Exterior Complex Scaling Method Research Articles

Calculation of DC Stark Resonances for the Ammonia Molecule.

A model potential previously developed for the ammonia molecule is treated in a single-center partial-wave approximation in analogy with a self-consistent field method developed by Moccia. The latter was used in a number of collision studies. The model potential is used to calculate DC Stark resonance parameters, i.e., resonance positions and shifts using the exterior complex scaling method for the radial coordinate. Three molecular valence orbitals are investigated for fields along the three Cartesian coordinates, i.e., along the molecular axis and in two perpendicular directions. The work extends previous work on the planar-geometry water molecule for which non-monotonic shifts were observed. We find such non-monotonic shifts for fields along the molecular axis. For perpendicular fields, we report the splitting of the 1e orbitals into a fast- and a slow-ionizing orbital.

Partial-wave approach to the Stark resonance problem of the water molecule

A partial-wave method is developed to deal with small molecules dominated by a central atom as an extension of earlier single-center methods. In particular, a model potential for the water molecule is expanded over a basis of spherical harmonics. A finite-element method is employed to generate local polynomial functions in subintervals to represent the radial part of the wave function. The angular parts of the wave function are represented by spherical harmonics. The problem of Stark resonances is treated with the exterior complex scaling method which incorporates a wave-function discontinuity at the scaling radius. The resultant non-Hermitian matrix eigenvalue problem yields resonance positions and widths (decay rates). We present these dc Stark shifts and exponential decay rates for the valence orbitals $1{b}_{1}, 3{a}_{1}$, and the bonding orbital $1{b}_{2}$. Furthermore, comparison is made with total molecular decay rates and dc shifts obtained recently within the Hartree-Fock and coupled-cluster approaches.

Resonance studies using the contour deformation method in the complex momentum plane

A contour deformation method (CDM) in the complex momentum plane has been successfully extended and implemented to probe resonances in atomic and molecular systems. Specifically, solution of the Schrödinger equation is performed in momentum space with momentum deformed on a contour in the complex plane. The bound, resonant, and complex continuum states could be directly revealed from the eigenvalues of the Schrödinger equation in the complex momentum plane. The calculations of shape resonances in electron scattering with Na+ in Debye plasmas (one channel), and in the charge transfer process H−(1s2)+Li(1s22s) (12Σ+) →H(1s)+Li−(1s22s2) (22Σ+) (coupled channels) are given as illustrative examples. It is shown that calculated results from CDM agree very well with those extracted from the eigenphase sum of scattering theories. The effectiveness of CDM is also demonstrated by comparing its results with those obtained by the complex rotation scaling and exterior complex scaling methods. The convergence of CDM results can be obtained by increasing the momentum integration region and the number of integration points. The studied examples demonstrate that CDM could be a powerful tool for studies of resonances in complex atomic and molecular systems.

Dissociative recombination by frame transformation to Siegert pseudostates: A comparison with a numerically solvable model

We present a simple two-dimensional model of the indirect dissociative recombination process. The model has one electronic and one nuclear degree of freedom and it can be solved to high precision, without making any physically motivated approximations, by employing the exterior complex scaling method together with the finite-elements method and discrete variable representation. The approach is applied to solve a model for dissociative recombination of H$_2^+$ and the results serve as a benchmark to test validity of several physical approximations commonly used in the computational modeling of dissociative recombination for real molecular targets. The second, approximate, set of calculations employs a combination of multi-channel quantum defect theory and frame transformation into a basis of Siegert pseudostates. The cross sections computed with the two methods are compared in detail for collision energies from 0 to 2 eV.

