3.1.1 The Quest for the Optimal DFA
In the search for an efficient and accurate DFA that should be able to handle correctly
the spin-state splittings of transition-metal complexes, different strategies were
followed. Reiher proposed to lower the amount of HF exchange to 15% in
B3LYP* [67] but later reported [68] that for other SCO complexes, it should be
lowered even more. Last year, Pinter and co-workers followed up on this and
showed that “optimal” percentages of HF exchange for a small set ranged from
+179% to À12% (!!) [82]. In my group we explored the use of the OPTX functional
[71] by Handy and Cohen and found the best performance when combined with PBE
[63] correlation [69]. Surprisingly, this OPBE functional also drastically improved
the performance of pure DFAs for the energy surfaces of nucleophilic substitution
(S N 2) reactions [83]; subsequently, we observed that the same region of the
exchange enhancement factor F x (s) (the region with s < 1) is responsible for this
good performance in both cases [84]. Ultimately, this led to the design [85] of the
spin-state consistent DFAs SSB-D [73] and S12g [74], which give accurate descriptions for weak interactions through the inclusion of Grimme’s D 2 /D 3 models [86–88]
for dispersion.
Perdew proposed a wide range of DFAs over the past 40 years, following the
rungs of Jacob’s ladder toward the heaven of chemical accuracy [89]: (1) the local
density approximation (LDA) with the electron (spin) density (ρ) as only ingredient;
(2) generalized gradient approximations (GGA), which adds the gradient of the
density (∇ρ) as ingredient; (3) meta-generalized gradient approximations (MGGA),
which can take either the Laplacian of the density (∇
2
ρ) or the kinetic energy density
(τ) into account; (4) hybrid or hyper-GGA functionals that include a portion of HF
exchange; and (5) random phase approximation (RPA), which includes explicitly
virtual orbitals in the energy expressions. In most of his DFAs, Perdew tried to
satisfy physical constraints for finding parameters that are used in these DFAs, with
the strongly correlated and appropriately normed (SCAN) MGGA as recent achievement. Nevertheless, in the same year, Perdew and co-workers reported the “made
very simple” (MVS) MGGA, which was found to drastically improve the barriers for
organic chemistry, unlike SCAN that gave deviations typical of standard pure DFAs.
Previously, my group had shown that if a pure DFA reduces the error for barriers of
organic chemistry, also spin states would be properly described; indeed, we did find
this good performance of MVS for spin states (but see Sect. 3.4) [85]. Since the MVS
functional at zero gradient (or rather at value s ¼ 0, where s is a dimensionless
reduced-density gradient) satisfies the LDA limit (unlike most DFAs that work well
for chemistry), and LDA is known to work well for physics, MVS is therefore a
promising advancement for the design of a DFA that is working well for both
chemistry and physics.
As Cohen and co-workers reported in 2008 [90]: “The beauty of DFT is that its
formalism is exact yet efficient, with one determinant describing the electron
density – all of the complexity is hidden in one term, the exchange-correlation
functional.” Describing this complexity well has led other research groups to follow
Dealing with Spin States in Computational Organometallic Catalysis
199
In the search for an efficient and accurate DFA that should be able to handle correctly
the spin-state splittings of transition-metal complexes, different strategies were
followed. Reiher proposed to lower the amount of HF exchange to 15% in
B3LYP* [67] but later reported [68] that for other SCO complexes, it should be
lowered even more. Last year, Pinter and co-workers followed up on this and
showed that “optimal” percentages of HF exchange for a small set ranged from
+179% to À12% (!!) [82]. In my group we explored the use of the OPTX functional
[71] by Handy and Cohen and found the best performance when combined with PBE
[63] correlation [69]. Surprisingly, this OPBE functional also drastically improved
the performance of pure DFAs for the energy surfaces of nucleophilic substitution
(S N 2) reactions [83]; subsequently, we observed that the same region of the
exchange enhancement factor F x (s) (the region with s < 1) is responsible for this
good performance in both cases [84]. Ultimately, this led to the design [85] of the
spin-state consistent DFAs SSB-D [73] and S12g [74], which give accurate descriptions for weak interactions through the inclusion of Grimme’s D 2 /D 3 models [86–88]
for dispersion.
Perdew proposed a wide range of DFAs over the past 40 years, following the
rungs of Jacob’s ladder toward the heaven of chemical accuracy [89]: (1) the local
density approximation (LDA) with the electron (spin) density (ρ) as only ingredient;
(2) generalized gradient approximations (GGA), which adds the gradient of the
density (∇ρ) as ingredient; (3) meta-generalized gradient approximations (MGGA),
which can take either the Laplacian of the density (∇
2
ρ) or the kinetic energy density
(τ) into account; (4) hybrid or hyper-GGA functionals that include a portion of HF
exchange; and (5) random phase approximation (RPA), which includes explicitly
virtual orbitals in the energy expressions. In most of his DFAs, Perdew tried to
satisfy physical constraints for finding parameters that are used in these DFAs, with
the strongly correlated and appropriately normed (SCAN) MGGA as recent achievement. Nevertheless, in the same year, Perdew and co-workers reported the “made
very simple” (MVS) MGGA, which was found to drastically improve the barriers for
organic chemistry, unlike SCAN that gave deviations typical of standard pure DFAs.
Previously, my group had shown that if a pure DFA reduces the error for barriers of
organic chemistry, also spin states would be properly described; indeed, we did find
this good performance of MVS for spin states (but see Sect. 3.4) [85]. Since the MVS
functional at zero gradient (or rather at value s ¼ 0, where s is a dimensionless
reduced-density gradient) satisfies the LDA limit (unlike most DFAs that work well
for chemistry), and LDA is known to work well for physics, MVS is therefore a
promising advancement for the design of a DFA that is working well for both
chemistry and physics.
As Cohen and co-workers reported in 2008 [90]: “The beauty of DFT is that its
formalism is exact yet efficient, with one determinant describing the electron
density – all of the complexity is hidden in one term, the exchange-correlation
functional.” Describing this complexity well has led other research groups to follow
Dealing with Spin States in Computational Organometallic Catalysis
199