Rotational Decoupling Approximations

A fundamental quantity often measured by experimentalists is the collision cross section which determines the effective size of the colliding particles. In an inelastic collision of an atom with a diatomic molecule many vibrational and rotational states are strongly coupled together. The equations are difficult to solve and hence requires massive amounts of computer time. We became interested in these processes and showed that an Infinite Order Sudden Approximation (IOSA) was accurate enough for many purposes. We then showed that one can drastically simplify the expressions for various cross sections. It was demonstrated that within this approximation one could view the collision of an atom with a diatomic molecule as fixed orientation collisions. That is, the collision can occur before the molecule has time to rotate. Hence, these simplified cross section formulae are angular averages of the fixed orientation results. One of my papers in this area was the most cited article in the {\it Journal of Chemical Physics} and was the second most cited article in the area of Atomic and Molecular Physics during 1978 and 1979.\footnote{$\dagger$} {Eugene Garfied - Current Contents -- Articles Most cited in 1978 and 1979 -- G. A. Parker and R. T Pack, ``Rotationally and Vibrationally Inelastic Scattering in the Rotational IOS Approximation. Ultrasimple Calculation of Total (Differential, Integral, and Transport) Cross Sections for Nonspherical Molecules'', {\it J. Chem. Phys.} {\bf 68}, 1585-1601 (1978).} Derived simplified differential cross section formulas and identified the partial-wave parameter in the J$_z$--CCS approximation for molecular scattering. Developed ultrasimple methods for calculation of Total (Differential, Integral and Transport) cross sections for nonspherical molecules in the Infinite Order Sudden Approximation (IOS).

• T. P. Tsein, G. A. Parker and R. T Pack, ``Rotationally Inelastic Molecular Scattering. Computational Tests of
  Some Simple Solutions of the Strong Coupling Problem'', {\it J. Chem. Phys.} {\bf 59}, 5373-5381 (1973).
• G. A. Parker and R. T Pack, ``Identification of the Partial-Wave Parameter and Simplification of the Differential Cross Section in the J$_{z}$-CCS Approximation in Molecular Scattering'', {\it J. Chem. Phys.} {\bf 66}, 2850-2853 (1977).
• G. A. Parker and R. T Pack, ``Rotationally and Vibrationally Inelastic Scattering in the Rotational IOS Approximation. Ultrasimple Calculation of Total (Differential, Integral, and Transport) Cross Sections for Nonspherical Molecules'', {\it J. Chem. Phys.} {\bf 68}, 1585-1601 (1978).

Intermolecular Potentials

Developed an electron-gas method for calculating intermolecular potentials from electron densities. This method assumed that the electrons formed a uniform electron gas and gave reasonable results for short range interactions. The long range interactions were obtained from van der Waals interactions obtained from frequency dependent polarizabilities using Pade approximates.

• G. A. Parker, R. L. Snow and R. T Pack, ``Calculation of Molecule-Molecule Intermolecular Potentials Using Electron Gas Methods'', {\it Chem. Phys Lett.} {\bf 33}, 399-403 (1975).
• G. A. Parker and R. T Pack, ``van der Waals Interactions of Carbon Monoxide'', {\it J. Chem. Phys.} {\bf 64}, 2010-2012 (1976).
• G. A. Parker, R. L. Snow and R. T Pack, ``Intermolecular Potential Surfaces from Electron Gas Methods. I. Angle and Distance Dependence of the $He+CO_{2}$ and $Ar+CO_{2}$ Interactions'', {\it J. Chem Phys.} {\bf 64}, 1668-1678 (1976).

Infinite Order Sudden Analysis of Differential Cross Section Data

Showed that the central field potentials fit to experimental data for highly anisotropic collision partners {\it do not} correspond to the spherical average of the intermolecular potential. Developed a simplified procedure using the IOSA for determining reliable and accurate intermolecular potentials from crossed molecular beam experiments. This method is now routinely used by experimentalist throughout the world for analysis of total differential cross section data. Using our multi-property fitting method one can obtain accurate anisotropic intermolecular potentials from experimental data.

• M. Keil, G. A. Parker and A. Kuppermann, ``An Empirical Anisotropic Intermolecular Potential for $He+CO_{2}$'' {\it Chem. Phys. Lett.} {\bf 59}, 443-448 (1978).
• G. A. Parker, M. Keil and A. Kuppermann, ``Scattering of Thermal He Beams by Crossed Atomic and Molecular Beams.V. Anisotropic Intermolecular Potentials for $He+ CO_{2},N_{2}O$ and $C_{2}N_{2}$'',
• {\it J. Chem. Phys.} {\bf 78}, 1145-1162 (1983). M. Keil and G. A. Parker, ``Empirical Potential for the $He+CO_{2}$ Interaction: Multi-property Fitting in the Infinite-Order Sudden Approximation.'', {\it J. Chem. Phys.} {\bf 82}, 1947-1966 (1985).

