Discrete Variable Representation Basis Research Articles

Quantum Dynamical Investigation of Dihydrogen-Hydride Exchange in a Transition-Metal Polyhydride Complex.

The ongoing shift toward clean, sustainable energy is a primary driving force behind hydrogen fuel research. Safe and effective storage of hydrogen is a major challenge (particularly for mobile applications) and requires a detailed understanding of the atomic level interactions of hydrogen with its host materials. The light mass of hydrogen, however, implies that quantum effects are important, so a quantum dynamical treatment is required to properly account for these effects in computational simulations. As one such example, we describe herein the hydrogen exchange dynamics between a hydride and a dihydrogen ligand in the [FeH(H2)(PH3)4]+ model complex. A global three-dimensional (3D) potential energy surface (PES) was constructed by fitting to and interpolating from a discrete set of grid points computed using density functional theory; exact quantum dynamical calculations were then carried out on the 3D PES using discrete variable representation basis sets. Energy levels and their quantum tunneling splittings were computed up to 3000 cm-1 above the ground state. Within that energy range, all three fundamentals have been identified using wave function plots, as well as the first three overtones of the exchange (reaction coordinate) motion and several of its combination bands. From the tunneling splittings, the Boltzmann-averaged tunneling rates were computed. The Arrhenius plot of the total exchange rate shows a clear transition around 150 K, below which the activation energy is essentially zero and above which it is less than half of the electronic structure barrier. This indicates that exchange rates are governed by quantum tunneling throughout the relevant temperature range with the low-temperature regime dominated by a single quantum (ground) state. This work is the first-ever fully quantum dynamical study to investigate the hydrogen exchange dynamics between hydride and dihydrogen ligands coordinated to a transition-metal complex.

Full-dimensional quantum dynamics study of isotope effects for the H2 + NH2/ND2/NHD and H2/D2/HD + NH2 reactions.

A full-dimensional quantum dynamical study for the bimolecular reactions of hydrogen molecules with amino radicals for different isotopologues is reported. The nonreactive amino radical is described by two Radau vectors that are very close to the valence bond coordinates. Potential-optimized discrete variable representation basis is used for the vibrational coordinates of the amino radical. Starting from the reaction H2 + NH2, we study the isotope effects for the two reagents separately, i.e., H2 + NH2/ND2/NHD and H2/D2/HD + NH2. The effects of different vibrational mode excitations of the reagents on the reactivities are studied. Physical explanations about the isotope effects are also provided thoroughly including the influence of vibrational energy differences between the different isotopologues and the impact of the tunneling effect.

Full-dimensional quantum mechanical calculations of the reaction probability of the H + CH4 reaction based on a mixed Jacobi and Radau description.

A full-dimensional time-dependent wave packet study using mixed polyspherical Jacobi and Radau coordinates for the title reaction has been reported. The non-reactive moiety CH3 has been described using three Radau vectors, whereas two Jacobi vectors have been used for the bond breaking/formation process. A potential-optimized discrete variable representation basis has been employed to describe the vibrational coordinates of the reagent CH4. About one hundred billion basis functions have been necessary to achieve converged results. The reaction probabilities for some initial vibrational states are given. A comparison between the present approach and other methods, including reduced and full-dimensional ones, is also presented.

Full dimensional quantum mechanical calculations of the reaction probability of the H + NH3 collision based on a mixed Jacobi and Radau description.

The collision between hydrogen and ammonia is a benchmark system to study chemical elementary reactions with five atoms. In this work, we present a description of the system based on mixed Jacobi and Radau coordinates combined with the time-dependent wave packet method to study the H + NH3 reaction. The Radau coordinates are used to describe the reactive moiety NH2. A salient feature of this approach is that the present coordinates have a great advantage that a very small number of basis set functions can be used to describe the NH2 group. Potential-optimized discrete variable representation basis is applied for the vibrational coordinates of the reagent NH3. The reaction probabilities for several initial vibrational states are presented in this paper. The role of the different vibrational excitations on the reactivity is thoroughly described.

Implementation of the infinite-range exterior complex scaling to the time-dependent complete-active-space self-consistent-field method

We present a numerical implementation of the infinite-range exterior complex scaling (irECS) [Phys. Rev. A 81, 053845 (2010)] as an efficient absorbing boundary to the time-dependent complete-active-space self-consistent field (TD-CASSCF) method [Phys. Rev. A 94, 023405 (2016)] for multielectron atoms subject to an intense laser pulse. We introduce Gauss-Laguerre-Radau quadrature points to construct discrete variable representation basis functions in the last radial finite element extending to infinity. This implementation is applied to strong-field ionization and high-harmonic generation in He, Be, and Ne atoms. It efficiently prevents unphysical reflection of photoelectron wave packets at the simulation boundary, enabling accurate simulations with substantially reduced computational cost, even under significant (~ 50%) double ionization. For the case of a simulation of high-harmonic generation from Ne, for example, 80% cost reduction is achieved, compared to a mask-function absorption boundary.

