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Keywords:

  • non-covalent interactions;
  • DFT benchmark study;
  • dispersion;
  • dipole-dipole systems;
  • dipole-induced dipole systems;
  • CCSD;
  • exchange-correlation functions;
  • long-range correction;
  • M06L

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. Supporting Information

A benchmark study on all possible density functional theory (DFT) methods in Gaussian09 is done to locate functionals that agree well with CCSD/aug-cc-pVTZ geometry and Ave-CCSD(T)/(Q-T) interaction energy (Eint) for small non-covalently interacting molecular dimers in “dispersion-dominated” (class 1), “dipole-induced dipole” (class 2), and “dipole-dipole” (class 3) classes. A DFT method is recommended acceptable if the geometry showed close agreement to CCSD result (RMSD < 0.045) and Eint was within 80–120% accuracy. Among 382 tested functionals, 1–46% gave good geometry, 13–44% gave good Eint, while 1–33% satisfied geometry and energy criteria. Further screening to locate the best performing functionals for all the three classes was made by counting the acceptable values of energy and geometry given by each functionals. The meta-generalized gradient approximation (GGA) functional M06L was the best performer with total 14 hits; seven acceptable energies and seven acceptable geometries. This was the only functional “recommended” for at least two dimers in each class. The functionals M05, B2PLYPD, B971, mPW2PLYPD, PBEB95, and CAM-B3LYP gave 11 hits while PBEhB95, PW91B95, Wb97x, BRxVP86, BRxP86, HSE2PBE, HSEh1PBE, PBE1PBE, PBEh1PBE, and PW91TPSS gave 10 hits. Among these, M05, B971, mPW2PLYPD, Wb97x, and PW91TPSS were among the “recommended” list of at least one dimer from each class. Long-range correction (LC) of Hirao and coworkers to exchange-correlation functionals showed massive improvement in geometry and Eint. The best performing LC-functionals were LC-G96KCIS and LC-PKZBPKZB. Our results predict that M06L is the most trustworthy DFT method in Gaussian09 to study small non-covalently interacting systems. © 2013 Wiley Periodicals, Inc.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. Supporting Information

Non-covalent interactions are very important in the field of biology (in stabilizing molecules like DNA, proteins, etc), nanomaterials (in the formation of molecular aggregates), and supra molecular chemistry (in the binding of molecular constituents).[1] For computational study of non-covalently interacting systems, Kohn-Sham density functional theory (DFT) methods are attractive for an effective and computationally less demanding research.[2, 3] So far, a large number of density functionals which are applicable to different kinds of systems have been developed and are assigned to different rungs of “Jacob's ladder”[4–6] wherein the first three rungs describes local density approximation (LDA), generalized gradient approximation (GGA), and meta-GGA density functionals. LDA includes the local electron spin density calculations while GGA functionals consider the electron spin density gradient Δρ along with the spin density. The meta-GGA functionals include the spin kinetic energy density in addition to these two factors. The next two rungs, the hybrid and double hybrid functionals, add some amount of Hartree-Fock exchange to the density functional.[4–7] With these constituents, many of the DFT methods are very good in describing the strong interactions among atoms and molecules, but many of them cannot give good descriptions of weak interactions like dispersion and van der Waals interactions. A lot of efforts have been conducted by many researchers in order to find out proper density functionals which give accurate energy, geometry, and thermo chemical properties of non-covalently interacting systems.[2, 7–16] Wu et al. studied a few typical weakly bound systems using both pure and hybrid density functionals. These include four systems of increasing binding strength, namely the Ar2 and Kr2 dimers, the benzene dimer, the water dimer, and a few metal carbonyls. For pure functionals, dispersion dominated systems (noble gases and benzene dimers) and carbonyls showed a strong dependence of results on the choice of functionals while relatively accurate, functional independent results were obtained for systems dominated by electrostatic interaction (water dimers).[13] Zhaho and Truhlar presented four benchmark databases on binding energies of hydrogen bonding, charge transfer, dipole interactions, and weak interactions. Among the 44 tested DFT methods, the best performance for dipole interactions was given by MPW3LYP, B97-1, PBE1KCIS, B98, and PBE1PBE and that for weak interactions was given by B97-1, MPWB1K, PBE1KCIS, and MPW1B95 functionals.[14] Zhaho and Truhler also tested their M05-2X and M06 class of functionals (M06L, M06, M06HF, and M06-2X) against some popular functionals to study non-covalent interactions and found that the performance of M06-2X, M05-2X, M06-HF, M06, and M06L was good.[7] Gkionis et al. studied the performance of BHandH functional for the description of non-covalent interactions using high level ab initio results as benchmarks.[9] Binding energies were well reproduced for dispersion dominated and “mixed” systems, whereas significant overestimation was found in the case of hydrogen bonded systems. Changing the proportion of exact and Slater exchange did not improve the overall performance. Sun et al. assessed the accuracy of nine DFT exchange-correlation functionals in calculating metal-dihydrogen binding energies using MP2 and CCSD(T) results using various high accuracy basis sets as benchmarks and showed that PBE and PW91 functionals give good results while LDA functional overestimating the dihydrogen binding.[12] Georigk and Grimme[2] carried out a benchmark study of 47 density functionals, DFT-D3 corrected forms of most of them, the double hybrids, and the M05 and M06 class of functionals on GMTKN30 database for main group thermo chemistry, kinetics, and non-covalent interactions. They recommended B97-D3 and PBE-D3 in the GGA and TPSS-D3 in the meta-GGA class of functionals and also found that PW6B95 in combination with DFT-D3 was the most robust hybrid functional.[2] Eshuis and Furche assessed the performance of random phase approximation (RPA), a parameter free 5th rung functional for reaction energies governed by changes in medium and long-range non-covalent interactions. The RPA results were found to be more accurate than those of PBE or B3LYP and more systematic than those of B2PLYPD or M06-2X.[10] Burns et al. studied a variety of density functionals against interaction energies at the CCSD(T)/CBS level databases including hydrogen bonded and dispersion dominated systems and concluded that the tested functionals elicit the best performance among the double zeta or triple zeta basis set regimes. They have also recommended suitable model chemistries for systems including different types of non-covalent interactions.[8] Hobza and coworkers have extensively studied non-covalently interacting systems.[17–21]

