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1 Table 1: Bond lengths (in Å) of rare-gas dimers without counterpoise correction for BSSE. a Method He 2 Ne 2 Ar 2 Kr 2 HeNe HeAr HeKr NeAr NeKr ArKr MSE MUE Reference 2.97 b 3.09 b 3.76 b 4.01 b 3.03 c 3.48 c 3.69 c 3.49 c 3.62 c 3.88 c B PBEPW B PBE PBE PW6B M05-2X PBE1W PW PWB6K MPW1B MPWB1K M TPSS TPSSh LSDA MPW1K mpw1pw Average a The aug-cc-pvtz basis set are employed in all calculations in this table. b Ref. 1 c Ref. 2
2 Table 2: Binding energies (in kcal/mol) of rare-gas dimers without counterpoise correction for BSSE a Method He 2 Ne 2 Ar 2 Kr 2 HeNe HeAr HeKr NeAr NeKr ArKr MSE MUE Reference b b b b c c c c c c M05-2X B B PBE PBEPW M PBE1W PBE PWB6K TPSS PW6B MPWB1K TPSSh MPW1B mpw1pw MPW1K PW LSDA Average a The aug-cc-pvtz basis set are employed in all calculations in this table. b Ref. 1 c Ref. 2
3 Table 3: Binding energies (in kcal/mol) of rare-gas dimers with counterpoise correction for BSSE a b Method He 2 Ne 2 Ar 2 Kr 2 HeNe HeAr HeKr NeAr NeKr ArKr MSE MUE Reference b b b b c c c c c c M05-2X B PBEPW PBE B M PBE1W PWB6K PBE PW6B TPSS MPWB1K MPW1B TPSSh mpw1pw MPW1K PW LSDA Average a The aug-cc-pvtz basis set are employed in all calculations in this table. Ref. 1 c Ref. 2
4 Table 4: Bond lengths (in Å) of alkaline metal dimers, Zn 2, and Zn-rare-gas dimers without counterpoise correction for BSSE a b Method Be 2 Mg 2 Ca 2 Zn 2 ZnNe ZnAr ZnKr MSE MUE Reference PWB6K M05-2X B PW6B B MPWB1K PBE MPW1B M PBE1W PBE PBEPW PW TPSSh TPSS MPW1K mpw1pw LSDA Average a The aug-cv(t+2d)z (see text) is employed for Ca, and the aug-cc-pvtz basis set is employed for all other elements. b The reference data for Be 2 : ref. 3 ; Mg 2 : ref. 4 ;Ca 2 : ref. 5 ; Zn 2 : 6 ; ZnNe: ref. 7 ; ZnAr: ref. 8 ; ZnKr: ref. 9.
5 Table 5: Binding energies (in kcal/mol) of alkaline metal dimers and Zn 2 without counterpoise correction for BSSE a Method X b Be 2 Mg 2 Ca 2 Zn 2 MSE MUE Reference 2.72 c 1.21 c 3.13 c 0.80 c M05-2X PWB6K MPWB1K PW6B MPW1B M MPW1K B mpw1pw B PBE TPSSh PBE1W TPSS PBEPW PBE PW LSDA Average a The aug-cv(t+2d)z (see text) is employed for Ca, and the aug-cc-pvtz basis set is employed for all other elements. b X denotes the percentage of Hartree-Fock exchange in the functional. c The reference data for Be 2 : ref. 3 ; Mg 2 : ref. 4 ;Ca 2 : ref. 5 ; Zn 2 : ref. 6.
6 Table 6: Binding energies (in kcal/mol) of alkaline metal dimers and Zn 2 with counterpoise correction for BSSE a Method X b Be 2 Mg 2 Ca 2 Zn 2 MSE MUE Reference 2.72 c 1.21 c 3.13 c 0.80 c M05-2X PWB6K MPWB1K PW6B M MPW1B MPW1K B mpw1pw B PBE TPSSh PBE1W TPSS PBEPW PBE PW LSDA Average a The aug-cv(t+2d)z (see text) is employed for Ca, and the aug-cc-pvtz basis set is employed for all other elements. b X denotes the percentage of Hartree-Fock exchange in the functional. c The reference data for Be 2 : ref. 3 ; Mg 2 : ref. 4 ;Ca 2 : ref. 5 ; Zn 2 : ref. 6.
