Rydberg States of Atoms and Molecules. Basic Group Theoretical and Topological Analysis

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.

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"gurations 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 "! . (1) 2k r L 2n Due to the speci"c Coulomb interaction there are two vector integrals of motion: J } the angular momentum vector, and X } the Laplace}Runge}Lenz vector, X"p;J!rr\ . (2) In other words, the Hamiltonian operator H commutes with the vector operators J and X and all their functions. After the energy-dependent scaled transformation " ! \" K X( 2EL) Xn , (3)

Table 1 Physical examples of the geometric invariance subgroups of Rydberg atoms and molecules

O(3) Rydberg atoms with the closed shell core CT Rydberg atoms in the presence of a weak electric "eld Heteronuclear diatomic Rydberg molecule AB CF Rydberg atoms in the presence of a weak magnetic "eld C Rydberg atoms in the presence of parallel magnetic and electric "elds DF Homonuclear diatomic Rydberg molecules A Point group G Polyatomic Rydberg molecules with the point group symmetry G ¹ H (DF), NH ( B),2 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 177 it was shown by Hulthen (1933) and Fock (1936), that J and K form a Lie algebra with de"ning commutation relations: " e [J?, J@] i ?@AJA , (4) " e [J?, K@] i ?@AKA , (5) " e [K?, K@] i ?@AJA . (6) We shall show in Section 2.1 that this Lie algebra is the one of the group O(4) or SO(4) (or several other groups). In the hydrogen atom the operators J, K satisfy moreover the two important relations: J ) K"K ) J"0 , (7) J#K"n!1 , (8) which give the value of the two SO(4) Casimir operators for the Hilbert space of vector states with the energy EL. Instead of J and K we can introduce the two linear combinations: " # J (J K)/2 , (9) " ! J (J K)/2 . (10) Using only relations (4) and (5) one veri"es that these operators satisfy " e " [JI?, JI@] i ?@AJIA, k 1, 2 , (11) " [J?, J@] 0 . (12) That shows that the Lie algebra of SO(4) is isomorphic to the Lie algebra of the direct product S;(2);S;(2) of two groups S;(2). Therefore, an irreducible representation of SO(4) can be labeled # # ! " ) by a pair ( j, j); its dimension is (2j 1)(2j 1). Relations (9) and (10) imply J J J K. Then (7) is equivalent to " J J . (13) This last relation proves that the relevant SO(4) irreducible representations for the Rydberg problems are of the form ( j, j) with n"2j#1; hence its dimension is n. The richness of the Rydberg problem makes its `qualitativea study very interesting and powerful (Bander and Itzykson, 1966; Boiteux, 1973, 1982; Brown and Steiner, 1966; Coulson and Joseph, 1967; Cushman and Bates, 1997; Engle"eld, 1972; Guillemin and Sternberg, 1990; Iwai, 1981a,b; Iwai and Uwano, 1986; McIntosh and Cisneros, 1970; Stiefel and Scheifele, 1971). We recall that many fascinating basic tools needed for the general approach are summarized in Chapter I and used for molecular problems in Chapter II where we introduce the general approach by using examples from molecular physics which are less complicated than Rydberg state problem from the point of view of the mathematical technique and give to the reader the opportunity to get the uni"ed qualitative approach developed recently for molecular rotation, vibration, and rovibra- tion problems (Pavlichenkov and Zhilinskii, 1988, 1993a,b; Sadovskii and Zhilinskii, 1988, 1993a,b; Zhilinskii, 1989a,b, 1996; Zhilinskii et al., 1993). 178 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Section 2 will give a more thorough study of the symmetry we have to use. The main result of this section is the description of the symmetry group action on the phase space, construction of the rings of the polynomial invariant functions and the orbifolds for all cases of the continuous invariant symmetry groups of the Rydberg problem and for "nite point symmetry groups. Section 3 gives the construction and properties of phenomenological Rydberg Hamiltonians based on the integrity bases introduced in Section 2. Section 4 will study some examples of qualitative e!ects. We restrict ourselves with the application of the technique developed to the problem of the hydrogen atom in external electric and magnetic "elds. In spite of the fact that this problem seems to be well understood [see, for example, recent reviews by Friedrich and Wintgen (1989), Hasegawa, Robnik and Wunner (1989), and Braun (1993)] it is still possible to "nd new features and to give di!erent (and we hope useful) interpretations of some qualitative modi"cations of the dynamical behavior. In particular, the simple geometrical description of the collapse phenomenon is given recently by Sadovskii et al. (1996). It is based on the orbifold construction discussed in detail in this chapter. The conclusion summarizes brie#y further steps to apply the technique developed in this chapter to a much wider class of molecular problems. The appendices give more details about the geometrical representation of Rydberg orbifolds, explain more technical mathematical constructions like Molien functions, and collect some auxiliary tables needed for further application of the qualitative approach to molecular problems with "nite point symmetry group.

1.1. Dynamical symmetry of Rydberg states

Now, we can formulate the classical limit construction for the Rydberg problem studied. General scheme of the classical limit construction is based on the method of generalized coherent states (Perelomov, 1986; Cavalli et al., 1985; Zhang et al., 1990). This formal construction starts with introducing a dynamical algebra g whose generators play the role of the dynamic variables of the problem. The Hamiltonian itself in this case is considered as an operator in the enveloping algebra. For the Rydberg problem the dynamic algebra is the so(4) algebra with J and K generators (see Section 1, Eqs. (4)}(6)). An equivalent way to represent the same algebra is to use two commuting vector operators J and J. In any of these representations two important relations in Eqs. (7) and (8) restrict the space of the dynamic variables variation to four-dimensional space. The geometrical signi"cance of this space is clearly seen in the J, J representation. Di!erent points of the classical limit phase space are in one-to-one correspondence with orientations of two vectors J, J. This ; space is the topological product of two two-dimensional spheres S: S S. We will denote this space as R. It plays the essential role in all the subsequent analysis. An arbitrary e!ective Hamiltonian which gives the description of the internal structure of Rydberg multiplets may be written in the classical limit as a function de"ned over R (the classical phase space for the Rydberg problem). Following steps of the qualitative analysis of the Rydberg problem may be formulated now as follows:

(i) Study of the action of the invariance group on R, classi"cation of orbits and strata. Finding the critical orbits (see Section 2). (ii) Application of Morse theory to the complexity classi"cation of functions de"ned over R in the presence of the invariance symmetry group (see Section 3). L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 179

(iii) Construction of an arbitrary e!ective Hamiltonian in terms of expansions over integrity basis. Representation of classical Hamiltonians near critical manifolds and the qualitative description of the corresponding quantum energy level patterns and wave functions (see Sections 3 and 4).

2. Groups and their actions appropriate for the Rydberg problem

The dynamical symmetry group of hydrogen atom Rydberg states is O(4)'T, where T is the time reversal. For other physical systems this group can be used as "rst approximation: only a subgroup of it is the exact symmetry. In the beginning of this section we build the group O(4)'T and the phase space R of Rydberg problems. The rest of the section is devoted to the study of the action of the O(4)'T subgroups on R.

2.1. Structure of O(4)

As we have seen in the introduction, the quantum Hamiltonian of the hydrogen atom commutes with the six-dimensional Lie algebra described in Eqs. (4)}(6) and recognized under the form (11) and (12) as the Lie algebra of S;(2);S;(2). But many non-isomorphic Lie groups have the same Lie algebra, e.g. SO(3);SO(3), O(3);O(3), S;(2)!2 (de"ned in Eq. (26)), SO(4) and O(4) as shown in Eqs. (22), (23) and (28). There is some ambiguity in choosing between these di!erent groups; but a non-connected one is de"nitely richer and better adapted to the physics, so we choose here O(4) as an approximate symmetry of Rydberg states (time reversal will be added in Section 2.2.1). We begin by studying SO(4). The connected orthogonal groups SO(n) for n'2 have a double covering called Spin(n). For n"3, SO(3) is the group of rotations in the three-dimensional space and its spin group Spin(3)"S;(2), the group of 2;2 unitary matrices with determinant 1. The relation between these two groups is the homomorphism S;(2)PN SO(3). The image of this + ! , homomorphism is the full SO(3) group and its kernel is the center I, I of S;(2).We denote it ! by Z( I). This whole information can be expressed in one-line way: P ! P PN P 1 Z( I) S;(2) SO(3) 1 . (14) " ! This notation is an exact sequence. A less explicit notation is SO(3) S;(2)/Z( I). To understand better the interrelations between di!erent Lie groups having the same su(2);su(2) algebra we give below their realization as groups of transformations of the quaternions. The elements q30, the quaternion "eld, can be represented by the 2;2 matrices (e.g. Duval, 1964): " # p # p # p q qI iq iq iq , (15)

The symbol ' is a shorthand for de"ning the groups generated by the groups or group element written on each side of this symbol. The exact sequences (see Appendix A to Chapter I) used here are all of the type 1PAPBPCP1. They mean A is the invariant subgroup of B and C"B/A. This notation is now currently used even in undergraduate studies. For more details see e.g. Michel (1980). 180 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

3 a" p "pH q? R, 1, 2, 3, 4, where the three Hermitian matrices I I are the Pauli matrices: 01 0 !i 10 p " , p " , p " . (16) A10B Ai0B A0 !1B Notice that q is not Hermitian. Indeed qH"q with " "! "! "! q q, q q, q q, q q . (17) We verify H" H "" " " "" + qq q q q I with q q? (18) ? where "q"""qH" is called the norm of the quaternion q. Notice that "q"'0 when qO0 and "qq"""q""q". That last property can be checked from Eq. (18) or from "q""det q . (19) Eq. (18) shows that the set of quaternions forms a dimension 4 orthogonal space with the scalar product: "1 H"1 H" + (q, q ) tr qq tr q q q?q? . (20) 2 2 ? By de"nition, the group S;(2) is the multiplicative group of 2;2 unitary matrices of determinant 1. That group is realized by the multiplication of the quaternions of norm "q""1; indeed Eq. (19) shows that det q"1 and, from Eq. (18), q is unitary. Moreover, the manifold of S;(2) elements, i.e. 0 the quaternions of norm 1, is S, the unit sphere in the dimension 4 orthogonal space of . The group S;(2);S;(2) acts linearly on the quaternions by (u, v)3S;(2);S;(2), (u, v) ) q"$%& uqvH . (21) This action preserves the quaternion norm. Moreover, it is transitive on the set of quaternions of a given norm: indeed the quaternion "q"I is transformed into the arbitrary quaternion q of the same norm by u"q"q"\, v"I. This shows the existence of a group homomorphism S;(2);S;(2)PF O(4) , (22) h" h" ! Im SO(4), Ker Z(I, I) . (23) Indeed, h is continuous so its image is connected; the transitivity property we have just established requires Im h"SO(4). By de"nition Ker h is the set of group elements acting trivially on 0; they are u"v"$ I. " The stabilizer of the quaternions of the form qI is made of the elements v u; one calls it the diagonal subgroup S;(2)BLS;(2);S;(2). Moreover, S;(2)B transforms into itself the three- dimensional subspace of quaternions orthogonal to I (their trace vanishes); that establishes the P " well-known homomorphism S;(2) SO(3), [S;(2)/Z SO(3)]. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 181

We shall consider the 4;4 orthogonal matrix !I 0 s" , s"I, det s"! 1 . (24) A 01B It is the inversion through the origin for the subgroup O(3)LO(4) which leaves "xed the coordinate H q; as Eq. (17) shows, it transforms q into q . Every element of O(4) not in SO(4) is of the form h ; s (u, v). Let us study the induced action of these elements on S; S; from their known action on every quaternion q: sh(u, v)s\ ) q"sh(u, v) ) qH"s ) (uqHvH)"vquH , (25) ; it is the permutation of the two factors in S; S;. So we are led to consider the wreath product ! S; 2 which is de"ned as the semi-direct product: ! & ; ) S; 2 (S; S;) Z(P) , (26) ; where Z(P) denotes the group of permutations of the two factors of S; S;. To summarize, we have established the two exact sequences (see Appendix A to Chapter I) P B ! ! P ; PF P 1 Z( I, I) S; S; SO(4) 1 , (27) P B ! ! P ! FY P 1 Z( I, I) S; 2P O(4) 1 . (28) ! Notice that SO(4) and O(4) have a two-element center generated by the matrix I. Eq. (27) shows ; " 3 that S;(2) S;(2) Spin(4). Its elements can be labeled by a pair of indices u, u S;(2). Then h 3 (u, u)de"nes an orthogonal matrix g SO(4) U &h SO(4) g (u, u) . (29) The elements of O(4) are either of the form g or sg with s de"ned in Eq. (24). The irreducible representations of a direct product of groups are the tensor product of their irreducible representations. It is customary to label the irreducible representations of S;(2) by j, 4 3 ; 0 2j Z; so the irreducible representations of S;(2) S;(2) are usually labeled by the pair ( j, j). # # ! They have the dimension (2j 1)(2j 1). The representations of S;(2) 2 are labeled by # # O " ( j, j) ( j, j) (of dimension 2(2j 1)(2j 1)) when j j. When j j this representation becomes reducible into the direct sum of two reducible representations ( j, j)! which are, respectively, symmetric and antisymmetric for the permutation of the two S;(2) factors. Among all $ 3 representations those with j j Z form the set of irreducible representations of O(4). They are faithful when j, j are both half-integers; when j, j are both integers their image is that of the ` a ! adjoint group of O(4), that is the quotient O(4)/Z( I)ofO(4) by its center. It is easy to see that it is isomorphic to SO(3)!2. This can be summarized by P ! P P0 ! P 1 Z( I) O(4) SO(3) 2 1 . (30) Eq. (13) shows that the only irreducible representations carried by the n-dimensional space of state " ! " # vectors of a Rydberg multiplet have j j. As we shall show they are ( j, j) with n 2j 1. It is why we say that it is O(4) and not S;(2)!2 which is the symmetry group of the Rydberg problem. 182 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

> Remark that these representations are faithful only when j is half integer. That is the case of (, ) , the vector representation (i.e. that of dimension 4) of O(4) that we have studied in this subsection.