Low-energy electron elastic scattering and impact ionization with hydrogenlike helium in Debye plasmas

Low-energy electron elastic scattering and impact ionization with hydrogenlike helium in Debye plasmas have been investigated by employing the exterior complex scaling method. The interactions between charged particles in the plasmas have been represented by Debye-H\"uckel potentials. The $1s\ensuremath{-}1s$ elastic collision strengths below the $n=2$ excitation threshold of ${\mathrm{He}}^{+}$ dominated by resonance structures are calculated for different screening lengths. As the screening strength increases, the resonance peaks studied ${[}_{2}{{(1,0)}_{2}}^{+}{}^{1}{S}^{e},{}^{3}{P}^{o},{}^{1}{D}^{e}$, and ${}_{2}{{(0,1)}_{2}}^{+}{}^{1}{P}^{o}]$ exhibit blueshifts and then redshifts with a further increase of the screening strength, which results in dramatic changes of the collision strengths. It is found that these dynamic variation features of the resonances are related to the changes of energy levels of ${\mathrm{He}}^{+}$ in the screened potential and geometric configurations of resonances. Triple-differential-ionization cross sections in coplanar geometries at 6-Ry incident electron energy are also reported, significant changes are observed with varying screening length.

Resonances in Cut-Off Potentials Determined by the Exterior Complex-Scaling Method

The resonances of one-dimensional short-range potentials defined by a cutoff are analyzed in terms of a modification of the well known complex-scaling method. Such a modification consists in transforming the eigenfunctions only in the zone where the potential is identically zero. As usual, the complex-scaled Hamiltonian is non-Hermitian and the functions that represent resonances become square-integrable. The method is illustrated by calculating the resonances of a concrete cut-off potential.

Time-dependent formulation of the two-dimensional model of resonant electron collisions with diatomic molecules and interpretation of the vibrational excitation cross sections

A two-dimensional model of the resonant electron-molecule collision processes with one nuclear and one electronic degree of freedom introduced by K. Houfek, T. N. Rescigno, and C. W. McCurdy [Phys. Rev. A 73, 032721 (2006)] is reformulated within the time-dependent framework and solved numerically using the finite-element method with the discrete variable representation basis, the exterior complex scaling method, and the generalized Crank-Nicolson method. On this model we illustrate how the time-dependent calculations can provide deep insight into the origin of oscillatory structures in the vibrational excitation cross sections if one evaluates the cross sections not only at sufficiently large time to obtain the final cross sections, but also at several characteristic times which are given by the evolution of the system. It is shown that all details of these structures, especially asymmetrical peaks, can be understood as quantum interference of several experimentally indistinguishable processes separated in time due to a resonant capture of the electron and the subsequent vibrational motion of the negative molecular ion. Numerical results are presented for the ${\mathrm{N}}_{2}$-like, NO-like, and ${\mathrm{F}}_{2}$-like models and compared with ones obtained within the time-independent approach and within the local complex potential approximation.

Low energy electron-impact ionization of hydrogen atom for coplanar equal-energy-sharing kinematics in Debye plasmas

Low energy electron-impact ionization of hydrogen atom in Debye plasmas has been investigated by employing the exterior complex scaling method. The interactions between the charged particles in the plasma have been represented by Debye-Huckel potentials. Triple differential cross sections (TDCS) in the coplanar equal-energy-sharing geometry at an incident energy of 15.6 eV for different screening lengths are reported. As the screening strength increases, TDCS change significantly. The evolutions of dominant typical peak structures of the TDCS are studied in detail for different screening lengths and for different coplanar equal-energy-sharing geometries.

Resonances in an S-wave model of electron scattering off hydrogen-like ions

Irregularities in the cross section of electron scattering off hydrogen and hydrogen-like He+, Li++, and Be+++ ions are studied using an s-wave model. The resonance structure and irregularities in the scattering data are compared. A unified approach based on an exterior complex scaling method is used in performing calculations. The potential splitting approach is used for calculations of scattering in systems with asymptotic Coulomb interactions.

