Recursive Residue Generation Method Research Articles

Isotopic effects in vibrational relaxation dynamics of H on a Si(100) surface

In a recent paper [U. Lorenz and P. Saalfrank, Chem. Phys. 482, 69 (2017)], we proposed a robust scheme to set up a system-bath model Hamiltonian, describing the coupling of adsorbate vibrations (system) to surface phonons (bath), from first principles. The method is based on an embedded cluster approach, using orthogonal coordinates for system and bath modes, and an anharmonic phononic expansion of the system-bath interaction up to second order. In this contribution, we use this model Hamiltonian to calculate vibrational relaxation rates of H-Si and D-Si bending modes, coupled to a fully H(D)-covered Si(100)-(2×1) surface, at zero temperature. The D-Si bending mode has an anharmonic frequency lying inside the bath frequency spectrum, whereas the H-Si bending mode frequency is outside the bath Debye band. Therefore, in the present calculations, we only take into account one-phonon system-bath couplings for the D-Si system and both one- and two-phonon interaction terms in the case of H-Si. The computation of vibrational lifetimes is performed with two different approaches, namely, Fermi's golden rule, and a generalized Bixon-Jortner model built in a restricted vibrational space of the adsorbate-surface zeroth-order Hamiltonian. For D-Si, the Bixon-Jortner Hamiltonian can be solved by exact diagonalization, serving as a benchmark, whereas for H-Si, an iterative scheme based on the recursive residue generation method is applied, with excellent convergence properties. We found that the lifetimes obtained with perturbation theory, albeit having almost the same order of magnitude-a few hundred fs for D-Si and a couple of ps for H-Si-, are strongly dependent on the discretized numerical representation of the bath spectral density. On the other hand, the Bixon-Jortner model is free of such numerical deficiencies, therefore providing better estimates of vibrational relaxation rates, at a very low computational cost. The results obtained with this model clearly show a net exponential decay of the time-dependent survival probability for the H-Si initial vibrational state, allowing an easy extraction of the bending mode "lifetime." This is in contrast with the D-Si system, whose survival probability exhibits a non-monotonic decay, making it difficult to define such a lifetime. This different behavior of the vibrational decay is rationalized in terms of the power spectrum of the adsorbate-surface system. In the case of D-Si, it consists of several, non-uniformly distributed peaks around the bending mode frequency, whereas the H-Si spectrum exhibits a single Lorentzian lineshape, whose width corresponds to the calculated lifetime. The present work gives some insight into mechanisms of vibration-phonon coupling at surfaces. It also serves as a benchmark for multidimensional system-bath quantum dynamics, for comparison with approximate schemes such as reduced, open-system density matrix theory (where the bath is traced out and a Liouville-von Neumann equation is solved) or approximate wavefunction methods to solve the combined system-bath Schrödinger equation.

Multi-layer Lanczos iteration approach to calculations of vibrational energies and dipole transition intensities for polyatomic molecules.

We report a rigorous full dimensional quantum dynamics algorithm, the multi-layer Lanczos method, for computing vibrational energies and dipole transition intensities of polyatomic molecules without any dynamics approximation. The multi-layer Lanczos method is developed by using a few advanced techniques including the guided spectral transform Lanczos method, multi-layer Lanczos iteration approach, recursive residue generation method, and dipole-wavefunction contraction. The quantum molecular Hamiltonian at the total angular momentum J = 0 is represented in a set of orthogonal polyspherical coordinates so that the large amplitude motions of vibrations are naturally described. In particular, the algorithm is general and problem-independent. An application is illustrated by calculating the infrared vibrational dipole transition spectrum of CH4 based on the ab initio T8 potential energy surface of Schwenke and Partridge [Spectrochimica Acta, Part A 57, 887 (2001)] and the low-order truncated ab initio dipole moment surfaces of Yurchenko et al. [J. Mol. Spectrosc. 291, 69 (2013)]. A comparison with experiments is made. The algorithm is also applicable for Raman polarizability active spectra.

