Feshbach Projection Operator Research Articles

Exchange symmetric effective interaction operators for coupled-reaction-channel equations and optical potentials

We derive formally exact expressions for the effective interactions to be used in coupled-reaction-channel equations. Exchange symmetry is explicitly taken into account. Our result is a generalization of the Feshbach projection operator formalism optical potential to include the effects of open rearrangement channels as well as the effects of exchange symmetry. The optical potential for elastic scattering is a special case of the effective interaction. Our result is simpler than previous multipartition treatments because it is based on the special Chandler-Gibson partition coupling scheme which is a completely symmetric arrangement of the Lippmann-Schwinger multiparticle scattering equations. We find that one version of our formalism for the effective interactions is similar to that of Birse when exchange effects are neglected. On the other hand, in the case that multipartition effects are neglected our optical-potential formalism reduces to that of Picklesimer and Thaler when the conventional post-interaction transition operator is used, and it reduces to the Kowalski and Picklesimer optical-potential formalism when the AGS off-shell extension of that operator is used. This reveals that these optical potentials are based on the single-partition approximation. We emphasize the distinction between multipartition and single-partition formalisms. Only the former can be exact.

Theoretical study of autoionising states of molecules; H2and He22+

The adiabatic potential energy curves of the lowest 1 Sigma g+, 1 Pi g, and 1 Delta g autoionising states for H2 are obtained in the region of internuclear distance 1.0<or=R<or=5.0 au, using Feshbach's projection operator techniques. The energy shift and the energy width of the lowest 1 Sigma g+ autoionising state are calculated for 1.0<or=R<or=2.6 au. It is found that near the stabilisation point (R=2.53 au), the shift is considerable and its effect on the width is not negligible. The revised results for He22+ are also presented.

Polarization potentials in heavy-ion scattering

A polarization potential is defined in terms of the Feshbach projection operator formalism to represent the effect upon the elastic channel of the coupling to non-elastic channels in heavy-ion scattering. The polarization potential represents coupling to specific surface degrees of freedom of the particular reaction considered and it is contrasted to the complementary global approaches for the volume potential such as the folding model and the proximity potential. The coupled channels method is used both as a source of exact model solutions for comparison with the various approximate potential forms and also as a numerical means of constructing trivially equivalent local potentials. The imaginary Coulomb polarization potential is due in lowest order to quadrupole coupling to the lowest collective 2 + state of a nucleus. It is considered in detail since it provides the insight of closed analytical forms in various approximations. Multistep coupling to higher states, energy loss and off-energy shell effects are also considered analytically. The real Coulomb polarization potential due to the virtual excitation of multipole giant resonances, and the polarization potential arising from relativistic corrections, are investigated in detail. Polarization potential components due to nuclear coupling are investigated numerically. Analytical cross section approaches are contrasted with the polarization potential approach and with coupled channels.

Nuclear dynamics in resonant electron-molecule scattering beyond the local approximation: model calculations on dissociative attachment and vibrational excitation

Within the framework of Feshbach's projection operator formalism, the calculation of cross sections for resonant electron-molecule scattering and dissociative attachment can be reduced to the treatment of nuclear dynamics in an energy-dependent, complex and non-local potential. In this work, a new method to solve the difficult problem of nuclear motion in the non-local potential is developed. The method is based on the parametrisation of the potential energy curves of the target molecule and the projected resonance state by Morse functions. The non-local part of the potential is represented in a separable form, using the Lanczos, (1950) basis of the Morse Hamiltonian as the basis set for the expansion. Numerically exact non-local calculations are performed, and the accuracy of the local approximation is critically assessed for a model representing dissociative attachment via a d-wave shape resonance. For the examples considered it is found that the most important corrections to the local approximation result from the energy dependence of the entry and exit amplitudes.

