Applications of the Density Matrix Renormalization Group …
113
As a corollary, extraction of a hypothetical exchange coupling constant J based
on only two computed states using state-specific calculations is unreliable because
the assumption of a Landé pattern may not hold at all. With the energies of only two
states, whether they are assumed to be the two lowest energy ones or the states with
highest and lowest spin multiplicity for a given spin-coupling situation, one could
still remain blind to potentially fundamental deficiencies in the description of the
electronic structure of the system. State-averaged orbital optimization can obviously
be more expensive than a state-specific approach, but as a counterweight it usually has
the advantage of more efficient convergence of the CASSCF procedure, associated
with the treatment of all states arising from a given electronic configuration. In
this respect, it should be realized that both the inability to adequately approximate an
experimental exchange coupling constant and the encounter of convergence problems
for specific magnetic states may not be due to active space selection but due to the
choice of the orbital optimization procedure. This may relate, for example, to the
observation of Harris et al. that certain spin states of the chromium dimer could not be
converged (in state-specific calculations) for specific choices of M and active space
[22].
State-averaged orbital optimization does not automatically eliminate Landé deviations; there is still a strong dependence of the relative energies of the spin states
on the number of retained states M, as shown in Table 1. Therefore, state-averaged
orbital optimization should be combined with careful examination of the convergence
with M of pairwise energy differences between the different spin states in order to
determine the point where converged values for the exchange coupling problem can
be obtained. It is important to note that at small values of M, the results on the Mn
dimer can be considered numerically unstable [23], but results obtained with such
small M values have been used in extrapolating spin-state energies in the study of
Harris et al. [22]. It will be interesting to see the effect of state-averaged orbital optimization in the case of the Fe and Cr dimers. It is clear in both studies that regardless
of the method employed for orbital optimization, different spin states converge at
different rates with increasing M. The energies of individually optimized spin states
can in principle be extrapolated to infinite M (see Fig. 3), whereas it is not clear
how this can be performed in the case of state-averaged results, especially when the
discarded weight for all states becomes negligible at high enough M values [23].
Further studies of exchange-coupled systems will be required to better evaluate the
various methodological parameters relating to spin state selection, orbital optimization, and convergence of relative energetics with M. An important question to clarify
for large-active-space DMRG-SCF calculations is whether it will be possible to avoid
or relax the requirement for state-averaged orbital optimization over the complete
span of the spin ladder, because this seriously limits the nuclearity of the complexes
that can be successfully treated.
113
As a corollary, extraction of a hypothetical exchange coupling constant J based
on only two computed states using state-specific calculations is unreliable because
the assumption of a Landé pattern may not hold at all. With the energies of only two
states, whether they are assumed to be the two lowest energy ones or the states with
highest and lowest spin multiplicity for a given spin-coupling situation, one could
still remain blind to potentially fundamental deficiencies in the description of the
electronic structure of the system. State-averaged orbital optimization can obviously
be more expensive than a state-specific approach, but as a counterweight it usually has
the advantage of more efficient convergence of the CASSCF procedure, associated
with the treatment of all states arising from a given electronic configuration. In
this respect, it should be realized that both the inability to adequately approximate an
experimental exchange coupling constant and the encounter of convergence problems
for specific magnetic states may not be due to active space selection but due to the
choice of the orbital optimization procedure. This may relate, for example, to the
observation of Harris et al. that certain spin states of the chromium dimer could not be
converged (in state-specific calculations) for specific choices of M and active space
[22].
State-averaged orbital optimization does not automatically eliminate Landé deviations; there is still a strong dependence of the relative energies of the spin states
on the number of retained states M, as shown in Table 1. Therefore, state-averaged
orbital optimization should be combined with careful examination of the convergence
with M of pairwise energy differences between the different spin states in order to
determine the point where converged values for the exchange coupling problem can
be obtained. It is important to note that at small values of M, the results on the Mn
dimer can be considered numerically unstable [23], but results obtained with such
small M values have been used in extrapolating spin-state energies in the study of
Harris et al. [22]. It will be interesting to see the effect of state-averaged orbital optimization in the case of the Fe and Cr dimers. It is clear in both studies that regardless
of the method employed for orbital optimization, different spin states converge at
different rates with increasing M. The energies of individually optimized spin states
can in principle be extrapolated to infinite M (see Fig. 3), whereas it is not clear
how this can be performed in the case of state-averaged results, especially when the
discarded weight for all states becomes negligible at high enough M values [23].
Further studies of exchange-coupled systems will be required to better evaluate the
various methodological parameters relating to spin state selection, orbital optimization, and convergence of relative energetics with M. An important question to clarify
for large-active-space DMRG-SCF calculations is whether it will be possible to avoid
or relax the requirement for state-averaged orbital optimization over the complete
span of the spin ladder, because this seriously limits the nuclearity of the complexes
that can be successfully treated.