Reduced-Order Modeling, Error Estimation, and the Role of the Start-Vector: The Recursive Residue Generation Method Revisited

The recursive residue generation method (RRGM) [Wyatt, R. E. Adv. Chem. Phys. 1989, 73, 231] is re-derived using the formalism of reduced-order modeling [Bai, Z. Appl. Numerical Math. 2002, 43, 9]. A stopping criteria for the RRGM recursions is proposed, on the basis of an expression for an upper bound to the absolute error [Bai, Z.; Ye, Q. Electron. Trans. Numerical Anal. 1998, 7, 1]. It is further pointed out that, in general, the start-vector has a negligible effect on the convergence of the RRGM.

Cross correlation functions Cnm(E) via Lanczos algorithms without diagonalization

It is shown how the quasiminimal residual algorithm (QMR), based on the Lanczos algorithm, can be modified to compute cross-correlation functions Cnm(E)=〈Ψn|(E−H)−1|Ψm〉 without any diagonalization by recursively updating a small number of scalars. Only three Lanczos vectors need to be stored. Several left-hand side vectors 〈Ψn| and multiple shifts E can be considered simultaneously. The new method is termed the quasiminimal recursive residue generation method (QM-RRGM) and is applied to the collinear H+H2 problem to illustrate its convergence behavior. The properties of two different formulations of the Lanczos algorithm, the usual three-term and a coupled two-term recursion, are also discussed. The QM-RRGM exhibits smooth convergence behavior, and it is shown that the stopping criteria used in the QMR algorithm can also be used for computing correlation functions.

Quantum mechanical study of the CH(v=2) overtone in 30-mode benzene

The results of large-scale quantum mechanical calculations of the CH(v=2) 1st overtone spectrum for 30-mode benzene are reported. This overtone was chosen for investigation because of its high degree of fragmentation and resulting complexity compared to spectra for the fundamental and higher overtones. These calculations use the best available ab initio force field supplemented by higher-order terms for the CH stretch–wag interaction. The dynamical calculations were conducted in large active spaces with 12 000 or 16 000 vibrational basis functions. The recursive residue generation method was used to compute residues (intensities) and eigenvalues. From these quantities, the lineshape function, survival probabilities, and vibrograms were computed. Wherever possible, these results were compared to experimental overtone spectra and to other computational results.

Calculation of vibrational (J=0) excitation energies and band intensities of formaldehyde using the recursive residue generation method

We use the recursive residue generation method (RRGM) with an exact kinetic energy operator to calculate vibrational excitation energies and band intensities for formaldehyde. The basis is a product of one-dimensional potential optimized discrete variable representation (PO-DVR) functions for each coordinate. We exploit the symmetry by using symmetry adapted basis functions obtained by taking linear combinations of PO-DVR functions. Our largest basis set consists of 798 600 functions (per symmetry block). The Lanczos tridiagonal representation of the Hamiltonian is generated iteratively (without constructing matrix elements explicitly) by sequential transformations. We determine a six-dimensional dipole moment function from the ab initio dipole moment values computed at the QCISD level with a 6-311++G(d,p) basis set. We converged all A1, B2 and B1 vibrational states up to the combination band with two quanta in the C–O stretch and one quantum in a C–H stretch at about 6 350 cm−1 above zero point energy. We present a simulated (J=0) infrared spectrum of CH2O for transitions from the ground state.

