Rydberg States of Atoms and Molecules. Basic Group Theoretical and Topological Analysis
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Physics Reports 341 (2001) 173}264 Symmetry, invariants, topology. III Rydberg states of atoms and molecules. Basic group theoretical and topological analysis L. Michel! , B.I. ZhilinskimH " * !Institut des Hautes E! tudes Scientixques, 91440 Bures-sur-Yvette, France "Universite& du Littoral, BP 5526, 59379 Dunkerque Ce& dex, France Contents 1. Introduction 175 3.2. Qualitative description of e!ective 1.1. Dynamical symmetry of Rydberg states 178 Hamiltonians invariant with respect to 2. Groups and their actions appropriate for the continuous subgroups of O(3) 205 Rydberg problem 179 3.3. Qualitative description of e!ective 2.1. Structure of O(4) 179 Hamiltonians invariant with respect to 2.2. The adjoint representation of O(4); induced "nite subgroups of O(3) 210 R" ; action on S S 182 4. Manifestation of qualitative e!ects in physical 2.3. Action of O(3), SO(3), O(3)?T, systems. Hydrogen atom in magnetic and ? ? R SO(3) T, and SO(3) TQ on ; their electric "eld 213 strata, orbits and invariants 185 4.1. Di!erent "eld con"gurations and their 2.4. Invariants of the one-dimensional Lie symmetry 213 subgroups of O(3) acting on R 188 4.2. Quadratic Zeeman e!ect in hydrogen 2.5. One-dimensional Lie subgroups of atom 215 O(3)?T and their invariants 191 4.3. Hydrogen atom in parallel electric and 2.6. Orbits, strata and orbit spaces of the one- magnetic "elds 217 dimensional Lie subgroups of O(3) acting 4.4. Hydrogen atom in orthogonal electric and on R 192 magnetic "elds 223 2.7. Invariants of "nite subgroups of O(3) 4.5. Where to look for bifurcations? 227 acting on R 196 5. Conclusions and perspectives 228 2.8. Orbits, strata and orbit spaces of "nite Appendix A. Geometrical representation 231 subgroups of O(3) acting on R 197 A.1. O(3) or SO(3) invariant Hamiltonian 231 2.9. Orbits, strata and orbit spaces for A.2. C invariant Hamiltonian 233 ? T-dependent subgroups of O(3) T 198 A.3. CT invariant Hamiltonian 235 3. Construction and analysis of Rydberg A.4. CF invariant Hamiltonian 236 Hamiltonians 203 A.5. D invariant Hamiltonian 238 3.1. E!ective Hamiltonians 203 A.6. DF invariant Hamiltonian 239 * Corresponding author. E-mail address: [email protected] (B.I. ZhilinskimH). Deceased 30 December 1999. 0370-1573/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 573(00)00090-9 174 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 Appendix B. Molien functions for point group Appendix C. Strata and orbits for point groups 250 invariants 240 Appendix D. Qualitative description of e!ective B.1. C group 241 Hamiltonians based on equivariant Morse}Bott B.2. C point group 242 theory 255 B.3. CG point group 243 D.1. SO(3) continuous subgroup 255 B.4. CQ point group 243 D.2. C continuous subgroup 256 B.5. CT point group 244 References 258 B.6. DF group 245 ¹ B.7. B point group 248 Abstract Rydberg states of atoms and molecules are studied within the qualitative approach-based primarily on topological and group theoretical analysis. The correspondence between classical and quantum mechanics is explored to apply the results of qualitative (topological) approach to classical mechanics developed by PoincareH , Lyapounov and Smale to quantum problems. The study of the action of the symmetry group of the problems considered on the classical phase space enables us to predict qualitative features of the energy level patterns for quantum Rydberg operators. ( 2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Fd; 31.15.Md; 32.80.Rm; 33.80.Rv Keywords: Atoms in "elds; Rydberg problem; Hydrogen atom L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 175 1. Introduction This chapter is devoted to the qualitative analysis of Rydberg states of atoms and molecules based on the extensive use of group actions combined with topological arguments introduced in Chapter I. Some results of such applications are published (Sadovskii et al., 1996; Sadovskii and Zhilinskii, 1998; Cushman and Sadovskii, 1999) but there are still many open problems to study within the formalism developed here. Rydberg states of atoms and molecules are very slightly bound quantum states of an electron and a positively charged ion. In the approximation one can neglect the excitations of the ionic core and all spin e!ects, the spectrum of states has a structure typical of that of the hydrogen atom: grouping in multiplets of n states for the large principal quantum number n. In the following, we shall restrict ourselves to the analysis of the internal structure of these large Rydberg multiplets when their splitting is small