Feshbach Projection Operator Research Articles

Nuclear Excitation by Free Muon Capture

Efficient excitation of nuclei via exchange of a real or virtual photon has a fundamental importance for nuclear science and technology development. Here, we present a mechanism of nuclear excitation based on the capture of a free muon into the atomic orbits (NEμC). The cross section of such a proposed process is evaluated using the Feshbach projection operator formalism and compared to other known excitation phenomena, i.e., photoexcitation and nuclear excitation by electron capture (NEEC), showing up to 10 orders of magnitude increase in cross section. NEμC is particularly interesting for MeV excitations that become accessible thanks to the stronger binding of muons to the nucleus. The binding energies of muonic atoms have been calculated introducing a state of the art modification to the Flexible Atomic Code. An analysis of experimental scenarios in the context of modern muon production facilities shows that the effect can be detectable for selected isotopes. The total probability of NEμC is predicted to be P≈1×10^{-6} per incident muon in a beam-based scenario. Given the high transition energy provided by muons, NEμC can have important consequences for isomer feeding and particle-induced fission.

A Unique QP Partitioning and Siegert Width Using Real-Valued Continuum-Remover Potential.

A simple, practical quantum chemical procedure is presented for computing the energy position and the decay width of autoionization resonances. It combines the L2-stabilized resonance wave function obtained using the real-valued continuum-remover (CR) potential [Y. Sajeev Chem. Phys. Lett. 2013, 587, 105-112] and the Feshbach projection operator (FPO) partitioning technique. Unlike the conventional FPO partitioning of the total wave function into its resonant space and background space components, an explicit partitioning of the total wave function into its interaction region and noninteraction region components is obtained with the help of real-valued continuum-remover potential. The molecular system is initially confined inside a CR potential which removes the electronic continuum of the molecular system in which its resonance state is embedded and, thus, unravels the space component of the resonance wave function as a bound, localized eigenstate of the confined system. The eigenfunctions of the molecular Hamiltonian represented in the space constitute a complementary, orthogonal space. A unique partition is obtained when the level-shift of the space function due to its coupling with the space is zero, and the resonance width is computed using these unique partitioned spaces. This new procedure, which we refer to as CR-FPO formalism, is formally very simple and straightforward to implement, yet its applications to the resonance state of a model Hamiltonian and to the doubly excited resonance states of atomic and molecular systems at the full-CI level are very accurate as compared to the alternative, very precise L2 methods. In addition, the CR-FPO formalism is implemented in the multireference configuration interaction (MRCI) method, and uses it for calculating the energy position and the autionization decay width of 2Πg shape resonance in N2-.

Emergence of N-Body Tunable Interactions in Universal Few-Atom Systems

A three-atom molecule AAB, formed by two identical bosons A and a distinct one B, is studied by considering coupled channels close to a Feshbach resonance. It is assumed that the subsystems AB and AA have, respectively, one and two channels, where, in this case, AA has open and closed channels separated by an energy gap. The induced three-body interaction appearing in the single channel description is derived using the Feshbach projection operators for the open and closed channels. An effective three-body interaction is revealed in the limit where the trap setup is tuned to vanishing scattering lengths . The corresponding homogeneous coupled Faddeev integral equations are derived in the unitarity limit. The s-wave transition matrix for the AA subsystem is obtained with a zero-range potential by a subtractive renormalization scheme with the introduction of two finite parameters, besides the energy gap. The effect of the coupling between the channels in the coupled equations is identified with the energy gap, which essentially provides an ultraviolet scale that competes with the van der Waals radius - this sets the short-range physics of the system in the open channel. The competition occurring at short distances exemplifies the violation of the ``van der Waals universality" for narrow Feshbach resonances in cold atomic setups. In this sense, the active role of the energy gap drives the short-range three-body physics.

Antisymmetrized, translationally invariant theory of the nucleon optical potential

Earlier work showed how a nucleon optical model wave function could be defined as a projection of a many-nucleon scattering state within a translationally invariant second quantised many-body theory. In this paper an optical potential operator that generates this optical model wave function is defined through a particular off-shell extension of the elastic transition operator. The theory is express explicitly in terms of the many-nucleon Hamiltonian in a mixed representation in which localised target nucleus states feature. No reference to a mean-field concept is involved in the definition. It is shown that the resulting optical model operator satisfies the requirements of rotational and translational invariance and has standard behaviour under the time reversal transformation. The contributions to the optical potential from two different exchange mechanisms are expressed in terms of an effective Hamiltonian involving a nucleon-number conserving one-body interaction. In the weak-binding limit the method reduces to a a version of Feshbach's projection operator formulation of the optical potential with a truncated nucleon-nucleon potential including exchange terms and recoil corrections. Definitions of the nucleon single-particle Green's function and the corresponding Dyson self-energy modified by corrections for translational invariance are presented and different definitions of the optical potential operator are compared.