In this study, our aim is to scan all the density functionals implemented in Gaussian09[22] suite of programs—one of the most widely used quantum chemical programs—to find out those functionals giving a good value for both geometry and interaction energy for non-covalently bonded dimer molecular systems. For screening the DFT results, we have used the CCSD/aug-cc-pVTZ level structures and Ave-CCSD(T)/(Q-T) level non-covalent interaction energies reported by Mackie and DiLabio[23] for nine non-covalently bonded systems as reference values. Tight screening parameters on geometry and interaction energy have been applied to assess the methods. This type of a wide screening of DFT methods based on a strategy that rely on the goodness of both geometry and energy is not yet discussed in the literature.

Computational Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. Supporting Information

A total of 382 density functionals have been selected which consist of LDA, GGA, meta-GGA, hybrid and double hybrid functionals as well as long-range corrected and some dispersion-corrected functionals. The exchange-correlation functionals are made by combining any of the 13 exchange functionals (S, XA,[24–26] B,[27] PW91,[28–32] mPW,[33] G96,[33, 34] PBE,[35, 36] O,[37, 38] TPSS,[39] BRx,[40] PKZB,[41] wPBEh,[42–44] and PBEh[45]) with any of the 13 correlation functionals (VWN,[46] VWN5,[46] LYP,[47, 48] PL,[49] P86,[50] PW91, B95,[51] PBE, TPSS, KCIS,[52–55] PKZB, VP86, and V5LYP) (total 169 combinations). Furthermore, long-range correction (LC) of Hirao and coworkers[56] is applied to 156 out of 169 exchange-correlation functionals (the functionals with wPBEh exchange cannot be done). The prefix notation LC is used exclusively for LC of Hirao and coworkers. Pure functionals selected for this study are B97D,[3] HCTH, HCTH147, HCTH407, HCTH93,[57–59] M06L,[60] tHCTH,[61] and VSXC.[62] LC of Hirao and coworkers is also applied for six pure functionals (LC cannot be applied for VSXC and HCTH). The selected hybrid functionals are B1B95[51], B1LYP[51, 63], B3LYP, B3P86, B3PW91,[64] B971,[59] B972,[65] B98,[66, 67] BHandH, BhandHLYP,[68] BMK,[69] HSE2PBE, HSEh1PBE,[42, 44, 70–74] M05,[75] M052X,[76] M06, M062X,[77] M06HF,[78, 79] mPW1LYP, mPW1PBE, mPW1PW91,[33, 51, 63] mPW3PBE, O3LYP,[80] PBE1PBE,[81] PBEh1PBE,[45] tHCTHhyb,[61] TPSSh,[39] and X3LYP.[82] Gaussian09 also supports the long range-corrected hybrid functionals LC-wPBE,[83–86] CAM-B3LYP[87] (long-range corrected version of B3LYP by Handy and coworkers), wB97, wB97X[88] and wB97XD[89] (functionals with LC from Head-Gordon and coworkers). Also included in the study are three exchange only functionals (HFB,[27] HFS, and Xalpha[24–26]), their LC-corrected forms and the double hybrid functionals B2PLYP, B2PLYPD, mPW2PLYP, and mPW2PLYPD.[90–92]

