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V. Krewald and D. A. Pantazis
is that the interaction between the spin of unpaired electrons, often called magnetic
or exchange coupling, gives rise to unique magnetic and spectroscopic properties
that could not arise from the isolated transition metal centers. The magnitude and
the nature of the coupling, for example, ferromagnetic or antiferromagnetic, have
profound impact on the properties and reactivity of these systems. One of the great
challenges for quantum chemistry is to understand these interactions in the context
of electronic structure theory, connect the fundamental description with the phenomenological models often employed in the analysis of experiments, and finally
predict the relevant parameters that describe the properties of such exchange-coupled
systems with high accuracy and reliability.
The spin states associated with this situation (magnetic levels) typically arise
from a single electronic configuration, but can be formally described only with linear combinations of multiple determinants. In contrast to spin states that correspond
to distinct configurations of d electrons, such as, a low-spin and high-spin configuration of a transition metal ion, the magnetic levels of an exchange-coupled system span
a narrow energy range of a few tens or hundreds of wavenumbers [1]. The demands
imposed on quantum chemical calculations that target magnetically coupled states
are therefore of the order of a wavenumber, and hence much higher than the usual
definitions of “chemical accuracy” related to the prediction of common thermodynamic properties (1 cm
−1
0.00286 kcal/mol 0.01196 kJ/mol, or 1 kJ/mol
0.239 kcal/mol 83.593 cm
−1 ). The computational method of choice must therefore be able to predict the energies of all spin states of the magnetically coupled
system equally well and converge them to the same accuracy.
Several quantum chemical approaches have been proposed to achieve qualitative
and quantitative descriptions of magnetic coupling in molecular complexes with
open-shell transition metal ions [2–6]. Density functional theory (DFT) based on the
broken-symmetry approach [7–10] has been used for a wide range of systems over
several decades with varying levels of success. However, the problem of exchange
coupling is inherently a multireference problem that should be formally treated with
multireference methods. These have also a long history in the field of exchangecoupled transition metal systems, but their applicability has been severely limited to
small dinuclear systems with very few unpaired electrons, for example, Cu(II) dimers
[2, 11]. This is due to the steeply increasing cost of multireference calculations for
problems with more than a few electrons in a few orbitals. The key challenge of
how to enable treatment of large active spaces, for example, in complete active space
self-consistent field (CASSCF) calculations, is of direct relevance for the treatment
of exchange-coupled transition metal systems, where the presence of more than two
metal ions, of many unpaired electrons, or the necessity to include electrons and
orbitals of bridging ligands in the active space quickly renders such calculations
entirely impossible.
The focus of this chapter is on a method that was introduced relatively recently
to the theoretical chemistry community, the density matrix renormalization group
(DMRG) [12, 13]. From the point of view of applied quantum chemistry, DMRG
can be considered as a method that enables the use of large active spaces in multireference calculations. It has already been employed in configuration interaction
V. Krewald and D. A. Pantazis
is that the interaction between the spin of unpaired electrons, often called magnetic
or exchange coupling, gives rise to unique magnetic and spectroscopic properties
that could not arise from the isolated transition metal centers. The magnitude and
the nature of the coupling, for example, ferromagnetic or antiferromagnetic, have
profound impact on the properties and reactivity of these systems. One of the great
challenges for quantum chemistry is to understand these interactions in the context
of electronic structure theory, connect the fundamental description with the phenomenological models often employed in the analysis of experiments, and finally
predict the relevant parameters that describe the properties of such exchange-coupled
systems with high accuracy and reliability.
The spin states associated with this situation (magnetic levels) typically arise
from a single electronic configuration, but can be formally described only with linear combinations of multiple determinants. In contrast to spin states that correspond
to distinct configurations of d electrons, such as, a low-spin and high-spin configuration of a transition metal ion, the magnetic levels of an exchange-coupled system span
a narrow energy range of a few tens or hundreds of wavenumbers [1]. The demands
imposed on quantum chemical calculations that target magnetically coupled states
are therefore of the order of a wavenumber, and hence much higher than the usual
definitions of “chemical accuracy” related to the prediction of common thermodynamic properties (1 cm
−1
0.00286 kcal/mol 0.01196 kJ/mol, or 1 kJ/mol
0.239 kcal/mol 83.593 cm
−1 ). The computational method of choice must therefore be able to predict the energies of all spin states of the magnetically coupled
system equally well and converge them to the same accuracy.
Several quantum chemical approaches have been proposed to achieve qualitative
and quantitative descriptions of magnetic coupling in molecular complexes with
open-shell transition metal ions [2–6]. Density functional theory (DFT) based on the
broken-symmetry approach [7–10] has been used for a wide range of systems over
several decades with varying levels of success. However, the problem of exchange
coupling is inherently a multireference problem that should be formally treated with
multireference methods. These have also a long history in the field of exchangecoupled transition metal systems, but their applicability has been severely limited to
small dinuclear systems with very few unpaired electrons, for example, Cu(II) dimers
[2, 11]. This is due to the steeply increasing cost of multireference calculations for
problems with more than a few electrons in a few orbitals. The key challenge of
how to enable treatment of large active spaces, for example, in complete active space
self-consistent field (CASSCF) calculations, is of direct relevance for the treatment
of exchange-coupled transition metal systems, where the presence of more than two
metal ions, of many unpaired electrons, or the necessity to include electrons and
orbitals of bridging ligands in the active space quickly renders such calculations
entirely impossible.
The focus of this chapter is on a method that was introduced relatively recently
to the theoretical chemistry community, the density matrix renormalization group
(DMRG) [12, 13]. From the point of view of applied quantum chemistry, DMRG
can be considered as a method that enables the use of large active spaces in multireference calculations. It has already been employed in configuration interaction