Larry Eno

Further developments and applications of sensitivity analysis to collisional energy transfer

This paper considers the sensitivity of collision cross sections to the variation of intermolecular potential parameters. The study is restricted to the scattering of an atom and a linear rigid rotor, and to an atom and a breathing sphere. Attention is focused upon first order sensitivity coefficients (i.e., the gradient of cross sections with respect to potential parameters) from which an entire family of derived sensitivity coefficients may be obtained. Within this family a special class of coefficients is shown to be particularly important in determining the extent to which a set of measurements is able to define the parameters of an assumed potential. Finally, the global behavior of cross sections in parameter space is examined, and a nonlinear interpolation formula is suggested which utilizes sensitivity information.

Generalized sensitivity analysis in quantum collision theory

This paper considers the sensitivity of the scattering matrix with respect to variations of input parameters including those that persist asymptotically. The analysis shows that the first order sensitivity coefficients (i.e., the partial derivatives with respect to input parameters) of the scattering matrix may be expressed in terms of the available solution of the Schrodinger equation. This work encompasses earlier results which appear as special cases. As an application sensitivity with respect to the total energy is considered and the relevance of the result to the energy behavior of observables is discussed. In a second application, we show how sensitivity analysis may be used to estimate corrections to effective Hamiltonian results and also to examine their accuracy. The coupled states and infinite order sudden methods are chosen to illustrate the approach. Finally, the scope of sensitivity analysis in quantum collision theory is discussed.

Sensitivity analysis of rotational energy transfer processes to the intermolecular potential

This paper considers the sensitivity of rotational energy transfer processes to the variation of parameters within an assumed model intermolecular potential. The following cross sections are considered here: integral state to state, pressure broadening, effective diffusion and viscosity, and final state summed integral cross sections. In order to simplify the calculation of cross sections, attention is restricted to the scattering of an atom and linear rigid rotor. Furthermore, the collision dynamics are approximated by using the infinite order sudden (IOS) method. It is shown that use of the IOS method allows for the very simple generation of first order sensitivity coefficients (i.e., the partial derivative of cross sections with respect to potential parameters). Particular attention is focused upon the sensitivities of different cross sections and combinations of cross sections to the various parameters. The first order sensitivities are also used to derive new coefficients which describe how the potential parameters correlate given a limited set of cross section measurements. These coefficients are shown to be particularly important in determining the degree to which a set of measurements is able to define various parameters of the assumed potential.

Sensitivity analysis of differential cross sections to the intermolecular potential

This paper considers the sensitivity of both final state summed and state to state differential cross sections to the variation of parameters within a model intermolecular potential. In order to simplify the calculation of the cross sections and first order sensitivity coefficients (i.e., the partial derivative of cross sections with respect to potential parameters) attention is restricted to the scattering of an atom and rigid rotor. Furthermore, the collision dynamics are approximated by using the infinite order sudden (IOS) approximation. Particular emphasis is given to an examination of the sensitivity of angular features of the cross sections to potential parameter variations. This is facilitated by fitting the cross sections to functional forms which contain several adjustable parameters, each of which controls a particular feature. First order sensitivity coefficients are then used to derive quantitities which measure the sensitivity of a ’’feature parameter’’ with respect to the variation of a pot...

Sensitivity analysis of experimental data

This paper examines how a number of sensitivity techniques may be adapted to deal with the analysis of real experiments. We show that the fundamental quantities in this context are the partial derivatives of model input parameters with respect to experimental observables. They appear as a consequence of the procedure used to fit the model to the experiment. These so-called experimental elementary sensitivities are then combined with the usual model elementary sensitivities to yield coefficients which relate experimental and model observables. The importance of such coefficients for the planning of experiments is, in particular, discussed. Based on a linear analysis, we also derive simple expressions (containing the experimental elementary sensitivities) for the degree of parameter deviation arising from uncertainties in and discrepancies between model and measured observables. Finally, an overview of the role of sensitivity theory in analyzing experimental data is given.