The non-covalent dimers under the investigation are selected from Ref. [23] which are categorized in to three classes. Class 1 includes “dispersion” dominated systems ((C2H2)2, (C2H6)2, and (CO2)2). Class 2 systems (CH4…NH3, CH4…HF, and C2H4…HF) are characterized by “dipole-induced dipole” interactions while class 3 dimers ((CH3CN)2, (HCHO)2, and (CH3F)2) possess “dipole-dipole” interactions. In Ref. [23], the optimization of these systems were done using the high accuracy ab initio method CCSD with aug-cc-pVTZ basis sets while the interaction energies were computed using a rigorous procedure, the Ave-CCSD(T)/(Q-T) level which yielded results at very high accuracy. We have optimized all the nine dimers belonging to the three classes using the selected 382 functionals. The RMSD of the DFT level optimized geometry from the CCSD/aug-cc-pVTZ geometry as well as the percentage deviation of the DFT interaction energy from the Ave-CCSD(T)/(Q-T) interaction energy was calculated. To calculate root mean square deviation (RMSD), one dimer is superimposed on the other in the best possible way and this accounts for the deviations of both inter and intramolecular geometry parameters. RMSD value less than 0.045 was selected as a criterion for “good geometry” and interaction energy (Eint) satisfying the condition that 0.8Eint0 < Eint < 1.2Eint0, where Eint0 is the Ave-CCSD(T)/(Q-T) level interaction energy, was considered as “good interaction energy.” Functionals satisfying both the criteria were listed as “recommended” for the non-covalent dimers. Eint is calculated using the supermolecule approach, viz. Eint = Edimer − (Emonomer1 + Emonomer2). BSSE correction to the interaction energy was found to be very small (0.0–0.4 kcal/mol) for all the systems at all levels of theory and hence this correction is not included in further discussions.

Results and Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. Supporting Information

Dispersion dominated systems

Acetylene dimer

Eint0 of acetylene dimer is −1.38 kcal/mol (Ave-CCSD(T)/(Q-T) value reported in Ref. [23]). Among the 382 DFT methods, 129 gave a good geometry (RMSD < 0.045) and 104 gave a good value of Eint (0.8Eint0 < Eint < 1.2Eint0). However, only 49 of them gave both good geometry and good Eint. Out of these 48 functionals, 23 were exchange-correlation functionals, 22 were LC-exchange-correlation functionals, two were pure functionals (M06L, B97D), and two were hybrid functionals (M05, B971) (Fig. 1). Interestingly, the correlation part of 20 out of 22 recommended LC-functionals were either LYP (10 cases) or V5LYP (10 cases). Among the “recommended” functionals, the best geometry was given by M06L (RMSD = 0.002, Fig. 2). The best value of Eint was given by two LC-functionals, LC-OLYP, and LC-OV5LYP (−1.376 kcal/mol for both).Further the functionals PBEhV5LYP, PBEhLYP, wPBEhV5LYP, wPBEhLYP, PBEV5LYP, and PBELYP performed exceedingly well for reproducing both the CCSD geometry and Eint (Fig. 1). The Eint values are −1.362 kcal/mol for PBEhV5LYP, PBEhLYP, wPBEhV5LYP, and wPBEhLYP and −1.375 kcal/mol for PBEV5LYP, and PBELYP.

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Figure 1. (A) and (B) The recommended functionals given in the increasing order of RMSD for acetylene dimer. Exchange-correlation (bold), LC-exchange-correlation (normal font), pure (Italic) and hybrid (bold Italic) functionals are also indicated. # indicates recommended functionals common for with and without LC.

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Figure 2. M06L/6-311++g(d,p) level optimized geometry of acetylene dimer. Distances are given in Å. CCSD/aug-cc-pVTZ level values are in parenthesis.

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In general, among the recommended functionals, most of the exchange-correlation functionals performed well for geometry than the LC-functionals while most of the LC-functionals yielded better Eint than the exchange-correlation functionals. A noteworthy feature is that many of the functionals having poor performance on both geometry and Eint showed big improvement (Supporting Information Table S2) with LC and many of them even satisfied the recommended criteria with LC (Fig. 1). Notable among them are exchange-correlation functionals with LDA exchange S and those with GGA exchange B, BRx, G96, O, and XA. For example, the OLYP and OV5LYP functionals showed Eint = −0.504 kcal/mol and RMSD = 0.370 while their LC-incorporated forms gave the best Eint among all the recommended functionals as well as a good RMSD (0.044). But such strong improvement was not observed in the case of exchange-only and pure functionals. With LC, the poor results given by the exchange only functionals Xalpha and HFS became worse and those given by HFB showed a slight improvement (Supporting Information Table S2). Among the pure functionals, the results given by the HCTH functionals with LC and without LC were poor except the fact that HCTH407 gave a good value for Eint (−1.355 kcal/mol). The pure functionals M06L and B97D gave very good results for both geometry and energy while with LC, they gave the worst results for Eint.

