Direct calculation of resonant states in reactive scattering. Application to linear triatomic systems

A direct method for calculating resonant states in reactive scattering is suggested, permitting us to obtain the characteristics of multichannel resonances (partial width amplitudes). The method is based on the construction of a Laurent expansion of the scattering matrix S(? ?iΓ/2) in the complex plane. The position of the poles of the S matrix are derived by solving the dynamical problem with complex energy values. The residue at the pole gives all the information concerning the partial widths. The method is applied to a linear triatomic reactive scattering problem. The properties of the resonant states in the H + H₂ system are calculated as an example. Two broad resonances are found which have not been reported in previous calculations. The interference of overlapping resonances is shown to have a profound effect on the energy dependence of the transition probabilities.     

Interacting resonances in the F+H2 reaction revisited: complex terms, Riemann surfaces, and angular distributions

We study the effect of overlapping resonances on the angular distributions of the reaction F+H2(v=0,j=0)-->HF(v=2,j=0)+H in the collision energy range from 5 to 65 meV, i.e., under the reaction barrier. Reactive scattering calculations were performed using the hyperquantization algorithm on the potential energy surface of Stark and Werner [J. Chem. Phys. 104, 6515 (1996)]. The positions of the Regge and complex energy poles are obtained by Pade reconstruction of the scattering matrix element. The Sturmian theory is invoked to relate the Regge and complex energy terms. For two interacting resonances, a two-sheet Riemann surface is contracted and inverted. The semiclassical complex angular momentum analysis is used to decompose the scattering amplitude into the direct and resonance contributions.     

Variational Solutions for Resonances by a Finite-Difference Grid Method

We demonstrate that the finite difference grid method (FDM) can be simply modified to satisfy the variational principle and enable calculations of both real and complex poles of the scattering matrix. These complex poles are known as resonances and provide the energies and inverse lifetimes of the system under study (e.g., molecules) in metastable states. This approach allows incorporating finite grid methods in the study of resonance phenomena in chemistry. Possible applications include the calculation of electronic autoionization resonances which occur when ionization takes place as the bond lengths of the molecule are varied. Alternatively, the method can be applied to calculate nuclear predissociation resonances which are associated with activated complexes with finite lifetimes.     

Classical theory of collisional entropy production

A classical dynamical theory of elementary collision processes is formulated in analogy to the quantum theory of the dynamical scattering matrix, which can be defined for a pure quantum stationary scattering state. The elements of this matrix are probability amplitudes for transitions between internal states defined for given values of a reaction coordinate. The squared magnitudes of these amplitudes, modeled in the proposed classical theory, define normalized internal state population distributions suitable for information theoretical analysis. Statistical entropy and surprisal are defined as dynamical functions of a reaction coordinate. This formalism differs fundamentally from concepts based on the classical Liouville equation.     

The dependence of the branching ratio in the F+HD reaction on the initial rotational state of HD

Both classical trajectory and quantal scattering calculations indicate that the branching ratio in the F+HD reaction varies considerably with the initial rotational state of HD. Information theory argues that this variation must be reflected in the distribution of the reaction products. Hence, given the (normalized) product distribution for each reaction path one should be able to predict the dependence of the branching ratio on the state of the reagents. The trajectory computations of Muckerman are used to illustrate the procedure. First the dynamic constraint is identified and then the reaction probability matrix is constructed. The determination (“synthesis”) of the matrix, in terms of the given constraint invokes information theory and, in particular, the procedure of maximising the entropy. The branching ratio is readily computed from the elements of the probability matrix. Very good agreement is obtained between the trajectory-computed and the synthetic branching ratio for all initial rotational states of HD.The F+HD reaction has three possible final arrangement channels (one nonreactive and two reactive ones) and is used to illustrate the structure of the reaction probability matrix and the associated entropy measures.     

A new method for the direct calculation of resonance parameters with application to the quasibound states of the H2X1Σ g + system

A new method for the direct calculation of resonance parameters is presented. It is based upon searching for poles of the scattering matrix at complex energies. This search is expedited by the use of analytic derivatives of the scattering matrix with respect to the total energy. This procedure is applied initially to a single channel problem, but is generalizable to more complicated systems. Using the most accurate available potential energy data, we calculate resonance parameters for all of the physically important quasibound states of the ground electronic state of the hydrogen molecule. Corrections to the Born-Oppenheimer potential are included and assessed. The new method has no difficulty locating resonances with widths greater than about 1×10^–7 cm^–1. It is easier to find narrow resonances by monitoring the dependence of the imaginary part of the reactance matrix on the real part of a complex energy than to monitor the dependence of the eigenphase sum on energy at real energies.     

