functionals .These dispersion correction methods are importantfor our present study. There are several studies onthe performance of the density functionals with disper-sion corrections . Several recent, accu-rate meta-GGA density functionals suitable for quantumchemical calculations are also proposed and tested for abroad range of systems . However, these function-als are not benchmarked for a wide range of molecularproperties. The motivation of the present study followsfrom the very accurate performance of the different vari-ants of TM semilocal functionals (TM and revTM )for quantum chemistry. Here, we combine the D3(0),D3(BJ), and D3(OP) with the TM and revTM function-als to assess their performance for non-covalent interac-tion test sets and H bonded water systems. We observethat the combination of the TM and revTM with D3(OP)gives improvements over various other combinations pro-posed so far. Most importantly, the TM+D3(OP) andrevTM+D3(OP) do not deteriorate much the H-bond en-ergies compared to their base functional accuracy. Topresent the functionals performance, we arrange our pa-per as follows: In the following, we briefly review theTM and revTM meta-GGA functionals, and we constructtheir dispersion corrected terms. Next, we test the pro-posed functionals concerning different non-covalent inter-action test sets. Lastly, we conclude and summarize ourresults based on insightful analysis. s F x ( s , α = ) TMrevTM
R(Å) -1001020304050 B i nd i ng E n e r gy ( m e V ) TMrevTMCCSD(T)
FIG. 1. (Upper panel) Exchange enhancement factors of TMand revTM functionals. (Lower panel) The binding energyof Ar dimer obtained from TM and revTM functionals. ForXC integrals the 99 points radial grid and 590 points angularLebedev grid are used II. THEORYA. Review of TM and revTM functionals
The construction of TM and revTM functionals havealready been reviewed in Ref. . Here, we only focuson the key differences between these two functionals, asoutlined below.(i) Firstly, the significant difference between the TMand revTM exchange functionals comes from model-ing the reduced Laplacian ˜ q in the slowly varying (sc)fourth-order gradient approximation (GE4) of the ex-change enhancement factor. Thus, the revTM uses˜ q b = α − bα ( α − / + p (with b = 0 .
40) instead of˜ q = ( α −
1) + p as found in the TM functional. As aresult, in the overlapping closed shells ( α >> s ≈ F T Mx and F revT Mx differ from each other drastically. Thisis shown in Fig. 1, where, in the upper panel, we haveplotted the exchange enhancement factors of both func-tionals for α = 10. Note that this modification affectsthe lattice constants of the alkali metals, ionic solids andlayered materials .The TM functional is very accurate for several solid- state and molecular properties . Specially, the bestperformance of the TM functional is evident from thelattice constants of the ionic solids and hydrogen-bonded complexes .In refs. it has been ar-gued that the TM exchange enhancement factor sharesslightly oscillatory behavior to some extent as it is shownin Voorhis-Scuseria (VSXC) and M06-L function-als . On the other-hand the revTM exchange enhance-ment factor behaves differently in the region α >> s ≈ /r ij form or the long-range interaction orcorrect dispersion physics .To further elaborate on this point, and distinguish thedifferent behavior of the TM and revTM for weekly inter-acting systems, we also plot the binding energy curve ofAr dimer for both the functionals (shown in the lowerpanel of Fig. 1). From the figure, we observe that thebare revTM functional unbound the Ar dimer because˜ q b < ˜ q in the middle of the bonding region. The behav-ior of the α and ˜ q b can be found in Fig. 4 and Fig. 5of the ref. .The difference in capturing the interactionby both the functionals are important for non-covalentbonded molecules. Note that bare TM functional is al-ready quite good without including any vdW correction.The behavior of the TM and revTM functionals canalso be understood from the recent investigation of thefunctionals performance for the water clusters . Inref. it is shown that both the TM and revTM predictscorrectly the ordering stability of the water hexamers,whereas, the revTM is quite good for overall performanceof water and ice structures.(ii) Secondly, the other important difference is aris-ing due to the correlation content of both the function-als. In revTM, the linear response parameter β has beengeneralized to the form of the exact, density-dependentsecond-order gradient expansion (GE2) parameter pro-posed in the revTPSS meta-GGA correlation energyfunctional. We recall that TM correlation functional usesthe high-density GE2 parameter (a.i. β = 0 . . The change in correlation energy functional im-proves the jellium surface XC energies , which are rele-vant for the surface energies of simple metals. Note thatthe change in the correlation does not affect the non-covalent interaction systems. B. Dispersion corrected TM and revTM functionals
To construct the dispersion corrected functionals, wecombine D3(0), D3(BJ), and D3(OP) dispersion correc-tions with the TM and revTM functionals. To determinethe dispersion parameters associated with the function-als, one needs to fit the functional with appropriate non- ij (Å)00.20.40.60.81 f d m p D3 (0)D3 (BJ)D3 (OP) -1.2e-05-8e-06-4e-060 E d i s p E d i s p (a)(b) FIG. 2. The lower panel shows the behavior of damping func-tions with respect to inter-nuclear distance r ij . The upperpanel shows the contribution to the dispersion energy of Ne for three types of damping functions. For all the figures, wehave used r ,ij = 3 . , C =6.35. The inset ofthe upper panel is for D3(0), as the scale of energy is different.TABLE I. Parameters used in the calculations.Parameters TM revTM D3(0)(Zero damping) s r, D3(BJ)(Becke and Johnson damping) s D3(OP)(Optimized Power damping) s α α β covalent interaction test set. The most preferred choice isto use the S22 test set of Jurecka et. al. . This test setis chosen wisely as it consists of hydrogen bonded com-plexes, dispersion bonded complexes, and mixed com-plexes (having both interaction types). In our calcula-tions, we consider the new benchmark CCSD(T) valuesof Marshall et. al. , along with the geometries availablefrom GMTKN55 test set . The optimized parameters ofthe respective functionals are summarized in Table I. Forall the functionals, we consider the standard s = 1, andwe obtain s = 0, as any other value of s increases themean absolute error (MAE) of the S22 test set. Also,the SCAN+D3(0) has been also proposed by considering s = 0. Though the revTM functional proposed fromthe TM functional, we do not observe any improvementin the error statistics by incorporating the s -term in TABLE II. Tabulated are the test sets used in our presentcalculations. All geometries are taken from Ref. , with ex-ception of the L7, DSCONF, and MG8 test sets, where weused the geometries from the respective reference articles. Test Set DescriptionS22 22 non-covalent interactive complexes
L7 7 large molecular binding energies
S66 66 non-covalent interactive complexes
ADIM6 6 n-alkane dimers interaction energies
AHB21 21 neutral anion dimers interaction energies
CARBHB12 12 hydrogen-bonded complexes CHB6 6 cation-neutral dimers interaction energies
HAL59 59 halogenated dimers interaction energies
HEAVY28 28 heavy element hydrides interaction energies
IL16 6 anion-cation dimers interaction energies
PNICO23 23 pnicogen-containing dimers interaction energies
RG18 18 rare-gas complexes interaction energies
ACONF alkane conformers interaction energies
Amino20 × BUT14DIOL 14 butane-1,4-diol conformers interaction energies
ICONF inorganic systems IDISP Intramolecular dispersion interactions
MCONF Melatonin conformers interaction energies
PCONF21 tri and tetrapeptide conformers
SCONF sugar conformers
UPU23 RNA-backbone conformers
WATER27 27 charged/neutral water clusters binding energies
DSCONF 30 conformers of Lactose, Maltose, and Sucrose
MG8 64 small representative thermochemical test set the revTM+D3 functionals. Therefore, we keep only the s -term, and the dispersion parameters are fixed by mini-mizing the MAE of the S22 test set. Note that the revTMfunctional demands more dispersion interaction than theTM functional due to its more unbound nature for dis-persion bonded systems.To understand the role played by and impact of differ-ent parameters on the damping function as well as en-ergy component, in Fig. 2, we plot the damping function(lower panel) and E vdW of Eq. (1) (upper panel), in caseof the Ne dimer for which the C coefficient is known.By construction, the DFT+D3(BJ) damping approachshows constant value at small inter-atomic separation,while the D3(OP) works within D3(0) and D3(BJ).It is noteworthy to mention that in this work the 3-body term is used with all the D3 schemes, being , E D − body = − triples X A,B,C C ABC (1 + 3 cos φ A cos φ B cos φ C ) r ABC × f d ( r ABC ) , (7)where the damping function f d is related to the D3 dis-persion interaction coupled with the correlation part ofthe semilocal density functional. Here φ A , φ B and φ C are the angles formed of by the three atoms A , B and C , and r ABC is the geometric mean distance. We recallthat the 3-body term represents only a small fraction ofthe total dispersion interaction, being analyzed in severalworks . TABLE III. Interaction energies (in kcal/mol) of S22 data set. The mean error (ME) and mean absolute error (MAE) are alsoreported. The best values are marked with bold style.
