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Journal ArticleDOI

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

19 Aug 1988-Chemical Physics Letters (North-Holland)-Vol. 149, Iss: 3, pp 257-264
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 (2 min read)

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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Citations
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Journal ArticleDOI
TL;DR: In this paper, a new radial basis set is devised which is both (a) ideally suited to the log derivative boundary value problem, and (b) directly amenable to a discrete representation based on Gauss-Lobatto quadrature.

178 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a theoretical approach for calculating the rate constant of a chemical reaction that avoids the need to solve the complete state-to-state reactive scattering problem, with no explicit information about reactant and product states being required.
Abstract: Transition-state theory (TST) provides a simple and useful way to understand and estimate the rates of chemical reactions. The fundamental assumption of transition-state theory, however, is based inherently on classical mechanics, so the theory must be quantized if it is to provide a quantitative description of chemical reaction rates. Unlike classical mechanics, though, there seems to be no way to construct a rigorous quantum mechanical theory that contains as its only approximation the transition-state assumption of [open quotes]direct dynamics[close quotes]. Pechukas has discussed this quite clearly: as soon as one tries to rid a quantum mechanical version of transition-state theory of all approximations (e.g., separability of a one-dimensional reaction coordinate) beyond the basic transition-state assumption itself, one is faced with having to solve the full (multidimensional) quantum reaction dynamics problem. But a correct treatment of the full quantum dynamics must yield the exact rate constant and is no longer a transition-state [open quotes]theory[close quotes]. Though there is no rigorous quantum prescription for determining the rate constant of a chemical reaction that avoids, the necessity of solving the Schroedinger equation, there is nevertheless a rigorous theoretical approach that avoids having to solve the complete state-to-state reactive scattering problem; one does notmore » avoid solving the Schroedinger equation, but needs to solve it only locally, in the vicinity of the transition state, with no explicit information about reactant and product states being required. After reviewing some of the notions alluded to above, the purpose of the Account is to describe this [open quotes]direct[close quotes] theoretical approach for calculating chemical reaction rates, the logical conclusion in the quest for a [open quotes]rigorous[close quotes] quantum mechanical version of transition-state theory. 20 refs., 3 figs.« less

175 citations

Journal ArticleDOI
TL;DR: In this article, a set of anharmonic vibrational constants was derived, unifying the SEP data reported here with previous infrared and overtone data, which is expected to be able to predict the position of normal mode states below 19 000 cm−1 with an accuracy within 3 cm −1.
Abstract: Stimulated-emission pumping (SEP) spectra of HCN have been measured by using a pulsed, tunable argon fluoride laser with a frequency-doubled, pulsed dye laser. Sixty-seven vibrational states of the ground electronic state between 8 900 and 18 900 cm−1 have been observed. Eighty percent of the states can be described within a traditional normal mode context. A full set of anharmonic vibrational constants was derived, unifying the SEP data reported here with previous infrared and overtone data. This set of molecular constants is expected to be able to predict the position of normal mode states below 19 000 cm−1 with an accuracy within 3 cm−1. Twenty percent of the states could not be assigned to unperturbed normal mode states, and a systematic analysis was performed in an attempt to find a simple explanation for them based on possible perturbations. Except for the lowest energies, no simple explanation was found, suggesting that delocalized isomerizing vibrational states are playing a role in the observed vibrational structure at higher energy. Direct comparison with assigned normal mode states derived from quantum-mechanical vibrational-structure calculations on the only available three-dimensional potential energy surface were made possible by these experiments. The deviation between experiment and theory as a function of the number of bending quanta, the vibrational motion that couples strongly to the isomerization reaction coordinate, makes clear that the isomerization barrier height is too low on this surface. The present state of experimental characterization of the HCN system should be good enough to permit a high-quality potential energy surface to be derived for highly vibrationally excited HCN.

69 citations

Journal ArticleDOI
TL;DR: In this article, the log derivative version of the Kohn variational principle is reviewed in the context of a general inelastic molecular collision, and the possibility of solving the resulting linear equations iteratively for a single initial state column of the scattering matrix, S, is made.
Abstract: The log derivative version of the Kohn variational principle is reviewed in the context of a general inelastic molecular collision. Particular emphasis is placed on the possibility of solving the resulting linear equations iteratively for a single initial state column of the scattering matrix, S, and several important practical observations are made in this regard. In particular, it is argued that the discrete variable representation proposed by Light and coworkers leads to an extremely sparse coefficient matrix, and so has distinct advantages over the more obvious variational basis approach. The resulting methodology is applied to the diffractive scattering of a beam of helium atoms from the (001) face of LiF. Here test calculations with up to 2601 coupled channels and 216 translational grid points clearly demonstrate the practical potential of the iterative technique. The implications of these tests for more general scattering problems are also briefly discussed.

