Theory and Computation of Nonstationary States of Polyelectronic Atoms and Molecules

  • Cleanthes A. Nicolaides

Abstract

I present a theory of polyelectronic atomic and molecular nonstationary states decaying into free particle continua, which shows how to define and compute efficiently correlated wavefunctions, energies, energy shifts and transition rates for phenomena such as autoionization, multiphoton ionization, static and dynamic polarization, predissociation etc. The problem is formulated in a unified manner as a complex eigenvalue Schrodinger equation (CESE) which is derived rigorously for each state of interest starting from Fano’s basic equation of discrete -continuous spectra mixing. The form of the resulting resonance wave-function is ψ=aψo+bXas, where ψo is the maximum square-integrable projec-tion of ψ, excluding the open channels. Knowledge of the exact asymptotic form of the resonance wavefunctions thus derived, allows the imposition of suitable perturbations of the asymptotic boundary conditions in the form of coordinate transformations, in order to render the resonance wavefunction normalizable. Thus, for short-range or Coulomb potentials, the corresponding coordinate transformation is r → ρ = re, first introduced for short-range potentials by Dykhne and Chaplik in 1961. For the LoSurdo-Stark potential, the herein derived resonance asymptotic form and subsequent analysis justifies previous choices of the coordinate rotation and translation transformations as well as of their combination r → ρ = re − zo/F, where zo is the complex eigenvalue and F is the field strength. Now, ψo and Xas are represented by different function spaces which are optimized separately, the first on the real axis once, yielding a real energy, Eo, and the second partly on the real axis for its bound part and partly in the complex energy plane. The second part of the computation depends only on the structure of the continuous spectrum and yields the energy shift, Δ, and the width, Γ. For Coulomb autoionization where the interaction operator is nonseparable, the state-specific calculation of ψo is based on the existence of a localized, self-consistent multiconfigurational zeroth order function satisfying the virial theorem and the physically correct orbital nodal structure. The remaining localized electron correlation is obtained variationally. Both the zeroth order and the localized correlation part exclude the open channels by construction and by satisfying orthogonality constraints to electronic structure dependent core orbitals. It is shown how this scheme can be used for the construction of diabatic or quasidiabatic states. Xas incorporates the interchannel couplings and a simple expression yields the partial widths to all orders. When the nonstationary state is caused by an external ac-field, the present theory is adapted to solving the ensuing many-electron, many-photon (MEMP) problem as a time-independent CESE where the computed multiphoton ionization rates and frequency-dependent hyperpolarizabilities constitute the Floquet averages over the field cycle. The following, field-free, examples are discussed: The inner-hole Auger state Ne+ 1s2s22p6 2S; the triply excited He 2s2p2 2D and 4P resonances; the \( He_2^ + {\mkern 1mu} {}^2\sum _g^ + \) diabatic spectrum. The formulation of the MEMP theory and its applications are relegated to the references.

Keywords

Continuous Spectrum Zeroth Order Multiphoton Ionization Autoionizing State Asymptotic Boundary Condition 
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References

  1. 1.
    W. Heitler, “The Quantum Theory of Radiation” 3rd Ed., Oxford 1954.Google Scholar
  2. 2.
    M. L. Goldberger and K. M. Watson, “Collision Theory”, J. Wiley N. Y. (1964).Google Scholar
  3. 3.
    C. A. Nicolaides and D. R. Beck, Int. J. Qu. Chem. 4 475 (1978).Google Scholar
  4. 4.
    R. G. Newton, “Scattering Theory of Waves and Particles” 2nd Ed., Springer-Verlag, N. Y. (1982).Google Scholar
  5. 5.
    C. Mahaux and H. A. Weidenmüller, “Shell Model Approach to Nuclear Reactions”, North Holland, Amsterdam (1969).Google Scholar
  6. 6.
    G. Gamow, Z. f. Physik, 204 (1928).Google Scholar
  7. 7.
    A. F. J. Siegert, Phys. Rev., 750 (1939).Google Scholar
  8. 8.