Glory and thresholds effects in H+D2 reactive angular scattering

We analyse H+D₂ reactive angular scattering using the S-matrix elements obtained by Aoiz et al. and Althorpe et al. Enhancement of small angle scattering in the v^′=3←v=0 H+D₂ delayed reaction is attributed to a glory effect caused by threshold resonances in the v=3 vibrationally adiabatic channel. The oscillatory structures in the reactive angular distributions are shown to be of nearside–farside (NP) origin and are likely to arise from capture in a number of relatively short-lived barrier Regge states at large angular momenta. Padé reconstruction of the reactive matrix element is discussed in detail.     

Frequency distribution of radiation emitted from excited atomic systems

General expressions for the time-dependent probability amplitudes of the quantum states of two arbitrary, interacting atoms are calculated when one atom is initially in an excited p state and the other atom is in an s ground state. The lifetimes of the excited states and the line shape of the emitted radiation are obtained as functions of both the atomic separation and the energy difference between the excited states of the two atoms. The emission line shape is shown to be doubly peaked and to agree with the line shape of the radiation scattered by a system of two interacting atoms. The expressions for the lifetimes of the excited states are found to be identical to those obtained for the radiation scattering situation.     

Resonances in the elastic electron-SF6 molecule scattering

In agreement with previous calculations by the authors, three definite resonances in the elastic electron-SF₆ molecule scattering in the energy range 5–40 eV have been obtained with a multiple scattering method, elaborated originally for molecular bound states and modified here for scattering states. Results with various more-reliable one-electron potentials have been compared with previous calculations and available experimental data.     

A simple classical prediction of quantal resonances in collinear reactive scattering

Quantal collinear reactive scattering computations have shown that in the vicinity of thresholds of reactant or product vibrational states, one finds resonances in the state to state reaction probability. We find that these resonances can be explained classically in terms of energy transfer between adiabatic reactant and product channels. This transfer is attributable to resonant periodic orbits, resonating between reactants and products. The classical condition for a quantal resonance is that the action of the orbit over one period be an integer (in units of h) and that the energy at which this occurs be lower than the adiabatic barrier heights of the resonating states. These conditions suffice for a prediction of the location of the quantal resonance within a 1% accuracy!     

Extraction of inelastic scattering information from approximate wavefunctions constructed in an L2 basis

It is shown that one can calculate the elements of the reactance matrix from square-integrable approximations to the exact scattering wavefunctions at energies where no resonances occur. In cases where isolated resonances dominate the scattering, the computational procedure yields the resonance parameters directly.     

Lineshape theory for three-level double resonance in the presence of infrared laser radiation

A set of coupled optical Bloch equations has been solved to obtain the absorption lineshape for both copropagating and counterpropagating three-level double resonances in the presence of a strong infrared laser source. The resonance conditions and lineshapes derived for the Lamb dip and three-level double resonances are confirmed by recent experimental findings in ¹⁵NH₃.     

Role of resonances in building cross sections: Comparison between the Mittag–Leffler and the T‐matrix Green function expansion approaches

Peaks in collision cross sections are often interpreted as resonances. The complex dilation method, as well as other methods relying on analytic continuation of the scattering formalism, can be used to clarify whether these structures are true resonances in the sense that they are poles of the S‐matrix and the associated Green function. The performance of the Mittag–Leffler expansion and T‐matrix Green function expansion methods are formally and computationally compared. The two methods are applied to two model potentials. Eigenenergies, s‐wave residues, and cross sections are computed with both methods. The resonance contributions to the cross sections are further analyzed by removing the residue contributions from the Mittag–Leffler and Green function expansion sums, respectively. It is suggested that the contribution of a resonance to a cross section should be defined through its S‐matrix residue. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

New lithium ion conducting polymer electrolytes based on polysiloxane grafted with Si-tripodand centers