New version of hex-ecs, the B-spline implementation of exterior complex scaling method for solution of electron–hydrogen scattering

We provide an updated version of the program hex-ecs originally presented in Comput. Phys. Commun. 185 (2014) 2903–2912. The original version used an iterative method preconditioned by the incomplete LU factorization (ILU), which–though very stable and predictable–requires a large amount of working memory. In the new version we implemented a “separated electrons” (or “Kronecker product approximation”, KPA) preconditioner as suggested by Bar-On et al., Appl. Num. Math. 33 (2000) 95–104. This preconditioner has much lower memory requirements, though in return it requires more iterations to reach converged results. By careful choice between ILU and KPA preconditioners one is able to extend the computational feasibility to larger calculations.Secondly, we added the option to run the KPA preconditioner on an OpenCL device (e.g. GPU). GPUs have generally better memory access times, which speeds up particularly the sparse matrix multiplication. New version program summaryProgram title: hex-ecsCatalogue identifier: AETI_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETI_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: MIT LicenseNo. of lines in distributed program, including test data, etc.: 73693No. of bytes in distributed program, including test data, etc.: 520475Distribution format: tar.gzProgramming language: C++11.Computer: Any recent CPU, preferably 64-bit. Computationally intensive parts can be run on GPU (tested on AMD Tahiti and NVidia TitanX models).Operating system: Tested on Windows 10 and various Linux distributions.RAM: Depends on the problem solved and particular setup; KPA test run uses apx. 300 MiB.Classification: 2.4.Catalogue identifier of previous version: AETI_v2_0Journal reference of previous version: Comput. Phys. Comm. 185 (2014) 2903External routines: GSL [1], UMFPACK [2], BLAS and LAPACK (ideally threaded OpenBLAS [3]).Does the new version supersede the previous version?: YesNature of problem: Solution of the two-particle Schrödinger equation in central field.Solution method: The two-electron states are expanded into angular momentum eigenstates, which gives rise to the coupled bi-radial equations. The bi-radially dependent solution is then represented in a B-spline product basis, which transforms the set of equations into a large matrix equation in this basis. The boundary condition is of Dirichlet type, thanks to the use of the exterior complex scaling method, which extends the coordinates into the complex plane. The matrix equation is then solved by preconditioned conjugated orthogonal conjugate gradient method (PCOCG) [4].Reasons for new version: The original program has been updated to achieve better performance. Also, some external dependencies have been removed (HDF5, FFTW3), which simplifies deployment.Summary of revisions: We implemented a new preconditioner introduced in [5], both for general CPU and also for an arbitrary OpenCL device (e.g. GPU) conforming to the OpenCL 2.0 specification. Furthermore, many other minor improvements have been made, particularly with the intention of reducing the memory requirements. With appropriate switches the program now does not precompute the used matrices and only calculates their elements on the fly. This is aided also by the vectorized B-spline evaluation function, which can now make use of AVX instructions when a single B-spline is being evaluated at several points. The accompanying tools hex-db and hex-dwba [6] have been also updated to use the shared code base.Running time: KPA test run — apx. 2 minutes on Intel i7-4790K (4 threads)

Collisions of electrons with hydrogen atoms II. Low-energy program using the method of the exterior complex scaling

While collisions of electrons with hydrogen atoms pose a well studied and in some sense closed problem, there is still no free computer code ready for “production use”, that would enable applied researchers to generate necessary data for arbitrary impact energies and scattering transitions directly if absent in on-line scattering databases. This is the second article on the Hex program package, which describes a new computer code that is, with a little setup, capable of solving the scattering equations for energies ranging from a fraction of the ionization threshold to approximately 100 eV or more, depending on the available computational resources. The program implements the exterior complex scaling method in the B-spline basis. Program summaryProgram title: hex-ecsCatalogue identifier: AETI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETI_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 44440No. of bytes in distributed program, including test data, etc.: 322643Distribution format: tar.gzProgramming language: C++11.Computer: Any.Operating system: Any system with a C++11 compiler (e.g. GCC 4.8.1; tested on OpenSUSE 13.1 and Windows 8).Has the code been vectorized or parallelized?: Parallelized by OpenMP and MPI.RAM: Depending on input; 4.9 GiB for the test run.Classification: 2.4.External routines: GSL [1], HDF5 [2], UMFPACK [3], FFTW3 [4], optionally with OpenBLAS [5].Nature of problem:Solution of the two-particle Schrödinger equation in central field.Solution method:The two-electron states are expanded into angular momentum eigenstates, which gives rise to the coupled bi-radial equations. The bi-radially dependent solution is then represented in a B-spline basis, which transforms the set of equations into a large matrix equation in this basis. The boundary condition is of Dirichlet type, thanks to the use of the exterior complex scaling method, which extends the coordinates into the complex plane. The matrix equation is then solved by preconditioned conjugate gradients.Running time:Depending on input; 16 min for the test run on Intel i3 3.07 GHz processor (2 threads).