S22 complex CCSD(T) TM+D3(0) TM+D3(BJ) TM+D3(OP) revTM+D3(0) revTM+D3(BJ) revTM+D3(OP)
Hydrogen bonded complexes NH dimer ( C h ) 3.133 3.397 3.403 3.352 3.257 3.303 3.150H O dimer ( C s ) 4.989 5.360 5.361 5.317 5.373 5.394 5.247Formic acid dimer ( C h ) 18.753 18.771 18.852 18.717 19.033 19.258 18.801Formamide dimer ( C h ) 16.062 15.740 15.782 15.658 15.968 16.139 15.791Uracil dimer ( C h ) 20.641 19.708 19.677 19.623 20.068 20.211 19.8902-pyridone-2-aminopyridine ( C ) 16.934 16.608 16.583 16.521 17.134 17.250 16.922Adenine-thymine WC ( C ) 16.660 15.931 15.905 15.840 16.395 16.462 16.132ME − -0.23 -0.22 -0.30 − CH dimer (D d ) 0.527 0.578 0.568 0.517 0.635 0.470 0.479C H dimer (D d ) 1.472 1.633 1.691 1.599 1.418 1.453 1.269Benzene-CH (C ) 1.448 1.560 1.482 1.460 1.470 1.459 1.413Benzene dimer (C h ) 2.654 2.670 2.663 2.531 2.772 2.908 3.267Pyrazine dimer (C s ) 4.255 3.999 4.076 3.847 4.018 4.196 4.317Uracil dimer (C ) 9.805 10.028 10.028 9.730 9.756 9.704 9.700Indole-benzene (C ) 4.524 4.432 4.467 4.267 4.406 4.723 5.078Adenine-thymine (C ) 11.730 12.127 12.179 11.792 11.397 11.644 11.524ME − − Hydrogen + dispersion (mixed) bonded complexes C H -C H (C ν ) 1.496 1.486 1.486 1.458 1.598 1.588 1.537Benzene-H O (C s ) 3.275 3.886 3.863 3.754 3.874 3.781 3.646Benzene-NH (C s ) 2.312 2.600 2.543 2.480 2.564 2.498 2.427Benzene-HCN (C s ) 4.541 4.657 4.685 4.597 4.420 4.741 4.531Benzene dimer (C ν ) 2.717 2.684 2.614 2.623 2.609 2.711 2.673Indole-benzene (C s ) 5.627 5.626 5.566 5.552 5.595 5.661 5.521Phenol dimer (C ) 7.097 6.825 6.761 6.756 6.723 6.706 6.475ME − − − -0.01 -0.01 -0.12 -0.00 − III. RESULTS
All the calculations are done with the developer ver-sion of Q-CHEM simulation package . For XC integralsthe 99 points radial grid and 590 points angular Lebedevgrid are used. Note that the non-bonded systems bindingenergies are very sensitive on the choice of the grid. Thepresent choice of the grid is adequate and highly recom-mended for the complete energy convergence of the non-bonded systems . The test sets used in our calcula-tions and the corresponding basis sets are mentioned inTable II. All calculations are performed with def2-QZVPbasis set except the AHB21, IL16, WATER27, DSCONF,and MG8 test sets, where the calculations are performedwith the def2-QZVPD basis set. It is shown that the useof diffuse basis set drastically improves the results forthose test sets . A. S22 test set
To start with, we consider the S22 test set. As men-tioned before it contains important non-covalent inter- acting molecules, that are often used for the bench-mark calculations. The details of the different functionalperformance for the individual molecules are presentedin Table III. For reference values those obtained fromCCSD(T)/CBS calculations by Sherrill et. al. are con-sidered. Regarding the performance of individual disper-sion corrected functionals, we observe that all functionalsperform in a impressive way. Regarding the H-bondedmolecules, which consist of different complexes havingbiological interests, the NH and H O dimer energies areoverestimated by the -D3(0) and -D3(BJ) dispersion cor-rections, while for -D3(OP) the overestimation tendencyis less evident. For other H-bonded systems, we also ob-serve same tendency as -D3(OP), indicating its balancedperformance for H-bonded systems.In case of the dispersion bonded systems, we observe asystematic slight underestimation of -D3(OP) function-als compared to the -D3(0) and -D3(BJ) ones. Overallboth the -D3(OP) corrected functionals underestimatethe interaction energies.Next, for the mixed interaction, we observe underesti-mation or overestimation in the interaction energies from-D3(OP) functional based on the interaction strength.
TABLE IV. ME and MAE (in kcal/mol) of different function-als for the S22 data set. The best values are marked with boldstyle.