P-Wave Positron-Hydrogen Scattering, Annihilation, and Positronium Formation

In a previous paper (Bhatia A.K. 2016), a hybrid theory for the scattering of positrons from hydrogen atoms was applied to calculate S-wave phase shifts, annihilation, and positronium formation cross sections. This approach is now being applied to calculate P-wave positron-hydrogen scattering. The present results, obtained using short-range correlation functions along with long-range correlations in the Schrödinger equation at the same time, agree very well with the results obtained in an earlier calculation by Bhatia et al. (1974), using the Feshbach projection operator formalism. In these earlier calculations, the correction due to the long-range correlations was applied to the variational results. In spite of the fact that this ad hoc correction destroyed the variational bound, the final results have been considered accurate. Annihilation cross-sections, positronium formation cross-sections, calculated in the distorted-wave approximation, are also presented.

Positron-Hydrogen Scattering, Annihilation, and Positronium Formation

In previous papers (Bhatia A.K. 2007, 2012) a hybrid theory for the scattering of electrons from a hydrogenic system was developed and applied to calculate scattering phase shifts, Feshbach resonances, and photoabsorption processes. This approach is now being applied to the scattering of positrons from hydrogen atoms. Very accurate phase shifts, using the Feshbach projection operator formalism, were calculated previously (Bhatia A.K. et al. 1971 and Bhatia et al. 1974a). The present results, obtained using shorter expansions in the correlation function, along with long-range correlations in the Schrödinger equation, agree very well with the results obtained earlier. The scattering length is also calculated and the present results are compared with the previous results. Annihilation cross-sections, and positronium formation cross-sections, calculated in the distorted-wave approximation, are also presented.

Positron-molecule interactions: theory and computation

SynopsisWe discuss the connection between the Feshbach Projection Operator approach to resonant positron annihilation on molecules and the Any-Particle Molecular Orbital Method to positronic molecules. This combination of formal and computational tools is a promising approach to the ab initio description of vibrationally-enhanced annihilation in molecular gases.

Single- or double-electron emission within the Keldysh nonequilibrium Green's function and Feshbach projection operator techniques

This work provides a unified theoretical treatment of the single and correlated double-electron emission from a general electronic system. Using Feshbach projection method, the states of interest are selected by the projection operator; the Feshbach-Schur map determines the effective Hamiltonian and the optical potential for the emitted electrons. On the other hand, the nonequilibrium Green's functions method is demonstrated to be a complementary approach and an explicit correspondence between both methods is established. For a self-contained exposition some results on single electron emission are re-derived using both formalisms. New insights and results are obtained for the correlated electron-pair emission: This includes the effective two-electron Hamiltonian, the explicit form of the Feshbach self-energy in terms of the many-body self-energies, and the diagrammatic expansion of the two-particle current. As an illustration of the diagrammatic technique the process of the two-particle emission assisted by the excitation of plasmons is explicitly worked out.

State selective photo-recombination cross sections in Be-like C and Al ions

We report results from detailed state selective photo-recombination study along with the doubly excited autoionizing resonances in Be-like C2+ and Al9+ ions. In the present investigation, the primary focus is on detailed energy profiles of the individual photo-recombination cross sections. The calculation was carried out for the excited Rydberg states of type 1s22sns(1Se) which interact with the odd-parity continua up to the C3+ and Al10+ 2p threshold limit. The numerical evaluation has been performed at a fine energy mesh across all the autoionizing Rydberg series of resonances 1s22pns(1P0) converging to Li-like ion 2p threshold. The method of calculation keeps the essential ingredients of the Feshbach projection operator approximation. The photo-ionization cross sections have been evaluated with and without relativistic effects included into the R-matrix numerical procedures, while the allowance for both quantum interference between dielectronic and radiative recombination, and overlapping resonances has been done utilizing results from the earlier R-matrix Floquet calculation. We discuss all these results with respect to the effect of quantum interference term on the energy dependence profile of photo-recombination cross section for studied transitions.

Positron interactions with molecules

In this work, we report elastic integral cross sections for low energy positron scattering by carbon monoxide and model calculations for vibrationally enhanced positron annhilation on molecules. The former studies employed the Schwinger Multichannel method in the static plus polarization approximation with two different basis sets to access numerical convergence. In the annihilation studies, the role of multimode vibrational couplings was surveyed with the help of an analytically solvable model based on the Feshbach projection operator formalism.