R" ; 2.2. The adjoint representation of O(4); induced action on S S

The adjoint representation of O(4) is (1, 0)(0, 1); indeed, by de"nition, it is the representation of " O(4) on the six-dimensional vector space of its Lie algebra. We denote this space by < < < direct sum of two three-dimensional vector spaces. The operators J, J de"ned in Eqs. (9) and (10) act e!ectively on their respective space <, < and trivially on the other one; that yields the representation of Eqs. (11) and (12). A vector of the six-dimensional space < < is naturally V decomposed in a direct sum that we denote by (W). Moreover, < < is an orthogonal space with the scalar product obtained as the matrix product

x (x y) "x ) x #y ) y . (31) Ay B

Each three-dimensional subspace is invariant by a SO(3). For the group SO(4), its matrix &h ; g (u, u) [see Eq. (29)] is represented by the matrix A(g) (written in 3 3 blocks): p(u )0 A(g)" , (32) A p B 0 (u) where p has been de"ned in Eq. (14). In the preceding section we have shown that s3O(4) (de"ned in Eq. (24)) corresponds to the permutation of the two three-dimensional subspaces (see Eq. (25)):

0 I A(s)"A B . (33) I 0 So the adjoint representation of the group O(4) is irreducible. " Instead of using for the vector space < of the Lie algebra of O(4) the decomposition < < < " corresponding to the operators J, J, it is interesting to use the decomposition <

1 I I S" . (34) A ! B 2 I I The conjugation by S diagonalizes A(s):

I 0 SA(s)S\"A(s)" , (35) A ! B 0 I 1 p(u )#p(u ) p(u )!p(u ) SA(g)S\"A(g)" . (36) Ap !p p #p B 2 (u) (u) (u) (u) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 183

In this new decomposition we write the component vectors as j x x#y x!y "S , i.e. j" , k" . (37) AkB AyB (2 (2 B " " We denote by g the matrices of SO(4) which correspond through Eq. (57) to the pairs u u u. The matrices A(gB) which represent them in the adjoint representation are block diagonal p(u)0 A(gB)" . (38) A 0 p(u)B They are the matrices of SO(4) which commute with A(s). They form the subgroup SO(3)B acting e!ectively on the axes 1, 2, 3 of the vector representation of O(4) and leaving the axis 4 "x. The orthogonal group O(3)B"SO(3)B's is the orthogonal group of the physical space; it transforms 3 3 j

2.2.1. Extension of O(4) by including the time reversal In classical mechanics of point particles, the operation T which leaves the position of the particles "xed and reverses their velocity is called time reversal. So T also changes the sign of angular momenta. More generally in physics, T changes the sign of axial vectors (e.g. the magnetic induction) and leaves the polar vectors "xed (electric "eld, Laplace}Runge}Lenz vector, etc.) (Wigner, 1959). Wigner has shown that in quantum physics T is represented by an anti-unitary operator. T is an exact symmetry of atomic and molecular physics (except for tiny e!ects induced by weak interaction!) (Sakurai, 1964). We denote by O(4)'T the group generated by O(4) and T;it is the full approximate dynamical symmetry group for the Rydberg states. The action of T on <, the space of the adjoint representation, is represented by the matrix !A(s) where A(s) (generally called P, the parity operator) is de"ned in Eq. (35). Remark that O(4)'T is not the direct product of O(4) by Z(T) since T does not commute with the elements of O(4) not in the subgroup B; ! O(3) Z( I) (the last factor is the center of O(4)); it is a semi-direct product. The image of the adjoint representation of O(4)'T is isomorphic to O(4).

2.2.2. The action of O(4)'T on the vector space < of the adjoint representation and on the phase space R We begin by studing the action of SO(4) on <.Itisde"ned by the image 0[SO(4)]" SO(3);SO(3) (30) of SO(4) in the adjoint representation. There are two strata in the action of O(3): the origin with the stabilizer O(3), and the rest of space with the stabilizer O(2). There is a unique invariant, the vector norm x ) x50 which de"nes the orbit space; the two strata are de"ned by x ) x"0 and x ) x'0. The invariant, the orbit space, and the orbits in < of index 2 subgroup SO(3) are identical. The corresponding stabilizers are, respectively, SO(3)5O(2)"SO(2), which are index 2 subgroups of the stabilizers of O(3). Note that the vector representation of SO(3) coincides with its adjoint representation. The adjoint representation of SO(3);SO(3) is the direct sum of two adjoint representations of SO(3). " We obtain the strata on the space < < < by using the general fact (see Chapter I): In a reducible representation of a compact group, the stabilizer of a vector is the intersection of the stabilizers of its components in the irreducible subspaces. 184 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 2 Strata of the SO(4) adjoint representation

Stablizer! Invariants dim.str." Topology#

SO(3);SO(3) x ) x"0"y ) y 0 One point ; ) ' " ) SO(2) SO(3) x x 0 y y 3 S ; ) " ( ) SO(3) SO(2) x x 0 y y 3 S ; ) ' ( ) ; SO(2) SO(2) x x 0 y y 6 S S

! We give the image by 0 from the stabilizer. " dim.str. means dimension of the stratum. # The topological structure of each orbit is given.

The two invariants x ) x and y ) y label the orbits. The orbit space is the semi-algebraic domain x ) x50, y ) y50. Its inside corresponds to the generic stratum. Its boundary is the union of the three strata x ) x'0"y ) y, x ) x"0(y ) y, and x ) x"0"y ) y. The results are given in Table 2. To obtain the stabilizers in SO(4) one must take the inverse image 0\ of those in SO(3);SO(3). The group O(4)"SO(4)'s is generated by SO(4) and s whose action on < is given in Eq. (33): it exchanges the vectors x and y. To write the invariants of O(4) (or of its image SO(3)!2) we use the symmetric and anti-symmetric combinations of those of SO(4)

i"$%& x ) x#y ) y"j ) j#k ) k, i'0 , (39)

o"$%& x ) x!y ) y"2j ) k , (40)

!i4o4i50 (41)

(the second form in j, k is obtained from Eq. (37)). Since o changes sign by the exchange x % y the invariants of the O(4) action on < are i, o and the orbit space is de"ned by restrictions i50, i5o50. The inside of the orbit space (obtained by replacing 5 by ') corresponds to the generic stratum. The three non-generic strata form the boundary. Their equations are, respectively, i'0, o"0; i'0, i"o'0; and i"0(No"0). The corresponding orbits in the image SO(3)!2 are SO(2)!2, SO(3);SO(2), SO(3)!2. Since i and o are also T invariant, O(4)'T has same orbits and same orbit space as O(4). The ! ' corresponding stabilizers [for the image (SO(3) 2) T] are given in Table 3 where TQ is the product Ts"sT. Taking into account the physical relation in Eq. (13) we have found that the phase space of ; R Rydberg problem has the topology S S and we have denoted it by .In< this phase space is de"ned by the equations R& ; i" o" S S: 1, 0 . (42) It is an orbit of O(4)'T [and even of SO(4)] belonging to the stratum o"0 of stabilizer (SO(2)!2)'T. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 185

Table 3 Strata of the O(4)'T and O(4) adjoint representation. Both groups have the same set of strata but their stabilizers are di!erent

Stablizer! Invariants" dim.str.# Topology$

(SO(3)!2)'T i"0"o 0 One point ; ' i"o' (SO(2) SO(3)) TQ 03 2S ! ' iOo" ; (SO(2) 2) T 03 S S ; ' i!o' ; (SO(2) SO(2)) TQ 06 2(S S) ! The image of the stabilizer is given. For the O(4) adjoint representation T or TQ should be omitted. " i50 is implicit. # dim.str. means the dimension of the stratum. $ The topological structure of each orbit is given.

' ' ' R 2.3. Action of O(3), SO(3), O(3) T, SO(3) T, and SO(3) TQ on ; their strata, orbits and invariants

In the four-dimensional space, the subgroup SO(3) leaving invariant the fourth coordinate is characterized in Eqs. (32) and (36) 3 % " % BL ; r SO(3) u u SO(3) SO(3) SO(3) . (43) " ; In Eq. (24) we have de"ned s, the space inversion of O(3) SO(3) Z(s). The restriction of the adjoint representation of O(4) to O(3) is reducible. The matrix S reduces this representation (the reduction is given explicitly in Section 2.2, see Eqs. (34)}(37)). This reducible representation of O(3) can be denoted by 1>1\ where 1! is, respectively, the axial and polar vector representation. In our problem, the axial and polar vectors j, k correspond, respectively, to the angular momentum and the Runge}Lenz vector. They satisfy the relations in Eqs. (39) and (40) which completely characterize the manifold R. We need also the new invariant m of SO(3) which we de"ne as

!14m"$%& j ) j!k ) k41 . (44)

The stabilizers of the action of O(3) on R are the intersection of O(3) with the stabilizers of O(4) that we have determined in Table 3. This gives immediately the classi"cation of the strata in the action of O(3) on R. Indeed, the stabilizers in the three-dimensional subspaces of j and k are, respectively, O O R O(3) for the null vector, CF( j) for j 0 and CT(k) for k 0. So for the points of , when both j, k ) " are non-zero vectors (they satisfy j k 0) the stabilizer is CQ( j), the group generated by the re#ection through the plane orthogonal to j. Table 4 summarizes the data of the three strata which appear in the action of O(3) on R. The stabilizers appearing in the action on R of the subgroup SO(3) are obtained by taking the intersection of SO(3) with the stabilizers of the O(3) action. This shows that on R the orbits are the same for O(3) and SO(3). However, the SO(3) orbits form only two strata: the same open dense m"$ stratum with trivial symmetry and one stratum with symmetry C formed by two orbits 1. Any SO(3)-invariant function on R is invariant by O(3) (see Table 4). 186 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 4 R Strata and orbits in the actions of O(3) and SO(3) on . The orbits of shape S are critical

G Stabilizer dim.! Equation n.o." topol.#

m"! O(3) CT(k)2 11 S m" CF( j)2 11 S ! (m( R CQ( j)4 1 1 RP "m"" SO(3) C 2 12 S 14 !1(m(1 R RP

! dim."dimension of the stratum. " n.o."number of orbits. # topol."topological structure of each orbit.

It is a general theorem of Weyl (1939) that the invariants of a direct sum of representations of SO(n)(n53) are made from all distinct scalar products and determinants. Its application to the action of SO(3) on the six-dimensional space < proves that i, m, o form a minimal set of generators of the invariant polynomials since they are algebraically independent. As a scalar product of an axial vector j and a polar vector k, o is a pseudo-scalar for O(3). For this group, i, m, o form an integrity basis. There is no di!erence in R between the P-() and P1-() invariant polynomial rings since o"0 (and i"1). Taking into account the fact that the values of polynomials of an integrity basis label the orbits, we can conclude that O(3) and SO(3) have the same orbits on R.Soon<, when oO0, the O(3) orbits split into two orbits of SO(3) with opposite values of o. Since i, m, o are algebraically independent polynomials on the six-dimensional space <, the ring of invariant polynomials of SO(3) and O(3) are the polynomial rings:

1-()" i m o P P[ , , ] , (45) -()" i m o P P[ , , ] . (46) Table 4 shows that any function on R, invariant under O(3) or SO(3), depends only on the variable m de"ned in Eq. (44); indeed this parameter labels completely the orbits (common to O(3) and SO(3)). The geometrical representation of the orbifold (which is a 1D-segment) is given in Appendix A. Table 4 also shows that the two orbits de"ned by m"$1 are isolated in their strata; therefore they are critical: these two orbits are orbits of extrema of any O or SO(3) invariant real function de"ned on R. We give a direct and explicit proof of this property:

Lemma. Any O(3)-invariant function f (m) on R has at least two orbits of extrema, dexned by m"$1. Indeed, at the point

j j 3R, m"2 . (47) AkB A!kB L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 187

For a function on R, we have to project m on the tangent plane to R. Two normal vectors at H 3R i" o" R (I) are obtained by di!erentiating the equations 1, 0de"ning , i.e. Ri/Rj j Ro/Rj k v " " , v " " , v v "d . (48) ARi/RkB A kB ARo/RkB A jB G H GH The rank one orthogonal projectors on these two vectors are, respectively: j21 jj21k k21kk21 j P " , P " . (49) A k21 jk21kB A j21kj21 jB " " m H 3R Since PP 0 PP the gradient of on the tangent plane at the point (I) is (1!m)j R m"(I !P !P ) m" . (50) A!(1#m)kB To end the proof of this lemma, we simply verify that this vector vanishes for m"1 (so k"0) and for m"! 1 (so j"0). Remark that this vector never vanishes on the interval !1(m(1. So the only other orbits of extrema of f (m) occur for the values of m on this interval, satisfying f (m),df/dm"0. The extension to the case of three di!erent two-dimensional Lie groups including T-dependent symmetry transformations can be easily done because the space of orbits remains the same and the same invariant m can be used to distinguish orbits and strata. These groups are SO(3)'T, ' ' O(3) T, and SO(3) TQ. Special care should be taken only to specify the stabilizers of di!erent strata. They are given in Table 5. As explained in Section 2.2.1 the action of T on the adjoint space < is represented by !A(s), so " ! i m o TQ sT is represented by I. Hence TQ leaves , , invariant and the orbits and strata of

Table 5 ' ' R Strata and orbits in the actions of O(3) T and SO(3) T on . The orbits of shape S are critical

G Stabilizer Equation

' ' m"! O(3) T CT(k) T 1 ' ! m" CF( j) TQ 1 ' " ! (m( CQ( j) TQY 1 1 ' ' m"! SO(3) T C T 1 ' # m" C T 1 $ ! (m( T 1 1 ' ' "m"" SO(3) TQ C TQ 1 ! (m( TQ 1 1 ! TQ is the product of time reversal and re#ection in plane orthogonal to j. " This group is generated by the re#ection in plane orthogonal to the vector j and by the product Ts of time reversal T and the re#ection s in plane of j and k. # T is the product of time reversal T by C rotation around axis orthogonal to vector j. $ T is the product of time reversal T by C rotation around k vector. 188 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

' ' ' SO(3) TQ are those of SO(3). Similarly, the orbits and strata of O(3), O(3) T, and SO(3) T are identical. Invariants i, m, o characterize orbits unambiguously. So we have

1-()" 1-();TQ P P , (51) -()" -();T" 1-();T P P P . (52) An orbit, common to di!erent groups, has di!erent conjugacy classes of stabilizers for each group. The stabilizers on R for these "ve groups are given in Tables 4 and 5.

