Discrete Variable Representation Approach Research Articles

Near-infrared spectroscopy of H3O+⋯Xn (X = Ar, N2, and CO, n = 1-3).

Near-infrared (NIR) spectra of H3O+⋯Xn (X = Ar, N2, and CO, n = 1-3) in the first overtone region of OH-stretching vibrations (4800-7000 cm-1) were measured. Not only OH-stretching overtones but also several combination bands are major features in this region, and assignments of these observed bands are not obvious at a glance. High-precision anharmonic vibrational simulations based on the discrete variable representation approach were performed. The simulated spectra show good agreement with the observed ones and provide firm assignments of the observed bands, except in the case of X = CO, in which higher order vibrational mode couplings seem significant. This agreement demonstrates that the present system can be a benchmark for high precision anharmonic vibrational computations of NIR spectra. Band broadening in the observed spectra becomes remarkable with an increase of the interaction with the solvent molecule (X). The origin of the band broadening is explored by rare gas tagging experiments and anharmonic vibrational simulations of hot bands.

Symmetry Analysis of the Square Well Potential

Symmetry considerations are taken into account when a particle in a square well potential is studied. This system may display natural degeneracy, accidental degeneracy or systematic accidental degeneracy depending on the depth of the potential. In order to obtain the solutions associated with an arbitrary potential an algebraic discrete variable representation approach based on Pöschl-Teller functions is proposed. It is proved that the geometrical group C 4v is the symmetry group of the system for the case of a finite potential barrier. A similar analysis is carried out for the rectangular square well potential with commensurate sides. In both cases the symmetry projection is crucial to simplify the calculations.

Study of the Lennard-Jones and H2 potentials in the framework of the Morse based algebraic DVR approach

We propose a method based on an algebraic discrete variable representation (DVR) of the coordinate and momentum in the scheme of complete basis related to the 1D Morse potential, with this approach, a Hamiltonian associated to a 1D system can be expressed in terms of diagonal matrices using the transformation coefficients and the diagonalization of the matrix representation of the coordinate and momentum. We probe the scope of our method obtaining the solutions associated of the Lennard-Jones potential, also we apply the method to a practical example considering the ab initio potential of the H2 molecule. For both cases, the vibrational energies and the wave functions were obtained with good accuracy and low computational cost.

Algebraic discrete variable representation approach applied to Lennard-Jones and H2 potentials

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.

Application of the Discrete Variable Representation Approach to Formation of Amplitude–Phase Field Distributions on a Vertical Array in a Waveguide

Application of the Discrete Variable Representation Approach to Formation of Amplitude–Phase Field Distributions on a Vertical Array in a Waveguide

Vibrational spectroscopic signatures of hydrogen bond induced NH stretch-bend Fermi-resonance in amines: The methylamine clusters and other N-H⋯N hydrogen-bonded complexes.

The appearance of multiple bands in the N-H stretching region of the infrared spectra of the neutral methylamine dimer and trimer is a sign of NH bend-stretch anharmonic coupling. Ab initio anharmonic calculations were carried out in a step-wise manner to reveal the origin of various bands observed in the spectrum of the methylamine dimer. A seven-dimensional potential energy surface involving symmetric and asymmetric stretching and bending vibrations of both the hydrogen bond donor and the acceptor along intermolecular-translational modes was constructed using the discrete variable representation approach. The resulting spectrum of the dimer shows five bands that can be attributed to the symmetric stretching (νsym D), asymmetric stretchin (νasym D), and bending overtone (2νbend D) of the donor moiety. These appear along with the combination band arising out of bending vibrations of the donor and acceptor (νbend D + νbend A) and with the combination of the intermolecular translational mode over the donor bending overtone (νtrans + 2νbend D). The spectrum of the trimer essentially consists of all the features seen in the dimer with marginal changes in band positions. The analysis of the experimental spectra based on the two-state deperturbation model and ab initio anharmonic calculations yield a matrix element of about 40 cm-1 for the N-H bend-stretch Fermi resonance coupling. In general, the IR spectra of the hydrogen-bonded amino group depict three sets of bands that arise due to bend-stretch Fermi resonance coupling.

