4,180 research outputs found
Algebraic tools for dealing with the atomic shell model. I. Wavefunctions and integrals for hydrogen--like ions
Today, the 'hydrogen atom model' is known to play its role not only in
teaching the basic elements of quantum mechanics but also for building up
effective theories in atomic and molecular physics, quantum optics, plasma
physics, or even in the design of semiconductor devices. Therefore, the
analytical as well as numerical solutions of the hydrogen--like ions are
frequently required both, for analyzing experimental data and for carrying out
quite advanced theoretical studies. In order to support a fast and consistent
access to these (Coulomb--field) solutions, here we present the Dirac program
which has been developed originally for studying the properties and dynamical
behaviour of the (hydrogen--like) ions. In the present version, a set of Maple
procedures is provided for the Coulomb wave and Green's functions by applying
the (wave) equations from both, the nonrelativistic and relativistic theory.
Apart from the interactive access to these functions, moreover, a number of
radial integrals are also implemented in the Dirac program which may help the
user to construct transition amplitudes and cross sections as they occur
frequently in the theory of ion--atom and ion--photon collisions.Comment: 23 pages, 1 figur
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
An efficient implementation of Slater-Condon rules
Slater-Condon rules are at the heart of any quantum chemistry method as they
allow to simplify -dimensional integrals as sums of 3- or 6-dimensional
integrals. In this paper, we propose an efficient implementation of those rules
in order to identify very rapidly which integrals are involved in a matrix
element expressed in the determinant basis set. This implementation takes
advantage of the bit manipulation instructions on x86 architectures that were
introduced in 2008 with the SSE4.2 instruction set. Finding which spin-orbitals
are involved in the calculation of a matrix element doesn't depend on the
number of electrons of the system.Comment: 8 pages, 5 figure
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