2.4. Invariants of the one-dimensional Lie subgroups of O(3) acting on R

Up to a conjugation, there are "ve one-dimensional Lie subgroups of O(3) whose traditional notations in molecular physics are

C, CF, CT, D, DF . (53) We denote unit vectors by a hat, e.g. n( , and by C(n( ) the group of rotations around the axis de"ned by n( . The stabilizers for the action of these groups on R and on the six-dimensional space < are the intersections with the stabilizers of O(3). The generic orbits are one dimensional, so we need at least "ve polynomial invariants to label the orbits on R, the vector space of the O(4) adjoint representation. With the two coordinate vectors j, k and the vector n("xing the C-axis, we can form "ve scalar products which are "ve algebraically independent invariant homogeneous polynomial for i m o C; we label them by Greek letters ( , , have already been de"ned): i"j ) j#k ) k , (54) m"j ) j!k ) k , (55) o"2 j ) k, (56) k"n( ) j , (57) l"n( ) k , (58) the "rst three and the last two are, respectively, of degree two, one in the coordinates. ! However, this is not enough for generating the ring P of C polynomial invariants and for labeling all the C-orbits. Indeed, the Molien function is 1 p dh M (j)" ! p !j ! j h#j 2 (1 ) (1 2 cos ) 1#j " . (59) (1!j)(1!j) The degrees of j in the "ve factors of the denominator are equal to the degrees of the "ve homogeneous polynomials of variables jG, kH of Eqs. (54)}(58). But the numerator requires a sixth

k" l" We could have chosen for the direction n( the third coordinate axis; then j, k. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 189 generator of P! of degree 2. The square of this invariant should be a polynomial function of six basic invariant polynomials. This statement can also be veri"ed geometrically: indeed any set of values satisfying Schwarz inequalities for the scalar products: o4i!m , (60) k4 i#m ( ) , (61) l4 i!m ( ) (62) " de"nes two circles C-orbits which can be distinguished by the sign of p"(n( , j, k),n( ) ( j;k) . (63) Moreover, 1 i 1 1 p"(i!m)#okl! o# (k#l)! mk# ml 50 . (64) A4 2 2 2 B

So the module of C-invariant functions on <, the six-dimensional vector space of the O(4) adjoint representation may be represented as

! " i m o k l ⅷ p P P[ , , , , ] (1, ) . (65) To each point of the "ve-dimensional domain de"ned by the inequalities in Eqs. (60)}(62) and (64) " corresponds a unique C orbit and conversely: this is the orbit space < C. The four other one-dimensional subgroups have C as invariant subgroup. We can build their orbit spaces from the one of C.LetG be any of the three groups CF, CT, D. Then G is the " 6 union of two C cosets: G C C6C where C6 denotes, respectively, the re#ections CT, CF, n and C a rotation by around an axis (in the plane h) orthogonal to the rotation axis of C. The action of the quotient group G/C on the C orbit space can be obtained by computing the action o k l p i m of these three operations C6 on the C invariants , , , (the O-invariants , are invariants for its "ve one-dimensional subgroups). The result is given in Table 6. We call those pseudo-invariants which are multiplied by !1 or invariant under the action of the symmetry elements (more properly speaking pseudo-invariants transform according to one- dimensional real representation of the symmetry group). Their squares are invariants and we will obtain the set of denominator invariants of G by replacing the pseudo-invariants among o, k, l by their squares. The product of any two pseudo-invariants transforming according to the same one-dimensional representation of the symmetry group is an invariant which will be a numerator invariant. We can treat similarly the group DF which is made of the four cosets of C which are in the union of the three groups G's. So its denominator invariants are i, m, k, l, o.The"rst three lines of Table 6 show that lp is the only non-trivial numerator invariant. The structure we found for the module of the invariant polynomials on < of these four other one-dimensional subgroups of O(3), can be compared with their Molien functions. To compute

4a4 We use the property that the square of the determinant on mm-component vectors v?,1 m is the determinant of the matrix of elements v?.v@. 190 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 6 ' i m Action of C and discrete symmetry generators of the group O(3) T on a set of C-invariants. and are invariant of the total group O(3)'T. They are omitted from the table. The ! sign indicates the multiplication by !1

Element oklp

! #### C !!## CT !#!! CF #!!! C

T !!#! ###! TT #!!# TF !#!# T ! This line is introduced to show explicitly that we deal with invariants of C group.

them it is useful to compute the (incomplete) Molien functions M% of the C cosets not containing 1 of the "rst three groups. We obtain

1 p dh 1 M " " , (66) !T p !j !j 2 (1 ) (1 ) 1 p dh M " !F p !j !j ! j h 2 (1 )((1 ) 4 (cos ) ) 1 " , (67) (1!j)(1!j)

1 p dh 1 M " " . (68) " p #j !j #j !j 2 (1 ) (1 ) (1 ) (1 ) We can express the complete Molien function for the invariants in terms of these M:

M "(M #M ) , (69) !T ! !T M "(M #M ) , (70) !F ! !F M "(M #M ) , (71) " ! " M "(M #M #M #M ) . (72) "F ! !T !F " Then we obtain for the expression of the Molien functions:

1#j#j#j 1#j M " " , (73) !T (1!j)(1!j)(1!j) A (1!j)(1!j)B L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 191

1#2j#j M " , (74) !F (1!j)(1!j)(1!j) 1#j#2j M " , (75) " (1!j) 1#j#j#j M " . (76) "F (1!j)(1!j) We verify again that Molien functions are rational fractions. However, for Eq. (73) the form we have used does not correspond to the reduced one (which is given between brackets). The situation is di!erent from the Sloane counter example referred to in Chapter I (Sloane, 1977). The reduced form of the CT Molien function does correspond to a module given in Eq. (78) while the module corresponding to nonreduced form (given in Eq. (73) without parenthesis) is given in Eq. (77):

!T " i m o k l ⅷ p ok pok P P[ , , , , ] (1, , , ) , (77)

!T " i m p k l ⅷ ok P P[ , , , , ] (1, ) . (78) To prove the equivalence of these two modules we replace p by its square in the denominator invariants of Eq. (78) and add to the numerator invariants the numerator invariants multiplied by p; the two modules have same basis. Eq. (64) shows that p is a function of o and of i, m, k, l, ok; since the latter are other invariants of Eq. (78), we can replace p by o; that ends the transformation of Eq. (78) into Eq. (77). We prefer to use the "rst form because o"0 in one of the equations de"ning the manifold R and our future qualitative analysis uses basically G-invariant functions on R (de"ned by i"1, o"0). Table 7 enables us to write explicitly the structure of the "ve modules of G-invariant functions on R:

! " " m k l ⅷ p P R P[ , , ] (1, ) , (79)

!T " " m k l ⅷ p P R P[ , , ] (1, ) , (80)

!F " " m k l ⅷ lp P R P[ , , ] (1, ) , (81)

"" " m k l ⅷ kl kp lp P R P[ , , ] (1, , , ) , (82)

"F" " m k l ⅷ lp P R P[ , , ] (1, ) . (83)

2.5. One-dimensional Lie subgroups of O(3)'T and their invariants

Extension to groups including time-reversal symmetry operation was discussed in Section 7 of Chapter I. We summarize here in Table 7 the system of invariants of all 16 one-dimensional subgroups of the group O(3)'T. The restriction of the system of invariant polynomials on R becomes especially simple for four ' ' ' ' symmetry groups C TT, CT T, CF T, and DF T. For all these groups we have on R only three denominator invariants:

! ;TT " " m k l P R P[ , , ] , (84) 192 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 7 ' h "i Denominator and numerator invariants in the action on R of the 16 one-dimensional Lie subgroups of O(3) T. h "m and are present for all subgroups and therefore omitted from the table. Invariants which have a "xed value on R, ; o" S S (due to 0) are between ( ). Four parts of the table correspond, respectively, to connected subgroup C,to seven subgroups with two connected components, to seven subgroups with four connected components and to the ' DF T group with eight connected components h h h u u u G o klp C ( ) o k lp ok okp CT ( ) ( )() o kl lp ol op CF ( ) ( )() o k l kl kp lp D ( ) ' o k lkp ok op C T ( ) ( )() ' o kl C TT ( ) ' o k l pklklp C TF ( ) ( ) ' o kl p lo lop C T ( ) ( )() o k l lp kop klo DF ( ) ( )() ' o k l ko CT T ( ) ( ) ' o k l klp po klo CF T ( ) ( )() ' o k l kp lko lpo D T ( ) ( )() ' o k l p lko lkpo CT TF ( ) ( )() ' o kl lo CF TT ( ) ( ) ' o k l kl D TT ( ) ' o k l okl DF T ( ) ( )

!T ;T" " m k l P R P[ , , ] , (85)

!F ;T" " m k l P R P[ , , ] , (86)

"F;T" " m k l P R P[ , , ] . (87)

It is curious to note that especially these groups have the most natural physical interpretation. ' CT T is the symmetry group for the atom in a static electric "eld or for Rydberg states of a heteronuclear diatomic molecule. ' CF TT is the symmetry group of an atom in a constant magnetic "eld (Zeeman e!ect). ' ' ' C TT is the maximal common subgroup of CT T and CF TT. Thus, it corresponds to the symmetry of an atom in the simultaneous presence of two parallel magnetic and electric "elds. ' The DF T group is the symmetry group for Rydberg states of a homonuclear diatomic molecule, or of the quadratic Zeeman e!ect.

2.6. Orbits, strata and orbit spaces of the one-dimensional Lie subgroups of O(3) acting on R

The stabilizers of O(3) have been determined in Table 4; they belong to three O(3) conjugation classes: those of CT(k), CF( j), CQ( j). The stabilizers of the "ve groups G of Eq. (53) are their L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 193

Table 8 Strata and orbits in the action of the "ve one-dimensional Lie subgroups of O(3) on R

G! Stabilizer" gc# dim.$ Strata equations% Number& Nature' cr?) R"G strat. and inequalities orbits orbits

k" l" C C c0 1, 14A, B, C, D cr k( l( R S 1g4 1, 1 S l" CT CT c0 12C, D cr k" C c0 11AB cr k" l( R B CQ( j )30, 1 S (k( Nl( R 1g401 1 2S k" CF CF c0 12A, B cr l" C c0 11CD cr 1#m S C (n)204 "k(1Nl"0 R S Q 2 m" k( R CG 2 1, 1 S m( (k( Nl( R 1g41, 0 1 1 2S k#l" kl" D C c0 1, 02AB, CD cr m" k"l" C C Susp. C c1 1, 02, cr m" (k( (l( R RP 1g4(1, 0 1, 0 1) 2S k" DF CF c0 11AB cr l" CT c0 11CD cr m" k"l" C CF( j)c1 1, 01 cr m"! k"l" C CT(k)c1 1, 01 cr 1#m B C (n)204 "k(1Nl"0 R 2S Q 2 m" (k( R CG 2 1, 0 1 2S k" l( m( #l R CQ( j )30, 1, 1 2S 1#m 1!m 1g40(k( (1, l( (1, R 4S 2 2 (m!1)#k#l'0

!Column 1: Below the group G is given the topological nature of the orbit space R"G (Susp. is for suspension). " " " ; Column 2: Stabilizer of the stratum. CQ( j ) CQ( j) except when j 0; then it is CQ(n( k). #Column 3: c is for closed stratum and g for generic stratum. $Column 4: Dimension of the stratum. %Column 5: The stratum is a semi-algebraic set; its de"ning polynomial equations and inequalities are given here. &Column 6: Number of orbits in the stratum. 'Column 7: Geometry of each orbit in the stratum. )Column 8: Critical orbits have a `cra in this column. intersections with the stabilizers of O(3). So, for these group G, we can directly make the list of strata, and give the invariant equations (for closed strata) and/or inequalities which de"ne them. We are interested only about the orbits and strata on R. This information is given in Table 8. 194 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

First, let us write again all equations on the C invariants when they are restricted on the manifold R de"ned by i"1 and o"0. This corresponds to the study of the three-dimensional R" orbifold C. Eqs. (60)}(62) and (64) become k4 #m (1 ) , (88) l4 !m (1 ) , (89) !4p4 (90) and 044p"(1!m)!2(1!m)k!2(1#m)l41 . (91) This last equation shows explicitly that p is determined up to a sign by m, k, l. From the two previous equations we deduce k"$1Nm"1, l"0"p , (92) l"$1Nm"!1, k"0"p . (93) R This means that for a given n( there are four points A, B, C, D of "xed by C(n( ): A: j"n( , k"0, ,k"1, m"1, l"0 , (94) B: j"!n( , k"0, ,k"!1, m"1, l"0 , (95) C: k"n( , j"0, , l"1, m"!1, k"0 , (96) D: j"0, k"!n( , ,l"!1, m"!1, k"0 . (97) ; On S S the four points correspond to the pairs of poles: NN, SS, NS, SN, respectively. As we shall see any function invariant by one of the "ve one-dimensional Lie subgroups of O(3), has an extremum on each of these four points of R. Similarly, we have two circles on R: C " ) " Nm" k"l" : k 0, n j 0 1, 0 , (98) C " ) " Nm"! k"l" : j 0, n k 0 1, 0 . (99) ; They correspond, on the two equators of S S, to pairs of identical points, diametrically opposed C C points, respectively. Each of these two circles , form a critical orbit of D or DF. Finally, we should consider the particular cases where one of the vectors j, k vanishes, i.e. m"$1. When m"1 then k"0; since any axial vector j is invariant by the symmetry through the origin (while any non-trivial polar vector changes of sign), m"1 contains a stratum with stabilizer L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 195

