Recursive Residue Generation Method Research Articles

Exact time evolution methods for large bound systems

We present a general purpose program, based on exact quantum methods, which allows one to study the spectroscopical properties of polyatomic molecules and to follow the time evolution of single initial states. The program displays no restriction concerning the molecule or basis set size, and the Hamiltonian specifications are done in a flexible, symbolic way, similar to the usual normal (or local) mode expansion. Also it makes full use of the vector and parallel facilities available on the present supercomputers. Three different recent techniques have been implemented, namely the recursive residue generation method, the Chebyshev time-propagator, and an active space method. The possibilities of the program are illustrated in the case of the CD 3H molecule, including all vibrational degrees of freedom. We first present an infrared spectrum determination, with special emphasis on the overtones of the C-H stretch mode, up to the fifth one (circa 16000 cm -1). We then show how the intramolecular vibrational relaxation from these highly excited local modes can be computed. The basis sets used in these studies range from 10 5 to several millions states.

Solving the discretized time-independent Schrödinger equation with the Lanczos procedure

A new method is presented to find bound state solutions of the one-, two-, or three-dimensional Schrödinger equation. The equation is turned into a sparse matrix eigenvalue problem by representing the potential energy surface and the wave function on a grid. The Laplacian is represented by a high (10th) order finite difference formula. Eigenvalues are found by the Lanczos procedure [J. Cullum and R. A. Willoughby, J. Comp. Phys. 44, 329 (1981)] and transition probabilities (Franck–Condon factors) are found by the recursive residue generation method [A. Nauts and R. E. Wyatt, Phys. Rev. Lett. 51, 2238 (1983)]. Examples are given for the 1D Morse oscillator and the 2D Hénon-Heiles potential. Numerical convergence can be checked easily and highly accurate results can be obtained. The algorithm is fast, easy to implement, and vectorizable.

Excitation trapping on two-dimensional percolating clusters

The trapping problem in disordered condensed phase systems is addressed by calculating the survival probability of a random walker on two-dimensional percolating clusters containing a random distribution of trap sites. The observable is directly determined by numerically solving the Pauli master equation via the Lanczos algorithm and the recursive residue generation method. From the results, we delimit the time and density regimes over which approximations developed for the corresponding statistic on regular lattices and deterministic fractals may be extended. In addition, recent predictions regarding the survival probability's long-time behavior are confirmed for moderately high trap concentrations; it is also shown that on this time scale, the fluctuations in the survival probability become enormous.

Residue decompositions of the propagator for time-independent Hamiltonian operators.

The formal equivalence and algorithmic differences between the Leverrier-Bateman resolvent method and the recursive residue generation method for constructing the evolution operator of time-independent Hamiltonian operators are discussed.

Solution of the Pauli master equation via the lanczos algorithm

We present a numerical Green's function approach to solving the Pauli master equation which utilizes the recursive residue generation method (RRGM) developed by Wyatt and co-workers. In most practical applications, only a few Green's function matrix elements are required to express all physical quantities of interest. For such systems, computation time reductions of 10 2–10 3 over direct diagonalization algorithms are achieved even for systems of moderate size. An application to a simple model for trapping in polymers is given.

Natural expansion of vibrational wave functions: RRGM with residue algebra

The natural expansion (NE) of vibrational eigenstates is useful for identifying the optimum local coordinates for any vibrational energy and it provides a positive test for regular (nonstochastic) behavior. In previous NE analyses, both eigenvalues and eigenvectors of the Hamiltonian matrix were required. However, through use of the recursive residue generation method (RRGM), we will illustrate how to perform the NE analysis without the need to compute eigenvectors of the N×N Hamiltonian matrix. In addition, a new computational method to obtain all transition amplitudes among a set of states is developed. The method, based upon residue algebra, reduces the CPU requirement by a factor of N/2. To illustrate these procedures, the A1 symmetry eigenfunctions in the classically chaotic regime (where the modes are strongly coupled) of a 2D model Hamiltonian are analyzed with the modified RRGM.

A generalized version of the recursive residue generation method for vector computers

After decomposing a multidimensional potential into a sum of products of simpler terms, an algorithm is presented for computing HU, where H is a sparse broad-banded Hamiltonian matrix and U is an N-vector. Unlike other methods, the new algorithm is explicitly vectorizable. When implemented on a vector processor, the algorithm leads to greatly reduced computation times in applications of the recursive residue generation method to quantum dynamics problems.

