Applications of the Density Matrix Renormalization Group …
115
above, i.e. S 0, 1, 2, 3, 4, and 5. The reduced dimer formally contains one high-spin
Fe(III) ion (S A 5/2) and one Fe(II) (S B 2). This is a particularly challenging
case because the additional electron can delocalize between the two iron sites. In the
classic picture, this leads to the splitting of the individual Heisenberg spin levels S
by a “double-exchange” term ±B(S + 1/2). Sharma et al. investigated the low-energy
spectrum for this complex, albeit without performing spin-state averaging of orbitals.
They noticed that the low-lying states could not be adequately described by the classical double-exchange Hamiltonian because their number was greater than what is
accounted for by the classical phenomenological model as a result of multi-orbital
double-exchange processes [20]. It remains to be seen how the choice of orbital optimization affects the conclusions in such electronic situations, electronically more
complex than the exchange-coupled dimers discussed by Harris et al. and Roemelt
et al., but potentially also more sensitive to methodological choices. At least in the
more straightforward examples of the dimers discussed as case studies herein, where
the spin manifold is sufficiently separated from excited electronic configurations, the
quality of the results is adversely affected by the use of spin-state-specific energies.
5.4 Analysis of Exchange Coupling
The investigation by Roemelt et al. into the effect of different bridging ligands in the
case of the Mn dimer by including subsets of bridge-localized orbitals in the active
space established that the oxo-bridges mediate exchange coupling but that the acetato
bridge plays only a structural and not a magnetic role [23, 45]. This application serves
to demonstrate an important use of DMRG in exchange-coupled systems, namely that
by selective inclusion of localized orbital subspaces in the multireference treatment
one can systematically evaluate the contribution of distinct magnetic pathways. Thus,
even if the absolute value of the computed exchange coupling constants is not in
quantitative agreement with experiment, this approach enables the mapping of the
“magnetic topology” of a complex. Analysis of broken-symmetry determinants along
the lines of the Amos–Hall corresponding orbital transformation have long been used
for qualitative analysis of superexchange pathways in DFT studies of exchangecoupled systems [33, 45], but is not directly applicable to systems with more than
two spin sites [47]. By contrast, the DMRG-driven analysis based on selective active
space inclusion of valence orbitals of specific groups is general and can be applied in
any chemical context. Evidently, this approach is not unique to DMRG approaches.
For example, Domingo et al. [126] have used standard CASPT2 calculations with
localized orbitals to investigate the influence of different parts of the molecule on
the overall magnetism of a series of dinuclear molecules. Still, DMRG enables this
treatment to be applied to much larger systems than previously possible in terms
of size, nuclearity, magnetic topology, type, and number of bridging ligands. This
type of treatment should nevertheless not be viewed as “quantitative,” because of the
missing contributions from dynamic correlation and possible cross-interactions.
115
above, i.e. S 0, 1, 2, 3, 4, and 5. The reduced dimer formally contains one high-spin
Fe(III) ion (S A 5/2) and one Fe(II) (S B 2). This is a particularly challenging
case because the additional electron can delocalize between the two iron sites. In the
classic picture, this leads to the splitting of the individual Heisenberg spin levels S
by a “double-exchange” term ±B(S + 1/2). Sharma et al. investigated the low-energy
spectrum for this complex, albeit without performing spin-state averaging of orbitals.
They noticed that the low-lying states could not be adequately described by the classical double-exchange Hamiltonian because their number was greater than what is
accounted for by the classical phenomenological model as a result of multi-orbital
double-exchange processes [20]. It remains to be seen how the choice of orbital optimization affects the conclusions in such electronic situations, electronically more
complex than the exchange-coupled dimers discussed by Harris et al. and Roemelt
et al., but potentially also more sensitive to methodological choices. At least in the
more straightforward examples of the dimers discussed as case studies herein, where
the spin manifold is sufficiently separated from excited electronic configurations, the
quality of the results is adversely affected by the use of spin-state-specific energies.
5.4 Analysis of Exchange Coupling
The investigation by Roemelt et al. into the effect of different bridging ligands in the
case of the Mn dimer by including subsets of bridge-localized orbitals in the active
space established that the oxo-bridges mediate exchange coupling but that the acetato
bridge plays only a structural and not a magnetic role [23, 45]. This application serves
to demonstrate an important use of DMRG in exchange-coupled systems, namely that
by selective inclusion of localized orbital subspaces in the multireference treatment
one can systematically evaluate the contribution of distinct magnetic pathways. Thus,
even if the absolute value of the computed exchange coupling constants is not in
quantitative agreement with experiment, this approach enables the mapping of the
“magnetic topology” of a complex. Analysis of broken-symmetry determinants along
the lines of the Amos–Hall corresponding orbital transformation have long been used
for qualitative analysis of superexchange pathways in DFT studies of exchangecoupled systems [33, 45], but is not directly applicable to systems with more than
two spin sites [47]. By contrast, the DMRG-driven analysis based on selective active
space inclusion of valence orbitals of specific groups is general and can be applied in
any chemical context. Evidently, this approach is not unique to DMRG approaches.
For example, Domingo et al. [126] have used standard CASPT2 calculations with
localized orbitals to investigate the influence of different parts of the molecule on
the overall magnetism of a series of dinuclear molecules. Still, DMRG enables this
treatment to be applied to much larger systems than previously possible in terms
of size, nuclearity, magnetic topology, type, and number of bridging ligands. This
type of treatment should nevertheless not be viewed as “quantitative,” because of the
missing contributions from dynamic correlation and possible cross-interactions.