# Theory and Computation of Nonstationary States of Polyelectronic Atoms and Molecules

## 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=aψ_{o}+bX_{as}, where ψ_{o} 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^{iθ}, 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^{iθ} − z_{o}/F, where z_{o} is the complex eigenvalue and F is the field strength. Now, ψ_{o} and X_{as} are represented by different function spaces which are optimized separately, the first on the real axis once, yielding a real energy, E_{o}, 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 ψ_{o} 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. X_{as} 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^{+} 1s2s^{2}2p^{6 2}S; the triply excited He^{−} 2s2p^{2 2}D and ^{4}P 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## Preview

Unable to display preview. Download preview PDF.

## References

- 1.W. Heitler, “The Quantum Theory of Radiation” 3rd Ed., Oxford 1954.Google Scholar
- 2.M. L. Goldberger and K. M. Watson, “Collision Theory”, J. Wiley N. Y. (1964).Google Scholar
- 3.C. A. Nicolaides and D. R. Beck, Int. J. Qu. Chem. 4 475 (1978).Google Scholar
- 4.R. G. Newton, “Scattering Theory of Waves and Particles” 2nd Ed., Springer-Verlag, N. Y. (1982).Google Scholar
- 5.C. Mahaux and H. A. Weidenmüller, “Shell Model Approach to Nuclear Reactions”, North Holland, Amsterdam (1969).Google Scholar
- 6.G. Gamow, Z. f. Physik, 204 (1928).Google Scholar
- 7.A. F. J. Siegert, Phys. Rev., 750 (1939).Google Scholar
- 8.U. Fano, Phys. Rev. 124 1866 (1961).CrossRefGoogle Scholar
- 9.U. Fano and F. Prats, J. Natl. Acad. Sci. (India) A33 533 (1963).Google Scholar
- 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
- Phys. Rev. A34 1995 (1986).Google Scholar
- 11.C. A. Nicolaides, Y. Komninos and Th. Mercouris, Int. J. Qu. Chem. S15, 355 (1981).Google Scholar
- 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
- 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. H
_{o}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 - 14.H. Feshbach, Ann. Phys. (N. Y. ) 357 (1958 ) íQ 287 (1962).Google Scholar
- 15.P. A. M. Dirac, “The Principles of Quantum Mechanics”, Oxford Univ. Pr. , 4th Ed. , (1957), chapter 8.Google Scholar
- 16.E. C. Kemble, “The Fundamental Principles of Quantum Mechanics” Dover, (1958).Google Scholar
- 17.Ya. B. Zeldovich, Sov. Phys. (JETP) 12 542 (1961).Google Scholar
- 18.A. M. Dykhne and A. V. Chaplik, Sov. Phys. (JETP) 1002 (1961).Google Scholar
- 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
- 20.J. N. Bardsley and B. R. Junker, J. Phys.. L178 (1972).Google Scholar
- 21.J. Aguilar and J. M. Combes, Commun. Math. Phys. 22. 269 (1972)CrossRefGoogle Scholar
- E. Balslev and J. M. Combes, Commun. Math. Phys. 22 280 (1972)CrossRefGoogle Scholar
- B. Simon, Ann. Math.. j 247 (1973).Google Scholar
- 22.G. Doolen, J. Nuttall and R. W. Stagat, Phys. Rev. A10 1612 (1974 )Google Scholar
- G. Doolen, J. Phys. . 525 (1975 )Google Scholar
- R. A. Bain, J. N. Bardsley, B. R. JunkerGoogle Scholar
- and C. V. Sukumar, J. Phys. B7, 2189 (1974 )Google Scholar
- B. R. Junker, Int. J. Qu. Chem. 14 371 (1978 )Google Scholar
- N. Moiseyev, P. R. Certain andGoogle Scholar
- F. Weinhold, Phys. Rev. A24 1254 (1981).CrossRefGoogle Scholar
- 23.B. R. Junker, Adv. At. Mol. Phys. 207 (1982 )Google Scholar
- Y. K. Ho, Phys. Reports,.. 1 (1983).Google Scholar
- 24.W. P. Reinhardt, Ann. Rev. Phys. Chem. 3. 223 (1982 )Google Scholar
- C. Cerjan, R. Hedges, C. Holt, W. P. Reinhardt, K. Scheibner and J. J. Wendoloski, Int. J. Qu. Chem. 4 393 (1978).CrossRefGoogle Scholar
- 25.B. Simon, Phys. Lett. A71 211 (1979).CrossRefGoogle Scholar
- 26.C. A. Nicolaides and D. R. Beck, Phys. Lett. A65 11 (1978).CrossRefGoogle Scholar
- 27.C. A. Nicolaides, H. J. Gotsis, M. Chrysos and Y. Komninos, Chem. Phys. Lett. 168 570 (1990) and refs. therein.Google Scholar
- 28.C. A. Nicolaides and S. Themelis, unpublished.Google Scholar
- 29.C. A. Nicolaides, Phys. Rev. 6 2078 (1972).Google Scholar
- 30.C. A. Nicolaides, Th. Mercouris and Y. Komninos, Int. J. Qu. Chem. 2. 1017 (1984 )Google Scholar
- C. A. Nicolaides and Th. Mercouris, Phys. Rev. A32 3247 (1985).CrossRefGoogle Scholar
