Bill Poirier

A general framework for discrete variable representation basis sets

A framework for discrete variable representation (DVR) basis sets is developed that is suitable for multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators in a central role and integrates semiclassical and phase space concepts into the basic framework. Rates of convergence of basis set expansions are emphasized, and it is shown that the DVR method gives exponential convergence, assuming conditions of analyticity and boundary conditions are met. A discussion of nonorthogonal generalizations of DVR functions is presented, in which it is shown that projected δ-functions and interpolating functions form a biorthogonal basis. It is also shown that one of the generalized DVR proposals due to Szalay [J. Chem. Phys. 105, 6940 (1996)] gives exponential convergence.

Accelerating the calculation of energy levels and wave functions using an efficient preconditioner with the inexact spectral transform method

In an earlier paper [J. Chem. Phys. 112, 8765 (2000)] our group introduced a preconditioned inexact spectral transform method for calculating energy levels and wave functions. Although we could calculate high-lying levels with far fewer matrix–vector products than with the filter diagonalization method of Mandelshtam and Taylor, even better performance can be achieved with a better preconditioner. In this paper, we develop an extremely efficient preconditioner consisting of two components: (1) transformation to an optimal separable basis, in which off-diagonal elements of the Hamiltonian matrix are minimized; and (2) removal of all off-diagonal coupling near the energies of interest. The new preconditioner works extremely well; it enables us to calculate high-lying vibrational states of H2O with orders of magnitude fewer matrix–vector products than for all other known methods. The new preconditioner should also accelerate the calculation of other quantities, such as photodissociation cross sections and ra...

A preconditioned inexact spectral transform method for calculating resonance energies and widths, as applied to HCO

We present a complex-symmetric version of the preconditioned inexact spectral transform (PIST) method, for calculating resonance energies and widths. The PIST method uses an iterative linear solver to compute inexact Lanczos vectors for (EI−H)−1, and then diagonalizes the Hamiltonian in the inexact Lanczos representation. Our new version requires complex-symmetric variants of: (1) the Lanczos algorithm, (2) the linear solver, (3) the preconditioner we introduced in a previous paper [J. Chem. Phys. 114, 9254 (2001)]. The new method works extremely well for HCO, enabling us to calculate the first 17 dissociative resonances in less then 90 second of CPU time.

Reconciling semiclassical and Bohmian mechanics. I. Stationary states

The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill behaved at caustics. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by divergent forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eigenstates in one degree of freedom are the primary focus, but more general applications are also anticipated.

USING WAVELETS TO EXTEND QUANTUM DYNAMICS CALCULATIONS TO TEN OR MORE DEGREES OF FREEDOM

A customizable, orthonormal basis for solving the multidimensional Schrodinger equation is constructed using modified Wilson–Daubechies wavelets, and a simple phase space truncation scheme. Unprecedented numerical efficiency is achieved, enabling a ten-dimensional direct calculation of nearly 600 eigenvalues to be performed. Higher dimensionalities are possible using more sophisticated linear algebra techniques. The new approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where elliptic partial differential equations are involved.

Semiclassically optimized complex absorbing potentials of polynomial form. I. Pure imaginary case

We present an optimal, pure imaginary complex absorbing potential (CAP) of polynomial form, for calculating resonance energies and widths, photodissociation cross sections, rate constants, etc. The optimal CAP is derived by minimizing reflection from, and transmission through, the CAP. Reflection and transmission are computed semiclassically. The optimal CAP is general, and can be used at any energy and with any absorbing region width. It significantly reduces the absorbing region width required to compute resonances of the one-dimensional Hazi–Taylor system. We also clearly discuss different types of reflection, and resolve apparent discrepancies relating to optimal CAPs.

ACCURATE AND HIGHLY EFFICIENT CALCULATION OF THE O(1D)HCl VIBRATIONAL BOUND STATES, USING A COMBINATION OF METHODS

Hypochlorous acid, HOCl, is an important intermediate in the O(1D)HCl reactive system. Due in part to a large number of vibrational bound states (over 800), extremely large direct product basis sets (around 300,000) are required to compute the energy levels just below the dissociation threshold. This situation, combined with a very high density of states, results in difficult convergence for iterative methods — e.g. Lanczos requires 50,000 iterations, and filter diagonalization uses 60,000 iterations. In contrast, using new methodologies, we are able to compute the highest-lying bound states with only 271 iterations, although the CPU cost per iteration is substantially greater. Lower lying states are also computed, for a fraction of the CPU cost of the highest energy calculation.

Multidimensional quantum trajectories: Applications of the derivative propagation method

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S=C(r,t) + iS(r,t)/Plancks over 2pi, where Psi(r,t)=exp(S). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrodinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.

Communication: Quantum mechanics without wavefunctions

We present a self-contained formulation of spin-free non-relativistic quantum mechanics that makes no use of wavefunctions or complex amplitudes of any kind. Quantum states are represented as ensembles of real-valued quantum trajectories, obtained by extremizing an action and satisfying energy conservation. The theory applies for arbitrary configuration spaces and system dimensionalities. Various beneficial ramifications-theoretical, computational, and interpretational-are discussed.

Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. II. Construction and optimization

In this paper, we extend and elaborate upon a wavelet method first presented in a previous publication [B. Poirier, J. Theo. Comput. Chem. 2, 65 (2003)]. In particular, we focus on construction and optimization of the wavelet functions, from theoretical and numerical viewpoints, and also examine their localization properties. The wavelets used are modified Wilson-Daubechies wavelets, which in conjunction with a simple phase space truncation scheme, enable one to solve the multidimensional Schrodinger equation. This approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where differential equations are involved.