square of correlation coefficient) of structures were used to evaluate the performance of DFT functionals versus the higher level MP2 results, shown in Figure 2, respectively, using the following expressions:
(similarly hereinafter).
, where , , and are the values of MP2 method , that of the related DFT method, the mean of ’s, and the mean of ’s, respectively; N = 8 for reactant and transition state, N = 7 for product (similarly hereinafter).
It is generally known that the R 2 more close 1 illustrates the data closer to the standard value while the RMSEvalue smaller data closer to the standard value. As shown in Figure 2a, for 1m , the values of R 2 are greater than 0.920 (except for B972 method with R 2 = 0.783). The functionals perform in the following order based on the RMSE errors:
(a) group 1 (RMSE ϵ [0.064 Å, 0.077 Å]): B1B95, PBE1PBE, mPW1PBE, HSEh1PBE, HSE1PBE, mPW1PW91, tHCTHhyb, M06, tHCTH, TPSSh;
(b) group 2 (RMSE ϵ [0.081 Å, 0.098 Å]): wB97XD, CAM-B3LYP, O3LYP, HCTH147, HCTH(407), B971, B98, HCTH93, M06-L, B3LYP;
(c) group 3 (RMSE ϵ [0.103 Å, 0.400 Å]): mPW1LYP, VSXC, BMK, B3PW91, M06-2X, wB97X, B972.
This indicates that several funcitonals (group 1) yield convincing structures with RMSE as small as 0.064 Å in case of the B1B95 functional. It is observed that most of the conventional hybrids including 9 HGGAs lie in this group as well as a GGA functional (tHCTH). The functionals in group 2 yield relatively good structures and the group consists of a collection of 4 GGAs, 4 HGGAs, and two LR-HGGAs (wB97XD, CAM-B3LYP). For group 3, which exhibits the RMSE greater than 0.100Å, consists of one GGA (VSXC), one LR-HGGA (wB97X), and 5HGGAs. From these investigations, it is apparent that all of the functionals designated in group 1 are suitable to reproduce the MP2 results.