Applications of the Density Matrix Renormalization Group …
95
Having obtained a value for the exchange coupling constant J, the spin-state ordering and the relative energies of all rungs of the spin ladder are deduced through the
HDvV Hamiltonian. The generalization to oligonuclear systems with N spin centers
is straightforward but of rapidly increasing complexity as one needs to determine
the values of up to N(N − 1)/2 pairwise exchange coupling constants J ij . These are
accessible through the computation of up to 2
N−1 distinct broken-symmetry determinants. For more than three non-symmetry-related spin centers, the number of possible
BS determinants can exceed the number of pairwise exchange coupling constants.
This leads to an overdetermined system of equations, which can be solved via singular value decomposition [37] to obtain a set of exchange coupling constants J ij
that is unique in the least-squares sense [29, 38]. It is noted that a generalized spin
projection method has also been introduced for oligonuclear systems [39].
Despite the extensive use of BS-DFT [40–52], the approach has significant limitations. In terms of energetics, the application of the method suffers by the pronounced
sensitivity on the density functional and relies on empirical benchmarking against
experimental data [4, 5, 53–55]. Although the charge density of the system described
by the broken-symmetry determinant is often reliable, the spin density of any state
other than the high-spin solution is qualitatively incorrect [4, 56]. The intermediate
spin states are not accessible at all by the broken-symmetry formalism; only their
energies relative to the HS or BS energy can be predicted, and this only indirectly
[4]. This necessitates the use of approximate spin projection methods for predicting
spin-dependent properties. Moreover, the interpretation of magnetic coupling based
on BS determinants is often limited to qualitative analysis or visualization of magnetic orbitals via the corresponding orbital transformation of Amos and Hall [33, 57]
which is not obviously extendable beyond dinuclear species [47].
Multireference wavefunction-based calculations present a distinct quantum chemical alternative, because they offer an opposite approach to the problem. Instead of
trying to approximate the HDvV solution space based on a much more limited and
approximate number of single-determinant solutions, one can work directly with the
(approximate) solutions of the Schrödinger equation. In this case, no assumptions are
required regarding the form and the terms of a phenomenological HDvV Hamiltonian, and hence the problem can be approached from the opposite direction than that
represented by BS-DFT. CI (if only the coefficients of distinct configuration state
functions are optimized) and CASSCF [58, 59] (if the orbitals are also optimized)
are examples of multireference methods by which all individual spin states of the
magnetically coupled system can be accessed directly.
To describe a magnetically coupled system at the very least, the magnetic orbitals
have to be considered in the construction of a minimal active space. A common expansion of the active space in systems with first-row transition metal ions is to include
unoccupied d orbitals. The so-called double shell, 3d
or 4d orbitals are important
for an adequate description of radial electron correlation [60, 61]. Considering the
Anderson model of superexchange, by which bridging ligands mediate the transfer
of spin between the individual spin sites, it is obvious that to describe magnetically
coupled systems larger active spaces are needed than in cases that are dominated by
the local properties of an individual transition metal ion. A logical extension of the
95
Having obtained a value for the exchange coupling constant J, the spin-state ordering and the relative energies of all rungs of the spin ladder are deduced through the
HDvV Hamiltonian. The generalization to oligonuclear systems with N spin centers
is straightforward but of rapidly increasing complexity as one needs to determine
the values of up to N(N − 1)/2 pairwise exchange coupling constants J ij . These are
accessible through the computation of up to 2
N−1 distinct broken-symmetry determinants. For more than three non-symmetry-related spin centers, the number of possible
BS determinants can exceed the number of pairwise exchange coupling constants.
This leads to an overdetermined system of equations, which can be solved via singular value decomposition [37] to obtain a set of exchange coupling constants J ij
that is unique in the least-squares sense [29, 38]. It is noted that a generalized spin
projection method has also been introduced for oligonuclear systems [39].
Despite the extensive use of BS-DFT [40–52], the approach has significant limitations. In terms of energetics, the application of the method suffers by the pronounced
sensitivity on the density functional and relies on empirical benchmarking against
experimental data [4, 5, 53–55]. Although the charge density of the system described
by the broken-symmetry determinant is often reliable, the spin density of any state
other than the high-spin solution is qualitatively incorrect [4, 56]. The intermediate
spin states are not accessible at all by the broken-symmetry formalism; only their
energies relative to the HS or BS energy can be predicted, and this only indirectly
[4]. This necessitates the use of approximate spin projection methods for predicting
spin-dependent properties. Moreover, the interpretation of magnetic coupling based
on BS determinants is often limited to qualitative analysis or visualization of magnetic orbitals via the corresponding orbital transformation of Amos and Hall [33, 57]
which is not obviously extendable beyond dinuclear species [47].
Multireference wavefunction-based calculations present a distinct quantum chemical alternative, because they offer an opposite approach to the problem. Instead of
trying to approximate the HDvV solution space based on a much more limited and
approximate number of single-determinant solutions, one can work directly with the
(approximate) solutions of the Schrödinger equation. In this case, no assumptions are
required regarding the form and the terms of a phenomenological HDvV Hamiltonian, and hence the problem can be approached from the opposite direction than that
represented by BS-DFT. CI (if only the coefficients of distinct configuration state
functions are optimized) and CASSCF [58, 59] (if the orbitals are also optimized)
are examples of multireference methods by which all individual spin states of the
magnetically coupled system can be accessed directly.
To describe a magnetically coupled system at the very least, the magnetic orbitals
have to be considered in the construction of a minimal active space. A common expansion of the active space in systems with first-row transition metal ions is to include
unoccupied d orbitals. The so-called double shell, 3d
or 4d orbitals are important
for an adequate description of radial electron correlation [60, 61]. Considering the
Anderson model of superexchange, by which bridging ligands mediate the transfer
of spin between the individual spin sites, it is obvious that to describe magnetically
coupled systems larger active spaces are needed than in cases that are dominated by
the local properties of an individual transition metal ion. A logical extension of the