Article pubs.acs.org/JPCA
Double-Hybrid Density Functionals Free of Dispersion and Counterpoise Corrections for Non-Covalent Interactions Feng Yu* Department of Physics, School of Science, Xi’an Technological University, No. 4 Jinhua North Road, Xi’an, Shaanxi 710032, China S Supporting Information *
ABSTRACT: We have optimized two double-hybrid density functionals (DHDFs) within the frameworks of B2PLYP and mPW2-PLYP against the S22B database. These two functionals are denoted as B2NC-PLYP and mPW2NC-PLYP, where “NC” represents noncovalent interaction. The DHDFs of B2NC-PLYP and mPW2NC-PLYP are optimized free of dispersion and counterpoise corrections with triple-ζ quality basis sets. Combined with the aug-cc-pVTZ basis set, these two functionals are further assessed with the S66 database. According to our computations, both the B2NC-PLYP and mPW2NC-PLYP functionals seem to be competent for investigating noncovalent interactions. Note that the triple-ζ quality basis sets with adequate polarization and diffuse functions should be employed for practical applications. However, different exchange and correlation functionals may be selected and/or modified to reduce the amount of the Fock-exchange in the future. improves the performances of the DHDFs.55 Without additional corrections, the XYG327 and XYGJ-OS28 are valuable functionals for studying noncovalent interactions.47 With the help of the localized molecular orbital energy decomposition analyses (LMO-EDA),62 two parameters of cHF = 0.81 and cMP2 = 0.52 for the framework of the B2PLYP functional were extrapolated with five HCOOH···C6H6 complexes.56 The reference molecular geometries and interaction energies were taken from ref 63. After they were examined with the noncovalent complexation energies database (NCCE31/ 05),8,64,65 we found that these two parameters for the framework of the B2PLYP functional were appropriate for investigating noncovalent interactions,56 at least suitable for the small and medium-sized noncovalent systems. And, thus, we denoted these two parameters for the framework of the B2PLYP functional as B2N-PLYP, where “N” stood for noncovalent interaction.56 However, the training set for the B2N-PLYP functional was very small, and the corresponding parameters should be improved with a well-designed training set. In this work, we have optimized the parameters of cHF and cMP2 for the frameworks of the B2PLYP and mPW2-PLYP functionals with the S22 database.66−69 Note that the S22B69 revision has been utilized as reference. Subsequently, the optimized parameters are further assessed with the S66 database.70,71 The main purpose of this work is to treat noncovalent interactions within the framework of the DHDF,
1. INTRODUCTION Different types of noncovalent interactions are fundamental topics in chemistry, because noncovalent interactions play significant roles in cluster chemistry, supramolecular chemistry, biochemistry, solution chemistry, surface chemistry, and so on.1,2 Therefore, developing and assessing ab initio methods and density functionals for noncovalent interactions are very important for theoretical and computational chemistry.1−15 However, elucidating the natures of noncovalent interactions is also important,1−3 and the favorable computational tools should perform well on various noncovalent interactions with different essences. Double-hybrid density functionals (DHDFs) have grown to be very powerful tools for thermochemistry, thermochemical kinetics, noncovalent interaction, etc.16−57 The B2PLYP18 functional has inspired rapid development of various DHDFs,19−57 and the exchange−correlation energy Exc for the B2PLYP functional was defined as Exc = (1 − c HF)ExGGA + c HFExHF + (1 − c MP2)EcGGA + c MP2EcPT2
(1)
To study noncovalent interactions, the dispersion corrections9,58−60 have been applied to the DHDFs.20,22,23,34−36,40 The DHDFs with dispersion corrections perform very well on different types of noncovalent interactions. For example, the B2PLYP-D3 20,60 functional is a reasonable choice for investigating noncovalent interactions based on systematic evaluations.10,61 Furthermore, the DHDFs with long-range corrections perform well on noncovalent interactions,25,26,54 and the nonlocal van der Waals method also significantly © 2014 American Chemical Society
Received: January 17, 2014 Revised: April 6, 2014 Published: April 10, 2014 3175
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3. RESULTS AND DISCUSSION 3.1. Assessment of Different (cHF, cMP2) Combinations for the Frameworks of the B2PLYP and mPW2-PLYP Functionals with the S22B Database. Different (cHF, cMP2) combinations for the framework of the B2PLYP functional have been assessed with the S22B database. The 6-311+G(2df,2p), i.e., MG3S for the S22B database, has been employed for single point energy computations. As listed in Table 1, the B2N-PLYP
and further corrections specifically designed for the DHDFs such as dispersion,20,22,23,34−36,40 long-range,25,26,54 and nonlocal van der Waals55 corrections could be ignored.
2. COMPUTATIONAL METHODS The databases of S22,66−69 S66,70,71 and ADIM6 (interaction energy database for six n-alkane dimers)40,72,73 for noncovalent interactions were employed in the present work. The S22B69 revision of the S22 database was used as the training set for the optimizations of the cHF and cMP2 parameters for the frameworks of the B2PLYP and mPW2-PLYP functionals. And then, the S6670,71 database was utilized as a validation set, and the corresponding reference interaction energies were taken from ref 71. As to the methane dimer in the S22 database, the dispersion component for the total interaction energy was −0.89 kcal/mol, whereas the corresponding electrostatic component was only −0.19 kcal/mol 74 from accurate symmetry-adapted perturbation theory (SAPT)75,76 computation. Therefore, we deduced that the dispersion components were extremely important for the dimers in the ADIM6 database. Accordingly, the ADIM6 database was tested in this work. Additionally, the interaction potential wells for some rare gas dimers77 were also tested. The basis sets of 6-31+G(d,p),78−82 6-311+G(d,p),82−84 6311+G(2df,2p), 82−84 6-311+G(2df,2pd), 82−84 6-311++G(3df,3pd),82−84 aug-cc-pVDZ,85−88 aug-cc-pVTZ,85−88 ccpVTZ,85 def2-TZVPP,89 ma-TZVPP,90 def2-QZVP,89 and maQZVP90 were employed for single point energy computations. The basis set superposition error (BSSE) was not corrected with the counterpoise (CP) method91 unless otherwise noted. For the S22 database, the 6-311+G(2df,2p) basis set was identical to the MG3S basis set defined by Truhlar and coworkers,92 which was accurate enough for relative energy computations. Therefore, the basis set of 6-311+G(2df,2p) was utilized in the optimization process of the cHF and cMP2 parameters within the framework of B2PLYP. In detail, the computations with the 6-31+G(d,p), 6-311+G(d,p), 6-311+G(2df,2p), and aug-cc-pVDZ basis sets were performed with the General Atomic and Molecular Electronic Structure System (GAMESS) program,93,94 and other computations with the 6311+G(2df,2pd), 6-311++G(3df,3pd), aug-cc-pVTZ, cc-pVTZ, def2-TZVPP, ma-TZVPP, def2-QZVP and ma-QZVP basis sets were carried out with the ORCA program.95 As to the computations related to the framework of mPW2-PLYP, only the ORCA program was employed. The corresponding cHF and cMP2 parameters were optimized with the 6-311+G(2df,2pd) basis set. Comparisons between the GAMESS and ORCA programs on single point energy computations were presented in the Supporting Information (Table S1). It should be emphasized that the chemical core electrons were frozen in the present work. In addition, the Lebedev (96,302) integration grid96 was employed for the computations performed with the GAMESS program, whereas the Grid4 was utilized for the computations carried out with the ORCA program. On the basis of our previous work,97 these two integration grids were suitable for single point energy computations. The performances of the DHDFs with different basis sets were evaluated with the signed mean deviation (MD), mean absolute deviation (MAD), and mean absolute percent deviation (MA%D). The corresponding definitions had been described in ref 10.
