114
V. Krewald and D. A. Pantazis
5.3 Deviations from Heisenberg Behavior
From the preceding discussion, it can be concluded that if a state-averaged approach
is required for the qualitatively correct description of magnetic levels, then the use of
state-specific DMRG-SCF calculations to predict deviations from Heisenberg behavior is unjustified. Such deviations can of course be perfectly physical: the isotropic
bilinear Heisenberg Hamiltonian is but an idealized approximation and further terms
can be invoked to account for specific observed deviations in the magnetic behavior of
a system, e.g., non-Hund states [2, 125]. As such, the reproduction of a Landé pattern
should not be considered as a desirable “target” for quantum chemical treatments of
exchange-coupled dimers. On the other hand, the analysis of Landé deviations for the
systems discussed above suggests that such deviations result from specific methodological choices, and at least in certain cases they can be viewed as artifacts of either
a small active space, state-specific orbital optimization, small M, or a combination of
the above, with state-averaged orbital optimization being the most important factor
in avoiding such artifacts.
The computed energetics in the study of Fe and Cr dimers suggested a progressive
compression of the spin ladder at higher S states (see Fig. 4) [22], which Harris et al.
considered to be a genuine and physically meaningful demonstration of nonHeisenberg behavior. Consequently, they fitted the computed energies with an additional biquadratic term in the Hamiltonian. However, de Graaf and coworkers did
not observe this type of deviation when they revisited one of the dimers of the Harris et al. study [118]. The critical difference in this subsequent study was the use
of state-averaged orbitals, which apparently eliminated the artificial compression
of the spin ladder. The direct comparison between state-specific and state-averaged
orbitals by Roemelt et al. for the Mn dimer quantified explicitly the significant differences between state-specific and state-averaged results, concluding that only the
latter afford a valid description of the spin ladder. One might argue that the use of
state-averaged orbitals could introduce a bias toward isotropic behavior. However,
this would be incorrect for two reasons: first, because the magnetic levels do normally arise from a single principle electronic configuration, and second, because the
use of state-averaged orbitals does not impose an idealized isotropic spacing of the
magnetic levels anyway, as demonstrated clearly from the results on the Mn dimer.
The question, therefore, is when should computed deviations be considered physically meaningful? One safe conclusion so far is that state-specific calculations are
inappropriate for this problem. This has wider implications for any study that attempts
to employ large-active-space DMRG calculations for the analysis of exchangecoupled systems, but even more so for studies that aspire to directly predict such deviations. An example of the latter is the investigation of double exchange in a series of
iron–sulfur systems by Sharma et al. [20], who employed DMRG to study a series of
iron–sulfur systems that can be considered models of the Fe/S cofactors in biological
electron transfer. Among the complexes investigated was the [Fe 2 S 2 (SCH 3 ) 4 ]
2−/3−
pair. In the oxidized form both ions are high-spin Fe(III), with local spins S A S B
5/2, which lead to the same set of total spin states as for the iron dimer discussed
V. Krewald and D. A. Pantazis
5.3 Deviations from Heisenberg Behavior
From the preceding discussion, it can be concluded that if a state-averaged approach
is required for the qualitatively correct description of magnetic levels, then the use of
state-specific DMRG-SCF calculations to predict deviations from Heisenberg behavior is unjustified. Such deviations can of course be perfectly physical: the isotropic
bilinear Heisenberg Hamiltonian is but an idealized approximation and further terms
can be invoked to account for specific observed deviations in the magnetic behavior of
a system, e.g., non-Hund states [2, 125]. As such, the reproduction of a Landé pattern
should not be considered as a desirable “target” for quantum chemical treatments of
exchange-coupled dimers. On the other hand, the analysis of Landé deviations for the
systems discussed above suggests that such deviations result from specific methodological choices, and at least in certain cases they can be viewed as artifacts of either
a small active space, state-specific orbital optimization, small M, or a combination of
the above, with state-averaged orbital optimization being the most important factor
in avoiding such artifacts.
The computed energetics in the study of Fe and Cr dimers suggested a progressive
compression of the spin ladder at higher S states (see Fig. 4) [22], which Harris et al.
considered to be a genuine and physically meaningful demonstration of nonHeisenberg behavior. Consequently, they fitted the computed energies with an additional biquadratic term in the Hamiltonian. However, de Graaf and coworkers did
not observe this type of deviation when they revisited one of the dimers of the Harris et al. study [118]. The critical difference in this subsequent study was the use
of state-averaged orbitals, which apparently eliminated the artificial compression
of the spin ladder. The direct comparison between state-specific and state-averaged
orbitals by Roemelt et al. for the Mn dimer quantified explicitly the significant differences between state-specific and state-averaged results, concluding that only the
latter afford a valid description of the spin ladder. One might argue that the use of
state-averaged orbitals could introduce a bias toward isotropic behavior. However,
this would be incorrect for two reasons: first, because the magnetic levels do normally arise from a single principle electronic configuration, and second, because the
use of state-averaged orbitals does not impose an idealized isotropic spacing of the
magnetic levels anyway, as demonstrated clearly from the results on the Mn dimer.
The question, therefore, is when should computed deviations be considered physically meaningful? One safe conclusion so far is that state-specific calculations are
inappropriate for this problem. This has wider implications for any study that attempts
to employ large-active-space DMRG calculations for the analysis of exchangecoupled systems, but even more so for studies that aspire to directly predict such deviations. An example of the latter is the investigation of double exchange in a series of
iron–sulfur systems by Sharma et al. [20], who employed DMRG to study a series of
iron–sulfur systems that can be considered models of the Fe/S cofactors in biological
electron transfer. Among the complexes investigated was the [Fe 2 S 2 (SCH 3 ) 4 ]
2−/3−
pair. In the oxidized form both ions are high-spin Fe(III), with local spins S A S B
5/2, which lead to the same set of total spin states as for the iron dimer discussed