Ethane and CO2 dimers

Eint0 of ethane dimer is −1.36 kcal/mol and that of CO2 dimer is −1.48 kcal/mol. For ethane dimer, among the 382 functionals, 83 functionals gave RMSD < 0.045 while 64 satisfied the condition 0.8Eint0 < Eint < 1.2Eint0. Only 19 satisfied geometry and energy criteria (Fig. 3a). Among the recommended 19 functionals, two exchange-correlation (PW91P86, PW91VP86), 16 LC-exchange-correlation, and one pure functional (M06L) were found. The correlation part of all the sixteen LC-exchange-correlation functionals were either LYP (8 cases) or V5LYP (8 cases).

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Figure 3. The recommended functionals given in the increasing order of RMSD for (A) ethane dimer and (B) CO2 dimer. Exchange-correlation (bold), LC-exchange-correlation (normal font), pure (Italic), hybrid (bold Italic) and double hybrid (in parenthesis) functionals are also indicated. # indicates recommended functionals common for with and without LC.

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In the case of CO2 dimer, 26 showed RMSD < 0.045 and 53 satisfied the Eint criterion while only 14 methods satisfied both criteria. Out of these 14 functionals, one exchange-correlation (PW91B95), 10 LC-exchange-correlation, one hybrid (M052X), one hybrid with LC of Head-Gordon (Wb97x), and one double hybrid (mPW2PLYPD) functionals were found. Interestingly, the correlation part of all the recommended LC- functionals and one exchange-correlation functional was B95.

Among the recommended functionals for ethane dimer, the best geometry was given by PW91P86 (RMSD = 0.035, Fig. 4a) and the best Eint was given by M06L (Eint= −1.492 kcal/mol). For CO2 dimer, the best geometry was given by PW91B95 functional (RMSD = 0.010, Fig. 4b) and the best Eint was given by LC-SB95 (Eint = −1.488 kcal/mol) functional.

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Figure 4. a) PW91P86/6-311++g(d,p) level optimized geometry of ethane dimer b) PW91B95/6-311++g(d,p) level optimized geometry of carbon dioxide dimer. CCSD/aug-cc-pVTZ level values are in parenthesis. Distances in Å.

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Functionals with B, G96, O, S, and XA exchange gave better results for both geometry and energy in their LC forms than their pure forms for ethane dimer. Though the S and XA exchange functionals gave the worst energies, substantial improvement was observed with LC. For instance, the RMSD and Eint values given by SLYP functional were 0.295 and −6.628 kcal/mol respectively while those of LC-SLYP were 0.045 and −1.161 kcal/mol, satisfying both geometry and energy criteria. Similarly, functionals with BRx, mPW, and TPSS exchange, with few exceptions, reproduced the CCSD results remarkably well in their LC forms. The functionals with PBE, PBEh, PKZB, and PW91 exchange parts showed irregular trends. In some cases, LC forms gave better results whereas in some other cases, non-LC forms performed better. Exchange-only functionals, with and without LC performed poorly for both geometry and energy. Among the pure functionals, M06L performed well to reproduce both energy and geometry close to the CCSD level accuracy. However, with LC, all pure functionals gave bad results for energies (Supporting Information Table S3).

In the case of CO2 dimer, functionals with B, BRx, G96, O, mPW, PKZB, S, TPSS, and XA exchange with all types of correlation gave good results in the LC forms. The geometries and energies given by some of them without LC were very poor, but after LC incorporation, some of them (BB95, G96B95, OB95, mPWB5, PKZBB95, SB95, and TPSSB95) were improved to the recommended levels. Functionals with PBE, PBEh, and PW91 exchange gave comparable results in LC and non-LC forms and in many cases; functionals without LC gave better results. Exchange-only functionals, neither with nor without LC gave acceptable results. Some of the pure functionals gave very good results without LC while they performed poorly in the LC forms (Supporting Information Table S4).

Dipole-induced dipole systems

Ammonia-methane complex

Ammonia-methane complex has Eint0 = −0.76 kcal/mol, the weakest interaction among all the complexes. Out of the 382 functionals, 68 gave RMSD < 0.045, and 68 gave Eint in the range 0.8Eint0 < Eint < 1.2Eint0. However, only 25 satisfied both criteria (Fig. 5). These 25 recommended functionals include 14 exchange-correlation, 10 LC- exchange-correlation, and one pure functional (HCTH147). All the 10 recommended LC-exchange-correlation functionals have VWN as their correlation part. PBEVWN functional gave the best geometry (RMSD = 0.005, Fig. 6) and mPWB95 gave the best interaction energy (Eint = −0.769 kcal/mol). Almost all the recommended functionals gave excellent geometry while the functionals PBEVWN, PBEhVWN, wPBEhVWN, PBEhPL, wPBEhPL, PBEVWN5, PBEPL, mPWB95, TPSSB95, and mPWKCIS performed very well for both geometry and interaction energy.