A quantum dynamical study of CH overtones in fluoroform. I. A nine-dimensional ab initio surface, vibrational spectra and dynamics

In this series, the characteristics of intramolecular vibrational-energy redistribution (IVR) present in the CH overtones of CHF3 are investigated. Particular attention is given to the multiple time scales and thus mechanisms present in the IVR dynamics. In Part I, a 9-dimensional ab initio potential energy surface is developed to adequately account for the vibrational couplings of all modes. Furthermore, all-mode vibrational state calculations, of large primitive space dimension, are performed using a recently developed wave operator sorting algorithm in tandem with the recursive residue generation method. All fundamentals, first overtones, and bimodal combination states with up to 3 quanta are presented. Also, the A1 and E-symmetry CH polyads are determined through the second overtone. Equilibrium geometry, rotational constants, and vibrational properties agree quantitatively with experiment in most cases. The error is systematic in origin and largely due to the error in the ab initio harmonic frequencies. New vibrational constants and resonance interactions are reported for the background modes. In contrast to the prominent CH stretch–bend Fermi resonance structure, responsible for ultrafast (t<50 fs) energy transfer, the CH polyads also exhibit vibrational fine structure of order 1 to 10 cm−1 due to background-mode coupling. This secondary coupling results in IVR on the picosecond time scale.

Autocorrelation functions in the photoionization of hydrogen in electric and magnetic fields.

The photoionization (PI) cross section of hydrogen in external electric and magnetic fields was studied by a direct version of the recursive residue generation method, combined with complex dilation and discretization. The PI cross section was computed for three different combinations of fields: an electric field only, perpendicular electric and magnetic fields, and, finally, an electric field with an angle of 80\ifmmode^\circ\else\textdegree\fi{} to the magnetic field. The field-induced resonances were shown to have Fano-type profiles, with a shape depending on the field combination. The fluctuations of the base line were studied using an autocorrelation function C(\ensuremath{\epsilon}). For the field combinations studied here, C(\ensuremath{\epsilon}) has for small \ensuremath{\epsilon} an initial Lorentzian shape. With increasing \ensuremath{\epsilon} different oscillatory structures in C(\ensuremath{\epsilon}) were observed. The correlation functions were studied for several energy ranges, achieving a common Lorentzian shape for the highest range, independent of the field combinations. A simple interpretation of the behavior for all cases in terms of chaotic ionization is possible.

Dynamics of hydrogenic wave packets in electric and magnetic fields

The survival probability P(t) of hydrogenic wave packets for some choices of strong external electric and magnetic fields is studied. The recursive residue generation method, in combination with complex dilation and a Laguerre basis, is used to calculate the local density of states from which P(t) is computed via a Fourier transform. P(t) is calculated both for a single initial ‖8p0〉 state and for an initial radial hydrogenic wave packet centered around n=15. In the first case only, in principle, a single n manifold is involved in the dynamics, while in the latter case several n manifolds contribute. The different choices of initial states lead to significantly different results. The choice of fields was made in order to exemplify various patterns of behavior. Results similar to those obtained here were found in recent experiments by Broers and co-workers [Phys. Rev. Lett. 71, 344 (1993); Phys. Rev. A 49, 2498 (1994)]. The results presented in this work are quantum-mechanical calculations of hydrogenic wave packets in strong fields that include the effect of finite lifetimes on P(t).

Evaluation of electronic matrix elements of long‐range electron transfer in proteins by the recursive residue generation method

AbstractAn efficient method for calculating electronic matrix elements for electron‐transfer reactions in biological systems is proposed. We use the tight‐binding model to describe the motion of one‐electron in an array representing the system: donor (D), acceptor (A), and all other atoms that compose the protein. The matrix elements |TDA|;2 is evaluated at the estimated electron energy. The essential part of the method lies in the evaluation of the off‐diagonal Green function 〈I|G|F〉 between the site I neighboring with the donor and the site F with the acceptor. The recursive residue generation method is used to evaluate 〈I|G|F|〉. As an application, these matrix elements in metal‐labeled myoglobin are calculated and the results are compared with those evaluated with the renormalized‐perturbation‐expansion approach. © 1994 John Wiley & Sons, Inc.

