Discrete Variable Representation Basis Research Articles

Structure of liquid water at ambient temperature from ab initio molecular dynamics performed in the complete basis set limit

Structural properties of liquid water at ambient temperature were studied using Car-Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] ab initio molecular dynamics (CPAIMD) simulations combined with the Kohn-Sham (KS) density functional theory and the BLYP exchange-correlation functional for the electronic structure. Unlike other recent work on the same subject, where plane-wave (PW) or hybrid Gaussian/plane-wave basis sets were employed, in the present paper, a discrete variable representation (DVR) basis set is used to expand the KS orbitals, so that with the real-space grid adapted in the present work, the properties of liquid water could be obtained very near the complete basis set limit. Structural properties of liquid water were extracted from a 30 ps CPAIMD-BLYP/DVR trajectory at 300 K. The radial distribution functions (RDFs), spatial distribution functions, and hydrogen bond geometry obtained from the CPAIMD-BLYP/DVR simulation are generally in good agreement with the most up to date experimental measurements. Compared to recent ab initio MD simulations based on PW basis sets, less significant overstructuring was found in the RDFs and the distributions of hydrogen bond angles, suggesting that previous plane-wave and Gaussian basis set calculations have exaggerated the tendency toward overstructuring.

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 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.

Quantum chemical calculation of infrared spectra of acidic groups in chabazite in the presence of water.

The changes in the spectra of the acidic group in chabazite are studied by quantum chemical calculations. The zeolite is modeled by two clusters consisting of eight tetrahedral atoms arranged in a ring and seven tetrahedral atoms coordinated around the zeolite OH group. The potential energy and dipole surfaces were constructed from the zeolite OH stretch, in-plane and out-of-plane bending coordinates, and the intermolecular stretch coordinate that corresponds to a movement of the water molecule as a whole. Both the anharmonicities of the potential energy and dipole were taken into account by calculation of the frequencies and intensities. The matrix elements of the vibrational Hamiltonian were calculated within the discrete variable representation basis set. We have assigned the experimentally observed frequencies at approximately 2900, approximately 2400, and approximately 1700 cm(-1) to the strongly perturbed zeolite OH vibrations caused by the hydrogen bonding with the water molecule. The ABC triplet is a Fermi resonance of the zeolite OH stretch mode with the overtone of the in-plane bending (the A band) and the overtone of the out-of-plane bending (the C band). In the B band the stretch is also coupled with the second overtone of the out-of-plane bending. The frequencies at approximately 3700 and approximately 3550 cm(-1) we have assigned to the OH stretch frequencies of a slightly perturbed water molecule.

Spectrum and vibrational predissociation of the HF dimer. I. Bound and quasibound states

We present full six-dimensional calculations of the bound states of the HF dimer for total angular momentum J=0,1 and of the quasibound states for J=0 that correspond with vibrational excitation of one of the HF monomers, either the donor or the acceptor in the hydrogen bond. Transition frequencies and rotational constants were calculated for all four molecular symmetry blocks. A contracted discrete variable representation basis was used for the dimer and monomer stretch coordinates R,rA,rB; the generation of the monomer basis in the dimer potential leads to significantly better convergence of the energies. We employed two different potential energy surfaces: the SQSBDE potential of Quack and Suhm and the SO-3 potential of Klopper, Quack, and Suhm. The frequencies calculated with the SO-3 potential agree very well with experimental data and are significantly better than those from the SQSBDE potential.

More on the Airy averaging method

The Airy averaging method illustrated in a previous paper [J. Math. Chem. 25 (1999) 93] is a simple and effective procedure for dealing with mathematical expressions/manipulations involving Airy functions. The potentiality of the method is emphasizd through a couple of applications drawn from recent research advances in the field of the discrete variable representation (DVR) basis sets.

An exact variational method to calculate vibrational energies of five atom molecules beyond the normal mode approach

A full dimensional variational algorithm to calculate vibrational energies of penta-atomic molecules is presented. The quantum mechanical Hamiltonian of the system for J=0 is derived in a set of orthogonal polyspherical coordinates in the body-fixed frame without any dynamical approximation. The vibrational Hamiltonian has been obtained in an explicitly Hermitian form. Variational calculations are performed in a direct product discrete variable representation basis set. The sine functions are used for the radial coordinates, whereas the Legendre polynomials are employed for the polar angles. For the azimuthal angles, the symmetrically adapted Fourier–Chebyshev basis functions are utilized. The eigenvalue problem is solved by a Lanczos iterative diagonalization algorithm. The preliminary application to methane is given. A comparison with previous results is made.

