Additions to the class of symmetric-antisymmetric multiwavelets: derivation and use as quantum basis functions

Multiwavelet bases have been shown recently to apply to a variety of quantum problems. There are, however, only a few multiwavelet families that have been defined to date. Chui-Lian-type symmetric and antisymmetric multiwavelets are derived here that equal and exceed the polynomial interpolating power of previously available examples. Adaptations to domain edges are made with a view to use in curvilinear coordinate molecular calculations. The new highest-order multiwavelet family is shown to provide uniformly better performance for (i) basis representation of terms such as 1r(2) in near approach to the singularity at r=0 and (ii) eigenvalue calculation of a bending Hamiltonian taken from a curvilinear model of the ground-state vibrations of nitrosyl chloride.     

Multidimensional reactive scattering with quantum trajectories: dynamics with 50-200 vibrational modes

The dynamics of ensembles containing thousands of quantum trajectories are studied for multidimensional systems undergoing reactive scattering. The Hamiltonian and equations of motion are formulated in curvilinear reaction path coordinates, for the case of a planar (zero-torsion) reaction path. In order to enhance the computational efficiency, an improved least squares fitting procedure is introduced. This scheme involves contracted basis sets and the use of inner and outer stencils around points where fitting is performed. This method is applied to reactive systems with 50-200 harmonic vibrational modes which are coupled to motion along the reaction coordinate. Dynamical results, including trajectory evolution and time-dependent reaction probabilities, are presented and power law scaling of computation time with the number of vibrational modes is described.     

Multiscale quantum propagation using compact-support wavelets in space and time

Orthogonal compact-support Daubechies wavelets are employed as bases for both space and time variables in the solution of the time-dependent Schrodinger equation. Initial value conditions are enforced using special early-time wavelets analogous to edge wavelets used in boundary-value problems. It is shown that the quantum equations may be solved directly and accurately in the discrete wavelet representation, an important finding for the eventual goal of highly adaptive multiresolution Schrodinger equation solvers. While the temporal part of the basis is not sharp in either time or frequency, the Chebyshev method used for pure time-domain propagations is adapted to use in the mixed domain and is able to take advantage of Hamiltonian matrix sparseness. The orthogonal separation into different time scales is determined theoretically to persist throughout the evolution and is demonstrated numerically in a partially adaptive treatment of scattering from an asymmetric Eckart barrier.     

Two-dimensional quantum propagation using wavelets in space and time

A recent method for solving the time-dependent Schrodinger equation has been developed using expansions in compact-support wavelet bases in both space and time [H. Wang et al., J. Chem. Phys. 121, 7647 (2004)]. This method represents an exact quantum mixed time-frequency approach, with special initial temporal wavelets used to solve the initial value problem. The present work is a first extension of the method to multiple spatial dimensions applied to a simple two-dimensional (2D) coupled anharmonic oscillator problem. A wavelet-discretized version of norm preservation for time-independent Hamiltonians discovered in the earlier one-dimensional investigation is verified to hold as well in 2D and, by implication, in higher numbers of spatial dimensions. The wavelet bases are not restricted to rectangular domains, a fact which is exploited here in a 2D adaptive version of the algorithm.     

An efficient route to thermal rate constants in reduced dimensional quantum scattering simulations: applications to the abstraction of hydrogen from alkanes

We present an efficient approach to the determination of two-dimensional potential energy surfaces for use in quantum reactive scattering simulations. Our method involves first determining the minimum energy path (MEP) for the reaction by means of an ab initio intrinsic reaction coordinate calculation. This one-dimensional potential is then corrected to take into account the zero point energies of the spectator modes. These are determined from Hessians in curvilinear coordinates after projecting out the modes to be explicitly treated in quantum scattering calculations. The final (1+1)-dimensional potential is constructed by harmonic expansion about each point along the MEP before transforming the whole surface to hyperspherical coordinates for use in the two-dimensional scattering simulations. This new method is applied to H-atom abstraction from methane, ethane and propane. For the latter, both reactive channels (producing i-C(3)H(7) or n-C(3)H(7)) are investigated. For all reactions, electronic structure calculations are performed using an efficient, explicitly correlated, coupled cluster methodology (CCSD(T)-F12). Calculated thermal rate constants are compared to experimental and previous theoretical results.     