Application ofP-wave hybrid theory to the scattering of electrons from He+and resonances in He and H−

The $P$-wave hybrid theory of electron-hydrogen elastic scattering [Bhatia, Phys. Rev. A 85, 052708 (2012)] is applied to the $P$-wave scattering from He ion. In this method, both short-range and long-range correlations are included in the Schr\"odinger equation at the same time, by using a combination of a modified method of polarized orbitals and the optical potential formalism. The short-range-correlation functions are of Hylleraas type. It is found that the phase shifts are not significantly affected by the modification of the target function by a method similar to the method of polarized orbitals and they are close to the phase shifts calculated earlier by Bhatia [Phys. Rev. A 69, 032714 (2004)]. This indicates that the correlation function is general enough to include the target distortion (polarization) in the presence of the incident electron. The important fact is that in the present calculation, to obtain similar results only a 20-term correlation function is needed in the wave function compared to the 220-term wave function required in the above-mentioned calculation. Results for the phase shifts, obtained in the present hybrid formalism, are rigorous lower bounds to the exact phase shifts. The lowest $P$-wave resonances in He atom and hydrogen ion have also been calculated and compared with the results obtained using the Feshbach projection operator formalism [Bhatia and Temkin, Phys. Rev. A 11, 2018 (1975)] and also with the results of other calculations. It is concluded that accurate resonance parameters can be obtained by the present method, which has the advantage of including corrections due to neighboring resonances, bound states, and the continuum in which these resonances are embedded.

Extended Brueckner-Hartree-Fock theory with pionic correlation in finite nuclei

An extended Brueckner-Hartree-Fock (EBHF) theory is constructed for the description of nuclear structure in order to use a bare interaction among constituent particles. The nuclear interaction is characterized by the strong tensor force induced by pion exchange interaction. To handle the strong tensor force based on the single-particle picture, the Hartree-Fock variational model space is extended to include 2-particle 2-hole (2p-2h) states with all possible configurations, which are able to describe high momentum components originating from pseudo-scalar nature of the pion. We take a variational principle of the total energy in this extended model space. We obtain an equation for single-particle states in the Fermi sea with inclusion of the effect of the pion exchange and short-range repulsive interaction. We elucidate the nature of the EBHF theory by comparing with the Brueckner-Hartree-Fock (BHF) theory and the Feshbach projection operator method. The EBHF theory has a similar structure as the BHF theory except for the inclusion of the concept of the energy of the total system. The Feshbach projection operator method completely agrees with our framework when the P-space projection corresponds to the Hartree-Fock state and the Q-space projection corresponds to 2p-2h states with all possible configurations.

Feshbach projection-operator partitioning for quantum open systems: Stochastic approach

The dynamics of a state of interest coupled to a non-Markovian environment is studied by concatenating the non-Markovian quantum state diffusion equation and the Feshbach projection-operator partitioning technique. An exact one-dimensional stochastic master equation is derived as a general tool for controlling an arbitrary component of the system. We show that the exact one-dimensional stochastic master equation can be efficiently solved beyond the widely adapted second-order master equations. The generality and applicability of this hybrid approach is justified and exemplified by several coherence control problems concerning quantum state protection against leakage and decoherence.

Overlapping resonances in nuclei coupling to the atomic shells

The resonant process of nuclear excitation by electron transition (NEET) in highly charged ions is investigated. In NEET, a bound electronic decay transition occurs with the simultaneous excitation of the nucleus, provided that the energies of the atomic and nuclear transition match. By varying the atomic charge state, the atomic transition energy can be tuned to a better match of the nuclear transition energy. We propose a new way to create the atomic hole in highly charged ions by dielectronic capture of a free electron. In order to derive the cross section for the three-step process composed by dielectronic capture, NEET, and the subsequent nuclear decay, a Feshbach projection operator formalism is developed. With the help of this formalism, the resonances in nuclei coupling to the atomic shell and the possible interference between several direct and resonant electronic and nuclear processes such as radiative recombination, dielectronic recombination and nuclear excitation by electron capture are described.

New full-wave phase-shift approach to solve the Helmholtz acoustic wave equation for modeling

We have developed a method of solving the Helmholtz equation based on a new way to generalize the “one-way” wave equation, impose correct boundary conditions, and eliminate exponentially growing evanescent waves. The full two-way nature of the Helmholtz equation is included, but the equation is converted into a pseudo one-way form in the framework of a generalized phase-shift structure consisting of two coupled first-order partial differential equations for wave propagation with depth. A new algorithm, based on the particular structure of the coupling between [Formula: see text] and [Formula: see text], is introduced to treat this problem by an explicit approach. More precisely, in a depth-marching strategy, the wave operator is decomposed into the sum of two matrices: The first one is a propagator in a reference velocity medium, whereas the second one is a perturbation term which takes into account the vertical and lateral variation of the velocity. The initial conditions are generated by solving the Lippmann-Schwinger integral equation formally, in a noniterative fashion. The approach corresponds essentially to “factoring out” the physical boundary conditions, thereby converting the inhomogeneous Lippmann-Schwinger integral equation of the second kind into a Volterra integral equation of the second kind. This procedure supplies artificial boundary conditions, along with a rigorous method for converting these solutions to those satisfying the correct, Lippmann-Scwinger (physical) boundary conditions. To make the solution numerically stable, the Feshbach projection operator technique is used to remove only the nonphysical exponentially growing evanescent waves, while retaining the exponentially decaying evanescent waves, along with all propagating waves. Suitable absorbing boundary conditions are implemented to deal with reflection or wraparound from boundaries. At the end, the Lippmann-Schwinger solutions are superposed to produce time snapshots of the propagating wave.