CG (the group generated by the symmetry through the origin) for the groups CF and DF. When m"!1 then j"0 and the polar vector k is invariant by the symmetry through any plane containing it. The symmetry through the plane which contains also n( belongs to a stabilizer of O ; CT and DF and this is still true when j 0 and collinear to n( k, see Table 8. The "ve orbit spaces R"G, are of dimension three. We will "rst make a direct study of their R" & topology. We shall "rst prove that topologically C S. Then, as suggested by the } signs in k l p the columns of , , of Table 6, the orbit spaces of CT, CF, D are obtained by identifying the points symmetrical through, respectively, a three-, two-, one-dimensional linear manifold containing the center of the sphere S. In the four-dimensional space of parameters m, k, l, p, when !1(m(1, Eq. (91) shows that the R" & m" & section of C S by a hyper-plane constant is an ellipsoid S: !1(m(1, p# !m k# #m l" !m 2 (1 ) (1 ) (1 ) . (100) Indeed this is true because each coe$cient (function of m) is strictly positive. Furthermore, this equation implies Eqs. (88)}(90). We are left to study the particularly two cases: m"$1. m"1Nl"p"0 (101) and from Eq. (88): !14k41. Similarly, m"!1Nk"p"0 (102) and from Eq. (89): !14l41. These two segments of line are the limits when mP$1of R the ellipsoids de"ned by Eq. (100). This proves that the orbifold for the C action on is the 3D-sphere R" & C S (103) with four marked points. Its four points A, B, C, D represent four critical orbits of one point each, and the complement is the image of the generic stratum. Remark that in the invariant space of ` a p m k l R" p" orthogonal coordinates , , , , the orbit space C has three symmetry hyper-planes 0, k"0, l"0, mutually orthogonal. R" R" Table 7 shows that CT is obtained from C by identifying its points of coordinates m p l $k R" k5 , , , . We can represent it by the intersection of CT with the closed half-space 0. This R" orbit space CT is therefore topologically equivalent to a hemisphere, i.e. a ball B: R" 5 k5 "R" & C ( 0) CT B . (104) R" k" This is also the topological nature of the projection of CT on its symmetry hyper-plane 0; indeed from (1!m)k50 we deduce from Eq. (91) that p#m# #m l4 2 (1 ) , (105)

Some general results are known on the topology of three-manifolds (a short for three-dimensional manifolds): (Fomenko, 1983; Thurston, 1969). They may be used to "nd the topology of the orbifolds on the basis of the geometrical representation (see Appendix A). 196 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 but one must remark that there is a unique point of this projection which does not correspond to R" m" p"k"l" a unique point of the orbit space CT: indeed 1, 0 if the projection of the whole m" p"l" ! 4k4 R" segment 1, 0, 1 1. To summarize, Table 8 shows that the orbit space CT contains four strata represented by

(1) two points C, D de"ned in Eqs. (97) and (98): each corresponds to a one point CT-orbit; m" "k p" "l + , (2) one point X: 1 , 0 , representing a two point C-orbit, A, B (see Eqs. (95) and (96)); R & (3) the boundary ( B S) minus the two points C, D representing the CQ stratum; (4) the interior of B minus the point X representing the generic stratum. R" & R" We now prove CF S. Indeed, from Table 7 in the parameter space one obtains CF from R" m k m k C by identifying the points symmetrical through the two-plane P( , ) of the coordinates , (it l" p" is the intersection of the two symmetry hyper-planes 0, 0). Consider the sphere S: m#k#l#p" R" 1 topologically equivalent to C. By identi"cation of its points in the pair " m k $l $p m k &R" s! ( , , , (which are not in P( , ))), one obtains a topological space CF. The m k m#k4 R " 5 m k projection on the two-plane P( , ) is the disk B: 1. Note that B S P( , ) while m k C each point p( , ) of the interior of B is the projection of a circle N of equation l#p"1!m!k in the two-plane perpendicular to P(m, k). By the identi"cation of its points C C symmetric through its center the circle N is transformed into a circle N; this holds for each p in the interior of B. Correspondingly this point identi"cation transforms S into S. R" R" We note from Table 7 that one transforms CF into DF by identifying the points of k R" R" opposite coordinate. As for the transformation of C into CT study above, we obtain that R" & DF B. R" R" We are left with the study of D. It is obtained from C by identifying the points in each pair " m $k $l $p s! ( , , , ), i.e. the points symmetric through the intersection axis of the three symmetry hyper-planes k"0, l"0, p"0. This axis is the normal to the hyper-plane H: m"0. R" & $ We remark that C S is topologically equivalent to the double cone of vertices 1, 0, 0, 0 " R" 5 & and basis B ( C) H S; this is also called a suspension of B. The point identi"cation transforms B into B&RP. So: R" & D suspension (RP ) . (106) Table 8 gives in its "rst column the topological nature of the "ve orbit spaces studied in this section. The geometrical representation of the orbit spaces for one-dimensional Lie subgroups studied in this section is discussed in more detail in Appendix A.

2.7. Invariants of xnite subgroups of O(3) acting on R

The construction of invariant functions on the R manifold is based on the preliminary construction of the integrity basis on the six-dimensional space where the action of the symmetry group of the problem is linear. The Molien function and the invariants themselves for the six-dimensional space xy or kj may be found from known expressions for Molien functions and integrity bases for irreducible representations. Next step includes the restriction of the polynomial algebra on the sub-manifold R of the 6D-space. The general procedure of the restriction of the polynomial ring L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 197 de"ned on the manifold to the sub-manifold was outlined in Section 5.2 of Chapter I. The sub-manifold R is de"ned in the six-dimensional space xy by the polynomial equations " " x , y . If the point symmetry group does not include improper rotations (inversion or re#ections) these polynomials may always be considered as denominator invariants. In such a case we just eliminate them from the integrity basis constructed for the 6D-space and the resulting integrity basis gives the basis for the 4D sub-manifold. For point groups which are not the subgroups of SO(3) we start with the consideration of the similar problem for the proper rotation subgroup and after that take into account the e!ect of improper symmetry elements working directly on the 4D-sub-manifold R. The detailed realization of this procedure is given in Appendix B on several examples. Below we just summarize the results for the case of the C point group as the invariance group of the problem. Let us take x and y to coincide with the C-axis. In such a case x, x, y, y transform according to the anti-symmetric representation B and x, y according to the totally symmetric representation A of the C point group. We are interested in the module of invariant functions de"ned on the sub-manifold R given by two polynomial equations: # # " x x x , (107) # # " y y y . (108) After the restriction on R the Molien function for invariants has the form 1#6j#j M "R" . (109) ! (1!j)(1!j) The explicit form of the module of invariant functions on R may be easily given as well:

!" " ⅷ u u u u u u u P R P[x, x, y, y] (1, , , , , , , ) , (110) where u " u " u " u " xy, xy, xy, xy , (111) u " u " u " xx, yy, xxyy . (112) More condensed notation for the set of numerator invariants may be used

!" " ⅷ P R P[x, x, y, y] ((1, xx)(1, yy), (x, x)(y, y)) . (113) Instead of listing explicitly [as in Eq. (110)] all eight numerator invariants, we show in Eq. (113) that they may be reconstructed as products of slightly simpler monomials. One should remark that the choice of denominator and numerator invariants is not unique and the choice proposed here is just one of the possible ones. Much more information may be found in Appendix B.

2.8. Orbits, strata and orbit spaces of xnite subgroups of O(3) acting on R

The strata for any "nite subgroup G of O(3) for its six-dimensional representation (kj which is the sum of polar and axial vector representations) follow immediately from the well-known results 198 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 9 R Strata and orbits in the action of CT on

Stabilizer gc! dim." Number# Nature$

CT c 02 1 C c 01 2 T R CQ 2 2 B R CQ 2 2 1 g 4 R 4

!g is for generic stratum; c is for closed stratum. "Dimension of the stratum. #Number of orbits in the stratum. RL stands for the n-dimensional stratum. $Number of points in each orbit. for polar and axial vector representations. We list below strata for group actions on R manifold for several point groups which are the most interesting from the point of view of possible applications. ¹ These are examples of molecules with the CT, DF, and B point symmetry. Similar tables for all other possible group symmetries are given in Appendix C. Table 9 listed in this subsection show in column 1, the stabilizer of the stratum. In column 2 closed and generic strata are indicated. We remark once more that some strata are neither closed nor generic. In column 3 the dimension of the stratum is given. For the "nite group action on R the dimension of the generic stratum is always 4. Column 4 gives the number of orbits in the stratum. If the dimension of the stratum is zero the number of orbits in the stratum is "nite. If the dimension of the stratum is positive the number of orbits in the stratum is in"nite. We denote it as RL with n equal to the dimension of the stratum. Column 5 shows the number of points in each orbit. For "nite group actions on R this number is always "nite and for the generic stratum it is equal to the number of elements of the group.

2.9. Orbits, strata and orbit spaces for T-dependent subgroups of O(3)'T

We have shown that there are 16 one-dimensional Lie subgroups of the complete symmetry group O(3)'T of the problem. There are naturally many "nite subgroups which can be obtained in a way similar to our construction of one-dimensional subgroups. Extension of point subgroups of O(3) to points subgroups of O(3)'T is analogous in some sense to the construction of antisymmetry (Shubnikov, 1951) or magnetic symmetry groups (color groups) well known in solid-state physics (Hamermesh, 1964; Shubnikov and Belov, 1964). We give in Tables 10, 11 and 12 the analysis of the orbits, strata and orbit spaces for some groups including time reversal. We have chosen those groups which have simple physical realization as symmetry groups of the hydrogen atom in the presence of di!erent external "elds. This simple quantum system enables one to study quite a lot of di!erent invariance symmetry groups. Let us consider several physical situations together with more or less detailed description of corresponding symmetry groups:

E Hydrogen atom without any external "elds. The symmetry group includes the full orthogonal group O(3) and the time reversal. We can write the group as the direct product O(3)'T where L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 199

Table 10 R Strata and orbits in the action of DF on

Stabilizer gc! dim." Number# Nature$

CF c 01 2 CT c 01 2 CT c 02 3 C c 01 6 R CF 2 6 R CQ 2 6 1 g 4 R 12

!g is for generic stratum; c is for closed stratum. "Dimension of the stratum. #Number of orbits in the stratum. RL stands for the n-dimensional stratum. $Number of points in each orbit.

Table 11 ¹ R Strata and orbits in the action of B on

Stabilizer gc! dim." Number# Nature$

CT c 02 4 C c 01 8 S c 01 6 CT c 01 6 R CQ 2 12 1 g 4 R 24

!g is for generic stratum; c is for closed stratum. "Dimension of the stratum. #Number of orbits in the stratum. RL stands for the n-dimensional stratum. $Number of points in each orbit.

Table 12 ' R Invariant manifolds for the CF TT symmetry group action on (Hydrogen atom in magnetic "eld.)

Stabilizer dim. Equations

' k"$ CF TT 0 1 ' l" C TT 0 1 ' ! l" m" k! [CF TT] 2 0; 2 1 ' " l" m" [CG TT] 2 0; 1 p" TT 3 0 l" T 3 0 R C 4 ! p This group includes four elements: E, F, TT, T. " This group includes four elements: E, i } inversion, TT, T. 200 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

the two element group Z(T) is formed by the identity E and time reversal T. In fact, the dynamic symmetry group for hydrogen atom in the absence of external "elds is higher. We use this higher group to introduce the approximate quantum number corresponding to an individual n-shell of the Rydberg atom even in the case of external "elds (assuming that the splitting of the n-shell is small as compared to the inter-n-shell gap). Di!erent external "elds break this symmetry till various subgroups. E Hydrogen atom in the presence of constant magnetic "eld. This case corresponds to the Zeeman e!ect for the hydrogen atom. CF group is the invariance group of spatial geometric transforma- ' tions. The complete symmetry group including time-reversal operations is CF TT. It is the "+ , semi-direct product of CF and two element group TT E, TT including two elements: the identity E and the product of time reversal and space re#ection TT in the plane of symmetry axis. ' + , The group CF TT includes the following classes of conjugated elements: E is the identity, +p , + , F the re#ection in the plane orthogonal to the magnetic "eld axis, R( the rotation on angle + , + , around magnetic "eld, S( the rotation-re#ection on angle around magnetic "eld, TT( ) + , the time reversal followed by re#ection in a plane passing through the magnetic "eld, T( ) the product of time reversal and rotation by n around an axis orthogonal to the "eld axis. An operation T can be equivalently described as a product of TT and S( operations. Table 12 gives ' R invariant manifolds for the CF TT group action on . We give here invariant manifolds instead of strata to simplify the de"ning equations. E Hydrogen atom in the presence of constant electric "eld. This is the case of the Stark e!ect for the hydrogen atom. CT group is the invariance group of spatial geometric transformations. The invariance under the time reversal takes place in the absence of a magnetic "eld as well. ' Therefore, the complete invariance group can be written as CT T. This group includes the following symmetry operations: E is the identity, T the time reversal, R( the rotation on angle p around electric "eld, ( the re#ection in plane passing through the "eld axis, TR( the time

reversal followed by rotation, TQ( ) the time reversal followed by re#ection. The system of ' R invariant manifolds of the CT T action on is given in Table 14. E Collinear electric and magnetic "elds. The symmetry group includes the C subgroup of rotations around the common direction of the electric and magnetic "eld. The complete ' invariance group can be written as C TT. Naturally, this group is the subgroup of the symmetry group appropriate for the case of only electric or only magnetic "eld. The system of ' R invariant manifolds of the C TT action on is given in Table 13. E Orthogonal electric and magnetic "elds. The symmetry group is "nite in this case and includes only four symmetry elements: identity E,re#ection in the plane orthogonal to the magnetic "eld

Table 13 ' R Invariant manifolds for the C TT symmetry group action on (Hydrogen atom in parallel magnetic and electric "elds.)

