Hitting the Trifecta: How to Simultaneously Push the Limits of Schrödinger Solution with Respect to System Size, Convergence Accuracy, and Number of Computed States

Methods for solving the Schrödinger equation without approximation are in high demand but are notoriously computationally expensive. In practical terms, there are just three primary factors that currently limit what can be achieved: 1) system size/dimensionality; 2) energy level excitation; and 3) numerical convergence accuracy. Broadly speaking, current methods can deliver on any two of these three goals, but achieving all three at once remains an enormous challenge. In this paper, we shall demonstrate how to "hit the trifecta" in the context of molecular vibrational spectroscopy calculations. In particular, we compute the lowest 1000 vibrational states for the six-atom acetonitrile molecule (CH₃CN), to a numerical convergence of accuracy 10^-2 cm^-1 or better. These calculations encompass all vibrational states throughout most of the dynamically relevant range (i.e., up to ∼4250 cm^-1 above the ground state), computed in full quantum dimensionality (12 dimensions), to near spectroscopic accuracy. To our knowledge, no such vibrational spectroscopy calculation has ever previously been performed.

Double Proton Transfer in the Dimer of Formic Acid: An Efficient Quantum Mechanical Scheme

Double proton transfer plays an important role in biology and chemistry, such as with DNA base pairs, proteins and molecular clusters, and direct information about these processes can be obtained from tunneling splittings. Carboxylic acid dimers are prototypes for multiple proton transfer, of which the formic acid dimer is the simplest one. Here, we present efficient quantum dynamics calculations of ground-state and fundamental excitation tunneling splittings in the formic acid dimer and its deuterium isotopologues. These are achieved with a multidimensional scheme developed by us, in which the saddle-point normal coordinates are chosen, the basis functions are customized for the proton transfer process, and the preconditioned inexact spectral transform method is used to solve the resultant eigenvalue problem. Our computational results are in excellent agreement with the most recent experiments (Zhang et al., 2017; Li et al., 2019).

Exact bound rovibrational spectra of the neon tetramer

Exact quantum dynamics calculations are performed for the bound rovibrational states of the neon tetramer (Ne₄) in its ground electronic state, using pair-wise Lennard-Jones potentials and the ScalIT suite of parallel codes. The vibrational states separate into a low-lying group mostly localized to a single potential well and a higher-energy delocalized group lying above the isomerization threshold-with such a structure serving as a testament to the "intermediate" quantum nature of the Ne₄ system. To accurately and efficiently represent both groups of states, the phase-space optimized discrete variable representation (PSO-DVR) approach was used, as implemented in the ScalIT code. The resultant 1D PSO effective potentials also shed significant light on the quantum dynamics of the system. All vibrational states were computed well up into the isomerization band and labeled up to the classical isomerization threshold-defined as the addition of the classical energy of a single bond, ϵ = 24.7 cm^-1, to the quantum ground state energy. Rovibrational energy levels for all total angular momentum values in the range J = 1-5 were also computed, treating all Coriolis coupling exactly.

Quantum Mechanical Investigation of Mode-Specific Tunneling upon Fundamental Excitation in Malonaldehyde

We present a quantum mechanical study of mode-specific tunneling upon fundamental excitation in malonaldehyde with a multidimensional theory that utilizes the saddle-point normal coordinates. We find that a ring-deformation normal mode is as essential as the well-known imaginary-frequency normal mode in the multidimensional investigation. The changes in tunneling splittings upon fundamental excitation are calculated. The results are competitive with those from a recently developed mixed classical-quantum method. Moreover, the results are qualitatively consistent with experiment for about half of all the modes.

Mode-Specific Tunneling Splittings for a Sequential Double-Hydrogen Transfer Case: An Accurate Quantum Mechanical Scheme

We present the first accurate quantum dynamics calculations of mode-specific tunneling splittings in a sequential double-hydrogen transfer process. This is achieved in the vinylidene-acetylene system, the simplest molecular system of this kind, and by large-scale parallel computations with an efficient theoretical scheme developed by us. In our scheme, basis functions are customized for the hydrogen transfer process; a 4-dimensional basis contraction strategy is combined with the preconditioned inexact spectral transform method; efficient parallel implementation is achieved. Mode-specific permutation tunneling splittings of vinylidene states are reported and tremendous mode-specific promotion effects are revealed; in particular, the CH2 rock mode enhances the ground-state splitting by a factor of 10(3). We find that the ground-state vinylidene has a reversible-isomerization time of 622 ps, much longer than all previous estimates. Our calculations also shed light on the importance of the deep intermediate well and vibrational excitation in the double-hydrogen transfer processes.

