A multidimensional discrete variable representation basis obtained by simultaneous diagonalization

Direct product basis functions are frequently used in quantum dynamics calculations, but they are poor in the sense that many such functions are required to converge a spectrum, compute a rate constant, etc. Much better, contracted, basis functions, that account for coupling between coordinates, can be obtained by diagonalizing reduced dimension Hamiltonians. If a direct product basis is used, it is advantageous to use discrete variable representation (DVR) basis functions because matrix representations of functions of coordinates are diagonal in the DVR. By diagonalizing matrices representing coordinates it is straightforward to obtain the DVR that corresponds to any direct product basis. Because contracted basis functions are eigenfunctions of reduced dimension Hamiltonians that include coupling terms they are not direct product functions. The advantages of contracted basis functions and the advantages of the DVR therefore appear to be mutually exclusive. A DVR that corresponds to contracted functions is unknown. In this paper we propose such a DVR. It spans the same space as a contracted basis, but in it matrix representations of coordinates are diagonal. The DVR basis functions are chosen to achieve maximal diagonality of coordinate matrices. We assess the accuracy of this DVR by applying it to model four-dimensional problems.     

Constant pressure ab initio molecular dynamics with discrete variable representation basis sets

The use of discrete variable representation (DVR) basis sets within ab initio molecular dynamics calculations allows the latter to be performed with converged energies and, more importantly, converged forces. In this paper, we show how to carry out ab initio molecular dynamics calculations in the isothermal-isobaric ensemble with fully flexible simulation boxes within the DVR basis set framework. In particular, we derive the appropriate DVR based expression for the pressure tensor when the electronic structure is represented using Kohn-Sham density functional theory, and we examine the convergence of this expression as a function of the basis set size. An illustrative example using 64 silicon atoms in a fully flexible box using a combination of the Martyna-Tobias-Klein [Martyna et al., J. Chem. Phys. 101, 4177 (1994)] and Car-Parrinello [Car and Parinello, Phys. Rev. Lett. 55, 2471 (1985)] algorithms is presented to demonstrate the efficacy of the approach.     

Distributed Gaussian discrete variable representation

A discrete variable representation (DVR) made from distributed Gaussians g_n(x) = e, (n = ?∞, …, ∞) and its infinite grid limit is described. The infinite grid limit of the distributed Gaussian DVR (DGDVR) reduces to the sinc function DVR of Colbert and Miller in the limit c → 0. The numerical performance of both finite and infinite grid DGDVRs and the sinc function DVR is compared. If a small number of quadrature points are taken, the finite grid DGDVR performs much better than both infinite grid DGDVR and sinc function DVR. The infinite grid DVRs lose accuracy due to the truncation error. In contrast, the sinc function DVR is found to be superior to both finite and infinite grid DGDVRs if enough grid points are taken to eliminate the truncation error. In particular, the accuracy of DGDVRs does not get better than some limit when the distance between Gaussians d goes to zero with fixed c, whereas the accuracy of the sinc function DVR improves very quickly as d becomes smaller, and the results are exact in the limit d → 0. An analysis of the performance of distributed basis functions to represent a given function is presented in a recent publication. With this analysis, we explain why the sinc function DVR performs better than the infinite grid DGDVR. The analysis also traces the inability of Gaussians to yield exact results in the limit d → 0 to the incompleteness of this basis in this limit. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Variational properties of the discrete variable representation: Discrete variable representation via effective operators

A variational finite basis representation/discrete variable representation (FBR/DVR) Hamiltonian operator has been introduced. By calculating its matrix elements exactly one obtains, depending on the choice of the basis set, either a variational FBR or a variational DVR. The domain of grid points on which the FBR/DVR is variational has been shown to consist of the subsets of the set of grid points one obtains by diagonalizing commuting variational basis representations of the coordinate operators. The variational property implies that the optimal of the subsets of a fixed number of points, i.e., the subset which gives the possible highest accuracy eigenpairs, gives the DVR of the smallest trace. The symmetry properties of the variational FBR/DVR Hamiltonian operator are analyzed and methods to incorporate symmetry into FBR/DVR calculations are discussed. It is shown how the Fourier-basis FBR/DVR suitable to solving periodic systems arise within the theory presented. Numerical examples are given to illustrate the theoretical results. The use of variational effective Hamiltonian and coordinate operators has been instrumental in this study. They have been introduced in a novel way by exploiting quasi-Hermiticity.     

Ab initio molecular dynamics with discrete variable representation basis sets: techniques and application to liquid water

Finite temperature ab initio molecular dynamics (AIMD), in which forces are obtained from "on-the-fly" electronic structure calculations, is a widely used technique for studying structural and dynamical properties of chemically active systems. Recently, we introduced an AIMD scheme based on discrete variable representation (DVR) basis sets, which was shown to have improved convergence properties over the conventional plane wave (PW) basis set [Liu,Y.; et al. Phys. Rev. B 2003, 68, 125110]. In the present work, the numerical algorithms for the DVR based AIMD scheme (DVR/AIMD) are provided in detail, and the latest developments of the approach are presented. The accuracy and stability of the current implementation of the DVR/AIMD scheme are tested by performing a simulation of liquid water at ambient conditions. The structural information obtained from the present work is in good agreement with the result of recent AIMD simulations with a PW basis set (PW/AIMD). Advantages of using the DVR/AIMD scheme over the PW/AIMD method are discussed. In particular, it is shown that a DVR/AIMD simulation of liquid water in the complete basis set limit is possible with a relatively small number of grid points.     

