98
V. Krewald and D. A. Pantazis
have specific sweep schedules in which the number of renormalized basis states and
sweeps is adjusted until the user-defined energy thresholds are reached.
As noted above, the DMRG algorithm in quantum chemistry packages is used
as a CI-solver, and thus the result of a DMRG-SCF calculation, where DMRGCI and orbital optimization steps alternate until convergence is achieved, should in
principle be identical to that of a CASSCF calculation. The key difference between
a CASSCF and a DMRG calculation lies in an additional parameter that needs to be
carefully monitored and adjusted by the user in any DMRG calculation: the number of
renormalized block states M, closely connected to the discarded weight and thus the
accuracy of the calculation. Because a DMRG calculation does not simply converge
to a preset energy criterion as a DFT or CASSCF calculation normally would, the
user has to run several DMRG calculations with increasing values of M until the
energy has converged to the required accuracy. The ideal value of M depends on the
nature of the chemical species under investigation, on the size and character of the
active space, as well as on the number of roots requested, their spin states and the
type of state averaging required.
Depending on the chemical property that is targeted and the nature of the computed
energies, one may be able to apply extrapolation techniques, where the full CI energy
is linearly extrapolated from several calculations with increasing M [25]. It has to
be noted that extrapolation from DMRG wavefunctions obtained at small M can
be problematic due to the “noise” introduced deliberately in the algorithm’s initial
phases [25]. The applicability of extrapolation techniques for magnetically coupled
systems will be discussed in more detail in the context of the case studies presented
in this chapter.
The choice of orbitals to be included in the active space, the origin of these
orbitals, the localization or not, and their initial ordering are crucial decisions for
a DMRG-SCF calculation and influence the convergence of the DMRG algorithm
[80]. For the choice of orbitals, similar arguments can be followed as in CASSCF
calculations [92]. Reiher and coworkers noted that natural orbitals from a CASSCF
calculation may be better suited as starting orbitals than orbitals derived from a preceding Hartree–Fock calculation [26]. More powerful approaches rely on automated
selection procedures [84, 93, 94]. For example, Stein and Reiher proposed an algorithm that employs orbital entanglement or orbital entropy measures derived from a
low-accuracy, large-CAS calculation and selects the most highly entangled orbitals
for the active space of the production-level calculation [84]. The initial ordering of
orbitals is an important technical aspect. In general, orbitals that are more entangled
should be placed closely together, but unlike in chain-like systems the optimal way
of doing this is not obvious for complex non-linear molecules. Current implementations of DMRG software in quantum chemistry usually optimize and update the
order of orbitals through automated reordering procedures [25, 80, 95]. Similarly,
localized orbitals can improve the performance of DMRG as they help to reduce the
entanglement of the system, which implies that the number of renormalized basis
states to reach a certain accuracy will be lower.
DMRG enables CASCI and CASSCF calculations with active spaces containing
tens of orbitals; however, only a small part of dynamic electron correlation can
V. Krewald and D. A. Pantazis
have specific sweep schedules in which the number of renormalized basis states and
sweeps is adjusted until the user-defined energy thresholds are reached.
As noted above, the DMRG algorithm in quantum chemistry packages is used
as a CI-solver, and thus the result of a DMRG-SCF calculation, where DMRGCI and orbital optimization steps alternate until convergence is achieved, should in
principle be identical to that of a CASSCF calculation. The key difference between
a CASSCF and a DMRG calculation lies in an additional parameter that needs to be
carefully monitored and adjusted by the user in any DMRG calculation: the number of
renormalized block states M, closely connected to the discarded weight and thus the
accuracy of the calculation. Because a DMRG calculation does not simply converge
to a preset energy criterion as a DFT or CASSCF calculation normally would, the
user has to run several DMRG calculations with increasing values of M until the
energy has converged to the required accuracy. The ideal value of M depends on the
nature of the chemical species under investigation, on the size and character of the
active space, as well as on the number of roots requested, their spin states and the
type of state averaging required.
Depending on the chemical property that is targeted and the nature of the computed
energies, one may be able to apply extrapolation techniques, where the full CI energy
is linearly extrapolated from several calculations with increasing M [25]. It has to
be noted that extrapolation from DMRG wavefunctions obtained at small M can
be problematic due to the “noise” introduced deliberately in the algorithm’s initial
phases [25]. The applicability of extrapolation techniques for magnetically coupled
systems will be discussed in more detail in the context of the case studies presented
in this chapter.
The choice of orbitals to be included in the active space, the origin of these
orbitals, the localization or not, and their initial ordering are crucial decisions for
a DMRG-SCF calculation and influence the convergence of the DMRG algorithm
[80]. For the choice of orbitals, similar arguments can be followed as in CASSCF
calculations [92]. Reiher and coworkers noted that natural orbitals from a CASSCF
calculation may be better suited as starting orbitals than orbitals derived from a preceding Hartree–Fock calculation [26]. More powerful approaches rely on automated
selection procedures [84, 93, 94]. For example, Stein and Reiher proposed an algorithm that employs orbital entanglement or orbital entropy measures derived from a
low-accuracy, large-CAS calculation and selects the most highly entangled orbitals
for the active space of the production-level calculation [84]. The initial ordering of
orbitals is an important technical aspect. In general, orbitals that are more entangled
should be placed closely together, but unlike in chain-like systems the optimal way
of doing this is not obvious for complex non-linear molecules. Current implementations of DMRG software in quantum chemistry usually optimize and update the
order of orbitals through automated reordering procedures [25, 80, 95]. Similarly,
localized orbitals can improve the performance of DMRG as they help to reduce the
entanglement of the system, which implies that the number of renormalized basis
states to reach a certain accuracy will be lower.
DMRG enables CASCI and CASSCF calculations with active spaces containing
tens of orbitals; however, only a small part of dynamic electron correlation can