Table 1. Performances of Different (cHF, cMP2) Combinations for the Framework of the B2PLYP Functional with the S22B Databasea DHDF B2PLYP B2T-PLYP B2GP-PLYP B2K-PLYP B2N-PLYP
(cHF,cMP2) (0.53, (0.60, (0.65, (0.72, (0.81,
0.27) 0.31) 0.36) 0.42) 0.52)
MD
MAD
MA%D
1.62 1.26 0.96 0.57 0.05
1.64 1.29 0.99 0.63 0.18
39.3 31.8 25.1 17.0 6.4
a
The 6-311+G(2df,2p) basis set is utilized for single point energy computations, and the MD (kcal/mol), MAD (kcal/mol), and MA%D are listed as comparisons.
functional performs relatively well with the S22B database. On the basis of the performance of the B2N-PLYP functional, we have examined a total of 35 (cHF, cMP2) combinations around the B2N-PLYP functional against the S22B database. As shown in Figure 1a, the range of the MADs of the tested (cHF, cMP2) combinations is about 0.15−0.30 kcal/mol. As a result, the MAD of the (cHF = 0.81, cMP2 = 0.55) combination is only 0.15 kcal/mol, and the corresponding MA%D is 5.9%. Therefore, we denote the (cHF = 0.81, cMP2 = 0.55) combination for the framework of the B2PLYP functional as B2NC-PLYP, where “NC” represents noncovalent interaction. The B2NC-PLYP functional is an improved revision for the B2N-PLYP functional. Similarly, a total of 45 (cHF, cMP2) combinations for the framework of the mPW2-PLYP functional were also assessed with the 6-311+G(2df,2pd) basis set against the S22B database. As shown in Figure 1b, the MADs of the tested (cHF, cMP2) combinations float within about 0.17−0.36 kcal/mol. The MAD of the (cHF = 0.67, cMP2 = 0.49) combination is 0.17 kcal/ mol, and the corresponding MA%D is 4.8%. And hence, we denote the (cHF = 0.67, cMP2 = 0.49) combination for the framework of the mPW2-PLYP functional as mPW2NC-PLYP. 3.2. Basis Set Effects of the B2NC-PLYP and mPW2NCPLYP Functionals. As comparisons, the basis sets of 631+G(d,p), 6-311+G(d,p), 6-311+G(2df,2pd), 6-311++G(3df,3pd), aug-cc-pVDZ, aug-cc-pVTZ, cc-pVTZ, def2TZVPP, ma-TZVPP, def2-QZVP, and ma-QZVP are systematically assessed for the B2NC-PLYP and mPW2NC-PLYP functionals against the S22B database. As listed in Table 2, the 6-311+G(2df,2pd) and aug-cc-pVTZ basis sets are reasonable choices for the B2NC-PLYP and mPW2NC-PLYP functionals to investigate noncovalent interactions. As the basis set utilized in the optimization process of the B2NC-PLYP functional, the 6-311+G(2df,2p) basis set is definitely suitable for the B2NCPLYP functional. Furthermore, according to the performances of the cc-pVTZ and aug-cc-pVTZ basis sets, the corresponding diffuse functions play important roles for these two functionals to study noncovalent interactions. The performances of these two functionals with the def2-QZVP and ma-QZVP basis sets are worse than those with the def2-TZVPP and ma-TZVPP 3176
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Table 2. Basis Set Effects of the B2NC-PLYP and mPW2NCPLYP Functionals with the S22B Databasea basis set
MD
6-31+G(d,p) 6-311+G(d,p) 6-311+G(2df,2p) 6-311+G(2df,2pd) 6-311++G(3df,3pd) aug-cc-pVDZ aug-cc-pVTZ cc-pVTZ def2-TZVPP ma-TZVPP def2-QZVP ma-QZVP 6-31+G(d,p) 6-311+G(d,p) 6-311+G(2df,2pd) 6-311++G(3df,3pd) aug-cc-pVDZ aug-cc-pVTZ cc-pVTZ def2-TZVPP ma-TZVPP def2-QZVP ma-QZVP
MAD
MA%D
0.42 0.50 0.15 0.21 0.45 0.97 0.23 0.46 0.31 0.29 0.39 0.40
12.5 12.7 5.9 6.8 11.5 22.2 5.7 10.9 9.3 8.3 10.5 10.7
0.37 0.52 0.17 0.30 0.79 0.16 0.51 0.36 0.38 0.44 0.46
10.7 12.0 4.8 8.6 20.4 3.6 11.2 9.1 8.4 10.2 10.4
B2NC-PLYP −0.21 −0.16 −0.01 −0.12 −0.40 −0.97 −0.16 −0.19 0.13 0.25 0.31 0.35 mPW2NC-PLYP −0.09 −0.20 −0.01 −0.27 −0.79 −0.03 −0.21 0.17 0.34 0.38 0.43
a
The MD (kcal/mol), MAD (kcal/mol), and MA%D are employed to evaluate the performances of various basis sets.
Table 3. MADs (kcal/mol) of the B2NC-PLYP and mPW2NC-PLYP Functionals with Several Selected Basis Sets on the Subsets of the S22B and S66 Databasesa basis set (database)
HB
DD
B2NC-PLYP 0.21 0.18 0.30 0.20 0.31 0.17 0.18 0.27 0.89 0.33 0.32 0.46 0.24 0.46 0.24 0.64 0.24 0.64 mPW2NC-PLYP 6-311+G(2df,2pd) (S22B) 0.26 0.19 aug-cc-pVTZ (S22B) 0.18 0.19 aug-cc-pVTZ (S66) 0.16 0.28 cc-pVTZ (S22B) 0.97 0.35 def2-TZVPP (S22B) 0.36 0.52 ma-TZVPP (S22B) 0.35 0.56 def2-QZVP (S22B) 0.28 0.72 ma-QZVP (S22B) 0.27 0.74 6-311+G(2df,2p) (S22B) 6-311+G(2df,2pd) (S22B) aug-cc-pVTZ (S22B) aug-cc-pVTZ (S66) cc-pVTZ (S22B) def2-TZVPP (S22B) ma-TZVPP (S22B) def2-QZVP (S22B) ma-QZVP (S22B)
Figure 1. Contour maps for the MADs (kcal/mol) of the tested (cHF, cMP2) combinations for the frameworks of the B2PLYP (a) and mPW2-PLYP (b) functionals against the S22B database. The 6311+G(2df,2p) (a) and 6-311+G(2df,2pd) (b) basis sets are used for single point energy computations, respectively.