A coherent discrete variable representation method for multidimensional systems in physics

A coherent discrete variable representation (ZDVR) is proposed for constructing a multidimensional potential-optimized DVR basis. The multidimensional quadrature pivots are obtained by diagonalizing a complex coordinate operator matrix in a finite basis set, which is spanned by the lowest eigenstates of a two-dimensional reference Hamiltonian. Here a c-norm condition is used in the diagonalization procedure. The orthonormal eigenvectors define a collocation matrix connecting the localized ZDVR basis functions and the finite basis set. The method is applied to two vibrational models for computing the lowest bound states. Results show that the ZDVR method provides exponential convergence and accurate energies. Finally, a zeroth-order approximation method is also derived.     

A study of two-electron quantum dot spectrum using discrete variable representation method

A variational method called discrete variable representation is applied to study the energy spectra of two interacting electrons in a quantum dot with a three-dimensional anisotropic harmonic confinement potential. This method, applied originally to problems in molecular physics and theoretical chemistry, is here used to solve the eigenvalue equation to relative motion between the electrons. The two-electron quantum dot spectrum is determined then with a precision of at least six digits. Moreover, the electron correlation energies for various potential confinement parameters are investigated for singlet and triplet states. When possible, the present results are compared with the available theoretical values.     

Resonances of triatomic van der Waals molecules by the complex discrete variable representation

Summary The results of Light and co-workers [J. Chem. Phys. 85:4594 (1986); 86:3065 (1987); 92:2129 (1990)] for the Hamiltonian matrix of a triatomic van der Waals molecule in the discrete variable representation, DVR, is extended to complex-scaled Hamiltonians. As an illustrative numerical example theJ=1 resonances positions and widths of a van der Waals model system were obtained by the calculation of the complex-scaled Hamiltonian matrix in the DVR formalism.Supported in part by the Albert Einstein Research Fund, and the Fund for the Promotion of Research at the Technion     

Semi‐classical formulation of time‐dependent discrete variable representation method

In this article, we apply a novel time‐dependent discrete variable representation (TDDVR) method proposed by Barkakaty and Adhikari to investigate tunneling through an Eckart barrier. This semi‐classical method is theoretically rigorous and straightforward to implement. Among the TDDVR formulations, this report presents the first derivation of a rigorous form of quantum force (QF) for the present perspective. The validity of this semi‐classical approach is demanded based on the excellent agreement of the tunneling probability with the corresponding quantum results. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Excited States of weakly bound bosonic clusters: discrete variable representation and quantum Monte Carlo

We compare two approaches for the accurate calculation of the energy levels of weakly bound boson trimers. The first approach is based on correlation function Monte Carlo employing optimized trial functions, while the second approach is based on the discrete variable representation. A trimer with atoms of half the mass of neon is used as a test problem for benchmark calculations. The two approaches yield identical results, within error bars, for all the J = 0 energy levels below the dissociation threshold. The relative merits of the two techniques are discussed, and a perspective is given for extension to larger clusters.     

Vibrational energy levels with arbitrary potentials using the Eckart-Watson Hamiltonians and the discrete variable representation

An effective and general algorithm is suggested for variational vibrational calculations of N-atomic molecules using orthogonal, rectilinear internal coordinates. The protocol has three essential parts. First, it advocates the use of the Eckart-Watson Hamiltonians of nonlinear or linear reference configuration. Second, with the help of an exact expression of curvilinear internal coordinates (e.g., valence coordinates) in terms of orthogonal, rectilinear internal coordinates (e.g., normal coordinates), any high-accuracy potential or force field expressed in curvilinear internal coordinates can be used in the calculations. Third, the matrix representation of the appropriate Eckart-Watson Hamiltonian is constructed in a discrete variable representation, in which the matrix of the potential energy operator is always diagonal, whatever complicated form the potential function assumes, and the matrix of the kinetic energy operator is a sparse matrix of special structure. Details of the suggested algorithm as well as results obtained for linear and nonlinear test cases including H(2)O, H(3) (+), CO(2), HCNHNC, and CH(4) are presented.     

A discrete variable representation study of the dynamics of the double proton transfer in bicyclic oxalamidines

DFT electronic calculations are performed for the double proton transfer in bicyclic oxalamidines 2,2^′-bis-(3,4,5,6-tetrahydro-1,3-diazine) (OA6) and 2,2^′-bis-(4,5,6,7-tetrahydro-1,3-diazepine) (OA7). Bidimensional potential energy surfaces retaining the characteristics of the stationary points located in the whole potential are built to calculate the vibrational levels through the discrete variable representation (DVR) method. Zero-point tunneling splittings are similar for both systems but for a given energy there are much less tunneling doublets available in OA6, this being the key factor responsible for the different temperature dependence of the rate for the two systems.     