7 Table 7: Binding energies (in kcal/mol) of zinc-rare-gas dimers without counterpoise corrections for BSSE a Method ZnNe ZnAr ZnKr MSE MUE Reference b b b B M05-2X PWB6K PBEPW B PBE M PBE1W PW6B PBE MPWB1K MPW1B TPSS TPSSh mpw1pw PW MPW1K LSDA Average a The aug-cc-pvtz basis set are employed for all calculations. b The reference data for ZnNe: ref. 7 ; ZnAr: ref. 8 ; ZnKr: ref. 9
8 Table 8: Binding energies (in kcal/mol) of zinc-rare-gas dimers with counterpoise corrections for BSSE a Method ZnNe ZnAr ZnKr MSE MUE Reference b b b B M05-2X PWB6K B PBEPW M PBE PBE1W PW6B PBE MPWB1K MPW1B PW TPSS TPSSh mpw1pw MPW1K LSDA Average a The aug-cc-pvtz basis set are employed for all calculations. b The reference data for ZnNe: ref. 7 ; ZnAr: ref. 8 ; ZnKr: ref. 9
9 Table 9: Mean percentage unsigned errors (M%UE) a Energetics Geometry Method Rare Gas Dimers Metal Dimers Zn-rare-gas dimers Bond Length MMM%UE no-cp CP MM%UE no-cp CP MM%UE no-cp CP MM%UE no-cp CP MM%UE MMMM%UE M05-2X MPWB1K B MPW1B B PBE MPW1K mpw1pw PW6B TPSSh M PWB6K PBE1W TPSS PBEPW PBE PW LSDA Average
10 a see section 3.6 for the definitions of MM%UE, MMM%UE, and MMMM%UE. CP denotes the counterpoise correction for BSSE.
11 Table 10: Mean unsigned errors (kcal/mol) on a broad range of energetic databases. Functional WI17 a NCBE31 TMAE4/05 b MLBE4/05 b MGT135 HTK57 NHTBH38 AMUEXM c AMUE d M05-2X b b 1.0 b 1.7 a MPWB1K e f 1.3 b 1.5 g B e f 2.6 b 3.4 g MPW1B e e,f 1.9 b 2.3 g B e e 2.9 b,f 3.5 g PBE e e,f 2.8 b 3.5 g MPW1K e e,f 1.3 b 1.8 g mpw1pw e e,f 2.3 f 3.2 g PW6B e f 2.0 a 2.9 a TPSSh e f 4.3 b,f 6.9 g M h h 1.4 b,f 2.1 g PWB6K f e 1.4 b,f 1.4 a a present work b Ref. 10 c Average over mean errors for the WI17, NCBE31, MGT135, HTK57, and NHTBH databases. d Average over mean errors for all seven databases. e Ref. 11 f Ref. 12 g Ref. 13 h Ref. 14
12 Figure captions Figure 1. Potential energy curves for the Be 2 dimer. The aug-cc-pvtz basis set is used for all calculations in this figure, and no counterpoise corrections are applied. The reference curve is taken form Ref. 3. Figure 2. Potential energy curves for the Ar 2 dimer. The aug-cc-pvtz basis set is used for all calculations in this figure, and no counterpoise corrections are applied.
13 Figure 1. V (kcal/mol) HF M05-2X Reference B3LYP M05 B97-1 TPSS PBE R (angstrom)
14 Figure V (kcal/mol) B3LYP HF TPSS PBE B97-1 M05 M05-2X Exp R (angstrom)
15 (1) Ogilvie, J. F.; Wang, F. Y. H. J. Mol. Struct. 1992, 273, 277. (2) Ogilvie, J. F.; Wang, F. Y. H. J. Mol. Struct. 1993, 291, 313. (3) Roeggen, I.; Veseth, L. Int. J. Quantum Chem. 2005, 101, 201. (4) Balfour, W. J.; Douglas, A. E. Can. J. Phys. 1970, 48, 901. (5) Balfour, W. J.; Whitlock, R. F. Can. J. Chem. 1975, 53, 472. (6) Czajkowski, M. A.; Koperski, J. Spectrochim. Acta A 1999, 55, (7) Koperski, J.; Czajkowski, M. Phys. Rev. A 2000, 62, (8) Koperski, J.; Czajkowski, M. J. Mol. Spectrosc. 2002, 212, 162. (9) Wallace, I.; Ryter, J.; Breckenridge, W. H. J. Chem. Phys. 1992, 96, 136. (10) Zhao, Y.; Schultz, N. E.; Truhlar, D. G. J. Chem. Theory Comput. 2006, in press. (11) Zhao, Y.; Truhlar, D. G. J. Phys. Chem. A 2005, 109, (12) Zhao, Y.; Truhlar, D. G. J. Phys. Chem. A 2004, 108, (13) Zhao, Y.; González-García, N.; Truhlar, D. G. J. Phys. Chem. A 2005, 109, (14) Zhao, Y.; Schultz, N. E.; Truhlar, D. G. J. Chem. Phys. 2005, 123, Note that in this communication we interchanged c Cαβ, i and c Cσσ, i in Table 1. In addition, "reduced density x σ " before eq. (1) should read "reduced density gradient x σ ".
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