    U. Fano, Phys. Rev. 124 1866 (1961).CrossRefGoogle Scholar
  9. 9.
    U. Fano and F. Prats, J. Natl. Acad. Sci. (India) A33 533 (1963).Google Scholar
  10. 10.
    The Fano-Prats work (ref. 9) is limited to the case of only open channels. The general theory with the inclusion of closed channels, the derivation of multichannel quantum defect formulae independently of the properties of the Coulomb function and the implementation in terms of state-specific numerical and analytic functions for the many-electron computation of photoabsorption cross-sections to perturbed Rydberg states close to threshold and to doubly excited Rydberg series of resonances, was presented recently by Y. Komninos and C. A. Nicolaides, Z. Phys. B4 301 (1987 )Google Scholar
  11. Phys. Rev. A34 1995 (1986).Google Scholar
  12. 11.
    C. A. Nicolaides, Y. Komninos and Th. Mercouris, Int. J. Qu. Chem. S15, 355 (1981).Google Scholar
  13. 12.
    For a short historical account of the LoSurdo-Stark effect, see the article by H. J. Silverstone in “Atoms in Strong Fields”, eds. C. A. Nicolaides, C. W. Clark and M. H. Nayfeh, Plenum (1990), p. 295.Google Scholar
  14. 13.
    In the theory of resonances by Feshbach (ref. 14) and by Fano (ref. 8), the projection operators P and Q and the corresponding effective Hamiltonians and projected interaction operators, as well as the prediagonalized zeroth order Hamiltonians, are defined only formally. These theories are fundamental in explaining the phenomenology of resonances. However, for real atomic and molecular nonstationary states, it is equally important to have theories and methods which provide not only the framework for the definition of their properties but also for their systematic, electronic structure-dependent computation. For example, in order to demonstrate the dissolution of a discrete level into the continuous spectrum, the doubly excited states of the He atom have been used as a prototype example in the following way. Ho is taken to be the interactionless hydrogenic operator and V=1/r12. However, such a model is conceptually unsatisfactory since the interelectronic interactions are, in fact, never turned-off! Furthermore, it is obviously computationally naive and cannot lead to accurate results with a reasonable amount of effort. This difficulty is accentuated for polyelectronic atoms. The theory of this article aims at justifying those essential formal results which allow the understanding and practical computation of nonstationary states many-electron atoms and molecules.Google Scholar
  15. 14.
    H. Feshbach, Ann. Phys. (N. Y. ) 357 (1958 ) íQ 287 (1962).Google Scholar
  16. 15.
    P. A. M. Dirac, “The Principles of Quantum Mechanics”, Oxford Univ. Pr. , 4th Ed. , (1957), chapter 8.Google Scholar
  17. 16.
    E. C. Kemble, “The Fundamental Principles of Quantum Mechanics” Dover, (1958).Google Scholar
  18. 17.
    Ya. B. Zeldovich, Sov. Phys. (JETP) 12 542 (1961).Google Scholar
  19. 18.
    A. M. Dykhne and A. V. Chaplik, Sov. Phys. (JETP) 1002 (1961).Google Scholar
  20. 19.
    The Dykhne-Chaplik paper (ref. 18) was apparently first quoted in the literature of atomic and molecular physics in ref. 11, as soon as it was discovered. In the meantime, their transformation (eq. 26) had been used and had become known in the 70’s, both as a regularization technique (ref. 20,26,3) and as a formal means of studying the spectral properties of the atomic Coulomb Hamiltonian (ref. 21). The results of ref. 21 led to the so-called complex coordinate rotation (CCR) method (refs 2224 ) whereby H(re’0) is diagonalized repeatedly in a large real or complex square-integrable basis set and the resonances are identified by the regions of stability observed as a function of the rotation angle 6 or of the size of the basis sets. Upon rereading their one-page article for the purpose of writing this review, I saw that Dykhne and Chaplik also proposed the possibility of starting the complex integration beyond a point on the real axis to circumvent problems of nonanalyticity. In recent years this idea has been known as “exterior complex scaling” (refs. 25–27 ).Google Scholar
  21. 20.