ROVIBRATIONAL BOUND STATES OF THE Ar2Ne COMPLEX

We report exact quantum dynamics calculations of the eigenstate energy levels for the bound rovibrational states of the Ar2Ne complex, across the range of J values for which such states are observed (J = 0–35). All calculations have been carried out using the ScalIT suite of parallel codes. These codes employ a combination of highly efficient methods, including phase-space optimized discrete variable representation, optimal separable basis, and preconditioned inexact spectral transform (PIST) methods, together with an effective massive parallelization scheme. The Ar2Ne energy levels were computed using a pair-wise Aziz potential plus a three-body correction, in Jacobi co-ordinates. Effective potentials for the radial co-ordinates are constructed, which reveal important physical insight into the two distinct dissociation pathways, Ar2Ne → NeAr + Ar and Ar2Ne → Ar2 + Ne . A calculation of the bound vibrational (J = 0) levels, computed using the Tang–Toennies potential, is also performed for comparison with results from the previous literature.

QUANTUM DYNAMICAL CALCULATION OF ALL ROVIBRATIONAL STATES OF HO2 FOR TOTAL ANGULAR MOMENTUM J = 0–10

The energy levels and wavefunctions for all rovibrational bound states of HO2 are systematically computed, for all total angular momentum values J = 0–10. The calculations are performed using ScalIT, a suite of software modules designed to enable quantum dynamics and related calculations to be performed on massively parallel computing architectures. This is the first-ever application of ScalIT to a real (and very challenging) molecular application. The codes, and in particular, the algorithms (optimal separable basis, preconditioned inexact spectral transform, phase space optimized discrete variable representation basis) are so efficient that in fact, the entire calculation can be performed on a single CPU — although parallel scalability over a small number of CPUs is also evaluated, and found to be essentially perfect in this regime. For the lowest 11 vibrational states, the rotational levels for J = 0–10 fit fairly well to a rigid rotor model, with all vibrational-state-dependent rotational constants, B eff (v), close to values obtained from a previous calculation for J = 0 and 1 [J Chem Phys107:2705, 1997]. However, comparatively larger discrepancies with the rigid-rotor model are found at the higher J values, manifesting in the observed K-splitting (along the O–O bond) of rovibrational levels. This supports earlier work [J Chem Phys113:11055, 2000] suggesting that Coriolis coupling is quite important for this system.

USING DISCRETE VARIABLE REPRESENTATION PATH INTEGRAL MONTE CARLO WITH METROPOLIS SAMPLING TO COMPUTE GROUND STATE WAVEFUNCTIONS

The discrete variable representation (DVR) matrix dynamics formulation of the path integral Monte Carlo (PIMC) method, implemented numerically in a way that enables Metropolis sampling to be employed, is proposed as a means of computing ground state quantum wavefunctions. A key advantage of the DVR-PIMC approach is that customized marginal potentials may be employed, leading to significantly larger PIMC time step sizes, and substantial reductions in computational (CPU) effort. An additional key advantage of the present implementation is that the DVR provides a natural set of interpolant functions that can be used for accurate interpolation and extrapolation of function and tensor quantities away from predefined grid points. The new method is applied here to compute the ground state wavefunction of a model one degree-of-freedom (1 DOF) Morse oscillator system. A one-to-two order-of-magnitude reduction in CPU effort is observed, in comparison with a conventional PIMC simulation. The generalization for many DOFs is straightforward, and expected to result in even greater performance enhancement.

THE EFFECT OF VIBRATIONAL EXCITATION OF THE REACTION O(3P) + HCl → OH + Cl FOR THE 3A″ ELECTRONIC STATES

Using the quasi-classical trajectory (QCT) method, this paper has explored the product rotational polarization for the reaction O (3 P ) + HCl → OH + Cl on the 3 A ″ potential energy surface constructed by Ramachandran B et al. (Ramachandran B et al., J. Chem. Phys.119: 9590, 2003). The distributions of product polarization P(θr),P(φr) and the generalized polarization-dependent differential cross-sections (2π/σ)(dσ00/dσt) and (2π/σ)(dσ22+/dωt) have been calculated. The results indicate that the vibrational excitation of HCl has a considerable influence on the distribution of the k-j′ correlation and the k-k′-j′ correlation.

