112
V. Krewald and D. A. Pantazis
On the other hand, the question of whether virtual orbitals of the metal ions and
the bridging ligands are required in the active space, for example, the double shell of
4d orbitals in the case of 3d metal ions, does not appear to have a straightforward and
universal answer based on the existing studies. It is clear that these orbitals have an
effect, which can be attributed in part to further recovery of dynamic correlation or
incorporation of specific excitation classes, but the magnitude of this effect appears
to be system-dependent. Therefore, this type of active space extension should be
evaluated in combination with the estimated gains and in relation to the relative cost.
This point became obvious with the MRCI+Q results in the study of Harris et al.
[22] and was further discussed in the study of Roemelt et al. where it was judged
that the steep increase in cost outweighs the gains, particularly if one considers that
the “missing” part of the exchange coupling interaction is unlikely to be sufficiently
recovered by the double-shell extension of the active space and can be more conveniently accounted for with the DMRG-NEVPT2 approach [97].
However, this is likely not a general result. The conclusion may be in part due
to the fact that the d shell of the Mn ions is less than half-filled. In cases where the
double-shell effect is strong, it is expected that it has to be treated as part of the
static correlation [123] and included directly in the orbital optimization. In addition,
CASPT2/NEVPT2 energies in general react more sensitively to the double-shell
effect than CASSCF. Although no general guideline with respect to the treatment of
virtual orbitals can be given at this stage, we expect that the increasing use of automated or semi-automated active space selection procedures based on entanglement
measures [84] will enable a more efficient and systematic approach to determining active space composition for DMRG-SCF calculations on exchange-coupled
systems.
5.2 Orbital Optimization, State Selection and Convergence
Besides the definition of the active space, a decision that crucially determines
the nature of results and conclusions concerns the states targeted and the method
employed for orbital optimization, given that the form of the active orbitals has a
crucial effect on the prediction of magnetic properties [124]. Of the case studies discussed above, only the work of Roemelt et al. [23] contrasted explicitly the results
of state-specific versus state-averaged orbital optimization. The study of the Mn
dimer vividly demonstrated that state-specific CASSCF calculations with a metalonly active space lead to erratic results, while state-specific DMRG-SCF calculations
with a full-valence active space, even though not producing qualitatively unreasonable values, still lead to large and experimentally incompatible deviations from the
Landé spacing of spin states. It was concluded that state-averaging is the preferred
approach because it minimizes these artificial deviations from the Landé pattern.
This means that the orbital optimization procedure should produce simultaneously
orbitals equally good for all states of the spin ladder that describe the magnetic
interaction in the exchange-coupled system.
V. Krewald and D. A. Pantazis
On the other hand, the question of whether virtual orbitals of the metal ions and
the bridging ligands are required in the active space, for example, the double shell of
4d orbitals in the case of 3d metal ions, does not appear to have a straightforward and
universal answer based on the existing studies. It is clear that these orbitals have an
effect, which can be attributed in part to further recovery of dynamic correlation or
incorporation of specific excitation classes, but the magnitude of this effect appears
to be system-dependent. Therefore, this type of active space extension should be
evaluated in combination with the estimated gains and in relation to the relative cost.
This point became obvious with the MRCI+Q results in the study of Harris et al.
[22] and was further discussed in the study of Roemelt et al. where it was judged
that the steep increase in cost outweighs the gains, particularly if one considers that
the “missing” part of the exchange coupling interaction is unlikely to be sufficiently
recovered by the double-shell extension of the active space and can be more conveniently accounted for with the DMRG-NEVPT2 approach [97].
However, this is likely not a general result. The conclusion may be in part due
to the fact that the d shell of the Mn ions is less than half-filled. In cases where the
double-shell effect is strong, it is expected that it has to be treated as part of the
static correlation [123] and included directly in the orbital optimization. In addition,
CASPT2/NEVPT2 energies in general react more sensitively to the double-shell
effect than CASSCF. Although no general guideline with respect to the treatment of
virtual orbitals can be given at this stage, we expect that the increasing use of automated or semi-automated active space selection procedures based on entanglement
measures [84] will enable a more efficient and systematic approach to determining active space composition for DMRG-SCF calculations on exchange-coupled
systems.
5.2 Orbital Optimization, State Selection and Convergence
Besides the definition of the active space, a decision that crucially determines
the nature of results and conclusions concerns the states targeted and the method
employed for orbital optimization, given that the form of the active orbitals has a
crucial effect on the prediction of magnetic properties [124]. Of the case studies discussed above, only the work of Roemelt et al. [23] contrasted explicitly the results
of state-specific versus state-averaged orbital optimization. The study of the Mn
dimer vividly demonstrated that state-specific CASSCF calculations with a metalonly active space lead to erratic results, while state-specific DMRG-SCF calculations
with a full-valence active space, even though not producing qualitatively unreasonable values, still lead to large and experimentally incompatible deviations from the
Landé spacing of spin states. It was concluded that state-averaging is the preferred
approach because it minimizes these artificial deviations from the Landé pattern.
This means that the orbital optimization procedure should produce simultaneously
orbitals equally good for all states of the spin ladder that describe the magnetic
interaction in the exchange-coupled system.