Youhong Huang

General, energy‐separable Faber polynomial representation of operator functions: Theory and application in quantum scattering

A general, uniformly convergent series representation of operator‐valued functions in terms of Faber polynomials is presented. The method can be used to evaluate the action of any operator‐valued function which is analytic in a simply connected region enclosed by a curve, Lγ. The three most important examples include the time‐independent Green’s operator, G+(E)=1/[E−(H−ie)], where H may be Hermitian or may also contain a negative imaginary absorbing potential, the time‐dependent Green’s or evolution operator, exp(−iHt/ℏ), and the generalized collision operator from nonequilibrium statistical mechanics, 1/[E−(L−ie)], where L is the Liouvillian operator for the Hamiltonian. The particular uniformly convergent Faber polynomial expansion employed is determined by the conformal mapping between the simply connected region external to the curve Lγ, which encloses the spectrum of H−ie (or L−ie), and the region external to a disk of radius γ. A locally smoothed conformal mapping is introduced containing a finite n...

A general time-to-energy transform of wavepackets. Time-independent wavepacket-Schrödinger and wavepacket-Lippmann—Schwinger equations

Abstract Recently, new time-independent wavepacket-Schrodinger and wavepacket-Lippmann—Schwinger equations have been derived making use of absorbing potentials. We show that these equations, which are characterized by the occurrence of an initial ¢L2-wavepacket source of scattered waves, can be gotten without introducing the absorbing potential. We also show that a powerful method of solving the equations can be based on a Chebychev representation of the causal full Green function, combined with the distributed approximating function representation of the system Hamiltonian, and we present two example applications illustrating approach.

Orthogonal polynomial expansion of the spectral density operator and the calculation of bound state energies and eigenfunctions

Abstract An orthogonal polynomial expansion method is presented, and illustrated with calculations, for calculating δ( E – H ), the spectral density operator (SDO), the projection operator that projects out of any L 2 wavepacket the eigenstate (s) of H having energy E . If applied to an L 2 wavepacket which overlaps the interaction, it yields either scattering-type (improper) eigenstates or proper bound eigenstates. For negative energies, the exact SDO yields zero away from an eigenvalue, and yields the energy eigenstate (times a constant) when E equals an eigenvalue. The finite orthogonal polynomial expansion of the SDO, acting on an L 2 wavepacket, yields approximately zero for E not equal to an eigenvalue, and becomes nonzero in the neighborhood of an eigenvalue.

Variational principles for the time‐independent wave‐packet‐Schrödinger and wave‐packet‐Lippmann–Schwinger equations

Several variational principles, whose Euler equations are the recently derived time‐independent wave‐packet‐Schrodinger or wave‐packet‐Lippmann–Schwinger equations, are presented. A particularly attractive wave‐packet‐Kohn variational principle for either the T‐ or S‐matrix is given which yields inhomogeneous algebraic equations whose ‘‘universal inhomogeneity’’ does not depend explicitly on the collision energy. The validity of the approach is demonstrated with calculations for two simple one dimensional scattering problems and for the collinear H+H2 reactive scattering problem.

Further analysis of solutions to the time‐independent wave packet equations of quantum dynamics. II. Scattering as a continuous function of energy using finite, discrete approximate Hamiltonians

We consider further how scattering information (the S‐matrix) can be obtained, as a continuous function of energy, by studying wave packet dynamics on a finite grid of restricted size. Solutions are expanded using recursively generated basis functions for calculating Green’s functions and the spectral density operator. These basis functions allow one to construct a general solution to both the standard homogeneous Schrodinger’s equation and the time‐independent wave packet, inhomogeneous Schrodinger equation, in the non‐interacting region (away from the boundaries and the interaction region) from which the scattering solution obeying the desired boundary conditions can be constructed. In addition, we derive new expressions for a ‘‘remainder or error term,’’ which can hopefully be used to optimize the choice of grid points at which the scattering information is evaluated. Problems with reflections at finite boundaries are dealt with using a Hamiltonian which is damped in the boundary region as was done by ...