Stabilizer dim. Equations

' l" k" C TT 0 1; 1 p" TT 3 0 R C 4 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 201

Table 14 ' R Invariant manifolds for CT T symmetry group action on (Hydrogen atom in electric "eld.)

Stabilizer dim. Equations

' l"$ CT T 0 1 ' k" C TT 0 1 ' ! k" m" ! l [CQ TQY] 2 0; 1 2 ' k" m"! [CQ T]2 0; 1 k" CQ 3 0 p" TT 3 0 R C 4

!This group includes four elements: E is the identity, p the re#ection in the plane including the "eld axis and a vector orthogonal to the vector L, TQY the time reversal followed by re#ection in the plane formed by "eld axis and the vector L, p T the time reversal followed by rotation over around "eld axis.

p p p F, product T T of time reversal T and re#ection T in plane formed by two orthogonal "elds, and the symmetry operation TC which is the product of the time reversal T and the C rotation around the electric "eld direction. We will use the notation G for this symmetry group. This group has three subgroups of order two which will be useful below. The group p TT includes as non-trivial symmetry operation the product T T of time reversal T and p re#ection T in plane formed by two orthogonal "elds. The group CQ includes the re#ection in the plane orthogonal to the magnetic "eld (this plane includes the electric "eld vector). The group T includes as a non-trivial symmetry operation, the product of time reversal and the C rotation around the electric "eld direction. E Generic non-orthogonal non-collinear con"guration of two "elds. The symmetry group TQ has order two and includes one non-trivial symmetry operation Tp which is the product of the time reversal and the re#ection in plane formed by the two "elds. TQ is the subgroup of the symmetry ' group G for orthogonal "elds and the subgroup of the symmetry group CT TQ for collinear "elds. It is the only common subgroup for these two important limiting cases of parallel and orthogonal "elds. E To complete the list of interesting symmetry groups we add here the symmetry group of the quadratic Zeeman e!ect or of Rydberg states of homonuclear diatomic molecules. This is the ' ' DF T group which is the maximal one-dimensional subgroup of the O(3) T symmetry group. The set of invariant manifolds for this group is given in Table 15.

Inclusion of additional symmetry operations enables simpli"cations in such physically important cases as the Zeeman e!ect, Stark e!ect, hydrogen atom in parallel "elds or quadratic Zeeman e!ect the integrity basis and the geometrical representation of the orbifold. Going from the C to ' ' ' ' C TQ, from CT to CT T, from CF to CF TQ, and from DF to DF T leads to an integrity basis consisting of only basic `denominatora invariants. The space of orbits for extended groups possesses richer strati"cation (see Tables 12}15) but the geometrical form of the orbifold becomes simpler (see Figs. 1}3) as compared to the geometrical form of the orbifold for purely geometrical point group symmetry studied in Appendix A. 202 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 15 ' R Invariant manifolds for the DF T symmetry group action on (Complete symmetry group of the quadratic Zeeman e!ect.)

Stabilizer dim. Equations

' k" CF TQ 0 1 ' l" CT T 0 1 ' ! m" k"l" CF TQ 1 1, 0 ' m"! k"l" CT T 1 1, 0 ' " #m" k CQ(n) TQ 212 ' m" CG TQ 2 1 ' k" m"! [CQ T]2 0; 1 ' # k" m" ! l [CQ TQY] 2 0; 1 2 ' $ k" l" CQ( j) T 2 0, 0 k" CQ 3 0 p" T 3 0 l" TQ 3 0 R C 4 ! CF subgroup includes the symmetry plane orthogonal to the vector L; TQ is the time reversal followed by re#ection in plane formed by L and magnetic "eld. " CQ is the re#ection in plane orthogonal to the C axis; TQ is the time reversal followed by re#ection in plane of K and L. #This group includes four elements: E is the identity, p the re#ection in the plane including the "eld axis and a vector orthogonal to the vector L, TQY is the time reversal followed by re#ection in the plane formed by "eld axis and the vector p L, T the time reversal followed by rotation over around "eld axis. $ T is the time reversal followed by C rotation around K.

' R Fig. 1. Space of orbits for the CF TQ action on . (H atom in the presence of magnetic "eld.)

' R Fig. 2. Space of orbits for the C TQ action on . (H atom in the presence of parallel magnetic and electric "elds.) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 203

' R Fig. 3. Space of orbits for the C TQ action on . (H atom in the presence of electric "eld.)

' k Fig. 4. Three slices of DF T orbifolds for di!erent values. Di!erent strata are denoted by letters as follows: ! ' ! ' ! ' ! ' ! ' ! ' ! ! ' ! a CF TQ; b CQ( j) T; c CT T; d [CQ T]; e CT T; f [CQ TQY]; g CQ; h CG TQ; i T; ! ' ! ! ! ' j CQ(n) TQ; k TQ; l C; m CF TQ.De"ning equation for the strata are given in Table 15.

R ' To see better the strati"cation of the under the action of DF T symmetry group which includes 13 di!erent strata we give in Fig. 4 the schematic representation of the space of orbits through three sections corresponding to k"0, 0(k(1, and k"1. Similar extension of the geometrical point group symmetry can be done in cases of "nite symmetry groups; e.g. see the example of the G symmetry group for the hydrogen atom in two orthogonal "elds mentioned a little earlier in this section is analyzed in more details in Section 4.4 and in a separate publication (Sadovskii and Zhilinskii, 1998).

3. Construction and analysis of Rydberg Hamiltonians

3.1. Ewective Hamiltonians

Let us discuss now the simplest e!ective Hamiltonians invariant with respect to subgroups of the O(4) group and their relation with invariant functions de"ned on the R manifold. We start with the Hamiltonian for a hydrogen atom which is O(4) invariant. It was introduced in Eq. (1) and rewritten in terms of the angular momentum vector J and the transformed Laplace}Runge}Lenz vector, K Eq. (3) which satis"ed the commutation relations (4)}(6) and two relations (7) and (8). Further we want to study a slightly perturbed Hamiltonian which conserves essentially the presence of n multiplets typical for the hydrogen atom (this is precisely the situation with Rydberg atoms and molecules in the limit of small splitting of n multiplets). An e!ective Hamiltonian for a given n shell may be written as a phenomenological e!ective Hamiltonian constructed from J and K operators. Relations (7) and (8) are important to reach the correspondence between the quantum 204 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 operators J and K and their classical analogs, because the classical variables j, k satisfy the normalization condition (40), (39), (42) which is independent of the particular realization of the quantum operator for any concrete n shell. As a consequence, the correspondence between quantum operators and classical dynamic variables has the form %( ! J? n 1j? , (114) %( ! K? n 1k? . (115) Using the integrity basis for the classical representation we can now construct easily the basic polynomial invariants formed by the quantum operators and to indicate their classical analogs J!K"(n!1)m , (116) "( ! k JX n 1 , (117) "( ! l KX n 1 . (118) For higher-order invariants one needs to make the anti-symmetrization of the operators which do not commute. For example, the operators representing the auxiliary numerator invariants have the form + ,!+ ," ! p JV, KW JW, KV (n 1) , (119) + + ,!+ , ," ! lp KX,( JV, KW JW, KV ) (n 1) , (120) + ," ! kl JX, KX (n 1) , (121) + + ,!+ , ," ! kp JX,( JV, KW JW, KV ) (n 1) , (122) where + ," # A, B (AB BA) . (123) Sometimes it is useful to work in the x, y representation (40), (39) instead of j, k. We denote the corresponding quantum operators by " ! J (J K)/2 , (124) " # J (J K)/2 . (125) The Hilbert space of wave functions associated with the n multiplet may in such a case be formed by the basis " " " ! 2 J J (n 1)/2; M, M , (126) "! ! ! MG (n 1)/2,2,(n 1)/2 . (127) The correspondence between quantum operators J, J and classical dynamic variables follows directly from Eqs. (116)}(122): ) " ! m J J (n 1) /4 , (128) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 205

# " ( ! k JX JX 2 n 1 , (129) ! "( ! l JX JX n 1 , (130) ! " ! p JWJV JVJW (n 1) /2 , (131) + ! ! ," ! lp (JX JX), (JWJV JVJW) (n 1) , (132) ! " ! kl JX JX (n 1) , (133) + # ! ," ! kp (JX JX), (JWJV JVJW) (n 1) . (134) There are two ways leading to the construction of the e!ective operators. One is purely a phenomenological approach based on the utilization of all operators allowed by symmetry with their coe$cients being the adjustable parameters of the model. The other one uses the transformation of the initial operator to the e!ective one by applying some kind of perturbation theory or contact transformation, etc. This second procedure results in the e!ective operator with "xed coe$cients which depend on the initial Hamiltonian. Both approaches result in the Hamiltonian which may be written in terms of integrity basis. We remind that as soon as invariant polynomials are known, an arbitrary Hamiltonian may be written in the form of expansions (properly symmetrized to take into h account the non-commutativity of quantum operators) in terms of denominator invariants G and u numerator invariants Q H" + + C u hL hL 2hLI . (135) L L 2 LI_Q Q I L L 2 LI Q An explicit form of invariants for all continuous subgroups of O(3) was found in Section 2.4 (see Table 7) and for some point groups in Section 2.7 and in Appendix B. The purely phenomenological way to represent e!ective Hamiltonians supposes that all the C coe$cients are L L 2 LI_Q adjustable parameters, whereas perturbation treatment gives the C coe$cients in L L 2 LI_Q Eq. (135) as explicit functions of the parameters of the initial Hamiltonian. We will start by analyzing "rst the phenomenological approach to the construction of e!ective Hamiltonians taking into account their symmetry properties. The main idea of our approach is to introduce along with the phenomenological construction some kind of complexity classi"cation of e!ective Hamiltonians which is based on the quantitative measure } the number of orbits of extrema. We also call them the stationary orbits (points, or "nite number of points, or manifolds).

3.2. Qualitative description of ewective Hamiltonians invariant with respect to continuous subgroups of O(3)

We discuss here the qualitative classi"cation of e!ective Hamiltonians invariant with respect to di!erent symmetry groups. By qualitatively di!erent Hamiltonians we mean those operators which are characterized by di!erent sets of stationary orbits with each stationary orbit characterized by its stabilizer and Morse index. The number of stationary orbits may be used as a measure of the complexity of the Hamiltonian. The zero level of complexity corresponds to a set of Hamiltonians with the minimal possible 206 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 number of stationary orbits (compatible with the topological structure and the symmetry group action according to the equivariant Morse}Bott theory). We will list below qualitatively di!erent Morse}Bott Hamiltonians of some low level of complexity, whereas in Appendix D more general results are outlined.

3.2.1. O(3) invariant Hamiltonian Any O(3) invariant Hamiltonian may be written in the form

H"H(m), (136) where H(m) is an arbitrary (su$ciently good) function of one variable which is de"ned for !14m41. m is the only invariant polynomial which is not constant on the R manifold (see Section 2.3). For quantum operators instead of m we should use equivalent operators given by Eq. (116) (or (128)) in the J, K (or J, J) representation. So any e!ective Hamiltonian may be written as a power series in m or more generally as an arbitrary function of m as in Eq. (136). Any O(3) invariant classical Hamiltonian possesses two critical manifolds due to the presence of the two critical orbits. These two critical orbits have very simple physical meaning. CF critical orbit corresponds on the orbifold to the point m"1(j ) j"1, k ) k"0), i.e. the quantum state localized near this point has maximal (possible for a given n) value of the orbital momentum l"n!1. Another critical orbit corresponds to the point m"!1& ( j ) j"0, k ) k"1) i.e. for the quantum problem the quantum state localized in the phase space near this point has minimum possible value of the orbital momentum l"0. More complicated Hamiltonians can be classi"ed according to the number of stationary orbits (the RP manifolds) which belong to the generic CQ stratum. Qualitatively di!erent generic O(3) invariant Hamiltonians are given in Table 16. It is clear that any O(3) invariant Hamiltonian results in a system of energy levels which is completely characterized by one quantum number l, the weight of the irreducible representation of the group O(3). This one regular sequence may be rather complicated in the general case but for the simplest Morse}Bott-type Hamiltonian (p"0 in Table 16) the sequence should be monotonic. It is reasonable to assume that the number of extrema of the H"H(m) function in a general case is much smaller than the number n imposing limit on possible l values, l4n!1. In such a case the energy spectrum explicitly shows the regular behavior. In contrast, if the number of extrema on H(m) is larger or of the same order as n (assuming n is large), the energy spectrum may seem to be erratic. m" & ) " ) " Near an extremum, corresponding to the CF critical orbit ( 1 ( j j 1, k k 0)) for any (su$ciently good) Hamiltonian, there is a system of regular energy levels,

E(l)"const. l(l#1) (137) characterized by l"(n!1), (n!2),2, whereas near an extremum, corresponding to the m"! & ) " ) " CT critical orbit ( 1 ( j j 0, k k 1)) there is a regular system of energy levels, E(l)"const. l(l#1) (138) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 207

Table 16 Classi"cation of qualitatively di!erent O(3) invariant Hamiltonians due to the complexity level

! Level of CT CF N N complexity" stratum stratum with index 0 with index 1

0 max min no no 0 min max no no 1 max max no 1 1 min min 1 no 2 max min 1 1 2 min max 1 1 2p max min pp 2p min max pp 2p#1 max max pp#1 2p#1 min min p#1 p

! N is the number of stationary RP manifolds on CQ stratum. "p is a non-negative integer.

