A pointwise representation of the s-matrix Kohn variational principle for quantum scattering

TL;DR: In this article, a method for reducing the complexity of scattering calculations carried out using the Kohn variational principle is proposed, which is based upon the use of a pointwise representation for the L 2 part of the basis set and eliminates the need to explicitly evaluate any integrals involving such functions.

About: This article is published in Chemical Physics Letters.The article was published on 1988-08-19 and is currently open access. It has received 17 citations till now. The article focuses on the topics: Pointwise & Variational principle.

Summary

Introduction

• The technique is based upon the use of a pointwise representation for the L2 part of the basis set and eliminates the need to explicitly evaluate any integrals involving such functions.
• The final expression is found to have a similar structure to the original one except that the matrices involving the L* basis now occur in point space.
• In section 3 more general basis sets and quadrature schemes are considered, and the authors also address the problem of hermiticity of the Hamiltonian matrix when calculated by an approximate quadrature.
• Application to a heavy particle potential scattering problem shows that the technique provides an accurate description when compared with the method in which all integrals are evaluated exactly.

3, Application to 1D electron scattering

• In order to test the accuracy of this pointwise representation of the Kohn expression the authors first consider the electron scattering problem used by Staszewska and Truhlar [ 191.
• The potential is attractive and has the form V(u)=-exp( -r) .
• This scheme evaluates the kinetic energy matrix exactly since the second derivative of eq. (10) is a linear combination of the basis functions and the Gaussian quadrature evaluates all overlap integrals exactly.
• Table 1 shows how the error in the tangent of the phase shift varies with size of the real basis for the pointwise method and compares the results with those obtained by evaluating all the integrals exactly, the variational method.
• Also, many problems are not conveniently described by a set of functions for which a Gaussian quadrature is available.

4. Combining the pointwise description with local basis functions

• Perhaps the most obvious place to put the points is at the centres of the Gaussians.
• The use of such a simple quadrature scheme dictates that the kinetic energy matrix will no longer be evaluated exactly over the basis set.
• An alternative method, favoured by Light [ 121, is to calculate the kinetic energy matrix exactly in the basis set representation and then transform to the pointwise scheme using the matrix R. This Table 2 Fractional errors in tan 6 for He-H, elastic scattering.
• This is as expected on the grounds of increasing the accuracy of the trapezoidal rule quadrature, From table 2 it is seen that the pointwise method requires a basis set about 20% greater than the variational method to attain results with a 1% error.

5. Multidimensional problems

• The extension of the pointwise representation to systems with more than one degree of freedom is straightforward and so the authors simply summarise the basic formulae.
• Such structure allows the kinetic energy matrix to be constructed from smaller matrices which have been evaluated in a space of lower dimension.
• These properties are best illustrated by a simple example.
• The parameters used are the same as for the 1D case of table 2 and five H2 vibrational functions are used.
• With these values, all matrix elements of the type ( uo&,IH-EI~O#n,) could be neglected without changing the third figure of the inelastic transition probability.

6. Conclusions

• A method has been proposed for calculating the S-matrix via a pointwise representation of the Kohn variational expression.
• The method was implemented and found to be of comparable accuracy to the full variational form for degrees of freedom which are well described by functions for which a Gaussian quadrature scheme exists.
• The application to more general situations has also been considered.
• It thus appears that the method should work well for any inelastic scattering process.
• The pointwise scheme should be readily applicable to such systems and, if found to be of good accuracy, will be a powerful technique for calculating reaction rates.

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In this paper, a pointwise representation of the Kohn variational expression for the L2 part of the basis set is proposed.