140 research outputs found
Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm
We investigate tree tensor network states for quantum chemistry. Tree tensor
network states represent one of the simplest generalizations of matrix product
states and the density matrix renormalization group. While matrix product
states encode a one-dimensional entanglement structure, tree tensor network
states encode a tree entanglement structure, allowing for a more flexible
description of general molecules. We describe an optimal tree tensor network
state algorithm for quantum chemistry. We introduce the concept of
half-renormalization which greatly improves the efficiency of the calculations.
Using our efficient formulation we demonstrate the strengths and weaknesses of
tree tensor network states versus matrix product states. We carry out benchmark
calculations both on tree systems (hydrogen trees and \pi-conjugated
dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and
chromium dimer). In general, tree tensor network states require much fewer
renormalized states to achieve the same accuracy as matrix product states. In
non-tree molecules, whether this translates into a computational savings is
system dependent, due to the higher prefactor and computational scaling
associated with tree algorithms. In tree like molecules, tree network states
are easily superior to matrix product states. As an ilustration, our largest
dendrimer calculation with tree tensor network states correlates 110 electrons
in 110 active orbitals.Comment: 15 pages, 19 figure
Site-Occupation Embedding Theory using Bethe Ansatz Local Density Approximations
Site-occupation embedding theory (SOET) is an alternative formulation of
density-functional theory (DFT) for model Hamiltonians where the
fully-interacting Hubbard problem is mapped, in principle exactly, onto an
impurity-interacting (rather than a non-interacting) one. It provides a
rigorous framework for combining wavefunction (or Green function) based methods
with DFT. In this work, exact expressions for the per-site energy and double
occupation of the uniform Hubbard model are derived in the context of SOET. As
readily seen from these derivations, the so-called bath contribution to the
per-site correlation energy is, in addition to the latter, the key density
functional quantity to model in SOET. Various approximations based on Bethe
ansatz and perturbative solutions to the Hubbard and single impurity Anderson
models are constructed and tested on a one-dimensional ring. The
self-consistent calculation of the embedded impurity wavefunction has been
performed with the density matrix renormalization group method. It has been
shown that promising results are obtained in specific regimes of correlation
and density. Possible further developments have been proposed in order to
provide reliable embedding functionals and potentials.Comment: Regular article with 14 pages including 6 figure
The Thouless theorem for matrix product states and subsequent post-density matrix renormalization group methods
The similarities between Hartree-Fock (HF) theory and the density-matrix
renormalization group (DMRG) are explored. Both methods can be formulated as
the variational optimization of a wave-function ansatz. Linearization of the
time-dependent variational principle near a variational minimum allows to
derive the random phase approximation (RPA). We show that the non-redundant
parametrization of the matrix product state (MPS) tangent space [J. Haegeman et
al., Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for
MPS, i.e. an explicit non-redundant parametrization of the entire MPS manifold,
starting from a specific MPS reference. Excitation operators are identified,
which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation
(TDA), the configuration interaction (CI) expansion, and coupled cluster
theory. For a small one-dimensional Hubbard chain, we use a CI-MPS ansatz with
single and double excitations to improve on the ground state and to calculate
low-lying excitation energies. For a symmetry-broken ground state of this
model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also
discuss calculations of the RPA-MPS correlation energy. With the long-range
quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS
excitation energies for polyenes are obtained.Comment: 16 pages, 3 figures and 1 tabl
Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms
Current descriptions of the ab initio DMRG algorithm use two superficially
different languages: an older language of the renormalization group and
renormalized operators, and a more recent language of matrix product states and
matrix product operators. The same algorithm can appear dramatically different
when written in the two different vocabularies. In this work, we carefully
describe the translation between the two languages in several contexts. First,
we describe how to efficiently implement the ab-initio DMRG sweep using a
matrix product operator based code, and the equivalence to the original
renormalized operator implementation. Next we describe how to implement the
general matrix product operator/matrix product state algebra within a pure
renormalized operator-based DMRG code. Finally, we discuss two improvements of
the ab initio DMRG sweep algorithm motivated by matrix product operator
language: Hamiltonian compression, and a sum over operators representation that
allows for perfect computational parallelism. The connections and
correspondences described here serve to link the future developments with the
past, and are important in the efficient implementation of continuing advances
in ab initio DMRG and related algorithms.Comment: 35 pages, 10 figure
- …