" which is characterized by l 0, 1, 2,2 . Near an extremum corresponding to generic CQ orbit the system of energy levels may be described as a bent series of the form " # ! # E(l) const.[l(l 1) l(l 1)] (139) with the l value varying around some generally non-integer l value.

3.2.2. C invariant Hamiltonian The C invariant phenomenological Hamiltonian may be written as H "+C mL kLlL #+CN pmL kLlL (140) ! L L L L L L with m, k, l, p given in Eqs. (116)}(119), or in (128)}(131) with all necessary symmetrization. Any Hamiltonian (140) has four critical orbits (stationary points A, B, C, D, see Table 8 and Eq. (101)). There is only one type of the simplest C invariant Hamiltonian. It is characterized by the absence of critical manifolds of non-zero dimension (see Table 17). The simplest classical Hamiltonian possesses one minimum and one maximum on two critical orbits and two saddle points (with Morse index 2) on the other pairs of critical orbits. The description of qualitatively di!erent Hamiltonians of the "rst and second level of complexity is given in Table 17 as well. We use in Table 17 Morse counting polynomials to represent the system of stationary orbits for a given set of the qualitatively similar Hamiltonians. To reduce the number of di!erent classes we neglect in Table 17 all Hamiltonians which may be obtained by a simple transformation (H) % (!H). It is clear that the inversion of the sign of the Hamiltonian is associated with the transformation of the Morse indices of stationary points (k % (4!k)) and of one-dimensional stationary manifolds (k % (3!k)). For several classes both Hamiltonians H and (!H) are described by the same Morse counting polynomial (we add to the level of complexity the index 208 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 17 Classi"cation of qualitatively di!erent C invariant Hamiltonians ! " Level of C C complexity# stratum stratum

0! 1#2t#t no 11#3t t 11#t#2t t 2! 1#2t#t t#t 21#2t#t 1#t 2! 4t 1#t 2! 2#2t t#t 21#3t 2t 22#2t t#t

!Morse counting polynomial is given in this column to characterize the system of stationary orbits. " Morse counting polynomial for the S stationary orbits is given. #The superscript $ in the "rst column indicates those classes which are invariant under the sign inversion of the Hamiltonian.

$ in such a case, see Table 17). For all other classes the change of the sign of the Hamiltonian will modify the redistribution of the stationary orbits over Morse indices. For example, there are two extra classes of the Hamiltonians of the "rst level of complexity and three additional classes for the second level of complexity, but we omit them from the table. Near minimum and maximum the simplest Morse-type C-invariant Hamiltonian (zero level of complexity) may be approximated as 2D-isotropic harmonic oscillator. So it is characterized by a qualitative energy level pattern shown in Fig. 5. Lower and upper parts of the multiplet are formed by a sequence of polyads typical for 2D-isotropic harmonic oscillator.

3.2.3. CT invariant Hamiltonian The CT invariant phenomenological Hamiltonian may be written as

H "+C mL (k)LlL#+CN pmL(k)LlL (141) !T L L L L L L with m,(k), l as denominator invariants and p as one numerator invariant given in Eqs. (116)}(119) or in Eqs. (128)}(131) with all necessary symmetrization. Any Hamiltonian (141) has three critical orbits: one C orbit consisting of two stationary points (A, B) and two CT orbits consisting of one stationary point (C, D) each (stationary points A, B, C, D, see Table 8 and Eq. (101)). The most natural physical application of the CT symmetry concerns the Stark e!ect. In this case the symmetry group should be extended to include the time reversal as a symmetry operation. Taking into account the invariance of m, k, l with respect to time reversal and the alternation of the sign of p under the same operation the general form of Hamiltonian (141) becomes simpler. In the case of Stark e!ect the e!ective Hamiltonian (141) is independent of p. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 209

Fig. 5. Qualitative energy level pattern for e!ective C-invariant Hamiltonians of the simplest Morse}Bott type. In the neighborhood above the minimum and below the maximum the energy level pattern is similar to that of a two- dimensional isotropic harmonic oscillator.

3.2.4. CF invariant Hamiltonian The CF invariant phenomenological Hamiltonian may be written as H "+C mL kL (l)L#+CJN (lp)mL kL(l)L (142) !F L L L L L L with m, k,(l) as denominator invariants and (lp) as one numerator invariant as given in Eqs. (116)}(118), (120), or in (128)}(130), (132) with all necessary symmetrization. Any Hamiltonian (142) has three critical orbits: one C orbit consisting of two stationary points (C, D) and two CF orbits consisting of one stationary point (A, B) each (stationary points A, B, C, D, see Table 8 and Eq. (101)). ' In the case of the Zeeman e!ect after extending the symmetry group from CF to CF TQ (see Section 2.9) the general form of the e!ective Hamiltonian becomes independent of (lp).

3.2.5. D invariant Hamiltonian The D invariant phenomenological Hamiltonian may be written as H "+C mL(k)L(l)L#+CJN (lp)mL(k)L(l)L " L L L L L L # +CIN (kp)mL (k)L (l)L#+CJI (lk)mL(k)L (l)L (143) L L L L L L with m,(k), (l) as denominator invariants and (lp), (kp), and (lk) as three numerator invariants given in Eqs. (120)}(122), or (132)}(134) with all necessary symmetrization. Any Hamiltonian (143) has four critical orbits: two C orbits consisting each of two stationary points (A, B and C, D) and C C two C orbits (both being S stationary manifold , ) (stationary points A, B, C, D, and C C stationary circles , see Table 8 and Eqs. (101) and (102)).

3.2.6. DF invariant Hamiltonian The DF invariant phenomenological Hamiltonian may be written as H "+C mL(k)L (l)L #+CJN (lp)mL(k)L (l)L (144) "F L L L L L L 210 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 with m,(k), (l) as denominator invariants and (lp) as one numerator invariant given in Eqs. (120) or (132) with all necessary symmetrization. Any Hamiltonian (144) has four critical orbits: one CF orbit consisting of two stationary points (A, B), one CT orbit consisting of two stationary C points (C, D), one CF orbit consisting of a S stationary manifold , and one CT orbit consisting C C C of a S stationary manifold (stationary points A, B, C, D, and stationary circles , are given in Table 8 and in Eqs. (101) and (102)).

3.3. Qualitative description of ewective Hamiltonians invariant with respect to xnite subgroups of O(3)

We characterize below in Tables 18 and 19 the simplest Morse-type functions for all "nite point group symmetries. For "nite groups, orbits include a "nite number of isolated points and any Morse-type function possesses only isolated extrema. We can work within the initial Morse theory (without extension to the Morse}Bott approach). For each point group the possible sets of % stationary points are indicated up to trivial change of Morse indices cI c\I. Stationary points for each group are split into columns according to their Morse index k. The number of stationary points cI with the Morse index k within each orbit and their symmetry types are given. If there are several orbits of stationary points with the same Morse index it is indicated explicitly as a sum. It should be noted that for many point groups the stationary points are situated on zero-dimensional strata only and their positions are "xed (i.e. only critical orbits are present) (see Section 2.8 and Appendix C for the description of all strata for the "nite group action on R). For some low , symmetry groups (C, CQ, S CG) there are no orbits isolated within the stratum and the positions of stationary points are not "xed by symmetry. At the same time the presence of a closed 2D-stratum for the CQ and S groups indicates that a number of stationary points should lie on the closed stratum. For these groups it is necessary to verify further Morse inequalities for the restriction of the complete initial function on the close stratum. For OF and >F groups, only part of the simplest Morse functions is given in Table 19. To obtain the complete list it is necessary to interchange in all possible manners the CLT and CLF critical orbits. Near minima or maxima any Hamiltonian may be approximately represented as 2D-harmonic oscillator. Taking into account the presence of several, say k (k is the dimension of the orbit of stationary points) equivalent by symmetry minima the model problem appropriate for the description of internal dynamics near the extrema is the motion of a particle in a 2D-potential with k equivalent extrema (the potential can be anisotropic or isotropic and slightly anharmonic near each extremum) assuming small tunneling between di!erent extrema. We can introduce three characteristic parameters for such a problem: (i) anisotropy of the 2D-harmonic oscillator, (ii) anharmonicity of the 2D-oscillator, and (iii) splitting due to tunneling. Let us consider several simple limiting situations from the point of view of the energy level patterns for quantum problems near extrema. Let d!* be the anisotropy of the model Hamiltonian near an extremum d!*& "l !l " l #l h /( ) , (145) l " d where G, i 1, 2, are two harmonic frequencies of the model operator, be the characteristic energy splitting due to tunneling, and d!) be the anharmonicity correction. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 211

Table 18 Simplest Morse type functions de"ned on the R manifold in the presence of point group symmetry (lower symmetry groups)

Point group c (c) c (c) c c (c) c (c)

C 1(C) no 2(C) no 1(C) CQ 1(CQ) no 2(C) no 1(CQ) # 1(CQ) no 1(CQ) 1(CQ) no 1(CQ) S 1(S) no 2(C) no 1(S) # 1(S) no 1(S) 1(S) no 1(S) SL 1(SL) no 2(CL) no 1(SL) n52 # CL 1(CL) no 1(CL) 1(CL) no 1(CL) n52

CLF 1(CLF) no 2(CL) no 1(CLF) n52

CLT 1(CLT) no 2(CL) no 1(CLT) n52 ! # D 2(C)2C 2(C) 2(C)2(C)2(C) " # DL 2(CL) n(C) n(C) n(C) n(C)2(CL) 5 # n 32(CL) n(C) n(C) 2(CL) n(C) n(C) # n(C) n(C)2(CL) 2(CL) n(C) n(C) " # n 3, 5 only 2(CL) 2(CL) n(C) n(C) n(C) n(C) DLF (DLB)2(CLF) n(CT)2n(C) n(CT)2(CLT) n53, odd # DLF (DLB)2(CLF) n(CF) n(CT) n(CF) n(CT)2(CLT) 5 # n 2, even 2(CLF) n(CF) n(CT) n(CT) n(CF)2(CLT) # # 2(CLF) n(CT) n(CT) n(CF) n(CF)2(CLT) # 2(CLF) n(CT) n(CF) n(CF) n(CT)2(CLT) " # n 4 only 2(CF) 2(CT)4(CT)4(CF)4(CT)4(CF) " # n 4 only 2(CF) 2(CT)4(CT)4(CF)4(CF)4(CT) " # n 4 only 2(CF) 2(CT)4(CT)4(CT)4(CF)4(CF) " # n 4 only 2(CF) 2(CT)4(CF)4(CF)4(CT)4(CT) " # n 4 only 2(CF) 2(CT)4(CF)4(CT)4(CT)4(CF) " # n 4 only 2(CF) 2(CT)4(CF)4(CT)4(CF)4(CT) ! There are three di!erent C strata for the D group. Each stratum includes two orbits. To reach complete description of qualitatively di!erent Morse-type functions it is necessary to specify the stratum for all stationary points. " 5 There are two di!erent C strata for even n 4, whereas there is only one stratum for odd n. # Further speci"cation of strata is needed. There are three di!erent CT and three di!erent CF strata for the DF (DB) 5 group. There are two di!erent CT and two di!erent CF strata for the DLF (DLB) group for n 4, even. 212 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Table 19 Simplest Morse-type functions de"ned on the R manifold in the presence of point group symmetry (higher symmetry groups). To complete the list of Morse functions for OF and >F groups it is necessary to add other sets of stationary points resulting from di!erent permutations within pairs of CLT and CLF strata

Point group c (c) c (c) c c (c) c (c) ¹ # 4(C)6(C)4(C) 4(C)6(C)4(C) ¹ F 4(CF)6(CF)8(C)6(CT)4(CF) ¹ B 4(CT)6(S)8(C)6(CT)4(CT) # O 6(C) 12(C)6(C) 8(C) 12(C)8(C) # 6(C) 12(C)8(C) 8(C) 12(C)6(C) # 8(C) 12(C)6(C) 6(C) 12(C)8(C) # 6(C) 6(C) 12(C)8(C) 12(C)8(C) # OF 6(CF) 12(CF)6(CT) 8(CF) 12(CT)8(CT) ? # ( )6(CF) 12(CF)8(CF) 8(CT) 12(CT)6(CT) # 8(CF) 12(CF)6(CF) 6(CT) 12(CT)8(CT) # 6(CF) 6(CT) 12(CF)8(CF) 12(CT)8(CT) # > 12(C) 30(C) 12(C) 20(C) 30(C) 20(C) # 12(C) 30(C) 20(C) 20(C) 30(C) 12(C) # 20(C) 30(C) 12(C) 12(C) 30(C) 20(C) # 12(C) 12(C) 30(C) 20(C) 30(C) 20(C) # >F 12(CF) 30(CF) 12(CT) 20(CF) 30(CT) 20(CT) # 12(CF) 30(CF) 20(CF) 20(CT) 30(CT) 12(CT) # 20(CF) 30(CF) 12(CF) 12(CT) 30(CT) 20(CT) # 12(CF) 12(CT) 30(CF) 20(CF) 30(CT) 20(CT)

In some cases the harmonic approximation of the Hamiltonian near an extremal orbit should be isotropic due to symmetry (extremal orbit is a critical one with its stabilizer being a group with high symmetry). In such a case we have only the anharmonicity parameter d!) and the tunneling splitting parameter d. Typically d!) is su$ciently small but nevertheless at the same time d!)'d. In such a case the energy spectrum of the quantum problem near the extremum may be represented as a spectrum of a k-fold 2D-dimensional isotropic (slightly anharmonic) oscillator. It means that each polyad of a 2D-isotropic (slightly anharmonic) harmonic oscillator is replaced by a k-fold cluster of similar polyads. Typically for the model problem with an anisotropic harmonic oscillator we can neglect the anharmonicity correction and assume equally that d(d!*. In such a case the energy spectrum of the quantum problem near the minimum or maximum may be represented as a spectrum of a k-fold 2D anisotropic harmonic oscillator. It means that each non-degenerate level of the 2D-anisotropic harmonic oscillator is replaced by a k-fold cluster of energy levels. If both d and d!) are small and have the same order of magnitude, then polyads are formed by energy levels of each almost isotropic harmonic oscillator and the total energy level pattern is a system of k-fold clusters of vibrational polyads. Internal structure of each cluster depends on the relation between d and L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 213 d!) and may be complicated and vary signi"cantly from one polyad to another because the ratio between anisotropic and tunneling splittings changes with the vibrational energy increase.

