M. L. Koszykowski

Semiclassical calculation of eigenvalues for a three‐dimensional system

A method utilizing integration along invariant curves on Poincares surfaces of section is described for the semiclassical calculation of eigenvalues for three and higher dimensional systems, supplementing thereby our previous work in two dimensions. The eigenvalues calculated for anharmonically coupled oscillators agree well with the exact quantum eigenvalues.

Mechanism reduction via principal component analysis

Principal component analysis, an advanced technique of sensitivity analysis, has been used to determine reduced mechanisms that can model species and temperature profiles in Plug Flow Reactors (PFR), Premixed Laminar Flames (PLF), and Perfectly Stirred Reactors (PSR) for two H2/air and two CH4/air mechanisms over a range of input parameters including initial temperature, equivalence ratio, and residence time. The results show that principal component analysis can be used effectively to reduce a comprehensive mechanism that contains unimportant reactions to a reduced mechanism that contains necessary and sufficient reactions. The accuracy of a reduced mechanism determined from principal component analysis can be easily controlled by carefully selecting reduction criteria. For the conditions chosen here, namely the requirement that radical profiles computed with reduced and comprehensive mechanisms agree to within 5%, substantial reductions were not achieved. Principal component analysis is able to resolve the influence of stoichiometry, combustor type, and mechanism on mechanism reduction. The two H2/air mechanisms were each reduced to mechanisms that can model all the cases considered, and the extent of reduction in each was very similar and modest. For H2/air chemistry, equivalence ratio had little effect on reduction. Combustor type was slightly more influential with the number of required reactions decreasing from PFR to PLF to PSR combustion. Relative to the H2/air system, principal component analysis of the CH4/air system is more difficult because of mechanism size. For CH4/air combustion, if we consider all equivalence ratios, PLFs require the most reactions, if individual equivalence ratios are examined, PFRs require the greatest number of reactions. Combustor type influences mechanism reduction substantially because of the different couplings between the fluid mechanics and chemistry. In H2/air combustion rich combustion required the fewest reactions and in CH4/air, it required the most. Reduction must be achieved by considering the entire mechanism since reactions interact in concert, for example, reactions unimportant in one CH4 mechanism are often important in the other.

Comparison of rotationally inelastic collision models for Q-branch Raman spectra of N2

Abstract We compare the ability of two fitting laws, one based on exponential energy-gap scaling and the other on power-law scaling of rotationally inelastic rates, to predict inverse Raman spectra of the N 2 Q-branch. Although both fitting laws have previously been shown to predict self-broadened linewidths over a wide temperature range, we find that the exponential energy-gap fitting law more accurately describes collisional narrowing.

Comparison of quantal, classical, and semiclassical behavior at an isolated avoided crossing

The quantal and classical/semiclassical behavior at an isloated avoided crossing are compared. While the quantum mechanical eigenvalue perturbation parameter plots exhibit the avoided crossing, the corresponding primitive semiclassical eigenvalue plots pass through the intersection. Otherwise, the eigenvalues agree well with the quantum mechanical values. The semiclassical splitting at the intersection is calculated from an appropriate Fourier transform. In the quasiperiodic regime, a quantum state near an avoided crossing is seen to exhibit typically more delocalization than the classical state. However, trajectories near the ‘‘separatrix’’ display a quasiperiodic ‘‘transition’’ between two zeroth order classical states.

Calculations related to quantum stochasticity, an example of overlapping avoided crossings

In plots of eigenvalues of the Schrodinger equation versus a perturbation parameter, many avoided crossings are found in the classically stochastic regime for the system studied. None were observed in the classically quasi-periodic regime. Overlapping avoided crossings are suggested as a mechanism for making the vibrational wavefunction a “statistical” one.

Semiclassical calculation of compound state resonances

Abstract The compound state resonances in a model collinear collision are shown to arise from semiclassically quasiperiodic quasibound states. These quasibound state energies are compared with previous exact quantum-mechanical and stabilization method calculations of the resonance energies and excellent agreement is found.

Random‐walk approach to mapping nodal regions of N‐body wave functions: Ground‐state Hartree–Fock wave functions for Li–C

Despite the widespread acceptance of the relevance of the nodes of one‐body electronic wave functions (atomic or molecular orbitals) in determining chemical properties, relatively little is known about the corresponding nodes of many‐body wave functions. As an alternative to mapping the nodal surfaces present in the ground states of many‐electron systems, we have focused instead on the structural domains implied by these surfaces. In the spirit of Monte Carlo techniques, the nodal hypervolumes of a series of atomic N‐body Hartree–Fock level electronic wave functions have been mapped using a random‐walk simulation in 3N dimensional configuration space. The basic structural elements of the domain of atomic or molecular wave functions are identified as nodal regions (continuous volumes of the same sign) and permutational cells (identical building blocks). Our algorithm determines both the relationships among nodal regions or cells (topology) as well as the geometric properties within each structural domain. Our results indicate that ground‐state Hartree–Fock wave functions generally consist of four equivalent nodal regions (two positive and two negative), each constructed from one or more permutational cells. We have developed an operational method to distinguish otherwise identical permutational cells. The limitations and most probable sources of error associated with this numerical method are discussed as are directions for future research.

Kinetic isotope effects for hydrogen diffusion in bulk nickel and on nickel surfaces

Diffusion coefficients for H, D, and T on a Ni(100) surface and in bulk Ni are calculated using variational transition state theory with semiclassical ground‐state transmission coefficients using two potential energy surfaces obtained by the embedded atom method (EAM). The original EAM potential reproduces experimental bulk diffusion coefficients, but greatly overestimates the diffusion coefficients for H and D on Ni(100). By refining the empirical potential parameters, a new EAM potential is developed that accurately reproduces both the bulk and surface diffusion coefficients. The variational transition state theory calculations are used to analyze the unusually low (compared to gas phase) H/D kinetic isotope effects for diffusion in bulk and on the Ni(100) surface. For the temperature range for which experiments have been carried out, quantum mechanical tunneling contributes negligibly to the diffusion and, in these cases, the kinetic isotope effect is determined largely by the change in zero‐point ener...

On correlation functions and the onset of chaotic motion

A variety of correlation functions computed over a microcanonical ensemble for the Henon–Heiles system are investigated. We find the general trend is that of a gradual change to some form of decaying behavior as the motion becomes predominantly chaotic. The decay of a mode energy correlation function indicates a time scale for intramolecular energy redistribution.

Cross‐correlation trajectory study of V–V energy transfer in HF–HF and DF–DF

Results of a fully three‐dimensional classical trajectory calculation of vibrational energy transfer are presented for the collision of HF(v=1) with HF(v=1) and its deuterium analog. A cross‐correlation method, together with quasiclassical trajectories, is introduced to relate the changes in vibrational states of the two molecules to probabilities and rate constants. Multiple collisions are found to make an important contribution to the vibrational energy transfer cross‐sections for the present potential surface. Vibrational anharmonicity is shown to decrease the energy transfer rate constant by a factor of ten, by causing the process to be further from exact resonance. Excellent agreement with experiment is obtained for the HF–HF and DF–DF systems.