Stationary approaches for solving the Schrödinger equation with time-dependent Hamiltonians

Accurate and efficient computational schemes are utilized for solving the Schrödinger equation with time-dependent Hamiltonians. These schemes, based on an extended Hilbert space in which time is treated like a space coordinate, allow essentially all of the computational strategies for stationary problems to be equally applicable to explicitly time-dependent problems. In particular, variational principles, the discrete variable representation in time, the fast Fourier transform in time, the recursive residue generation method (RRGM), and the Chebyshev propagator can be employed. Some of these methods are implemented and tested for a rigid rotor interacting with various laser pulses. The fast exponential convergence with respect to the number of time grid points is illustrated, which accounts for the high accuracy of the methods (limited only by the precision of computer), a feature which is hardly achievable by other methods.

Numerical study of Frenkel excitons in random chain conformations.

This work deals with the simulation of excitonic absorption line shapes (ALS's) of random aromatic chains based upon a description in terms of a Frenkel Hamiltonian. By averaging the ALS's of single configurations over an ensemble of chain conformations, generated by means of a Pivot algorithm on a square lattice, a type of off-diagonal disorder is examined, which originates from the randomness of non-nearest-neighbor contacts in polymer coils. Dimensions of up to several thousand chromophores are studied by utilizing the recursive residue generation method [A. Nauts and R. E. Wyatt, Phys. Rev. Lett. 51, 2238 (1983)], which resorts to the Lanczos tridiagonalization. With this method ALS's of pure donor chains and binary chains are investigated and the effects of diagonal and off-diagonal disorder, as well as convergence pattern of the recursive residue generation method in application to such systems is demonstrated.

A direct recursive residue generation method: application to photoionization of hydrogen in static electric fields

In studies of hydrogenic systems via the recursive residue generation method (RRGM) the major bottleneck is the matrix vector product HC, between the Hamiltonian matrix H and a Lanczos vector C. For highly excited states and/or strong perturbations the size of H grows fast leading to storage problems. By making use of direct methods, i.e. avoidance of explicit construction of large Hamiltonian matrices, such problems can be overcome. Utilizing the underlying analytical properties of the Laguerre basis e- lambda rLk2l+2(2 lambda r) a direct RRGM (D-RRGM) for the unperturbed hydrogenic Hamiltonian is derived, changing the storage needs from scaling as N2 to 4N where N is the number of radial functions for each factorized H0(l,m) block with the possibility of parallel processing. A further computational simplification is introduced by putting the expression for the photoionization (PI) cross section in the rational form conventionally used in the representation of density of states (DOS). This allows the construction of the PI cross section directly from the tridiagonal Lanczos matrix avoiding the explicit calculation of individual eigenvalues and eigenvectors. To illustrate and verify the method the PI cross section for a hydrogen atom in a static electric field, for both pi and sigma polarization, was calculated for an electric field strength of F=5714 V cm-1. Sufficiently large basis sets could be employed so that good comparison with experiment and other theoretical work was obtained, including the field-induced modulations near the zero-field ionization limit.

Adiabatic pseudospectral methods for multidimensional vibrational potentials

We describe a new algorithm for computing eigenvalues, spectral intensities, and selected eigenvectors of multidimensional vibrational potential surfaces. The method involves a synthesis of pseudospectral and sequential adiabatic reduction methods and merges the storage and computational advantages of the former with the improved basis set generated by the latter. The recursive residue generation method, which utilizes a Lanczos-based diagonalization procedure, is employed to calculate the observables. As a test case, we apply the method to computation of the infrared and stimulated emission pumping spectra for the HCN molecule and demonstrate a very large (one to three orders of magnitude) reduction in computational effort (for comparable accuracy) as compared to discrete variable representation (DVR)/adiabatic reduction or standard collocation approaches. We expect that this advantage will be increased considerably for larger (e.g., tetra-atomic) systems and will permit accurate basis set calculations on such systems to be carried out in a straightforward fashion.

