44
M. Stein
codes is given in [28] and [40]. The various computational approaches to calculate
EPR parameters have been thoroughly reviewed in [23, 28, 30, 32, 41] and are not
repeated here. They range from first [42–44] to more recent ab initio computations
[45–47], and from early work on DFT [48, 49] to more recent implementations
[50–56], multireference spin–orbit configuration (MRSOCI) [57], CASPT2 [58],
MRCI [59], coupled-cluster theory [60] and DMRG [61].
Despite all efforts, the calculation of g-tensors for transition metal complexes is
still more challenging than for organic systems since the spin–orbit coupling term is
dominating. It is therefore clear that an appropriate treatment of spin–orbit coupling
is a pre-requisite for accurate calculations of g. An accuracy of 30% for the calculated ‘g-shift’ is already considered to be satisfying [53]. Spin–orbit coupling has to
be included either into the ground state calculation self-consistently or as a perturbation to the non-relativistic solution. A non-relativistic ground state DFT calculation
with a perturbative approach to introduce spin–orbit coupling into the Breit-Pauli
Hamiltonian was first formulated by [48, 62, 63]. One approach to consider SOC
self-consistently is the zero-order approximation (ZORA; see Chap. 5) which also
enables the calculations of g-tensors [49] and A-tensors [64]. ZORA was shown to
be a good approximation to the fully relativistic treatment, in particular for properties
that involve valence orbitals. To a certain extent, the ZORA and the DKH method
to lowest order yield comparable results for many molecular properties. An efficient
approximation to the full spin–orbit-coupling operator is the spin–orbit mean-field
(SOMF) approximation developed by Hess et al. [65] and extended and implemented
recently [66].
An accurate molecular geometry is an absolute necessity for a reliable relation
between molecule structure and calculated magnetic resonance parameters. Density
Functional Theory (DFT) is one computational method for the optimizations of
large systems with 100–200 atoms and usually gives reliable structural parameters
for transition metal systems. A correct description of the electron density in the
system is also required for the calculation of spectroscopic parameters and other
properties that rely on the electron density. It is thus always necessary to employ an
all-electron and sufficiently large basis set with scalar relativistic effects included for
the calculation of magnetic resonance parameters of transition metal systems.
In literature, quantum chemically calculated EPR g-tensors very often report only
the principal values g 1 , g 2 , g 3 or an average g iso value. There is, however, much more
information in the principal axes system x, y, z which reveals spatially directed
magnetic interactions and defines the magnetic axes of the system. In the presence
of molecular symmetry, they may coincide with the symmetry axes. In biological
systems, in the absence of molecular symmetry, the experimental assignment of principal axes and anisotropic magnetic interactions is more difficult and computational
means are used to resolve ambiguities (see below).
M. Stein
codes is given in [28] and [40]. The various computational approaches to calculate
EPR parameters have been thoroughly reviewed in [23, 28, 30, 32, 41] and are not
repeated here. They range from first [42–44] to more recent ab initio computations
[45–47], and from early work on DFT [48, 49] to more recent implementations
[50–56], multireference spin–orbit configuration (MRSOCI) [57], CASPT2 [58],
MRCI [59], coupled-cluster theory [60] and DMRG [61].
Despite all efforts, the calculation of g-tensors for transition metal complexes is
still more challenging than for organic systems since the spin–orbit coupling term is
dominating. It is therefore clear that an appropriate treatment of spin–orbit coupling
is a pre-requisite for accurate calculations of g. An accuracy of 30% for the calculated ‘g-shift’ is already considered to be satisfying [53]. Spin–orbit coupling has to
be included either into the ground state calculation self-consistently or as a perturbation to the non-relativistic solution. A non-relativistic ground state DFT calculation
with a perturbative approach to introduce spin–orbit coupling into the Breit-Pauli
Hamiltonian was first formulated by [48, 62, 63]. One approach to consider SOC
self-consistently is the zero-order approximation (ZORA; see Chap. 5) which also
enables the calculations of g-tensors [49] and A-tensors [64]. ZORA was shown to
be a good approximation to the fully relativistic treatment, in particular for properties
that involve valence orbitals. To a certain extent, the ZORA and the DKH method
to lowest order yield comparable results for many molecular properties. An efficient
approximation to the full spin–orbit-coupling operator is the spin–orbit mean-field
(SOMF) approximation developed by Hess et al. [65] and extended and implemented
recently [66].
An accurate molecular geometry is an absolute necessity for a reliable relation
between molecule structure and calculated magnetic resonance parameters. Density
Functional Theory (DFT) is one computational method for the optimizations of
large systems with 100–200 atoms and usually gives reliable structural parameters
for transition metal systems. A correct description of the electron density in the
system is also required for the calculation of spectroscopic parameters and other
properties that rely on the electron density. It is thus always necessary to employ an
all-electron and sufficiently large basis set with scalar relativistic effects included for
the calculation of magnetic resonance parameters of transition metal systems.
In literature, quantum chemically calculated EPR g-tensors very often report only
the principal values g 1 , g 2 , g 3 or an average g iso value. There is, however, much more
information in the principal axes system x, y, z which reveals spatially directed
magnetic interactions and defines the magnetic axes of the system. In the presence
of molecular symmetry, they may coincide with the symmetry axes. In biological
systems, in the absence of molecular symmetry, the experimental assignment of principal axes and anisotropic magnetic interactions is more difficult and computational
means are used to resolve ambiguities (see below).