Time-dependent formulation of the two-dimensional model of resonant electron collisions with diatomic molecules and interpretation of the vibrational excitation cross sections

A two-dimensional model of the resonant electron-molecule collision processes with one nuclear and one electronic degree of freedom introduced by K. Houfek, T. N. Rescigno, and C. W. McCurdy [Phys. Rev. A 73, 032721 (2006)] is reformulated within the time-dependent framework and solved numerically using the finite-element method with the discrete variable representation basis, the exterior complex scaling method, and the generalized Crank-Nicolson method. On this model we illustrate how the time-dependent calculations can provide deep insight into the origin of oscillatory structures in the vibrational excitation cross sections if one evaluates the cross sections not only at sufficiently large time to obtain the final cross sections, but also at several characteristic times which are given by the evolution of the system. It is shown that all details of these structures, especially asymmetrical peaks, can be understood as quantum interference of several experimentally indistinguishable processes separated in time due to a resonant capture of the electron and the subsequent vibrational motion of the negative molecular ion. Numerical results are presented for the ${\mathrm{N}}_{2}$-like, NO-like, and ${\mathrm{F}}_{2}$-like models and compared with ones obtained within the time-independent approach and within the local complex potential approximation.

Free rotor model or rigid rotor model for CH3F-Ne complex and comparison with other CH3F-rare gas systems

The assignment of the rovibrational spectra of molecule-Ne complexes is always a challenge to study van der Waals systems, since they usually exhibit behavior intermediate between free rotor and rigid rotor. In this paper, the microwave and infrared spectra of CH3F-Ne, a model system for symmetric-top-atom dimer, were firstly predicted and analyzed based on the four-dimensional ab initio intermolecular potential energy surfaces(PESs), which explicitly incorporate the v3(C—F) stretch normal model coordinate of the CH3F monomer. Analytic three-dimensional PESs were obtained by least-squares fitting vibrationally averaged interaction energies for v3(CH3F)=0 and 1 to the Morse/long-range(MLR) potential function for symmetry top impurity with atom model. These PESs fitting to 2340 points have root-mean-square(RMS) deviations of 0.07 cm–1, and require only 167 parameters. Based on the analytical vibrationally averaged PESs, the rovibrational energy levels were calculated by employing Lanczos algorithm, with combined radial discrete variable representation and parity-adapted angular finite basis representation. Based on the wavefunction analysis and comparison of CH3F-Ne with CH3F-He and CH3F-Ar complexes, the bound states were assigned. Spectral parameters for CH3F-Rg(Rg: rare gas, Rg=He, Ne, Ar) complexes were fitted and discussed. Temperature dependent transition intensities for CH3F-Ne were also reported and analyzed. The complete microwave and infrared spectra information for CH3F-Ne made it possible to provide important guidance for future experimental spectroscopic assignments.

Using an iterative eigensolver to compute vibrational energies with phase-spaced localized basis functions.

Although phase-space localized Gaussians are themselves poor basis functions, they can be used to effectively contract a discrete variable representation basis [A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. 109, 070402 (2012)]. This works despite the fact that elements of the Hamiltonian and overlap matrices labelled by discarded Gaussians are not small. By formulating the matrix problem as a regular (i.e., not a generalized) matrix eigenvalue problem, we show that it is possible to use an iterative eigensolver to compute vibrational energy levels in the Gaussian basis.

‘Slow’ time discretization: a versatile time propagator for the time-dependent Schrödinger equation

We present the slow time discretization (STD) method for solving the time-dependent Schrödinger equation. The method is an extension of the slow variable discretization method for solving the stationary Schrödinger equation (Tolstikhin et al 1996 J. Phys. B: At. Mol. Opt. Phys. 29 L389), with time treated as the ‘slow’ variable. It is based on an expansion of the state vector in a discrete variable representation basis, in time, and an adiabatic basis, in Hilbert space. This approach is much more efficient in implementation than a direct solution of the Born–Fock equations. The versatility of the STD time propagator is illustrated through calculations for one-dimensional models of the ionization of hydrogen by an intense laser pulse and resonance charge transfer in proton–hydrogen collisions. The method is shown to perform well in the broad dynamical range considered, from adiabatic to nonadiabatic regimes.

