Discrete variable representation for singular Hamiltonians

A discrete representation of vibrational molecular wavefunctions in coordinate space offers many advantages for the solution of problems in quantum dynamics, in particular if realistic, multidimensional potential energy surfaces are involved. This leads to the problem of how to construct localized representations of operators. While any set of grid points might be defined for a discrete variable representation (DVR) of local operators this is not true for the Hamiltonian operator as a whole. As a consequence of the non-local nature of the kinetic energy operator a correspondence between the localized picture and a delocalized variational basis representation (VBR) or momentum space, respectively, must be established. The arrangement of grid points is therefore determined by the properties of the VBR basis functions. It is shown that linear matrix transformations between localized and delocalized representations and the so-called Fourier method for the evaluation of non-local operations can essentially be reduced to the same idea: localization transformations of orthogonal functions. Gaussian quadrature is a classical way to obtain localized basis functions derived from polynomials. We investigate the properties of the localized functions related to important polynomials and obtain a new relation for Legendre polynomials. VBR basis functions related to equidistant quadrature points, like particle-in-a-box or Fourier functions, are frequently applied and the associated localized basis functions are shown to possess interesting symmetry features. We give a mathematically consistent theory of matrix transformations between localized and delocalized pictures: a new representation, called nearly discrete variable representation (NDVR), is introduced, that is equivalent to the VBR, in analogy to the correspondence of DVR and finite basis representation (FBR). DVR basis functions are defined in a new way via delta sequences. Finally, a brief analysis of the error made in calculations with a DVR potential matrix is given for two simple one-dimensional models.

The formal definition of the generalized discrete variable representation (DVR) for quantum mechanics and its connection to the usual variational basis representation (VBR) is given. Using the one dimensional Morse oscillator example, we compare the ‘‘Gaussian quadrature’’ DVR, more general DVR’s, and other ‘‘pointwise’’ representations such as the finite difference method and a Simpson’s rule quadrature. The DVR is shown to be accurate in itself, and an efficient representation for optimizing basis set parameters. Extensions to multidimensional problems are discussed.

In developing a pseudospectral transform between a nondirect product basis of spherical harmonics and a direct product grid, Corey and Lemoine [J. Chem. Phys. 97, 4115 (1992)] generalized the Fourier method of Kosloff and the discrete variable representation (DVR) of Light by introducing more grid points than spectral basis functions. Assuming that the potential energy matrix is diagonal on the grid destroys the variational principle in the Fourier and DVR methods. In the present article we (1) demonstrate that the extra grid points in the generalized discrete variable representation (GDVR) act as dealiasing functions that restore the variational principle and make a pseudospectral calculation equivalent to a purely spectral one, (2) describe the general principles for extending the GDVR to other nondirect product spectral bases of orthogonal polynomials, and (3) establish the relation between the GDVR and the least squares method exploited in the pseudospectral electronic structure and adiabatic pseudospectral bound state calculations of Friesner and collaborators.

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.

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 derivation of discrete variable representations (DVR’s) is given, with emphasis on formal properties and relations to the classical orthogonal polynomials. These “Gaussian quadrature” DVR’s are compared with several different DVR’s proposed recently. Finally, some cautions and brief comparisons of accuracy are noted for quantum solutions for large amplitude motions of triatomic molecules (H2O in this case).

A new discrete variable representation (DVR) in generalized vibrational coordinates is proposed together with a new mixed diabatic/adiabatic contraction technique for the treatment of multidimensional vibrational problems up to high vibrational excitations. Formally based on the equidistant Chebyshev DVR in the grid index the new formulation is particularly suitable for multidimensional minimum energy paths. The new Z-matrix DVR proposed in this paper encompasses usual valence coordinates as well as nonlinear maps of coordinates on optimal nonequidistant grids. The pointwise numerical calculation of all kinetic energy terms avoids the algebraic derivation of specialized analytical forms of the kinetic energy adding to the flexibility of the method. With efficient truncation schemes the generalized DVR allows for a compact representation of the time-dependent wave-packet dynamics in up to six dimensions. Vibrationally adiabatic approaches to the detailed modelling of multidimensional quantum-dynamics usually are hampered by the typically large number of (avoided) crossings in dense spectra. This problem is particularly severe for discrete variable representations. A solution is provided by the new technique of diabatic rotations leading to a systematic construction of locally diabatic channels. This allows the treatment of very dense spectra where conventional truncation techniques fail. Applying the new approach to the vibrational problem of tetratomic molecules demonstrates its flexibility and efficiency. The examples of formaldehyde, ammonia, and hydrogen peroxide cover the whole range from semirigid (CH2O) to large amplitude inversion (NH3) and torsional tunnelling dynamics (H2O2). In solving the full six-dimensional vibrational eigenvalue problems for CH2O and NH3 the Z-matrix DVR shows at least comparable if not superior numerical efficiency compared with specialized techniques. In the case of H2O2 the technique of diabatic rotations and adiabatic contraction for the first time allows the treatment of the tunneling dynamics significantly above the dissociation threshold up to the fifth OH stretch overtone. The calculated decrease of the tunneling rate by about one order of magnitude agrees well with experimental observations.