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Figure 5. The recommended functionals given in the increasing order of RMSD for ammonia-methane complex. Exchange-correlation (bold), LC- exchange-correlation (normal font) and pure (Italic) functionals are also indicated. # indicates recommended functionals common for with and without LC.

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Figure 6. PBEVWN/6-311++g(d,p) level optimized geometry of ammonia-methane dimer. CCSD/aug-cc-pVTZ level values are in parenthesis. Distances in Å. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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As observed in the case of acetylene dimer, very poor results were obtained for both geometry and interaction energy with B, BRx, G96, S, O, and XA as exchange part in many cases while they all showed massive improvement, particularly on the Eint with LC incorporation (Supporting Information Table S5) and some of them with VWN correlation (LC-OVWN, LC-BVWN, LC-G96VWN, LC-SVWN, LC-BRxVWN) even qualified to the recommended levels (Fig. 5). All the pure functionals showed poor behavior with LC incorporation, particularly Eint was the worst compared to other methods. In many cases, functionals with PBE, PBEh, and PW91 exchange gave good geometries compared to many other functionals. In general, a good geometry produced by an exchange-correlation functional was not improved by further incorporation of LC. However, with LC-functionals, overall the estimation of Eint was significantly improved in many cases. In the case of exchange only functionals, improvement in geometry and energy was observed with LC, though not to the recommended levels.

CH4…HF and C2H4…HF

Eint0 is −1.64 kcal/mol for CH4…HF and −4.50 kcal/mol for C2H4…HF. For CH4…HF, 121 functionals gave RMSD < 0.045 and 113 gave Eint in the range 0.8Eint0 < Eint < 1.2Eint0 while 54 satisfied both criteria (Fig. 7). These 54 recommended functionals include 32 exchange-correlation, three LC-exchange-correlation, one pure (M06L), 16 hybrid, and two double hybrid functionals (B2PLYPD and mPW2PLYPD). In the case of C2H4…HF, 177 functionals gave RMSD < 0.045 and 131 satisfied the condition on Eint. Furthermore, 127 functionals satisfied both criteria (Fig. 8) and these include 54 exchange-correlation, 36 LC-exchange-correlation, five pure, 28 hybrid, and four double hybrid functionals. The correlation part of the LC-exchange-correlation functionals was PL (12 out of 36), VWN (11 out of 36) or VWN5 (13 out of 36).

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Figure 7. The recommended functionals given in the increasing order of RMSD for CH4…HF. Exchange-correlation (bold), LC- exchange-correlation (normal font) pure (Italic), hybrid (bold Italic) and double hybrid (in parenthesis) functionals are also indicated.

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Figure 8. The recommended functionals given in the increasing order of RMSD for C2H4…HF. Exchange-correlation (bold), LC- exchange-correlation (normal font) pure (Italic), hybrid (bold Italic) and double hybrid (in parenthesis) functionals are also indicated. # indicates recommended functionals common for with and without LC.

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In the case of CH4…HF, the best geometry was given by the hybrid functional mPW1LYP (RMSD = 0.002, Fig. 9a), while the best Eint was given by LC-TPSSKCIS (Eint = −1.634 kcal/mol). In the case of C2H4…HF, the best geometry was obtained with B1B95 functional (RMSD = 0.002, Fig. 9b) whereas the best Eint was given by wPBEhVWN and PBEhVWN (Eint = −4.495 kcal/mol for both) methods. In the case of C2H4…HF, several LC methods performed exceedingly well for reproducing both the geometry and Eint at the CCSD level accuracy while in the case of CH4...HF, only three LC methods (LC-PKZBKCIS, LC-PKZBKCIS, and LC-TPSSKCIS) performed to the recommended levels of accuracy.

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Figure 9. (a) mPW1LYP/6-311++g(d,p) optimized geometry of CH4…HF complex. (b) B1B95/6-311++g(d,p) optimized geometry of C2H4…HF complex. CCSD/aug-cc-pVTZ level values are in parenthesis. Distances in Å. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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In the case of CH4...HF, exchange-correlation functionals with B (with few exceptions), G96, O, S, and XA exchange showed very good improvement with LC for reproducing both geometry and energy. Functionals having BRx, mPW, PBE, PBEh PW91, and TPSS exchange, except with PL, VWN, and VWN5 correlation, gave better geometries without LC. But energies showed dissimilar and irregular trends. LC-functionals gave good energies in some cases, whereas in some other cases non LC forms gave better results. For PBE and PBEh exchange, both non-LC and LC forms showed similar results. Exchange-only functionals were slightly improved on LC incorporation, but not to the “recommended” levels. Pure functionals, especially M06L produced good results while all of them performed badly with LC incorporation (Supporting Information Table S6).