Quantum dynamics of hydrogen interacting with single-walled carbon nanotubes: multiple H-atom adsorbates

In a previous paper [J. L. McAfee and B. Poirier, J. Chem. Phys. 130, 064701 (2009)], using spin-polarized density functional theory (DFT), the authors reported a binding energy of 0.755 eV, for a single hydrogen atom adsorbed on a pristine (unrelaxed) (5,5) single-walled carbon nanotube (SWNT) substrate. A full three-dimensional (3D) potential energy surface (PES) for the SWNT-H system was also developed, and used in a quantum dynamics calculation to compute all rovibrational bound states, and associated equatorial and longitudinal adsorbate migration rates. A highly pronounced preference for the latter migration pathway at ambient temperatures was observed. In this work, we extend the aforementioned study to include multiple H-atom adsorbates. Extensive DFT calculations are performed, in order to ascertain the most relevant dynamical pathways. For two adsorbates, the SWNT-H-H system is found to exhibit highly site-specific binding, as well as long-range correlation and pronounced binding energy enhancement. The latter effect is even more pronounced in the full-hydrogenation limit, increasing the per-adsorbate binding energy to 2.6 eV. To study migration dynamics, a single-hole model is developed, for which the binding energy drops to 2.11 eV. A global 3D PES is developed for the hole migration model, using 40 radial × 18 cylindrical ab initio geometries, fit to a Fourier basis with radially dependent expansion coefficients (rms error 4.9 meV). As compared with the single-adsorbate case, the hole migration PES does not exhibit separate chemisorption and physisorption wells. The barrier to longitudinal migration is also found to be much lower. Quantum dynamics calculations for all rovibrational states are then performed (using a mixed spectral basis/phase-space optimized discrete variable representation), and used to compute longitudinal migration rates. Ramifications for the use of SWNTs as potential hydrogen storage materials are discussed.

Variational Calculation of Specific, Highly Excited Vibrational States in DFCO: Comparison with Experimental Data

A stimulated emission pumping spectra of jet-cooled DFCO performed by Crane et al. (J. Mol. Spectrosc. 1997, 183, 273) has provided a great number of ro-vibrational lines up to 9000 cm(-1) of excitation energy. By combining a Jacobi-Wilson (JW) approach with a Davidson scheme, we calculate the lines provided by the experiment up to 9000 cm(-1) using an ab initio global potential energy surface (PES) developed by Kato et al. (J. Chem. Phys. 1997, 107, 6114). Comparisons between experimental and calculated data provide a critical test of the quality of the PES used. We show that the variational calculated energies can be efficiently corrected by taking into account the error observed for the A' fundamental transitions nu(i) (i = 1, ..., 5) and the first overtone 2nu(6). A detailed analysis of the eigenstates obtained by the calculation allows one to quantify the coupling between the different modes. Such an information is essential to understand and predict the energy flow through a DFCO molecule that is initially excited.

Improving the accuracy of Weyl-Heisenberg wavelet and symmetrized Gaussian representations using customized phase-space-region operators

A particular basis set method developed by one of the authors, involving maximally localized orthogonal Weyl-Heisenberg wavelets (or "weylets") and a phase space truncation scheme, has been successfully applied to exact quantum calculations for many degrees of freedom (DOF's) [B. Poirier and A. Salam, J. Chem. Phys. 121, 1740 (2004)]. However, limitations in accuracy arise in the many-DOF case, owing to memory limits on conventional computers. This paper addresses this accuracy limitation by introducing phase space region operators (PSRO's) that customize individual weylet basis functions for the problem of interest. The construction of the PSRO's is straightforward, and does not require a priori knowledge of the desired eigenstates. The PSRO, when applied to weylets, as well as to simple phase space Gaussian basis functions, exhibits remarkable improvements in accuracy, reducing computed eigenvalue errors by orders of magnitude. The method is applied to various model systems at varying DOF's.