basis sets, respectively. These abnormal results will be discussed in section 3.5. The S22 database contains three different types of complexes. They are hydrogen-bonded (HB) complexes, dispersion-dominated (DD) complexes, and complexes with mixed (MX) influence. The detailed categories used in this work accord with the original ref 66. The performances of the B2NC-PLYP and mPW2NC-PLYP functionals with the 6311+G(2df,2pd), aug-cc-pVTZ, cc-pVTZ, def2-TZVPP, maTZVPP, def2-QZVP, and ma-QZVP basis sets on these three subsets are presented in Table 3. With the 6-311+G(2df,2pd) and aug-cc-pVTZ basis sets, these two functionals perform relatively well on each subset of the S22B database. However, these two functionals perform worse on the HB subset than other two subsets with the cc-pVTZ basis set, whereas they perform worse on the DD subset with the def2-QZVP and maQZVP basis sets. As further validations, the B2NC-PLYP/augcc-pVTZ and mPW2NC-PLYP/aug-cc-pVTZ levels of theory perform very well on the S66 database and corresponding subsets.
MX
total
0.07 0.12 0.21 0.16 0.17 0.13 0.15 0.25 0.28
0.15 0.21 0.23 0.21 0.46 0.31 0.29 0.39 0.40
0.05 0.11 0.15 0.24 0.17 0.21 0.29 0.34
0.17 0.16 0.20 0.51 0.36 0.38 0.44 0.46
a
The corresponding subsets are hydrogen-bonded (HB) complexes, dispersion-dominated (DD) complexes, and complexes with mixed (MX) influence.
3.3. Assessment of the B2NC-PLYP and mPW2NCPLYP functionals with the ADIM6 database and six rare gas dimers. As to the methane dimer in the S22B database, we have found that the absolute percentage deviations of the 3177
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B2NC-PLYP functional with the 6-311+G(2df,2p), 6-311+G(2df,2pd), and aug-cc-pVTZ basis sets are 53.9%, 54.6%, and 34.1%, respectively. However, the corresponding absolute deviations are relatively small, because the reference total interaction energy of the methane dimer is only −0.527 kcal/ mol in the S22B database. On the basis of the SAPT computation,74 the dispersion component is extremely significant for the total interaction energy of the methane dimer. Similarly, the dispersion energies are very important for the total interaction energies of the dimers in the ADIM6 database. Therefore, we have assessed the B2N-PLYP, B2NCPLYP, and mPW2NC-PLYP functionals with the aug-cc-pVTZ basis set against the ADIM6 database. As listed in Table 4, the Table 4. Performances of the B2N-PLYP, B2NC-PLYP, and mPW2NC-PLYP Functionals with the aug-cc-pVTZ Basis Set against the ADIM6 Databasea functional
MD
MAD
MA%D
B2N-PLYP B2NC-PLYP mPW2NC-PLYP
−0.75 −0.65 −0.57
0.75 0.65 0.57
22.9 19.9 16.4
Figure 2. Interaction energy curves for the Ar···Ar dimer at the B2NCPLYP/aug-cc-pVTZ and mPW2NC-PLYP/aug-cc-pVTZ levels of theory, respectively. The reference interaction energies obtained at the CCSD(T)/aug-cc-pV5Z+(3s3p2d2f1g) level of theory are taken from ref 77.
a
The MD (kcal/mol), MAD (kcal/mol), and MA%D are utilized to evaluate the performances.
performances of the B2N-PLYP, B2NC-PLYP, and mPW2NCPLYP functionals with the aug-cc-pVTZ basis set are less satisfactory. And, thus, we define the noncovalent interactions between alkanes as problematic cases for these functionals. As listed in Table 5, the B2NC-PLYP and mPW2NC-PLYP functionals with the aug-cc-pVTZ basis set are able to locate the
Table 6. MADs (kcal/mol) of Various Computational Levels against the S22 and S66 Databases level of theory S22 B2NC-PLYP/6-311+G(2df,2p) B2NC-PLYP/6-311+G(2df,2pd) B2NC-PLYP/aug-cc-pVTZ mPW2NC-PLYP/6-311+G(2df,2pd) mPW2NC-PLYP/aug-cc-pVTZ B2PLYP-D3/aug-cc-pVTZa ωB97X-D/aug-cc-pVTZa S66 B2NC-PLYP/aug-cc-pVTZ mPW2NC-PLYP/aug-cc-pVTZ PW6B95-D3/def2-QZVPc B2PLYP-D3/def2-QZVPc B2GP-PLYP-D3/def2-QZVPc DSD-BLYP-D3/def2-QZVPc PWPB95-D3/def2-QZVPc MP2/CBSf MP2.5/CBSf SCS-MI-CCSD/CBSf
Table 5. Interaction Potential Wells (kcal/mol) for Six Rare Gas Dimersa rare gas dimer
referenceb
B2NC-PLYP
mPW2NC-PLYP
He2 Ne2 Ar2 He···Ne He···Ar Ne···Ar
−0.02 −0.08 −0.28 −0.04 −0.06 −0.13
−0.01 −0.09 −0.19 −0.03 −0.03 −0.11
−0.07 −0.18 −0.27 −0.12 −0.10 −0.21
a
These interaction potential wells are determined by potential energy scans at the B2NC-PLYP/aug-cc-pVTZ and mPW2NC-PLYP/aug-ccpVTZ levels of theory. bThe reference interaction potential wells obtained at the CCSD(T)/aug-cc-pV5Z+(3s3p2d2f1g) level of theory are taken from ref 77.
MAD 0.15 0.21 0.23 0.17 0.16 0.38 0.33, 0.18b 0.21 0.20 0.23,d 0.26,d 0.26,d 0.21,d 0.19,d 0.45 0.12 0.06
0.18e 0.21e 0.24e 0.23e 0.22e
a
Reference 10. bCP corrected value. cReference 61. dDFT-D3. eDFTD3(BJ). fReference 70.