In the case of C2H4...HF, many exchange-correlation functionals with and without LC gave very good results for both geometry and energy (Fig. 8). The exchange-correlation functionals made up of B, BRx, G96, mPW, PBE, PBEh, PW91, and TPSS exchange and PL, VWN, and VWN5 correlation are particularly noteworthy as they showed very good improvement in geometry with LC incorporation while the non-LC forms fared better when these exchange functions where combined with other correlation functions. Though the energies did not show a regular trend, the difference between the LC and non-LC functionals were not very high compared to other dimers. Functionals with S and XA exchange gave very poor results while their LC-forms showed acceptable values. Similarly, the exchange-only functionals though showed good improvement for geometry and Eint with LC, none of them satisfied the recommended values. The good results given by pure functionals became worse with LC, particularly the interaction energy (Supporting Information Table S7).

Dipole-dipole systems

Acetonitrile dimer

Among all the systems, acetonitrile has the highest Eint0 (−6.28 kcal/mol). For this systems, 159 gave good geometry (RMSD < 0.045) and 170 gave good Eint (0.8Eint0 < Eint < 1.2Eint0) while 104 satisfied both criteria. Among the 104 functionals, 31 exchange-correlation, 58 LC-exchange-correlation, two pure (B97D and M06L), 11 hybrid, and two double hybrid functionals (B2PLYPD and wPW2PLYPD) were located (Fig. 10). The best geometry was given by the long range-corrected hybrid functional CAM-B3LYP (RMSD = 0.014, Fig. 11) and the best value for Eint was given by M06L (Eint = −6.242 kcal/mol).

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Figure 10. The recommended functionals given in the increasing order of RMSD for acetonitrile dimer. Exchange-correlation (bold), LC- exchange-correlation (normal font), pure (Italic), hybrid (bold Italic) and double hybrid (in parenthesis) functionals are also indicated. # indicates recommended functionals common for with and without LC.

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Figure 11. CAM-B3LYP/6-311++g(d,p) optimized geometry of acetonitrile dimer. CCSD/aug-cc-pVTZ level values are in parenthesis. Distances in Å. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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Among the recommended methods given in Figure 10, many of the methods without LC gave better geometries (Figs. 10a and 10d) than those with LC (Figs. 10b and 10c). However, the bad geometries as well as bad interaction energies given by the functionals with G96, O, PKZB, S, TPSS, and XA exchange showed massive improvement with LC (Supporting Information Table S8). For instance, the (RMSD, Eint) for G96LYP, OLYP, PKZBLYP, XALYP, and SLYP were (0.253–1.277 kcal/mol), (0.290–2.766 kcal/mol), (0.157–4.322 kcal/mol), (0.206–14.460 kcal/mol), and (0.177–13.572 kcal/mol), respectively, whereas in their LC forms, these were improved to (0.032–6.681 kcal/mol), (0.037–6.628 kcal/mol), (0.021–6.481 kcal/mol), (0.050–11.142 kcal/mol), and (0.034–6.849 kcal/mol), respectively. Some of these methods, particularly the SLYP satisfied the criteria for both geometry and Eint. Similarly, the bad geometries and bad interaction energies given by functionals with B and mPW exchange showed good improvement with LC. In many cases, the functionals with BRx, PBE, and PBEh exchange gave good values for Eint and they were only slightly improved with LC.

Formaldehyde and fluoromethane dimers

Eint0 of (HCHO)2 is −3.72 kcal/mol and that of (CH3F)2 is −2.43 kcal/mol. For (HCHO)2, 108 functionals gave RMSD < 0.045 and 159 gave Eint in the range 0.8Eint0 < Eint < 1.2Eint0 while 75 satisfied both criteria (Figs. 12a–12c). They include 25 exchange-correlation, 35 LC-exchange-correlation, 12 hybrid, and three double hybrid functionals. The correlation part of the LC-functionals showed some similarities; 13 out of 35 had VWN as their correlation part while others were VWN5 (11 cases) and PL (11 cases). In the case of (CH3F)2, a large number of functionals (181) gave Eint in the range 0.8Eint0 < Eint < 1.2Eint0 while only five gave RMSD < 0.045. Four functionals satisfied both geometry and energy criteria (Fig. 12d). These four functionals include one exchange-correlation (mPWP86), two hybrid (wB97XD and M06), and one pure (M06L) functionals. The (RMSD, Eint) values of mPWP86, wB97XD, M06, and M06L are (0.045, −2.204 kcal/mol), (0.015, −2.758 kcal/mol), (0.043, −2.290 kcal/mol), and (0.005, −1.992 kcal/mol), respectively.

