Variational many-channel nuclear reaction formalism

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 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.

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

We establish a connection between the Kohn variational principle, with (complex) outgoing-wave boundary conditions, and the Kapur-Peierls form of the R-matrix theory. We show that the complex Kohn method, unlike the usual Kohn method, does not suffer from the problem of spurious singularities. We also discuss a generalization that allows the calculation of scattering cross sections over a continuous range of energies from a single diagonalization of the Hamiltonian. Several numerical examples are presented.

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 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.

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 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.

Quantum scattering using a novel implementation based on the variational R matrix formalism and the finite element method: a comparative study

The effect of the inclusion of different models of three nucleon (3N) forces in p − 3He elastic scattering at low energies is studied. Two models have been considered: one derived from effective field theory at next-to-next-to-leading order and one derived from a more phenomenological point of view—the so-called Illinois model. The four nucleon scattering observables are calculated using the Kohn variational principle and the hyperspherical harmonic technique and the results are compared with available experimental data. We have found that with the inclusion of both 3N force models the agreement with the experimental data is improved, in particular for the proton vector analyzing power A y .

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,

We present a detailed study of the effect of different three-nucleon interactions in p-(3)He elastic scattering at low energies. In particular, two interactions have been considered: one derived from effective field theory at next-to-next-to-leading order and one derived from a more phenomenological point of view-the so-called Illinois model. The four-nucleon scattering observables are calculated by using the Kohn variational principle and the hyperspherical harmonics technique, and the results are compared with available experimental data. We have found that the inclusion of both interactions improves the agreement with the experimental data, in particular, for the proton vector analyzing power.