- 31.Th. Mercouris and C. A. Nicolaides, J. Phys. B17 4127 (1984).Google Scholar
- 32.C. A. Nicolaides and Th. Mercouris, Phys. Rev. A36 390 (1987).CrossRefGoogle Scholar
- Th. Mercouris and C. A. Nicolaides, Z. Phys.
_{Q.}1 (1987).Google Scholar - 34.M. Chrysos, Y. Komninos, Th. Mercouris and C. A. Nicolaides, Phys. Rev. A42 2634 (1990).CrossRefGoogle Scholar
- 35.Th. Mercouris and C. A. Nicolaides, J. Phys. B21 L285 (1988).Google Scholar
- 36.C. A. Nicolaides and Th. Mercouris, Chem. Phys. Lett. 159 45 (1989).CrossRefGoogle Scholar
- 37.Th. Mercouris and C. A. Nicolaides, J. Phys. B23 2037 (1990).Google Scholar
- 38.Th. Mercouris and C. A. Nicolaides, J. Phys. B24 L 57 and L165 (1991).Google Scholar
- 39.C. A. Nicolaides, Th. Mercouris and G. Aspromallis, J. Opt. Soc. Am. B. Z. 494 (1990).Google Scholar
- 40.C. A. Nicolaides, Th. Mercouris and N. A. Piangos, J. Phys. B23 L669 (1990).Google Scholar
- 41.I. D. Petsalakis, Th. Mercouris, G. Theodorakopoulos and C. A. Nicolaides, J. Phys. B23 L89 (1990 )Google Scholar
- I. D. Petsalakis, Th. Mercouris, G. Theodorakopoulos and C. A. Nicolaides, J. Chem. Phys. a3. . 6642 (1990 )Google Scholar
- I. D. Petsalakis, Th. Mercouris, G. Theodorakopoulos and C. A. Nicolaides, Chem. Phys. Lett. (1991).Google Scholar
- 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
- 43.Even this picture breaks down in principle, when the strength of the external field increases to the point that the W
_{o}cannot represent only the unperturbed, free atomic or molecular state.Google Scholar - 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 HeH_{2}, NeH_{2}and ArH_{2}(ref. 45).Google Scholar - C. A. Nicolaides and A. Zdetsis, J. Chem. Phys.
_{Q.}1900 (1984).Google Scholar - 46.Y. Komninos, N. Makri and C. A. Nicolaides, Z. Phys. D2 105 (1986).Google Scholar
- 47.Y. Komninos and C. A. Nicolaides, J. Phys. B19 1701 (1986).Google Scholar
- 48.Y. Komninos, G. Aspromallis and C. A. Nicolaides, Phys. Rev. A27 1865 (1983).CrossRefGoogle Scholar
- 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
- 50.C. A. Nicolaides, in “Quantum Chemistry - Basic Aspects, Actual Trends”, ed., R. Carbo, Elservier (1989).Google Scholar
- 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
- 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
- 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
- 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
- 55.M. Bylicki, Phys. Rev. A, in press (1991).Google Scholar
- 56.C. A. Nicolaides, Y. Komninos and D. R. Beck, Phys. Rev. A24 1103 (1981).Google Scholar
- 57.C. E. Kuyatt, J. A. Simson and S. R. Mielczarek, Phys. Rev. 138 A385 (1965 )Google Scholar
- P. J. Hicks, C. Cvejanovic, J. Comer, F. H. Read and J. M. Sharp, Vacuum 24 573 (1974).Google Scholar
- 58.G. J. Schulz, Rev. Mod. Phys. 45 378 (1973).CrossRefGoogle Scholar
- 59.U. Fano and J. W. Cooper, Phys. Rev. 138 A400 (1965).CrossRefGoogle Scholar
- 60.K. Smith, D. E. Golden, S. Ormonde, B. W. Torres and A. R. Davis, Phys. Rev. ALI3001 (1973).Google Scholar
- 61.W. Lichten, Phys. Rev. 131 229 (1963).CrossRefGoogle Scholar
- 62.F. T. Smith, Phys. Rev. 179 111 (1969).CrossRefGoogle Scholar
- 63.T. F. O’Malley, Adv. At. Mol. Phys. 7 223 (1971).Google Scholar
- 64.C. A. Mead and D. G. Truhlar, J. Chem. Phys.. ZZ6090 (1982).Google Scholar
- 65.T. F. O. Malley, Phys. Rev. 162 98 (1967).CrossRefGoogle Scholar
- 66.T. F. O. Malley, J. Chem. Phys. 322 (1969).Google Scholar
- 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 T
_{o}for a V state is straight forward (For example, the 1 s22p2^{1}S valence excited state is represented mainly by a(1s^{2}2p2)+b(1s^{2}2s2)). 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 s^{2}2p21 S V state. For Z=4 (Be) it lies in the continuous spectrum. For Z=5 (B^{k}), it lies below the 1 s^{2}2sns^{1}S series and above the ground state 1 s^{2}2s21 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^{i}S channel, an “atomic diabatic state” is calculated.Google Scholar - 68.N. Bacalis, Y. Komninos and C. A. Nicolaides, unpublished.Google Scholar
- 69.E. A. McCullough, J. Chem. Phys. fa3991 (1975).Google Scholar
- 70.C. A. Nicolaides, Chem. Phys. Lett. 161 547 (1989 )Google Scholar
- 71.A. Metropoulos, C. A. Nicolaides and R. J. Buenker, Chem. Phys. 114 1 (1987).CrossRefGoogle Scholar