94
V. Krewald and D. A. Pantazis
electron paramagnetic resonance spectroscopy, or polarized neutron diffraction, are
fitted with such a phenomenological Hamiltonian as appropriate to the chemical system at hand, yielding numerical values for the terms introduced above. In order to
make the fitting problem tractable and reasonably defined, simplifying assumptions
are often made regarding the relative magnitudes of particular coupling constants
and the magnetic topology of a compound. Importantly, for a sufficiently complex
system the fitting cannot be unique, not even if the Hamiltonian is restricted to a single term [29]. Instead of treating the quantities that appear in the HDvV Hamiltonian
as merely numerical parameters to be fitted, quantum chemistry attempts to assign
physical meaning to these parameters by connecting them with fundamental aspects
of the electronic structure, thus enabling both interpretation and prediction by first
principles.
The magnetic coupling problem is inherently a multireference problem: Even if
the ground state of an exchange-coupled system is described by a single electronic
configuration, that is, a unique distribution of electrons among a set of metal-based
orbitals, the resulting spin states are multideterminantal. Nevertheless, the use of
approximate treatments based on single-determinant methods has a long tradition
in computational studies of exchange-coupled transition metal systems. With the
exception of approaches that allow local spins to be non-collinear, single-reference
treatments are mostly restricted to broken-symmetry DFT (BS-DFT). A Kohn–Sham
determinant can formally represent only the magnetically coupled state with maximum total spin multiplicity (e.g., for a dinuclear complex with local spins S A and
S B , S max S A + S B ), referred to as the high-spin (HS) solution. For all other rungs
of the spin ladder with S < S max , more than one determinant is required. The brokensymmetry (BS) formalism was introduced to circumvent this problem [6–8, 30, 31].
Here, an unrestricted determinant is constructed with an M S value equal to that of the
antiferromagnetically coupled state (S min |S A − S B |). In the BS determinant, the
singly occupied orbitals of opposite spin (“magnetic orbitals”) are allowed to localize
at the spin centers while retaining overlap “tails” [7, 32, 33]. The BS determinant is
not a spin eigenfunction, and hence, it has no defined spin quantum number S; it can
be seen as a weighted mixture [34] of all spin states that contain magnetic sublevels
with the same magnetic quantum number M S .
A central question is how to interpret the energy of the BS solution. Several mapping procedures have been proposed and they all use the energy difference between
the HS and BS determinants, relying on assumptions regarding a valid form of a
phenomenological Hamiltonian, focusing chiefly on isotropic bilinear exchange [7,
10, 35, 36]. A popular expression for two-spin systems was proposed by Yamaguchi,
who used the total spin angular momentum expectation values of the HS and BS
determinants to provide a consistent description for weakly to strongly coupled systems [10, 36]:
J −
E HS − E BS
S 2
HS
−
S 2
BS
V. Krewald and D. A. Pantazis
electron paramagnetic resonance spectroscopy, or polarized neutron diffraction, are
fitted with such a phenomenological Hamiltonian as appropriate to the chemical system at hand, yielding numerical values for the terms introduced above. In order to
make the fitting problem tractable and reasonably defined, simplifying assumptions
are often made regarding the relative magnitudes of particular coupling constants
and the magnetic topology of a compound. Importantly, for a sufficiently complex
system the fitting cannot be unique, not even if the Hamiltonian is restricted to a single term [29]. Instead of treating the quantities that appear in the HDvV Hamiltonian
as merely numerical parameters to be fitted, quantum chemistry attempts to assign
physical meaning to these parameters by connecting them with fundamental aspects
of the electronic structure, thus enabling both interpretation and prediction by first
principles.
The magnetic coupling problem is inherently a multireference problem: Even if
the ground state of an exchange-coupled system is described by a single electronic
configuration, that is, a unique distribution of electrons among a set of metal-based
orbitals, the resulting spin states are multideterminantal. Nevertheless, the use of
approximate treatments based on single-determinant methods has a long tradition
in computational studies of exchange-coupled transition metal systems. With the
exception of approaches that allow local spins to be non-collinear, single-reference
treatments are mostly restricted to broken-symmetry DFT (BS-DFT). A Kohn–Sham
determinant can formally represent only the magnetically coupled state with maximum total spin multiplicity (e.g., for a dinuclear complex with local spins S A and
S B , S max S A + S B ), referred to as the high-spin (HS) solution. For all other rungs
of the spin ladder with S < S max , more than one determinant is required. The brokensymmetry (BS) formalism was introduced to circumvent this problem [6–8, 30, 31].
Here, an unrestricted determinant is constructed with an M S value equal to that of the
antiferromagnetically coupled state (S min |S A − S B |). In the BS determinant, the
singly occupied orbitals of opposite spin (“magnetic orbitals”) are allowed to localize
at the spin centers while retaining overlap “tails” [7, 32, 33]. The BS determinant is
not a spin eigenfunction, and hence, it has no defined spin quantum number S; it can
be seen as a weighted mixture [34] of all spin states that contain magnetic sublevels
with the same magnetic quantum number M S .
A central question is how to interpret the energy of the BS solution. Several mapping procedures have been proposed and they all use the energy difference between
the HS and BS determinants, relying on assumptions regarding a valid form of a
phenomenological Hamiltonian, focusing chiefly on isotropic bilinear exchange [7,
10, 35, 36]. A popular expression for two-spin systems was proposed by Yamaguchi,
who used the total spin angular momentum expectation values of the HS and BS
determinants to provide a consistent description for weakly to strongly coupled systems [10, 36]:
J −
E HS − E BS
S 2
HS
−
S 2
BS