5,392 research outputs found
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
Finite elements and the discrete variable representation in nonequilibrium Green's function calculations. Atomic and molecular models
In this contribution, we discuss the finite-element discrete variable
representation (FE-DVR) of the nonequilibrium Green's function and its
implications on the description of strongly inhomogeneous quantum systems. In
detail, we show that the complementary features of FEs and the DVR allows for a
notably more efficient solution of the two-time Schwinger/Keldysh/Kadanoff-Baym
equations compared to a general basis approach. Particularly, the use of the
FE-DVR leads to an essential speedup in computing the self-energies.
As atomic and molecular examples we consider the He atom and the linear
version of H in one spatial dimension. For these closed-shell models we,
in Hartree-Fock and second Born approximation, compute the ground-state
properties and compare with the exact findings obtained from the solution of
the few-particle time-dependent Schr\"odinger equation.Comment: 12 pages, 3 figures, submitted as proceedings of conference "PNGF IV
Efficient grid-based method in nonequilibrium Green's function calculations. Application to model atoms and molecules
We propose and apply the finite-element discrete variable representation to
express the nonequilibrium Green's function for strongly inhomogeneous quantum
systems. This method is highly favorable against a general basis approach with
regard to numerical complexity, memory resources, and computation time. Its
flexibility also allows for an accurate representation of spatially extended
hamiltonians, and thus opens the way towards a direct solution of the two-time
Schwinger/Keldysh/Kadanoff-Baym equations on spatial grids, including e.g. the
description of highly excited states in atoms. As first benchmarks, we compute
and characterize, in Hartree-Fock and second Born approximation, the ground
states of the He atom, the H molecule and the LiH molecule in one spatial
dimension. Thereby, the ground-state/binding energies, densities and
bond-lengths are compared with the direct solution of the time-dependent
Schr\"odinger equation.Comment: 11 pages, 5 figures, submitted to Physical Review
Variational Thomas-Fermi Theory of a Nonuniform Bose Condensate at Zero Temperature
We derive a description of the spatially inhomogeneous Bose-Einstein
condensate which treats the system locally as a homogeneous system. This
approach, similar to the Thomas-Fermi model for the inhomogeneous many-particle
fermion system, is well-suited to describe the atomic Bose-Einstein condensates
that have recently been obtained experimentally through atomic trapping and
cooling. In this paper, we confine our attention to the zero temperature case,
although the treatment can be generalized to finite temperatures, as we shall
discuss elsewhere.Comment: 24 pages, latex, 6 ps figures, BoxedEPS include
A New Approach To Relativistic Gaussian Basis Functions: Theory And Applications
We present a new hybrid method to solve the relativistic Hartree-Fock-Roothan
equations where the one- and two-electron radial integrals are evaluated
numerically by defining the basis functions on a grid. This procedure reduces
the computational costs in the evaluation of two-electron radial integrals. The
orbitals generated by this method are employed to compute the ionization
potentials, excitation energies and oscillator strengths of alkali-metal atoms
and elements of group IIIA through second order many-body perturbation theor
and other correlated theories.Comment: RevTex (15 pages) one figur
Strong-coupling approach for strongly correlated electron systems
A perturbation theory scheme in terms of electron hopping, which is based on
the Wick theorem for Hubbard operators, is developed. Diagrammatic series
contain single-site vertices connected by hopping lines and it is shown that
for each vertex the problem splits into the subspaces with ``vacuum states''
determined by the diagonal Hubbard operators and only excitations around these
vacuum states are allowed. The rules to construct diagrams are proposed. In the
limit of infinite spatial dimensions the total auxiliary single-site problem
exactly splits into subspaces that allows to build an analytical
thermodynamically consistent approach for a Hubbard model. Some analytical
results are given for the simple approximations when the two-pole
(alloy-analogy solution) and four-pole (Hartree-Fock approximation) structure
for Green's function is obtained. Two poles describe contribution from the
Fermi-liquid component, which is dominant for small electron and hole
concentrations (``overdoped case'' of high-'s), whereas other two describe
contribution from the non-Fermi liquid and are dominant close to half-filling
(``underdoped case'').Comment: 14 pages, revtex, feynmf, 5 EPS figures, two-column PRB style,
published in PR
Collective excitations in trapped boson-fermion mixtures: from demixing to collapse
We calculate the spectrum of low-lying collective excitations in a gaseous
cloud formed by a Bose-Einstein condensate and a spin-polarized Fermi gas over
a range of the boson-fermion coupling strength extending from strongly
repulsive to strongly attractive. Increasing boson-fermion repulsions drive the
system towards spatial separation of its components (``demixing''), whereas
boson-fermion attractions drive it towards implosion (``collapse''). The
dynamics of the system is treated in the experimentally relevant collisionless
regime by means of a Random-Phase approximation and the behavior of a
mesoscopic cloud under isotropic harmonic confinement is contrasted with that
of a macroscopic mixture at given average particle densities. In the latter
case the locations of both the demixing and the collapse phase transitions are
sharply defined by the same stability condition, which is determined by the
softening of an eigenmode of either fermionic or bosonic origin. In contrast,
the transitions to either demixing or collapse in a mesoscopic cloud at fixed
confinement and particle numbers are spread out over a range of boson-fermion
coupling strength, and some initial decrease of the frequencies of a set of
collective modes is followed by hardening as evidenced by blue shifts of most
eigenmodes. The spectral hardening can serve as a signal of the impending
transition and is most evident when the number of bosons in the cloud is
relatively large. We propose physical interpretations for these dynamical
behaviors with the help of suitably defined partial compressibilities for the
gaseous cloud under confinement.Comment: 16 pages, 7 figures, revtex
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