interaction potential wells for the six tested rare gas dimers. In addition, the interaction energy curves for the Ar···Ar dimer are shown in Figure 2. The interaction energies of the rare gas dimers are rather small, and it is very challenging to compute them accurately with density functionals. 3.4. Comparisons with Previous Benchmark Results. In this section, we compare the performances of various computational levels on the S22 and S66 databases. The corresponding results are mainly cited from refs 10, 61, and 70. In fact, the reference interaction energies for the S22 and S66 databases are slightly different in different revisions. However, these differences will not affect the assessment of various computational methods, especially the density functionals. As shown in Table 6, the performances of the B2NC-PLYP and mPW2NC-PLYP functionals combined with the 6-
311+G(2df,2pd) and aug-cc-pVTZ basis sets are close to those of the computational levels involving the density functionals with dispersion corrections. The B2NC-PLYP and mPW2NC-PLYP functionals are free of dispersion correction, and moreover, the CP correction is not needed with the triple-ζ quality basis sets including adequate polarization and diffuse functions. 3.5. Comments on the B2NC-PLYP and mPW2NCPLYP Functionals. The cHF and cMP2 parameters for the B2NC-PLYP functional are 0.81 and 0.55, respectively. This PT2 implies that the amounts of EHF are very large in the x and Ec exchange−correlation energy Exc. According to our pervious 3178
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work,56 the large amount of EHF in the Exc will reduce the x exchange-repulsion components for the total interaction energies of the HCOOH···C6H6 complexes. Therefore, we consider that the large amount of EHF contributes to the x performance of the B2NC-PLYP functional on noncovalent interactions. Furthermore, the large amount of EPT2 well c describes the dispersion components for noncovalent interactions and also contributes to the performance of B2NC-PLYP functional, especially for the DD and MX complexes. As to the mPW2NC-PLYP functional, the cHF and cMP2 parameters are 0.67 and 0.49, respectively. Combined with the LYP correlation,98 the mPW exchange99 can reduce the amounts PT2 of EHF compared with the B88 exchange.100 This is x and Ec due to the improved long-range behavior of the mPW exchange.99 As mentioned above, the performances of the B2NC-PLYP and mPW2NC-PLYP functionals with the aug-cc-pVTZ basis set against the ADIM6 database are less satisfactory. Furthermore, some radical−molecule noncovalent interactions may also be problematic cases for these two functionals because of the large amount of EHF x . Various density functionals should be assessed against the open-shell noncovalent interactions in the future.101 Additionally, we have assessed the performances of the B2NC-PLYP and mPW2NC-PLYP functionals with the 6311+G(2df,2p) basis set on other subsets of the GMTKN30 database.40,73 The W4-08,22,40,73 BH76, and BH76RC40,73,102,103 subsets are employed. The corresponding computations using the B2NC-PLYP and mPW2NC-PLYP functionals were performed with the GAMESS93,94 and Firefly93,104 programs, respectively. The MDs and MADs of these two levels against the W4-08, BH76, and BH76RC subsets are listed in Table 7. As a result, the mPW2NC-PLYP
another issue. In this work, the basis set incompleteness is absorbed by the optimized parameters; i.e., the corresponding BSSEs are incorporated in the B2NC-PLYP and mPW2NCPLYP functionals. Therefore, these two functionals perform relatively well with the triple-ζ quality basis sets. And, moreover, the MADs using the nearly saturated def2-QZVP and ma-QZVP basis sets are larger than those with the def2TZVPP and ma-TZVPP basis sets, respectively. In subsequent work, the basis set such as def2-QZVP and/or extrapolation to the complete basis set (CBS) limit26 will be used to address this issue. We have also examined the CP correction for the basis sets of aug-cc-pVDZ and def2-TZVPP with the B2NC-PLYP and mPW2NC-PLYP functionals. As shown in Table 8, the half Table 8. Comparisons of CP and Half CP Corrected Results for the aug-cc-pVDZ and def2-TZVPP Basis Setsa basis set aug-cc-pVDZ (CP) aug-cc-pVDZ (half CP) def2-TZVPP (CP) def2-TZVPP (half CP) aug-cc-pVDZ (CP) aug-cc-pVDZ (half CP) def2-TZVPP (CP) def2-TZVPP (half CP)
B2NC-PLYP MAD
MD
MAD
W4-08 BH76 BH76RC
−7.91 1.79 −0.15
8.03 1.87 1.11
0.03 −1.93 0.27
3.97 2.69 1.57
MA%D
1.07 0.23 0.99 0.59
22.5 5.4 22.0 14.8
1.13 0.24 1.05 0.66
21.8 3.9 21.3 14.4
The MD (kcal/mol), MAD (kcal/mol), and MA%D are used to assess the performances.
CP22,36,108 correction for the aug-cc-pVDZ basis set performs relatively well against the S22B database. As mentioned above, the BSSEs of the triple-ζ quality 6-311+G(2df,2p) and 6311+G(2df,2pd) basis sets have been incorporated in the B2NC-PLYP and mPW2NC-PLYP functionals, respectively. And thus, the CP and half CP corrections degrade the corresponding performances with the def2-TZVPP basis set.
mPW2NC-PLYP
MD
MAD
a
Table 7. Performances of the B2NC-PLYP and mPW2NCPLYP Functionals with the 6-311+G(2df,2p) Basis Set against the W4-08, BH76, and BH76RC Subsets of the GMTKN30 Databasea subset
MD B2NC-PLYP 1.07 0.05 0.99 0.56 mPW2NC-PLYP 1.13 0.17 1.05 0.61
4. CONCLUSIONS Two DHDFs of B2NC-PLYP and mPW2NC-PLYP have been obtained for noncovalent interactions in this work. These two functionals are derived free of dispersion and CP corrections against the S22B database. On the basis of our benchmark computations, we recommend the B2NC-PLYP and mPW2NC-PLYP functionals with the triple-ζ quality basis sets including adequate polarization and diffuse functions to study noncovalent interactions. The B2PLYP-D3 and mPW2PLYP-D3 functionals may be utilized for further validation. However, different exchange and correlation functionals will be tested as the B2NC-PLYP and mPW2NC-PLYP functionals in the future. We hope to reduce the amount of the Fockexchange further and improve the corresponding performance on the ADIM6 database.
a
The MD (in kcal/mol) and MAD (in kcal/mol) are utilized to evaluate the performances.
functional performs better than the B2NC-PLYP functional on the W4-08 subset. On the contrary, the B2NC-PLYP functional performs better on the BH76 and BH76RC subsets. Comparisons with other computational levels are presented in the Supporting Information (Table S2). However, the B2TPLYP,21 B2K-PLYP,21 mPW2K-PLYP,21 and B2GP-PLYP22 functionals are reasonable choices for thermochemistry and/or thermochemical kinetics. Moreover, the performances of the B2NC-PLYP/aug-cc-pVTZ level of theory against the small representative atomization energies (AE6) and barrier heights (BH6) databases105−107 are given in the Supporting Information (Table S3). Both the MAD per bond with the AE6 database and MAD against the BH6 database are close to 1 kcal/mol. The incompleteness of the Pople’s split-valence triple-ζ basis set in the optimization of the cHF and cMP2 parameters is
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ASSOCIATED CONTENT
S Supporting Information *
Input files for the B2NC-PLYP and mPW2NC-PLYP functionals; total energies of three examples in the S22B database computed at the B2NC-PLYP/aug-cc-pVTZ level of theory with the GAMESS and ORCA programs (Table S1); MADs of various computational levels against the W4-08, BH76, and 3179
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BH76RC databases (Table S2); and assessment of the B2NCPLYP functional with the aug-cc-pVTZ basis set against the AE6 and BH6 databases (Table S3). This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Feng Yu gratefully acknowledges the Xi’an Technological University for the financial support (No. XAGDXJJ1030). Partial computations were performed at the Supercomputing Center of Qingdao Institute of Bioenergy and Bioprocess Technology.
