Journal of Theoretical and Computational Chemistry

It is shown how various ideas that are well established for the solution of Poisson's equation using plane wave and multigrid methods, can be combined with wavelet concepts. The combination of wavelet concepts and multigrid techniques turns out to be particularly fruitful. We propose a modified multigrid V cycle scheme that is not only much simpler, but also more efficient than the standard V cycle. Whereas in the traditional V cycle the residue is passed to the coarser grid levels, this new scheme does not require the calculation of a residue. Instead it works with copies of the charge density on the different grid levels that were obtained from the underlying charge density on the finest grid by wavelet transformations. This scheme is not limited to the pure wavelet setting, where it is faster than the preconditioned conjugate gradient method, but equally well applicable for finite difference discretizations.

Harmonic inversion of Chebyshev correlation and cross-correlation functions by the filter diagonalization method (FDM) is one of the most efficient ways to accurately compute the complex spectra of low dimensional quantum molecular systems. This explains the growing popularity of the FDM in the past several years. Some of its most attractive features are the predictable convergence properties and the lack of adjusting parameters. These issues however are often misunderstood and mystified. We discuss the questions relevant to the optimal choices for the FDM parameters, such as the window size and the number of basis functions. We also demonstrate that the cross-correlation approach (using multiple initial states) is significantly more effective than the conventional autocorrelation approach (single initial state) for the common case of a non-uniform eigenvalue distribution.

In this paper the derivation of exact and constrained kinetic energy operators in curvilinear coordinates is presented in a very general context. The polyspherical approach allows us to provide simple and explicit expressions of these operators which are well adapted to the numerical approaches used to solve the Schrödinger equation. These expressions can be exploited to treat a lot of problems such as the calculation of infrared or photoabsorption spectra as well as the study of intramolecular vibrational relaxation of molecules or the study of reactive scattering systems. Special emphasis is placed on concrete applications.

Cross correlation (CC) functions C_if(E) play an important role in chemical physics. They appear in the description of reactive scattering, photo-dissociation, photo-electron spectroscopy and electron transfer to mention a few. In this paper, we discuss two methods based on the Lanczos algorithm to compute the CC function for several initial and final states at the same time, without diagonalization. The property of the coupled two-term recursions variant of the Lanczos algorithm that yields a decomposition of the tridiagonal Lanczos matrix is crucial. The first method, the quasi minimal-recursive residue generation method (QM-RRGM) is based on solving a set of linear equations whereas the second method is based on a band-Lanczos method. The computational cost is of the same order of magnitude for both methods and is given by the number of matrix-vector multiplications in the underlying Lanczos method. Only a small set of scalars needs to be updated each recursion. The methods are compared for a model problem, the continuum resonance Raman cross section for a collinear model of CH₂IBr. Both methods shows similar convergence properties. By adding a pre-conditioner, the rate of convergence can be increased dramatically.

We develop a formalism for efficient iterative solutions of scattering problems involving the Maxwell equations. The methods, borrowed from recent advancements in chemical reaction dynamics, represent the scattering wavefunctions on two grids; one used for the initial wave and is one-dimensional; the other is a small three-dimensional grid padded with absorbing-potentials on which the scattered function is represented. The formalism is automatically suitable for scattering studies of transmission, reflection and scattering components of a wave. The simulations can be done with time-dependent wavepackets or direct iterative solution for the Green's function, but the results are rigorous time-independent (frequency-dependent) scattering amplitudes. Model time-dependent simulations involving up to a 100×100×100 grid for the inner wavefunction were numerically done on a simple PC.

We have investigated the dynamics of the title reaction with the Gray and Balint-Kurti approach, which propagates real wave packets (WP) under an arccos mapping of a scaled and shifted Hamiltonian. We have considered H₂ rotational quanta j=0 and 1 and obtained reaction probabilities using reactant coordinates and the flux analysis. We have calculated accurate reaction probabilities for total angular momentum quantum number J=0, centrifugal-sudden probabilities for J>0, cross sections, and the room temperature rate constant. The present cross sections are in good agreement with previous quasiclassical trajectory (QCT) results and the theoretical rate constant compares rather well with that observed. WP snapshots show that the reaction occurs via a C_2v insertion mechanism, confirming previous QCT calculations.