Solution of Coupled Differential Equations

Developed the VIVAS method which is an efficient and accurate method for solving the coupled differential equations encountered in quantum scattering.

• G. A. Parker, T. G. Schmalz and J. C. Light, ``A Variable Interval Variable Step Method for the Solution of Coupled Second Order Differential Equations'', {\it J. Chem. Phys.} {\bf 73}, 1757-1764 (1980).
• G. A. Parker, J. C. Light and B. R. Johnson, ``The Logarithmic Derivative-Variable Interval Variable Step Hybrid Method for the Solution of Coupled Linear Second Order Differential Equations'', {\it Chem. Phys. Lett.} {\bf 73}, 572-575 (1980).

Discrete Variable Representation

Contributed to the development of the Discrete Variable Representataion (DVR) method which is widely used in molecular scattering. In this representation the potential is diagonal and the kinetic energy operator provides the coupling between different quantum states. The Hamiltonian can be easily constructed and efficiently stored.

• J. V. Lill, G. A. Parker and J. C. Light, ``Discrete Variable Representations and Sudden Models in Quantum Scattering Theory'', {\it Chem. Phys. Lett.} {\bf 89}, 483-489 (1982).
• J. V. Lill, G. A. Parker and J. C. Light, ``The Discrete Variable-Finite Basis Approach to Quantum Scattering'' {\it J. Chem. Phys.} {\bf 85}, 900-910 (1986).

Angular Correlation

Derived expressions for analysis of angular correlation experiments.

Reactive Scattering

Reactive scattering processes form the heart of chemistry and the quantum theory thereof is one of the most important problems in theoretical chemistry or theoretical atomic and molecular physics. We developed the theory and computer code for reactive scattering using Adiabatically-adjusting Principal-axis Hyperspherical (APH) coordinates. We produce essentially exact bench mark results on a variety of chemically interesting reactions. We have contributed significantly to current understanding of reactive scattering. We have seen that quantum resonances are very common and may dominate most chemical reactions. We have been the first to calculate state-to-state results for the following systems:

• $$F+H_2 \rightleftharpoons H+FH $$
• $$e^++H \rightleftharpoons p^++Ps$$
• $$He+H_2^+ \rightleftharpoons HeH^++H$$
• $$Li+FH \rightleftharpoons LiF+H$$
• $$H+O_2 \rightleftharpoons O+OH$$

References

• G. A. Parker, R. T Pack, B. J. Archer and R. B. Walker, ``Quantum Reactive Scattering in Three Dimensions using Hyperspherical (APH) Coordinates. Test on $H+H_{2}$ and $D+H_{2}$.'' {\it Chem. Phys. Lett.} {\bf 137}, 564-568 (1987).
• R. T Pack and G. A. Parker, ``Quantum Reactive Scattering in Three Dimensions Using Hyperspherical (APH) Coordinates. II. Theory.'' {\it J. Chem. Phys.} {\bf 87}, 3888-3921 (1987).
• J. D. Kress, Z. Bacic, G. A. Parker and R. T. Pack, ``Quantum Effects in the $F+H_{2} \rightleftharpoons HF+H $ Reaction. Accurate 3D Calculations with a Realistic Potential Energy Surface.'' {\it Chem. Phys. Lett.} {\bf 157}, 484-490 (1989).
• B. J. Archer, G. A. Parker and R. T Pack, ``Positron-Hydrogen Atom S-Wave Coupled Channel Scattering at Low Energies''. {\it Phys. Rev. Lett.} {\bf 41}, 1303-1310 (1990).
• Z. Bacic, J. D. Kress, G. A. Parker and R. T Pack, ``Quantum Reactive Scattering in Three Dimensions Using Hyperspherical (APH) Coordinates. IV. Discrete Variable Representation (DVR) Basis Functions and the Analysis of Accurate Results for $F+H_{2}$ ''. {\it J. Chem. Phys. } {\bf 92}, 2344-2361 (1990).
• A. Lagan\`a, R. T Pack, and G. A. Parker, ``Li+FH Reactive Cross Sections From J=0 Accurate Quantum Reactivity'' {\it J. Chem. Phys.} {\bf 99}, 2269-2270 (1993).
• R. T Pack, E. A. Butcher and G. A. Parker, ``Accurate Quantum Probabilities and Threshold Behavior of the $H+O_{2}$ Combustion Reaction''. {\it J. Chem. Phys.} {\bf 99}, 9310-9313 (1993) (Rapid Communication).
• C. Y. Yang, S. J. Klippenstein, J. D. Kress, R. T Pack, G. A. Parker, and A. Lagan\`a, ``Comparison of Transition State Theory with Quantum Scattering Theory for the Reaction $Li+HF \rightleftharpoons LiF+H$, {\it J. Chem. Phys.}, {\bf 100}, 4917-4924 (1994).
• G. A. Parker, A. Lagan\`a, S. Croccianti, and R. T Pack, ``A Detailed 3D Quantum Study of the $Li+HF$ Reaction.'' {\it J. Chem. Phys.}, {\bf 102} 1238-1250 (1995).
• R. T Pack, E. A. Butcher, and G. A. Parker, ``Accurate 3D Quantum Probabilities and Collision Lifetimes of the $H+O_2$ Combustion Reaction,'' {\it J. Chem. Phys.}, {\bf 102} 5998-6012 (1995).