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REFERENCES
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Double-Hybrid Density Functionals Free of Dispersion and Counterpoise Corrections for Non-Covalent Interactions Feng Yu* Department of Physics, School of Science, Xi’an Technological University, No. 4 Jinhua North Road, Xi’an, Shaanxi 710032, China S Supporting Information *
ABSTRACT: We have optimized two double-hybrid density functionals (DHDFs) within the frameworks of B2PLYP and mPW2-PLYP against the S22B database. These two functionals are denoted as B2NC-PLYP and mPW2NC-PLYP, where “NC” represents noncovalent interaction. The DHDFs of B2NC-PLYP and mPW2NC-PLYP are optimized free of dispersion and counterpoise corrections with triple-ζ quality basis sets. Combined with the aug-cc-pVTZ basis set, these two functionals are further assessed with the S66 database. According to our computations, both the B2NC-PLYP and mPW2NC-PLYP functionals seem to be competent for investigating noncovalent interactions. Note that the triple-ζ quality basis sets with adequate polarization and diffuse functions should be employed for practical applications. However, different exchange and correlation functionals may be selected and/or modified to reduce the amount of the Fock-exchange in the future. improves the performances of the DHDFs.55 Without additional corrections, the XYG327 and XYGJ-OS28 are valuable functionals for studying noncovalent interactions.47 With the help of the localized molecular orbital energy decomposition analyses (LMO-EDA),62 two parameters of cHF = 0.81 and cMP2 = 0.52 for the framework of the B2PLYP functional were extrapolated with five HCOOH···C6H6 complexes.56 The reference molecular geometries and interaction energies were taken from ref 63. After they were examined with the noncovalent complexation energies database (NCCE31/ 05),8,64,65 we found that these two parameters for the framework of the B2PLYP functional were appropriate for investigating noncovalent interactions,56 at least suitable for the small and medium-sized noncovalent systems. And, thus, we denoted these two parameters for the framework of the B2PLYP functional as B2N-PLYP, where “N” stood for noncovalent interaction.56 However, the training set for the B2N-PLYP functional was very small, and the corresponding parameters should be improved with a well-designed training set. In this work, we have optimized the parameters of cHF and cMP2 for the frameworks of the B2PLYP and mPW2-PLYP functionals with the S22 database.66−69 Note that the S22B69 revision has been utilized as reference. Subsequently, the optimized parameters are further assessed with the S66 database.70,71 The main purpose of this work is to treat noncovalent interactions within the framework of the DHDF,
1. INTRODUCTION Different types of noncovalent interactions are fundamental topics in chemistry, because noncovalent interactions play significant roles in cluster chemistry, supramolecular chemistry, biochemistry, solution chemistry, surface chemistry, and so on.1,2 Therefore, developing and assessing ab initio methods and density functionals for noncovalent interactions are very important for theoretical and computational chemistry.1−15 However, elucidating the natures of noncovalent interactions is also important,1−3 and the favorable computational tools should perform well on various noncovalent interactions with different essences. Double-hybrid density functionals (DHDFs) have grown to be very powerful tools for thermochemistry, thermochemical kinetics, noncovalent interaction, etc.16−57 The B2PLYP18 functional has inspired rapid development of various DHDFs,19−57 and the exchange−correlation energy Exc for the B2PLYP functional was defined as Exc = (1 − c HF)ExGGA + c HFExHF + (1 − c MP2)EcGGA + c MP2EcPT2
(1)
To study noncovalent interactions, the dispersion corrections9,58−60 have been applied to the DHDFs.20,22,23,34−36,40 The DHDFs with dispersion corrections perform very well on different types of noncovalent interactions. For example, the B2PLYP-D3 20,60 functional is a reasonable choice for investigating noncovalent interactions based on systematic evaluations.10,61 Furthermore, the DHDFs with long-range corrections perform well on noncovalent interactions,25,26,54 and the nonlocal van der Waals method also significantly © 2014 American Chemical Society
Received: January 17, 2014 Revised: April 6, 2014 Published: April 10, 2014 3175
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3. RESULTS AND DISCUSSION 3.1. Assessment of Different (cHF, cMP2) Combinations for the Frameworks of the B2PLYP and mPW2-PLYP Functionals with the S22B Database. Different (cHF, cMP2) combinations for the framework of the B2PLYP functional have been assessed with the S22B database. The 6-311+G(2df,2p), i.e., MG3S for the S22B database, has been employed for single point energy computations. As listed in Table 1, the B2N-PLYP
and further corrections specifically designed for the DHDFs such as dispersion,20,22,23,34−36,40 long-range,25,26,54 and nonlocal van der Waals55 corrections could be ignored.
2. COMPUTATIONAL METHODS The databases of S22,66−69 S66,70,71 and ADIM6 (interaction energy database for six n-alkane dimers)40,72,73 for noncovalent interactions were employed in the present work. The S22B69 revision of the S22 database was used as the training set for the optimizations of the cHF and cMP2 parameters for the frameworks of the B2PLYP and mPW2-PLYP functionals. And then, the S6670,71 database was utilized as a validation set, and the corresponding reference interaction energies were taken from ref 71. As to the methane dimer in the S22 database, the dispersion component for the total interaction energy was −0.89 kcal/mol, whereas the corresponding electrostatic component was only −0.19 kcal/mol 74 from accurate symmetry-adapted perturbation theory (SAPT)75,76 computation. Therefore, we deduced that the dispersion components were extremely important for the dimers in the ADIM6 database. Accordingly, the ADIM6 database was tested in this work. Additionally, the interaction potential wells for some rare gas dimers77 were also tested. The basis sets of 6-31+G(d,p),78−82 6-311+G(d,p),82−84 6311+G(2df,2p), 82−84 6-311+G(2df,2pd), 82−84 6-311++G(3df,3pd),82−84 aug-cc-pVDZ,85−88 aug-cc-pVTZ,85−88 ccpVTZ,85 def2-TZVPP,89 ma-TZVPP,90 def2-QZVP,89 and maQZVP90 were employed for single point energy computations. The basis set superposition error (BSSE) was not corrected with the counterpoise (CP) method91 unless otherwise noted. For the S22 database, the 6-311+G(2df,2p) basis set was identical to the MG3S basis set defined by Truhlar and coworkers,92 which was accurate enough for relative energy computations. Therefore, the basis set of 6-311+G(2df,2p) was utilized in the optimization process of the cHF and cMP2 parameters within the framework of B2PLYP. In detail, the computations with the 6-31+G(d,p), 6-311+G(d,p), 6-311+G(2df,2p), and aug-cc-pVDZ basis sets were performed with the General Atomic and Molecular Electronic Structure System (GAMESS) program,93,94 and other computations with the 6311+G(2df,2pd), 6-311++G(3df,3pd), aug-cc-pVTZ, cc-pVTZ, def2-TZVPP, ma-TZVPP, def2-QZVP and ma-QZVP basis sets were carried out with the ORCA program.95 As to the computations related to the framework of mPW2-PLYP, only the ORCA program was employed. The corresponding cHF and cMP2 parameters were optimized with the 6-311+G(2df,2pd) basis set. Comparisons between the GAMESS and ORCA programs on single point energy computations were presented in the Supporting Information (Table S1). It should be emphasized that the chemical core electrons were frozen in the present work. In addition, the Lebedev (96,302) integration grid96 was employed for the computations performed with the GAMESS program, whereas the Grid4 was utilized for the computations carried out with the ORCA program. On the basis of our previous work,97 these two integration grids were suitable for single point energy computations. The performances of the DHDFs with different basis sets were evaluated with the signed mean deviation (MD), mean absolute deviation (MAD), and mean absolute percent deviation (MA%D). The corresponding definitions had been described in ref 10.
