Exchange Energy for Two-Active-Electron Diatomic Systems Within the Surface Integral Method

  • T.C. Scott
  • M. Aubert-Frécon
  • D. Andrae
  • J. Grotendorst
  • J.D. Morgan III
  • M.L. Glasser


We have analyzed and reduced a general (quantum-mechanical) expression for the atom-atom exchange energy formulated as a five-dimensional surface integral, which arises in studying the charge exchange processes in diatomic molecules. It is shown that this five-dimensional surface integral can be decoupled into a three-dimensional integral and a two-dimensional angular integral which can be solved analytically using a special decomposition. Exact solutions of the two-dimensional angular integrals are presented and generalized. Algebraic aspects, invariance properties and exact solutions of integrals involving Legendre and Chebyshev polynomials are also discussed.


Symbolic integration Numerical integration Molecular physics Special functions and sums Asymptotic series 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions (9th printing). New York: Dover, 1972Google Scholar
  2. 2.
    Aubert-Frécon, M., Scott, T.C., Hadinger, G., Andrae, D., Grotendorst, J., Morgan III, J.D.: Asymptotically Exact Calculation of the Exchange Energies of One-Active-Electron Diatomic Ions with the Surface Integral Method. In preparation, 2004Google Scholar
  3. 3.
    Forrey, R.C.: Computing the Hypergeometric Function. J. Comput. Phys. 137, 79–100 (1997)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Geddes, K.O., Glasser, M.L., Moore, R.A., Scott, T.C.: Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions. AAECC 1, 149–165 (1990)MathSciNetMATHGoogle Scholar
  5. 5.
    Gomez, C., Scott, T.: Maple Programs for Generating Efficient FORTRAN Code for Serial and Vectorised Machines. Comput. Phys. Commun. 115, 548–562 (1998)CrossRefGoogle Scholar
  6. 6.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products (5th ed.). Boston: Academic Press, 1994Google Scholar
  7. 7.
    Grotendorst, J.: A Maple Package for Transforming Series, Sequences and Functions. Comput. Phys. Commun. 67, 325–342 (1991)CrossRefMATHGoogle Scholar
  8. 8.
    Hadinger, G., Hadinger, G., Bouty, O., Aubert-Frécon, M.: Asymptotic calculation of the exchange interaction between two long-range interacting atoms with open valence shells of any type. Phys. Rev. A 50, 1927–1930 (1994)CrossRefGoogle Scholar
  9. 9.
    (a) Holstein, T.: Mobilities of positive ions in their parent gases. J. Phys. Chem. 56, 832–836 (1952); (b) Holstein, T.: Westinghouse Research Report 60-94698-3-R9 (1955, unpublished); (c) Herring, C.: Critique of the Heitler-London Method of Calculating Spin Couplings at Large Distances. Rev. Mod. Phys. 34, 631–645 (1962); (d) Bardsley, J.N., Holstein, T., Junker, B.R., Sinha, S.: Calculations of ion-atom interactions relating to resonant charge-transfer collisions. Phys. Rev. A 11, 1911–1920 (1975); (e) Herring, C., Flicker, M.: Asymptotic Exchange Coupling of Two Hydrogen Atoms. Phys. Rev. [Sect.] A 134, 362–366 (1964)Google Scholar
  10. 10.
    Kołos, W., Wolniewicz, L.: Variational calculation of the long-range interaction between two ground-state hydrogen atoms. Chem. Phys. Lett. 24, 457–460 (1974)CrossRefGoogle Scholar
  11. 11.
    Levin, D.: Development of Nonlinear Transformations for Improving Convergence of Sequences. Int. J. Comput. Math. B 3, 371–388 (1973)MATHGoogle Scholar
  12. 12.
    Preuß, H.: Zur Behandlung der Zweizentren-Wechselwirkungsintegrale. Z. Naturforsch. A 10, 211–215 (1955)Google Scholar
  13. 13.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, Vol. 1 (4th printing). New York: Gordon & Breach, 1998Google Scholar
  14. 14.
    Sack, R.A.: Generalization of Laplace’s Expansion to Arbitrary Powers and Functions of the Distance between Two Points. J. Math. Phys. 5, 245–251 (1964)MATHGoogle Scholar
  15. 15.
    Scott, T.C., Aubert-Frécon, M., Andrae, D.: Asymptotics of Quantum Mechanical Atom-Ion Systems. AAECC 13, 233–255 (2002)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Scott, T.C., Dalgarno, A., Morgan III, J.D.: Exchange Energy of H2+ Calculated from Polarization Perturbation Theory and the Holstein-Herring Method. Phys. Rev. Lett. 67, 1419–1422 (1991)Google Scholar
  17. 17.
    Scott, T.C., Babb, J.F., Dalgarno, A., Morgan III, J.D.: The Calculation of Exchange Forces: General Results and Specific Models. J. Chem. Phys. 99, 2841–2854 (1993)CrossRefGoogle Scholar
  18. 18.
    Scott, T.C., Babb, J.F., Dalgarno, A., Morgan III, J.D.: Resolution of a Paradox in the Calculation of Exchange Forces for H2+. Chem. Phys. Lett. 203, 175–183 (1993)CrossRefGoogle Scholar
  19. 19.
    Scott, T.C., Aubert-Frécon, M., Hadinger, G., Andrae, D., Morgan III, J.D.: Exchange Energy for Two-Active-Electron Diatomic Systems in the Surface Integral Method. In preparation, 2004Google Scholar
  20. 20.
    Smirnov, B.M., Chibisov, M.I.: Electron exchange and changes in the hyperfine state of colliding alkaline metal atoms. Sov. Phys. JETP 21, 624–628 (1965)Google Scholar
  21. 21.
    Spanier, J., Oldham, K. B.: An Atlas of Functions. Berlin: Springer, 1987Google Scholar
  22. 22.
    Umanskii, Ya.S., Hadinger, G., Aubert-Frécon, M.: Nonadiabatic formulation of the slow-atomic-collision problem in the finite electronic basis. Phys. Rev. A 49, 2651–2666 (1994)CrossRefGoogle Scholar
  23. 23.
    Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. Singapore: World Scientific, 1988Google Scholar
  24. 24.
    Weniger, E.J.: Nonlinear Sequence Transformations for the Acceleration of Convergence and the Summation of Divergent Series. Comput. Phys. Rep. 10, 189–371 (1989)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • T.C. Scott
    • 1
    • 3
  • M. Aubert-Frécon
    • 2
  • D. Andrae
    • 3
  • J. Grotendorst
    • 1
  • J.D. Morgan III
    • 4
  • M.L. Glasser
    • 5
  1. 1.Zentralinstitut für Angewandte Mathematik (ZAM)Forschungszentrum Jülich GmbHJülichGermany
  2. 2.LASIMCNRS et UniversitéBâtiment Alfred KastlerFrance
  3. 3.Theoretische ChemieFakultät für Chemie, Universität BielefeldBielefeldGermany
  4. 4.Department of Physics and AstronomyUniversity of DelawareNewarkUSA
  5. 5.Department of PhysicsClarkson UniversityPotsdamUSA

Personalised recommendations