A complex guided spectral transform Lanczos method for studying quantum resonance states
Abstract
A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths and wavefunctions. The algorithm comprises of two layers of complexsymmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos (cFOL) polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigenpairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonian in the spectral range of interest. Therefore the energies and/or widths of bound or resonance states can be easily computed with their wavefunctions or by using a rootsearching method from the guided spectral transform surface. The new cGSTL algorithm is applied to bound and resonance states of HO₂, and compared to previous calculations.
 Authors:
 Brookhaven National Lab. (BNL), Upton, NY (United States). Dept. of Chemistry
 Publication Date:
 Research Org.:
 Brookhaven National Laboratory (BNL), Upton, NY (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22)
 OSTI Identifier:
 1182481
 Alternate Identifier(s):
 OSTI ID: 1226656
 Report Number(s):
 BNL1073102014JA
Journal ID: ISSN 00219606; JCPSA6; R&D Project: 04540; 04548; TRN: US1500504
 Grant/Contract Number:
 SC00112704
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Journal of Chemical Physics
 Additional Journal Information:
 Journal Volume: 141; Journal Issue: 24; Journal ID: ISSN 00219606
 Publisher:
 American Institute of Physics (AIP)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 74 ATOMIC AND MOLECULAR PHYSICS; iterative diagonalization; complex symmetric matrix; spectral transform Lanczos
Citation Formats
Yu, HuaGen. A complex guided spectral transform Lanczos method for studying quantum resonance states. United States: N. p., 2014.
Web. doi:10.1063/1.4905083.
Yu, HuaGen. A complex guided spectral transform Lanczos method for studying quantum resonance states. United States. doi:10.1063/1.4905083.
Yu, HuaGen. 2014.
"A complex guided spectral transform Lanczos method for studying quantum resonance states". United States.
doi:10.1063/1.4905083. https://www.osti.gov/servlets/purl/1182481.
@article{osti_1182481,
title = {A complex guided spectral transform Lanczos method for studying quantum resonance states},
author = {Yu, HuaGen},
abstractNote = {A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths and wavefunctions. The algorithm comprises of two layers of complexsymmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos (cFOL) polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigenpairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonian in the spectral range of interest. Therefore the energies and/or widths of bound or resonance states can be easily computed with their wavefunctions or by using a rootsearching method from the guided spectral transform surface. The new cGSTL algorithm is applied to bound and resonance states of HO₂, and compared to previous calculations.},
doi = {10.1063/1.4905083},
journal = {Journal of Chemical Physics},
number = 24,
volume = 141,
place = {United States},
year = 2014,
month =
}
Web of Science

A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths, and wavefunctions. The algorithm comprises of two layers of complexsymmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigenpairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonianmore »

Spectral transform Lanczos for Electronic states from density functional computation
Planewave, density functional computation successfully determines crystal structures, via minimization of an energy functional. This may fail when several minima are related by small changes of physical parameters. We present an alternative approach, minimizing the wave function error using the Lanczos algorithm. Coding for efficient communications is necessary for distributedmemory machines, specifically the Intel iPSC860. Rather than decomposing the matrix algebra, we apply the Lanczos algorithm to different but related problems on each processor, simplifying the communications. The set of smaller but related matrix problems are generated using spectral transform of the Hamiltonian operator together with projections onto subHilbert spaces. 
Lanczostype algorithm for excited states of verylargescale quantum systems
We provide a very efficient procedure for obtaining the excited states of a quantum operator {ital H}, in any arbitrary chosen energy range, independently from the knowledge of the states at lower energies. Our procedure consists in determining, within the Lanczos algorithm, the ground state of the auxiliary operator {ital A}=({ital H}{minus}{ital E}{sub {ital t}}){sup 2}, and hence the eigenvalue of {ital H} nearest in energy to {ital E}{sub {ital t}}, where {ital E}{sub {ital t}} is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2{times}2 Lanczos matrices, combinedmore »