Feshbach Formalism Research Articles

A new method to calculate lifetimes of atomic and molecular autoionizing states

We give a summary of a new method to calculate positions and widths of two-electron atomic (molecular) doubly excited states. This procedure, based on a formal analogy between the Feshbach formalism and the Phillips-Kleinman pseudopotential approach, yields results in very good agreement with those obtained with the standard Feshbach formalism, and requires a considerably smaller computational effort in cases where the resonant states of interest lie above several ionization thresholds. This alternative method can be easily generalized and has been applied to the treatment of doubly excited states of systems with a closed shell core.

Coupled continuum perturbative Feshbach approach to calculate total and partial widths: 1,3Se(3lnl) resonant states of He, C4+, N5+, and O6+

We apply a perturbative method, which takes into account the interaction between different open channels, to calculate total and partial widths of atomic resonances in the framework of the Feshbach formalism. The method requires the calculation of elastic, e.g., noncoupled open-channel wave functions, as in the case in which only one channel is open, thus avoiding the solution of complicated systems of coupled differential equations. As an illustration we have calculated energy positions, total and partial widths of $^{1}$,3${\mathrm{S}}^{\mathrm{e}}$(3lnl) resonant states of He, ${\mathrm{C}}^{4+}$, ${\mathrm{N}}^{5+}$, and ${\mathrm{O}}^{6+}$. Our results show that the accuracy reached with this simple approach is comparable to other different non-Feshbach methods and that neglect of the coupling between different open channels can be considered as a reasonable approximation to obtain total and partial widths.

Interference effects in electron-ion recombination. II. Resonance and direct channels.

The Feshbach formalism is employed in a study of the effects of interference between direct radiative recombination and dielectronic recombination. A general result is obtained, valid for an arbitrary number of interacting resonances and an arbitrary number of coupled continua. The results of explicit model calculations, based on the pole approximation, are described, and comparison is made with earlier, related work.

Interference effects in electron-ion recombination. I. Resonance channels only.

The Feshbach formalism is employed in investigations of the effects of interacting and/or overlapping resonances on dielectronic recombination. Model systems are considered, and the problem of singly charged target ions is discussed. Small reductions in dielectronic recombination cross sections are found to occur for these systems. Comparison is made with the results of multichannel quantum-defect theory calculations.

Simple discretization method for autoionization widths. III. Molecules.

We apply a new method to calculate widths of two-electron Feshbach resonances, which was described in detail and applied to atomic systems in preceding articles (this issue), to molecular and quasimolecular autoionizing states. For simplicity in the programming effort, we restrict our calculations to the small-R region where one-centered expansions are sufficiently accurate to describe the wave functions. As test cases, positions and widths for the ${\mathrm{H}}_{2}$, ${\mathrm{He}}_{2}$${\mathrm{}}^{2+}$, ${\mathrm{HeH}}^{+}$, and ${\mathrm{LiHe}}^{3+}$ resonances of lowest energy are computed for R0.6 a.u. The advantage of using block-diagonalization techniques to define diabatic resonant states instead of generalizing the Feshbach formalism is pointed out.

Autoionization from stark-mixed states

The Stark mixing of autoionizing states of a doubly excited ion placed in the field of a distant charge is considered in the frame work of the Feshbach formalism. The system N 5+ ∗∗ (3l, 3l′) + He 2+ . produced by double charge transfer from helium to N 7+, illustrates the importance of the mixing as a post collisionnal interaction effect (PCI).

Projection-operator calculations of the lowest e--He resonance.

The 1s (2s)2:2S Schulz resonance of He(-) is investigated theoretically, applying the full projection-operator formalism developed by Temkin and Bhatia (1985) in a Rayleigh-Ritz variational calculation. The technique is described in detail, and results for five different approximations of the He target state are presented in a table. Good convergence is obtained, but it is found that even the best calculated value of the resonance is about 130 meV higher than the experimentally measured value of 19.367 + or - 0.007 eV (Brunt et al., 1977), a discrepancy attributed to the contribution of the shift in the Feshbach formalism.