Methods ME MAE semilocal/hybrid
PBE a -2.55 2.55TPSS a -3.44 3.44SCAN a -0.57 0.91TM b -0.53 0.61revTM b -1.80 1.82M06-L a -0.77 0.81B3LYP a -3.78 3.78PBE0 a -2.33 2.37TPSS0 a -3.06 3.06 semilocal+dispersion PW86R-VV10 d r VV10 c r VV10 c a a b -0.01 0.25TM+D3(BJ) b -0.01 0.26TM+D3(OP) b -0.12 0.25M06-L+D3(0) a e − b -0.00 b revTM+D3(OP) b -0.03 0.22 (range-separated)hybrid+dispersion B3LYP+D3 a a a a a − a a ω B97X-D3 a ω B97X-V a -0.10 0.22a-Ref. b-present workc-Ref. d-Ref. e-Ref. Note that for this case the -D3(OP) balances more theinteraction energies for individual molecules compared tothe other two dispersion interactions.To complete our analysis, in Table IV, we compare theME and MAE of several popular GGA, meta-GGA andhybrid density functionals (global and range-separated).The dispersion corrected functionals are consistentlyimproving their performance compared to the corre-sponding bare functionals. Note that revTM+D3(BJ)achieves the the best accuracy among the dispersion cor-rected semilocal functionals with MAE=0.19 kcal/mol,being significantly better than other dispersion correctedsemilocal functionals. Within hybrid functionals, the ω B97X-D3 is close to that of the revTM+D3(BJ).
B. L7 test set
The L7 test set consists of large sized complexes hav-ing dispersion dominated non-covalent bonds. Due to the computational efficiency, dispersion corrected semilocalXC functionals are very promising in case of such largecomplexes. Now, to test the accuracy of the above dis-cussed methods, we apply both bare semilocal, and D3corrected semilocal functionals to the optimized struc-tures (TPSS-D/TZVP) of the complexes present in theL7 test set . This data set includes mixed hydrogenbonded complexes along with aliphatic, and strong aro-matic dispersion bonded complexes. The binding ener-gies of all the seven large complexes are shown in Ta-ble V considering all D3 corrected functionals and theCCSD(T) reference data . Among all the six disper-sion corrected methods, revTM+D3(OP) has the least er-ror with more accurately description of aromatic disper-sion interactions(C3A, C3GC, C2C2PD) and hydrogenbonds (PHE). However, all the methods underbind thestacked Watson-Crick H-bonded guanine-cytosine dimer(GCGC) significantly. Such underestimation by TMbased functionals is also reported in literature . Wealso show the errors excluding the GCGC base pair fromL7 data set in the lower panel of Table V. A drasticdrop of the MAE for all the cases can be seen and therevTM+D3(OP) is the best method with MAE=0.86kcal/mol. Now, it is necessary to compare our meth-ods with contemporary dispersion corrected methods tounderstand the hierarchy of development. So, we listthe errors of L7 data set for above discussed methodsalong with errors of some available functionals in TableVI. The TPSS+D3 method is proved to be best havingleast MAE value of 1.1 kcal/mol. Note that the S30Lbenchmark set proposed in ref. is more realistic thanL7. We will consider these test cases in our future study.
C. Inter and intra-molecular non-covalent interactions
The inter-molecular binding energies of the dispersionbonded molecular complexes, arise from atoms of the twoseparate molecular systems. All the test sets and ge-ometries are taken from the GMTKN55 database, wherewe do not include the WATER27, which is discussedseparately within the hydrogen bonded complexes. Ta-ble VII reports MAE of all the constructed dispersioncorrected functionals, along with the best dispersion cor-rected semilocal and the overall best method.To start with, the RG18 test set contains the rare-gas dimers, trimers, tetramers, hexamers and complexesof rare gas with HF, ethyne, ethane and benzene. Weobtain the best MAE from revTM+D3(OP) within theconsidered functionals with MAE=0.15 kcal/mol. In allcases, the -D3(OP) improves over -D3(0) and -D3(BJ)functionals. The ADIM6 test set consists of six alkanedimers binding energies. We observe revTM+D3(BJ)achieves the best accuracy among the semilocal D3 cor-rected functionals with MAE=0.06 kcal/mol, performingas the best semilocal-D3 result found from the OLYP-D3(BJ) functional. Therefore, for alkane dimers bind-ing energies, revTM+D3(BJ) is quite a good candidate.
TABLE V. Interaction energies (in kcal/mol) of L7 data set. The CCSD(T) reference values are given in the first column.The best values are marked with bold style.
L7 Complexes CCSD(T) TM+D3(0) TM+D3(BJ) TM+D3(OP) revTM+D3(0) revTM+D3(BJ) revTM+D3(OP)Octadecane dimer (CBH) -11.6 -11.33 -10.09 -10.40 -12.08 -10.23 -10.91Guanine trimer (GGG) -1.9 -2.08 -1.87 -1.85 -2.09 -1.71 -2.22Circumcoronene-Adenine dimer (C3A) -17.0 -14.20 -13.86 -14.14 -14.71 -14.74 -15.81Circumcoronene-Guanine-cytosine dimer (C3GC) -29.1 -25.12 -24.50 -24.76 -25.57 -25.58 -27.43Phenylalanine trimer (PHE) -23.0 -24.90 -24.26 -24.58 -24.72 -24.27 -24.19Coronene dimer (C2C2PD) -21.2 -16.70 -16.73 -17.02 -17.73 -18.66 -21.05Guanine-cytosine dimer (GCGC) -12.8 -3.73 -3.15 -2.97 -2.95 -2.00 -3.14ME − -2.64 -3.15 -2.98 -2.39 -2.77 -1.69 MAE − Errors for L6 (removing GCGC from L7)ME − -1.59 -2.07 -1.84 -1.15 -1.43 -0.36 MAE − TABLE VI. The ME and MAE (in kcal/mol) of different func-tionals for the L7 data set.
