Applications of the Density Matrix Renormalization Group …
99
be recovered by extending the active space. Therefore, additional treatments must
still be applied to the DMRG-SCF wavefunction, such as second-order perturbation
theory [96–99]. An example of the effect of perturbative treatment of a DMRG-SCF
wavefunction in the case of a magnetically coupled system will be discussed in one
of the case studies below.
DMRG has already seen a number of applications in transition metal chemistry
[15, 16, 19, 20, 100–107] and its ability to handle large active spaces in exchangecoupled transition metal systems has been showcased in two papers that deal with
tetramanganese cluster complexes (Fig. 1). The first one studied a minimal model of
the oxygen-evolving complex of Photosystem II, a tetramanganese–calcium cluster
with five oxo-bridges embedded in a protein environment composed mainly of carboxylate ligands [19]. An active space of 44 electrons in 35 orbitals was constructed
from all Mn 3d orbitals and the oxygen 2p orbitals of all five bridges. A single root
with the experimentally known spin multiplicity was calculated with this (44, 35)
active space using DMRG-SCF and its energy was converged to 0.16 kJ mol
−1 . Furthermore, the quantum entanglement of the cluster [83] was analyzed [19]. In another
example, Paul et al. [21] studied a synthetic tetramanganese–calcium complex [108]
that is considered a structural mimic [109] of the oxygen-evolving complex in Photosystem II [110], albeit lacking one flexible oxo-bridge in the center of the inorganic
core [111, 112]. DMRG-SCF with a (37, 32) active space containing all Mn 3d and
O 2p orbitals was used to distinguish between two isomeric forms of the complex. A
single root was calculated for each isomer and the energies were converged to 10
−4
kcal mol
−1 . It is also worth mentioning a DMRG-SCF study by Sharma et al. [20]
of biologically ubiquitous [113] iron–sulfur systems, specifically Fe 2 S 2 dimers and
Fe 4 S 4 clusters. DMRG allowed the use of a (32, 30) active space for the dimers, i.e.,
including all Fe 3d, 4s, 4d and S 3p orbitals, and energies of individual spin states
were converged to 0.1 kcal mol
−1 (35 cm
−1 ). For the Fe 4 S 4 cluster, a Fe 3d and S
3p (54, 36) active space could be used for specific roots in DMRG-CI calculations.
These three studies either did not attempt or did not conclusively address the problem
of magnetic coupling in the tetranuclear systems, but the impressive feat of performing multireference calculations on systems of this size nevertheless demonstrates the
impressive new possibilities offered by DMRG. At the time of this writing, only two
detailed studies of the performance of DMRG-SCF for the exchange coupling problem per se exist in the literature, both on exchange-coupled transition metal dimers.
In the remainder of this chapter, we present and discuss the content and insights
gained from these studies.
4 Case Studies: Magnetic Coupling in Dinuclear
Complexes
Studies of exchange coupling in transition metal complexes using DMRG-based
multireference approaches are still rare. Consequently, the optimal ways of constructing and handling the large active spaces enabled by the DMRG approach, as
99
be recovered by extending the active space. Therefore, additional treatments must
still be applied to the DMRG-SCF wavefunction, such as second-order perturbation
theory [96–99]. An example of the effect of perturbative treatment of a DMRG-SCF
wavefunction in the case of a magnetically coupled system will be discussed in one
of the case studies below.
DMRG has already seen a number of applications in transition metal chemistry
[15, 16, 19, 20, 100–107] and its ability to handle large active spaces in exchangecoupled transition metal systems has been showcased in two papers that deal with
tetramanganese cluster complexes (Fig. 1). The first one studied a minimal model of
the oxygen-evolving complex of Photosystem II, a tetramanganese–calcium cluster
with five oxo-bridges embedded in a protein environment composed mainly of carboxylate ligands [19]. An active space of 44 electrons in 35 orbitals was constructed
from all Mn 3d orbitals and the oxygen 2p orbitals of all five bridges. A single root
with the experimentally known spin multiplicity was calculated with this (44, 35)
active space using DMRG-SCF and its energy was converged to 0.16 kJ mol
−1 . Furthermore, the quantum entanglement of the cluster [83] was analyzed [19]. In another
example, Paul et al. [21] studied a synthetic tetramanganese–calcium complex [108]
that is considered a structural mimic [109] of the oxygen-evolving complex in Photosystem II [110], albeit lacking one flexible oxo-bridge in the center of the inorganic
core [111, 112]. DMRG-SCF with a (37, 32) active space containing all Mn 3d and
O 2p orbitals was used to distinguish between two isomeric forms of the complex. A
single root was calculated for each isomer and the energies were converged to 10
−4
kcal mol
−1 . It is also worth mentioning a DMRG-SCF study by Sharma et al. [20]
of biologically ubiquitous [113] iron–sulfur systems, specifically Fe 2 S 2 dimers and
Fe 4 S 4 clusters. DMRG allowed the use of a (32, 30) active space for the dimers, i.e.,
including all Fe 3d, 4s, 4d and S 3p orbitals, and energies of individual spin states
were converged to 0.1 kcal mol
−1 (35 cm
−1 ). For the Fe 4 S 4 cluster, a Fe 3d and S
3p (54, 36) active space could be used for specific roots in DMRG-CI calculations.
These three studies either did not attempt or did not conclusively address the problem
of magnetic coupling in the tetranuclear systems, but the impressive feat of performing multireference calculations on systems of this size nevertheless demonstrates the
impressive new possibilities offered by DMRG. At the time of this writing, only two
detailed studies of the performance of DMRG-SCF for the exchange coupling problem per se exist in the literature, both on exchange-coupled transition metal dimers.
In the remainder of this chapter, we present and discuss the content and insights
gained from these studies.
4 Case Studies: Magnetic Coupling in Dinuclear
Complexes
Studies of exchange coupling in transition metal complexes using DMRG-based
multireference approaches are still rare. Consequently, the optimal ways of constructing and handling the large active spaces enabled by the DMRG approach, as