Interrelation between variational principles for scattering amplitudes and generalized R-matrix theory.

The functionals of the Kohn and Sil variational principles of atomic collision theory contain second-order spatial derivatives of the wave-function. Ipso facto the "second-order" generalised Euler-Lagrange differential equation is a necessary condition for these functionals to yield a stationary value. To illustrate this we provide a classical prescription for the time-dependent and time-independent quantum potential scattering problems, generalising the latter to the three-body problem. As practical examples where the presence of second-order derivatives in the Lagrangian density makes a non-trivial contribution to the equations of motion, we consider the optimisation of translation factors in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principle. As further examples of Lagrangian densities where second-order derivatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.

In this article, I consider the use of the Kohn variational principle in obtaining the reaction matrix, which determines the positive energy solution to the partial wave Schrodinger equation in coordinate space. The formulation considered is appropriate for electron scattering from an atom or molecule. In its usual form, the Kohn variational principle provides the relationship between the exact reaction matrix, independent of first-order errors in the trial wavefunction, and the trial reaction matrix. This relationship is usually called the Kato identity. It is obtained using the second form of Green's theorem for a sphere of infinite radius. I show how a Kohn-like variational principle, using Green's theorem for a sphere of arbitrary radius, provides an expression for the exact, coordinate space, integral equation amplitudes in terms of the trial amplitudes. The results are again exact to second order in the trial wavefunction. Applications of the present results to electron-molecule frame transformations and algebraic scattering theory are then suggested.

It is proved that if a scaling parameter alpha contained in suitably chosen trial functions F1T and F2T is optimized using Kohn's variational principle, then the virial theorem is satisfied by the optimized variational functions and the variationally determined scattering amplitude. A generalization of the theorem to arbitrary values of the scaling parameter is also established. In addition the virial theorem for the scattering amplitude is applied to the elastic scattering of electrons by atomic hydrogen.

We propose new basis sets for linear algebraic variational calculations of transition amplitudes for quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained even for basis sets with the majority of the members independent of energy.

The role of basis set expansions in the relative performance of the schwinger and the Newton variation principles

The form of hypervirial theorem which is appropriate in scattering theory is discussed in general terms. It is shown that variational wave functions which are optimized in accordance with Kohn's variational principle do satisfy hypervirial theorems. Thus such theorems may be useful in selecting approximate wave functions to give accurate phase shifts or scattering amplitudes. The situation is analogous to that of energy-optimized wave functions for bound-state systems.

The extension of the Kohn-Hulthen variational principle to the multi-channel reaction problem is discussed. It is shown that this method leads to a unitaryS-matrix. As an application of the method a four channel calculation for the 3/2+ state of5Li is presented.

We show explicitly how the commonly adopted prescription for calculating effective mode volumes is wrong and leads to uncontrolled errors. Instead, we introduce a generalized mode volume that can be easily evaluated based on the mode calculation methods typically applied in the literature, and which allows one to compute the Purcell effect and other interesting optical phenomena in a rigorous and unambiguous way.

The Kohn variational principle and the correlated Hyperspherical Harmonics technique are applied to study the n — 3H and p — 3 He low energy scattering. The calculated n — 3H total cross section at zero energy agrees well with the measured value, while a small discrepancy is found for the coherent scattering length. For the p — 3He scattering process, a comparison between the calculated and measured observables is performed at 3 MeV, where accurate experimental data are available. A reasonable agreement is found for the differential cross section, whereas clear discrepancies are observed for the polarization observables.

We present multichannel nuclear reaction formalisms based on the Schwinger and Kohn variational principles. Our formalisms seem somewhat simpler in form than those heretofore proposed. The necessity of using the entire set of asymptotic boundary conditions to constrain the trial wave function is stressed. We show how this may be done by employing the coupled integral equations formalism discussed by Baer and Kouri, Kouri and Levin, and by Tobocman. Several explicit expressions for the transition amplitude are presented which appear to provide a promising basis for doing many-body system scattering reaction calculations.

