J. V. Lill

Bio: J. V. Lill is an academic researcher from University of Chicago. The author has contributed to research in topics: Scattering & Scattering theory. The author has an hindex of 3, co-authored 3 publications receiving 1947 citations.

Generalized discrete variable approximation in quantum mechanics

Abstract: 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.

The discrete variable–finite basis approach to quantum scattering

Abstract: A discrete variable representation for scattering problems is developed. In this representation the potential matrix is diagonal, with elements being the potential evaluated at the proper quadrature points. The angular momentum operators may be treated exactly up to truncation of the basis set and provide the coupling in the coordinate‐labeled discrete variable representation. The definition of the inner product over the internal coordinates as quadratures rather than integrations allows a discrete matrix transformation to be used to diagonalize any potential matrix. This framework allows one to obtain approximate solutions in a particularly simple and efficient manner and is presented in detail for atom–diatom collisions. At large values of the scattering distance the coupled equations may be solved to a high degree of accuracy using the distorted wave approximation in the finite basis representation. At small values of the scattering distance an exactly analogous technique may be used to obtain an approximate solution in the discrete variable representation. The solutions may be matched exactly and the exact scattering boundary conditions applied. Numerical results from atom–rigid rotor collisions are presented.

Measurement and simulation of temperature-dependent spontaneous Raman scattering of O2 including P and R branches

Abstract: Evaluation methods for species and temperature determination in gaseous mixtures using spontaneous Raman scattering require detailed information on the spectra of the involved species. For most diatomic and some triatomic molecules that are relevant in combustion processes (H2, N2, O2, CO, CO2, H2O) these spectra can be simulated based on the underlying quantum mechanical processes. In contrast to the other diatomic molecules, the electronic ground state of oxygen has an electronic spin of S=1 which leads to the tripling of transitions and the occurrence of P and R branches. Though being neglected so far due to their small effect size, these additional transitions change the spectral shape and the integrated signal intensity which can lead to inaccuracies in evaluation methods such as the hybrid matrix inversion or full spectral fit. In this paper, P and R branches were simulated and their effect on the ro-vibrational oxygen spectrum evaluated by comparison to high-resolution experimental spectra in temperatures up to over 2000 K. Spectral fitting of O2 using this simulation allows for temperature determination of gaseous mixtures with a uncertainty better than 10 K and no significant difference to temperatures determined with the more established fitting of N2. Fitted temperatures deviate by 4 K or less when P and R transitions are considered but fitting quality improves significantly when including them in the simulation. More importantly, neglecting P and R transitions leads to an overestimation of the temperature-dependent Raman cross section of O2 which causes underestimations of O2 concentration measurements using the hybrid matrix inversion or full spectral fit method.

Generalized discrete variable approximation in quantum mechanics

Abstract: 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.

The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions

Abstract: A new method for the calculation of bound state eigenvalues and eigenfunctions of the Schrodinger equation is presented. The Fourier grid Hamiltonian method is derived from the discrete Fourier transform algorithm. Its implementation and use is extremely simple, requiring the evaluation of the potential only at certain grid points and yielding directly the amplitude of the eigenfunctions at the same grid points.

Beyond Born-Oppenheimer: molecular dynamics through a conical intersection.

Abstract: Nonadiabatic effects play an important role in many areas of physics and chemistry. The coupling between electrons and nuclei may, for example, lead to the formation of a conical intersection between potential energy surfaces, which provides an efficient pathway for radiationless decay between electronic states. At such intersections the Born-Oppenheimer approximation breaks down, and unexpected dynamical processes result, which can be observed spectroscopically. We review the basic theory required to understand and describe conical, and related, intersections. A simple model is presented, which can be used to classify the different types of intersections known. An example is also given using wavepacket dynamics simulations to demonstrate the prototypical features of how a molecular system passes through a conical intersection.

An overlap fitted chain of spheres exchange method.

Abstract: The “chain of spheres” (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, Q, which leads the numerical and analytically evaluated overlap matrices to coincide. Using Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calculations and geometry optimizations. The fitting procedure results in a reduction of grid sizes necessary for achieving chemical accuracy. We demonstrate this by testing a number of grids and comparing results to the fully analytic and the earlier COS approximations. This turns out to be favourable for total and reaction energies, for which chemical accuracy can now be reached with a corresponding ∼30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favourable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theoretical framework.