A general, energy-separable polynomial representation of the time-independent full Green operator with application to time-independent wavepacket forms of Schrödinger and Lippmann—Schwinger equations

Abstract A general, energy-separable Faber polynomial representation of the full time-independent Green operator is presented. Non-Hermitian Hamiltonians are included, allowing treatment of negative imaginary absorbing potentials. A connection between the Faber polynomial expansion and our earlier Chebychev polynomial expansion (Chem. Phys. Letters 206 (1993) 96) is established, thereby generalizing the Chebychev expansion to the complex energy plane. The method is applied to collinear H + H2 reactive scattering.

Analytic continuation of the polynomial representation of the full, interacting time-independent Green function

Abstract We present an analytic continuation of a polynomial representation of the full, interacting time-independent Green function, thereby enabling the use of negative, imaginary absorbing potentials to shorten the grid necessary to treat scattering problems. The approach retains the clean separation of the energy and Hamiltonian dependences characteristic of our earlier orthogonal polynomial representation of the operator ( E - H +i0 + ) −1 . This treatment, combined with our time-independent wavepacket Lippmann-Schwinger equation method, leads to a computational approach in which all of the energy dependence resides in known analytical expansion coefficients. The Hamiltonian operator appears as the argument of other orthogonal polynomials. These act solely on an initial wavepacket which provides a “universal source” of scattered waves, independent of the particular energies of interest. This energy independence, combined with highly truncated grids, results in an extremely efficient procedure for scattering calculations.

Matrix pseudo-spectroscopy : iterative calculation of matrix eigenvalues and eigenvectors of large matrices using a polynomial expansion of the Dirac delta function

Abstract The method of diagonalizing Hermitian matrices based on a polynomial expansion of the Dirac delta function δ(E − H) is further refined so as to accelerate the convergence. Improved choices of the bases used for subspace diagonalization of the matrix, along with accuracy controls and estimates, are introduced. It is shown that the improved method can accurately deliver eigenvalues and eigenvectors in any region of the spectrum, including cases where the spacings are very small for “interior” eigenvalues. In addition, accurate values can be obtained for as many states as are desired. The method is illustrated for a model problem introduced recently in a study of another type approach.

Distributed approximating function approach to time-dependent wavepacket propagation in 3-dimensions: atom-surface scattering

Abstract The theoretical formalism of the distributed approximating functions (DAF) is applied to solve accurately 3D-atom-surface scattering problems. Formulated in coordinate space, the DAF approach starts from an entirely new idea: providing a “uniform” approximation everywhere to a wavepacket, and results naturally in a near-local or banded free propagator. The banded Toeplitz structure of the DAF free propagator matrix on a uniform grid makes possible the application of the most efficient codes in the matrix-vector multiplication in evolving the wavefunction of a quantum system in time, and with extremely small memory requirements. The numerical study conducted in this paper demonstrates that the DAF method outperforms the most powerful available FFT method both in CPU time and storage requirements. The DAF approach gives the same accurate results as the FFT does, and, in some cases, yields more accurate results.

Further analysis of solutions to the time‐independent wave packet equations for quantum dynamics: General initial wave packets

In this paper we reexamine, and analyze solutions to, the recently derived time‐independent wave packet‐Schrodinger (TIWS) and time‐independent wave packet‐Lippmann–Schwinger (TIWLS) equations. These equations are so named because they are inhomogeneous, with the inhomogeneity being the initial L2 wave packet from an underlying time‐dependent treatment of the dynamics. We explicitly show that a particular solution of the homogeneous Schrodinger equation can be constructed out of two particular solutions of the inhomogeneous TIW equation satisfying causal and anticausal boundary conditions. The structure of this solution of the homogeneous equation is shown to depend sensitively on the nature of the initial wave packet inhomogeneity, but, as we demonstrate, correct scattering information can be obtained even when the initial wave packet is nonzero only in the target region. It thus becomes possible to carry out quantum scattering calculations in which one need not propagate the wave packet from the noninte...