A study of one-dimensional transport of Bose–Einstein condensates using exterior complex scaling

We numerically investigate the one-dimensional transport of Bose–Einstein condensates in the context of guided atom lasers using a mean-field description of the condensate in terms of a spatially discretized Gross–Pitaevskii equation. We specifically consider a waveguide configuration in which spatial inhomogeneities and nonvanishing atom–atom interactions are restricted to a spatially localized scattering region of finite extent. We show how the method of smooth exterior complex scaling can be implemented for this particular configuration in order to efficiently absorb the outgoing flux within the waveguide. A numerical comparison with the introduction of a complex absorbing potential as well as with the analytically exact elimination of the dynamics of the free non-interacting motion outside the scattering region, giving rise to transparent boundary conditions, clearly confirms the accuracy and efficiency of the smooth exterior complex scaling method.

Exterior complex scaling method in TDDFT: HHG of Ar atoms in intense laser fields

The exterior complex scaling (ECS) method is applied in the framework of time-dependent density-functional theory (TDDFT) to study high-order harmonic generation (HHG) of multielectron atoms in intense laser fields. With the help of ECS, correct outgoing-wave boundary conditions can be imposed on the wave functions at large distances. In our implementation, ECS is combined with the time-dependent generalized pseudospectral method for accurate and efficient solution of the time-dependent Kohn-Sham equations. We make use of LB94 exchange-correlation potential which appears quite accurate in calculations of unperturbed electronic structure of Ar. Calculations of HHG are performed for the laser fields with the wavelength of 800 nm and several peak intensities. The HHG spectrum exhibits an intensity-independent minimum corresponding to the photon energy of about 51 eV which is closely related to the Cooper minimum in the photoionization cross section of Ar. We found that HHG spectra calculated with the frozen-core potential (not including dynamic response of the electron density) differ significantly from those obtained by TDDFT.

Exterior complex scaling method in TDDFT: HHG of Ar atoms in intense laser fields

Exterior complex scaling (ECS) method is applied in the framework of the time-dependent density functional theory to study high-order harmonic generation (HHG) of multielectron atoms in intense laser fields. ECS allows to impose correct (outgoing-wave) boundary conditions on the wave functions at large distances. In the present work, ECS is combined with the time-dependent generalized pseudospectral method for accurate and efficient solution of the time-dependent Kohn-Sham equations. Results for HHG of Ar atoms are presented.

On the non-equivalence of perfectly matched layers and exterior complex scaling

The perfectly matched layers (PML) and exterior complex scaling (ECS) methods for absorbing boundary conditions are analyzed using spectral decomposition. Both methods are derived through analytical continuations from unitary to contractive transformations. We find that the methods are mathematically and numerically distinct: ECS is complex stretching that rotates the operator's spectrum into the complex plane, whereas PML is a complex gauge transform which shifts the spectrum. Consequently, the schemes differ in their time-stability. Numerical examples are given.

Exterior complex scaling method in time-dependent density-functional theory: Multiphoton ionization and high-order-harmonic generation of Ar atoms

The exterior complex scaling (ECS) method is applied in the framework of time-dependent density-functional theory (TDDFT) to study multiphoton ionization (MPI) and high-order-harmonic generation (HHG) of multielectron atoms in intense laser fields. ECS allows one to impose correct (outgoing-wave) boundary conditions on the wave functions at large distances. In our implementation, ECS is combined with the time-dependent generalized pseudospectral method for accurate and efficient solution of the time-dependent Kohn-Sham equations. We make use of LB94 exchange-correlation potential which proved accurate in calculations of unperturbed electronic structure of Ar. Calculations of MPI and HHG are performed for the laser pulses with the wavelength of 800 nm and several peak intensities. The HHG spectrum exhibits an intensity-independent minimum corresponding to the photon energy of about 51 eV which is closely related to the Cooper minimum observed in the photoionization cross section of Ar. We found that MPI probabilities and HHG spectra calculated with the frozen-core potential (that is, not including dynamic response of the electron density) differ significantly from those obtained by TDDFT.

