Kohn Method Research Articles

Disappearance of continuum exchange integrals from algebraic variational calculations of electron scattering.

We show how it is possible to systematically eliminate, with little resulting error, all ``free-free'' and ``bound-free'' exchange integrals needed to implement electron scattering calculations with algebraic variational techniques, such as the Kohn method. This elimination is rigorous within the framework of the separable approximation for exchange interactions and follows from certain invariance properties of the scattering wave function. The procedure is illustrated for the case of ${e}^{\mathrm{\ensuremath{-}}}$+He and ${e}^{\mathrm{\ensuremath{-}}}$+H scattering in the static-exchange approximation.

The inclusion of $\pi$ functions in the treatment of low-energy positron - hydrogen-molecule scattering by a generalisation of the Kohn method

In the application of a generalisation of the Kohn method to low-energy positron-hydrogen-molecule scattering which was described in a recent letter, up to 31 short-range correlation functions, made up of one-particle functions of sigma symmetry, were included in the trial function. In the calculation described in this paper, the flexibility of the trial function is greatly improved by the inclusion of up to 64 short-range correlation functions of which 32 contain products of one-particle functions of pi symmetry. The inclusion of the pi functions has a very big effect on the calculated results, especially at very low energies. The behaviour of the phaseshift values with increasing incident energy is qualitatively similar to the behaviour of the corresponding S-wave phaseshifts in low-energy positron-helium scattering. Comparison with experiment indicates that the results reproduce qualitatively the experimental trend at very low energies but higher partial waves must make a significant contribution to the total cross section above about 0.1 eV. The improved results can be attributed to the much more satisfactory description of the polarisation of the hydrogen molecule made possible by the inclusion of pi functions.

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

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.

A new basis set method for quantum scattering calculations

A new basis set approach for quantum scattering calculations is described and tested on model problems of elastic and inelastic collisions. The approach is essentially the Kohn variational method, but applied to the S or T matrix directly rather than to the K matrix as is normally done; it is seen that the result of the present approach is not equivalent to the usual Kohn method (i.e., for the K matrix) and is indeed preferable to it. The present approach is seen to have the same structure as the complex scaling/coordinate rotation expressions for the T matrix (but with some added features). Its potential advantage over the Schwinger variational method, another useful basis set technique, is that matrix elements of the Green’s function for some reference Hamiltonian are not required; the present method requires only matrix elements of the Hamiltonian itself between the basis functions. The essential reason for all of these desirable features is that the basis set which is used incorporates the correct scattering boundary conditions.

The inclusion of π functions in the treatment of low-energy positron-hydrogen-molecule scattering by a generalisation of the Kohn method

In the application of a generalisation of the Kohn method to low-energy positron-hydrogen-molecule scattering which was described in a recent letter, up to 31 short-range correlation functions, made up of one-particle functions of sigma symmetry, were included in the trial function. In the calculation described in this paper, the flexibility of the trial function is greatly improved by the inclusion of up to 64 short-range correlation functions of which 32 contain products of one-particle functions of pi symmetry. The inclusion of the pi functions has a very big effect on the calculated results, especially at very low energies. The behaviour of the phaseshift values with increasing incident energy is qualitatively similar to the behaviour of the corresponding S-wave phaseshifts in low-energy positron-helium scattering. Comparison with experiment indicates that the results reproduce qualitatively the experimental trend at very low energies but higher partial waves must make a significant contribution to the total cross section above about 0.1 eV. The improved results can be attributed to the much more satisfactory description of the polarisation of the hydrogen molecule made possible by the inclusion of pi functions.

Reexamination of the optimized Kohn method

The optimized Kohn method proposed by Takatsuka and Fueno [Phys. Rev. A 19, 1011 (1979)] is reexamined. The approach is clarified and, in particular, it is shown that the "unavoidable" singularities appearing in their treatment can also be eliminated and, therefore, have nothing to do with true resonances. The strong points of the optimized procedure are discussed and reemphasized in the light of the present formulation.

Comparison of convergence for the schwinger, optimized anomaly-free, and optimized minimum-norm variational methods for potential scattering

We apply the Schwinger method, two Kohn-type methods, and three Harris-Michels-type methods to electron scattering by the same potential with the same sets of basis functions We also test a polynomial basis in the Schwinger method. We use Nesbet and Oberoi's method to avoid spurious singularities of the Kohn method Kohn and Harris-Michels methods show much better convergence to the accurate result than the Schwinger method

Resonance states in scattering: some variational methods revisited

The authors have studied two variational methods of scattering, namely the Kohn method and the Harris modification of it and applied them to resonance. As a case study, they consider a coulombic acceptor impurity potential in a zero-gap semiconductor like mercury telluride in which, on physical grounds, an impurity resonance is expected and also the free particle Hamiltonian has a fairly complex structure due to the strong spin orbit coupling. The advantages and the limitations of the two methods are discussed with an emphasis on resonance. They have made a significant improvement in the algorithms of both methods over their previous applications in the literature and have emphasised the role of the nonlinear parameter which in effect has been used as a variational parameter in the theory.