in some sense. The experimental study of Rydberg states is a very active "eld of physics (Aymar, 1984; Stebbings and Dunning, 1983). There are many works on Rydberg atoms isolated or in di!erent con"gura- tions of magnetic and electric "elds (Beims and Alber, 1993; Boris et al., 1993; Cacciani et al., 1988, 1989, 1992; Fabre et al., 1977, 1984; Flothmann et al., 1994; Frey et al., 1996; Fujii and Morita, 1994; Hulet and Kleppner, 1983; Jacobson et al., 1996; Jones, 1996; Konig et al., 1988; Lahaye and Hogervorst, 1989; Raithel et al., 1991, 1993a,b; Raithel and Fauth, 1995; Rinneberg et al., 1985; Rothery et al., 1995; Seipp et al., 1996; van der Veldt et al., 1993; Zimmerman et al., 1979). The study of Rydberg molecules is beginning (Bordas et al., 1985, 1991; Bordas and Helm, 1991, 1992; Broyer et al., 1986; Dabrowski et al., 1992; Dabrowski and Sadovskii, 1994; Davies et al., 1990; Dietrich et al., 1996; Dodhy et al., 1988; Helm, 1988; Herzberg, 1987; Herzberg and Jungen, 1972; Hiskes, 1996; Jungen, 1988; Jungen et al., 1989, 1990; Ketterle et al., 1989; Labastie et al., 1984; Lembo et al., 1989, 1990; Mayer and Grant, 1995; Merkt et al., 1995, 1996; Schwarz et al., 1988; Sturrus et al., 1988; Weber et al., 1996). Theoretical studies also exist for each type of experiments (Aymar et al., 1996; Bander and Itzykson, 1966; Bixon and Jortner, 1996; Braun, 1993; Braun and Solov'ev, 1984; Chiu, 1986; Clark et al., 1996; Delande and Gay, 1986, 1988, 1991; Delande et al., 1994; Delos et al., 1983; Engle"eld, 1972; Gourlay et al., 1993; Greene and Jungen, 1985; Herrick, 1982; Howard and Wilkerson, 1995; Huppner et al., 1996; Iken et al., 1994; Kalinski and Eberly 1996a,b; Kazanskii and Ostrovskii, 1989, 1990; Kelleher and Saloman, 1987; King and Morokuma, 1979; Kuwata et al., 1990; Laughlin, 1995; Lombardi et al., 1988; Lombardi and Seligman, 1993; Mao and Delos 1992; Pan and Lu, 1988; Rabani and Levine, 1996; Rau and Zhang, 1990; Remacle and Levine, 1996a,b; Robnik and Schrufer, 1985; Solov'ev, 1981; Tanner et al., 1996; Thoss and Domcke, 1995; Uzer, 1990; Zakrzewski et al., 1995). It is time to establish some general physical laws applying to the Rydberg multiplets. They can be obtained by a general qualitative analysis of the relevant problems based on general methods that we have introduced in the initial Chapter I of this issue. Further applications to atomic and molecular problems are currently in progress. Two features of Rydberg physics make a general approach attractive: First, when the external "elds are small enough or vanish, each Rydberg multiplet is labeled by the value n of the principal quantum number and the splitting inside each multiplet is small compared to the splitting between them. The dynamical system which describes the internal structure of an individual n multiplet has two degrees of freedom. For large n, the set of n quantum 176 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 levels is su$ciently large to be well studied by classical analysis, as a Hamiltonian dynamical system de"ned on a four-dimensional phase space with non-trivial topology and symmetry that we shall soon make precise. Second, the isolated hydrogen atom has a large dynamical symmetry: that of the orthogonal group O(4) and time reversal. As we shall recall, that is an exact symmetry for the quantum mechanical study, in the non-relativistic approximation, of an electron in the static Coulomb potential of a point-like nucleus; then the energy of bound states depends only on the principal quantum number n; that exceptional energy degeneracy between the states of di!erent orbital 4 4 ! + # " momenta of value l,0 l n 1 contains l (2l 1) n states whose state vectors form the space of an irreducible linear representation of O(4). For atomic or molecular ions the O(4) dynamical symmetry is only an initial approximation for the Rydberg states, which becomes exact at the asymptotic limit nPR when there are no external "elds. Their geometric symmetry is that of the positive ion: it is at most O(3) (case of the isolated hydrogenoid atom) and it is a "nite subgroup of O(3) in the case of non-aligned molecules. Several important physical examples of geometric symmetry are listed in Table 1. Further extension of geometric invariance groups to higher symmetry groups including time-reversal operation will be discussed as well (Section 2.9). As it is well known, the quantum mechanical study of the hydrogen atom, in the non- relativistic approximation, was "rst made by Pauli (1926) just before the appearance of the ShroK dinger equation. In a convenient unit system, the Hamiltonian for a hydrogen atom can be written 1 1 1 H" p! , E "! .