Table 1. Performances of Different (cHF, cMP2) Combinations for the Framework of the B2PLYP Functional with the S22B Databasea DHDF B2PLYP B2T-PLYP B2GP-PLYP B2K-PLYP B2N-PLYP
(cHF,cMP2) (0.53, (0.60, (0.65, (0.72, (0.81,
0.27) 0.31) 0.36) 0.42) 0.52)
MD
MAD
MA%D
1.62 1.26 0.96 0.57 0.05
1.64 1.29 0.99 0.63 0.18
39.3 31.8 25.1 17.0 6.4
a
The 6-311+G(2df,2p) basis set is utilized for single point energy computations, and the MD (kcal/mol), MAD (kcal/mol), and MA%D are listed as comparisons.
functional performs relatively well with the S22B database. On the basis of the performance of the B2N-PLYP functional, we have examined a total of 35 (cHF, cMP2) combinations around the B2N-PLYP functional against the S22B database. As shown in Figure 1a, the range of the MADs of the tested (cHF, cMP2) combinations is about 0.15−0.30 kcal/mol. As a result, the MAD of the (cHF = 0.81, cMP2 = 0.55) combination is only 0.15 kcal/mol, and the corresponding MA%D is 5.9%. Therefore, we denote the (cHF = 0.81, cMP2 = 0.55) combination for the framework of the B2PLYP functional as B2NC-PLYP, where “NC” represents noncovalent interaction. The B2NC-PLYP functional is an improved revision for the B2N-PLYP functional. Similarly, a total of 45 (cHF, cMP2) combinations for the framework of the mPW2-PLYP functional were also assessed with the 6-311+G(2df,2pd) basis set against the S22B database. As shown in Figure 1b, the MADs of the tested (cHF, cMP2) combinations float within about 0.17−0.36 kcal/mol. The MAD of the (cHF = 0.67, cMP2 = 0.49) combination is 0.17 kcal/ mol, and the corresponding MA%D is 4.8%. And hence, we denote the (cHF = 0.67, cMP2 = 0.49) combination for the framework of the mPW2-PLYP functional as mPW2NC-PLYP. 3.2. Basis Set Effects of the B2NC-PLYP and mPW2NCPLYP Functionals. As comparisons, the basis sets of 631+G(d,p), 6-311+G(d,p), 6-311+G(2df,2pd), 6-311++G(3df,3pd), aug-cc-pVDZ, aug-cc-pVTZ, cc-pVTZ, def2TZVPP, ma-TZVPP, def2-QZVP, and ma-QZVP are systematically assessed for the B2NC-PLYP and mPW2NC-PLYP functionals against the S22B database. As listed in Table 2, the 6-311+G(2df,2pd) and aug-cc-pVTZ basis sets are reasonable choices for the B2NC-PLYP and mPW2NC-PLYP functionals to investigate noncovalent interactions. As the basis set utilized in the optimization process of the B2NC-PLYP functional, the 6-311+G(2df,2p) basis set is definitely suitable for the B2NCPLYP functional. Furthermore, according to the performances of the cc-pVTZ and aug-cc-pVTZ basis sets, the corresponding diffuse functions play important roles for these two functionals to study noncovalent interactions. The performances of these two functionals with the def2-QZVP and ma-QZVP basis sets are worse than those with the def2-TZVPP and ma-TZVPP 3176
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Table 2. Basis Set Effects of the B2NC-PLYP and mPW2NCPLYP Functionals with the S22B Databasea basis set
MD
6-31+G(d,p) 6-311+G(d,p) 6-311+G(2df,2p) 6-311+G(2df,2pd) 6-311++G(3df,3pd) aug-cc-pVDZ aug-cc-pVTZ cc-pVTZ def2-TZVPP ma-TZVPP def2-QZVP ma-QZVP 6-31+G(d,p) 6-311+G(d,p) 6-311+G(2df,2pd) 6-311++G(3df,3pd) aug-cc-pVDZ aug-cc-pVTZ cc-pVTZ def2-TZVPP ma-TZVPP def2-QZVP ma-QZVP
MAD
MA%D
0.42 0.50 0.15 0.21 0.45 0.97 0.23 0.46 0.31 0.29 0.39 0.40
12.5 12.7 5.9 6.8 11.5 22.2 5.7 10.9 9.3 8.3 10.5 10.7
0.37 0.52 0.17 0.30 0.79 0.16 0.51 0.36 0.38 0.44 0.46
10.7 12.0 4.8 8.6 20.4 3.6 11.2 9.1 8.4 10.2 10.4
B2NC-PLYP −0.21 −0.16 −0.01 −0.12 −0.40 −0.97 −0.16 −0.19 0.13 0.25 0.31 0.35 mPW2NC-PLYP −0.09 −0.20 −0.01 −0.27 −0.79 −0.03 −0.21 0.17 0.34 0.38 0.43
a
The MD (kcal/mol), MAD (kcal/mol), and MA%D are employed to evaluate the performances of various basis sets.
Table 3. MADs (kcal/mol) of the B2NC-PLYP and mPW2NC-PLYP Functionals with Several Selected Basis Sets on the Subsets of the S22B and S66 Databasesa basis set (database)
HB
DD
B2NC-PLYP 0.21 0.18 0.30 0.20 0.31 0.17 0.18 0.27 0.89 0.33 0.32 0.46 0.24 0.46 0.24 0.64 0.24 0.64 mPW2NC-PLYP 6-311+G(2df,2pd) (S22B) 0.26 0.19 aug-cc-pVTZ (S22B) 0.18 0.19 aug-cc-pVTZ (S66) 0.16 0.28 cc-pVTZ (S22B) 0.97 0.35 def2-TZVPP (S22B) 0.36 0.52 ma-TZVPP (S22B) 0.35 0.56 def2-QZVP (S22B) 0.28 0.72 ma-QZVP (S22B) 0.27 0.74 6-311+G(2df,2p) (S22B) 6-311+G(2df,2pd) (S22B) aug-cc-pVTZ (S22B) aug-cc-pVTZ (S66) cc-pVTZ (S22B) def2-TZVPP (S22B) ma-TZVPP (S22B) def2-QZVP (S22B) ma-QZVP (S22B)
Figure 1. Contour maps for the MADs (kcal/mol) of the tested (cHF, cMP2) combinations for the frameworks of the B2PLYP (a) and mPW2-PLYP (b) functionals against the S22B database. The 6311+G(2df,2p) (a) and 6-311+G(2df,2pd) (b) basis sets are used for single point energy computations, respectively.
