5.3 Analysis with Single Determinant Methods
157
5.3 Analysis with Single Determinant Methods
The multiconfigurational wave function is a natural starting point for a decomposition of the magnetic coupling in polynuclear complexes. The relative magnitude
of the coefficients of the determinants gives a straightforward strategy to analyze
the importance of the different mechanisms. The disadvantage of this approach is
of course the rather heavy computational burden, which limits the applicability to
medium-sized systems. Alternatively, DFT can be used for larger systems but the
analysis of the results cannot be made in the same way as discussed above, since
there is only one Slater determinant that contains all the physics. Instead, one can
decompose the coupling by a series of partial orbital optimizations of the high-spin
and broken symmetry determinants [5]. The intermediate energy differences can be
related to the direct and kinetic exchange, and the spin polarization as will be shown
below.
The analysis starts with a restricted open-shell Kohn-Sham (ROKS) calculation on
the HS state. If necessary, the magnetic orbitals are transformed to the representation
with local orthogonal orbitals a and b, as shown in the first column of Fig. 5.11.
Staying within the spin-restricted formalism makes that for each α orbital a β orbital
can be found which has the same spatial part. In the first step, the direct exchange
is estimated from the energy difference of the HS(ROKS) and a BS determinant in
which only the spin of one of the unpaired electrons is inverted, but neither the core
nor the magnetic orbitals are optimized.
E(BS-ROKS) = E(HS-ROKS) + K ab
(5.22)
With the help of Yamaguchi’s expression (Eq. 4.65), one can define the direct exchange contribution (J DE ) to the total magnetic coupling as
Fig. 5.11 Schematic representation of the decomposition of the magnetic coupling for computational schemes based on a single determinant. The orbitals and electrons marked in red are subject
to changes with respect to the previous step, the rest is kept frozen
157
5.3 Analysis with Single Determinant Methods
The multiconfigurational wave function is a natural starting point for a decomposition of the magnetic coupling in polynuclear complexes. The relative magnitude
of the coefficients of the determinants gives a straightforward strategy to analyze
the importance of the different mechanisms. The disadvantage of this approach is
of course the rather heavy computational burden, which limits the applicability to
medium-sized systems. Alternatively, DFT can be used for larger systems but the
analysis of the results cannot be made in the same way as discussed above, since
there is only one Slater determinant that contains all the physics. Instead, one can
decompose the coupling by a series of partial orbital optimizations of the high-spin
and broken symmetry determinants [5]. The intermediate energy differences can be
related to the direct and kinetic exchange, and the spin polarization as will be shown
below.
The analysis starts with a restricted open-shell Kohn-Sham (ROKS) calculation on
the HS state. If necessary, the magnetic orbitals are transformed to the representation
with local orthogonal orbitals a and b, as shown in the first column of Fig. 5.11.
Staying within the spin-restricted formalism makes that for each α orbital a β orbital
can be found which has the same spatial part. In the first step, the direct exchange
is estimated from the energy difference of the HS(ROKS) and a BS determinant in
which only the spin of one of the unpaired electrons is inverted, but neither the core
nor the magnetic orbitals are optimized.
E(BS-ROKS) = E(HS-ROKS) + K ab
(5.22)
With the help of Yamaguchi’s expression (Eq. 4.65), one can define the direct exchange contribution (J DE ) to the total magnetic coupling as
Fig. 5.11 Schematic representation of the decomposition of the magnetic coupling for computational schemes based on a single determinant. The orbitals and electrons marked in red are subject
to changes with respect to the previous step, the rest is kept frozen