Application of discrete variable representation to planar H(2) (+) in strong xuv laser fields

We present an efficient and accurate grid method to study the strong field dynamics of planar H(2) (+) under Born-Oppenheimer approximation. After introducing the elliptical coordinates to the planar H(2) (+), we show that the Coulomb singularities at the nuclei can be successfully overcome so that both bound and continuum states can be accurately calculated by the method of separation of variables. The time-dependent Schro?dinger equation (TDSE) can be accurately solved by a two-dimensional discrete variable representation (DVR) method, where the radial coordinate is discretized with the finite-element discrete variable representation for easy parallel computation and the angular coordinate with the trigonometric DVR which can describe the periodicity in this direction. The bound states energies can be accurately calculated by the imaginary time propagation of TDSE, which agree very well with those computed by the separation of variables. We apply the TDSE to study the ionization dynamics of the planar H(2) (+) by short extreme ultra-violet (xuv) pulses, in which case the differential momentum distributions from both the length and the velocity gauge agree very well with those calculated by the lowest order perturbation theory.     

Using a nondirect product discrete variable representation for angular coordinates to compute vibrational levels of polyatomic molecules

In this paper we test a nondirect product discrete variable representation (DVR) method for solving the bend vibration problem and compare it with well-established direct product DVR and finite basis representation approaches.     

Optimal grids for generalized finite basis and discrete variable representations: definition and method of calculation

The method of optimal generalized finite basis and discrete variable representations (FBR and DVR) generalizes the standard, Gaussian quadrature grid-classical orthonormal polynomial basis-based FBR/DVR method to general sets of grid points and to general, nondirect product, and/or nonpolynomial bases. Here, it is shown how an optimal set of grid points can be obtained for an optimal generalized FBR/DVR calculation with a given truncated basis. Basis set optimized and potential optimized grids are defined. The optimized grids are shown to minimize a function of grid points derived by relating the optimal generalized FBR of a Hamiltonian operator to a non-Hermitian effective Hamiltonian matrix. Locating the global minimum of this function can be reduced to finding the zeros of a function in the case of one dimensional problems and to solving a system of D nonlinear equations repeatedly in the case of D>1 dimensional problems when there is an equal number of grid points and basis functions. Gaussian quadrature grids are shown to be basis optimized grids. It is demonstrated by a numerical example that an optimal generalized FBR/DVR calculation of the eigenvalues of a Hamiltonian operator with potential optimized grids can have orders of magnitude higher accuracy than a variational calculation employing the same truncated basis. Nevertheless, for numerical integration with the optimal generalized FBR quadrature rule basis optimized grids are the best among grids of the same number of points. The notions of Gaussian quadrature and Gaussian quadrature accuracy are extended to general, multivariable basis functions.     

Reduced dimension discrete variable representation study of cis-trans isomerization in the S1 state of C2H2

Isomerization between the cis and trans conformers of the S(1) state of acetylene is studied using a reduced dimension discrete variable representation (DVR) calculation. Existing DVR techniques are combined with a high accuracy potential energy surface and a kinetic energy operator derived from FG theory to yield an effective but simple Hamiltonian for treating large amplitude motions. The spectroscopic signatures of the S(1) isomerization are discussed, with emphasis on the vibrational aspects. The presence of a low barrier to isomerization causes distortion of the trans vibrational level structure and the appearance of nominally electronically forbidden A? (1)A(2)←X? (1)Σ(g)(+) transitions to vibrational levels of the cis conformer. Both of these effects are modeled in agreement with experimental results, and the underlying mechanisms of tunneling and state mixing are elucidated by use of the calculated vibrational wavefunctions.     

Calculating multidimensional discrete variable representations from cubature formulas

Finding multidimensional nondirect product discrete variable representations (DVRs) of Hamiltonian operators is one of the long standing challenges in computational quantum mechanics. The concept of a "DVR set" was introduced as a general framework for treating this problem by R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A. Mitchell, and B. Poirier (J. Chem. Phys. 2002, 116, 8691). We present a general solution of the problem of calculating multidimensional DVR sets whose points are those of a known cubature formula. As an illustration, we calculate several new nondirect product cubature DVRs on the plane and on the sphere with up to 110 points. We also discuss simple and potentially very useful finite basis representations (FBRs), based on general (nonproduct) cubatures. Connections are drawn to a novel view on cubature presented by I. Degani, J. Schiff, and D. J. Tannor (Num. Math. 2005, 101, 479), in which commuting extensions of coordinate matrices play a central role. Our construction of DVR sets answers a problem left unresolved in the latter paper, namely, the problem of interpreting as function spaces the vector spaces on which commuting extensions act.