basis sets, respectively. These abnormal results will be discussed in section 3.5. The S22 database contains three different types of complexes. They are hydrogen-bonded (HB) complexes, dispersion-dominated (DD) complexes, and complexes with mixed (MX) influence. The detailed categories used in this work accord with the original ref 66. The performances of the B2NC-PLYP and mPW2NC-PLYP functionals with the 6311+G(2df,2pd), aug-cc-pVTZ, cc-pVTZ, def2-TZVPP, maTZVPP, def2-QZVP, and ma-QZVP basis sets on these three subsets are presented in Table 3. With the 6-311+G(2df,2pd) and aug-cc-pVTZ basis sets, these two functionals perform relatively well on each subset of the S22B database. However, these two functionals perform worse on the HB subset than other two subsets with the cc-pVTZ basis set, whereas they perform worse on the DD subset with the def2-QZVP and maQZVP basis sets. As further validations, the B2NC-PLYP/augcc-pVTZ and mPW2NC-PLYP/aug-cc-pVTZ levels of theory perform very well on the S66 database and corresponding subsets.
MX
total
0.07 0.12 0.21 0.16 0.17 0.13 0.15 0.25 0.28
0.15 0.21 0.23 0.21 0.46 0.31 0.29 0.39 0.40
0.05 0.11 0.15 0.24 0.17 0.21 0.29 0.34
0.17 0.16 0.20 0.51 0.36 0.38 0.44 0.46
a
The corresponding subsets are hydrogen-bonded (HB) complexes, dispersion-dominated (DD) complexes, and complexes with mixed (MX) influence.
3.3. Assessment of the B2NC-PLYP and mPW2NCPLYP functionals with the ADIM6 database and six rare gas dimers. As to the methane dimer in the S22B database, we have found that the absolute percentage deviations of the 3177
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B2NC-PLYP functional with the 6-311+G(2df,2p), 6-311+G(2df,2pd), and aug-cc-pVTZ basis sets are 53.9%, 54.6%, and 34.1%, respectively. However, the corresponding absolute deviations are relatively small, because the reference total interaction energy of the methane dimer is only −0.527 kcal/ mol in the S22B database. On the basis of the SAPT computation,74 the dispersion component is extremely significant for the total interaction energy of the methane dimer. Similarly, the dispersion energies are very important for the total interaction energies of the dimers in the ADIM6 database. Therefore, we have assessed the B2N-PLYP, B2NCPLYP, and mPW2NC-PLYP functionals with the aug-cc-pVTZ basis set against the ADIM6 database. As listed in Table 4, the Table 4. Performances of the B2N-PLYP, B2NC-PLYP, and mPW2NC-PLYP Functionals with the aug-cc-pVTZ Basis Set against the ADIM6 Databasea functional
MD
MAD
MA%D
B2N-PLYP B2NC-PLYP mPW2NC-PLYP
−0.75 −0.65 −0.57
0.75 0.65 0.57
22.9 19.9 16.4
Figure 2. Interaction energy curves for the Ar···Ar dimer at the B2NCPLYP/aug-cc-pVTZ and mPW2NC-PLYP/aug-cc-pVTZ levels of theory, respectively. The reference interaction energies obtained at the CCSD(T)/aug-cc-pV5Z+(3s3p2d2f1g) level of theory are taken from ref 77.
a
The MD (kcal/mol), MAD (kcal/mol), and MA%D are utilized to evaluate the performances.
performances of the B2N-PLYP, B2NC-PLYP, and mPW2NCPLYP functionals with the aug-cc-pVTZ basis set are less satisfactory. And, thus, we define the noncovalent interactions between alkanes as problematic cases for these functionals. As listed in Table 5, the B2NC-PLYP and mPW2NC-PLYP functionals with the aug-cc-pVTZ basis set are able to locate the
Table 6. MADs (kcal/mol) of Various Computational Levels against the S22 and S66 Databases level of theory S22 B2NC-PLYP/6-311+G(2df,2p) B2NC-PLYP/6-311+G(2df,2pd) B2NC-PLYP/aug-cc-pVTZ mPW2NC-PLYP/6-311+G(2df,2pd) mPW2NC-PLYP/aug-cc-pVTZ B2PLYP-D3/aug-cc-pVTZa ωB97X-D/aug-cc-pVTZa S66 B2NC-PLYP/aug-cc-pVTZ mPW2NC-PLYP/aug-cc-pVTZ PW6B95-D3/def2-QZVPc B2PLYP-D3/def2-QZVPc B2GP-PLYP-D3/def2-QZVPc DSD-BLYP-D3/def2-QZVPc PWPB95-D3/def2-QZVPc MP2/CBSf MP2.5/CBSf SCS-MI-CCSD/CBSf
Table 5. Interaction Potential Wells (kcal/mol) for Six Rare Gas Dimersa rare gas dimer
referenceb
B2NC-PLYP
mPW2NC-PLYP
He2 Ne2 Ar2 He···Ne He···Ar Ne···Ar
−0.02 −0.08 −0.28 −0.04 −0.06 −0.13
−0.01 −0.09 −0.19 −0.03 −0.03 −0.11
−0.07 −0.18 −0.27 −0.12 −0.10 −0.21
a
These interaction potential wells are determined by potential energy scans at the B2NC-PLYP/aug-cc-pVTZ and mPW2NC-PLYP/aug-ccpVTZ levels of theory. bThe reference interaction potential wells obtained at the CCSD(T)/aug-cc-pV5Z+(3s3p2d2f1g) level of theory are taken from ref 77.
MAD 0.15 0.21 0.23 0.17 0.16 0.38 0.33, 0.18b 0.21 0.20 0.23,d 0.26,d 0.26,d 0.21,d 0.19,d 0.45 0.12 0.06
0.18e 0.21e 0.24e 0.23e 0.22e
a
Reference 10. bCP corrected value. cReference 61. dDFT-D3. eDFTD3(BJ). fReference 70.
