2,410 research outputs found
On the ground states and dynamics of space fractional nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions
In this paper, we propose some efficient and robust numerical methods to
compute the ground states and dynamics of Fractional Schr\"{o}dinger Equation
(FSE) with a rotation term and nonlocal nonlinear interactions. In particular,
a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal
interaction evaluation \cite{EMZ2015}. To compute the ground states, we
integrate the preconditioned Krylov subspace pseudo-spectral method \cite{AD1}
and the GauSum solver. For the dynamics simulation, using the rotating
Lagrangian coordinates transform \cite{BMTZ2013}, we first reformulate the FSE
into a new equation without rotation. Then, a time-splitting pseudo-spectral
scheme incorporated with the GauSum solver is proposed to simulate the new FSE
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
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