Multilevel augmentation method with wavelet bases for singularly perturbed problem

Multilevel augmentation method with wavelet bases is demonstrated to show as the fast technique for solving singularly perturbed problems. Linear and quadratic wavelet bases are employed for constructing the full form of matrix system. To reduce the size of matrix coefficients, the multilevel augmented technique is applied at each current basis level. It is found that the multilevel augmentation method is faster than the standard multilevel method at the same order of accuracy. Convergent rates for linear and quadratic bases are 2 and 4 respectively. By the application of wavelet bases, numerical accuracy can be easily improved by increasing just desired levels in the multilevel augmentation process.     

Using semiclassical surface hopping for coupled partial wave calculations on systems with non-spherically symmetric potentials

It is shown that a semiclassical surface hopping (SH) approach provides a simple and efficient method for scattering calculations with non-spherically symmetric potentials. The calculations are performed by expanding the wave function in an angular momentum state basis. Since the potential is not spherically symmetric, the different angular states are coupled. The semiclassical SH method, which is typically used for problems with coupled electronic states, can, in principle, be employed for any coupled state problem. The particular SH method employed is known to provide highly accurate results for coupled electronic state problems. The method is tested on model two angular state problems using potential surfaces and couplings arising from a non-spherically symmetric scattering problem. The results for these model problems are in excellent agreement with exact quantum calculations. Full calculations, which are converged with regard to the number of angular basis states, are also performed for the non-spherically symmetric problem. It is shown that an approximation to the surface hopping amplitudes that simplifies the numerical implementation of the method provides results in excellent agreement with the full surface hopping calculation.     

Vibrational dynamics of DNA. I. Vibrational basis modes and couplings

Carrying out density functional theory calculations of four DNA bases, base derivatives, Watson-Crick (WC) base pairs, and multiple-layer base pair stacks, we studied vibrational dynamics of delocalized modes with frequency ranging from 1400 to 1800 cm(-1). These modes have been found to be highly sensitive to structure fluctuation and base pair conformation of DNA. By identifying eight fundamental basis modes, it is shown that the normal modes of base pairs and multilayer base pair stacks can be described by linear combinations of these vibrational basis modes. By using the Hessian matrix reconstruction method, vibrational coupling constants between the basis modes are determined for WC base pairs and multilayer systems and are found to be most strongly affected by the hydrogen bonding interaction between bases. It is also found that the propeller twist and buckle motions do not strongly affect vibrational couplings and basis mode frequencies. Numerically simulated IR spectra of guanine-cytosine and adenine-thymine bases pairs as well as of multilayer base pair stacks are presented and described in terms of coupled basis modes. It turns out that, due to the small interlayer base-base vibrational interactions, the IR absorption spectrum of multilayer base pair system does not strongly depend on the number of base pairs.     

Simulation of inversion motion and N-H stretching overtone spectra of aniline

A curvilinear internal coordinate Hamiltonian is used to simulate the N-H stretching overtone spectra and the associated inversion splittings in aniline. A simple local mode type model is applied to the N-H stretching and H-N-H bending modes. Geometric algebra is employed to derive the kinetic energy operator for the large amplitude inversion motion. Electronic structure calculations at the Moller-Plesset second order perturbation theory and correlation consistent aug-cc-pVTZ basis set level are used to obtain model parameters, some of which have been optimized with the least-squares method using experimental vibrational term values as data. The observed N-H stretching overtone vibrational levels and the inversional tunneling splittings are well reproduced with our approach.     

Spectroscopic consequences of a mixed valence excited state: quantitative treatment of a dihydrazine diradical dication