    J. N. Bardsley and B. R. Junker, J. Phys.. L178 (1972).Google Scholar
  22. 21.
    J. Aguilar and J. M. Combes, Commun. Math. Phys. 22. 269 (1972)CrossRefGoogle Scholar
  23. E. Balslev and J. M. Combes, Commun. Math. Phys. 22 280 (1972)CrossRefGoogle Scholar
  24. B. Simon, Ann. Math.. j 247 (1973).Google Scholar
  25. 22.
    G. Doolen, J. Nuttall and R. W. Stagat, Phys. Rev. A10 1612 (1974 )Google Scholar
  26. G. Doolen, J. Phys. . 525 (1975 )Google Scholar
  27. R. A. Bain, J. N. Bardsley, B. R. JunkerGoogle Scholar
  28. and C. V. Sukumar, J. Phys. B7, 2189 (1974 )Google Scholar
  29. B. R. Junker, Int. J. Qu. Chem. 14 371 (1978 )Google Scholar
  30. N. Moiseyev, P. R. Certain andGoogle Scholar
  31. F. Weinhold, Phys. Rev. A24 1254 (1981).CrossRefGoogle Scholar
  32. 23.
    B. R. Junker, Adv. At. Mol. Phys. 207 (1982 )Google Scholar
  33. Y. K. Ho, Phys. Reports,.. 1 (1983).Google Scholar
  34. 24.
    W. P. Reinhardt, Ann. Rev. Phys. Chem. 3. 223 (1982 )Google Scholar
  35. C. Cerjan, R. Hedges, C. Holt, W. P. Reinhardt, K. Scheibner and J. J. Wendoloski, Int. J. Qu. Chem. 4 393 (1978).CrossRefGoogle Scholar
  36. 25.
    B. Simon, Phys. Lett. A71 211 (1979).CrossRefGoogle Scholar
  37. 26.
    C. A. Nicolaides and D. R. Beck, Phys. Lett. A65 11 (1978).CrossRefGoogle Scholar
  38. 27.
    C. A. Nicolaides, H. J. Gotsis, M. Chrysos and Y. Komninos, Chem. Phys. Lett. 168 570 (1990) and refs. therein.Google Scholar
  39. 28.
    C. A. Nicolaides and S. Themelis, unpublished.Google Scholar
  40. 29.
    C. A. Nicolaides, Phys. Rev. 6 2078 (1972).Google Scholar
  41. 30.
    C. A. Nicolaides, Th. Mercouris and Y. Komninos, Int. J. Qu. Chem. 2. 1017 (1984 )Google Scholar
  42. C. A. Nicolaides and Th. Mercouris, Phys. Rev. A32 3247 (1985).CrossRefGoogle Scholar
  43. 31.
    Th. Mercouris and C. A. Nicolaides, J. Phys. B17 4127 (1984).Google Scholar
  44. 32.
    C. A. Nicolaides and Th. Mercouris, Phys. Rev. A36 390 (1987).CrossRefGoogle Scholar
  45. Th. Mercouris and C. A. Nicolaides, Z. Phys. Q. 1 (1987).Google Scholar
  46. 34.
    M. Chrysos, Y. Komninos, Th. Mercouris and C. A. Nicolaides, Phys. Rev. A42 2634 (1990).CrossRefGoogle Scholar
  47. 35.
    Th. Mercouris and C. A. Nicolaides, J. Phys. B21 L285 (1988).Google Scholar
  48. 36.
    C. A. Nicolaides and Th. Mercouris, Chem. Phys. Lett. 159 45 (1989).CrossRefGoogle Scholar
  49. 37.
    Th. Mercouris and C. A. Nicolaides, J. Phys. B23 2037 (1990).Google Scholar
  50. 38.
    Th. Mercouris and C. A. Nicolaides, J. Phys. B24 L 57 and L165 (1991).Google Scholar
  51. 39.
    C. A. Nicolaides, Th. Mercouris and G. Aspromallis, J. Opt. Soc. Am. B. Z. 494 (1990).Google Scholar
  52. 40.
