2.1 Spin-Orbit Coupling
Molecules that contain heavy elements (in particular 5d transition metals) play an
important role in the photochemistry and photophysics of coordination compounds
with regard to their luminescent properties and their implication in catalysis and
energy/electron transfer processes. Whereas molecular properties and electronic
spectroscopy of light molecules can be studied in a non-relativistic quantum
chemical framework, one has to consider the theory of relativity when dealing
with elements that belong to the lower region of the periodic table. As far as
transition metal complexes are concerned, one has to distinguish between different
manifestations of relativity. Important but not directly observable manifestations of
relativity are the mass velocity correction and the Darwin correction. These terms
lead to the so-called relativistic contraction of the s- and p-shells and to the
relativistic expansion of the d- and f-shells. A chemical consequence of this is,
for instance, a destabilization of the 5d shells with respect to the 3d shells in
transition metals.
Other important evidence of relativity in electronic spectroscopy, photophysics,
and photochemistry is the spin-orbit coupling between states of different multiplicities. Indeed, most light-induced processes in transition metal complexes involve a
change of spin state and are not allowed in non-relativistic quantum formalism. A
fully relativistic treatment based on the Dirac equation [55] and a four-component
Hamiltonian including scalar and spin-orbit contributions for many electrons systems is unrealistic and beyond the scope of the systems and problems of interest in
this review. The reduction of the Dirac equation and its extension to many-electron
problems has opened the route to several relativistic approaches based on approximate Hamiltonians applied with success to chemistry [56]. The spin-orbit coupling
terms arise from one- and two-electron operators developed within the
two-component formalism of the relativistic theory obtained by transformation of
the four-component equation. The Douglas–Kroll (DK) [57, 58] and Breit–Pauli
(BP) [59–62] forms are the most popular relativistic two-component operators.
In most of today’s applications the SOC effects in large transition metal complexes are included by means of two approaches: (1) the restricted active space state
interaction (RASSI) including SOC [63] developed on the basis of a one-electron
Fock-type spin-orbit Hamiltonian [64] within the atomic mean field approximation
(AMFI) [65]; and (2) the zeroth-order regular approximation (ZORA) to the full
relativistic Hamiltonian based on a one effective two-component regular Hamiltonian developed at the zeroth-order [66–68]. Both methods are derived from the BP
spin-orbit Hamiltonian and are a good approximation to the BP theory. Whereas
RASSI-SOC formalism has been developed for correlated wave functions, the
ZORA operator is better adapted to perturbation and Kohn–Sham (KS) theories.
Most ZORA applications are performed within the framework of DFT despite some
limitations [35, 69]. SOC has recently been evaluated by means of a full BP
Hamiltonian applied to multi-reference CI wave functions or combined with
Absorption Spectroscopy, Emissive Properties, and Ultrafast Intersystem. . .
381
Molecules that contain heavy elements (in particular 5d transition metals) play an
important role in the photochemistry and photophysics of coordination compounds
with regard to their luminescent properties and their implication in catalysis and
energy/electron transfer processes. Whereas molecular properties and electronic
spectroscopy of light molecules can be studied in a non-relativistic quantum
chemical framework, one has to consider the theory of relativity when dealing
with elements that belong to the lower region of the periodic table. As far as
transition metal complexes are concerned, one has to distinguish between different
manifestations of relativity. Important but not directly observable manifestations of
relativity are the mass velocity correction and the Darwin correction. These terms
lead to the so-called relativistic contraction of the s- and p-shells and to the
relativistic expansion of the d- and f-shells. A chemical consequence of this is,
for instance, a destabilization of the 5d shells with respect to the 3d shells in
transition metals.
Other important evidence of relativity in electronic spectroscopy, photophysics,
and photochemistry is the spin-orbit coupling between states of different multiplicities. Indeed, most light-induced processes in transition metal complexes involve a
change of spin state and are not allowed in non-relativistic quantum formalism. A
fully relativistic treatment based on the Dirac equation [55] and a four-component
Hamiltonian including scalar and spin-orbit contributions for many electrons systems is unrealistic and beyond the scope of the systems and problems of interest in
this review. The reduction of the Dirac equation and its extension to many-electron
problems has opened the route to several relativistic approaches based on approximate Hamiltonians applied with success to chemistry [56]. The spin-orbit coupling
terms arise from one- and two-electron operators developed within the
two-component formalism of the relativistic theory obtained by transformation of
the four-component equation. The Douglas–Kroll (DK) [57, 58] and Breit–Pauli
(BP) [59–62] forms are the most popular relativistic two-component operators.
In most of today’s applications the SOC effects in large transition metal complexes are included by means of two approaches: (1) the restricted active space state
interaction (RASSI) including SOC [63] developed on the basis of a one-electron
Fock-type spin-orbit Hamiltonian [64] within the atomic mean field approximation
(AMFI) [65]; and (2) the zeroth-order regular approximation (ZORA) to the full
relativistic Hamiltonian based on a one effective two-component regular Hamiltonian developed at the zeroth-order [66–68]. Both methods are derived from the BP
spin-orbit Hamiltonian and are a good approximation to the BP theory. Whereas
RASSI-SOC formalism has been developed for correlated wave functions, the
ZORA operator is better adapted to perturbation and Kohn–Sham (KS) theories.
Most ZORA applications are performed within the framework of DFT despite some
limitations [35, 69]. SOC has recently been evaluated by means of a full BP
Hamiltonian applied to multi-reference CI wave functions or combined with
Absorption Spectroscopy, Emissive Properties, and Ultrafast Intersystem. . .
381