Vladimir A. Mandelshtam

Reconstruction of Abel-transformable images: The Gaussian basis-set expansion Abel transform method

In this article we present a new method for reconstructing three-dimensional (3D) images with cylindrical symmetry from their two-dimensional projections. The method is based on expanding the projection in a basis set of functions that are analytical projections of known well-behaved functions. The original 3D image can then be reconstructed as a linear combination of these well-behaved functions, which have a Gaussian-like shape, with the same expansion coefficients as the projection. In the process of finding the expansion coefficients, regularization is used to achieve a more reliable reconstruction of noisy projections. The method is efficient and computationally cheap and is particularly well suited for transforming projections obtained in photoion and photoelectron imaging experiments. It can be used for any image with cylindrical symmetry, requires minimal user’s input, and provides a reliable reconstruction in certain cases when the commonly used Fourier–Hankel Abel transform method fails.

Harmonic inversion of time signals and its applications

New methods of high resolution spectral analysis of short time signals are presented. These methods utilize the filter-diagonalization approach of Wall and Neuhauser [J. Chem. Phys. 102, 8011 (1995)] that extracts the complex frequencies ωk and amplitudes dk from a signal C(t)=∑kdke−itωk in a small frequency interval by recasting the harmonic inversion problem as the one of a small matrix diagonalization. The present methods are rigorously adapted to the conventional case of the signal available on a sparse equidistant time grid and use a more efficient boxlike filter. Various applications are discussed, such as iterative diagonalization of large Hamiltonian matrices for calculating bound and resonance states, scattering calculations in the presence of narrow resonances, etc. For the scattering problem the harmonic inversion is directly applied to the signal cn=(χf,Tn(Ĥ)χi), generated by the dynamical system governed by a modified Chebyshev recursion, avoiding the usual recasting the problem to the time d...

A simple recursion polynomial expansion of the Green’s function with absorbing boundary conditions. Application to the reactive scattering

The new recently introduced [J. Chem. Phys 102, 7390 (1995)] empirical recursion formula for the scattering solution is here proved to yield an exact polynomial expansion of the operator [E−(H+Γ)]−1, Γ being a simple complex optical potential. The expansion is energy separable and converges uniformly in the real energy domain. The scaling of the Hamiltonian is trivial and does not involve complex analysis. Formal use of the energy‐to‐time Fourier transform of the ABC (absorbing boundary conditions) Green’s function leads to a recursion polynomial expansion of the ABC time evolution operator that is global in time. Results at any energy and any time can be accumulated simultaneously from a single iterative procedure; no actual Fourier transform is needed since the expansion coefficients are known analytically. The approach can be also used to obtain a perturbation series for the Green’s function. The new iterative methods should be of a great use in the area of the reactive scattering calculations and o...

Spectral projection approach to the quantum scattering calculations

A new method of implementing scattering calculations is presented. For the S‐matrix computation it produces a complete set of solutions of the wave equation that need be valid only inside the interaction region. For problems with small sizes the method is one of several that are practical in the sense that it involves merely a real symmetric Hamiltonian represented in a minimal L2 basis set. For more challenging larger systems it lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. A modified Chebyshev polynomial expansion of (E−H)−1 is used. This acts on a set of energy independent wave packets located on the edge of the interaction region. The procedure requires minimal storage and is shown to converge rapidly in a manner that is uniform in energy.

A LOW-STORAGE FILTER DIAGONALIZATION METHOD FOR QUANTUM EIGENENERGY CALCULATION OR FOR SPECTRAL ANALYSIS OF TIME SIGNALS

A new version of the filter diagonalization method of diagonalizing large real symmetric Hamiltonian matrices is presented. Our previous version would first produce a small set of adapted basis functions by applying the Chebyshev polynomial expansion of the Green’s function on a generic initial vector χ. The small Hamiltonian, H, and overlap, S, matrices would then be evaluated in this adapted basis and the corresponding generalized eigenvalue problem would be solved yielding the desired spectral information. Here in analogy to a recent work by Wall and Neuhauser [J. Chem. Phys. 102, 8011 (1995)] H and S are computed directly using only the Chebyshev coefficients cn=〈χ|Tn(Ĥ)|χ〉, calculation of which requires a minimal storage if the Ĥ matrix is sparse. The expressions for H and S are analytically simple, computationally very inexpensive and stable. The method can be used to obtain all the eigenvalues of Ĥ using the same sequence {cn}. We present an application of the method to a realistic quantum dynamics...