4. Manifestation of qualitative e4ects in physical systems. Hydrogen atom in magnetic and electric 5eld

Hydrogen atom in magnetic and electric "elds gives us an opportunity to study the dependence of the Rydberg n multiplets on variable parameters such as strength of electric and magnetic "elds. To apply directly our approach we should remind once more that external "elds are su$ciently low to ensure the small splitting of the n multiplets with respect to the splitting between multiplets. In the case of non-zero electric "eld the ionization is always possible and consequently the strict non-relativistic Hamiltonian of the hydrogen atom in an external electric "eld possesses every- where the continuous spectrum. We will neglect here the e!ect of ionization and restrict ourselves with the analysis of e!ective Hamiltonians diagonal in n. To see the limits of the applicability of the present treatment in the case of a Rydberg atom in an external magnetic "eld, for example, we "nd here conditions which enable one to treat n-multiplets separately. It is su$cient that the diamagnetic shift which roughly varies as Gn (G is a magnetic "eld strength in atomic units, 1 a.u."2.35;10 T) remains smaller than the energy di!erence * " ! # & ( between two consecutive multiplets ( E n (n 1) 1/n ), i.e. Gn 1. This estimation shows that for a su$ciently low magnetic "eld G we can always "nd a relatively high n shell (to ensure the applicability of the classical approach) which can be analyzed in terms of e!ective Hamiltonians for an isolated n.

4.1. Diwerent xeld conxgurations and their symmetry

The qualitative description of the hydrogen atom in two external "elds depends strongly on the symmetry of the problem created by the two "elds. To specify the symmetry we de"ne "rst the absolute con"guration of two "elds in the laboratory "xed frame. It is given by two vectors: F is the electric "eld (polar) vector and G the magnetic "eld (axial) vector. Any two absolute con"gurations which can be mutually transformed by some rotation of the laboratory frame should be considered as physically equivalent and form one (relative) con"guration of two "elds. This means that the classi"cation of di!erent relative con"gurations of two "elds is equivalent to the classi"cation of orbits of the O(3) group on the six-dimensional space generated by one polar and one axial vector. Three parameters, F, G, and the scalar product (FG) with a natural relation between them (FG)4FG, are needed to characterize completely all relative con"gurations of two "elds. Thus, a one-to-one correspondence exists between di!erent relative con"gurations of two "elds and points of the "lled cone in the 3D-space (see Fig. 6). The cone shown in Fig. 6 is the orbifold of the O(3) group action on the space of absolute "eld con"gurations. Qualitatively di!erent relative con"gurations are characterized by di!erent symmetry groups. There are six di!erent types of "eld con"gurations which are listed in Table 20 together with their symmetry groups. Let us specify now the symmetry groups introduced in Table 20 to characterize di!erent types of "eld con"gurations. In each case the complete symmetry group includes two kinds of symmetry 214 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Fig. 6. Geometrical representations of di!erent "eld con"gurations. F is the electric "eld vector and G the magnetic "eld vector.

Table 20 Symmetry classi"cation of relative con"gurations of the electric "eld F and the magnetic "eld G

De"ning equations Geometrical description Symmetry group Physical problem

G"F"0 Point OO(3)'T No "elds " ' ' G 0, F 0 Ray OF CT T Stark e!ect ' " ' G 0, F 0 Ray OG CF TQ Zeeman e!ect ' ' " ' G 0, F 0, (GF) G F Conical surface C TQ Parallel "elds ' ' " G 0, F 0, (GF) 0 Plane G OF G Orthogonal "elds ' ' G 0, F 0, Interior of the cone TQ Generic (GF)(GF, con"guration (GF)O0

operations: purely geometrical spatial transformations forming one of the standard point symmetry groups and symmetry operations related to time reversal. We use the following notation. The group T includes two elements, identity E and time reversal ¹ ¹p . The group TQ includes also two elements, identity E and the symmetry operation ( ) which is the product of the time reversal and the re#ection in a plane including both "elds. Group G includes four symmetry elements: identity E,re#ection in the plane orthogonal to the magnetic p ¹p ¹ p "eld F, product of time reversal and re#ection in plane formed by two orthogonal "elds, ¹ ¹ and the symmetry operation C which is the product of the time reversal and the C rotation " p " ¹p around the electric "eld direction. G has three subgroups of order two: CQ (E, F), TQ (E, ), " ¹ and T (E, C). We can study now the dependence of the dynamics on the ratio of the "eld strengths assuming that the total e!ect of both "elds is kept to be more or less the same even if we change the "eld con"guration from that corresponding to pure Zeeman e!ect till that of pure Stark e!ect. These "eld con"gurations lie in some section of the cone in Fig. 6. To make an interesting section of it we take into account that the energy correction to the hydrogen atom in the linear Zeeman limit is L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 215

* +$ * +$ EL Gn and in the linear Stark limit is EL 3Fn where n is the principal quantum number for a perturbed hydrogen atom n+(!1/(2E). Thus for given n it is natural to "x S"((Gn)#(3Fn) and to vary only the relative strength of the two "elds: " Gn GQ , (146) ((Gn)#(3Fn) " 3Fn FQ . (147) ((Gn)#(3Fn)

We consider the case of small S but we allow large variations of the relative strengths of the two 4 4 # " "elds 0 FQ, GQ 1 with the restriction FQ GQ 1. A very interesting question concerns the qualitative behavior of the n-shell dynamics under the variation of "eld parameters (assuming S to be small enough). Are e!ective Hamiltonians for the n-shell always of the simplest Morse type? Or there are some regions in the space of parameters where the e!ective Hamiltonian should possess more than a minimal number of stationary points. If such a region exists it means that under the variation of "eld parameters (even for a low "eld limit) some bifurcations are present and the qualitative modi"cations of the dynamics take place. Two important cases to study are the case of two parallel "elds (con"gurations of "elds represented by points on the surface of cone in Fig. 6) and the case of orthogonal "elds (con"gurations represented by points on the bisectral plane of the cone in Fig. 6). But before looking on these cases we will brie#y apply the qualitative analysis to the limiting case of the Zeeman e!ect.

4.2. Quadratic Zeeman ewect in hydrogen atom

Let us consider the Hamiltonian for the quadratic Zeeman e!ect

p 1 G H" ! # (x#y) . (148) 2 r 8

This Hamiltonian is very popular from the point of view of theoretical investigations of di!erent dynamical regimes in quasi-regular and chaotic regions (Braun, 1993; Delande and Gay, 1986; Delos et al., 1983; Fano and Sidky, 1992; Farrelly and Krantzman, 1991, Farrelly and Milligan, 1992; Friedrich and Wintgen, 1989; Herrick, 1982; Huppner et al., 1996; Krantzman et al., 1992, Kuwata et al., 1990; Liu et al., 1996; Mao and Delos, 1992; Robnik and Schrufer, 1985; Sadovskii et al., 1995; Solov'ev, 1981,1982; Tanner et al., 1996; Uzer, 1990). ' The Hamiltonian in Eq. (148) is DF T invariant. Remark that its symmetry is higher than that ' of an atom in the presence of magnetic "eld (which is CF TQ) because we neglect the terms linear in angular momentum. Consequently any e!ective diagonal in n operator can be rewritten in terms m k l of operators corresponding to the DF polynomial invariants, , , (see Eqs. (116)}(118)). The auxiliary invariant lp does not enter in Hamiltonian (148) due to additional time-reversal invariance. We replace Hamiltonian (148) by the diagonal in n e!ective operator just by projecting it on the manifold of the non-perturbed hydrogen atom wave functions with a given n quantum number. Such procedure is physically meaningful in the low magnetic "eld limit. 216 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

The explicit expressions for the diagonal in n matrix elements of the diamagnetic terms are either diagonal in l or non-diagonal in l with *l"$2 (they are diagonal in m). It is necessary to "nd such combinations of diagonal in n operators which give the same matrix elements as the diamagnetic perturbation. The diagonal in n perturbation of the hydrogen atom may be equivalently represented in the form of the e!ective operator identical to that used by Solov'ev (1981), Herrick (1982), Delande and Gay (1986) and many others. In terms of invariant polynomials the Hamiltonian for the n shell has the following form:

n(n!1) H "const.#G [k!2m!5l] , (149) L 16 where the const. is n-dependent. This Hamiltonian in Eq. (149) is invariant with respect to the ' DF T group. Its action on the phase space of the reduced problem (n-shell e!ective Hamil- tonian) is summarized in Table 15. Energy levels of this Hamiltonian can be trivially visualized on ' the DF T orbifold because the energy function is linear in invariant polynomials used to construct the orbifold. Let us analyze now the essential part of the Hamiltonian:

k!2m!5l . (150)

There are four critical orbits and the topological structure of the energy levels varies by passing "! ' through the critical orbits only. Corresponding energies are (in increased order) E 3, CT T "! ' "! ' } critical orbit; E 2, CF T } critical orbit (one-dimensional manifold); E 1, CF T " ' } critical orbit; E 2, CT T } critical orbit (one-dimensional manifold). So, the Rydberg multiplet in relative units lies between !3(E(2. It is split into three di!erent regions. The lowest one occupies 1/5 of the multiplet width. The highest one occupies 3/5 of the multiplet width (see Fig. 7). Let us consider the lowest part of the multiplet. The energy surface near the minimum may be represented as the energy surface for two equivalent 2D slightly anharmonic oscillators. Energy levels form polyads. The nth polyad consists of 2n levels. There are two e!ects which lead to the splitting of the energy levels within a polyad: anharmonicity of the isotropic oscillator and the tunneling between two equivalent wells. The splitting of polyad due to anharmonicity e!ects

Fig. 7. Qualitative description of critical orbits versus energy for e!ective DF-invariant Hamiltonian for quadratic Zeeman e!ect. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 217

" " preserves the C symmetry. As a consequence, the degeneracy of energy levels with the same m , the projection of the angular momentum is present. If we neglect the tunneling splitting, for any "m"O0 there are four degenerate energy levels. The numerical results for the quantum problem clearly show this patterns. We can estimate the position of the classical energy minimum for the CT orbit from quantum results by assuming the harmonic oscillator model and taking the energy of the zero level equal to the fundamental frequency (this is the linear approximation). Thus for example, for n"45 the harmonic frequency (the fundamental transition) is 0.09442 and the estimated energy of the classical energy minimum found from the "! ! entirely quantum calculations is E *(CT) 2.99797 as compared to ( 3) for the purely classical model. At the other end of the energy multiplet (near CT orbit) the energy level pattern is similar to that of a one-dimensional rotator plus one-dimensional oscillator. The one-dimensional harmonic oscillator describes energy levels with m"0. The harmonic frequency (for n"45) can be again estimated from the quantum calculations as a fundamental transition. It is equal to 0.194414. Adding the half quanta of the corresponding harmonic frequency to the highest (extremal) quantum energy level gives the estimation for the classical energy of the (CT) critical orbit: " " E *(CT) 1.99987 as compared with E 2 in the classical limit. This numerical comparison is completely satisfactory taking into account the extreme simplicity of the classical model. It is clearly seen from the numerical results that the m"0 energy levels form the regular sequence of non-degenerate energy levels within the energy interval 25E5!1 and the regular sequence of doublets within the energy interval !25E5!3. This fact well correlates with the "! energy of the critical orbit CF characterizing by the energy E(CF) 2. It is important to note that the existence of four critical orbits is the consequence of the symmetry of the Hamiltonian and does not depend on the concrete form of the Hamiltonian. At the same time the relative positions in energy of critical orbits strictly depend on the concrete form of the operator.