Tetrahedrally invariant discrete variable representation basis on the sphere

This paper explores the difficulties of constructing multidimensional discrete variable representation (DVR) basis sets and the strategies that can be used to overcome them. A parameter count shows that the conditions on a DVR basis set cannot be satisfied on most spaces of wave functions. One-dimensional, orthogonal polynomials are an exception, but the Ylm’s on the sphere only go 3/4 of the way, in a certain sense, toward supplying enough parameters to satisfy the DVR conditions. It is shown that DVR sets involving rotationally invariant subspaces of wave functions on the sphere (consisting of complete subshells only) exist only for small values of the angular momentum cutoff. However, an exploration of DVR sets invariant under subgroups of the full rotation group leads to the discovery of a 12-point DVR set that is invariant under the tetrahedral group, whose grid points are the vertices of an icosahedron.

An Airy discrete variable representation basis

A new discrete variable representation (DVR) basis is presented, one that is based on Airy functions. That is, the functions of the “finite basis representation” (actually infinite in this case) are energy eigenfunctions in a constant force field. The exact matrix elements of the kinetic energy are computed. The use of the Airy DVR functions in diagonalizing a one-dimensional Hamiltonian is illustrated.

Bessel discrete variable representation bases

Discrete variable representation (DVR) basis sets on the radial half-line, based on Bessel functions, are presented. These are Hankel transforms of the eigenfunctions of the particle in a spherical box in k space, but there is no box or bound on the radial variable r. The grid points extend to infinity on the r axis. The DVR functions are exactly orthonormal and exactly satisfy the interpolation properties usually associated with DVR functions. The exact matrix elements of the kinetic energy are computed, and the use of the Bessel DVR functions in radial eigenvalue problems is illustrated. The phase space or semiclassical interpretation of the Bessel DVR functions is presented, and variations on these functions, corresponding to alternative boundary conditions in k space, are discussed. An interesting feature of Bessel DVR functions is that they are based on a finite basis representation that is continuously infinite.

A general framework for discrete variable representation basis sets

A framework for discrete variable representation (DVR) basis sets is developed that is suitable for multidimensional generalizations. Those generalizations will be presented in future publications. The new axiomatization of the DVR construction places projection operators in a central role and integrates semiclassical and phase space concepts into the basic framework. Rates of convergence of basis set expansions are emphasized, and it is shown that the DVR method gives exponential convergence, assuming conditions of analyticity and boundary conditions are met. A discussion of nonorthogonal generalizations of DVR functions is presented, in which it is shown that projected δ-functions and interpolating functions form a biorthogonal basis. It is also shown that one of the generalized DVR proposals due to Szalay [J. Chem. Phys. 105, 6940 (1996)] gives exponential convergence.

Multidimensional discrete variable representation bases: Sinc functions and group theory

Examples of nontrivial (that is, non-Cartesian product) multidimensional discrete variable representation (DVR) basis sets are presented that are generalizations of sinc functions in one dimension. Their use in solving quantum problems in two dimensions is illustrated. Unlike all standard (one-dimensional) examples of DVR bases, these bases cannot be created by dividing out the roots of a generating function. It is argued that the difficulty of constructing nontrivial, multidimensional DVR bases is due to the restrictive nature of the DVR conditions, which cannot be satisfied on most subspaces of wave functions. The bases considered in this paper, however, are invariant under translations on a lattice in n-dimensional space, and the properties of the Abelian group of lattice translations allow the DVR conditions to be satisfied. More generally, the question of the relation between group theory and the conditions necessary to qualify a set of basis functions as a DVR basis is considered. It is shown how to construct orthonormal states that are related by the action of some group, and, in the case of Abelian groups, the extra conditions required to qualify the basis as a DVR set are discussed.

A time-dependent discrete variable representation method

We have developed a novel discrete variable representation (DVR) method where not only the amplitudes of the wave function at the DVR grid points can change but also the positions of these grid points can move as a function of time. Since the Gauss–Hermite basis set is used as the primitive basis functions (PBF) to construct the DVR basis set, the method appears as a semiclassical one with a small number of PBF but converges very fast to the quantum with an increasing PBF. We have investigated the dynamics of a reaction coordinate with or without coupling to a heat bath of harmonic oscillators to demonstrate the validity of the proposed method. The excellent agreement of the calculated tunneling probabilities with numbers obtained by traditional quantum grid method (FFT) and the fast computability of the present method compared to the latter are remarkable.

Six-dimensional calculation of the vibrational spectrum of the HFCO molecule

Bound energy levels, up to 5000 cm−1 of internal excitation, have been computed for the HFCO molecule. An exact six-dimensional (6D) quantum Hamiltonian, expressed in terms of Jacobi vectors, has been used. It is shown to lead to a compact form of the kinetic energy operator, easy to implement in the calculations. The primary discrete variable representation (DVR) basis set has been contracted by means of the adiabatic pseudospectral method of Friesner et al. [J. Chem. Phys. 99, 324 (1993)]. Two different, global, potential energy surfaces have been considered. The calculated energy levels have been successfully assigned by an automatic labeling procedure. These levels have been compared to the experimental results, providing a test of the accuracy of the existing surfaces.