New kind of polymer host for lithium cations has been synthesized by catalyzed hydrosilylation reaction involving hydrogen atoms of a polysiloxane and double bonds of vinyl tris-2-methoxyethoxy silane. The obtained macromolecule can be regarded as siloxane backbone grafted with silicon tripodand elements with very short polyether chains. A family of Li ion conducting polymer electrolyte membranes have been prepared by dissolving LiPF₆ in thus obtained polymer matrix. Exceptionally high room temperature specific conductivities, exceeding 10⁻³ S/cm at 25 °C, have been measured for the studied polymer electrolytes. It is proposed that polyether chains tend to self-assembly in the presence of Li cations and this highly organized arrangement of Li coordination sites creates pathways of high lithium conductivity along the polysiloxane backbones. In addition to that, strong shielding of Li-cations suppresses the formation of ion pairs, thus increasing the charge carrier concentration. The electrolytes can be easily formed into dimensionally stable, flexible membranes.     

Direct evaluation of the lifetime matrix by the hyperquantization algorithm: narrow resonances in the F + H2 reaction dynamics and their splitting for nonzero angular momentum

We propose a new method for the direct and efficient evaluation of the Felix Smith's lifetime Q matrix for reactive scattering problems. Simultaneous propagation of the solution to a set of close-coupled equations together with its energy derivative allows one to avoid common problems pertinent to the finite-difference approach. The procedure is implemented on a reactive scattering code which employs the hyperquantization algorithm and the Johnson-Manolopoulos [J. Comput. Phys. 13, 455 (1973); J. Chem. Phys 85, 6425 (1986)] propagation to obtain the complete S matrix and scattering observables. As an application of the developed formalism, we focus on the total angular momentum dependence of narrow under-barrier resonances supported by van der Waals wells of the title reaction. Using our method, we fully characterize these metastable states obtaining their positions and lifetimes from Lorentzian fits to the largest eigenvalue of the lifetime matrix. Remarkable splittings of the resonances observed at J>0 are rationalized in terms of a hyperspherical model. In order to provide an insight on the decay mechanism, the Q-matrix eigenvectors are analyzed and the dominant channels populated during the decomposition of metastable states are determined. Possible relevance of the present results to reactive scattering experiments is discussed.     

Ring-breaking electron attachment to uracil: following bond dissociations via evolving resonances

Calculations are carried out at various distinct energies to obtain both elastic cross sections and S-matrix resonance indicators (poles) from a quantum treatment of the electron scattering from gas-phase uracil. The low-energy region confirms the presence of pi(*) resonances as revealed by earlier calculations and experiments which are compared with the present findings. They turn out to be little affected by bond deformation, while the transient negative ions (TNIs) associated with sigma(*) resonances in the higher energy region ( approximately 8 eV) indeed show that ring deformations which allow vibrational redistribution of the excess electron energy into the molecular target strongly affect these shape resonances: They therefore evolve along different dissociative pathways and stabilize different fragment anions. The calculations further show that the occurrence of conical intersections between sigma(*) and pi(*)-type potential energy surfaces (real parts) is a very likely mechanism responsible for energy transfers between different TNIs. The excess electron wavefunctions for such scattering states, once mapped over the molecular space, provide nanoscopic reasons for the selective breaking of different bonds in the ring region.     

Exponential perturbation theories for inelastic scattering

An Exponential Perturbation Theory (EPT) is derived whereby one calculates a phase-shift matrix by an nth order perturbation theory and then exponentiates it to obtain the scattering matrix. The theory has been developed to include high-order terms, closed channels and resonances. The radial wavefunctions used are WKB solutions which are generalized to cases where there are multiple turning points. The orbital angular momentum may be treated exactly or in the classical or sudden limits. Calculations are done for the rotationally inelastic scattering in He + H₂, Ar + N₂ and Ar + HCl. The first two systems give fair to good agreement with accurate calculations; the last case gives poor agreement. The first-order EPT is very much better than the first-order distorted-wave approximation.     

Unimolecular dissociation rates by a semiclassical analysis of quantum resonance lifetimes

A semiclassical analysis of quantum-mechanical resonance lifetimes provides a framework for describing unimolecular dissociation rates near or below the adiabatic thresholds associated with saddles on potential energy surfaces. A formula which can be used when only one channel is effectively open is derived. The behaviour of resonances supported by two interacting states is briefly studied as a function of the coupling strength.