Reactive Scattering Simulations.

In collaborations with Molecular Beam Experimentalist Prof. Mark Keil we have implemented a computer program for fully simulating the experimental characteristics in his laboratory. This enables direct comparisons of theoretically calculated differential cross sections to measured laboratory frame angular distributions. Thus it is no longer necessary to de-convolute the measured angular distributions.

• G. Dharmasena, T. R. Phillips, K. N. Shokhirev, G. A. Parker and M. Keil,``Vibrationally and Rotationally Resolved Angular Distributions for $F+H_2\rightleftharpoons HF(\nu^\prime,j^\prime)+H$ Reactive Scattering.'' {\it J. Chem. Phys.}, {\bf 106}, 9950-9953, (1997). (Rapid Communication)
• G. Dharmasena, K. A. Copeland, J. A. Young, R. A. Lasell, T. R. Phillips, G. A. Parker and M. Keil, ``Angular Dependence for the $\nu^\prime j^\prime$ -- Resolved States in $F+H_2\rightleftharpoons HF+H$ Scattering.'' {\it J. Phys. Chem.}, {\bf 101}, 6429-6440, (1997).
• G. Dharmasena, T. R. Phillips, G. A. Parker and M. Keil, ``Angular Distributions for Specific $HF$ Vib-Rotational States from $F+H_2\rightleftharpoons HF+H$ Reactive Scattering.'' To be submitted to the Journal of Chemical Physics.

Distributed Approximating Functionals

Contributed to the understanding of Distributed Approximating Functionals.

• W. Zhu, Y. Huang, G. A. Parker, D. J. Kouri, and D. K. Hoffman, ``Application of Distributed Approximating Functionals for Atom-Rigid Rotor Inelastic Scattering: Body Frame Close-Coupling Time-Dependent and Time-Independent Wavepacket Approaches.'' {\it J. Phys. Chem.}, {\bf 98}, 12516-12520 (1994).
• D. J. Kouri, W. Zhu, G. A. Parker and D. K. Hoffman, ``Accelleration of Convergence in the Polynomial-Expanded Spectral Density Approach to Bound and Resonance State Calculations.''
• S. S. Iyengar, G. A. Parker, D. J. Kouri, and D. K. Hoffman ``Symmetry Adapted Distributed Approximating Functionals:
  Theory and Application to the ro-vibrational states of $H_3^+$.'' {\it J. Chem. Phys.}, {\bf 110}, 10283-10298, (1999).
• K. Zhang, G. A. Parker, D. J. Kouri, D. K. Hoffman, S. S. Iyengar ``Quantum Reactive Scattering in Three Dimensions using Hyperspherical (APH) Coordinates: Periodic Distributed Approximating Functional (PDAF) Method for Surface Functions.'' {\it J. Chem. Phys.} {\bf }, 569-581 (2003).
  

Iterative Diagonalizations of Large Matrices

Developed the Spectral Density method for iterative solution of the matrix eigenvalue problem.

• D. J. Kouri, W. Zhu, G. A. Parker and D. K. Hoffman, ``Accelleration of Convergence in the Polynomial-Expanded Spectral Density Approach to Bound and Resonance State Calculations.'' {\it Chem. Phys. Lett.}, {\bf 238} 395-403 (1995).
• G. A. Parker, W. Zhu, Y. H. Huang, D. K. Hoffman, and D. J. Kouri, ``Matrix Pseudo-Spectroscopy -- Iterative Calculation of Matrix Eigenvalues and Eigenvectors of Large Matrices Using a Polynomial Expansion of the Dirac Delta-Function.'' {\it Comp. Phys. Comm.}, {\bf 96} 27-35 (1996).