Methods ME MAEM06-L a -3.0 3.0M062X b -3.2 3.3SCAN a -7.9 7.9TM -8.0 8.0revTM -15.0 15.0PBE+D3 a -2.1 2.6BLYP+D3 b a -0.9 SCAN+D3 a -1.2 2.5M062X-D3 b -0.1 a b b-Ref. Considering the S22 test set, it was already discussedin the previous section. In this case, revTM+D3(BJ)achieves the best accuracy among the semilocal D3 the-ory, better than so far best BLYP-D3(BJ). Similar ac-curacy is also observed for the S66 test set with therevTM+D3(BJ) functional. However, in this case we ob-serve revTM+D3(OP) bit better than revTM+D3(BJ).This is due to the better performance of revTM+D3(OP)for H-bonded systems. The HEAVY28 test set consistsof non-covalent binding energies of 28 heavy-element-hydride dimers. In this case also, revTM+D3(OP)outperforms other dispersion corrected functionals withMAE=0.18 kcal/mol. The CARBHB12 test set repre-sents 12 hydrogen-bonded complexes of carbene boundwith CClCH , SiH , and 1,3-dimethylimidazol-2-ylidene.Though the TM+D3(OP) gives MAE=0.65 kcal/mol,still M06-L-D3(0) is the best dispersion corrected semilo-cal functional with MAE=0.44 kcal/mol. For PNICO23test set, all considered functionals overestimate thebinding energies corresponding to the most accuratesemilocal D3 approach MN12L-D3(BJ). The HAL59 test M A E f o r i n t r a m o l ec u l a r non - c ov a l e n t i n t e r ac ti on s ( k ca l/ m o l ) r e v T M + D ( O P ) ω B X - D r e v T M + D ω B X - V M - L + D ( ) B L Y P + D B L Y P + D ( B J ) S C AN + D S C AN + D ( B J ) T M + D P B E + D T PSS + D ( B J ) r e v P B E + D ( B J ) P B E + D ( B J ) T M + D ( B J ) revTM+D3(BJ) r e v P B E + D TM+D3(OP)
TPSS+D3 D S D - B L Y P + D D S D - B L Y P + D ( B J ) FIG. 3. Shown is the MAE (in kcal/mol) of inter-molecularnon-covalent interactions versus the MAE (in kcal/mol) ofintra-molecular non-covalent interactions for various func-tionals. Red-squares represent the meta-GGA+D3 function-als, black-crosses are GGA+D3 functionals, blue-x-shapes areglobal hybrids + D3, green-triangles are vdW -corrected long-range screened hybrids, and orange-stars are double hybrids.The WATER27 test set is not considered and the reference er-ror of the different funcionals (except TM and revTM basedfunctionals) are taken from ref. . set represents non-covalent binding energies of halo-genated dimers, being constructed from the combina-tion of XB51 and X40 test sets. In this case alsothe slight overestimation is observed from all the dis-persion corrected functionals, and the lowest MAE of1.10 kcal/mol is obtained from revTM+D3(0), while thebest semilocal D3 corrected functional is the M06-L-D3(0) with MAE=0.49 kcal/mol. The AHB21 test setcontains the interaction energies of 21 anionic and neu-tral dimers. The TM+D3(OP) and revTM+D3(OP)are performing better compared to the others func-tionals, because -D3(OP) performs in a more balancedway for H-bonded and dipole-interacting systems. TherevTM+D3(OP) is also performing comprehensively forthe six cationic − neutral dimers test set CHB6. Nextfor the IL16 test set which consists of 16 cation − anionnon-covalently bonded model dimers, revTM+D3(OP) TABLE VII. Mean errors and mean absolute errors (in kcal/mol) for benchmark test sets, using the D3-corrected semilocal XCfunctionals. For a better evaluation, we also provide the best semilocal+D3 and overall results for each test, taken from ref. .The best values within TM and revTM based dispersion methods are marked with bold style. Test sets Errors TM TM TM revTM revTM revTM Best Best+D3(0) +D3(BJ) +D3(OP) +D3(0) +D3(BJ) +D3(OP) Semilocal+D3 OverallIntermolecular non-covalent interactions (kcal/mol)RG18 ME 0.01 0.01 -0.02 -0.10 -0.14 -0.13MAE 0.21 0.19 0.19 ω B97X-V)HEAVY28 ME -0.07 -0.05 -0.08 -0.19 0.15 0.08MAE 0.25 0.24 0.25 0.32 0.22 ω B97X-V)(revTPSS0-D3(BJ))Amino20x4 ME 0.11 0.11 0.11 0.06 0.10 0.08MAE 0.25 0.27 0.24 0.22 0.22 ω B97X-V)TME 0.04 0.17 0.15 0.10 0.28 0.23 − −
TMAE 0.79 0.68 0.64 0.61 0.59 − −
TABLE VIII. Mean errors and mean absolute errors (in kcal/mol) for the WATER27 benchmark test set, using the studiedsemilocal functionals along with their dispersion corrected counterparts. The best semilocal+D3 and overall results are takenfrom ref. . TM and revTM results are from ref. .Errors TM revTM TM TM TM revTM revTM revTM M06-L-D3(0) DSD-BLYP-D3(BJ)+D3(0) +D3(BJ) +D3(OP) +D3(0) +D3(BJ) +D3(OP)ME 1.32 -1.24 2.71 2.38 1.94 3.21 2.88 1.21MAE 1.44 . Errors TM revTM TM+D3(0) TM+D3(BJ) TM+D3(OP) revTM+D3(0) revTM+D3(BJ) revTM+D3(OP) B-P86
DSD-PBE-P86
ME 0.15 -0.24 0.23 0.23 0.20 0.12 0.19 -0.02MAE 0.93
TABLE X. Mean absolute errors (in kcal/mol) for MG8 test set as calculated using different methods. The details of the testset and reference values are provided in ref. . Group description
TM revTM TM+D3(0) TM+D3(BJ) TM+D3(OP) revTM+D3(0) revTM+D3(BJ) revTM+D3(OP)NCED noncovalent interaction (easy, cluster) 0.3 1.2 0.2 0.2 0.2 0.1 0.1 0.1NCEC noncovalent interaction (easy, dimer) 3.5 15.1 3.0 1.3 0.1 4.4 2.3 4.3NCD noncovalent interaction(difficult) 2.9 2.6 2.9 2.9 2.9 2.7 3.1 2.7IE isomerization energy (easy) 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3ID isomerization energy (difficult) 18.0 17.5 17.5 17.7 17.4 17.0 17.1 17.6TCE thermochemistry (easy) 6.9 6.1 6.9 6.8 6.9 5.9 5.8 6.0TCD thermochemistry (difficult) 14.3 14.1 14.4 14.4 14.4 14.1 14.8 14.2BH barrier height 7.7 7.7 7.7 7.7 7.7 7.8 7.9 7.7MGCDB82 2.3 2.7 2.2 2.2 2.2 2.1 2.1 2.1 n -butane, n -pentane and n -hexane conform-ers. It also gives the very similar accuracy as that ofthe so far best dispersion corrected semilocal methodOLYP-D3(BJ). The revTM+D3(OP) is also very ac-curate for AMINO20 × − dipole, aromatic − amide, and hydro-gen bond interactions important for biomolecules. TheSCONF test set consists of 14 and 3 relative ener-gies of 3,6-anhydro-4-O-methyl-D-galactitol and b-D-glucopyranose conformers, respectively. In this case also,revTM+D3(OP) performs better than the other disper-sion corrected functionals, while the best semilocal+D3method is the M11L-D3(0). For UPU23 test set, alldispersion corrected revTM perform with almost sameaccuracy. Same is true for the TM based dispersioncorrected methods. This test set consists of nucleicacids and biomolecules which are the main constituentsof RNA. Finally, for the BUT14DIOL test set, whichconsists of strong intra-molecular hydrogen bonds, therevTM+D3(OP) is the best within the semilocal+D3methods.To make our comparison of the accuracy of differentpopular functionals in a more competitive manner, inFig. 3, we plot the MAE of the inter-molecular non-covalent interactions versus the MAE of intra-molecularnon-covalent interactions. It is noticed that, at thesemilocal level, revTM+D3(OP) achieves the best accu-racy. Moreover, the revTM+D3(OP) functional is evenbetter than the ω B97X-V and ω B97X-D for the intra-molecular non-covalent interactions, where both func-tionals are the range-separated hybrids and quite ex- pensive for large molecular systems. Note also thatrevTM+D3(OP) is better than well known hybrid+D3functionals like PBE0+D3(BJ) in both cases.