Bound states and resonances of the hydroperoxyl radical HO2: An accurate quantum mechanical calculation using filter diagonalization

An accurate calculation of bound and resonance spectra of the non‐rotating odd O2 exchange symmetry HO2 radical is presented. The calculation has been carried out by a recently developed iterative technique which uses filter diagonalization of a sparse matrix of the system Hamiltonian with absorbing boundary conditions. We were able to obtain 361 bound states and some 232 isolatable resonances (Γ<0.01 eV) in a wide energy range corresponding to the HO2→H+O2 unimolecular decomposition reaction. It is shown that all resonances found have the same nature as the bound states in that they all are localized in the same region of space over the deep potential well, and moreover the extrapolated smoothed density of the bound states merges easily with the smoothed density of the resonance states. The level statistics for both bound and resonance states indicates a highly chaotic regime consistent with the random matrix theory. Strong mode mixing makes assignments of most bound and resonance states impossible becau...

Recursion polynomial expansion of the Green’s function with absorbing boundary conditions: Calculations of resonances of HCO by filter diagonalization

An iterative method for calculating resonance positions and widths is developed. The system Hamiltonian with an asymptotic complex absorbing potential is represented by a large and sparse matrix. A small set of ‘‘good’’ basis functions suitable for diagonalizing the Hamiltonian matrix in a given energy window is generated by acting with a polynomial expansion of the imaginary part of the system Green’s function onto a generic initial wave packet. As an application to a realistic three‐dimensional system, the calculation of 65 resonances of the nonrotating HCO molecule up to the energy 9000 cm−1 is presented. The method is shown to be rapidly convergent and accurate, especially for narrow resonances.

Quantum statistical mechanics with Gaussians: Equilibrium properties of van der Waals clusters

The variational Gaussian wave-packet method for computation of equilibrium density matrices of quantum many-body systems is further developed. The density matrix is expressed in terms of Gaussian resolution, in which each Gaussian is propagated independently in imaginary time beta=(k(B)T)(-1) starting at the classical limit beta=0. For an N-particle system a Gaussian exp[(r-q)(T)G(r-q)+gamma] is represented by its center qinR(3N), the width matrix GinR(3Nx3N), and the scale gammainR, all treated as dynamical variables. Evaluation of observables is done by Monte Carlo sampling of the initial Gaussian positions. As demonstrated previously at not-very-low temperatures the method is surprisingly accurate for a range of model systems including the case of double-well potential. Ideally, a single Gaussian propagation requires numerical effort comparable to the propagation of a single classical trajectory for a system with 9(N(2)+N)/2 degrees of freedom. Furthermore, an approximation based on a direct product of single-particle Gaussians, rather than a fully coupled Gaussian, reduces the number of dynamical variables to 9N. The success of the methodology depends on whether various Gaussian integrals needed for calculation of, e.g., the potential matrix elements or pair correlation functions could be evaluated efficiently. We present techniques to accomplish these goals and apply the method to compute the heat capacity and radial pair correlation function of Ne(13) Lennard-Jones cluster. Our results agree very well with the available path-integral Monte Carlo calculations.

Accurate variational calculations and analysis of the HOCl vibrational energy spectrum

Large scale variational calculations for the vibrational states of HOCl are performed using a recently developed, accurate ab initio potential energy surface. Three different approaches for obtaining vibrational states are employed and contrasted; a truncation/recoupling scheme with direct diagonalization, the Lanczos method, and Chebyshev iteration with filter diagonalization. The complete spectrum of bound states for nonrotating HOCl is computed and analyzed within a random matrix theory framework. This analysis indicates almost entirely regular dynamics with only a small degree of chaos. The nearly regular spectral structure allows us to make assignments for the most significant part of the spectrum, based on analysis of coordinate expectation values and eigenfunctions. Ground state dipole moments and dipole transition probabilities are also calculated using accurate ab initio data. Computed values are in good agreement with available experimental data. Some exact rovibrational calculations for J=1, in...

Harmonic inversion of time cross-correlation functions: The optimal way to perform quantum or semiclassical dynamics calculations

We explore two new applications of the filter-diagonalization method (FDM) for harmonic inversion of time cross-correlation functions arising in various contexts in molecular dynamics calculations. We show that the Chebyshev cross-correlation functions ciα(n)=(Φα|Tn(Ĥ)Φi) obtained by propagation of a single initial wave packet Φi correlated with a set of final states Φα, can be harmonically inverted to yield a complete description of the system dynamics in terms of the spectral parameters. In particular, all S-matrix elements can be obtained in such a way. Compared to the conventional way of spectral analysis, when only a column of the S-matrix is extracted from a single wave packet propagation, this approach leads to a significant numerical saving especially for resonance dominated multichannel scattering. The second application of FDM is based on the harmonic inversion of semiclassically computed time cross-correlation matrices. The main assumption is that for a not-too-long time semiclassical propagato...