The most detailed observables in theoretical studies of chemical reactions are the quantum state-to-state reaction probabilities determined as the modulus squared of the scattering matrix (S matrix) elements. Quantum mechanical methodology for S matrix calculation is well developed, especially in the level of Born-Oppenheimer approximation used for separating the nuclear dynamics from electronic motions. One of such numerical methods adopts boundary value Bloch operator in setting up the basis set representation of Schrodinger equation for the nuclear dynamical solution in the continuum region of energy (scattering energy) such as in the log derivative Kohn variational principle (Y-KVP) method of Manolopoulos and Wyatt. It has virtues of “totally energy-independent and real-valued basis functions/intermediate integrals” up to the point just before the final complex-valued S matrix evaluation. Since one must investigate the scattering solutions for many energies in the general practice of scattering problem, these properties substantially increase the computational efficiency than otherwise. However, when applied for general reactive multi-arrangement scattering on non-orthogonal coordinate system (e.g., the well-known collinear H2+H reactive scattering), it turns out to be inevitable to adopt wasteful redundant basis sets, each of which suitable for entrance and exit channels (at least, in the primitive direct product basis set description), and the subsequent cumbersome exchange-type integrations between the functions defined on different coordinate systems are unavoidable (about half of the integrals are of this type for the above example). Such formal inefficiency in the YKVP method can be avoided in the S-matrix version of Kohn variational principle (S-KVP) method by replacing the real-valued and energy-independent continuum-type basis functions with complex-valued and energy-dependent ones. However, it accompanies adverse effects, i.e., making a small rectangular part of Hamiltonian matrix to be complexvalued and energy-dependent. This work was motivated by noting that a delta function is included in the definition of conventional Bloch operator, consequently, it becomes a surface operator in multi-dimensional problem. By changing the effective region of the Bloch operator from surface to volume, we might hope to remove such unfavorable features of the Y-KVP method in a different way, and we intend to pursue this idea further. In the long run, no redundant basis sets are needed and all basis functions are real-valued and energy-independent, subsequently most of the intermediate integrals are real-valued and energy-independent except Nopen (the number of open channels) integrals which are complex-valued and energydependent in the present approach. All such features add up to suggest better performance over both of the Y-KVP and S-KVP methods, at least formally. Unfortunately, a small rectangular Bloch-operator-related matrix becomes complexvalued (though it can be evaluated from energy-independent intermediate integrals) just like in the S-KVP method. Perhaps, more importantly, it is a pity that this method is not benefited from the variational property, thus the calculated numerical S matrix is not guaranteed to be symmetric, unlike in the Y-KVP and S-KVP methods, as a result, it might be suffered from unstable features. Now we detail the derivation of the present approach. Scattering wave function ψ is chosen to satisfy the usual Smatrix boundary condition in the asymptotic region as a linear combination of incoming (I) and outgoing (O) waves with S matrix as the expansion coefficient,

The analysis of multiple elastic scattering by a system of fixed scatterers in terms of two-body collision matrices is extended to large bodies. The dynamical theory of particle wave propagation in crystals is formulated and discussed from this viewpoint, as an extension of Ewald's theory to scatterers of finite size. A theory of wave propagation in a random medium, including disordered crystals, is obtained as the limiting case for a crystal with infinite unit cell. Contrary to the usual approach, this analysis considers a definite "frozen" configuration of the medium; statistical averaging over configurations may be used on the intensities obtained.In crystals and dense liquids, radiation damping is compensated by the reaction of surrounding scatterers. Therefore, it is found convenient to restate the basic equations of multiple scattering in terms of a matrix $\ensuremath{\beta}$ which is essentially the Green's function for the single scattering event, with "standing wave" rather than "outgoing wave" boundary conditions. This has the advantage of eliminating the spurious radiation damping terms at the outset; one obtains nonattenuated waves in nondissipative crystals or dense liquids by a simple expansion method.Application is made to the following problems: the equivalent of the Lorentz-Lorenz formula for matter waves; index of refraction for an interaction of the form of a one-level Breit-Wigner formula; attenuation of long waves in a disordered crystal containing protons with random spins; double refraction effect in a liquid or crystal containing protons with partially oriented spin.

Two first-order differential operators are introduced to generate recursion formulae for hypergeometric functions. These operators, on the one hand, factorise the associated second-order operator and, on the other hand, reproduce the equations resulting from the action of the supersymmetry generators on the positive-energy solution of the Coulomb field. The results for the confluent hypergeometric functions also provide a natural basis for constructing ladder operator recursion relations for the Coulomb Green function with the outgoing wave boundary condition.

The Correlated Hyperspherical Harmonic Method is used to describe bound and scattering states in few-body systems. Special attention is given to the three-nucleon problem in which the presence of correlation factors substantially accelerate the convergence of the expansion. For scattering states the complex form of the Kohn variational principle is used to determine the S- or T-matrix. Using this formalism the method can be extended to energies above the three-body breakup threshold. Using realistic NN interactions comparisons to experimental data are given.

We present a preliminary study of the effect of a three-nucleon force (3NF) in \(p{\text{-}}^3\mathrm{H}\) and \(n{\text{-}}^3\mathrm{He}\) scattering at low energies. The used 3NF is derived from effective field theory at next-to-next-to-leading order. The four-nucleon scattering observables are calculated using the Kohn variational principle and the hyperspherical harmonics technique and the results are compared with available experimental data. We have found that the effect of introducing this type of 3NF is tiny, and sometimes worsens the agreement with the experimental data.