168
5 Towards a Quantitative Understanding
The comparison of this expression with the one derived for the ordinary twocenter coupling (4t 2 /U ) shows that J r is expected to be significantly smaller than
the interactions described in the standard Heisenberg Hamiltonian, dividing by U 3
instead of U makes the interaction much smaller. However, the very large prefactor
in the perturbative estimate makes that the ring exchange is not necessarily negligible
in all cases. As long as U is not too large and t sizeable, one can expect significant
four-center interactions when the geometry of the system is square-like.
5.4.3 Complex Interactions with Single Determinant
Approaches
Biquadratic exchange: The isotropic linear magnetic exchange can be calculated
in a rather straightforward way with single determinant spin unrestricted methods.
Assuming that the BS determinant is a linear combination of the spin states with lowest and highest possible spin moment, the Yamaguchi equation (Eq. 4.85) relates the
energies of the BS and HS determinants with J in a straightforward way, independent
of the number of unpaired electrons on the magnetic sites involved in the coupling.
The biquadratic exchange can however not be addressed from energy differences
only, simply because we have only access to one energy difference, obviously too
few to determine two parameters.
Instead one can estimate the strength of the biquadratic exchange in an indirect way
via the electronic structure parameters U , t and K . To derive the relevant equations
we need to compare the expressions of the singlet and triplet states in terms of
J and λ given in Eq. 3.75 with their fourth-order perturbation estimates using the
matrix elements derived in Sect. 5.4.1. In the first place, we need a common zero of
energy. This is easily achieved by putting the energy of the quintet state to zero. The
expressions of singlet, triplet and quintet states in terms of J and λ then become
E(Q) = 0
E(T ) = 2J
E(S) = 3J + 3λ
(5.46)
The fourth-order perturbation estimates give us expressions in terms of t, U and
K , and hence, we can relate λ to these electronic structure parameters, which can
be calculated with spin-unrestricted single determinant methods. To simplify the
perturbation estimates we neglect α, β and γ , and all intersite exchange integrals
in the interaction matrices derived in Sect. 5.4.1. The interaction matrices for triplet
and singlet states then become
5 Towards a Quantitative Understanding
The comparison of this expression with the one derived for the ordinary twocenter coupling (4t 2 /U ) shows that J r is expected to be significantly smaller than
the interactions described in the standard Heisenberg Hamiltonian, dividing by U 3
instead of U makes the interaction much smaller. However, the very large prefactor
in the perturbative estimate makes that the ring exchange is not necessarily negligible
in all cases. As long as U is not too large and t sizeable, one can expect significant
four-center interactions when the geometry of the system is square-like.
5.4.3 Complex Interactions with Single Determinant
Approaches
Biquadratic exchange: The isotropic linear magnetic exchange can be calculated
in a rather straightforward way with single determinant spin unrestricted methods.
Assuming that the BS determinant is a linear combination of the spin states with lowest and highest possible spin moment, the Yamaguchi equation (Eq. 4.85) relates the
energies of the BS and HS determinants with J in a straightforward way, independent
of the number of unpaired electrons on the magnetic sites involved in the coupling.
The biquadratic exchange can however not be addressed from energy differences
only, simply because we have only access to one energy difference, obviously too
few to determine two parameters.
Instead one can estimate the strength of the biquadratic exchange in an indirect way
via the electronic structure parameters U , t and K . To derive the relevant equations
we need to compare the expressions of the singlet and triplet states in terms of
J and λ given in Eq. 3.75 with their fourth-order perturbation estimates using the
matrix elements derived in Sect. 5.4.1. In the first place, we need a common zero of
energy. This is easily achieved by putting the energy of the quintet state to zero. The
expressions of singlet, triplet and quintet states in terms of J and λ then become
E(Q) = 0
E(T ) = 2J
E(S) = 3J + 3λ
(5.46)
The fourth-order perturbation estimates give us expressions in terms of t, U and
K , and hence, we can relate λ to these electronic structure parameters, which can
be calculated with spin-unrestricted single determinant methods. To simplify the
perturbation estimates we neglect α, β and γ , and all intersite exchange integrals
in the interaction matrices derived in Sect. 5.4.1. The interaction matrices for triplet
and singlet states then become