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Figure 12. The recommended functionals given in the increasing order of RMSD. (A), (B) and (C) for formaldehyde dimer and (D) for fluoromethane dimer. Exchange-correlation (bold), LC- exchange-correlation (normal font), pure (Italic), hybrid (bold Italic) and double hybrid (in parenthesis) functionals are also indicated.

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The best geometry for formaldehyde dimer was given by mPW2PLYP (RMSD = 0.002, Fig. 13a) and the best interaction energy was given by PBELYP and PBEV5LYP (Eint = −3.707 kcal/mol). For fluoromethane dimer, the best geometry was given by M06L (RMSD = 0.005, Fig. 13b) and the best interaction energy was given by M06 functional (Eint = −2.290 kcal/mol).

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Figure 13. a) mPW2PLYP/6-311++g(d,p) optimized geometry of formaldehyde dimer. b) M06L/6-311++g(d,p) optimized geometry of fluoromethane dimer. CCSD/aug-cc-pVTZ level values are in parenthesis. Distances in Å. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

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In the case of formaldehyde dimer, functionals with G96, O, PKZB, S, and XA exchange gave poor results. LC incorporation showed very good improvement in geometry and energy and some LC forms even appeared in the recommended list (Fig. 12). Remaining functionals showed mixed trends. Functionals with PBE, PBEh, and PW91 exchange gave good geometries without LC in most of the cases. Exchange-only functionals neither with nor without LC gave good results (Supporting Information Table S9).

In the case of fluoromethane dimer, exchange-correlation functionals with G96, O, S, and XA exchange showed very good improvement with LC. With PBE, PBEh, and PW91 exchange, the difference between LC and non-LC values were not very significant. Others showed irregular tendencies for energies and geometries. Similarly, the exchange-only functionals showed irregular trends and none of them appeared among the recommended functionals. Pure functionals gave good values for energy and geometry in the non-LC forms (Supporting Information Table S10).

Best performing functionals for the nine dimers

In order to find out the “best performing” functionals to study majority of the nine systems, we counted the number of acceptable values given by each functional for energy and geometry separately. A functional was selected in the best performing list if it provided at least five hits for acceptable geometry and five hits for acceptable energy. There were 17 functionals satisfying these criteria (Table 1). In this list, M06L functional emerged as the most outstanding with total hit count of 14 out of a maximum of 18. In fact, this was the only functional which showed good agreement on both geometry and energy to at least two systems in each class. Also, this was the functional recommended for maximum number of dimers (6 dimers). None of the energies given by this functional showed an error greater than 25% compared to the Ave-CCSD(T)/(Q-T) value and the maximum RMSD showed by an M06L optimized geometry was 0.109. Other functionals in the best performing category were M05, B2PLYPD, B971, mPW2PLYPD, PBEB95, and CAM-B3LYP giving 11 hit counts. Out of these, M05, B971, and mPW2PLYPD were recommended for five dimers and there were at least one dimer from each class. Remaining functionals in the best performing list were PBEhB95, PW91B95, Wb97x, BRxVP86, BRxP86, HSE2PBE, HSEh1PBE, PBE1PBE, PBEh1PBE, and PW91TPSS giving 10 hit counts. Among these, Wb97x and PW91TPSS were recommended for four dimers and there were at least one dimer from each class. CAM-B3LYP, HSE2PBE, HSEh1PBE, PBE1PBE, and PBEh1PBE functionals were also recommended for four systems but they did not appear among the “recommended” functionals of any of the class 1 dimers. We also note that functionals such as PBEhB95, PW91B95, BRxP86, and BRxVP86 have never been discussed in the literature and hence further studies may be required to validate their use as recommended methods for non-covalent interactions.

Table 1. The list of best performing DFT methods.
 No. of acceptable RMSDNo. of acceptable EintTotal hit countsAve. RMSD (of all 9 systems)Ave.% ΔEint (of all 9 systems)
M06L77140.04213
M0556110.040−5
B2PLYPD65110.04322
B97156110.05112
mPW2PLYPD56110.04616
PBEB9556110.04920
CAM-B3LYP56110.05525
PBEhB9555100.04921
PW91B9555100.05224
Wb97x55100.05621
BRxVP8655100.04932
BRxP8655100.05132
HSE2PBE55100.06926
HSEh1PBE55100.07126
PBE1PBE55100.07127
PBEh1PBE55100.07227
PW91TPSS55100.08728

Thus the best performing list consists of six exchange-correlation functionals, one pure functional (M06L), six hybrid functionals, long-range corrected version of B3LYP by Handy and coworkers (CAM-B3LYP), one long-range corrected functional from Head-Gordon and coworkers (Wb97x), and two double hybrid functionals with dispersion correction (B2PLYPD, mPW2PLYPD). It may be noted that these functionals were screened from a total of 169 exchange-correlation, eight pure, 33 hybrid, and four double hybrid functionals.