Rovibrational spectroscopy calculations of neon dimer using a phase space truncated Weyl-Heisenberg wavelet basis

In a series of earlier articles [B. Poirier J. Theor. Comput. Chem. 2, 65 (2003); B. Poirier and A. Salam J. Chem. Phys. 121, 1690 (2004); B. Poirier and A. Salam J. Chem. Phys. 121, 1740 (2004)], a new method was introduced for performing exact quantum dynamics calculations in a manner that formally defeats exponential scaling with system dimensionality. The method combines an optimally localized, orthogonal Weyl-Heisenberg wavelet basis set with a simple phase space truncation scheme, and has already been applied to model systems up to 17 degrees of freedom (DOF's). In this paper, the approach is applied for the first time to a real molecular system (neon dimer), necessitating the development of an efficient numerical scheme for representing arbitrary potential energy functions in the wavelet representation. All bound rovibrational energy levels of neon dimer are computed, using both one DOF radial coordinate calculations and a three DOF Cartesian coordinate calculation. Even at such low dimensionalities, the approach is found to be competitive with another state-of-the-art method applied to the same system [J. Montgomery and B. Poirier J. Chem. Phys. 119, 6609 (2003)].

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.

Fully Quantum Rovibrational Calculation of the He(H2) Bound and Resonance States

In a recent paper [J. Chem. Phys. 2005, 122, 124318], a full-dimensional quantum method, designed to efficiently compute the rovibrational states of triatomic systems with long-range interactions, was applied to the benchmark Li-(H2) ion-molecule system. The method incorporates several key features in order to accurately represent the rovibrational Hamiltonian using only modestly sized basis sets: (1) exact analytical treatment of Coriolis coupling; (2) a single bend-angle basis for all rotational states; (3) phase space optimization of the vibrational basis; (4) G(4) symmetry adaptation of the rovibrational basis. In this paper, the same methodology is applied for the first time to a van der Waals complex system, He(H2). As in the Li-(H2) study, all of the rovibrational bound states, and a number of resonance states, are computed to very high accuracy (1/10,000 of a wavenumber or better). Three different isotopologues are considered, all of which are found to have a single bound state with a very low binding energy. Several extremely long-lived Feshbach resonances are also reported.

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.

Improving the calculation of rovibrational spectra of five-atom molecules with three identical atoms by using a C3υ(G6) symmetry-adapted grid: Applied to CH3D and CHD3

In this paper we report two improvements on the approach we have used to compute rovibrational levels of methane and apply the new ideas to calculate rovibrational levels of two methane isotopomers CH3D and CHD3. Both improvements make the bend calculation better. The first improvement is a G6-invariant (or C3upsilon-invariant) grid which is designed such that each point on the grid is mapped to another point on the grid by any of the G6 operations. The second improvement is the use of fast Fourier transform (FFT) to compute the bend potential matrix-vector products. The FFT matrix-vector product is about three and ten times faster than the previous sequential summation method for the J=0 and J>0 cases, respectively. The calculated J=1 rovibrational levels of CH3D and CHD3 on the Schwenke and Partridge [Spectrochim. Acta, Part A 57, 887 (2001)] ab initio potential are in good agreement (within 6 cm(-1) for the levels up to 3000 cm(-1)) with the experimental data. The agreement is even better (within 0.1 cm(-1) for the levels up to 6000 cm(-1)) if the associated J=0 energies are subtracted.

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.

Theoretical and experimental studies of the infrared rovibrational spectrum of He2–N2O

Rovibrational spectra of the He(2)-N(2)O complex in the nu(1) fundamental band of N(2)O (2224 cm(-1)) have been observed using a tunable infrared laser to probe a pulsed supersonic jet expansion, and calculated using five coordinates that specify the positions of the He atoms with respect to the NNO molecule, a product basis, and a Lanczos eigensolver. Vibrational dynamics of the complex are dominated by the torsional motion of the two He atoms on a ring encircling the N(2)O molecule. The resulting torsional states could be readily identified, and they are relatively uncoupled to other He motions up to at least upsilon(t) = 7. Good agreement between experiment and theory was obtained with only one adjustable parameter, the band origin. The calculated results were crucial in assigning many weaker observed transitions because the effective rotational constants depend strongly on the torsional state. The observed spectra had effective temperatures around 0.7 K and involved transitions with J < or =3, with upsilon(t) = 0 and 1, and (with one possible exception) with Deltaupsilon(t)=0. Mixing of the torsion-rotation states is small but significant: some transitions with Deltaupsilon(t) not equal 0 were predicted to have appreciable intensity even assuming that the dipole transition moment coincides perfectly with the NNO axis. One such transition was tentatively assigned in the observed spectra, but confirmation will require further work.