Photoionization of helium by attosecond pulses: Extraction of spectra from correlated wave functions

We investigate the photoionization spectrum of helium by attosecond XUV pulses both in the spectral region of doubly excited resonances as well as above the double ionization threshold. In order to probe for convergence, we compare three techniques to extract photoelectron spectra from the wavepacket resulting from the integration of the time-dependent Schroedinger equation in a finite-element discrete variable representation basis. These techniques are: projection on products of hydrogenic bound and continuum states, projection onto multi-channel scattering states computed in a B-spline close-coupling basis, and a technique based on exterior complex scaling (ECS) implemented in the same basis used for the time propagation. These methods allow to monitor the population of continuum states in wavepackets created with ultrashort pulses in different regimes. Applications include photo cross sections and anisotropy parameters in the spectral region of doubly excited resonances, time-resolved photoexcitation of autoionizing resonances in an attosecond pump-probe setting, and the energy and angular distribution of correlated wavepackets for two-photon double ionization.

Use of the discrete variable representation basis in nuclear physics

The discrete variable representation (DVR) basis is nearly optimal for numerically representing wave functions in nuclear physics: Suitable problems enjoy exponential convergence, yet the Hamiltonian remains sparse. We show that one can often use smaller basis sets than with the traditional harmonic oscillator basis, and still benefit from the simple analytic properties of the DVR bases which requires no overlap integrals, simply permit using various Jacobi coordinates, and admit straightforward analyses of the ultraviolet and infrared convergence properties.

Ab initio molecular dynamics study of water at constant pressure using converged basis sets and empirical dispersion corrections

It is generally believed that studies of liquid water using the generalized gradient approximation to density functional theory require dispersion corrections in order to obtain reasonably accurate structural and dynamical properties. Here, we report on an ab initio molecular dynamics study of water in the isothermal-isobaric ensemble using a converged discrete variable representation basis set and an empirical dispersion correction due to Grimme [J. Comp. Chem. 27, 1787 (2006)]. At 300 K and an applied pressure of 1 bar, the density obtained without dispersion corrections is approximately 0.92 g/cm(3) while that obtained with dispersion corrections is 1.07 g/cm(3), indicating that the empirical dispersion correction overestimates the density by almost as much as it is underestimated without the correction for this converged basis. Radial distribution functions exhibit a loss of structure in the second solvation shell. Comparison of our results with other studies using the same empirical correction suggests the cause of the discrepancy: the Grimme dispersion correction is parameterized for use with a particular basis set; this parameterization is sensitive to this choice and, therefore, is not transferable to other basis sets.

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 new basis set for molecular bending degrees of freedom

We present a new basis set as an alternative to Legendre polynomials for the variational treatment of bending vibrational degrees of freedom in order to highly reduce the number of basis functions. This basis set is inspired from the harmonic oscillator eigenfunctions but is defined for a bending angle in the range theta in [0:pi]. The aim is to bring the basis functions closer to the final (ro)vibronic wave functions nature. Our methodology is extended to complicated potential energy surfaces, such as quasilinearity or multiequilibrium geometries, by using several free parameters in the basis functions. These parameters allow several density maxima, linear or not, around which the basis functions will be mainly located. Divergences at linearity in integral computations are resolved as generalized Legendre polynomials. All integral computations required for the evaluation of molecular Hamiltonian matrix elements are given for both discrete variable representation and finite basis representation. Convergence tests for the low energy vibronic states of HCCH(++), HCCH(+), and HCCS are presented.

Discrete Variable Representation Implementation of the One-Electron Polarization Model.

A discrete variable representation (DVR) implementation of an one-electron polarization model (OPEM) for characterizing (H2O)n− clusters is described. For the (H2O)90− cluster, evaluation of the energy and gradient using a suitable DVR basis sets is about a 2 orders of magnitude faster than corresponding calculations using a Gaussian orbital basis set. The DVR version of the code has been parallelized using OpenMP to enable molecular dynamics (MD) simulations of large (H2O)n− clusters.

QUANTUM DYNAMICAL CALCULATION OF ALL ROVIBRATIONAL STATES OF HO2 FOR TOTAL ANGULAR MOMENTUM J = 0–10

The energy levels and wavefunctions for all rovibrational bound states of HO2 are systematically computed, for all total angular momentum values J = 0–10. The calculations are performed using ScalIT, a suite of software modules designed to enable quantum dynamics and related calculations to be performed on massively parallel computing architectures. This is the first-ever application of ScalIT to a real (and very challenging) molecular application. The codes, and in particular, the algorithms (optimal separable basis, preconditioned inexact spectral transform, phase space optimized discrete variable representation basis) are so efficient that in fact, the entire calculation can be performed on a single CPU — although parallel scalability over a small number of CPUs is also evaluated, and found to be essentially perfect in this regime. For the lowest 11 vibrational states, the rotational levels for J = 0–10 fit fairly well to a rigid rotor model, with all vibrational-state-dependent rotational constants, B eff (v), close to values obtained from a previous calculation for J = 0 and 1 [J Chem Phys107:2705, 1997]. However, comparatively larger discrepancies with the rigid-rotor model are found at the higher J values, manifesting in the observed K-splitting (along the O–O bond) of rovibrational levels. This supports earlier work [J Chem Phys113:11055, 2000] suggesting that Coriolis coupling is quite important for this system.