A DVR analysis of some vibrational modes in the elongated dihydrogen complex [Ru(η 2-H 2)(C 5H 5)(H 2PCH 2PH 2)] +

Nuclear motion quantum calculations have been performed on mono- and bidimensional versions of the potential energy surface for complex [Ru(η 2-H 2)(C 5H 5)(H 2PCH 2PH 2)] +, both with the Discrete Variable Representation (DVR) and Finite Basis Representation (FBR) methodologies. It is found that the DVR methodology produces accurate vibrational energy levels and wave functions at a much lower CPU cost. The DVR results are then used to carry out a study of the vibrational modes. Remarkable agreement between the calculated and the experimental transitions for this complex is found, for both the (η 2-H 2) and (η 2-D 2) isotopomers. It is found that the H–H and Ru–H 2 stretchings are thoroughly mixed in this complex. Finally, it is seen that the experimental assignments of the vibrational transitions as H–H and Ru–H 2 symmetric stretch should be revised.

Strong-field optical control of vibrational dynamics: Vibrational Stark effect in planar acetylene

We perform quantum mechanical simulations of vibrational excitation of planar (5D) acetylene (HCCH) with linearly polarized, intense but nonionizing, infrared laser pulses, exploring one particular pathway for exciting the HCCH molecule into a CC-stretching state via the fundamental excitation in a two-pulse scheme. We optimize the pulse widths, time centers, and carrier frequencies of the two pulses to achieve the maximal projection onto the target CC-stretching state, (0,3,0,0,0) A1, subject to penalties related to peak electric field and pulse duration. The influences of Fermi resonance, the vibrational Stark effect, and avoided crossings on the selective excitation are discussed. Different sizes of “essential-states” representation are used and checked against the underlying 299 475-point discrete variable representation (DVR) basis. We find that an essential-states basis consisting of 362 A1 and B2 eigenstates represents the results of a full-grid calculation for the excitation process under study. Moreover, we demonstrate that despite the complications associated with the higher dimensionality of the 5D model, we can nevertheless exert infrared laser control over the vibrational dynamics of selective excitation.

A spectral transform Krylov subspace iteration approach to quantum scattering

A spectral transform Lanczos iteration method is suggested to find the eigenvalues and eigenvectors of the coupling matrix appearing in quantum scattering calculations. Three main attributes which improve previous implementations are (a) partial reorthogonalization, (b) exponential filter, and (c) discrete variable representation basis. The method is applied to the 3D rotating line umbrella calculations of the reaction Cl+CH4⇌ HCl+CH3 which shows that it is substantially faster than the direct diagonalization method. Calculated state-to-state reaction probabilities as well as adiabats are in good agreement with previous results.

Localization transformations of orthogonal functions

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.

DVR study of the absorption spectrum of NO2

Using available potential energy surfaces, a theoretical calculation of the absorption spectrum of NO2 is presented. Energy levels and nuclear wave functions are calculated, using a discrete variable representation (DVR) basis expansion to solve the nuclear Schrödinger equation. Various energy regions of the absorption spectrum are considered. In the low-energy region, below the onset of the excited state, accurate results are recovered in a one-electronic-state calculation. It is also shown that using a decoupled potential energy surface derived from Schryber potential, accurate results are also obtained in the two-coupled-state case. Calculations are also performed in the higher energy part of the spectrum up to 16,000 cm−1. Analysis of the calculated wave functions makes the assignment of vibrational quantum numbers possible for both electronic components to the various excited levels. Comparison with experimental results are presented. Reasonable agreement is obtained for the transition frequencies. An overall reliable reproduction of experimental data, including the intensities through Franck–Condon calculation, requires, however, the transition moment function as additional information. General conclusions concerning the limitation of the model are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 475–489, 1998

HO 2 rovibrational eigenvalue studies for nonzero angular momentum

An efficient parallel algorithm is reported for determining all bound rovibrational energy levels for the HO2 molecule for nonzero angular momentum values, J=1, 2, and 3. Performance tests on the CRAY T3D indicate that the algorithm scales almost linearly when up to 128 processors are used. Sustained performance levels of up to 3.8 Gflops have been achieved using 128 processors for J=3. The algorithm uses a direct product discrete variable representation (DVR) basis and the implicitly restarted Lanczos method (IRLM) of Sorensen to compute the eigenvalues of the polyatomic Hamiltonian. Since the IRLM is an iterative method, it does not require storage of the full Hamiltonian matrix—it only requires the multiplication of the Hamiltonian matrix by a vector. When the IRLM is combined with a formulation such as DVR, which produces a very sparse matrix, both memory and computation times can be reduced dramatically. This algorithm has the potential to achieve even higher performance levels for larger values of the total angular momentum.