25 citations

Journal ArticleDOI
TL;DR: In this article, a collocation approach based on the S-matrix version of the Kohn variational principle with a different linear expansion used for the two wave functions is presented, where one is a linear combination of basis functions and the other is a pointwise representation with proper asymptotic conditions imposed.
Abstract: A collocation approach to quantum scattering is presented. The method is based on the S‐matrix version of the Kohn variational principle with a different linear expansion used for the two wave functions—one is a linear combination of basis functions and the other is a pointwise representation with proper asymptotic conditions imposed. The resulting equations are similar in structure to the usual version of the Kohn variational principle, however, in the present approach there are no integrals between the square integrable (L2) basis functions. In addition, the method does not require the knowledge of quadrature weights associated with the collocation points as was the case in a previous pointwise method for quantum scattering. This property means that the method is readily applicable to reactive scattering problems which use different sets of coordinates for reactants and products. Appliction to a simple inelastic test problem (collinear He–H2 vibrationally inelastic scattering) shows the accuracy of the ...

20 citations

References
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Journal ArticleDOI
TL;DR: In this article, an exact formalism in which the scattering problem may be described by sets of coupled equations labeled either by basis functions or quadrature points is presented, and the simply evaluated unitary transformation which connects them results in an efficient procedure for performing quantum scattering calculations.

416 citations

Journal ArticleDOI
TL;DR: In this article, a discrete variable representation (DVR) for the angular, bend coordinate is combined with the distributed (real) Gaussian basis for the expansion of other, radial coordinates.
Abstract: A novel, efficient, and accurate quantum method for the calculation of highly excited vibrational levels of triatomic molecules is presented. The method is particularly well suited for applications to ‘‘floppy’’ molecules, having large amplitude motion, on potential surfaces which may have more than one local minimum. The discrete variable representation (DVR) for the angular, bend coordinate is combined with the distributed (real) Gaussian basis (DGB) for the expansion of other, radial coordinates. The DGB is tailored to the potential, covering only those regions where V(r)

380 citations

Journal ArticleDOI
TL;DR: The distributed Gaussian bases are defined and used to calculate eigenvalues for one and multidimensional potentials and are shown to be accurate, flexible, and efficient.
Abstract: Distributed Gaussian bases (DGB) are defined and used to calculate eigenvalues for one and multidimensional potentials. Comparisons are made with calculations using other bases. The DGB is shown to be accurate, flexible, and efficient. In addition, the localized nature of the basis requires only very low order numerical quadrature for the evaluation of potential matrix elements.

375 citations

Journal ArticleDOI
TL;DR: In this article, the first accurate quantum calculation of the delocalized, large amplitude motion vibrational (J=0) levels of HCN/HNC, lying above the isomerization barrier, is presented.
Abstract: Results of the first accurate quantum calculation of the delocalized, large amplitude motion vibrational (J=0) levels of HCN/HNC, lying above the isomerization barrier, are presented. The recently developed DVR‐DGB quantum method [Z. Bacic and J. C. Light, J. Chem. Phys. 85, 4594 (1986)] is employed in this work. A model, empirical surface by Murrell et al. is used. All modes are included; the energy level calculation does not involve any approximations. Over a hundred vibrational levels are calculated accurately for this model surface. A number of them lie above the isomerization barrier; some are extensively delocalized over both HCN and HNC minima. Analysis shows that for HCN/HNC the threshold for significant delocalization is determined by the height of the vibrationally adiabatic bending barrier. In addition, the nearest neighbor level spacing distribution is obtained and compared to that of LiCN/LiNC. Various computational aspects of the DVR‐DGB approach, which is applicable to any triatomic molecul...

273 citations

Journal ArticleDOI
TL;DR: In this paper, a method for solving self-consistent electronic structure equations which offers orders of magnitude reductions in computation time and storage space with no obvious loss of accuracy is described and applied to the restricted Hartree-Fock equations for the neon atom.

246 citations

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Q1. What contributions have the authors mentioned in the paper "A pointwise representation of the s-matrix kohn variational principle for quantum scattering" ?

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