Applications of the Density Matrix Renormalization Group …
109
demonstrate that antiferromagnetic interaction in the Mn dimer arises exclusively
through superexchange via the oxo-bridges and that the acetato ligand plays no role
in mediating the spin coupling.
A crucial methodological point that was encountered again, even with the more
extended active spaces, concerns the use of state-specific versus state-averaged orbital
optimization procedures. In contrast to the state-specific calculations with the metalonly (7, 10) active space, which did not produce a qualitatively correct order of states,
spate-specific DMRG-SCF(19, 16) calculations correctly produce the ordering of
spin states from S 1/2 as the lowest to S 7/2 as the highest. Nevertheless, the spinstate energy spacings deviate strongly from the Landé pattern that is approximated
very closely by state-averaged calculations. State-specific calculations produce a
strong compression of the spin ladder at progressively higher spin levels, which is
particularly exaggerated at low values of M. A crucial observation was that even with
fully individually converged absolute energy values (at M 2000) state-specific
calculations underestimate the stability of the intermediate S 3/2 and S 5/2
states and overestimate the stability of the high-spin S 7/2 state. The result is that
at the level of accuracy required for description of magnetic levels, no meaningful
exchange coupling constant can be extracted because the J values computed from
energy differences between adjacent levels range from −75 cm
−1 for the energy
difference between the two lowest spin states to −44 cm
−1 for the energy difference
between the two highest spin states.
Importantly, by simply using the energy difference between the two lowest states
from state-specific calculations, a deceptively “good” value for J would result, and
this effect would be exaggerated at low M values (<1000). These results clearly
demonstrate that the method of orbital optimization and the careful examination
of convergence with M for all states of the spin ladder are essential for successful
applications of DMRG-based approaches to exchange coupling problems.
A second methodological point concerns the convergence with M of the exchange
coupling values obtained by state-averaged DMRG-SCF(19, 16) calculations. Table 1
reproduces some of the results from the Roemelt et al. study [23], showing the
evolution of the energy levels with the number of renormalized states. The smallest
value reported was M 250 because smaller values either led to numerically scattered
results or failed to converge. The M 250 results are not physically meaningful as
they show no reasonable relation between the spin levels, strongly underestimating
the stability of the intermediate S 3/2 and S 5/2 states. M 500 is an improvement
but must be considered similarly unusable because of the large differences of the J
value obtained for different pairs of states. For M ≥ 1000, the average J value is
converged, but further small improvements are observed up to the highest tested M
3000 with respect to the energy of individual spin states, particularly the S 3/2
state.
In an attempt to further improve the numerical result for the exchange coupling
constant, virtual orbitals of the Mn ions (the “double shell” of 4d orbitals) were
included in the active space. DMRG-CI(19, 26) calculations demonstrated a small
increase in the antiferromagnetic interaction (J −65 cm
−1 ). This improvement
was however negated by the further expansion of the active space to include the 3p
109
demonstrate that antiferromagnetic interaction in the Mn dimer arises exclusively
through superexchange via the oxo-bridges and that the acetato ligand plays no role
in mediating the spin coupling.
A crucial methodological point that was encountered again, even with the more
extended active spaces, concerns the use of state-specific versus state-averaged orbital
optimization procedures. In contrast to the state-specific calculations with the metalonly (7, 10) active space, which did not produce a qualitatively correct order of states,
spate-specific DMRG-SCF(19, 16) calculations correctly produce the ordering of
spin states from S 1/2 as the lowest to S 7/2 as the highest. Nevertheless, the spinstate energy spacings deviate strongly from the Landé pattern that is approximated
very closely by state-averaged calculations. State-specific calculations produce a
strong compression of the spin ladder at progressively higher spin levels, which is
particularly exaggerated at low values of M. A crucial observation was that even with
fully individually converged absolute energy values (at M 2000) state-specific
calculations underestimate the stability of the intermediate S 3/2 and S 5/2
states and overestimate the stability of the high-spin S 7/2 state. The result is that
at the level of accuracy required for description of magnetic levels, no meaningful
exchange coupling constant can be extracted because the J values computed from
energy differences between adjacent levels range from −75 cm
−1 for the energy
difference between the two lowest spin states to −44 cm
−1 for the energy difference
between the two highest spin states.
Importantly, by simply using the energy difference between the two lowest states
from state-specific calculations, a deceptively “good” value for J would result, and
this effect would be exaggerated at low M values (<1000). These results clearly
demonstrate that the method of orbital optimization and the careful examination
of convergence with M for all states of the spin ladder are essential for successful
applications of DMRG-based approaches to exchange coupling problems.
A second methodological point concerns the convergence with M of the exchange
coupling values obtained by state-averaged DMRG-SCF(19, 16) calculations. Table 1
reproduces some of the results from the Roemelt et al. study [23], showing the
evolution of the energy levels with the number of renormalized states. The smallest
value reported was M 250 because smaller values either led to numerically scattered
results or failed to converge. The M 250 results are not physically meaningful as
they show no reasonable relation between the spin levels, strongly underestimating
the stability of the intermediate S 3/2 and S 5/2 states. M 500 is an improvement
but must be considered similarly unusable because of the large differences of the J
value obtained for different pairs of states. For M ≥ 1000, the average J value is
converged, but further small improvements are observed up to the highest tested M
3000 with respect to the energy of individual spin states, particularly the S 3/2
state.
In an attempt to further improve the numerical result for the exchange coupling
constant, virtual orbitals of the Mn ions (the “double shell” of 4d orbitals) were
included in the active space. DMRG-CI(19, 26) calculations demonstrated a small
increase in the antiferromagnetic interaction (J −65 cm
−1 ). This improvement
was however negated by the further expansion of the active space to include the 3p