Iterative multiscale methods for electronic structure calculations offer several advantages for large-scale problems. Here we examine a nonlinear full approximation scheme (FAS) multigrid method for solving fixed potential and self-consistent eigenvalue problems. In principle, the expensive orthogonalization and Ritz projection operations can be moved to coarse levels, thus substantially reducing the overall computational expense. Results of the nonlinear multiscale approach are presented for simple fixed potential problems and for self-consistent pseudopotential calculations on large molecules. It is shown that, while excellent efficiencies can be obtained for problems with small numbers of states or well-defined eigenvalue cluster structure, the algorithm in its original form stalls for large-molecule problems with tens of occupied levels. Work is in progress to attempt to alleviate those difficulties.

We have recently developed a scaleable Artificial Boundary Inhomogeneity (ABI) method [Chem. Phys. Lett.366, 390–397 (2002)] based on the utilization of the Lanczos algorithm, and in this work explore an alternative iterative implementation based on the Chebyshev algorithm. Detailed comparisons between the two iterative methods have been made in terms of efficiency as well as convergence behavior. The Lanczos subspace ABI method was also further improved by the use of a simpler three-term backward recursion algorithm to solve the subspace linear system. The two different iterative methods are tested on the model collinear H+H₂ reactive state-to-state scattering.

The rovibrational energy levels of HXeOH and its isotopomers have been calculated using a two-layer Lanczos algorithm in Radau-diatom-Jacobi coordinates based on a high level ab initio potential energy surface. The surface is obtained by fitting to 1229 RCCSD(T)/SDB-cc-pVQZ energy points. The equilibrium geometry of HXeOH is determined to have a trans configuration with a nearly collinear HXeO bond angle of 177.32 degrees. The well depth of this minimum is only 0.6123 eV with respect to the OH+Xe+H dissociation limit. The results show that the H–Xe stretching frequency of HXeOH and HXeOD has an anomalous isotopic shift upon ¹⁸O isotope substitution, whereas the D–Xe stretch in DXeOH and DXeOD displays a normal isotopic shift. This trend is consistent with the experimental observations in a Xe solid matrix. The present results predict a lower frequency for the H–Xe stretch than was observed in the solid matrix experiment. Either these results are too low or there is a strong blue shift due to the matrix in the experimental values.

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.

We calculate energy levels of a six-dimensional bending Hamiltonian for HF trimer using a finite basis representation (FBR) in conjunction with the Lanczos eigensolver. We improve on our previous method [J. Chem. Phys.115, 9781 (2001)] using three techniques: (1) Lebedev's quadrature scheme is used to reduce the size of quadrature grid by a factor of 3.4. (2) Since the barrier separating the two equivalent versions of HF trimer is high and wide, it is a good approximation to confine the bending motion to one well by using sine spherical harmonics basis functions (this reduces the size of the basis by a factor of 8). (3) The sine spherical harmonic basis is contracted for each monomer to generate a very efficient basis. It is shown that the best approach is to combine all the three techniques.

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.

A model for H₂ inside single-walled carbon nanotubes is outlined. ARPACK (the Arnoldi package), a robust iterative matrix-vector eigenvalue software library, is used to determine the allowed quantum states of H₂ inside various carbon nanotubes. This information is used to construct the equilibrium constants for H₂ adsorption as a function of temperature for a variety of CNTs.

We further develop our recent direct method for the optimized effective potential (OEP) in density functional theory (DFT) [Yang and Wu, Phys. Rev. Lett.89, 143002 (2002)]. First, we show that the stationary condition in our optimization approach leads to a proper nonlinear algebraic equation for the OEP in a finite basis set, which differs from other finite basis set approaches. Then by constructing an approximate second derivative matrix of the energy functional in conjunction with the use of the Newton method, we significantly accelerate the convergence of the iterative optimization for OEP. Enhancement of the method is made in using the Tikhonov regularization method for the inversion of the second derivative matrix when it is singular or nearly singular and the direct inversion in the iterative space. It is shown that under a fixed stepsize condition, the optimization approach is equivalent to the self-consistent solution to the nonlinear algebraic equation for OEP. Because the approximate second derivatives are easy to compute and the iteration numbers are small now, the computation costs of OEP become comparable to that of regular DFT calculations as shown by calculations of some molecules, small and larger ones. We show how to find balanced results between energies and potentials when choosing a basis set for potentials.

Using an efficient single Lanczos propagation method, we report the resonance emission spectra of HCN and DCN from a number of low-lying vibrational levels of the Ã-state manifold. Our calculations represent the first such undertaking in which a high-quality ab initio based potential energy surface of the excited (ù A″) state and a transition dipole surface were used. The results show a significant improvement over previous theoretical work in reproducing experimental stimulated emission pumping spectra of HCN. The improved theory-experiment agreement is attributed to the accurate Ã-state potential energy surface, while the impact of the transition dipole function was found to be relatively minor.