Wave operator and artificial intelligence contraction algorithms in quantum dynamics: Application to CD3H and C6H6

We have established in this study the capabilities of the wave operator (WO) algorithm to extract from a huge primitive space a smaller subspace (the active space) containing all of the zero order states which play an active role during the intramolecular vibrational energy redistribution (IVR) from an initial state ‖i≳0. While exact methods such as the recursive residue generation method (RRGM) or the Chebychev algorithms can only be applied in a primitive space containing less than about 200 000 states, the WO algorithm can be used efficiently in ultralarge basis sets containing billions of states. The recursive residue generation method (RRGM) or Chebychev methods can then be applied in this active space which typically contains less than 10 000 states. In order to draw general conclusions on the efficiency of such a method and on the main features of IVR phenomena, we have concurrently studied IVR from the fifth CH overtone in the nine mode CD3H molecule and from the second CH overtone in the 16 mode C6H6 system. We have analyzed the main features of the active space and have shown that the WO algorithm selects the important states. A very broad energy distribution of states in the active space has been obtained for these two systems. We have also shown that C6H6 is a very complex system to study; while only a few hundred states are effectively populated during the IVR from the fifth CH overtone in CD3H, about 8000 states have to be considered in order to accurately study IVR from the second CH overtone in C6H6. However, we have shown that the WO method is able to reproduce correctly both the survival probability of the initial state and the intricate energy flow through the molecule during the first picosecond. Finally, we have shown that the WO algorithm builds a far more efficient active space than a more traditional artificial intelligence (Al) tree pruning procedure.

Theoretical study of intramolecular vibrational relaxation of acetylenic CH vibration for v=1 and 2 in large polyatomic molecules (CX3)3YCCH, where X=H or D and Y=C or Si

Quantum calculations are reported for the intramolecular vibrational energy redistribution and absorption spectra of the first two excited states of the acetylenic CH stretch vibration in the polyatomic molecules (CX3)3YCCH, where X=H or D and Y=C or Si. Using approximate potential energy surfaces, comparison is made with the corresponding recent experimental spectra. It is found that a model of intramolecular vibrational relaxation based on the assumption of sequential off-resonance transitions via third and fourth order vibrational couplings (as opposed to direct high order couplings) is in agreement with experimental results on spectral linewidths. In a semiclassical limit this type of relaxation corresponds to a dynamic tunneling in phase space. It is shown that the local density of resonances of third and fourth order, rather than the total density of states, plays a central role for the relaxation. It is found that in the Si molecule an accidental absence of appropriate resonances results in a bottleneck in the initial stages of relaxation. As a result, an almost complete localization of the initially prepared excitation occurs. It is shown that an increase of the mass alone of the central atom from C to Si cannot explain the observed difference in the C and Si molecules. The spectral linewidths were calculated with the Golden Rule formula after prediagonalization of the relevant vibrational states which are coupled in the molecule to the CH vibration, directly or indirectly. For the spectral calculations, in addition to the direct diagonalization, a modified recursive residue generation method was used, allowing one to avoid diagonalization of the transformed Lanczos Hamiltonian. With this method up to 30 000 coupled states could be analyzed on a computer with relatively small memory. The efficiency of C programming language for the problem is discussed.

Quantum dynamics of overtone relaxation in benzene. III. Spectra and dynamics for relaxation from CH(v=3)

This series is concerned with the quantum dynamics of overtone relaxation in benzene and in reduced mode benzene fragments. In part III of this series, emphasis is placed on the CH(v=3) overtone spectrum and the survival probability for both 16 mode (five CH stretch modes are forced to remain inactive out of 21 planar modes) and 21 mode planar benzene models. Through use of the wave operator contraction algorithm, primitive vibrational basis sets containing up to 9×109 states are contracted to active spaces containing 2000–7000 states. The exact dynamics within the active space is developed with the recursive residue generation method (RRGM). Specific results reported in this study include the following: (a) a comparison is made between dipole spectra and residue spectra, the former explicitly involving the CH dipole function; (b) for the 16 mode model, the following quantities are displayed: overtone spectrum, survival probability of the initial state, complex-valued autocorrelation function, number of phase space cells explored as a function of time, and the rate of exploring phase space cells; (c) sensitivity of the overtone spectrum to the size and composition of the active space built with the wave operator contraction algorithm; (d) for 21 mode benzene, we consider the sensitivity of the overtone spectrum to ‘‘softening’’ the anharmonic potential; (e) comparisons are made with recent jet cooled beam experimental overtone spectra.