Improved convergence of scattering calculations in the oscillator representation

The Schrödinger equation for two and tree-body problems is solved for scattering states in a hybrid representation where solutions are expanded in the eigenstates of the harmonic oscillator in the interaction region and on a finite difference grid in the near- and far-field. The two representations are coupled through a high-order asymptotic formula that takes into account the function values and the third derivative in the classical turning points. For various examples the convergence is analyzed for various physics problems that use an expansion in a large number of oscillator states. The results show significant improvement over the JM-ECS method [Y. Bidasyuk, W. Vanroose, J. Broeckhove, F. Arickx, V. Vasilevsky, Hybrid method (JM-ECS) combining the J-matrix and exterior complex scaling methods for scattering calculations, Phys. Rev. C 82 (6) (2010) 064603].

Time dependent wave packet treatment of 2ΠgN2− and 3Σ−NO− shape resonances using two‐dimensional surfaces for electron‐N2 and NO interactions

AbstractThe 2ΠgN and 3Σ−NO− resonances in electron‐N2 and NO collisions have been treated using both nuclear and electronic degrees of freedom and a two‐dimensional (2D) time dependent wave packet approach to ascertain the importance of nonlocality in electron–nuclear interaction. The results so obtained are compared with vibrational excitation cross‐sections obtained experimentally and those from other theoretical/numerical approaches using 1D local complex potential, 2D model with a combination of the exterior complex scaling method and a finite‐element implementation of the discrete‐variable representation. The results obtained provide detailed insight into the nuclear dynamics induced by electron–molecule collision and reveal that while for resonant excitation of lower vibrational modes, the nonlocal effect may not be as critical but importance of nonlocal effects may increase with increase in quanta of resonant vibrational excitation. © 2012 Wiley Periodicals, Inc.

On the applicability of the exterior complex scaling method for scattering problems including Coulombic potentials

We study some formal aspects of the exterior complex scaling (ECS) approach when implemented for both short and long-range potentials. In particular, we focus on the inconsistencies related to the requirement of an artificial cut-off of the potential in order to avoid exponential divergencies due to the complex rotation. For the pure two-body Coulomb potential we demonstrate analytically and numerically that the ECS inner solution is indeed the correct one, thus reinforcing the method; the extraction of the transition amplitude, however, remains problematic. We also show that a consistent application of the ECS method requires a distorted wave formulation, and two variants are proposed. Finally, we will propose an approach equivalent to the original ECS but that avoids all formal difficulties. It is based on performing the complex rotation on the basis functions rather than on the driven equation itself, and makes use of Sturmian functions with appropriately chosen outgoing boundary conditions. Our proposal differs from one of the original versions of the ECS method, through the use of physically based basis functions rather than pure numeric ones.

A spectral approach based on generalized Sturmian functions for two- and three-body scattering problems

A methodology based on generalized Sturmian functions is put forward to solve two- and three-body scattering problems. It uses a spectral method which allows for the inclusion of the correct asymptotic behavior when solving the associated driven Schrödinger equation. For the two-body case, we demonstrate the equivalence between the exterior complex scaling (ECS) and the Sturmian approaches and illustrate the latter by using Hulthén Sturmian functions. Contrary to the ECS approach, no artificial cut-off of the potential is required in the Surmian approach. For the three-body scattering problem, the theoretical framework is presented in hyperspherical coordinates and a set of hyperspherical generalized Sturmian functions possessing outgoing asymptotic behavior is introduced. The Sturmian procedure is a direct generalization of the method discussed for the two-body problem; thus, the comparison with the ECS method is similar. For both the two- and three-body cases, Sturmian bases are efficient as they possess the correct outgoing behavior, diagonalize part of the potentials involved and are essentially localized in the region where the unsolved interaction is not negligible. Moreover, with the Sturmian basis, the operator (H − E) is represented by a diagonal matrix whose elements are simply the Sturmian eigenvalues.