Unitary approximations to transition probabilities: Generalized impact parameter theory and the time dependent projection operator method

A general unitary approximation scheme is presented for transition probabilities based upon the application of the time dependent version of the Zwanzig–Feshbach projection operator method to the generalized impact parameter approximation. The overall approximation scheme is based upon the time dependent picture of the binary collision process. To begin with, the translational and internal motions are decoupled with the translational motion being treated in one of three ways, namely, as a straight line trajectory, as a single curved reference classical trajectory, or as a collection of such trajectories, each of which is associated with a pair of internal states. The internal dynamics is then parameterized by the set of given trajectories which are functions of the impact parameter. This motion is then formally solved using the Zwanzig–Feshbach projection operator method. A set of unitary approximations to these solutions are then presented, based upon a time disordered approximation. These latter approximations are applicable to each of the three treatments of the translational motion. As well, the basis used to describe the internal degrees of freedom may be either atomic or molecular (diabatic or adiabatic). Thus this combined approximation scheme provides a systematic theory for testing various effects associated with either the treatment of the translational motion or the internal motion.

Nuclear dynamics in resonant electron-molecule scattering beyond the local approximation: The 2.3-eV shape resonance inN2

Resonant electron scattering from the ${\mathrm{N}}_{2}$ molecule is treated within the framework of Feshbach's projection-operator formalism. The problem of nuclear motion in the complex, nonlocal, and energy-dependent potential of the resonance state is solved with the use of basis-set expansion methods. The width function $\ensuremath{\Gamma}(E,R)$ of the 2.3-eV shape resonance is taken from previous ab initio calculations A. U. Hazi, T. N. Rescigno, and M. Kurilla, [Phys. Rev. A 23, 1089 (1981)]. The potential-energy curve ${V}_{1}(R)$ of the discrete electronic state giving rise to the resonance is adjusted to obtain quantitative agreement with experiment for the $v=0\ensuremath{\rightarrow}1$ vibrational excitation channel. Calculated cross sections are reported for many vibrational excitation channels including scattering from vibrationally excited target molecules and are compared with experiment. The results are in good agreement with experiment as far as experimental data are available. The information on the 2.3-eV shape resonance in ${\mathrm{N}}_{2}$ obtained in this way (potential-energy curve, width function, complex pole of the $S$ matrix, phase shift, etc.) is discussed and compared to the results of other ab initio and empirical calculations. Particular emphasis is laid on the assessment of the accuracy of the local complex potential approximation (boomerang model). It is shown that the local approximation is excellent for the 2.3-eV resonance in ${\mathrm{N}}_{2}$ as far as the calculation of vibrationally inelastic cross sections is concerned. The local complex potential emerging from the present study differs significantly from the potential obtained empirically by Dub\'e and Herzenberg [Phys. Rev. A 20 194 (1979)], although the calculated cross sections are virtually identical for a large number of channels. We discuss the precision with which a local complex potential can be defined.

Molecular study of outer K-L shell interactions: the Na-He collisional system

A Feshbach projection operator technique involving self-consistent field (SCF) calculations in orthogonal subspaces is used to obtain diabatic orbitals of the Na-He collisional system. Analysis of the molecular orbital wavefunctions and their delta / delta R couplings show that the Barat-Lichten correlation rule for the promoted orbital is incompatible with the diabatic smoothness criterion. The previously admitted conjecture that the strong asymmetry of the system would lead to an adiabatic behaviour of the sigma 1sHe- sigma 3sNa molecular orbital (MO) crossing is not supported by the presently calculated coupling matrix elements. The molecular mechanism for Na(3s to 3p) excitation appears to be induced indirectly (as for Na-Ne collisions) by two successive curve crossings via the core-core K-L shell interaction.