Switch of Conducting Orbital by Bias-Induced Electronic Contact Asymmetry in a Bipyrimidinyl-biphenyl Diblock Molecule: Mechanism to Achieve a pn Directional Molecular Diode

We have both experimentally and theoretically studied low voltage rectification phenomena in an asymmetric single diblock molecular junction and have analyzed its mechanism. First-principles nonequilibrium Green’s function (NEGF) theory was applied to electric current calculations. Then we identified a conducting molecular orbital based on the Breit–Wigner formula combined with the effective molecular projected state Hamiltonian (MPSH) analysis, where real and complex energy terms are renormalized explicitly into the standard MPSH by a Feshbach projection operator technique. The analysis discusses voltage dependences of conducting molecular orbitals as well as of site energy alignment defined in terms of molecular fragments. We found quantitative results of (i) electronic switching of the conducting molecular orbital caused by bias-dependent intrinsic electronic couplings with electrodes and (ii) change of site energy alignment driven by bias voltage sweep without significant additional charge injection. ...

Quantum mechanical generalized phase-shift approach to atom-surface scattering: A Feshbach projection approach to dealing with closed channel effects

We have developed a new method for solving quantum dynamical scattering problems, using the time-independent Schrödinger equation (TISE), based on a novel method to generalize a "one-way" quantum mechanical wave equation, impose correct boundary conditions, and eliminate exponentially growing closed channel solutions. The approach is readily parallelized to achieve approximate N(2) scaling, where N is the number of coupled equations. The full two-way nature of the TISE is included while propagating the wave function in the scattering variable and the full S-matrix is obtained. The new algorithm is based on a "Modified Cayley" operator splitting approach, generalizing earlier work where the method was applied to the time-dependent Schrödinger equation. All scattering variable propagation approaches to solving the TISE involve solving a Helmholtz-type equation, and for more than one degree of freedom, these are notoriously ill-behaved, due to the unavoidable presence of exponentially growing contributions to the numerical solution. Traditionally, the method used to eliminate exponential growth has posed a major obstacle to the full parallelization of such propagation algorithms. We stabilize by using the Feshbach projection operator technique to remove all the nonphysical exponentially growing closed channels, while retaining all of the propagating open channel components, as well as exponentially decaying closed channel components.

Coherent site-directed transport in complex molecular networks: An effective Hamiltonian approach

Defining the conditions for coherent site-directed transport from an electron donor to a specific acceptor through tunneling barriers in a network of multiple donor/acceptors sites is an important step toward controlling electronic processes in molecular networks. The required analysis is most challenging since the entire network in essentially involved in coherent transport. In this work we introduce an efficient approach for formulating an effective donor/acceptor coupling in terms of the entire network parameters. The approach is based on implementation of Feshbach projection operators to map the entire network Hamiltonian onto a subspace defined by two specific donor and acceptor sites. This nonperturbative approach enables to define regimes of network parameters in which the effective donor-acceptor coupling is optimal. This is demonstrated numerically for simple models of molecular networks.

Near-threshold vibrational excitation of acetylene by positron impact

We report vibrational excitation cross sections for C-C and C-H symmetric stretch modes of acetylene by positron impact. The contribution of these infrared inactive modes to the annihilation parameter is also addressed. The Feshbach projection operator approach was employed to vibrationally resolve e{sup +}-acetylene scattering phase shifts obtained with the Schwinger multichannel method. The present results point out a virtual state pole at the equilibrium geometry of acetylene that becomes a bound state as either bond is stretched, in qualitative agreement with previous calculations for small hydrocarbons. The vibrational couplings are stronger for the C-C mode, giving rise to a bound state pole within the Franck-Condon region of the vibrational ground state. These bound and virtual states give rise to sharp threshold structures (vibrational resonances) in both the vibrational excitation cross sections and the annihilation parameter (Z{sub eff}). We found fair agreement between the present calculations and previously reported e{sup +}-acetylene vibrational excitation cross sections.

Feshbach projection operator approach to positron annihilation

We present a theory of vibrationally enhanced positron annihilation on molecules based on the projection operator formalism of Feshbach. Results are shown for a simple semiempirical model derived from the formal Zeff expression.