A model for the quantitative treatment of molecular systems possessing mixed valence excited states is introduced and used to explain observed spectroscopic consequences. The specific example studied in this paper is 1,4-bis(2-tert-butyl-2,3-diazabicyclo[2.2.2]oct-3-yl)-2,3,5,6-tetramethylbenzene-1,4-diyl dication. The lowest energy excited state of this molecule arises from a transition from the ground state where one positive charge is associated with each of the hydrazine units, to an excited state where both charges are associated with one of the hydrazine units, that is, a Hy-to-Hy charge transfer. The resulting excited state is a Class II mixed valence molecule. The electronic emission and absorption spectra, and resonance Raman spectra, of this molecule are reported. The lowest energy absorption band is asymmetric with a weak low-energy shoulder and an intense higher energy peak. Emission is observed at low temperature. The details of the absorption and emission spectra are calculated for the coupled surfaces by using the time-dependent theory of spectroscopy. The calculations are carried out in the diabatic basis, but the nuclear kinetic energy is explicitly included and the calculations are exact quantum calculations of the model Hamiltonian. Because the transition involves the transfer of an electron from the hydrazine on one side of the molecule to the hydrazine on the other side and vice versa, the two transitions are antiparallel and the transition dipole moments have opposite signs. Upon transformation to the adiabatic basis, the dipole moment for the transition to the highest energy adiabatic surface is nonzero, but that for the transition to the lowest surface changes sign at the origin. The energy separation between the two components of the absorption spectrum is twice the coupling between the diabatic basis states. The bandwidths of the electronic spectra are caused by progressions in totally symmetric modes as well as progressions in the modes along the coupled coordinate. The totally symmetric modes are modeled as displaced harmonic oscillators; the frequencies and displacements are determined from resonance Raman spectra. The absorption, emission, and Raman spectra are fit simultaneously with one parameter set. The coupling in the excited electronic state H(ab)(ex) is 2000 cm(-1). Excited-state mixed valence is expected to be an important contributor to the electronic spectra of many organic and inorganic compounds. The energy separations and relative intensities enable the excited-state properties to be calculated as shown in this paper, and the spectra provide new information for probing and understanding coupling in mixed valence systems.     

Analytical treatment of Coriolis coupling for three-body systems

In a previous article [J. Chem. Phys. 108 (1998) 5216], an efficient method was presented for performing “exact” quantum calculations for the three-body rovibrational Hamiltonian, within the helicity-conserving approximation. This approach makes use of a certain three-body “effective potential,” enabling the same bend angle basis set to be employed for all values of the rotational quantum numbers, J, K and M. In the present work, the method is extended to incorporate Coriolis coupling, for which the relevant matrix elements are derived exactly. These can be used to solve the full three-body rovibrational problem, in the standard Jacobi coordinate vector embedding. Generalization of the method for coupled kinetic energy operators arising from other coordinate systems, embeddings, and/or system sizes, is also discussed.     

Wavelet algorithm for solving integral equations of molecular liquids. A test for the reference interaction site model

A new efficient method is developed for solving integral equations based on the reference interaction site model (RISM) of molecular liquids. The method proposes the expansion of site-site correlation functions into the wavelet series and further calculations of the approximating coefficients. To solve the integral equations we have applied the hybrid scheme in which the coarse part of the solution is calculated by wavelets with the use of the Newton-Raphson procedure, while the fine part is evaluated by the direct iterations. The Coifman 2 basis set is employed for the wavelet treatment of the coarse solution. This wavelet basis set provides compact and accurate approximation of site-site correlation functions so that the number of basis functions and the amplitude of the fine part of solution decrease sufficiently with respect to those obtained by the conventional scheme. The efficiency of the method is tested by calculations of SPC/E model of water. The results indicated that the total CPU time to obtain solution by the proposed procedure reduces to 20% of that required for the conventional hybrid method.     

Monge-Ampere grids and the multidimensional mapped Fourier method

The efficiency of a numerical method can be greatly improved by combining it with coordinate transformations tailored to a given problem. This is the basis for the mapped Fourier methods. However, obtaining "good" coordinate transformations is a major obstacle for this approach in multidimensions. Here, we calculate coordinate transformations based on solving the Monge-Ampere equation. These transformations are combined in the mapped Fourier method and applied to Schrodinger's equation in multidimensions. Dramatic improvements in accuracy compared to the standard Fourier method were observed in eigenvalue calculations for two-dimensional systems. This work indicates that the Monge-Ampere equation may serve as a useful tool for constructing efficient representations for problems in computational quantum mechanics and other fields.     

Anharmonic Force Fields and Perturbation Theory in the Interpretation of Vibrational Spectra of Polyatomic Molecules

The problem of describing real vibrational spectra of large molecules in terms of perturbation theory is considered. Equations necessary for presenting theoretical anharmonic force fields in various coordinate systems (Cartesian, normal, and internal curvilinear) are discussed. A review of second-order perturbation theory equations necessary for calculating certain spectroscopic values (anharmonicity constants, rotational-vibrational interaction, etc.) is given. A scheme for including resonances based on the construction of the interaction matrix between vibrational transitions of various types is described. This scheme can be used as a basis for anharmonic calculations of vibrations of medium-sized molecules.     

