Multidimensional time-dependent discrete variable representations in multiconfiguration Hartree calculations

In the multiconfiguration time-dependent Hartree (MCTDH) approach, the wave function is expanded in time-dependent basis functions, called single-particle functions, to increase the efficiency of the wave-packet propagation. The correlation discrete variable representation (CDVR) approach, which is based on a time-dependent discrete variable representation (DVR), can be employed to evaluate matrix elements of the potential energy. The efficiency of the MCTDH method can be further enhanced by using multidimensional single-particle functions. However, up to now the CDVR approach could not be used in MCTDH calculations employing multidimensional single-particle functions, since this would require a general multidimensional non-direct-product DVR scheme. Recently, Dawes and Carrington presented a practical scheme to implement general non-direct-product multidimensional DVRs [R. Dawes and T. Carrington, Jr., J. Chem. Phys. 121, 726 (2004)]. The present work utilizes their scheme in the MCTDH/CDVR approach. The accuracy is tested using the photodissociation of NOCl as example. The results show that the CDVR scheme based on multidimensional time-dependent DVRs allows for an accurate evaluation of the potential in MCTDH calculations with multidimensional single-particle functions.     

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.     

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.     

System-specific discrete variable representations for path integral calculations with quasi-adiabatic propagators

Discrete variable representations (DVRs), constructed numerically from eigenstates of the one-dimensional adiabatic potential, provide the optimal quadrature for evaluating quasi-adiabatic propagator path integrals (QUAPI) for a system coupled to a harmonic bath. Calculations of partition functions and reaction rates for a multiple-minimum potential in a dissipative environment illustrate the convergence characteristics of this approach. The small number of quadrature points required, along with the rapid convergence of QUAPI methods, results in a powerful numerical scheme, complementary to Monte Carlo methods, for performing condensed phase dynamics calculations over the entire temperature range of interest in chemical physics.     

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.     

Degeneracy in discrete variable representations: general considerations and application to the multiconfigurational time-dependent Hartree approach

Problems appear in discrete variable representations (DVRs) based on general basis sets when the coordinate matrix has degenerate eigenvalues. Then the DVR is not uniquely defined. This paper shows that this problem can be caused by symmetry. Taking the symmetry into account when constructing the DVR solves the problem. The symmetry effect can be particularly important for the time-dependent DVR used in multiconfigurational time-dependent Hartree calculations employing the correlation DVR (CDVR) approach. Problems reported previously for the initial-state selected treatment of the H+H(2) reaction can be attributed to this symmetry effect. They can be solved by using a symmetry-adapted approach to construct the time-dependent DVR. Thus, the present paper shows that the CDVR scheme can be employed also in initial-state selected scattering calculations if the symmetry of the system is properly taken into account in the construction of the time-dependent DVR.     

Bessel-zernike discrete variable representation basis

The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermore, the Zernike polynomials may also be used to efficiently evaluate the Hankel transform for rapidly decaying functions or functions with finite support.     

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     

Analytical Shannon information entropies for all discrete multidimensional hydrogenic states

The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work, the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydrogenic systems are obtained from first principles; that is, they are given in terms of the states' principal and magnetic hyperquantum numbers (n, μ₁, μ₂, …, μ _D−1), the system's dimensionality D and the nuclear charge Z in an analytical, compact form. Explicit expressions for the total Shannon entropies are given for the quasi-spherical states, which conform to a relevant class of specific states of the D-dimensional hydrogenic system characterized by the hyperquantum numbers μ₁ = μ₂ … = μ _D−1 = n − 1, including the ground state.     

Discrete variable representations and sudden models in quantum scattering theory

An exact formalism in which the scattering problem may be described by sets of coupled equations labeled either by basis functions or quadrature points is presented. Use of each frame and the simply evaluated unitary transformation which connects them results in an efficient procedure for performing quantum scattering calculations. Two approximations are compared with the IOS.     

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.     

Calculating rate constants with updated Hessians using variational transition state theory with multidimensional tunneling

Variational transition state theory with multidimensional tunneling (VTST/MT) has been used for calculating the rate constants of reactions. The updated Hessians have been used to reduce the computational costs for both geometry optimization and trajectory following procedures. In this paper, updated Hessians are used to reduce the computational costs while calculating the rate constants applying VTST/MT. Although we found that directly applying the updated Hessians will not generate good vibrational frequencies along the minimum energy path (MEP), however, we can either re-compute the full Hessian matrices at fixed intervals or calculate the Block Hessians, which is constructed by numerical one-side difference for the Hessian elements in the "critical" region and Bofill updating scheme for the rest of the Hessian elements. Due to the numerical instability of the Bofill update method near the saddle point region, we have suggested a simple strategy in which we follow the MEP until certain percentage of the classical barrier height from the barrier top with full Hessians computed and then performing rate constant calculation with the extended MEP using Block Hessians. This strategy results a mean unsigned percentage deviation (MUPD) around 10% with full Hessians computed till the point with 80% classical barrier height for four studied reactions. This proposed strategy is attractive not only it can be implemented as an automatic procedure but also speeds up the VTST/MT calculation via embarrassingly parallelization to a personal computer cluster.     