Molecular (Feshbach) treatment of charge exchange Li3++He collisions. I. Energies and couplings

We point out a fundamental difference between the molecular treatment of charge exchange Xn++H(1s) and Xn++He(1s2) collisions, which is that the latter process involves molecular states that are formally autoionizing. Then standard ab initio methods do not, in general, yield the relevant wave functions that are needed in the collision treatment, irrespective of whether quasimolecular autoionization be significant or not during the collision. We implement a particularly simple and useful form of the Feshbach formalism to calculate the energies of those two electron systems, and a method to evaluate the corresponding dynamical couplings is presented for the first time. Our implementation of this formalism together with the new computational techniques involved are presented in detail.

Radically simplified proofs of the Weyl and Stenger inequalities concerning eigenvalues of self-adjoint operators

Radically simplified proofs of the generalized Weyl inequality and the Stenger inequality for eigenvalues of self-adjoint operators is presented. The main application of these results lie in atomic and molecular physics. The generalized inequality has application in obtaining bounds on molecular energy levels in cases where the molecular hamiltonian can be written as a sum of several atomic ones and the Stenger inequality has application in putting bounds on energy levels of atomic and molecular resonances to which the Feshbach formalism applies.

Position and widths of autoionising states in the helium isoelectronic sequence above the N=2 continuum

Energies and widths of autoionising states in the helium isoelectronic sequence (Z=5-10) converging to the n=3 and n=4 hydrogenic thresholds are computed. States of symmetry 1Se, 1Po, 1De are considered. Using the Feshbach formalism the eigenvalues of QHQ are calculated by the truncated diagonalisation method of Altick and Moore (1965) or Lipsky et al. (1977) and the continuum functions by the static exchange approximation. The author computes both total widths and partial widths associated with the various open channels.

Line-shape parameters forP1Feshbach resonances in He andLi+

The line-shape parameter ($q$) for the three lowest $^{1}P$ autoionization states of He and ${\mathrm{Li}}^{+}$ below the first excited target (${\mathrm{He}}^{+}$, ${\mathrm{Li}}^{2+}$) level are accurately calculated. In order to do this we rederive a rigorous expression for $q$ in terms of the Feshbach formalism; the formula contains additional terms representing the effect of other resonances on the resonance being calculated. Hylleraas functions are used to calculate $Q\ensuremath{\Phi}$, and the exchange approximation is used for the nonresonant continuum. Results agree well with the experiment where available. Our previous calculations of positions and widths are not affected; however, it is noted that the energy of the newest experiment of Morgan and Ederer for the lowest $^{1}P$ autoionization state of He (60.151 \ifmmode\pm\else\textpm\fi{} 0.01 eV) is such that when it is combined with the older experimental value of Madden and Codling (60.133 \ifmmode\pm\else\textpm\fi{} 0.015 eV), the weighted mean of the two measurements (60.145 \ifmmode\pm\else\textpm\fi{} 0.008 eV) is in exceptional agreement with our previous value (60.145 eV).

Analysis of stripping to isolated analog resonances

The Feshbach projection formalism is used to calculate the form factors for the (d,n) stripping process to isolated analog resonances. These are used in a standard distorted-wave Born-approximation stripping calculation in which the radial integration over all space is accomplished by including outer space contributions evaluated along the complex contours of Vincent and Fortune. It turns out that the shape and magnitude of the predicted cross section are quite insensitive to the continuum proton wave emanating from the resonant residual state.[NUCLEAR REACTIONS Stripping to isolated analog resonance states. Calculation of DWBA form factor using Feshbach formalism. Demonstrated that continuum proton wave contributes little to cross section calculations.]

Feshbach formalism, coupled reaction channels, and effective interactions

The Feshbach effective Hamiltonian for unsymmetrized rearrangement channels is used to obtain coupled reaction channels equations with effective interactions. Operators are introduced which solve nonorthogonality to all orders. Two representations of the effective Schr\"odinger equation are obtained with solved nonorthogonality. These equations have unique solutions, even when the basis used is overcomplete. The effective interactions in these equations are compared with those defined by Goldflam and Kowalski. Implications for the coupled reaction channels method are discussed.NUCLEAR REACTIONS Feshbach formalism for rearrangement channels. Coupled reaction channels. Nonorthogonality and overcomplete bases. Effective interactions.

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.

