The variation method and the algebraic eigenvalue problem

The generalized algebraic eigenvalue problem ( A ? λ B )x = 0 arises in the use of the variation method in quantum mechanics. If, within the limitations of the computer word-length, the basis set used to expand the trial wave function is linearly dependent, the matrix B becomes singular. Three different algorithms designed to deal with this difficulty have been investigated, paying special attention to the problem of identifying which members of the basis set are effectively linearly dependent. The advantages and limitations of each method are discussed.     

An improved iterative solution to solve the electrostatic problem in the polarizable continuum model

 We present a discrete iterative interpolation scheme (DIIS) to improve the convergence rate of electrostatic calculations in the polarizable continuum model (PCM) to describe solvent effects on molecular solutes. The electrostatic calculations may easily become the bottleneck of the calculation when the solute size is large. For large molecules iterative procedures turn out to be computationally more convenient than matrix inversion or closure methods. The DIIS scheme is compared here to another iterative procedure (DAMP) and to the biconjugate gradient (BCG) method. The comparisons show that DIIS leads to a sizeable saving of computational time for the C-PCM and IEF-PCM methods (average 40%) compared to DAMP, and more than 50% with respect to the BCG method. Received: 5 October 2000 / Accepted: 13 November 2000 / Published online: 19 January 2001     

An analysis of the siegert eigenvalue problem for autoionizing states

We show that the divergent integrals which appear in a direct matrix solution to the Siegert problem for autoionizing (or electron scattering) state energies and widths can be cancelled exactly. When this is done the Siegert problem becomes essentially a bound state problem. We also show that the resulting non-hermitian secular equation which requires several non-hermitian diagonalizations in the iterative solution for the complex energy can be exactly reduced by a partitioning technique to a single hermitian diagonalization (for a single open channel) with the subsequent iterative solution of a simple algebraic equation.     

A new algebraic approach to the eigenvalue problems of linear differential operators without integrations

In this work, a new approximation scheme based on the evaluation of the pointwise expectation of the Hamiltonian (H) via a conveniently chosen basis set is proposed. This scheme does not necessitate integration; however, physical and mathematical considerations in choosing the basis set are considerably important when very precise and rapidly convergent results are desired. In this method, the best linear combination of “well-selected” basis functions are sought in a way such that H ψ / ψ is flat in the neighborhood of a conveniently chosen point in the domain of H. This yields an algebraic eigenvalue problem. Some concrete applications that have already been realized confirm the efficiency of this approach.     

An algebraic approach to the morse potential scattering

A new realization of an algebra SU(1.1), associated with one-dimensional Morse oscillator is introduced. The calculation of the Morse reflection amplitude is then formulated via a recently established algebraic approach to the scattering matrix. In this approach the invariance group of the free particle, The Euclidean group, is used to describe asymptotic behaviour.     

A discontinuous Galerkin method to solve chromatographic models

This article proposes a discontinuous Galerkin method for solving model equations describing isothermal non-reactive and reactive chromatography. The models contain a system of convection-diffusion-reaction partial differential equations with dominated convective terms. The suggested method has capability to capture sharp discontinuities and narrow peaks of the elution profiles. The accuracy of the method can be improved by introducing additional nodes in the same solution element and, hence, avoids the expansion of mesh stencils normally encountered in the high order finite volume schemes. Thus, the method can be uniformly applied up to boundary cells without loosing accuracy. The method is robust and well suited for large-scale time-dependent simulations of chromatographic processes where accuracy is highly demanding. Several test problems of isothermal non-reactive and reactive chromatographic processes are presented. The results of the current method are validated against flux-limiting finite volume schemes. The numerical results verify the efficiency and accuracy of the investigated method. The proposed scheme gives more resolved solutions than the high resolution finite volume schemes.     

An algebraic operator approach to electronic structure

In this paper, we introduce an algebraic approach to electronic structure calculations. Our approach constructs a Jordan algebra based on the second-quantized electronic Hamiltonian. From the structure factor of this algebra, we show that we can calculate the energy of the ground electronic state of the Hamiltonian operator. We apply our method to several generalized Hubbard models and show that we can usually obtain a significant fraction of the correlation energy for low-to-moderate values of the electronic repulsion parameter while still retaining the O(L(3)) scaling of the Hartree-Fock algorithm. This surprising result, along with several other observations, suggests that our algebraic approach represents a new paradigm for electronic structure calculations which opens up many new directions for research.     

On energy bounds derived from the conjugate eigenvalue problem

An upper bound for E₀, which has been derived from the conjugate eigenvalue problem by Hall, is discussed. It is emphasized that the bound is only guaranteed when V is negative-definite. An alternative bound is presented which is free from this restriction, and the underlying iterative procedure is given. Hall's result is generalized to admit internuclear distances, and the theory is illustrated by a one-dimensional system with delta-function potentials. Some disadvantages of the approach are mentioned.     

