Howard S. Taylor

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...

Theoretical Interpretation of the Optical and Electron Scattering Spectra of H2O

Energies and potential surface characteristics are assigned to the first eight excited states of the water molecule. This assignment is shown to be consistent with all data from optical spectra, electron scattering, rotational distributions of the OH fragment in photodissociation and associated data, and with semi‐empirical INDO calculations. Energies and potential surfaces are given for the lowest resonant states of H2O−. These are consistent within the explainable error of the INDO calculations, as well as with the data on dissociative attachment and associative detachment in which H2O− is an intemediate for species. Assignment‐confirming experiments are suggested.

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.

Green's Function Technique in Atomic and Molecular Physics

Publisher Summary This chapter seeks to familiarize the quantum chemist and molecular physicist with some of the ways one can apply the Greens function technique to the problems of calculating excitation energies, ionization energies, ground state energies, and transition matrix elements. Density matrices and natural orbitals can be calculated directly without prior calculation of the wavefunctions and therefore the Greens function technique is closely related to the density matrix methods already developed in quantum chemistry. The Greens function method is clearly the way to calculate the density matrix and the natural orbitals directly. For finite atomic and molecular systems, the branch cuts are replaced by a set of poles and adjoining branch cuts. Poles can exist on the physical sheet, as well as on the nonphysical sheets reached by continuing across the cut.

A quantum analog to the classical quasiperiodic motion

A quantum mechanical mnemonic which identifies the energy levels corresponding to those obtained by quantization of the classical quasiperiodic motion in coupled anaharmonic oscillator systems is presented. The mnemonic reproduces the known classical regular to irregular behavior as a function of the energy. The mnemonic is based on the definition of a suitable effective Hamiltonian for each level. Quasiperiodic levels are those in which the wave function has a weight greater than 50% on a degenerate subspace of any separable part of the Hamiltonian. This condition ensures that there exists an effective Hamiltonian defined on this degenerate subspace which commutes with the separable Hamiltonian; the latter is therefore an effective constant of motion for the total Hamiltonian. In some cases, depending on the degenerate subspace, other effective constants of motion exist. There is a correlation between the semiclassical quantization methods of quasiperiodic trajectories and the present effective Hamiltoni...

Qualitative Aspects of Resonances in Electron—Atom and Electron—Molecule Scattering, Excitation, and Reactions

A qualitative description of various types of resonant states of atoms and molecules is given. The classification is seen to be useful in explaining and discussing experimental phenomena. It is found that, as in nuclear physics, atoms and molecules exhibit both single‐particle and core‐excited resonant states. A general quasistationary method is presented which can be used to calculate the energies of the resonant states from intuitively selected trial functions. The Feshbach technique is used for formal justification of this method, and the relation of the quasistationary method to other published resonance calculations is discussed. Experiments are suggested to test the ideas presented.