An improved phase shift approach to the energy correction of the infinite order sudden approximation

A new method is presented for obtaining energy corrections to the infinite order sudden (IOS) approximation by incorporating the effect of the internal molecular Hamiltonian into the IOS wave function. This is done by utilizing the JWKB approximation to transform the Schrodinger equation into a differential equation for the phase. It is found that the internal Hamiltonian generates an effective potential from which a new improved phase shift is obtained. This phase shift is then used in place of the IOS phase shift to generate new transition probabilities. As an illustration the resulting improved phase shift (IPS) method is applied to the Secrest–Johnson model for the collinear collision of an atom and diatom. In the vicinity of the sudden limit, the IPS method gives results for transition probabilities, Pn→n+Δn, in significantly better agreement with the ’exact’ close coupling calculations than the IOS method, particularly for large Δn. However, when the IOS results are not even qualitatively correct, t...

An examination of the use of vibrationally adiabatic basis functions for the calculation of molecular collision cross sections

Comparison is made between the use of two different types of vibrational basis functions for the expansion of the total wave function in a vibrationally inelastic scattering problem. The calculations are performed within the framework of the sudden approximation for the rotational motion of the molecular fragments. The different basis functions that are compared are a vibrationally adiabatic set and the standardly used set of diabatic vibrational basis functions. The adiabatic vibrational basis functions are chosen so as to approximately diagonalize the matrix representation of the interaction potential at each value of the scattering coordinate. Nevertheless, they permit the formulation of analytic expressions for the nonadiabatic coupling terms of the kinetic energy operator that are present when an adiabatic basis is used. In order to provide a reference against which to judge the two different bases, the sets of coupled differential equations which arise in the rotational sudden approximation are solv...

Exact scattering solutions in an energy sudden (ES) representation

In this paper, we lay down the theoretical foundations for computing exact scattering wave functions in a reference frame which moves in unison with the system internal coordinates. In this frame the (internal) coordinates appear to be fixed and its adoption leads very naturally (in zeroth order) to the energy sudden (ES) approximation [and the related infinite order sudden (IOS) method]. For this reason we call the new representation for describing the exact dynamics of a many channel scattering problem, the ES representation. Exact scattering solutions are derived in both time dependent and time independent frameworks for the representation and many interesting results in these frames are established. It is shown, e.g., that in a time dependent frame the usual Schrodinger propagator factorizes into internal Hamiltonian, ES, and energy correcting propagators. We also show that in a time independent frame the full Green’s functions can be similarly factorize. Another important feature of the new representation is that it forms a firm foundation for seeking corrections to the ES approximation. Thus, for example, the singularity which arises in a conventional perturbative expansions of the full Green’s functions (with the ES Green’s function as the zeroth order solution) is avoided in the ES representation. Finally, a number of both time independent and time dependent ES correction schemes are suggested.

A simple model study of reactive collisions in an intense nonresonant laser field

A simple model study of laser induced transitions between electronic surfaces in reactive molecular collisions has been undertaken. The investigation is characterized by laser and nonadiabatic couplings which are turned on during the course of a collision. Transition probabilities are determined within an exact quantum‐mechanical framework, for switching between the model one‐dimensional potential curves as a function of various system parameters. Such parameters include the photon energy, the reactant collision energy, and the coordinate separation between the positions of potential barrier maxima. The processes studied involve not only laser switching but, also, cooperative laser and nonadiabatic effects. A number of features of the results are emphasized.

Scaling relations for multiplicative quantum mechanical operators

Scaling relations are derived for the matrix elements of a multiplicative quantum mechanical operator. While the relations have been set down earlier, the present derivation is, we believe, more transparent. In order to satisfy certain attractive properties, the coefficients in these relations have previously been identified as the principal values of improper integrals. We show here, however, that the scaling relations can be rewritten in terms of scaling coefficients which are given as proper integrals and which satisfy the same attractive properties.