A discrete variable representation method for studying the rovibrational quantum dynamics of molecules with more than three atoms

Program Synthesis of Sparse Algorithms for Wave Function and Energy Prediction in Grid-Based Quantum Simulations

We have recently shown how program synthesis (PS), or the concept of "self-writing code", can generate novel algorithms that solve the vibrational Schrödinger equation, providing approximations to the allowed wave functions for bound, one-dimensional (1-D) potential energy surfaces (PESs). The resulting algorithms use a grid-based representation of the underlying wave function ψ(x) and PES V(x), providing codes which represent approximations to standard discrete variable representation (DVR) methods. In this Article, we show how this inductive PS strategy can be improved and modified to enable prediction of both vibrational wave functions and energy eigenvalues of representative model PESs (both 1-D and multidimensional). We show that PS can generate algorithms that offer some improvements in energy eigenvalue accuracy over standard DVR schemes; however, we also demonstrate that PS can identify accurate numerical methods that exhibit desirable computational features, such as employing very sparse (tridiagonal) matrices. The resulting PS-generated algorithms are initially developed and tested for 1-D vibrational eigenproblems, before solution of multidimensional problems is demonstrated; we find that our new PS-generated algorithms can reduce calculation times for grid-based eigenvector computation by an order of magnitude or more. More generally, with further development and optimization, we anticipate that PS-generated algorithms based on effective Hamiltonian approximations, such as those proposed here, could be useful in direct simulations of quantum dynamics via wave function propagation and evaluation of molecular electronic structure.

Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy

Algebraic discrete variable representation (DVR) methods that have been recently proposed are applied to describe 1D and 2D piecewise potentials. First, it is shown that it is possible to use a DVR approach to describe 1D square well potentials testing the wave functions with exact results. Thereafter, Morse and Pöschl-Teller (PT) potentials are described with multistep piecewise potentials in order to explore the sensibility of the potential to reproduce the transition from a pure square well energy pattern to an anharmonic energy spectrum. Once the properties of the different algebraic DVR approaches are identified, the 2D square potential as a function of the potential depth is studied. We show that this system displays natural degeneracy, accidental degeneracy and systematic accidental degeneracy. The latter appears only for a confined potential, where the symmetry group is identified and irreducible representations are constructed. One particle confined in a rectangular well potential with commensurate sides is also analyzed. It is proved that the systematic accidental degeneracy appearing in this system is removed for finite potential depth.

Advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase

The purpose of this review is to demonstrate advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase. Molecular spectroscopy, particularly vibrational spectroscopy and electronic spectroscopy, has been used extensively for a wide range of areas of chemical sciences and materials science as well as nano- and biosciences because it provides valuable information about structure, functions, and reactions of molecules. In the meantime, quantum chemical approaches play crucial roles in the spectral analysis. They also yield important knowledge about molecular and electronic structures as well as electronic transitions. The combination of spectroscopic approaches and quantum chemical calculations is a powerful tool for science, in general. Thus, our article, which treats various spectroscopy and quantum chemical approaches, should have strong implications in the wider scientific community. This review covers a wide area of molecular spectroscopy from far-ultraviolet (FUV, 120-200 nm) to far-infrared (FIR, 400-10 cm-1)/terahertz and Raman spectroscopy. As quantum chemical approaches, we introduce several anharmonic approaches such as vibrational self-consistent field (VSCF) and the combination of periodic harmonic calculations with anharmonic corrections based on finite models, grid-based techniques like the Numerov approach, the Cartesian coordinate tensor transfer (CCT) method, Symmetry-Adapted Cluster Configuration-Interaction (SAC-CI), and the ZINDO (Semi-empirical calculations at Zerner's Intermediate Neglect of Differential Overlap). One can use anharmonic approaches and grid-based approaches for both infrared (IR) and near-infrared (NIR) spectroscopy, while CCT methods are employed for Raman, Raman optical activity (ROA), FIR/terahertz and low-frequency Raman spectroscopy. Therefore, this review overviews cross relations between molecular spectroscopy and quantum chemical approaches, and provides various kinds of close-reality advanced spectral simulation for condensed phases.