We propose a new numerically optimized discrete variable representation using eigenstates of diabatic Hamiltonians. This procedure provides an efficient method to solve non-adiabatic coupling problems since the generated basis sets take into account information on the diabatic potentials. The method is applied to the B1Σ+ − D′1Σ+ Rydberg-valence predissociation interaction in the CO molecule. Here we give an account of the discrete variable representation and present the procedure for the calculation of its optimized version, which we apply to obtain the total photodissociation cross sections of the CO molecule.

The general procedure for constructing a set of orthonormal polynomials is given for an arbitrary positive definite weight function, w(x), in the interval [a, b]. The Lanczos method is used to generate the three‐term recursion relation, which is then used to produce the polynomial coefficients. A discrete variable representation (DVR) is constructed from Gaussian nodes and weights that result from the three‐term recursion relation. These are termed “designer polynomials” and the associated “designer DVRs.” It will be shown by construction that every such set of “synthetic polynomials” carries an associated DVR. The term “designer” derives from the fact that the interval [a, b] and the weight function w(x) are arbitrary (except that w(x) must be positive definite on [a, b] and must have continuous derivatives except at a finite number of isolated discontinuities) and may be adapted to the physical problem of interest. The difficulties of applying a DVR to a “bare” Coulomb problem will be illustrated on a “toy” model in one dimension (1‐D hydrogen atom). A solution for the 1‐D Coulomb problem will be given, thereby motivating the need for designer DVRs. In doing so, a new set of polynomials is defined with a weight function w(x) = |x|kexp(−α|x|), (such that k = −1, 0, +1, +2, …) between the symmetrical limits [−∞, +∞]. These are called “synthetic Cartesian exponential polynomials (SCEP).” These polynomials are then used in a spectral and pseudospectral (DVR) representation to solve the 1‐D hydrogen atom problem. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Potential optimized discrete variable representation

An algebraic method based on a discrete variable representation (DVR) scheme is applied to describe 1D potentials. The approach is based on the algebraic realisation of the coordinate and momentum in the framework of a complete basis associated with the 1D Morse potential. The diagonalisation of the matrix representations of the coordinate and momentum in this scheme provides discrete bases that establish the DVR approach, allowing the matrix representation of the Hamiltonian to be expressed in a simple form in terms of diagonal matrices through the use of the transformation coefficients. The advantage of this approach is that it may be suitable for any potential using only algebraic means at an affordable computational cost. Application to obtain the solutions associated with the Lennard-Jones potential and the ab initio potential of the H2 molecule is presented. Reliable, accurate vibrational energies as well as wave functions are obtained in both cases.

An adaptable discrete variable representation based on coherent states

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.

The application of discrete singular convolution (DSC) algorithm in solving the Schrödinger equation of one-electron systems was attempted with a hydrogen atom as an example. Using the uniform discretization and Shannon kernel, the Schrödinger equations were solved in spherical coordinates. Compared with other methods such as discrete variable representation, Hartree–Fock, and density functional theory, the DSC algorithm is robust and efficient in numerical solutions for achieving accurate eigenvalues of excited states, and thus, especially suitable for problems in which lots of accurate eigenvalues of excited states are requested. The restrictions of boundary conditions on the discretization and the impact of singularities are studied. Both the boundary conditions and singularities were found to be critical in the numerical applications.

In the discrete variable representation (DVR) method the potential energy matrix \({\cal V}\) has a particular simple form. It is diagonal with values of the interaction potential at the discrete points. However, this simple form is obtained by making approximations in the calculation of the matrix elements of \({\cal V}\). As a consequence the results cannot be considered as variational estimates. We will show how to recover the variational character of the method using the discrete variable representation eigenvectors as trial functions and performing a variational calculation in a restricted Hilbert space.