Vibrationally adiabatic models for reactive tunneling

The approximation of vibrational adiabaticity in curvilinear natural collision coordinates is investigated for tunneling in three-atom collinear reactions. A validity criterion is derived which limits the adiabatic approximation to systems with small reaction-path curvature. A general formalism is developed for systems which satisfy this criterion. A one-dimensional Schrödinger equation is proposed which is sufficiently flexible so as to be adaptable to many different models of tunneling. We present three new methods for including reaction-path curvature effects on multidimensional tunneling in reactive systems: a method based on a quantum mechanical vibrational average (VA) over degrees of freedom transverse to the minimum-energy path; a method (called DA for dynamical-path vibrational- average) that includes internal centrifugal effects in the description of the transverse vibrational motion (in this method the vibrational average is approximated as a quantal vibrational average about the dynamical path along which the Born–Oppenheimer force cancels the internal centrifugal force); and a semiclassical optical potential (SOP) method based on the Feshbach formalism translated into an adiabatic representation with reaction-path curvature providing the coupling mechanism between the explicit and implicit spaces. These models are compared, both formally and numerically, to each other and to four other methods that have been proposed previously, including the small-curvature (SC) approximation that we have proposed in a recent communication. The VA and SOP methods are shown to provide generalizations of phase average (PA) and second-order (SO) methods proposed earlier by Miller and co-workers. The difference is that vibrations are treated quantum mechanically in the VA and SOP methods but classically and harmonically in the PA and SO methods; the quantum mechanical methods have the advantage that anharmonicity can be included more straightforwardly. The DA, SO, and SOP methods, although they include internal centrifugal effects more fully than the VA and PA methods, do not offer significant improvement in accuracy. The numerical results clearly support the physical interpretation of the collapse of the vibrational wave function about a least-action path. The most successful methods are the Marcus–Coltrin path (MCP) and SC approximations. These methods, especially the SC approximation because it is more general, are recommended for future applications.

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.

The equivalence of the Feshbach and J-matrix methods

It is shown, by takcing P to be the projection operator on the subspace of function space in which the potential is truncated, that the exact solution of the scattering problem for the truncated potential using the Feshbach formalism is identical to the J-matrix solution.

On the energy dependence of the imaginary optical-model potential for proton scattering from 40Ca

The energy dependence of the imaginary potentials for proton elastic scattering from 40Ca has been derived on the basis of the Feshbach formalism by taking into account the coupling with the inelastic and pick-up channels to the elastic channel. It is found that the calculated potential describes quite well the experimental differential and absorption cross sections for incident energies between 10 MeV and 50 MeV. However, at high energies above 50 MeV, the calculated imaginary potentials seem to be inadequate. We assume a Woods-Saxon form for the real part of the potential, the parameters of which are in good agreement with phenomenological values, if finite-range calculations are made for the p-d-p process. The local potentials equivalent to the non-local potentials are also calculated and are compared with the phenomenological imaginary potentials.

Antisymmetrized Lippmann-Schwinger equations and optical potentials

A recent study of the effect of the Pauli principle in elastic nuclear scattering uses the prior off-shell extension for the multichannel transition operators to derive antisymmetrized Lippmann-Schwinger equations, while such equations are shown not to exist if the off-shell extension due to Alt, Grassberger, and Sandhas is employed instead. We show that the limitations associated with this second off-shell extension are inconsequential for all physical considerations. New physically constrained antisymmetrized Lippmann-Schwinger equations are found in terms of a simple effective interaction which incorporates all of the effects of the Pauli principle and whose properties are explicit. These equations, which are based on the Alt et al. off-shell extension, are employed to obtain in a direct manner the antisymmetrized generalization of the Feshbach formalism for the two fragment optical potential.NUCLEAR REACTIONS Pauli principle in nuclear reaction theory. Comparison of different off-shell extensions for transition operators. Antisymmetrized Lippmann-Schwinger equations. Antisymmetrized optical potential.

Calculation of the Imaginary Part of the Optical Potential for Proton Scattering

The imaginary part of optical potentials has been calculated on the basis of Feshbach formalism by taking account of the coupling with the inelastic and pick-up channels to the elastic channel. Differential and absorption cross sections are calculated and are compared with the experiments for the elastic scattering of 30 MeV protons by Ca and 205Pb. It is found that the calculated potential describes quite well the experimental cross sections, if the real part of the potential with a vVoocls-Saxon form is suitably chosen. Effects of the propagator potential are also studied.