D. Water clusters
The remarkable accuracy of the -D3(OP) based semilo-cal fnctionals is also clearly evident from Table VIII,where we assess the dispersion corrected semilocal func-tionals for various water clusters. This test set includesH-bonded water clusters which are either neutral or pos-itively, and negatively charged. This test set is extractedfrom the GMTKN55 database as mentioned before, inorder to emphasize the performance of the functionalsfor H-bond within water molecules. It is seemingly quiteinteresting that the -D3(OP) does not deteriorate theperformance of TM and revTM functionals, unlike other-D3 methods. The bare TM and revTM give the MAEof about 1.44 kcal/mol and 1.31 kcal/mol, respectively,which are only slightly better than 2.02 kcal/mol and1.47 kcal/mol obtained upon addition of the -D3(OP)correction. These results motivate us to further studythe -D3(OP) corrected TM and revTM functionals forwater properties. Note that very recently the revTMfunctional is assessed for different water properties and found to be very accurate for different water prop-erties. In this case M06-L-D3(0) is the best dispersioncorrected semilocal functional with MAE=1.11 kcal/moland overall DSD-BLYP-D3(BJ) is the best functionalwith MAE=0.94 kcal/mol. E. Conformers for lactose, maltose, and sucrose
Energetic of the bio-molecular conformers are impor-tant in various applications of chemical and biologicalsystems. Being very large structures, the semilocal XCare the most preferred method to simulate those sys-tems. Here, we studied relative energies of the differ-ent conformers of the lactose, maltose, and sucrose us-ing the prescribed methods. This test set (DSCONF) isproposed recently . Note that the basic constituent ofthese conformers are the amino acids and peptides havinghydrogen bonds. Therefore, it is an interesting test casebecause a major factor of this test set is determined bythe relative conformer energies of OH-O hydrogen bond,similar to the WATER27 test set. The error statistics asobtained from different functionals are reported in Ta-ble IX. We observe the revTM becomes the most accu-rate functional with MAE=0.69 kcal/mol followed by thebare TM functional. Similar to the WATER27 test casethe D3-0 and D3-BJ variants work less well than D3-OP.1
F. Small representative MG8 thermochemical test set
Lastly, we assess the constructed functionals perfor-mance for the small representative MG8 thermochemi-cal test set. The MG8 test set is proposed recently and it represents statistically accurate depiction of theMGCDB84 test set . This test set contains 64 datapoints instead of the large 5000 data points of theMGCDB82 test set. Like MGCDB84 on which it is based,MG8 divides the data into different types of proper-ties like noncovalent interactions, isomerization energies,thermochemical properties, and barrier heights. The de-tails of the test set and its benchmark values can befound in ref. . The MAEs of the each test set as ob-tained form different functionals are listed in Table X.The MAEs for MGCDB82 are also calculated in Table Xusing the formula suggested in Eq.(1) of ref. . It is ob-vious that the isomerization energy and thermochemistryof difficult cases are particularly challenging, though thatis generally true for most functionals; for example, evenB97M-V has an MAE over 10 kcal/mol for isomeriza-tion energy . In this respect, the dispersion correctedsemilocal functionals show improvement in a systematicway than its bare functionals. Interestingly, the perfor-mance of the -OP corrected functionals is quite promis-ing.
IV. CONCLUSIONS
We have assessed the benchmark calculations of theD3-corrected TM and revTM meta-GGA XC function-als, for a large palette of molecular complexes, character-ized by various non-covalent interactions, such as inter-and intra-molecular dispersion, hydrogen, halogen, dihy-drogen, dipole-dipole and mixed bonded systems. Wehave constructed several forms of the D3-functionals, us-ing the zero, rational damping, and optimized param-eter damping functions. A total of six variants of dis-persion corrected functionals are tested for a wide rangeof interesting systems. Our primary focus has been tomeasure the accuracy and applicability of the proposedmethods for different kinds of dispersion interactions. Itturns out that within the vdW -corrected semilocal ap-proximations, the revTM+D3(OP) gives an outstandingperformance, outclassing many popular functionals, andcompeting with the expensive dispersion corrected range-separated hybrids ω B97X-D and ω B97X-V.For the energetic of the non-covalent binding ener-gies, the performance of revTM+D3(OP) is obtainedto be very good for S22, L7, various inter- and intra-molecular non-covalent interaction test sets of the well-known GMTKN55 database, and the H-bond interactionof charged moieties with neutral small molecules. Forcomparison purposes, we discuss separately the bindingenergy of the neutral and charged water clusters, wherethe inclusion of the -D3 within semilocal functional usu-ally over-binds the energies. Interestingly, we observe that TM+D3(OP) and revTM+D3(OP) do not deterio-rate much the accuracy of the bare functionals. The im-pressive performance of (rev)TM and (rev)TM+D3(OP)is also more evident from the relative conformer ener-gies is OH-O hydrogen bond of the lactose, maltose, andsucrose. Overall, for the small representative MG8 ther-mochemical test set also the “OP” corrected functionalsperformance in an impressive manner.Overall, revTM-D3 XC functional delivers awe-inspiring performance and acquire excellent accuracyclose to the computationally costly range-separated hy-brids and double-hybrid functionals. Importantly, it per-forms well for different interaction ranges of the non-covalent systems and can be considered as an importantdispersion corrected functional within the dispersion cor-rected density functional theory zoo. As a concludingremark it is also important to note that recently the -D4 dispersion correction of Grimme shows its productivepower over -D3, which we will consider in our future as-sessment.
V. ACKNOWLEDGEMENTS
A.P. would also like to acknowledge the financial sup-port from the Department of Atomic Energy, Govern-ment of India. S.J. would like to thank Prof. StefanGrimme for many useful and technical suggestions. S.J.would also like to thank Prof. Bun Chan for provid-ing much useful technical information about the calcu-lation of the DSCONF, MG8 test set. This work hasbeen performed in a high-performance computing facil-ity of NISER. PS would like to thank Q-Chem, Inc. anddevelopers for providing the code.