Wavelet based in eigenvalue problems in quantum mechanics

The use of Daubechies' compact support wavelets in quantum mechanical eigenvalue problems is investigated. It is shown that these orthogonal multiresolution functions provide an efficient basis for systems in which the potentials vary strongly in different regions.     

Breaking the carbon dimer: the challenges of multiple bond dissociation with full configuration interaction quantum Monte Carlo methods

The full configuration interaction quantum Monte Carlo (FCIQMC) method, as well as its "initiator" extension (i-FCIQMC), is used to tackle the complex electronic structure of the carbon dimer across the entire dissociation reaction coordinate, as a prototypical example of a strongly correlated molecular system. Various basis sets of increasing size up to the large cc-pVQZ are used, spanning a fully accessible N-electron basis of over 10(12) Slater determinants, and the accuracy of the method is demonstrated in each basis set. Convergence to the FCI limit is achieved in the largest basis with only O[10(7)] walkers within random errorbars of a few tenths of a millihartree across the binding curve, and extensive comparisons to FCI, CCSD(T), MRCI, and CEEIS results are made where possible. A detailed exposition of the convergence properties of the FCIQMC methods is provided, considering convergence with elapsed imaginary time, number of walkers and size of the basis. Various symmetries which can be incorporated into the stochastic dynamic, beyond the standard abelian point group symmetry and spin polarisation are also described. These can have significant benefit to the computational effort of the calculations, as well as the ability to converge to various excited states. The results presented demonstrate a new benchmark accuracy in basis-set energies for systems of this size, significantly improving on previous state of the art estimates.     

Alternative common bases and signal compression for wavelets application in chemometrics

Representation or compression of data sets in the wavelet space is usually performed to retain the maximum variance of the original or pretreated data, like in the compression by means of principal components. In order to represent together a number of objects in the wavelet space, a common basis is required, and this common basis is usually obtained by means of the variance spectrum or of the variance wavelet tree. In this study, the use of alternative common bases is suggested, both for classification and regression problems. In the case of classification or class-modeling, the suggested common bases are based on the spectrum of the Fisher weights (a measure of the between-class to within-class variance ratio) or on the spectrum of the SIMCA discriminant weights. In the case of regression, the suggested common bases are obtained by the correlation spectrum (the correlation coefficients of the predictor variables with a response variable) or by the PLS (Partial Least Squares regression) importance of the predictors (the product between the absolute value of the regression coefficient of the predictor in the PLS model and its standard deviation). Other alternative strategies apply the Gram–Schmidt supervised orthogonalization to the wavelet coefficients. The results indicate that, both in classification and regression, the information retained after compression in the wavelets space can be more efficient than that retained with a common basis obtained by variance.     

Mixed time slicing in path integral simulations

A simple and efficient scheme is presented for using different time slices for different degrees of freedom in path integral calculations. This method bridges the gap between full quantization and the standard mixed quantum-classical (MQC) scheme and, therefore, still provides quantum mechanical effects in the less-quantized variables. Underlying the algorithm is the notion that time slices (beads) may be "collapsed" in a manner that preserves quantization in the less quantum mechanical degrees of freedom. The method is shown to be analogous to multiple-time step integration techniques in classical molecular dynamics. The algorithm and its associated error are demonstrated on model systems containing coupled high- and low-frequency modes; results indicate that convergence of quantum mechanical observables can be achieved with disparate bead numbers in the different modes. Cost estimates indicate that this procedure, much like the MQC method, is most efficient for only a relatively few quantum mechanical degrees of freedom, such as proton transfer. In this regime, however, the cost of a fully quantum mechanical simulation is determined by the quantization of the least quantum mechanical degrees of freedom.     

An economic prediction of refinement coefficients in wavelet‐based adaptive methods for electron structure calculations

The wave function of a many electron system contains inhomogeneously distributed spatial details, which allows to reduce the number of fine detail wavelets in multiresolution analysis approximations. Finding a method for decimating the unnecessary basis functions plays an essential role in avoiding an exponential increase of computational demand in wavelet‐based calculations. We describe an effective prediction algorithm for the next resolution level wavelet coefficients, based on the approximate wave function expanded up to a given level. The prediction results in a reasonable approximation of the wave function and allows to sort out the unnecessary wavelets with a great reliability. © 2012 Wiley Periodicals, Inc.