    C. A. Nicolaides, Th. Mercouris and N. A. Piangos, J. Phys. B23 L669 (1990).Google Scholar
  53. 41.
    I. D. Petsalakis, Th. Mercouris, G. Theodorakopoulos and C. A. Nicolaides, J. Phys. B23 L89 (1990 )Google Scholar
  54. I. D. Petsalakis, Th. Mercouris, G. Theodorakopoulos and C. A. Nicolaides, J. Chem. Phys. a3. . 6642 (1990 )Google Scholar
  55. I. D. Petsalakis, Th. Mercouris, G. Theodorakopoulos and C. A. Nicolaides, Chem. Phys. Lett. (1991).Google Scholar
  56. 42.
    A systematic perturbative treatment of the rotated atomic Coulomb Hamiltonian, H(re’e) is also possible, based on the form of eq. 9. 3 of ref. 3. Formally, the infinite nonHermitian Hamiltonian matrix is written as H(8)=H(0)+K(8). K(8)=(e-2ie-1)T+(e-’e -1)V, where T and V are the kinetic and potential energy matrices respectively. This formulation is an expression of the idea that the calculation of the complex eigen-value, zo, should constitute a continuation from E0, the expectation value of H(0) on the real axis, and allow the possibly interesting study of the autoionization shift and width, of say a doubly excited state, in the complex plane via Cl-based small-or large-order perturbation theory (J. N. Silverman and C. A. Nicolaides, Chem. Phys. Lett. 153 61 (1988 ) in “Atoms in Strong Fields” eds. C. A. Nicolaides, C. W. Clark and M. H. Nayfeh, Plenum (1990), p. 309.Google Scholar
  57. 43.
    Even this picture breaks down in principle, when the strength of the external field increases to the point that the Wo cannot represent only the unperturbed, free atomic or molecular state.Google Scholar
  58. 44.
    Although the thrust of the work of ref. 29 was on N-electron autoioinizing states of arbitrary electronic structure, its concepts and methods are applicable to the subject of the a priori construction of correlated diabatic molecular states. This was pointed out in footnote 73 of ref. 29 but no such computations were possible at that time. Later on, an application of the idea of starting with the properly projected diabatic solution in the dissociated region and moving into the interaction region while exluding unwanted interacting configurations by maximizing the wavefunction of each state-specific solution at each geometry to the previous one, was opplied to She analysis of the potential energy surfaces of HeH2, NeH2 and ArH2 (ref. 45).Google Scholar
  59. C. A. Nicolaides and A. Zdetsis, J. Chem. Phys. Q. 1900 (1984).Google Scholar
  60. 46.
    Y. Komninos, N. Makri and C. A. Nicolaides, Z. Phys. D2 105 (1986).Google Scholar
  61. 47.
    Y. Komninos and C. A. Nicolaides, J. Phys. B19 1701 (1986).Google Scholar
  62. 48.
    Y. Komninos, G. Aspromallis and C. A. Nicolaides, Phys. Rev. A27 1865 (1983).CrossRefGoogle Scholar
  63. 49.
    C. A. Nicolaides in “Advanced Theories and Computational Approaches to the Electronic Structure of Molecules” ed. C. E. Dykstra, Reidel (1984), p. 161.Google Scholar
  64. 50.
    C. A. Nicolaides, in “Quantum Chemistry - Basic Aspects, Actual Trends”, ed., R. Carbo, Elservier (1989).Google Scholar
  65. 51.
    It is obvious from the structure of the theory that interacting scattering resonances as well as intermediate dressed bound states can be included in the formalism and the overall computation using the same methods for obtaining their correlated wavefunctions (see refs. 37, 46–48, 52).Google Scholar
  66. 52.
    C. A. Nicolaides and Th. Mercouris, in “Atoms in Strong Fields”, eds C. A. Nicolaides, C. W. Clark and M. H. Nayfeh, Plenum (1990), p. 353.Google Scholar
  67. 53.
    D. R. Beck and C. A. Nicolaides, in “Excited States in Quantum Chemistry” eds. C. A. Nicolaides and D. R. Beck, Reidel (1978), p. 105.Google Scholar
  68. 54.