Even though the LC of Hirao and coworkers applied on exchange-correlation functionals (LC-forms) showed massive improvement in describing the geometry and Eint of all molecular systems, none of the methods consistently gave good results. Furthermore, none of LC functionals could give at least five good geometries along with five good Eint. Hence, we looked for LC-methods which show at least four acceptable geometries, four acceptable energies and a total hit count nine to make the list of the best performing LC-methods (Table 2). Notable among them were LC-G96KCIS, LC-PKZBPKZB, LC-PKZBKCIS, LC-OKCIS, and LC-TPSSKCIS whose ave. RMSD and ave. %ΔEint were close to the acceptable values. Among these, LC-G96KCIS and LC-PKZBPKZB gave 10 hit counts with four acceptable geometries and six acceptable energies.

Table 2. The list of best performing LC-exchange-correlation functionals.
FunctionalNo. of acceptable RMSDNo. of acceptable EintTotal hit countsAve. RMSD (of all 9 systems)Ave.% ΔEint (of all 9 systems)
LC-G96KCIS46100.04818
LC-PKZBPKZB46100.04521
LC-PKZBKCIS4590.04519
LC-OKCIS4590.04620
LC-TPSSKCIS4590.04719
LC-BRxVWN5490.07727
LC-G96VWN5490.08231
LC-PW91VWN5490.08331
LC-PBEhVWN5490.08331
LC-mPWVWN5490.08331
LC-BVWN5490.08331
LC-SVWN5490.08429
LC-PBEVWN5490.08431
LC-OVWN5490.08432

Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. Supporting Information

Among the 382 density functionals studied, M06L outperformed all the other functionals in calculating the geometry and interaction energy of the dimers with accuracy close to that of CCSD. The performance of M05, B971, mPW2PLYPD, Wb97x, and PW91TPSS functionals was also good since these functionals gave acceptable energy and geometry for at least one system from each class. The effect of LC on exchange-correlation functionals, especially on those with S, G96, O, and XA exchange was remarkable. The use of LC is highly recommended for almost all exchange-correlation functionals. Among dispersion dominated class 1 dimers, exchange-correlation, and LC-exchange-correlation functionals performed well along with a few pure, hybrid, and double hybrid functionals. Only a few functionals were recommended for the CO2 dimer while several functionals could reproduce correct geometry for the C2H2 dimer. For C2H4…HF, a member of the “dipole-induced dipole” class 2, several functionals gave very good results. This was the only system for which most of the functionals with B and BRx exchange gave good geometry as well as good energy. All the dimers of this class showed good performance with exchange-correlation functionals. The CH3F dimer in the “dipole-dipole” class 3 was the toughest to deal with as only five functionals could give an acceptable geometry. Only for “dipole-induced dipole” systems, the performance of mPWB95 and TPSSB95 was exceptional as they gave good geometry and energy for all the three systems. In all the three classes, some functionals gave acceptable Eint while some others gave acceptable geometry (Supporting Information Fig. S1–S3) for all the three dimers. Although, a functional that satisfies geometry and energy criteria for all the molecules in three classes was not found, M06L gave the most satisfactory results with seven acceptable geometries and seven acceptable energies. Moreover, the average RMSD and %ΔEint values given by this functional were in the acceptable range for all the systems.

Among the “recommended” LC-exchange-correlation functionals, a dominance of LYP and V5LYP correlation is observed for (C2H2)2 and (C2H6)2 while B95 correlation showed good performance for (CO2)2. To describe CH4…NH3 and C2H4…HF in class 2 and (HCHO)2 in class 3, VWN and VWN5 have emerged as the best correlation functions with LC. PBE, PBEh, and PW91 exchange were not much influenced by LC whereas S, G96, O, and XA exchange-correlation functionals must always be used along with LC to get good values for geometry and interaction energy. On the basis of this study, we recommend that M06L is the best DFT method in Gaussian09 for the study of small non-covalently interacting systems. M05, B971, B2PLYPD, mPW2PLYPD, PBEB95, and CAM-B3LYP can also be recommended on the basis their total number of hit counts and the average of RMSD and %ΔEint. The best among the LC-exchange-correlation methods are LC-G96KCIS and LC-PKZBPKZB.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. Supporting Information

Support from Council of Scientific and Industrial Research (CSIR) through a network project (CSC0129) to CHS and research fellowship to KR is gratefully acknowledged. The authors are also grateful to Dr. Gino A. DiLabio at the National Institute for Nanotechnology, Alberta, Canada for providing us with the CCSD/ aug-cc-pVTZ level geometries of the dimer systems.

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Computational Methods
  5. Results and Discussion
  6. Conclusions
  7. Acknowledgements
  8. Supporting Information

Additional Supporting Information may be found in the online version of this article.

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