6D vibrational quantum dynamics: Generalized coordinate discrete variable representation and (a)diabatic contraction

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 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

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 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.

Potential optimized discrete variable representation

Vibrational overtone excitation of jet-cooled methanol, in combination with infrared laser assisted photofragment spectroscopy (IRLAPS) detection, reveals OH stretch bands that are significantly simplified with respect to room-temperature spectra. The simplification afforded by jet-cooling permits the observation of spectral splitting on the order of 50 cm−1 in the region of the 5ν1 OH stretch overtone band. Tracking this splitting as a function of OH stretch vibrational level in combination with isotopic substitution studies allows us to identify the perturbing state as the combination level involving four quanta of OH stretch and one quantum of CH asymmetric stretch, 4ν1+ν2. Careful examination of the spectra reveals that this strong interaction arises from a fourth-order anharmonic term in the Hamiltonian that couples the OH and CH ends of the molecule. These frequency domain results indicate that subsequent to coherent excitation of the 5ν1 band, methanol would undergo energy redistribution to the methyl part of the molecule on a time scale of ∼130 fs. This work also suggests that similar strong resonances may occur more generally in molecules that possess two different high-frequency oscillators in close proximity.

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.

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.

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).

Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.

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 photodissociation dynamics of methyl iodide is investigated on a three-dimensional grid using a time-dependent quantum-mechanical method. Two electronic states which correlate with I and I* fragments are explicitly included in the calculation. The potential-energy functions and the nonadiabatic coupling are adapted from a recent ab initio calculation by Morokuma and co-workers. The dynamically active degrees of freedom include the dissociation coordinate, the umbrella bend of the methyl group which is treated as a stretch between a hypothetic atom X (X=H3) and the carbon atom, and the H3–C–I bend. The discrete variable representation is used to describe the dynamics in the bending coordinate θ while the other two degrees of freedom are treated by a fast Fourier transform (FFT) based approach. The time propagation of the wave packet is carried out using the Chebychev expansion of the time propagator and the grid in the translational coordinate space is shifted during the propagation to avoid reflection at the end of the grid. Absorption spectra calculated from these ab initio surfaces are found to be blue shifted by a few thousand wave numbers and have broader widths when compared with experimental results. These discrepancies can be attributed to the topology of the ab initio potential-energy surfaces in the Franck–Condon region. Vibrational and rotational distributions for both the CH3 and CD3 fragments are calculated at several photon wavelengths. The agreement with experimental results is excellent. The vibration of the methyl fragment has a weak dependence on the photon wavelength and is found to be largely separable from its rotation. The rotational distribution of the methyl fragment is cold in the I* channel, but relatively hot for the lower I channel. The I* yield is strongly influenced by the initial excitation scheme. When both excited states are considered to be optically active, the calculated I* yield gives a better agreement with experimental observations. Our calculations are also compared with several previous theoretical works.

Computation of Rovibrational Eigenvalues of van der Waals Molecules on a CRAY T3D