Branching Ratios, Radiative Lifetimes, and Transition Dipole Moments for YbOH.

Medium resolution (Δν̃ ∼ 3 GHz) laser-induced fluorescence (LIF) excitation spectra of a rotationally cold sample of YbOH in the 17300-17950 cm-1 range have been recorded using two-dimensional (excitation and dispersed fluorescence) spectroscopy. High resolution (Δλ ∼ 0.65 nm) dispersed laser-induced fluorescence (DLIF) spectra and radiative decay curves of numerous bands detected in the medium resolution LIF excitation spectra were recorded. The vibronic energy levels of the X̃2Σ+ state were predicted using a discrete variable representation approach and compared with observations. The radiative decay curves were analyzed to produce fluorescence lifetimes. DLIF spectra resulting from high resolution (Δν̃ < 10 MHz) LIF excitation of individual low-rotational lines in the Ã2Π1/2(0,0,0)-X̃2Σ+(0,0,0), Ã2Π1/2(1,0,0)-X̃2Σ+(0,0,0), and [17.73]Ω = 0.5(0,0,0)-X̃2Σ+(0,0,0) bands were also recorded. The DLIF spectra were analyzed to determine branching ratios which were combined with radiative lifetimes to obtain transition dipole moments. The implications for laser cooling and trapping of YbOH are discussed.

Implementation of analytical gradients and of a mixed real and momentum space DVR method for excess electron systems described by a self-consistent polarization model.

This work presents two extensions of our self-consistent polarization model for treating non-valence excess electron systems. The first extension is the implementation of analytical gradients, and the second extension is the implementation of a mixed real space plus momentum space approach combined with fast Fourier transforms to reduce the computational time compared to a purely real space discrete variable representation approach. The performance of the new algorithms is assessed in calculations of the excess electron states of various size water clusters and of the non-valence correlation-bound anion of the C240 fullerene.

A quantum-classical simulation of the nuclear dynamics in NO 2 − and C6H 6 + with realistic model Hamiltonian

We perform the nuclear dynamics simulation to calculate the broad band as well as better resolved (a) photodetachment spectra of nitrogen dioxide anion (NO −2 ), and (b) photoelectron spectra of benzene radical cation (C6H +6 ) with degeneracy among the electronic states using our parallelized time dependent discrete variable representation approach. For this purpose, we first consider a theoretical model that describes the photodetachment spectrum originating due to the Franck-Condon (FC) transition from the anionic ground state to the conically intersecting \(\tilde X^2 A_1 - \tilde A^2 B_2\) electronic manifold of the neutral NO2, whereas two multi-state, multi-mode model Hamiltonians of C6H +6 with degeneracy among the electronic states are explored for its’ photoelectron spectra. The dynamics of C6H +6 has been carried out with two Hamiltonians: (a) one consists with four states and nine modes, and (b) the other is constituted with three states and seven modes. Since the electronic states of both the systems are interconnected by several conical intersections in the vicinity of the FC region, it would be challenging to persue the dynamical calculation due to the impact of non-adiabaticity among the electronic states. The spectral profiles obtained with the advent of TDDVR approach show reasonably good agreement with the results obtained by other theoretical methodologies and experimental measurements.

A restricted quantum reaction path Hamiltonian: Theory, discrete variable representation propagation algorithm, and applications

A derivation of a quantum reaction path Hamiltonian is proposed, which is based on a reformulation of the classical version of Gonzalez et al. [J. Phys. Chem. A 105, 5022 (2001)], and the resulting equations are solved by means of a discrete variable representation approach, leading to a well-suited algorithm for the calculation of quantum dynamics of chemical reactions involving polyatomic molecules. General expressions for any type of reaction path are presented with special interest in the intrinsic reaction coordinate, which have been used to study selected cases, including a one-dimensional Eckart barrier, for which results are shown to be exact, two bidimensional systems, namely, a Muller-Brown potential energy surface, which is characteristic of polyatomic isomerization processes, and the collinear H + H(2) chemical reaction, and finally the tridimensional, J = 0, F + H(2) reaction. Results for the specific chemical systems are shown to be in quite good agreement with exact two- and three-dimensional quantum calculations concerning autocorrelation functions as well as transmission factors as a function of total energy.