VI. DATA AVAILABLE ON REQUEST FROM THEAUTHORS
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest. P. Hohenberg and W. Kohn, Phys. Rev. , B864 (1964). W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965). J. P. Perdew and A. Zunger, Phys. Rev. B , 5048 (1981). J. P. Perdew and K. Schmidt, in
AIP Conference Proceedings (IOP INSTITUTE OF PHYSICS PUBLISHING LTD, 2001)pp. 1–20. G. E. Scuseria and V. N. Staroverov, in
Theory and Applicationof Computational Chemistry: The First 40 Years , edited byC. E. Dykstra, G. Frenking, K. S. Kim, and G. E. Scuseria(Elsevier: Amsterdam, 2005) pp. 669–724. A. D. Becke, Phys. Rev. A , 3098 (1988). C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B , 785 (1988). J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R.Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B , 6671(1992). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. lett. ,3865 (1996). R. Armiento and A. E. Mattsson, Phys. Rev. B , 085108(2005). L. A. Constantin, A. Ruzsinszky, and J. P. Perdew, Phys. Rev.B , 035125 (2009). Z. Wu and R. E. Cohen, Phys. Rev. B , 235116 (2006). Y. Zhao and D. G. Truhlar, J. Chem. Phys. , 184109 (2008). L. A. Constantin, J. C. Snyder, J. P. Perdew, and K. Burke, J.Chem Phys. , 241103 (2010). L. A. Constantin, A. Terentjevs, F. Della Sala, and E. Fabiano,Phys. Rev. B , 041120 (2015). L. A. Constantin, L. Chiodo, E. Fabiano, I. Bodrenko, andF. Della Sala, Phys. Rev. B , 045126 (2011). E. Fabiano, P. E. Trevisanutto, A. Terentjevs, and L. A. Con-stantin, J. Chem. Theory Comput. , 2016 (2014). L. A. Constantin, A. Terentjevs, F. Della Sala, P. Cortona, andE. Fabiano, Phys. Rev. B , 045126 (2016). A. Cancio, G. P. Chen, B. T. Krull, and K. Burke, J. Chem.Phys. , 084116 (2018). L. A. Constantin, Phys. Rev.B , 155106 (2008). J. Pˇrecechtˇelov´a, H. Bahmann, M. Kaupp, and M. Ernzerhof,J. Chem. Phys. , 111102 (2014). J. P. Pˇrecechtˇelov´a, H. Bahmann, M. Kaupp, and M. Ernzerhof,J. Chem. Phys. , 144102 (2015). T. M. Henderson, B. G. Janesko, and G. E. Scuseria, J. Chem.Phys. , 194105 (2008). J. Carmona-Esp´ındola, J. L. G´azquez, A. Vela, and S. Trickey,J. Chem. Theory Comput. , 303 (2018). R. Peverati and D. G. Truhlar, J. Chem. Theory Comput. ,2310 (2012). F. Della Sala, E. Fabiano, and L. A. Constantin, Int. J. Quan-tum Chem. , 1641 (2016). A. Becke and M. R. Roussel, Phys. Rev. A , 3761 (1989). T. Van Voorhis and G. E. Scuseria, J. Chem. Phys. , 400(1998). Y. Zhao and D. G. Truhlar, J. Chem. Phys. , 194101 (2006). J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev.letters , 2544 (1999). J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria,Phys. Rev. Lett. , 146401 (2003). L. A. Constantin, E. Fabiano, and F. Della Sala, Phys. Rev. B , 035130 (2012). J. Sun, J. P. Perdew, and A. Ruzsinszky, Proc. Natl. Acad. Sci. , 685 (2015). H. S. Yu, X. He, and D. G. Truhlar, Journal of Chemical Theoryand Computation , 1280 (2016). H. S. Yu, X. He, S. L. Li, and D. G. Truhlar, Chem. Sci. ,5032 (2016). A. Ruzsinszky, J. Sun, B. Xiao, and G. I. Csonka, J. Chem.Theory Comput. , 2078 (2012). L. A. Constantin, E. Fabiano, J. Pitarke, and F. Della Sala,Phys. Rev. B , 115127 (2016). J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. lett. ,036402 (2015). J. Tao and Y. Mo, Phys. Rev. lett. , 073001 (2016). Y. Wang, X. Jin, S. Y. Haoyu, D. G. Truhlar, and X. He, Proc.Natl. Acad. Sci. , 8487 (2017). Y. Wang, X. Jin, H. S. Yu, D. G. Truhlar, and X. He,Proceedings of the National Academy of Sciences , 8487 (2017). Y. Wang, P. Verma, X. Jin, D. G. Truhlar, and X. He,Proceedings of the National Academy of Sciences , 10257 (2018). P. D. Mezei, G. I. Csonka, and M. K´allay, J. Chem. TheoryComput. , 2469 (2018). S. Jana, K. Sharma, and P. Samal, J. Phys. Chem. A , 6356(2019). B. Patra, S. Jana, L. A. Constantin, and P. Samal, Phys. Rev.B , 045147 (2019). B. Patra, S. Jana, L. A. Constantin, and P. Samal, Phys. Rev.B , 155140 (2019). A. Patra, S. Jana, H. Myneni, and P. Samal, Phys. Chem.Chem. Phys. , 19639 (2019). T. Aschebrock and S. K¨ummel, Phys. Rev. Research , 033082(2019). J. Sun, B. Xiao, Y. Fang, R. Haunschild, P. Hao, A. Ruzsinszky,G. I. Csonka, G. E. Scuseria, and J. P. Perdew, Phys. Rev. lett. , 106401 (2013). P. Verma, Y. Wang, S. Ghosh, X. He, and D. G. Truhlar, TheJournal of Physical Chemistry A , 2966 (2019). B. G. Janesko and A. Aguero, J. Chem. Phys. , 024111(2012). B. G. Janesko, Int. J. Quantum Chem. , 83 (2013). B. G. Janesko, J. Chem. Phys. , 104103 (2010). B. G. Janesko, J. Chem. Phys. , 224110 (2012). B. G. Janesko, E. Proynov, G. Scalmani, and M. J. Frisch, J.Chem. Phys. , 104112 (2018). L. A. Constantin, E. Fabiano, and F. Della Sala, J. Chem. Phys. , 084110 (2016). L. A. Constantin, E. Fabiano, and F. Della Sala, J. Chem.Theory Comput. , 4228 (2017). J. P. Perdew, V. N. Staroverov, J. Tao, and G. E. Scuseria,Phys. Rev. A , 052513 (2008). J. P. Perdew, A. Ruzsinszky, J. Tao, V. N. Staroverov, G. E.Scuseria, and G. I. Csonka, J. Chem. Phys. , 062201 (2005). M. M. Odashima and K. Capelle, Phys. Rev. A , 062515(2009). A. V. Arbuznikov and M. Kaupp, Int. J. Quantum Chem. ,2625 (2011). J. Jaramillo, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. , 1068 (2003). S. K¨ummel and L. Kronik, Rev. Mod. Phys. , 3 (2008). A. D. Becke, J. Chem. Phys. , 064101 (2005). A. D. Becke and E. R. Johnson, J. Chem. Phys. , 124108(2007). A. D. Becke, J. Chem. Phys. , 2972 (2003). A. D. Becke, J. Chem. Phys. , 074109 (2013). E. Fabiano, L. A. Constantin, P. Cortona, and F. Della Sala,J. Chem. Theory Comput. , 122 (2014). J. Sun, R. Haunschild, B. Xiao, I. W. Bulik, G. E. Scuseria,and J. P. Perdew, J. Chem. Phys. , 044113 (2013). E. Fabiano, L. A. Constantin, and