Optimized Kohn method for scattering. I. Single-channel scattering

An algebraic variational procedure has been developed for the calculation of the phase shift $\ensuremath{\rho}$ of the radial wave function for a particle undergoing single-channel scattering. The method is essentially an optimization of Kohn's theory with respect to the phase parameter $\ensuremath{\theta}$ as involved in Kato's wave function. Specifically, the basis set has been transformed linearly so as to permit distinction between avoidable (spurious) and unavoidable (innate) singularities. On this basis, two new optimization procedures, which are termed the minimum-basis-dependence (MBD) and minimum-error (ME) methods, have been proposed. Various standard variational theories have also been reformulated in a unified manner. Sample basis-set calculations of $\ensuremath{\rho}$ have been carried out for the Hazi-Taylor model potential in order to demonstrate the relative merits of the MBD and ME methods.

Optimized Kohn method for scattering. II. Multichannel scattering

The minimum-basis-dependence method which has previously been proposed for the single-channel scattering is extended to the multichannel case. The extension entails solutions of a set of simultaneous equations in two matrices which directly give the reactance matrix. A general variational functional is proposed explicitly for any wave function which is determined by an algebraic method. The functional can be made use of as a correction formula for a reactance matrix which is obtained as the solution of the algebraic equation.

Coulombic resonance states in zero-gap semiconductors

We present a detailed theory of resonance states in HgTe due to charged acceptor impurities. This problem has previously been studied by Gel'mont and Dyakonov, but on the rather unrealistic assumptions of an unscreened Coulomb potential and zero electron mass. In a zero-gap (ZG) semiconductor, the screening is important and sensitive to the band parameters. Within the effective-mass formalism, we have obtained a variational solution for the lowest-order radial components in the partial-wave expansion of the wave function. Two variational methods for the phase shift are used and improved upon---the Kohn method and the Harris method, which is an interesting modification of the former and is more appropriate for the study of resonances. A simple Thomas-Fermi (TF) potential and the full random-phase approximation screened Coulomb potential have been used. We find that the resonance energy ${E}_{R}$ is very sensitive to screening, while the relative width $\frac{\ensuremath{\Gamma}}{{E}_{R}}$ is only weakly dependent upon it. The interband polarization, which is known to be substantial in a ZG semiconductor, dramatically reduces the resonance energy. Interestingly, within the reported range of band parameters, either the 0.7-meV or the 2.2-meV resonance can be reproduced, corresponding respectively to hole masses of $0.35{m}_{0}$ or $0.58{m}_{0}$. We also find that the resonance energy varies by at most 10% with the Fermi level ${E}_{F}$ in the range of interest, ${E}_{F}l{E}_{R}$.

"Gauge transformation" for the Kohn method

A simple example of a "gauge transformation" for trial functions used in the Kohn method is presented. Under this transformation the Kohn-method answer remains invariant while the optimized value for the phase shift of the trial function becomes completely arbitrary. Thus the only important quantity may be $ktan\ensuremath{\delta}\ensuremath{-}\ensuremath{\int}\ensuremath{\psi}(H\ensuremath{-}E)\ensuremath{\psi}dr$, and no special significance should be attached to $ktan\ensuremath{\delta}$.

A variational principle for scattering from an asymmetric non- local potential

A variational method, analogous to that of the Kohn method in potential scattering, is derived which solves integro-differential equations with asymmetric kernels. The method is applied to S-wave elastic scattering of electrons from hydrogen atoms in the polarized-orbital approximation.

Variational approach to the quantum theory of scattering. II. The Kohn method and Kato eigenvalue problem

Certain similarities between the equations arising in scattering theory and the variational approach to time dependent perturbation theory are used to show that the Kohn variational technique for solving scattering problems is equivalent to a variational method for calculating a special set of eigenfunctions, in terms of which the solution to the scattering problem is expanded. Contrary to the results of previous studies, the error term in the scattering problem is bound from below, and the Kohn method is equivalent to a minimum principle. Levinson's theorem is rederived, and extended to higher energies. On the basis of the theory presented in this study, a new method for calculating upper and lower bounds on the energies of resonance and Ramsauer-Townsend states is suggested.

