The Electronic Determinants of Spin Crossover Described …
21
because the exact exchange integrals are compensated by explicit pure non-exchange
correlation terms via second-order perturbation theory [36, 94]. Also, the optimized
exchange functional in the form of, e.g., OPBE [139, 140] or OLYP [141, 142],
which does not include exact HF exchange, performs much better and more like a
hybrid functional than other GGAs [132, 143]. The O exchange functional has been
estimated to have an effect that corresponds to ~15% HF exchange [94]. The accuracy
of OPBE supports previous findings by Swart [143, 144]. The O exchange functional
was made from B88 by parameterization toward HF unrestricted energies of atoms
of the first and second periods, and thus, this parameterization by design includes
HF-like energetics. It is very interesting that re-parametrization toward HF energies
or, as analyzed by Swart et al. [132] a leading s
4 term in the exchange functional, can
cause a GGA non-hybrid exchange functional to behave similarly to a hybrid with
15% HF exchange because it tells us that HF exchange is not a “universal” feature
by itself, but a pragmatic solution to a major problem of accuracy [127].
4.2 The Role of the Correlation Functional
Inspection of the original paper by Paulsen et al. [34] reveals that BLYP has the
same ~17 kJ/mol smaller bias toward LS than PW91 for two distinct systems. This
consistent difference could be coincidental, and even if not, it could be due to many
features of the two functionals. Systematic comparison of functional types such
as, e.g., BLYP versus BP86 (which use the same exchange functional) shows that
the correlation functional, perhaps surprisingly, also contributes systematically to
the spin-state balance [132]. Thus, for example, whereas BP86 and PBE give very
similar results for SCO energetics and can be considered to have the same spin-state
balance, the bias toward LS is reduced by typically 10–15 kJ/mol when using BLYP,
and this effect is thus explicitly due to the LYP correlation functional [36].
It is interesting here to comment on the analogy between the performance of DFT
applied to SCO and to the modeling of chemical bond strengths. The dissociation
energy of a chemical bond is arguably the most fundamental energy of chemistry,
as most chemical processes involve breaking and forming bonds with a net effect
resembling the involved BDEs. It has been shown many years ago that HF exchange
weakens the BDE of bonds, and correspondingly, that the LYP functional also lowers
bond energies relative to other correlation functionals [145]. This has been seen
repeatedly and is true for bonds involving strictly main-group elements [146] as well
as bonds involving transition metals [98, 147]. Thus, the experience with modeling
chemical bond strengths and spin-state energetics is intriguingly similar. The reason
for this similarity has been proposed [28] to be due to HF exchange generally favoring
the looser electron densities and higher spin quantum numbers reminiscent of both
the dissociated states upon bond breaking and the high-spin electronic state.
Finally, it is relevant to mention the recent observation by Kulik and coworkers
that also the additionally included gradient terms of the metafunctional, as shown
for the TPSS functional, contribute to the spin-state balance [52, 148].
21
because the exact exchange integrals are compensated by explicit pure non-exchange
correlation terms via second-order perturbation theory [36, 94]. Also, the optimized
exchange functional in the form of, e.g., OPBE [139, 140] or OLYP [141, 142],
which does not include exact HF exchange, performs much better and more like a
hybrid functional than other GGAs [132, 143]. The O exchange functional has been
estimated to have an effect that corresponds to ~15% HF exchange [94]. The accuracy
of OPBE supports previous findings by Swart [143, 144]. The O exchange functional
was made from B88 by parameterization toward HF unrestricted energies of atoms
of the first and second periods, and thus, this parameterization by design includes
HF-like energetics. It is very interesting that re-parametrization toward HF energies
or, as analyzed by Swart et al. [132] a leading s
4 term in the exchange functional, can
cause a GGA non-hybrid exchange functional to behave similarly to a hybrid with
15% HF exchange because it tells us that HF exchange is not a “universal” feature
by itself, but a pragmatic solution to a major problem of accuracy [127].
4.2 The Role of the Correlation Functional
Inspection of the original paper by Paulsen et al. [34] reveals that BLYP has the
same ~17 kJ/mol smaller bias toward LS than PW91 for two distinct systems. This
consistent difference could be coincidental, and even if not, it could be due to many
features of the two functionals. Systematic comparison of functional types such
as, e.g., BLYP versus BP86 (which use the same exchange functional) shows that
the correlation functional, perhaps surprisingly, also contributes systematically to
the spin-state balance [132]. Thus, for example, whereas BP86 and PBE give very
similar results for SCO energetics and can be considered to have the same spin-state
balance, the bias toward LS is reduced by typically 10–15 kJ/mol when using BLYP,
and this effect is thus explicitly due to the LYP correlation functional [36].
It is interesting here to comment on the analogy between the performance of DFT
applied to SCO and to the modeling of chemical bond strengths. The dissociation
energy of a chemical bond is arguably the most fundamental energy of chemistry,
as most chemical processes involve breaking and forming bonds with a net effect
resembling the involved BDEs. It has been shown many years ago that HF exchange
weakens the BDE of bonds, and correspondingly, that the LYP functional also lowers
bond energies relative to other correlation functionals [145]. This has been seen
repeatedly and is true for bonds involving strictly main-group elements [146] as well
as bonds involving transition metals [98, 147]. Thus, the experience with modeling
chemical bond strengths and spin-state energetics is intriguingly similar. The reason
for this similarity has been proposed [28] to be due to HF exchange generally favoring
the looser electron densities and higher spin quantum numbers reminiscent of both
the dissociated states upon bond breaking and the high-spin electronic state.
Finally, it is relevant to mention the recent observation by Kulik and coworkers
that also the additionally included gradient terms of the metafunctional, as shown
for the TPSS functional, contribute to the spin-state balance [52, 148].