6,328 research outputs found
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Quantitative wave function analysis for excited states of transition metal complexes
The character of an electronically excited state is one of the most important
descriptors employed to discuss the photophysics and photochemistry of
transition metal complexes. In transition metal complexes, the interaction
between the metal and the different ligands gives rise to a rich variety of
excited states, including metal-centered, intra-ligand, metal-to-ligand charge
transfer, ligand-to-metal charge transfer, and ligand-to-ligand charge transfer
states. Most often, these excited states are identified by considering the most
important wave function excitation coefficients and inspecting visually the
involved orbitals. This procedure is tedious, subjective, and imprecise.
Instead, automatic and quantitative techniques for excited-state
characterization are desirable. In this contribution we review the concept of
charge transfer numbers---as implemented in the TheoDORE package---and show its
wide applicability to characterize the excited states of transition metal
complexes. Charge transfer numbers are a formal way to analyze an excited state
in terms of electron transitions between groups of atoms based only on the
well-defined transition density matrix. Its advantages are many: it can be
fully automatized for many excited states, is objective and reproducible, and
provides quantitative data useful for the discussion of trends or patterns. We
also introduce a formalism for spin-orbit-mixed states and a method for
statistical analysis of charge transfer numbers. The potential of this
technique is demonstrated for a number of prototypical transition metal
complexes containing Ir, Ru, and Re. Topics discussed include orbital
delocalization between metal and carbonyl ligands, nonradiative decay through
metal-centered states, effect of spin-orbit couplings on state character, and
comparison among results obtained from different electronic structure methods.Comment: 47 pages, 19 figures, including supporting information (7 pages, 1
figure
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
Problem-orientable numerical algorithm for modelling multi-dimensional radiative MHD flows in astrophysics -- the hierarchical solution scenario
We present a hierarchical approach for enhancing the robustness of numerical
solvers for modelling radiative MHD flows in multi-dimensions. This approach is
based on clustering the entries of the global Jacobian in a hierarchical manner
that enables employing a variety of solution procedures ranging from a purely
explicit time-stepping up to fully implicit schemes. A gradual coupling of the
radiative MHD equation with the radiative transfer equation in higher
dimensions is possible. Using this approach, it is possible to follow the
evolution of strongly time-dependent flows with low/high accuracies and with
efficiency comparable to explicit methods, as well as searching
quasi-stationary solutions for highly viscous flows. In particular, it is shown
that the hierarchical approach is capable of modelling the formation of jets in
active galactic nuclei and reproduce the corresponding spectral energy
distribution with a reasonable accuracy.Comment: 28 pages, 9 figure
A hybrid stochastic hierarchy equations of motion approach to treat the low temperature dynamics of non-Markovian open quantum systems
The hierarchical equations of motion technique has found widespread success
as a tool to generate the numerically exact dynamics of non-Markovian open
quantum systems. However, its application to low temperature environments
remains a serious challenge due to the need for a deep hierarchy that arises
from the Matsubara expansion of the bath correlation function. Here we present
a hybrid stochastic hierarchical equation of motion (sHEOM) approach that
alleviates this bottleneck and leads to a numerical cost that is nearly
independent of temperature. Additionally, the sHEOM method generally converges
with fewer hierarchy tiers allowing for the treatment of larger systems.
Benchmark calculations are presented on the dynamics of two level systems at
both high and low temperatures to demonstrate the efficacy of the approach.
Then the hybrid method is used to generate the exact dynamics of systems that
are nearly impossible to treat by the standard hierarchy. First, exact energy
transfer rates are calculated across a broad range of temperatures revealing
the deviations from the Forster rates. This is followed by computations of the
entanglement dynamics in a system of two qubits at low temperature spanning the
weak to strong system-bath coupling regimes.Comment: 20 pages, 6 figure
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
A Unified Stochastic Formulation of Dissipative Quantum Dynamics. I. Generalized Hierarchical Equations
We extend a standard stochastic theory to study open quantum systems coupled
to generic quantum environments including the three fundamental classes of
noninteracting particles: bosons, fermions and spins. In this unified
stochastic approach, the generalized stochastic Liouville equation (SLE)
formally captures the exact quantum dissipations when noise variables with
appropriate statistics for different bath models are applied. Anharmonic
effects of a non-Gaussian bath are precisely encoded in the bath multi-time
correlation functions that noise variables have to satisfy. Staring from the
SLE, we devise a family of generalized hierarchical equations by averaging out
the noise variables and expand bath multi-time correlation functions in a
complete basis of orthonormal functions. The general hiearchical equations
constitute systems of linear equations that provide numerically exact
simulations of quantum dynamics. For bosonic bath models, our general
hierarchical equation of motion reduces exactly to an extended version of
hierarchical equation of motion which allows efficient simulation for arbitrary
spectral densities and temperature regimes. Similar efficiency and exibility
can be achieved for the fermionic bath models within our formalism. The spin
bath models can be simulated with two complementary approaches in the presetn
formalism. (I) They can be viewed as an example of non-Gaussian bath models and
be directly handled with the general hierarchical equation approach given their
multi-time correlation functions. (II) Alterantively, each bath spin can be
first mapped onto a pair of fermions and be treated as fermionic environments
within the present formalism.Comment: 31 pages, 2 figure
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