96
V. Krewald and D. A. Pantazis
minimal valence space is thus to include orbitals of the bridging ligands [2], which
turns on various types of charge-transfer excitations that contribute to charge and
spin polarization effects and adjust the weight of neutral and ionic determinants to
better describe the low-energy region of the spin ladder for the exchange-coupled
system.
The total size of the active space is commonly abbreviated with the (N electrons ,
N orbitals ) notation, e.g., (12, 10) denotes an active space with 12 electrons in 10
orbitals. Given that the upper limit for an active space size that can be practically
treated with CASSCF is around 16–18 orbitals, the method may be inapplicable even
for relatively simple dinuclear exchange-coupled systems. The selection of orbitals
that should enter the active space in addition to the magnetic orbitals, the details of
orbital preparation and optimization, the number of states targeted, and other technical choices are crucial factors for the design and ultimately for the success of a
computational study. Still, a CASSCF treatment does not afford quantitative predictions, and may even fail qualitatively, because despite the formally correct multideterminantal description of the states, dynamic electron correlation is absent. Some of
this may be recovered by applying second-order perturbation theory to the CASSCF
wavefunction (complete active space second-order perturbation theory, CASPT2 [62,
63], or N-electron valence second-order perturbation theory, NEVPT2 [64, 65]). In
contrast to these perturbational methods, difference-dedicated configuration interaction (DDCI) is a variational approach, in which particular classes of CT-excitations
are included explicitly in the wavefunction [2, 11, 66–71]. DDCI was suggested to
have considerably better performance and robustness for exchange-coupled systems
over CASPT2 [72], but its applicability remains severely restricted to minimalistic
problems because of its high computational cost.
Although by no means the only issue that has to be addressed, increasing the size of
the active space appears as the major obstacle to applications of multiconfigurational
SCF methods in exchange-coupled transition metal systems. One way of dealing with
this problem has been to use partitioning or truncation schemes, as represented for
example by the restricted active space (RAS) [73, 74], the generalized active space
(GAS) [75], and the split GAS [76, 77] approaches. Alternatively, the active space
limitations are attacked through novel algorithmic approaches, such as the stochastic
full configuration interaction quantum Monte Carlo (FCIQMC) [78, 79] technique,
and the DMRG approach that is the subject of this chapter.
3 The Density Matrix Renormalization Group Approach
The DMRG algorithm, its implementation, and the extraction of (chemical) observables have been discussed in many papers and reviews [12, 13, 25, 26, 80–88]. Its
importance and relevance in particular to inorganic complexes lies in enabling the
description of large active spaces in CASCI and CASSCF calculations. Here, we
present a qualitative description of the fundamental concepts and highlight practical
considerations for the application to open-shell transition metal complexes. DMRG
V. Krewald and D. A. Pantazis
minimal valence space is thus to include orbitals of the bridging ligands [2], which
turns on various types of charge-transfer excitations that contribute to charge and
spin polarization effects and adjust the weight of neutral and ionic determinants to
better describe the low-energy region of the spin ladder for the exchange-coupled
system.
The total size of the active space is commonly abbreviated with the (N electrons ,
N orbitals ) notation, e.g., (12, 10) denotes an active space with 12 electrons in 10
orbitals. Given that the upper limit for an active space size that can be practically
treated with CASSCF is around 16–18 orbitals, the method may be inapplicable even
for relatively simple dinuclear exchange-coupled systems. The selection of orbitals
that should enter the active space in addition to the magnetic orbitals, the details of
orbital preparation and optimization, the number of states targeted, and other technical choices are crucial factors for the design and ultimately for the success of a
computational study. Still, a CASSCF treatment does not afford quantitative predictions, and may even fail qualitatively, because despite the formally correct multideterminantal description of the states, dynamic electron correlation is absent. Some of
this may be recovered by applying second-order perturbation theory to the CASSCF
wavefunction (complete active space second-order perturbation theory, CASPT2 [62,
63], or N-electron valence second-order perturbation theory, NEVPT2 [64, 65]). In
contrast to these perturbational methods, difference-dedicated configuration interaction (DDCI) is a variational approach, in which particular classes of CT-excitations
are included explicitly in the wavefunction [2, 11, 66–71]. DDCI was suggested to
have considerably better performance and robustness for exchange-coupled systems
over CASPT2 [72], but its applicability remains severely restricted to minimalistic
problems because of its high computational cost.
Although by no means the only issue that has to be addressed, increasing the size of
the active space appears as the major obstacle to applications of multiconfigurational
SCF methods in exchange-coupled transition metal systems. One way of dealing with
this problem has been to use partitioning or truncation schemes, as represented for
example by the restricted active space (RAS) [73, 74], the generalized active space
(GAS) [75], and the split GAS [76, 77] approaches. Alternatively, the active space
limitations are attacked through novel algorithmic approaches, such as the stochastic
full configuration interaction quantum Monte Carlo (FCIQMC) [78, 79] technique,
and the DMRG approach that is the subject of this chapter.
3 The Density Matrix Renormalization Group Approach
The DMRG algorithm, its implementation, and the extraction of (chemical) observables have been discussed in many papers and reviews [12, 13, 25, 26, 80–88]. Its
importance and relevance in particular to inorganic complexes lies in enabling the
description of large active spaces in CASCI and CASSCF calculations. Here, we
present a qualitative description of the fundamental concepts and highlight practical
considerations for the application to open-shell transition metal complexes. DMRG