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K. P. Kepp
4.3 The Use of Quantum-Chemical Benchmarks
and the Post-HF Bias
Two very important requirements for further progress in the accurate description of
spin states is the use of adequate and reliable benchmark data either from high-level
quantum-chemical computations or experimental data, and the proper account of
systematic corrections to ensure that the actual property calculated corresponds to
the experimental observable. For many SCO systems, the H SCO is in fact available
as recently compiled in the SCOFE30 database [36]; these enthalpies can be accurately computed by DFT once ZPE, dispersion, relativistic, and solvent effects are
accounted for.
However, more generally, one has to invoke adequate high-level quantummechanical benchmarks. One such type of calculation is CCSD(T); another is
CASPT2; and a third is diffusion quantum Monte Carlo techniques (DMC). One
of the main worries of these benchmarks is that they depend on a single HF reference
which includes all of the exact exchange energy in a one-determinant basis but none
of the compensating correlation energy. Whereas this is probably not a problem for
the LS state, for the HS state, this single-determinant reference is heavily influenced
by the impact of exact HF exchange and the orbitals and electron density must reflect
this. A valid question is thus whether the correlated method is truly capable of bringing this overly spin-polarized reference state into complete spin balance by affording
most of the compensating correlation energy.
Knowing the HS–LS biases of these benchmark methods is obviously extremely
important in order to avoid false conclusions on the performance of functionals versus such a method. Notably, CASPT2 is biased toward configuration state functions
with more exchange integrals (i.e., higher spin states), as they have favorable interactions with the HF reference. A modified shifted reference state was introduced into
CASPT2 in 2004 [149] but a bias toward HS remained thereafter as shown from a
low-lying triplet in the first CASPT2 study of O 2 -binding to heme [150] or from the
study of other hemes [135]; this bias can be partly remedied by using CC computation
of the 3s3p correlation effects in combination with CASPT2 [151].
The CASPT2 example illustrates well a principle that may be a priori true but
depends in practice on the implemented correlation method: A single-reference postHF method that is not perfectly correlated (i.e., is not full-CI) will not compensate
completely the HF exchange of the reference state and will thus carry some bias
toward this state. Due to this “post-HF-bias”, such methods will tend to favor HS
too much and underbind metal–ligand bonds where more exchange integrals are
presented for the dissociated states than for the bound state. A bias with this type of
effect was in fact reported in the original paper on the shifted CASPT2 zeroth-order
Hamiltonian [149].
The simple mononuclear nitrogen-donor octahedral complex, [Fe(NH 3 ) 6 ]
2+ , is
ideally suited for comparing method performance as most methods can be applied
to this small system [152]. For this system with adequately optimized HS and LS
geometries, the adiabatic energy difference for B3LYP is roughly 60 kJ/mol in favor
K. P. Kepp
4.3 The Use of Quantum-Chemical Benchmarks
and the Post-HF Bias
Two very important requirements for further progress in the accurate description of
spin states is the use of adequate and reliable benchmark data either from high-level
quantum-chemical computations or experimental data, and the proper account of
systematic corrections to ensure that the actual property calculated corresponds to
the experimental observable. For many SCO systems, the H SCO is in fact available
as recently compiled in the SCOFE30 database [36]; these enthalpies can be accurately computed by DFT once ZPE, dispersion, relativistic, and solvent effects are
accounted for.
However, more generally, one has to invoke adequate high-level quantummechanical benchmarks. One such type of calculation is CCSD(T); another is
CASPT2; and a third is diffusion quantum Monte Carlo techniques (DMC). One
of the main worries of these benchmarks is that they depend on a single HF reference
which includes all of the exact exchange energy in a one-determinant basis but none
of the compensating correlation energy. Whereas this is probably not a problem for
the LS state, for the HS state, this single-determinant reference is heavily influenced
by the impact of exact HF exchange and the orbitals and electron density must reflect
this. A valid question is thus whether the correlated method is truly capable of bringing this overly spin-polarized reference state into complete spin balance by affording
most of the compensating correlation energy.
Knowing the HS–LS biases of these benchmark methods is obviously extremely
important in order to avoid false conclusions on the performance of functionals versus such a method. Notably, CASPT2 is biased toward configuration state functions
with more exchange integrals (i.e., higher spin states), as they have favorable interactions with the HF reference. A modified shifted reference state was introduced into
CASPT2 in 2004 [149] but a bias toward HS remained thereafter as shown from a
low-lying triplet in the first CASPT2 study of O 2 -binding to heme [150] or from the
study of other hemes [135]; this bias can be partly remedied by using CC computation
of the 3s3p correlation effects in combination with CASPT2 [151].
The CASPT2 example illustrates well a principle that may be a priori true but
depends in practice on the implemented correlation method: A single-reference postHF method that is not perfectly correlated (i.e., is not full-CI) will not compensate
completely the HF exchange of the reference state and will thus carry some bias
toward this state. Due to this “post-HF-bias”, such methods will tend to favor HS
too much and underbind metal–ligand bonds where more exchange integrals are
presented for the dissociated states than for the bound state. A bias with this type of
effect was in fact reported in the original paper on the shifted CASPT2 zeroth-order
Hamiltonian [149].
The simple mononuclear nitrogen-donor octahedral complex, [Fe(NH 3 ) 6 ]
2+ , is
ideally suited for comparing method performance as most methods can be applied
to this small system [152]. For this system with adequately optimized HS and LS
geometries, the adiabatic energy difference for B3LYP is roughly 60 kJ/mol in favor