Applications of the Density Matrix Renormalization Group …
97
was first used in physics to describe spin-spin correlations in a more refined way than
can be achieved with a mean-field approach [12]. The DMRG algorithm was originally devised for chain-like systems or lattices, and thus the first applications aimed at
predicting magnetically coupled systems involved linear chains of open-shell metals
[89–91]. However, in these studies the electronic states are not constructed as manyelectron wavefunctions as is the case in quantum chemical calculations. As such,
DMRG is not a fundamentally new method for describing molecules, but rather a
new algorithm that is effectively used as a CI-solver.
The algorithm differs from other CI approaches in that it stores the wavefunction
in a different numerical representation than that typically encountered in CASSCF
calculations. The striking feature of DMRG is that in principle the full CI solution can
be approximated to an accuracy typically required for chemically relevant systems
with a computational cost that is normally lower scaling than other multireference
methods. While the computational cost of conventional multireference approaches
is exponential, and DMRG can approximate the correct solution with polynomial
cost [26] for chemically relevant systems.
The DMRG algorithm benefits from proper choice of orbital shape and ordering
to achieve a desired level of accuracy while minimizing the computational cost in
solving the CI problem. A key aspect is that only a few orbitals are treated exactly
during each substep of the iterative procedure, and the other orbitals are either part of
the so-called active subsystem or the complementary subsystem. Thus, the orbitals
sequentially become part of what is known as the “exactly represented subsystem,”
a set of neighboring spatial orbitals. Once each orbital has been treated exactly, i.e.,
after a series of microiterations, a macroiteration or sweep is completed. The wavefunction is represented in the product space of all orbitals, restricted to the desired
total number of electrons and spin state. The number of basis states in the active and
complementary subsystems is denoted by M. For two spatial orbitals in the exactly
represented subsystem, the number of states is 16, as both orbitals can be in any of four
occupations (doubly occupied, singly occupied spin up, singly occupied spin down,
and unoccupied). The algorithm involves a step known as blocking, where the active
subsystem is enlarged by the adjacent orbital from the exactly represented system,
leading to a dimension of 4M for the increased active subsystem. To obtain a system
size of M again, the following step is to transform the system to a new many-particle
basis and thus reduce its size. This step, the transformation from a M × 4M matrix
to an M × M matrix, is called renormalization and involves the diagonalization of
the reduced density matrix. The choice of which elements to discard is based on the
weights of the corresponding eigenstates, and the effect is measured as the so-called
discarded weight. With the renormalized system, the next microiteration can start, in
which the active system is enlarged by one, the exactly represented system loses one
old member and gains one new member, and the complementary subsystem is diminished by one. Once the algorithm has reached the final pair of orbitals in the exactly
represented subsystem, one sweep is completed. Usually several sweeps in alternating directions are performed to improve the accuracy of the DMRG representation
by optimizing the representation of the complementary subsystem. The number of
sweeps at a given discarded weight can be adjusted. Most DMRG implementations
97
was first used in physics to describe spin-spin correlations in a more refined way than
can be achieved with a mean-field approach [12]. The DMRG algorithm was originally devised for chain-like systems or lattices, and thus the first applications aimed at
predicting magnetically coupled systems involved linear chains of open-shell metals
[89–91]. However, in these studies the electronic states are not constructed as manyelectron wavefunctions as is the case in quantum chemical calculations. As such,
DMRG is not a fundamentally new method for describing molecules, but rather a
new algorithm that is effectively used as a CI-solver.
The algorithm differs from other CI approaches in that it stores the wavefunction
in a different numerical representation than that typically encountered in CASSCF
calculations. The striking feature of DMRG is that in principle the full CI solution can
be approximated to an accuracy typically required for chemically relevant systems
with a computational cost that is normally lower scaling than other multireference
methods. While the computational cost of conventional multireference approaches
is exponential, and DMRG can approximate the correct solution with polynomial
cost [26] for chemically relevant systems.
The DMRG algorithm benefits from proper choice of orbital shape and ordering
to achieve a desired level of accuracy while minimizing the computational cost in
solving the CI problem. A key aspect is that only a few orbitals are treated exactly
during each substep of the iterative procedure, and the other orbitals are either part of
the so-called active subsystem or the complementary subsystem. Thus, the orbitals
sequentially become part of what is known as the “exactly represented subsystem,”
a set of neighboring spatial orbitals. Once each orbital has been treated exactly, i.e.,
after a series of microiterations, a macroiteration or sweep is completed. The wavefunction is represented in the product space of all orbitals, restricted to the desired
total number of electrons and spin state. The number of basis states in the active and
complementary subsystems is denoted by M. For two spatial orbitals in the exactly
represented subsystem, the number of states is 16, as both orbitals can be in any of four
occupations (doubly occupied, singly occupied spin up, singly occupied spin down,
and unoccupied). The algorithm involves a step known as blocking, where the active
subsystem is enlarged by the adjacent orbital from the exactly represented system,
leading to a dimension of 4M for the increased active subsystem. To obtain a system
size of M again, the following step is to transform the system to a new many-particle
basis and thus reduce its size. This step, the transformation from a M × 4M matrix
to an M × M matrix, is called renormalization and involves the diagonalization of
the reduced density matrix. The choice of which elements to discard is based on the
weights of the corresponding eigenstates, and the effect is measured as the so-called
discarded weight. With the renormalized system, the next microiteration can start, in
which the active system is enlarged by one, the exactly represented system loses one
old member and gains one new member, and the complementary subsystem is diminished by one. Once the algorithm has reached the final pair of orbitals in the exactly
represented subsystem, one sweep is completed. Usually several sweeps in alternating directions are performed to improve the accuracy of the DMRG representation
by optimizing the representation of the complementary subsystem. The number of
sweeps at a given discarded weight can be adjusted. Most DMRG implementations