interaction potential wells for the six tested rare gas dimers. In addition, the interaction energy curves for the Ar···Ar dimer are shown in Figure 2. The interaction energies of the rare gas dimers are rather small, and it is very challenging to compute them accurately with density functionals. 3.4. Comparisons with Previous Benchmark Results. In this section, we compare the performances of various computational levels on the S22 and S66 databases. The corresponding results are mainly cited from refs 10, 61, and 70. In fact, the reference interaction energies for the S22 and S66 databases are slightly different in different revisions. However, these differences will not affect the assessment of various computational methods, especially the density functionals. As shown in Table 6, the performances of the B2NC-PLYP and mPW2NC-PLYP functionals combined with the 6-
311+G(2df,2pd) and aug-cc-pVTZ basis sets are close to those of the computational levels involving the density functionals with dispersion corrections. The B2NC-PLYP and mPW2NC-PLYP functionals are free of dispersion correction, and moreover, the CP correction is not needed with the triple-ζ quality basis sets including adequate polarization and diffuse functions. 3.5. Comments on the B2NC-PLYP and mPW2NCPLYP Functionals. The cHF and cMP2 parameters for the B2NC-PLYP functional are 0.81 and 0.55, respectively. This PT2 implies that the amounts of EHF are very large in the x and Ec exchange−correlation energy Exc. According to our pervious 3178
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work,56 the large amount of EHF in the Exc will reduce the x exchange-repulsion components for the total interaction energies of the HCOOH···C6H6 complexes. Therefore, we consider that the large amount of EHF contributes to the x performance of the B2NC-PLYP functional on noncovalent interactions. Furthermore, the large amount of EPT2 well c describes the dispersion components for noncovalent interactions and also contributes to the performance of B2NC-PLYP functional, especially for the DD and MX complexes. As to the mPW2NC-PLYP functional, the cHF and cMP2 parameters are 0.67 and 0.49, respectively. Combined with the LYP correlation,98 the mPW exchange99 can reduce the amounts PT2 of EHF compared with the B88 exchange.100 This is x and Ec due to the improved long-range behavior of the mPW exchange.99 As mentioned above, the performances of the B2NC-PLYP and mPW2NC-PLYP functionals with the aug-cc-pVTZ basis set against the ADIM6 database are less satisfactory. Furthermore, some radical−molecule noncovalent interactions may also be problematic cases for these two functionals because of the large amount of EHF x . Various density functionals should be assessed against the open-shell noncovalent interactions in the future.101 Additionally, we have assessed the performances of the B2NC-PLYP and mPW2NC-PLYP functionals with the 6311+G(2df,2p) basis set on other subsets of the GMTKN30 database.40,73 The W4-08,22,40,73 BH76, and BH76RC40,73,102,103 subsets are employed. The corresponding computations using the B2NC-PLYP and mPW2NC-PLYP functionals were performed with the GAMESS93,94 and Firefly93,104 programs, respectively. The MDs and MADs of these two levels against the W4-08, BH76, and BH76RC subsets are listed in Table 7. As a result, the mPW2NC-PLYP
another issue. In this work, the basis set incompleteness is absorbed by the optimized parameters; i.e., the corresponding BSSEs are incorporated in the B2NC-PLYP and mPW2NCPLYP functionals. Therefore, these two functionals perform relatively well with the triple-ζ quality basis sets. And, moreover, the MADs using the nearly saturated def2-QZVP and ma-QZVP basis sets are larger than those with the def2TZVPP and ma-TZVPP basis sets, respectively. In subsequent work, the basis set such as def2-QZVP and/or extrapolation to the complete basis set (CBS) limit26 will be used to address this issue. We have also examined the CP correction for the basis sets of aug-cc-pVDZ and def2-TZVPP with the B2NC-PLYP and mPW2NC-PLYP functionals. As shown in Table 8, the half Table 8. Comparisons of CP and Half CP Corrected Results for the aug-cc-pVDZ and def2-TZVPP Basis Setsa basis set aug-cc-pVDZ (CP) aug-cc-pVDZ (half CP) def2-TZVPP (CP) def2-TZVPP (half CP) aug-cc-pVDZ (CP) aug-cc-pVDZ (half CP) def2-TZVPP (CP) def2-TZVPP (half CP)
B2NC-PLYP MAD
MD
MAD
W4-08 BH76 BH76RC
−7.91 1.79 −0.15
8.03 1.87 1.11
0.03 −1.93 0.27
3.97 2.69 1.57
MA%D
1.07 0.23 0.99 0.59
22.5 5.4 22.0 14.8
1.13 0.24 1.05 0.66
21.8 3.9 21.3 14.4
The MD (kcal/mol), MAD (kcal/mol), and MA%D are used to assess the performances.
CP22,36,108 correction for the aug-cc-pVDZ basis set performs relatively well against the S22B database. As mentioned above, the BSSEs of the triple-ζ quality 6-311+G(2df,2p) and 6311+G(2df,2pd) basis sets have been incorporated in the B2NC-PLYP and mPW2NC-PLYP functionals, respectively. And thus, the CP and half CP corrections degrade the corresponding performances with the def2-TZVPP basis set.
mPW2NC-PLYP
MD
MAD
a
Table 7. Performances of the B2NC-PLYP and mPW2NCPLYP Functionals with the 6-311+G(2df,2p) Basis Set against the W4-08, BH76, and BH76RC Subsets of the GMTKN30 Databasea subset
MD B2NC-PLYP 1.07 0.05 0.99 0.56 mPW2NC-PLYP 1.13 0.17 1.05 0.61
4. CONCLUSIONS Two DHDFs of B2NC-PLYP and mPW2NC-PLYP have been obtained for noncovalent interactions in this work. These two functionals are derived free of dispersion and CP corrections against the S22B database. On the basis of our benchmark computations, we recommend the B2NC-PLYP and mPW2NC-PLYP functionals with the triple-ζ quality basis sets including adequate polarization and diffuse functions to study noncovalent interactions. The B2PLYP-D3 and mPW2PLYP-D3 functionals may be utilized for further validation. However, different exchange and correlation functionals will be tested as the B2NC-PLYP and mPW2NC-PLYP functionals in the future. We hope to reduce the amount of the Fockexchange further and improve the corresponding performance on the ADIM6 database.
a
The MD (in kcal/mol) and MAD (in kcal/mol) are utilized to evaluate the performances.
functional performs better than the B2NC-PLYP functional on the W4-08 subset. On the contrary, the B2NC-PLYP functional performs better on the BH76 and BH76RC subsets. Comparisons with other computational levels are presented in the Supporting Information (Table S2). However, the B2TPLYP,21 B2K-PLYP,21 mPW2K-PLYP,21 and B2GP-PLYP22 functionals are reasonable choices for thermochemistry and/or thermochemical kinetics. Moreover, the performances of the B2NC-PLYP/aug-cc-pVTZ level of theory against the small representative atomization energies (AE6) and barrier heights (BH6) databases105−107 are given in the Supporting Information (Table S3). Both the MAD per bond with the AE6 database and MAD against the BH6 database are close to 1 kcal/mol. The incompleteness of the Pople’s split-valence triple-ζ basis set in the optimization of the cHF and cMP2 parameters is
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ASSOCIATED CONTENT
S Supporting Information *
Input files for the B2NC-PLYP and mPW2NC-PLYP functionals; total energies of three examples in the S22B database computed at the B2NC-PLYP/aug-cc-pVTZ level of theory with the GAMESS and ORCA programs (Table S1); MADs of various computational levels against the W4-08, BH76, and 3179
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BH76RC databases (Table S2); and assessment of the B2NCPLYP functional with the aug-cc-pVTZ basis set against the AE6 and BH6 databases (Table S3). This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Feng Yu gratefully acknowledges the Xi’an Technological University for the financial support (No. XAGDXJJ1030). Partial computations were performed at the Supercomputing Center of Qingdao Institute of Bioenergy and Bioprocess Technology.
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REFERENCES
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