Efficient calculation of matrix elements in low storage filter diagonalization

Full-dimensional quantum mechanical calculations for the tunneling behavior of HOCO dissociation to H + CO2

The tunneling behavior during HOCO dissociation to H + CO₂ was investigated by full-dimensional quantum mechanical calculations based on an accurate global potential energy surface. The tunneling lifetimes for the low-lying 1500 vibrational states were calculated using the low-storage filter diagonalization method after a 1 million-step Chebyshev propagation. In the calculated energy range, the lifetimes of different vibrational states with similar energy are found to differ by 3-4 orders of magnitude, and the lower limit for these tunneling lifetimes is consistent with the experimental results reported by Continetti et al. For the given vibrational progressions, the lifetime of the vibrational states decreases with the increasing energy level, which is consistent with the results of 1D simulation calculations reported by Bowman, but the declining curve obviously fluctuates, and the declining slope is significantly different from that obtained by 1D simulation. Due to a difference in the effective barrier width, the mode-specific behavior of the tunneling effect is manifested in that the O-C-O' and H-O-C bend modes lead to the largest enhancement and an inhibitory effect on the tunneling process, respectively.

A complex guided spectral transform Lanczos method for studying quantum resonance states

A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths, and wavefunctions. The algorithm comprises of two layers of complex-symmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigen-pairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonian in the spectral range of interest. Therefore, the energies and/or widths of bound or resonance states can be easily computed with their wavefunctions or by using a root-searching method from the guided spectral transform surface. The new cGSTL algorithm is applied to bound and resonance states of HO2, and compared to previous calculations.

CALCULATION OF TRANSITION AMPLITUDES WITH A SINGLE LANCZOS PROPAGATION

We review in this article a recently proposed energy-global method that is capable of calculating the entire transition amplitude matrix with a single Lanczos propagation. This method requires neither explicit computation nor storage of the eigenfunctions, rendering it extremely memory efficient. Procedures are proposed to handle situations where "spurious" eigenvalues aggregate around true eigenvalues due to round-off errors. This method is amenable to both real-symmetric and complex-symmetric Hamiltonians. Applications to molecular spectra and reactive scattering are presented. Its relationships with other methods are also discussed.

CALCULATION OF SELECTED HIGHLY EXCITED VIBRATIONAL STATES OF POLYATOMIC MOLECULES BY THE DAVIDSON ALGORITHM

We described an improved version of a modified Davidson scheme previously introduced (F. Ribeiro, C. Iung and C. Leforestier, Chem. Phys. Lett.362, 199 (2002)), aimed at computing highly excited energy levels of polyatomic molecules. The key ingredient is a prediagonalization-perturbation step performed on a subspace of a curvilinear normal modes basis set (including diagonal anharmonicities). The efficiency of the method is demonstrated by computing the lowest 350 vibrational states of A′ symmetry of the HFCO molecule. Also shown is the possibility to restrict the calculation to selected energy levels, based on their zero-order description. This State Filtered Diagonalization method is illustrated on a high overtone (7ν5) of the OCF bend, and on the few energy levels (20) which have been experimentally assigned up to 5000 cm -1 of excitation energy.

Revelation of non-statistical behavior in HO2 vibration by a newab initiopotential energy surface

The hydroperoxyl radical (HO2) has long been considered as a prototype for statistical vibrational dynamics. In this work, however, it is shown that the bound state energy levels (up to the dissociation threshold) and low-lying resonances of the HO2 system (J=0) obtained on a new ab initio potential energy surface exhibit surprisingly large regularity. The implications of the non-statistical behavior of the HO2 system in unimolecular and bimolecular reactions are discussed.

HOCl Ro-Vibrational Bound-State Calculations for Nonzero Total Angular Momentum

The Lanczos homogeneous filter diagonalization method has been employed to compute the HOCl ro-vibrational states for a range of total angular momenta (J = 0, 1, 5, 10, 11, 20, 30) on a newly developed ab initio potential energy surface by Nanbu et al. (J. Theor. Comput. Chem. 2002, 1, 263). For such computationally challenging calculations, a parallel computing strategy has been incorporated into our method to perform the matrix-vector multiplications. For the computed low bound states, a spectroscopic assignment has been made and the widely used approximate adiabatic rotation method has been tested for the broad range of total angular momenta for this deep-well system. Comparison of experimental results with exact quantum mechanical calculations for the selected far-infrared transitions involving the range of total angular momenta has been made possible for the first time.

HO2 Ro-Vibrational Bound-State Calculations for Large Angular Momentum: J = 30, 40, and 50

The Lanczos homogeneous filter diagonalization method and the real Chebyshev filter diagonalization scheme incorporating doubling of the autocorrelation functions have been employed to compute the HO2 ro-vibrational states for high total angular momenta, J = 30, 40, and 50. For such computationally challenging calculations, we have adopted a parallel computing strategy to perform the matrix-vector multiplications. Low-lying bound states and high-lying bound states close to the dissociation threshold are reported. For low-lying bound states, a spectroscopic assignment has been attempted and the widely used approximate J-shifting method has been tested for this deep-well system. For high-lying bound states, the attempted spectroscopic assignments as well as the J-shifting approximation fail because of very strong Coriolis mixing, indicating that the Coriolis couplings are important for this system.