Theoretical study of the lowestΣg+1doubly excited state ofH2

The energy and width of the $^{1}\ensuremath{\sigma}_{u}^{2}^{1}\ensuremath{\Sigma}_{g}^{+}$ state of ${\mathrm{H}}_{2}$ are calculated using Feshbach projection operators and the Stieltjes moment method. The results are compared with an analysis of the double minima in the potential-energy curves of the $^{1}\ensuremath{\Sigma}_{g}^{+}$ Rydberg states. The doubly excited $^{1}\ensuremath{\sigma}_{u}^{2}$ configuration is unstable with respect to autoionization for $R&lt;2.65{a}_{0}$, and the autoionization rate is sensitive to both the value of the internuclear separation and the energy of the emitted electron.

Analytic theory of resonances and bound states near Coulomb thresholds

The passage of an electronic resonance through the continuum threshold in the presence of a Coulomb potential is analysed. The approach is based on Feshbach's projection operator formalism combined with analytic continuation and a threshold expansion of the complex level shift function. Resonances and bound states are obtained as the poles of the analytically continued fixed-nuclei S matrix, S(k), for electron scattering. Close to threshold the behaviour of resonances and bound states is completely determined by the long-range Coulomb potential which causes an essential singularity of S(k) at k=0. Both the attractive and the repulsive Coulomb potentials are considered. The relationship between the Feshbach formalism and the quantum defect theory is established. The relevance of these results for the understanding of nuclear dynamics in electron-ion scattering is illustrated by a model calculation.

The doubly excited autoionizing states of H2

Ab initio potential curves have been determined for 24 doubly excited autoionizing states of H2 having primarily the lowest 2Σ+u and 2Πu states of H+2 as the core orbital. Fourteen of these states have not appeared previously in the literature. The configuration interaction wave functions are constructed with a Feshbach projection operator formalism. The accuracy of the calculations is assessed by comparison with calculations in the same bases on the lower bound states of H2. The potential curves for the doubly excited states fall into three distinct groups with an energy ordering related to the Hartree energy of the most important configuration of each state. The relative energies of several doubly excited states and the corresponding singly excited bound states of H2 are compared. The grouping of doubly excited states is in good agreement with the electron impact results of Kollman. Both the lowest two states of Q1 1Σ+u symmetry and a Q2 1Σ+u state with thresholds of 25.5, 27.8, and 33.3 eV, respectively, are important in the interpretation of dissociative photoionization and electron impact experiments. A Q2 1Σ+g state with a threshold at 30.2 eV will contribute to the fast H atoms seen in several electron impact experiments.

Influence of 1p- 1h-excitedαparticles on elasticα−αscattering

The influence of channels with 1p- 1h-excited $\ensuremath{\alpha}$ particles with $S=T=0$ on elastic $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ scattering is studied in the framework of the Feshbach projection operator formalism. Calculated phase shifts and transmission coefficients for $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ scattering are compared to experimental data for c.m. energies between 15 and 35 MeV. The effect of positive parity $\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}^{*}$ rotational states is discussed.NUCLEAR REACTIONS Excited $\ensuremath{\alpha}$ particles in elastic $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ scattering microscopic absorptive potential. Feshbach projection operator formalism. Phase shifts and transmission coefficients.

Theory of resonances in three-dimensional chemical reactions. II. Application to a model atom–diatom reaction

The Feshbach projection operator theory is applied to the study of resonances in model 3D atom–diatom chemical reactions. A resonance formalism based on the Feshbach theory along with the computational techniques described previously are used to solve the Schrödinger equation and interpret the numerical results. Resonance positions are formally explained by the interaction of open channel shape resonances and closed channel ‘‘zero-order resonance states.’’ Illustrations are provided by the numerical results. In particular, resonances occurring near channel thresholds are predicted formally and illustrated numerically. In addition, the role of closed channels is illustrated numerically and is explained in the context of the Feshbach theory. This work represents the first application of the Feshbach resonance theory to three-dimensional reactive molecular collisions.