Algebraic DVR Approaches Applied to Describe the Stark Effect

Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the operators r2 and p2 are diagonalized to define useful bases to obtain the matrix representation of the Hamiltonian in a simple form in terms of diagonal matrices. The second approach is based on the U(4) dynamical algebra which consists of the addition of a scalar boson to the 3D harmonic oscillator space keeping constant the total number of bosons. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations, whose involved branching rules define the discrete variable representation. Both methods, although originating from the harmonic oscillator basis, provide different convergence tests due to the fact that the associated discrete bases turn out to be different. These approaches provide powerful tools to obtain the matrix representation of 3D general Hamiltonians in a simple form. In particular, the Hydrogen atom interacting with a static electric field is described. To accomplish this task, the diagonalization of the exact matrix representation of the Hamiltonian is carried out. Particular attention is paid to the subspaces associated with the quantum numbers n=2,3 with m=0.

A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

One of the most accurate methods for solving the time-dependent Schrödinger equation uses a combination of the dynamic Fourier method with the split-operator algorithm on a tensor-product grid. To reduce the number of required grid points, we let the grid move together with the wavepacket but find that the naïve algorithm based on an alternate evolution of the wavefunction and grid destroys the time reversibility of the exact evolution. Yet, we show that the time reversibility is recovered if the wavefunction and grid are evolved simultaneously during each kinetic or potential step; this is achieved by using the Ehrenfest theorem together with the splitting method. The proposed algorithm is conditionally stable, symmetric, and time-reversible and conserves the norm of the wavefunction. The preservation of these geometric properties is shown analytically and demonstrated numerically on a three-dimensional harmonic model and collinear model of He-H₂ scattering. We also show that the proposed algorithm can be symmetrically composed to obtain time-reversible integrators of an arbitrary even order. We observed 10 000-fold speedup by using the tenth-order instead of the second-order method to obtain a solution with a time discretization error below 10^-9. Moreover, using the adaptive grid instead of the fixed grid resulted in a 64-fold reduction in the required number of grid points in the harmonic system and made it possible to simulate the He-H₂ scattering for six times longer while maintaining reasonable accuracy. Applicability of the algorithm to high-dimensional quantum dynamics is demonstrated using the strongly anharmonic eight-dimensional Hénon-Heiles model.

Communication: General variational approach to nuclear-quadrupole coupling in rovibrational spectra of polyatomic molecules

A general algorithm for computing the quadrupole-hyperfine effects in the rovibrational spectra of polyatomic molecules is presented for the case of ammonia (NH₃). The method extends the general variational approach TROVE [J. Mol. Spectrosc. 245, 126-140 (2007)] by adding the extra term in the Hamiltonian that describes the nuclear quadrupole coupling, with no inherent limitation on the number of quadrupolar nuclei in a molecule. We applied the new approach to compute the nitrogen-nuclear-quadrupole hyperfine structure in the rovibrational spectrum of NH314. These results agree very well with recent experimental spectroscopic data for the pure rotational transitions in the ground vibrational and ν₂ states and the rovibrational transitions in the ν₁, ν₃, 2ν₄, and ν₁ + ν₃ bands. The computed hyperfine-resolved rovibrational spectrum of ammonia will be beneficial for the assignment of experimental rovibrational spectra, further detection of ammonia in interstellar space, and studies of the proton-to-electron mass variation.

Ro-vibronic transition intensities for triatomic molecules from the exact kinetic energy operator; electronic spectrum for the C̃ 1B2 ← X̃ 1A1 transition in SO2

A procedure for calculating ro-vibronic transition intensities for triatomic molecules within the Born-Oppenheimer approximation is reported. Ro-vibrational energy levels and wavefunctions are obtained with the DVR3D suite, which solves the nuclear motion problem with an exact kinetic energy operator. Absolute transition intensities are calculated both with the Franck-Condon approximation and with a full transition dipole moment surface. The theoretical scheme is tested on C̃ ¹B₂ ← X̃ ¹A₁ ro-vibronic transitions of SO₂. Ab initio potential energy and dipole moment surfaces are generated for this purpose. The calculated ro-vibronic transition intensities and cross sections are compared with the available experimental and theoretical data.