Calculation of specific, highly excited vibrational states based on a Davidson scheme: Application to HFCO

We present the efficiency of a new modified Davidson scheme which yields selectively one high-energy vibrationally excited eigenstate or a series of eigenstates. The calculation of a highly vibrationally excited state psi located in a dense part of the spectrum requires a specific prediagonalization step before the Davidson scheme. It consists in building a small active space P containing the zero-order states which are coupled with the zero-order description of the eigenstate of interest. We propose a general way to define this active space P which plays a crucial role in the method. The efficiency of the method is illustrated by computing and analyzing the high-energy excited overtones of the out-of-plane mode [formula: see text] in HFCO. These overtone energies correspond to the 234th, 713th, and 1774th energy levels in our reference basis set which contains roughly 140,000 states. One of the main advantages of this Davidson scheme comes from the fact that the eigenstate and eigenvalue convergence can be assessed during the iterations by looking at the residual [formula: see text]. The maximum value epsilon allowed for this residual constitutes a very sensitive and efficient parameter which sets the accuracy of the eigenvalues and eigenstates, even when the studied states are highly excited and are localized in a dense part of the spectrum. The physical analysis of the eigenstates associated with the 5th, 7th, and 9th out-of-plane overtones in HFCO provides some interesting information on the energy localization in this mode and on the role played by the in-plane modes. Also, it provides some ideas on the numerical methods which should be developed in the future to tackle higher-energy excited states in polyatomics.

A Jacobi-Wilson description coupled to a block-Davidson algorithm: An efficient scheme to calculate highly excited vibrational levels

We present a new approach based on the block-Davidson scheme which provides eigenvalues and eigenvectors of highly excited (ro) vibrational states of polyatomic molecules. The key ingredient is a prediagonalized-perturbative scheme applied to a subspace of a curvilinear normal-mode basis set. This approach is coupled to the Jacobi vector description recently developed by our group [C. Leforestier, A. Viel, F. Gatti, C. Munoz, and C. Iung, J. Chem. Phys. 114, 2099 (2001)], and applied to the HFCO and H2CO molecules, which represent the main difficulties of such calculations for any available method. The first one presents a significant state density because of its low symmetry and the presence of a fluorine atom, while strong resonances and intermode couplings occur in H2CO. This study establishes the robustness, the numerical efficiency, and the versatility of the method which is compared to the regular Lanczos and Davidson schemes. It is also shown that the eigenvectors can be obtained within a given accuracy easily set by the user. This point constitutes one of the main advantages of the method as very few potential-energy surfaces achieve an accuracy of the order of a wave number for highly excited states. Furthermore, this method allows one to restrict the calculations to selected energy levels based on their zero-order descriptions.

Unimolecular rovibrational bound and resonance states for large angular momentum: J=20 calculations for HO2

We explore the calculation of unimolecular bound states and resonances for deep-well species at large angular momentum using a Chebychev filter diagonalization scheme incorporating doubling of the autocorrelation function as presented recently by Neumaier and Mandelshtam [Phys. Rev. Lett. 86, 5031 (2001)]. The method has been employed to compute the challenging J=20 bound and resonance states for the HO2 system. The methodology has firstly been tested for J=2 in comparison with previous calculations, and then extended to J=20 using a parallel computing strategy. The quantum J-specific unimolecular dissociation rates for HO2-->H+O2 in the energy range from 2.114 to 2.596 eV have been reported for the first time, and comparisons with the results of Troe and co-workers [J. Chem. Phys. 113, 11019 (2000) Phys. Chem. Chem. Phys. 2, 631 (2000)] from statistical adiabatic channel method/classical trajectory calculations have been made. For most of the energies, the reported statistical adiabatic channel method/classical trajectory rate constants agree well with the average of the fluctuating quantum-mechanical rates. Near the dissociation threshold, quantum rates fluctuate more severely, but their average is still in agreement with the statistical adiabatic channel method/classical trajectory results.

Computing resonance energies, widths, and wave functions using a Lanczos method in real arithmetic

We introduce new ideas for calculating resonance energies and widths. It is shown that a non-Hermitian-Lanczos approach can be used to compute eigenvalues of H+W, where H is the Hamiltonian and W is a complex absorbing potential (CAP), without evaluating complex matrix-vector products. This is done by exploiting the link between a CAP-modified Hamiltonian matrix and a real but nonsymmetric matrix U suggested by Mandelshtam and Neumaier [J. Theor. Comput. Chem. 1, 1 (2002)] and using a coupled-two-term Lanczos procedure. We use approximate resonance eigenvectors obtained from the non-Hermitian-Lanczos algorithm and a very good CAP to obtain very accurate energies and widths without solving eigenvalue problems for many values of the CAP strength parameter and searching for cusps. The method is applied to the resonances of HCO. We compare properties of the method with those of established approaches.

Converged quantum calculations of HO2 bound states and resonances for J=6 and 10

Bound and resonance states of HO(2) are calculated quantum mechanically using both the Lanczos homogeneous filter diagonalization method and the real Chebyshev filter diagonalization method for nonzero total angular momentum J=6 and 10, using a parallel computing strategy. For bound states, agreement between the two methods is quite satisfactory; for resonances, while the energies are in good agreement, the widths are in general agreement. The quantum nonzero-J specific unimolecular dissociation rates for HO(2) are also calculated.

Spectral density calculation by using the Chebyshev expansion

A method to calculate the spectral density of any state vectors with respect to a set of eigenstates of a Hamiltonian is presented. A spectral density operator, whose expectation value on the state vector gives the spectral density, is evaluated indirectly by using the Chebyshev expansion method. A spectral transformation function is introduced to improve resolution at the low energy region, at the expense of the one at the higher region. The predissociation spectrum of CO+ is calculated to demonstrate the method.