Application of B-splines in determining the eigenspectrum of diatomic molecules: robust numerical description of halo-state and Feshbach molecules

The B-spline basis-set method is applied to determining the rovibrational eigenspectrum of diatomic molecules. Particular attention is paid to a challenging numerical task of an accurate and efficient description of the vibrational levels near the dissociation limit (halo-state and Feshbach molecules). Advantages of using B-splines are highlighted by comparing the performance of the method with that of the commonly used discrete-variable representation (DVR) approach. Several model cases, including the Morse potential and realistic potentials with 1/R3and 1/R6long-range dependence of the internuclear separation are studied. We find that the B-spline method is superior to the DVR approach and it is robust enough to properly describe the Feshbach molecules. The developed numerical method is applied to studying the universal relation of the energy of the last bound state to the scattering length. We illustrate numerically the validity of the quantum-defect-theoretic formulation of such a relation for a 1/R6potential.PACS Nos.: 31.15.–p,34.50.Cx

Lower vibrational states of SO 2. Density functional and discrete variable representation approach

A set of lower vibrational levels of SO 2 ( 1A 1) has been calculated with inclusion of anharmonic effects. The discrete variable representation of bending coordinate and a two-dimensional distributed Gaussian basis expansion for stretching coordinates have been used. Complete quartic force fields in three dimensions have been constructed, based on density functional theory (BLYP, B3LYP, B3PW91) and CASSCF(14/10) calculations. Our results indicate the B3LYP approach as the best one for reproduction of the observed vibrational excitation energies and a set of Dunham-type spectroscopic constants for the molecule studied.

Molecules MH3 (M = Cr, Mo, W) Are Pyramidal: CCSD(T) Ab Initio Study of Structure and Vibrational Dynamics

Ab initio calculations were carried out on chromium, molybdenum, and tungsten trihydrides. The equilibrium parameters of the ground states, vibrational frequencies, IR intensities and the inversion barriers of the molecules were calculated at the CCSD(T) level with large basis sets. The relative energies to the excited states were evaluated by the EOM−CCSD method. All molecules were found to have pyramidal (C3v) equilibrium structures. The calculated inversion barriers of CrH3, MoH3, and WH3 are 1353, 1086, and 472 cm-1, respectively. The vibrational levels corresponding to the totally symmetric H−M−H bending mode were predicted using the discrete variable representation approach. The tunneling splitting and rates were also evaluated within the Wentzel−Kramers−Brillouin approximation. The available experimental data are discussed in light of results from the ab initio calculations.

A generalized discrete variable representation approach to interpolating or fitting potential energy surfaces

A general interpolation/fitting method is presented for constructing a smooth molecular potential energy surface from ab initio energy points. The approach is based on generalized discrete variable representation theory, where a damping term has been introduced. The method is illustrated by constructing a one-dimensional potential representing H 2 and a two-dimensional surface representing collinear H 3.