A general rigorous quantum dynamics algorithm to calculate vibrational energy levels of pentaatomic molecules

An exact variational algorithm is presented for calculating vibrational energy levels of pentaatomic molecules without any dynamical approximation. The quantum mechanical Hamiltonian of the system is expressed in a set of orthogonal coordinates defined by four scattering vectors in the body-fixed frame. The eigenvalue problem is solved using a two-layer Lanczos iterative diagonalization method in a mixed grid/basis set. A direct product potential-optimized discrete variable representation (PO-DVR) basis is used for the radial coordinates while a non-direct product finite basis representation (FBR) is employed for the angular variables. The two-layer Lanczos method requires only the actions of the Hamiltonian operator on the Lanczos vectors, where the potential-vector products are accomplished via a pseudo-spectral transform technique. By using Jacobi, Radau and orthogonal satellite vectors, we have proposed 21 types of orthogonal coordinate systems so that the algorithm is capable of describing most five-atom systems with small and/or large amplitude vibrational motions. Finally, an universal program ( PetroVib) has been developed. Its applications to the molecules CH 4 and H 3 O 2 - , and the van der Waals cluster He 3 Cl 2 are also discussed.

Efficient solution of Poisson’s equation using discrete variable representation basis sets for Car–Parrinello ab initio molecular dynamics simulations with cluster boundary conditions

An efficient computational approach to perform Car-Parrinello ab initio molecular dynamics (CPAIMD) simulations under cluster (free) boundary conditions is presented. The general approach builds upon a recent real-space CPAIMD formalism using discrete variable representation (DVR) basis sets [Y. Liu et al., Phys. Rev. B 12, 125110 (2003); H.-S. Lee and M. E. Tuckerman, J. Phys. Chem. A 110, 5549 (2006)]. In order to satisfy cluster boundary conditions, a DVR based on sinc functions is utilized to expand the Kohn-Sham orbitals and electron density. Poisson's equation is solved in order to calculate the Hartree potential via an integral representation of the 1/r singularity. Excellent convergence properties are achieved with respect to the number of grid points (or DVR functions) and the size of the simulation cell. A straightforward implementation of the present approach leads to near linear scaling [O(N(4/3))] of the computational cost with respect to the system size (N) for the solution of Poisson's equation. The accuracy and stability of CPAIMD simulations based on sinc DVR are tested for a model problem as well as for N(2) and a water dimer.

Efficient numerical method for calculating exciton states in quantum boxes

We have developed an efficient numerical method for exciton states confined in quantum boxes. The exciton wave function is expanded in terms of discrete variable representation basis functions. Our numerical approach has proved to be computationally much less demanding in comparison with the conventional configuration-interaction one.

Dynamical properties of liquid water from ab initio molecular dynamics performed in the complete basis set limit

Dynamical properties of liquid water were studied using Car-Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] ab initio molecular dynamics (AIMD) simulations within the Kohn-Sham (KS) density functional theory employing the Becke-Lee-Yang-Parr exchange-correlation functional for the electronic structure. The KS orbitals were expanded in a discrete variable representation basis set, wherein the complete basis set limit can be easily reached and which, therefore, provides complete convergence of ionic forces. In order to minimize possible nonergodic behavior of the simulated water system in a constant energy (NVE) ensemble, a long equilibration run (30 ps) preceded a 60 ps long production run. The temperature drift during the entire 60 ps trajectory was found to be minimal. The diffusion coefficient [0.055 A2/ps] obtained from the present work for 32 D2O molecules is a factor of 4 smaller than the most up to date experimental value, but significantly larger than those of other recent AIMD studies. Adjusting the experimental result so as to match the finite-sized system used in the present study brings the comparison between theory and experiment to within a factor of 3. More importantly, the system is not observed to become "glassy" as has been reported in previous AIMD studies. The computed infrared spectrum is in good agreement with experimental data, especially in the low frequency regime where the translational and librational motions of water are manifested. The long simulation length also made it possible to perform detailed studies of hydrogen bond dynamics. The relaxation dynamics of hydrogen bonds observed in the present AIMD simulation is slower than those of popular force fields, such as the TIP4P potential, but comparable to that of the TIP5P potential.