Note on the complex stabilization method (original) (raw)
Related papers
Computation of resonances by two methods involving the use of complex coordinates
Physical Review A, 1993
We have studied two different systems producing resonances, a highly excited multielectron Coulombic negative ion (the He 2s2p P state) and a hydrogen atom in a magnetic field, via the complexcoordinate rotation (CCR) and the state-specific complex-eigenvalue Schrodinger equation (CESE) approaches. For the He 2s2p P resonance, a series of large CCR calculations, up to 353 basis functions with explicit r;, dependence, were carried out to serve as benchmarks. For the magnetic-field problem, the CCR results were taken from the literature. Comparison shows that the state-specific CESE theory allows the physics of the problem to be incorporated systematically while keeping the overall size of the computation tractable regardless of the number of electrons.
The Journal of Chemical Physics, 1994
The discrete variable representation (DVR) formulation of the complex coordinate method as has been used for calculating several resonances of NeICl [J. Chem. Phys. 98, 1888], and a modified version of the recent developed stabilization method [Phys. Rev. Lett. 70, 1932] are used for calculating all 30 isolated narrow resonances of NeICl (B, v=2). The two Z* methods require a similar computational effort. The modified stabilization method requires the calculations of eigenvalues of real and symmetric Hamiltonian matrices in a sequence of ever larger enclosing boxes. The complex DVR method requires the use of complex arithmetic and calculations of eigenvalues of complex symmetrical matrices.
Stabilization methods for quantum mechanical resonance states of four-body systems
Computer Physics Communications, 2000
We describe the methods used in a computer code for quantum mechanical calculations of bound and resonance states of four-body systems. Three stabilization techniques for identifying the resonances and estimating resonance widths are discussed: calculation of coordinate-space moments, application of an imaginary optical potential, and analytic continuation of a real discrete spectrum using rational polynomial interpolation. Computational implementation of these methods, recent improvements to the computer code, and parallelization strategies are described in detail.
Extensions of the complex-coordinate method to the study of resonances in many-electron systems
Physical Review A, 1978
DiAiculties in the straightfoward application of the complex-coordinate method to the calculation of resonance states in many-electron systems are examined. For the case of shape resonances, it is shown that many of these difficulties can be avoided by using complex coordinates only after reduction of the system to an effective one-electron problem. Further simplifications are achieved by the use of an inner-projection technique to facilitate the computation of the complex Hamiltonian matrix elements. The method is first illustrated by application to a model-potential problem. Its suitability for studying many-electron problems is demonstrated by calculation of the position and width of a low-energy P' shape resonance in Be. We discuss the modifications necessary to study core-excited (Feshbach) resonances.
Determination of resonances by the optimized spectral approach
Journal of Physics A: Mathematical and Theoretical, 2013
The Rayleigh-Ritz procedure for determining bound-states of the Schrödinger equation relies on spectral representation of the solution as a linear combination of the basis functions. Several possible extensions of the method to resonance states have been considered in the literature. Here we propose the application of the optimized Rayleigh-Ritz method to this end. The method uses a basis of the functions containing adjustable nonlinear parameters, the values of which are fixed so as to make the trace of the variational matrix stationary. Generalization to resonances proceeds by allowing the parameters to be complex numbers. Using various basis sets, we demonstrate that the optimized Rayleigh-Ritz scheme with complex parameters provides an effective algorithm for the determination of both the energy and lifetime of the resonant states for various one-dimensional and spherically symmetric potentials. The method is computationally inexpensive since it does not require iterations or predetermined initial values. The convergence rate compares favorably to other approaches.
Efficient direct calculation of complex resonance (Siegert) energies
Physical Review A, 1995
An improved version of the Siegert method to calculate resonance state parameters directly is presented. A prediagonalization of the Hamiltonian in a real L basis and an equation partitioning technique are utilized along with an inverse iteration method (combined with a rational fraction root search) to find accurate positions and widths of narrow resonances. Model calculations suggest that the present method is efFicient and accurate especially for narrow resonances regardless of the potential range.
Journal of Physics B: Atomic, Molecular and Optical Physics, 2011
The resonance states of one-particle Hamiltonians are studied using variational expansions with real basis-set functions. The resonance energies, Er, and widths, Γ, are calculated using the density of states and an L 2 golden rule-like formula, respectively. We present a recipe to select adequately some solutions of the variational problem. The set of approximate energies obtained show a very regular behaviour with the basis-set size, N . Indeed, these particular variational eigenvalues show a quite simple scaling behaviour and convergence when N → ∞. Following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths. Using the scaling function that characterizes the behaviour of the approximate energies as a guide, it is possible to find a very good approximation to the actual value of the resonance width.
Calculation of resonance energies and widths using the complex absorbing potential method
Journal of Physics B: Atomic, Molecular and Optical Physics, 1993
The spectral properties of Hamilton operators perturbed by a complex absorbing potential (CAP) are studied. It is shown that for a wide class of CAPS proper eigenvalues of Ihe perturbed Hamilton opemar mnverge to Siegetl resonance eigenvalues of the unperturbed Hamiltonian with decreasing UP strength. The errors in the calculation of complex resonance energies caused by Ihe additional CAP and by finite bask set representation are examined. In order to minimize these errors a scheme of approximalions is provided. The application of this method allows for the use of real L1 basis seis. The feasibility and accuracy of the proposed method is demonstrated by calculations of resonance energies of a model potential and of the 'ifg shape resonance of N ;
Journal of Physical Chemistry B, 2008
We develop, implement, and apply a quadratically convergent complex multiconfigurational self-consistent field method (CMCSCF) that uses the complex scaling theorem of Aguilar, Balslev, and Combes within the framework of the multiconfigurational self-consistent field method (MCSCF) in order to theoretically investigate the resonances originated due to scattering of a low-energy electron off of a neutral or an ionic target (atomic or molecular). The need to scale the electronic coordinates of the Hamiltonian as prescribed in the complex scaling theorem requires the use of a modified second quantization algebra suitable for biorthonormal spin orbital bases. In order to control the convergence to a stationary point in the complex energy hypersurface, a modified step-length control algorithm is incorporated. The position and width of 2 P Beshape resonances are calculated by inspecting the continuum states of Be -. To our knowledge, this is the first time that CMCSCF has been directly used to determine electron-atom/molecule scattering resonances. We demonstrate that both relaxation and nondynamical correlation are important for accurately describing shape resonances. For all of the calculations, the quadratically convergent CMCSCF was found to converge to the correct stationary point with a tolerance of 1.0 × 10 -10 au for the energy gradient within 10 iterations or less.
Quasilinearization approach to the resonance calculations: the quartic oscillator
Physica Scripta, 2008
We pioneered the application of the quasilinearization method (QLM) to resonance calculations. The quartic anharmonic oscillator (kx 2 /2) + λx 4 with a negative coupling constant λ was chosen as the simplest example of the resonant potential. The QLM has been suggested recently for solving the bound state Schrödinger equation after conversion into Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. Comparison of our approximate analytic expressions for the resonance energies and wavefunctions obtained in the first QLM iteration with the exact numerical solutions demonstrate their high accuracy in the wide range of the negative coupling constant. The results enable accurate analytic estimates of the effects of the coupling constant variation on the positions and widths of the resonances.