F. Della Sala, Int. J. Quan-tum Chem. , 673 (2013). S. Jana, A. Patra, L. A. Constantin, H. Myneni, and P. Samal,Phys. Rev. A , 042515 (2019). O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. , 234109(2006). S. Jana and P. Samal, Phys. Chem. Chem. Phys. , 8999(2018). B. Patra, S. Jana, and P. Samal, Phys. Chem. Chem. Phys. ,8991 (2018). S. Jana and P. Samal, Phys. Chem. Chem. Phys. , 3002(2019). S. Jana, B. Patra, H. Myneni, and P. Samal, Chem. Phys. Lett. , 1 (2018). S. Jana, A. Patra, and P. Samal, J. Chem. Phys. , 094105(2018). L. Goerigk and S. Grimme, J. Chem. Theory Comput. , 291(2010). L. Goerigk, A. Hansen, C. Bauer, S. Ehrlich, A. Najibi, andS. Grimme, Phys. Chem. Chem. Phys. , 32184 (2017). S. ´Smiga, O. Franck, B. Mussard, A. Buksztel, I. Grabowski,E. Luppi, and J. Toulouse, J. Chem. Phys. , 144102 (2016),https://doi.org/10.1063/1.4964319. R. J. Bartlett, I. Grabowski, S. Hirata, and S. Ivanov, J. Chem.Phys. , 034104 (2005). R. J. Bartlett, V. F. Lotrich, and I. V. Schweigert, J. Chem.Phys. , 062205 (2005). I. Grabowski, E. Fabiano, and F. Della Sala, Phys. Rev. B ,075103 (2013). I. Grabowski, E. Fabiano, A. M. Teale, S. ´Smiga, A. Buksztel,and F. D. Sala, J. Chem. Phys. , 024113 (2014). M. Seidl, J. P. Perdew, and S. Kurth, Phys. Rev. Lett. , 5070(2000). Z.-F. Liu and K. Burke, Phys. Rev. A , 064503 (2009). J. Sun, J. Chem. Theory Comput. , 708 (2009). P. Gori-Giorgi, G. Vignale, and M. Seidl, J. Chem. TheoryComput. , 743 (2009). A. Mirtschink, M. Seidl, and P. Gori-Giorgi, J. Chem. TheoryComput. , 3097 (2012). S. Vuckovic, T. J. Irons, A. Savin, A. M. Teale, and P. Gori-Giorgi, J. Chem. Theory Comput. , 2598 (2016). E. Fabiano, P. Gori-Giorgi, M. Seidl, and F. Della Sala, J.Chem. Theory Comput. , 4885 (2016). S. Giarrusso, P. Gori-Giorgi, F. Della Sala, and E. Fabiano, J.Chem. Phys. , 134106 (2018). E. Fabiano, S. Smiga, S. Giarrusso, T. J. Daas,F. Della Sala, I. Grabowski, and P. Gori-Giorgi,J. Chem. Theory Comput. , 1006 (2019). L. A. Constantin, Phys. Rev. B , 085117 (2019). J. F. Dobson, J. Wang, and T. Gould, Phys. Rev. B , 081108(2002). L. A. Constantin and J. M. Pitarke, Phys. Rev. B , 245127(2007). A. V. Terentjev, L. A. Constantin, and J. M. Pitarke, Phys.Rev. B , 085123 (2018). L. A. Constantin, Phys. Rev. B , 121104 (2016). J. Toulouse, Phys. Rev. B , 035117 (2005). C. F. Richardson and N. W. Ashcroft, Phys. Rev. B , 8170(1994). J. E. Bates, S. Laricchia, and A. Ruzsinszky, Phys. Rev. B ,045119 (2016). J. E. Bates, J. Sensenig, and A. Ruzsinszky, Phys. Rev. B ,195158 (2017). A. Ruzsinszky, L. A. Constantin, and J. M. Pitarke, Phys. Rev.B , 165155 (2016). J. F. Dobson and J. Wang, Phys. Rev. B , 10038 (2000). J. Erhard, P. Bleiziffer, and A. G¨orling, Phys. Rev. Lett. ,143002 (2016).
C. E. Patrick and K. S. Thygesen, J. Chem. Phys. , 102802(2015).
M. Shahbaz and K. Szalewicz,Phys. Rev. Lett. , 113402 (2018).
M. Shahbaz and K. Szalewicz,Phys. Rev. Lett. , 213001 (2019).
N. Mardirossian and M. Head-Gordon, J. Chem. Phys. , 214110 (2016),https://doi.org/10.1063/1.4952647.
N. Mardirossian and M. Head-Gordon,Phys. Chem. Chem. Phys. , 9904 (2014). J.-D. Chai and M. Head-Gordon,J. Chem. Phys. , 084106 (2008),https://doi.org/10.1063/1.2834918.
J. G. Brandenburg, J. E. Bates, J. Sun, and J. P. Perdew,Phys. Rev. B , 115144 (2016). S. Grimme, J. Comput. Chem. , 1787 (2006),https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.20495. Q. Wu and W. Yang, J. Chem. Phys. , 515 (2002),https://doi.org/10.1063/1.1424928.
S. Grimme, J. Comput. Chem. , 1463 (2004),https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.20078. S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, J. Chem. Phys. , 154104 (2010).
A. D. Becke and E. R. Johnson, J. Chem. Phys. , 154101(2005).
E. R. Johnson and A. D. Becke,J. Chem. Phys. , 024101 (2005),https://doi.org/10.1063/1.1949201.
E. R. Johnson and A. D. Becke, J. Chem. Phys. , 174104(2006).
A. D. Becke and E. R. John-son, J. Chem. Phys. , 154104 (2005),https://doi.org/10.1063/1.1884601.
S. Grimme, S. Ehrlich, and L. Go-erigk, J. Comput. Chem. , 1456, https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.21759. E. R. Johnson and G. A. DiLabio,Chem. Phys. Lett. , 333 (2006).
H. Schrder, A. Creon, and T. Schwabe,J. Chem. Theory Comput. , 3163 (2015),https://doi.org/10.1021/acs.jctc.5b00400. J. Witte, N. Mardirossian, J. B. Neaton, and M. Head-Gordon, J. Chem. Theory Comput. , 2043 (2017),https://doi.org/10.1021/acs.jctc.7b00176. L. Goerigk and S. Grimme,J. Chem. Theory Comput. , 107 (2010),https://doi.org/10.1021/ct900489g. L. Goerigk and S. Grimme,Phys. Chem. Chem. Phys. , 6670 (2011). E. R. Johnson, I. D. Mackie, and G. A.DiLabio, J. Phys. Org. Chem. , 1127 (2009),https://onlinelibrary.wiley.com/doi/pdf/10.1002/poc.1606. P. Hao, J. Sun, B. Xiao, A. Ruzsinszky, G. I.Csonka, J. Tao, S. Glindmeyer, and J. P.Perdew, J. Chem. Theory Comput. , 355 (2013),https://doi.org/10.1021/ct300868x. J. Antony and S. Grimme, Phys. Chem. Chem. Phys. , 5287 (2006). N. Marom, A. Tkatchenko, M. Rossi, V. V.Gobre, O. Hod, M. Scheffler, and L. Kro-nik, J. Chem. Theory Comput. , 3944 (2011),https://doi.org/10.1021/ct2005616. H. Tang and J. Tao, Int. J. Mod. Phys. B , 1950300 (2019),https://doi.org/10.1142/S0217979219503004. A. Patra, S. Jana, and P. Samal, J. Phys. Chem. A , 10582.
Y. Mo, G. Tian, and J. Tao,Phys. Chem. Chem. Phys. , 21707 (2017). Y. Mo, R. Car, V. N. Staroverov, G. E. Scuseria, and J. Tao,Phys. Rev. B , 035118 (2017). Y. Mo, H. Tang, A. Bansil, andJ. Tao, AIP Advances , 095209 (2018),https://doi.org/10.1063/1.5050241. H. Tang and J. Tao, Mater. Res. Express , 076302 (2018). G. Tian, Y. Mo, and J. Tao, Computation , 27 (2017).