Resonances and Pseudoresonances in the Kohn Variational Method and the Feshbach Method

An operator formalism of the general single-channel Kohn method is developed, and Schwartz's theory of singularities in the Kohn method is generalized; any approximation method described in the framework of the Kohn method is shown to have, in general, singularities, some of which may correspond to true (either closed-channel or shape) resonances. Caution is urged against simple interpretation of “resonance results” of close-coupling calculations and other expansion methods. The Feshbach Q H Q eigenvalues are shown to be approximations to the positions of the Kohn-method singularities. A correction to the Schwartz theory leads to the Harris method. Precisely at the characteristic Harris eigenvalues, the Hulthen condition \(\langle\varPsi|E-H|\varPsi\rangle{=}0\) cannot be satisfied in general, though, as was shown previously, a well-defined limiting value (which is equal to the Harris value, and has a first-order error contrary to a variational phase shift) of the Hulthen phase shift exists. An elaborate ...

Variational Methods in Potential Scattering

A comparison of a variety of standard variational methods with a new method due to Harris is made with a view towards selection of the optimal method to be used in realistic many-body scattering calculations. Numerical results for two short-range potentials, the attractive exponential and the attractive Yukawa potential, are given and compared with exact results obtained analytically or by direct numerical integration. It is demonstrated by calculation that the source of an anomaly observed in earlier studies by Schwartz is not due to the attributed reason and, furthermore, we find the Kohn method significantly more accurate at the Harris eigenvalues than any of the other methods. Therefore, we propose the use of the Kohn method at the eigenvalues of the matrix Hamiltonian in the trial function subspace as the optimal method to be used in variational scattering calculations.

Surface states in semi-infinite crystals: A Green function method

In this paper we develop a Green function method, which can be considered as a generalization of the Kohn and Rostoker method, and which enables us to calculate the surface states in a semi-infinite three-dimensional crystal. We only consider the case of one atom in a unit cell, the extension to complex crystals being straightforward. We do not only demonstrate the existence of two-dimensional energy bands localized at the surface and superimposed on the three-dimensional bandstructure of the bulk, but we derive formulae which can be used for practical calculations if one has the disposal of high speed computers with large capacity.

Study of the Singularities in the Kohn Variational Method for Scattering Parameters

The Kohn variational method for calculating scattering parameters is formulated in an abstract vector space in terms of the extremization of a quadratic form whose coefficients represent matrix elements of $H\ensuremath{-}E$. This manifestly possesses certain symmetry and invariance properties not apparent in the usual formulation. Under certain conditions the method provides a bound for the exact answer if the number of variational parameters, $N$, is large enough. The extraneous singularities in the variationally determined value of $tan\ensuremath{\delta}$ noticed by Schwartz are analyzed. These are a direct consequence of the Kohn method and the particular normalization employed. It is shown that the pole strength (residue) of each such singularity is the product of two terms, each of which is small; these factors may be expected to decrease with increasing $N$.

On the variational method for nonrelativistic scattering: I. Differential form

(i) Hulthén's and Kohn's sets of equations for the functional of Hulthén are not enough to satisfy the Euler equation of the variational problem (even in case of infinite parameters) which is a necessary condition of the calculus of variation, because the phase shift found in either way usually fails to satisfy the identity obtained from the asymptotic behavior of the wave functions. (ii) Since the free boudary condition contains in itself the unknown phase shift, a self-consistent procedure is involved. A set of variational equations is suggested where a prescription is given to replace the self-consistency. This ensures that the Schrödinger equation for the scattering comes out as the Euler equation of the variational problem. (iii) In this case Hulthén's and Kohn's methods become equivalent and Kohn's condition is same as Eq. 11. (iv) Furthermore, this variational prescription automatically excludes such choices of trial wave functions which give rise to variational parameters which are discontinuous functions of energy as found by Schwartz (v) The importance of the sufficiency condition is stressed. It is then pointed out that in all those cases already calculated, where Hulthén's and Kohn's method gave good results, the prescription actually formulated here has been applied. For the specific case of the elastic scattering by He in the Hartree approximation, this prescription gave better results than Hulthén's or Kohn's method (Section III).

Variational Method for Scattering Length

The properties of the scattering length obtained by Kohn's method, which is one of Hulthen's variational methods, arc studied by assuming a linear trial function with n adjustable parameters. The scattering length A/sup (n)/ decreases monotoiiically as the number of adjustable parameters n increases, if there is no bound state in the system. This conclusion essentially comes from the upper bound theorem of Spruch and Rosenberg. When the system has m bound states, the scattering length increases in value only m times, and otherwise decreases monotonically. Therefore, after one verifies the presence of m increases, the calculated value is certain to give an upper bound on the scattering length. The connection between the result above and the condition of Rosenberg, Spruch, and O'Malley is considered. In the Appendix comparison is made of the scattering length A/sup (n)/ obtained by Hulthen's original method and Kohn's method when m bound states exist in general. (auth)