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.

How to choose one-dimensional basis functions so that a very efficient multidimensional basis may be extracted from a direct product of the one-dimensional functions: Energy levels of coupled systems with as many as 16 coordinates

In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the accuracy of computed energy levels. Two basic types of 1D basis functions are in common use: eigenfunctions of 1D Hamiltonians and discrete variable representation (DVR) functions. Both have advantages and disadvantages. The 1D functions we propose are intermediate between the 1D eigenfunction functions and the DVR functions. If the coupling is very weak, they are very nearly 1D eigenfunction functions. As the strength of the coupling is increased they resemble more closely DVR functions. We assess the usefulness of our basis by applying it to model 6D, 8D, and 16D Hamiltonians with various coupling strengths. We find approximately linear scaling.

Accurate quantum calculation of the bound and resonant rovibrational states of Li−(H2)

In a recent paper [B. Poirier, Chem. Phys. 308, 305 (2005)] a full-dimensional quantum method for computing the rovibrational dynamics of triatomic systems was presented, incorporating three key features: (1) exact analytical treatment of Coriolis coupling, (2) three-body "effective potential," and (3) a single bend angle basis for all rotational states. In this paper, these ideas are applied to the Li-(H2) electrostatic complex, to compute all of the rovibrational bound state energies, and a number of resonance energies and widths, to very high accuracy (thousandths of a wave number). This application is very challenging, owing to the long-range nature of the interaction and to narrow level spacings near dissociation. Nevertheless, by combining the present method with a G4 symmetry-adapted phase-space-optimized representation, only modest basis sizes are required for which the matrices are amenable to direct diagonalization. Several new bound levels are reported, as compared with a previous calculation [D. T. Chang, G. Surratt, G. Ristroff, and G. I. Gellene, J. Chem. Phys. 116, 9188 (2002)]. The resonances exhibit a clear-cut separation into shape and Feshbach varieties, with the latter characterized by extremely long lifetimes (microseconds or longer).

Accurate and highly efficient calculation of the highly excited pure OH stretching resonances of O(1D)HCl, using a combination of methods

Accurate calculation of the energies and widths of the resonances of HOCl--an important intermediate in the O(1D)HCl reactive system--poses a challenging benchmark for computational methods. The need for very large direct product basis sets, combined with an extremely high density of states, results in difficult convergence for iterative methods. A recent calculation of the highly excited OH stretch mode resonances using the filter diagonalization method, for example, required 462,000 basis functions, and 180,000 iterations. In contrast, using a combination of new methods, we are able to compute the same resonance states to higher accuracy with a basis less than half the size, using only a few hundred iterations-although the CPU cost per iteration is substantially greater. Similar performance enhancements are observed for calculations of the high-lying bound states, as reported in a previous paper [J. Theo. Comput. Chem. 2, 583 (2003)].

Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. III. Representations and calculations

In a previous paper [J. Theo. Comput. Chem. 2, 65 (2003)], one of the authors (B.P.) presented a method for solving the multidimensional Schrodinger equation, using modified Wilson-Daubechies wavelets, and a simple phase space truncation scheme. Unprecedented numerical efficiency was achieved, enabling a ten-dimensional calculation of nearly 600 eigenvalues to be performed using direct matrix diagonalization techniques. In a second paper [J. Chem. Phys. 121, 1690 (2004)], and in this paper, we extend and elaborate upon the previous work in several important ways. The second paper focuses on construction and optimization of the wavelength functions, from theoretical and numerical viewpoints, and also examines their localization. This paper deals with their use in representations and eigenproblem calculations, which are extended to 15-dimensional systems. Even higher dimensionalities are possible using more sophisticated linear algebra techniques. This approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where differential equations are involved.

Quantum dynamics calculations using symmetrized, orthogonal Weyl-Heisenberg wavelets with a phase space truncation scheme. II. Construction and optimization

In this paper, we extend and elaborate upon a wavelet method first presented in a previous publication [B. Poirier, J. Theo. Comput. Chem. 2, 65 (2003)]. In particular, we focus on construction and optimization of the wavelet functions, from theoretical and numerical viewpoints, and also examine their localization properties. The wavelets used are modified Wilson-Daubechies wavelets, which in conjunction with a simple phase space truncation scheme, enable one to solve the multidimensional Schrodinger equation. This approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where differential equations are involved.