    K. T. Chung and B. F. Davis, in “Autoionization”, ed. A. Temkin, Plenum, N. Y. (1985) p. 73; K. T. Chung, Phys. Rev. A22 1341 (1980).Google Scholar
  69. 55.
    M. Bylicki, Phys. Rev. A, in press (1991).Google Scholar
  70. 56.
    C. A. Nicolaides, Y. Komninos and D. R. Beck, Phys. Rev. A24 1103 (1981).Google Scholar
  71. 57.
    C. E. Kuyatt, J. A. Simson and S. R. Mielczarek, Phys. Rev. 138 A385 (1965 )Google Scholar
  72. P. J. Hicks, C. Cvejanovic, J. Comer, F. H. Read and J. M. Sharp, Vacuum 24 573 (1974).Google Scholar
  73. 58.
    G. J. Schulz, Rev. Mod. Phys. 45 378 (1973).CrossRefGoogle Scholar
  74. 59.
    U. Fano and J. W. Cooper, Phys. Rev. 138 A400 (1965).CrossRefGoogle Scholar
  75. 60.
    K. Smith, D. E. Golden, S. Ormonde, B. W. Torres and A. R. Davis, Phys. Rev. ALI3001 (1973).Google Scholar
  76. 61.
    W. Lichten, Phys. Rev. 131 229 (1963).CrossRefGoogle Scholar
  77. 62.
    F. T. Smith, Phys. Rev. 179 111 (1969).CrossRefGoogle Scholar
  78. 63.
    T. F. O’Malley, Adv. At. Mol. Phys. 7 223 (1971).Google Scholar
  79. 64.
    C. A. Mead and D. G. Truhlar, J. Chem. Phys.. ZZ6090 (1982).Google Scholar
  80. 65.
    T. F. O. Malley, Phys. Rev. 162 98 (1967).CrossRefGoogle Scholar
  81. 66.
    T. F. O. Malley, J. Chem. Phys. 322 (1969).Google Scholar
  82. 67.
    For a diatomic molecular electronic spectrum, the analogy with the atomic spectra as a function of Z, treated as a continuous parameter, is enlightening. Consider the mixing of a valence configuration (V) with a Rydberg (R) series and the scattering (S) states of the same channel. The V-R-S mixing is Z-dependent. For large Z, the V state is found below the R states which acquire more hydrogenic character and are raised in energy. Call the large Z region, the “dissociation” region. Here, the definition and computation of the To for a V state is straight forward (For example, the 1 s22p2 1 S valence excited state is represented mainly by a(1s22p2)+b(1s22s2)). As Z is decreased, the V state may start “crossing” the R states which start coming down. At the neutral or negative ion end, the V state may lie in the continuous spectrum, mixing with the scattering states of the same symmetry and configuration as those of the R states below the ionization threshold. This is indeed the case with the 1 s22p21 S V state. For Z=4 (Be) it lies in the continuous spectrum. For Z=5 (Bk), it lies below the 1 s22sns 1S series and above the ground state 1 s22s21 S. For 400Z005, it “crosses” the Rydberg states. If its W, which is defined unambiguously for Z=5, is optimized for each noninteger value of Z between 5 and 4 with its state-specific numerical zeroth order and analytic correlation functions excluding by construction or orthogonality the R-S iS channel, an “atomic diabatic state” is calculated.Google Scholar
  83. 68.
    N. Bacalis, Y. Komninos and C. A. Nicolaides, unpublished.Google Scholar
  84. 69.
    E. A. McCullough, J. Chem. Phys. fa3991 (1975).Google Scholar
  85. 70.
    C. A. Nicolaides, Chem. Phys. Lett. 161 547 (1989 )Google Scholar
  86. 71.
    A. Metropoulos, C. A. Nicolaides and R. J. Buenker, Chem. Phys. 114 1 (1987).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Cleanthes A. Nicolaides
    • 1
  1. 1.National Hellenic Research FoundationTheoretical and Physical Chemistry InstituteAthensGreece

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