2D porous honeycomb polymers versus discrete nanocubes from trigonal trinuclear complexes and ligands with variable topology

Trinuclear building block {Fe(2)NiO(Piv)(6)} (Piv = pivalate), which possessed pseudo-D(3h) symmetry, was linked by two ligands, pseudo-D(3h) ligand tris-(4-pyridyl)pyridine (L1) and C(2v) ligand 4-(N,N-dimethylamino)phenyl-2,6-bis(4-pyridyl)pyridine (L2) into two products with different topologies: 2D coordination polymer [Fe(2)NiO(Piv)(6)(L1)](n) (1), and discrete molecule [{Fe(2)NiO(Piv)(6)}(8) {L2}(12)], which had a nanocube structure (2). In compound 1, trinuclear {Fe(2)NiO(Piv)(6)} blocks were linked through ligand L1 into layers with honeycomb topology. In compound 2, eight trinuclear blocks were located in the vertices of the nanocube, with each L2 ligand linked to two {Fe(2)NiO(Piv)(6)} units. In the crystal structure, these nanocubes formed infinite catenated chains. Analysis of possible structures that could be assembled from these building blocks showed that compounds 1 and 2 corresponded to their respective predicted topologies. Compound [1?solvent] possessed a porous structure, in which the voids were filled by solvent molecules (DMF or DMSO). This structure was retained following desolvation, and compound 1 absorbed significant quantities of N(2) and H(2) at 78?K (S(BET) = 730?m(2) g(-1), H(2) sorption capacity: 0.9?% by weight at 865?Torr). Desolvation of [2?solvent] led to disorder of its crystal structure, and compound 2 only adsorbed negligible quantities of N(2) but adsorbed 0.27?% H(2) (by weight) at 855?Torr and 78?K. The magnetic properties of these complexes (temperature dependence of molar magnetic susceptibility) were governed by the magnetic properties of the trinuclear "building block".     

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.     

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.     

Quantum mechanical study of the correlation of attack and recoil angles for the Cl + H2 reaction using the stereodirected and discrete variable representations on two potential energy surfaces

The zero total angular momentum (J = 0) S matrix elements, calculated using a time-dependent wave packet method for the Cl (2P) + H2 reaction on two different potential energy surfaces, have been matrix transformed to the stereodirected and Gauss-Legendre discrete variable representations. Although the results in the two representations are (as expected) quantitatively different with respect to the angular selectivity and specificity of the reactive process, the qualitative similarity has allowed us to draw for the first time conclusions with respect to some characteristics of the potential energy surface.     

Bidentate ligands capable of variable bond angles in the self-assembly of discrete supramolecules

Flexible donor ligands like 1,2-bis(3-pyridyl)ethyne or 1,4-bis(3-pyridyl)-1,3-butadiyne self-assemble into discrete supramolecules instead of infinite networks upon combination with organoplatinum 90, 120, and 180 degree acceptor units. These systems are unique examples of versatile pyridine donors adjusting their bonding directionality to accommodate rigid platinum acceptors in the formation of closed macrocycles.     

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     

Tools for visualizing multidimensional images from living specimens

Over the last 50 years modern cell biology has been driven by the development of powerful imaging techniques. In particular, new developments in light microscopy that provide the potential to image the dynamics of biological events have had significant impact. Optical sectioning techniques allow three-dimensional information to be obtained from living specimens noninvasively. When used with multimodal fluorescence microscopy, advanced optical sectioning techniques provide multidimensional image data that can reveal information not only about the changing cytoarchitecture of a cell but also about its physiology. These additional dimensions of information, although providing powerful tools, also pose significant visualization challenges to the investigator. Particularly in the current postgenomic era there is a greater need than ever for the development of effective tools for image visualization and management. In this review we discuss the visualization challenges presented by multidimensional imaging and describe three open-source software programs being developed to help address these challenges: ImageJ, the Open Microscopy Environment, and VisBio.