Perturbed hydrogenic manifolds studied by the recursive residue generation method

A method for calculating the perturbation of hydrogenic manifolds, the emerging bound states and resonances, for arbitrary combinations of external fields, is presented. It requires the combined use of complex dilation, an orthonormal Laguerre basis e- lambda rL2l+2k( lambda r) rather than the non-orthogonal Sturmians e- lambda rL2l+1k( lambda r), and the recursive residue generation method (RRGM) version of the Lanczos algorithm. Generalized eigenvalue problems are avoided. Furthermore, direct computation of the residues of resolvents, transition amplitudes and sum rules is achieved. Comparison with other methods and with previous calculations, suitable for one perturbation at a time, indicates that high accuracy is achieved separately both for the 1s Stark resonance and for the 1s Zeeman effect. Accurate results for the 1s Stark-Zeeman resonance, for various combinations of fields, are given.

Quantum dynamics of overtone relaxation in benzene. I. 5 and 9 mode models for relaxation from CH(v=3)

This series is concerned with the quantum dynamics of overtone relaxation in planar benzene and in reduced mode planar benzene fragments. In these studies, ultralarge direct product primitive vibrational spaces (of dimension up to 1010) are contracted to active spaces of dimension 5000–10 000. The contractions are carried out via artificial intelligence tree pruning algorithms, or a new iterative wave operator pruning algorithm. The exact dynamics within the active space is then developed via the recursive residue generation method. In part I of this series, emphasis is placed upon v=3 CH overtone dynamics in the 5 and 9 mode benzene fragments C3H and C3H3. Neither system undergoes complete relaxation, but the survival probability in C3H undergoes large amplitude oscillations with a period characteristic of stretch–wag interaction in the CH chromophore. For C3H3, the two initially nonexcited CH stretch modes do not play a significant role in the dynamics for t<1 ps. However, modes in both systems that have a high degree of wag motion for the initially excited chromophore play a significant role at short times. Comparisons with earlier classical trajectory studies show good correspondence between the classical and quantum results only at short times, t<0.1 ps.

Quantum dynamics of overtone relaxation in benzene. II. 16 mode model for relaxation from CH(v=3)

This series is concerned with the quantum dynamics of overtone relaxation in benzene and in reduced mode benzene fragments. In part II of this series, emphasis is placed upon time-dependent studies of v=3 CH overtone relaxation in a 16 mode planar benzene model (5 CH stretch modes are inactive out of the 21 planar modes). Through use of the wave operator contraction algorithm, primitive vibrational basis sets containing up to 1.1×109 states are contracted to active spaces of dimension about 4000 states. The exact dynamics within the active space is developed with the recursive residue generation method. The time-dependent survival probability of the CH(v=3) overtone is compared with results from classical mechanics and with the Fourier transform of the experimental overtone spectra. The classical and quantum results are in good agreement only at short times (t<0.1 ps). At longer times multiple small recurrences are present in the quantum results. The quantum and experimental results share common features (rapid falloff at short times, followed by multiple small recurrences), but there are quantitative differences. The sensitivity of the quantum survival to alterations in the anharmonic potential is also studied. The short-time dynamics (t<0.5 ps) is shown to be accurately reproduced by reduced 4 and 5 mode models. Only a small subset of states in these carefully selected modes are significantly populated during the early decay of the v=3 overtone, even though the total density of states is very large.