Potential energy surface for linear triatomic molecules: An algebraic method

Recently, we proposed a new transformation between the angle of canonical coordinates and the bond angle to describe the bending motion in Potential Energy Surfaces (PES) of bent triatomic molecules. In this work we extend the transformation to include linear triatomic molecules. Results for the linear triatomic molecule N₂O are reported.     

Constrained solutions of the eigenvalue problem in truncated basis sets

It is shown that simple orthogonality constraints between some set of known approximate eigenfunctions and another set of functions which are to be determined as approximate eigensolutions need to be modified. The proposed modification introduces a measure of the approximate character of the known functions and leads to the reduction of the dimensionality of the eigenvalue problem for other solutions. The discussed method is fully variational and leads directly to a Hermitian eigenvalue problem. This approach is also independent of the choice of truncated basis sets for different classes of approximate solutions of the eigenvalue problem. © 1997 John Wiley & Sons, Inc.     

An improved iterative Extended Hückel method for conjugated molecules

The deficiencies of iterative extended Hückel theory as applied to conjugated molecules are discussed. An improved treatment is suggested which sacrifices complete rotational invariance but which is capable of a better representation of the t-electron energy levels and molecular ionization potentials.     

An iterative method for obtaining accurate atomic inner shell binding energies

A new method based on an iterative procedure for obtaining accurate atomic inner shell binding energies is described. The method makes use of the doubly modi-fied Moseley plot as the starting point. As an illustration accurate binding energies for the M ₁ shell in the atomic number range 57 ≤ Z ≤ 71 have been obtained. This has also resulted in the removal of the anomaly in the M ₁ binding energy of Z = 68. Our calculations point to a need for a fresh look at the theoretical calculations of inner shell binding energies for Z = 57 to 71 in which some unsuspected effects appear to occur when the 4f shell is opened, half filled and closed.     

An application of the dynamical Lie algebraic method to energy transfer of the collinear scattering system AB+CD

The dynamical Lie algebraic method has been applied to treat the V–V and T–V energy transfers in the collinear scattering system AB+CD. The expression for the vibrational transition probability, which contains the main dynamical parameters, is given analytically. By using this expression we probe into the V–V resonance and T–V resonance phenomena appearing in the process of energy transfer. We find that the transition probability of V–V resonance is in good agreement with that obtained using the resonant exchange hypothesis. Then the reliability of the resonant exchange hypothesis is confirmed. This revised version was published online in July 2006 with corrections to the Cover Date.     

An iterative process to calculate the SCF open shell orbitals

A method is described for calculating SCF open shell orbitals. In comparison with the coupling operator method, a greater velocity of convergency of the iterative process is obtained by taking into account not only the correct variational conditions, but also the best variations of orbitals step by step.     

An iterative approach to the synthesis of thiophene-based organic dyes

We developed an iterative synthetic method for oligo-aryl compounds using an organosilicon-based palladium-catalyzed cross-coupling reaction. Aryl compounds containing a benzyloxy(diisopropyl)silyl group (masked Si group) had sufficient chemical stability, and the unmasking step proceeded in a high yield under mild conditions. Both of the key unmasking/coupling steps required no strict anhydrous or degassed conditions. The developed procedure was used for the synthesis of thiophene-based organic dyes for dye-sensitized solar cells.     

Excited-state energy eigenvalue and wave-function evaluation of the Gaussian asymmetric double-well potential problem via numerical shooting method 2

This project aims at computation excited-state energy eigenvalues and wave-function of a particle under Gaussian asymmetric double-well potential using numerical shooting method and perturbation theory a method to deal with discrete-eigenvalue problems. We also compare the energy eigenvalue and wave-function with those obtained from other typical means popular among physics students, namely the numerical shooting method and perturbation theory. Show that the idea of program of the numerical shooting method and perturbation theory of this problem (see Sects. 2.1 and 4) The numerical shooting method is generally regarded as one of the most efficient methods that give very accurate results because it integrates the Schr?dinger equation directly, though in the numerical sense. The n = even case is shown in Figs. 4, 5 and 6. In this case, the wave-function has split up on asymmetric nodes under Gaussian asymmetric double-well potential. The n = odd case is shown in Fig. 7. In this case, the wave-function has not split up on asymmetric nodes under Gaussian asymmetric double-well potential.     

Excited-state energy eigenvalue and wave-function evaluation of the Gaussian symmetric double-well potential problem via numerical shooting method 1

This work aims at computing excited-state energy eigenvalues and wave-function of a particle under Gaussian symmetric double-wells potential using numerical shooting method and perturbation theory a method to deal with discrete-eigenvalue problems. We also compare the energy eigenvalue and wave-function with those obtained from other typical means popular among physics students, namely the numerical shooting method and perturbation theory. Show that the idea of program of the numerical shooting method and perturbation theory of this problem (see Sects. 2.2 and 3). The numerical shooting method is generally regarded as one of the most efficient methods that give very accurate results because it integrates the Schrödinger equation directly, though in the numerical sense. The n = even case is shown in Fig. 5. In this case, the wave-function has split up on symmetric nodes under Gaussian symmetric double-well potential.