Four-dimensional model calculation of torsional levels of cyclic water tetramer

Quantum four-dimensional model calculations of the coupled intermolecular torsional vibrations of the cyclic homodromic water tetramers (H2O)4 and (D2O)4 are presented, based on the analytical modEPEN4B potential energy surface [S. Graf and S. Leutwyler, J. Chem. Phys. 109, 5393 (1998), preceding paper] and a four-dimensional discrete variable representation approach. The lowest 50 torsional levels were calculated up to 420 and 500 cm−1 for (D2O)4 and (H2O)4, respectively. For both clusters, the torsional ground state is split by a synchronous O–H torsional inversion process, similar to inversion tunneling in ammonia, with calculated tunnel splittings of 21.8 and 0.000 12 MHz for (H2O)4 and (D2O)4, respectively. As for the cyclic water trimer and pentamer, the four torsional fundamentals of the tetramer lie above the torsional interconversion barriers, between 185–200 cm−1 for (D2O)4 and 229–242 cm−1 for (H2O)4, but also lie below the one-dimensional torsionally adiabatic barriers. The anharmonic fundamental frequencies lie both above and below the normal-mode frequencies, by up to 33%. Slightly above the fundamental torsional excitations, at 257–260 and 280–281 cm−1 for (H2O)4 and (D2O)4, respectively, lie four states corresponding to four versions of the {uudd} isomer, which form a pseudorotational manifold; the torsional interconversion occurs by a sequence of double O–H flips. Higher excited pseudorotational states are calculated up to a vibrational angular momentum of k=3. At ≈295 and ≈300 cm−1, a further group of eight states is found, corresponding to the eight permutationally equivalent versions of yet another isomer, the {uuud} structure. The four {uudd} and eight {uuud} states of (H2O)4 exhibit inverse isotope effects, and lie at lower energy than their (D2O)4 counterparts.

Three-dimensional model treatment of the torsional levels of isotopic water trimers

A coupled three-dimensional model calculation of the low-frequency intermolecular torsional levels of the isotopically substituted water trimers (H 2O) 2 · D 2O and (D 2O) 2 · H 2O is presented, using the modEPEN torsional potential energy surface and a three-dimensional discrete variable representation approach. Relative to the pure trimers (D 2O) 3 and (H 2O) 3, the dynamical symmetry breaking leads to shifts of all torsional levels, as well as splittings of the degenerate e u and e g torsional levels of up to 320 cm −1. The lowest nine (eight) levels of (H 2O) 2 · D 2O ((D 2O) 2 · H 2O) are pseudorotational states. The lowest torsional excitation of (D 2O) 2 · H 2O is predicted to lie at only 1.6 cm −1.

Three-dimensional model calculation of torsional levels of (H 2O) 3 and (D 2O) 3

A coupled three-dimensional model calculation of the low-frequency large-amplitude intermolecular torsional states in (H 2O) 3 and (D 2O) 3 is presented, based on the analytical modEPEN intermolecular potential surface and a three-dimensional discrete variable representation approach. The lowest torsional levels of both (H 2O) 3 and (D 2O) 3 lie above the sixfold (upd) torsional barrier. The first eight (eleven) torsions of (H 2O) 3 ((D 2O) 3) are pseudorotational states. The ‘radial’ and ‘polar’ torsional fundamental frequencies are predicted at 151 and 160 cm −1 for (D 2O) 3, and for (H 2O) 3 at 185.0 and 185.3 cm −1, respectively. Each of these in turn support a ladder of pseudorotational levels.

Toeplitz matrices within DVR formulation: Application to quantum scattering problems

It is shown that the matrix obtained from the infinite order discrete variable representation (DVR) of a scattering problem has the structure of a Toeplitz matrix. The resulting properties can be used to reduce the associated infinite system of algebraic equations to a finite (and relatively small) one. An example is worked out to show the efficiency of the combined Toeplitz DVR approach.

The Fourier-grid formalism: philosophy and application to scattering problems using R-matrix theory

The Fourier-grid (FG) method is a recent L2 variational treatment of the quantum mechanical eigenvalue problem that does not require the use of a set of basis functions: it is rather a discrete variable representation approach. The author restates the FG philosophy in more general terms, examine and compare this method with other approaches to the eigenvalue problem, and begin the development of an FG R-matrix method for scattering. The philosophy of the FG method is to use the simplest representation for each of the kinetic and potential energy operators of the Hamiltonian, and use a generalized Fourier transform to put the matrix elements of one of the above operators in the same representation as the other, so the Hamiltonian has a single representation. Thus, the Hamiltonian is represented at discrete points in either configuration or its reciprocal space.