Importance of linked diagrams in the recursive residue generation method

Previous studies on the use of diagrammatic moment methods in the recursive residue generation method [I. Schek and R. E. Wyatt, J. Chem. Phys. 83, 3028, 4650 (1985)] are extended, with emphasis upon the role played by linked diagrams. Equivalence between the Green function for the source state (which initiates the recursion method) in the zero order and the corresponding tridiagonal (Lanczos) representation is shown. Chain parameters in the tridiagonal representation are constructed by comparing linked diagrams in the two representations and the dynamics of the source state is introduced in terms of the linked moments. The equivalence between Lanczos states and generalized doorway states is shown. Finally, a closed expression for the level shift of the source state (due to its interaction with the rest of the states) is given, using a diagrammatic approach in the tridiagonal representation.

Direct computation of quantal rate constants: Recursive development of the flux autocorrelation function

The recursive residue generation method (RRGM) is used to develop controlled approximations to the flux autocorrelation function, from which the quantal reaction rate constant is obtained by integrating over time. Computations for the Eckart potential show the superiority of sine functions over harmonic oscillator functions in discretizing the continuum.

Diagrammatic representation in the recursive residue generation method: Multiphoton excitation of an active mode coupled to a molecular background

A diagrammatic representation for the chain elements in the tridiagonal representation of a Hamiltonian is developed for the mutliphoton excitation of a molecular active mode which is coupled to a background of radiatively inactive modes. Both harmonic and anharmonic models for the active mode are discussed. Statistical level congestion of the background modes is discussed and we show the relative importance of various diagrammatic paths for both strong and weak laser intensities.

Generation and interpretation of chain parameters in the recursive residue generation method

In the recursive residue generation method [A. Nauts and R. E. Wyatt, Phys. Rev. Lett. 51, 2238 (1983)] molecular multiphoton excitation is treated by first converting the molecule-field Hamiltonian matrix into tridiagonal form, using the Lanczos equations. A procedure is developed here for construction of generating operators for successive recursive basis functions in the chain (tridiagonal) representation. These operators are used to construct the self-energies and the linking terms from successive moments of the Hamiltonian with respect to a source vector. These self-energies and linking terms are interpreted in terms of masses and stiffness of members of a mechanical chain and the chain dynamics is explained. The self-energies and coupling terms are expressed in terms of linked diagrams and explicit connections to the Hamiltonian moments are developed.

How to extend the recursive residue generation method (RRGM) to the degenerate case?

Some features of the recursive residue generation method (RRGM) initially intended for the non-degenerate case are generalized and shown to provide the starting point for further extension to the degenerate case.

Dynamics of eigenstate transitions induced by external fields: A new approach

We present a new general method to obtain the dynamics of molecular eigenstates perturbed by an external field. The method uses the Davidson algorithm and the recursive residue generation method (RRGM) to obtain molecular states and dynamic information, respectively. For sparse matrices, as many as 106 basis states can be used to represent the molecule’s Hamiltonian and its perturbation.

Recursive residue generation method for laser-molecule interaction: utilization of structured sparsity

The recursive residue generation method (RRGM) permits the generation of time-dependent transition amplitudes in many-state quantum systems. An important step in the method utilizes the Lanczos algorithm to tridiagonalize the matrix representation of the Hamiltonian. Analysis of this sparse matrix, for a generic laser-molecule Hamiltonian, revealed structure (repeating blocks of elements in each off-diagonal) which allowed for greatly reduced storage. In addition, utilization of this structure for construction of a smart matrix-vector multiplier then permitted calculations on systems with very large bases (∼40,000). Time-dependent transition probabilities are shown for excitation into several bands of states in a system with over 3000 States.

Quantum statistical mechanics via the recursive residue generation method

Using the recursive residue generation method (RRGM), it is shown how Green’s function matrix elements calculated in large basis sets (N≫103) can be utilized to obtain controlled approximations to average values, correlation functions, and power spectra in multimode anharmonic systems. Convergence of the method is demonstrated through studies on five and seven mode model systems with basis sets containing up to about 8×103 states.