Y. Mo, G. Tian, R. Car, V. N. Staroverov, G. E. Scuseria, andJ. Tao, The Journal of Chemical Physics , 234306 (2016),https://doi.org/10.1063/1.4971853.
E. R. Johnson, A. D. Becke, C. D. Sherrill, and G. A.DiLabio, The Journal of Chemical Physics , 034111 (2009),https://doi.org/10.1063/1.3177061.
S. Jana, L. A. Constantin, and P. Samal,Journal of Chemical Theory and Computation , 974 (2020),pMID: 31910012, https://doi.org/10.1021/acs.jctc.9b01018. J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin,and J. Sun, Phys. Rev. Lett. , 026403 (2009).
P. Jurecka, J. Sponer, J. Cerny, and P. Hobza,Phys. Chem. Chem. Phys. , 1985 (2006). M. S. Marshall, L. A. Burns, and C. D. Sherrill,The Journal of Chemical Physics , 194102 (2011),https://doi.org/10.1063/1.3659142.
B. M. Axilrod and E. Teller, J. Chem. Phys. , 299 (1943),https://doi.org/10.1063/1.1723844. R. A. DiStasio, V. V. Gobre, and A. Tkatchenko,J. Phys.: Condens. Matter , 213202 (2014). J. F. Dobson, Int. J. Quantum Chem. , 1157 (2014),https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.24635.
M. R. Kennedy, A. R. McDonald, A. E. De-Prince, M. S. Marshall, R. Podeszwa, andC. D. Sherrill, J. Chem. Phys. , 121104 (2014),https://doi.org/10.1063/1.4869686.
R. Sedlak, T. Janowski, M. Pitonak, J. ˇRez´aˇc, P. Pu-lay, and P. Hobza, J. Chem. Theory Comput. , 3364 (2013),https://doi.org/10.1021/ct400036b. J. ˇRez´aˇc, K. E. Riley, and P. Hobza,J. Chem. Theory Comput. , 2427 (2011),https://doi.org/10.1021/ct2002946. K. U. Lao, R. Sch¨affer, G. Jansen, and J. M.Herbert, J. Chem. Theory Comput. , 2473 (2015),https://doi.org/10.1021/ct5010593. G. Knizia, T. B. Adler, and H.-J.Werner, J. Chem. Phys. , 054104 (2009),https://doi.org/10.1063/1.3054300.
K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. , 479 (1989).
J. ˇRez´aˇc, K. E. Riley, and P. Hobza,J. Chem. Theory Comput. , 4285 (2012),https://doi.org/10.1021/ct300647k. S. Kozuch and J. M. L. Martin,J. Chem. Theory Comput. , 1918 (2013),https://doi.org/10.1021/ct301064t. A. Karton and J. M. L. Mar-tin, J. Chem. Phys. , 124114 (2012),https://doi.org/10.1063/1.3697678.
D. Setiawan, E. Kraka, and D. Cre-mer, J. Phys. Chem. A , 1642 (2015),https://doi.org/10.1021/jp508270g.
S. Boys and F. Bernardi, Molecular Physics , 553 (1970),https://doi.org/10.1080/00268977000101561. S. Kozuch and J. M. L. Mar-tin, J. Comput. Chem. , 2327 (2013),https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.23391. D. Gruzman, A. Karton, and J. M. L.Martin, J. Phys. Chem. A , 11974 (2009),https://doi.org/10.1021/jp903640h.
M. K. Kesharwani, A. Karton, and J. M. L.Martin, J. Chem. Theory Comput. , 444 (2016),https://doi.org/10.1021/acs.jctc.5b01066. J. J. Wilke, M. C. Lind, H. F. Schaefer, A. G. Cs´asz´ar,and W. D. Allen, J. Chem. Theory Comput. , 1511 (2009),https://doi.org/10.1021/ct900005c. J. M. L. Martin, J. Phys. Chem. A , 3118 (2013),https://doi.org/10.1021/jp401429u.
S. Grimme, Angewandte Chemie International Edition , 4460 (2006),https://onlinelibrary.wiley.com/doi/pdf/10.1002/anie.200600448. T. Schwabe and S. Grimme,Phys. Chem. Chem. Phys. , 3397 (2007). S. Grimme, M. Steinmetz, andM. Korth, J. Org. Chem. , 2118 (2007), https://doi.org/10.1021/jo062446p. U. R. Fogueri, S. Kozuch, A. Karton, andJ. M. Martin, J. Phys. Chem. A , 2269 (2013),https://doi.org/10.1021/jp312644t.
L. Goerigk, A. Karton, J. M. L. Martin, and L. Radom,Phys. Chem. Chem. Phys. , 7028 (2013). D. Reha, H. Vald´es, J. Vondrasek, P. Hobza, A. Abu-Riziq,B. Crews, and M. S. de Vries, Chem.: Eur. J , 6803 (2005),https://onlinelibrary.wiley.com/doi/pdf/10.1002/chem.200500465. G. I. Csonka, A. D. French, G. P. Johnson, andC. A. Stortz, J. Chem. Theory Comput. , 679 (2009),https://doi.org/10.1021/ct8004479. H. Kruse, A. Mladek, K. Gkionis, A. Hansen, S. Grimme,and J. Sponer, J. Chem. Theory Comput. , 4972 (2015),https://doi.org/10.1021/acs.jctc.5b00515. V. S. Bryantsev, M. S. Diallo, A. C. T. van Duin, andW. A. Goddard, J. Chem. Theory Comput. , 1016 (2009),https://doi.org/10.1021/ct800549f. D. Manna, M. K. Kesharwani, N. Sylvetsky, andJ. M. L. Martin, J. Chem. Theory Comput. , 3136 (2017),https://doi.org/10.1021/acs.jctc.6b01046. B. Chan, The Journal of Physical Chemistry A , 582 (2020),pMID: 31927999, https://doi.org/10.1021/acs.jpca.9b10932.
B. Chan, Journal of Chemical Theory and Computation , 4254 (2018),pMID: 30004698, https://doi.org/10.1021/acs.jctc.8b00514. Y. Shao, L. F. Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld,S. T. Brown, A. T. Gilbert, L. V. Slipchenko, S. V. Levchenko,and D. P. O’Neill, Phys. Chem. Chem. Phys. , 3172 (2006). H. Peng, Z.-H. Yang, J. P. Perdew, and J. Sun,Phys. Rev. X , 041005 (2016). T. Bj¨orkman, Phys. Rev. B , 165109 (2012). J. G. Brandenburg, C. Bannwarth, A. Hansen,and S. Grimme, J. Chem. Phys. , 064104 (2018),https://doi.org/10.1063/1.5012601.
R. Sure and S. Grimme, Journal of Chemical Theory and Com-putation , 3785 (2015). N. Mardirossian and M. Head-Gordon, Molecular Physics115