Theory of resonances in three-dimensional chemical reactions. I. Feshbach analysis and computational techniques

An analysis of internal excitation resonances in three-dimensional, symmetric atom/diatom chemical reactions is presented. The description is based on the Feshbach projection operator theory and is formulated in terms of ‘‘natural collision coordinates’’ (NCC) and the NCC Hamiltonian appropriate for near-linear intermediates. The resonance theory is reviewed and developed using a computationally oriented matrix notation. Expressions are given for the level operator matrix, formation and decay amplitudes, and S-matrix elements. Computational techniques are emphasized. Symmetry relationships in the theory resulting from reaction symmetry are developed. Finally, resonance effects in cross sections for symmetric reactions are discussed. Numerical applications to a model reaction are presented and analyzed in the following paper.

Rydberg states of helium: An optical-potential analysis

The Feshbach projection-operator technique has been applied to the problem of electrostatic level shifts in $1sNL$ Rydberg states of helium. It is assumed at first that, for sufficiently high $L$, only long-range terms in the optical potential affect the energy of the outer electron. These terms up to ${x}^{\ensuremath{-}8}$ are carefully derived, used in a first-order perturbation calculation, and compared with recent accurate experimental results. Second-order effects due to the leading ${x}^{\ensuremath{-}4}$ potential and some short-range terms are also calculated in a few cases; these effects are generally not negligible for $L\ensuremath{\le}4$.

Generalized Feshbach projection operator method and the coupled reaction channel formalism

The multichannel generalizations of the Feshbach formalism are considered. These generalizations are nonunique. We demonstrate the equivalence of two such generalizations and show that they reduce to the familiar coupled reaction channel formalism under certain approximations. The relationship to the many-body formulations of the multichannel problem proposed recently is also discussed.

Microscopic absorptive interaction for composite finite nuclei

The theoretical framework for the microscopic calculation of an effective interaction of two composite nuclei is developed by applying Feshbach's projection operator formalism. Both translational invariance and the Pauli principle are taken into account exactly. The theory is based on model wavefunctions for finite nuclei and does not require nuclear matter approximations. The model Q-space consists of channels in which one or both nuclei are excited. A 1 p1 h-excitation mechanism is considered in some detail and the properties of an effective absorptive interaction are studied for the α-α-system as an example.

Theory of indirect dissociative recombination

Feshbach's projection operator technique as adapted in atomic collision theory is applied to obtain the T-matrix element for indirect dissociative recombination

Electron Cooling through Resonant Collisions with H2+ Molecular Ion

Electron cooling in H 2 + through resonant capture leading to (i) vibrational excitation and (ii) vibrational dissociation of H 2 + has been studied within the scope of Born-Oppenheimer approximation, using Feshbach's projection operator technique. Results are presented for (i) cross-sections for certain specific vibrational transitions ( v i → v f ), and for (ii) total cross-sections for vibrational dissociation. Variations in the rotational inelasticity effects as obtained for different types of vibrational excitations are discussed.

Feshbach projection operator method and few-body reaction models in Baer-Kouri-Levin-Tobocman many-body scattering theory

A generalization of the Feshbach projection operator method to include rearrangement in the context of the Baer, Kouri, Levin, and Tobocman many-body scattering theory has been proposed recently. The formalism provides a simplified set of connected kernal equations for an approximate transition operator matrix and an explicit procedure for relating the approximate transition operators to the exact ones. In this paper we extend the flexibility of the aforementioned generalization by making it possible to allow the approximate transition operators to couple fewer partitions than the exact ones. Particular attention is given to the case where the approximate transition operator matrix is the solution of coupled-reaction-channel--type equations for transitions between few cluster configurations of the system. It is also shown that all the equations derived here have connected kernels after iterations.

Feshbach projection-operator calculation of the resonant states of H2-

A formulation of the Feshbach projection-operator technique for a three-electron system is presented which renders practicable the calculation of the H2-resonance positions, and ultimately their widths, by standard configurational interaction methods. Striking and hitherto unidentified resonances in the 0 to 12 eV region have been discovered and these may lead to a more complete interpretation of the existing experimental results.