4.3. Hydrogen atom in parallel electric and magnetic xelds

We demonstrate shortly in this section one particular application of the qualitative analysis of Rydberg states by studying the transition from a Zeeman to a Stark structure of a weakly split Rydberg n-multiplet of the H atom in parallel magnetic and electric "elds (Sadovskii et al., 1996). The geometrical approach clearly shows the origin of the new phenomenon, the collapse of the energy levels. The use of classical mechanics, topology, and group theory provides detailed description of the modi"cations of dynamics due to the variation of the electric "eld. We focus on the point where the collapse of the Zeeman structure occurs, give the sequence of classical bifurcations responsible for the transition between di!erent dynamic regimes, and compare it to the quantum energy-level structure. In fact, Rydberg atoms in parallel magnetic and electric "elds have been extensively studied both theoretically and experimentally during the last decade. In particular, many studies have focused on the situation where the "elds are (relatively) weak and the dynamics can be analyzed in terms of additional approximate integrals of motion (Braun, 1993; Braun and Solov'ev, 1984; Cacciani et al., 1988, 1989, 1992; Delande and Gay, 1986; Farrelly et al., 1992; Iken et al., 1994; Seipp et al., 1996; van der Veldt et al., 1993). We use a similar idea to analyze several dynamic regimes that exist for 218 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Fig. 8. Collapse of the Zeeman structure for the magnetic "eld G"0.06 due to the increasing electric "eld. Quantum " I " I levels are calculated for the n 10 multiplet of the hydrogen atom. Scaled electric "eld strength is given in units F G/3 [see Eq. (153)]. di!erent strengths of the electric (F) and magnetic (G) "elds. These regimes clearly manifest themselves in the energy level pattern in Fig. 8. At very weak electric "eld, where most of the studies ¸ were done, the energy levels are grouped according to the value of X, the projection of the angular momentum on the "eld axis. The internal structure in this region is mainly due to quadratic Zeeman e!ect (see Fig. 9 and discussion in the Section 4.2). When F increases this structure quickly disappears. Instead we observe regular `resonancea structures at certain values of F (see Fig. 8). This culminates in an almost complete collapse of the internal structure. Surprisingly, and contrary to the F&0 case the dynamics near this collapse has not been earlier analyzed in detail. Neglecting the spin e!ects the Hamiltonian for the hydrogen atom in constant parallel magnetic G and electric F "elds (along the z-axis) has the form (in atomic units)

p 1 G G H" ! # ¸ # (x#y)!Fz (151) 2 r 2 X 8 with G and F in units of 2.35;10 T and 5.14;10 V/cm. We restrict ourselves to the low "eld case where the splitting of an n-shell caused by both "elds is small compared to the splitting between neighboring n-shells (see Fig. 10). As is well known, in the absence of electric "eld low-m submanifolds of the n-shell show characteristic pattern of the second order Zeeman e!ect. When the electric "eld e!ect is of the same order as the quadratic Zeeman e!ect (see Fig. 9), this pattern disappears and turns into a Stark structure for each m sub-manifold (Braun and Solov'ev, 1984; Delande and Gay, 1986). Much lesser attention has been paid to the region where the Stark splitting of the n-shell (J3Fn) is of the same order as the n-shell splitting due to magnetic "eld L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 219

Fig. 9. Deformation of the QZE structure of the n"10, m"0 multiplet of the hydrogen atom. Dashed lines show the energy in stationary points of the classical Hamiltonian restricted on k"0.

Fig. 10. Neighboring Rydberg multiplets n"9, 10, 11 of the hydrogen atom. 220 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

(JGn) (Fig. 8), in other words when " F G/(3n) . (152) Eq. (152) gives the collapse condition for shell n. To ensure that this collapse happens when the n-shell splitting remains small compared to the gap between n-shells, we take G(1/n. Under this assumption we can study an isolated n-shell and can use scaling I " I " I " # F Fn , G Gn , EL 2n EL 1 , (153) to remove n from the e!ective n-shell Hamiltonian [Eq. (157) below]. Our purpose is to study the dynamics under the variation of the electric "eld F in the neighborhood of its critical value. The natural parameter for this study is d"3Fn/G!1. The analysis is based on the transformation of the initial Hamiltonian (151) into an e!ective one for an individual n-shell. This can be done either by quantum or by classical perturbation theory (SoloveH v, 1981; Delande and Gay, 1986; Farrelly et al., 1992; Stiefel and Scheifele, 1971). An e!ective n-shell Hamiltonian can be expressed in terms of angular momentum L and Runge}Lenz vector A"p;L!r/r, or, alternatively, in terms of their linear combinations " # " ! J (L A)/2 and J (L A)/2. For the linear Stark}Zeeman e!ect in parallel "elds the e!ective Hamiltonian is 1 H" (!1#Gn¸ #3FnA ) . (154) 2n X X If we impose the relation between "eld strengths (152) this Hamiltonian becomes 1 H" (!1#3F n(J ) ) . (155) 2n X The n energy levels in the n-shell described by Eq. (155) form n-fold degenerate groups. The levels in each group are labeled by the same value of (J)X and by di!erent values of (J)X. Fig. 8 shows how the Zeeman structure of the n-shell at f"0 transforms into this highly degenerate structure at " I " I F F (F G/3). We call this e!ect the collapse of the Zeeman structure caused by electric "eld. To describe the "ne structure of each (J)X manifold of states the second-order e!ects should be taken into account. To develop the e!ective n-shell Hamiltonian to higher orders we consider n as an integral of motion and use the perturbation theory to reduce the initial problem (151) to two degrees of freedom. Naturally, the pair (¸ , ) describes one of these degrees; the other degree can be X *X described by A and (Farrelly et al., 1992). Of course, for Hamiltonian (151) ¸ is strictly X X X conserved and the n-shell Hamiltonian does not depend on . However, to study the collapse we *X should consider the energy level structure of the n-shell as a whole, and therefore, we should keep ¸ X as a dynamical variable. Hence our n-shell Hamiltonian is a function of dynamical variables (¸ , A , ) and parameters (n, F, G). X X X R ; The classical phase space for e!ective n-shell Hamiltonian is a 4D space with topology S S. Its parametrization can be done either using the L, A variables with L#A"n, and L ) A"0, or " " using the J, J variables with J J n /4. (In the classical limit n is su$ciently large and n+n!1.) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 221

For the qualitative analysis of the n-shell dynamics we use invariant polynomials l" k"¸ m" ¸! AX/n; X/n; ( A )/n (156) forming integrity basis which is used both to label the points of the phase space and to expand the Hamilton function as explained in Section 5.6 of Chapter I (see also Appendix A). Furthermore, the proper scaling in Eqs. (156) and (153) results in equations which do not depend on n. " ' R The symmetry group G C TQ of the problem and its action on are described in detail in Section 2.9. Consecutive steps in the qualitative analysis of an e!ective Hamiltonian in the presence of the symmetry group (Sadovskii and Zhilinskii, 1993a) which are summarized brie#y in Section 2 of Chapter II include the study of the action of the symmetry group on the classical phase space, construction of the space of orbits (orbifold), and the analysis of the system of stationary points (orbits) of the Hamilton function using the topological and group theoretical information about the phase space. The strati"cation of the R phase space under the action of the symmetry group ' C TQ is given in Tables 8 and 13. The Hamilton function can be expressed as a polynomial H"H(l, k, m) of invariant polynomials l, k, m. Up to quadratic in F and G terms the scaled energy EI has the form I " I k# I l! I m# I # I k# I ! I l# I ! I E G 3F G (9F G ) (3F 5G ) (3G 17F ) . (157) To qualitatively characterize classical and quantum dynamics we "nd the system of stationary points (manifolds) of the energy function on the phase space. Group theory asserts that four points A, B, C, D (critical orbits) must be stationary for any smooth function de"ned over the phase space (see Section 4 of Chapter I). Energy values (157) at these points are shown in Figs. 8 and 10. Morse inequalities con"rm that the simplest Morse-type functions possessing stationary points only on the four critical orbits really exist on R and have one minimum, one maximum, and two saddle points. For more complicated functions any other stationary points can be found by looking for those energy sections of the orbifold which correspond to the modi"cation of the topology of the energy section. ( Simple geometrical analysis shows that in the linear (in F and G) approximation for F F the energy function is of the simplest type with minimum in B, maximum in A, and two saddle points in ' C and D.ForF F the energy function is again of the simplest type with minimum in D, maximum in C, and two saddle points in A and B. Sudden transition from one simplest type of the energy function to another one in the linear model occurs due to the formation of the degenerate " stationary manifold at value F F corresponding to Hamiltonian (155). In the linear model the energy surface touches the orbifold through the whole interval [C, A]or[B, D]. Introduction of the F and G terms into the energy function removes this degeneracy. The energy surface (157) is the second-order surface in m, k, l variables. It can touch the orbifold O at some isolated points on the p"0 surface which are di!erent from the critical orbits A, B, C, D. If this happens, additional stationary orbits are present. The detailed analysis of a system of stationary points as a function of F near the collapse value F shows how the transformation from the Zeeman-type energy function (with only four stationary critical orbits having minimum and maximum in B and A) to the Stark-type energy function (with only four stationary critical orbits having minimum and maximum in D and C) occurs. Two sequences of bifurcations are present with two bifurcations in each sequence. As F increases, one sequence begins with a bifurcation at point B which creates a new 222 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

I R stationary S orbit of E on . The corresponding point on the surface of the orbifold moves from B to D and disappears at D after the second bifurcation. Another sequence of bifurcations proceeds in the similar way with two bifurcations at C and A and the new additional stationary orbit moving from C to A. Positions of all stationary orbits can be found by solving the Hamiltonian equations on R. Alternative way is to use the geometrical representation of the orbifold and of the energy surface. To "nd non-critical stationary orbits we "nd points where the energy surface touches the orbifold. In other words, we "nd points where the normal vector to the p"0 surface and the normal vector " to the energy surface k (kJ, kI, kK) are collinear. This geometric view gives us extremely simple conditions for bifurcations at points A, B, C, D: " ! d +! I ! I A:4kKkI kJ kI, G /8 G /16 , (158) " ! d + I # I C:4kKkJ kJ kI, ! 2G/3 G /72 , (159) " ! d +! I # I B:4kKkI kI kJ, G /8 G /16 , (160) " ! d +! I # I D:4kKkJ kI kJ, " 2G/3 G /72 . (161) d d d When varies between and ! an additional stationary orbit exists on the surface of the orbifold d d d and moves from the point A to the point C. Similarly, for between " and another additional stationary orbit moves from D to B. The energies of all stationary orbits near the bifurcation points and the quantum energy levels are shown in Fig. 11.

Fig. 11. Bifurcation diagram near the collapse region. Classical (left) versus quantum (right) representation. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 223

Simple quantum mechanical interpretation of the e!ect of the transformation of the Zeeman-type structure into Stark-type one can be done by looking at the two extremal states (with minimal and maximal energy) of the same n multiplet (Fig. 11). We can characterize each 1¸ 2 1 2 extremal state by two average values X and AX . From the positions of stationary points on 1¸ 2+ d(d the orbifold it follows immediately that for the state with maximal energy X n for , 1¸ 2+ d'd 1¸ 2 d d (d(d X 0 for !, whereas X varies almost linearly with for !. For the same 1 2+ d(d 1 2+ d'd 1 2 d state AX 0 for , AX n for !, whereas AX varies almost linearly with for d (d(d !. We conclude that important qualitative modi"cations of dynamics take place in the collapse region. This suggests new experimental investigations which can use the detailed information on many energy-level crossings in the collapse region to obtain quantum states with desired properties by "ne tuning of the "eld parameters. Existence of collapsed levels with di!erent projections of the orbital momentum m can be used in the experiment to selectively produce states with any possible m using adiabatic change of "eld parameters. This is specially important for formation of so-called `circulara states with very high m&n values (Delande and Gay, 1988; Germann et al., 1995; Hulet and Kleppner, 1983; Kalinski and Eberly, 1996a,b).

4.4. Hydrogen atom in orthogonal electric and magnetic xelds

Hydrogen atom in crossed (orthogonal) electric and magnetic "elds was the subject of many experimental (Flothmann et al., 1994; Raithel and Fauth, 1995; Raithel et al., 1993a,b,1991; Raithel, Fauth and Walther, 1993; Rinneberg et al., 1985) and theoretical (Farrelly, 1994; Gourlay et al., 1993; von Milczewski et al., 1994,1996) studies. The purpose of this section is to show what kind of qualitative information can be obtained taking into account only general topology and symmetry information. The hydrogen atom in the presence of two orthogonal "elds gives us an example of a system with the "nite symmetry group G (see Sections 2.9 and 4.1). The space of orbits in this case is four dimensional and it is naturally more di$cult to visualize its strati"cation and to represent the orbifold in a geometrical way. Nevertheless, very useful topological and group-theoretical information can be found by studying the invariant subspaces of di!erent symmetry. We remind on this example again major steps of the qualitative analysis realized in Chapter II for di!erent rovibrational problems. More details can be found in Sadovskii and Zhilinskii (1998). First, we construct the Molien function and the integrity basis for the ring of G invariant polynomials on R. The procedure we follow here is formally the same as that used in Appendix B for molecular point group symmetry. The simplest way to write the Molien function is to work in the J, J representation, to consider the ring of all invariant functions constructed from six dynamic variables and to restrict this ring to 4D classical phase space, R. We start with the situation without any additional symmetry a b" (the symmetry group is trivial, C). On the six-dimensional space (J)?,(J)@,( , x, y, z) the Molien function for invariants has a trivial form

1 M " . (162) ! (1!j) 224 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

! The ring of the invariant polynomials on the 6D-space P is the ring of all polynomials !" P P[(J)V,(J)W,(J)X,(J)V,(J)W,(J)X] . (163) To restrict the polynomial ring on the sub-manifold h " # # " (J)V (J)W (J)X const. , (164) h " # # " (J)V (J)W (J)X const. , (165) h h we are obliged to introduce two second-order polynomials, , given by Eqs. (164) and (165) as denominator invariants. This may be done by multiplying both the numerator and denominator of the Molien function (162) by (1#j). The new form of the Molien function 1#2j#j M " (166) ! (1!j)(1!j)

! corresponds now to another description of the ring of invariants P which is considered as a free module

!" h h ⅷ P P[(J)V,(J)W, ,(J)V,(J)W, ] (1, (J)X)(1, (J)X) (167) u " u " u " u " with four auxiliary invariants 1, (J)X, (J)X, (J)X(J)X. We use the notation (a, b)(c, d)"(ac, ad, bc, bd) to show that four numerator invariants are represented as products of more simple terms. Having the form (166) of the Molien function and the form (167) of the module of invariant functions it is easy to make the restriction to the sub-manifold R. We just h h R eliminate two denominator invariants, , , corresponding to the equations de"ning . The resulting Molien function and the module of invariants on R are written as follows: 1#2j#j M "R" , (168) ! (1!j)

!" " ⅷ P R P[(J)V,(J)W,(J)V,(J)W] (1, (J)X)(1, (J)X) . (169) The choice of basic and auxiliary invariants is ambiguous and the ones proposed here is just an example. The important point is that the integrity basis may be constructed which includes four numerator invariants (including 1) and four denominator invariants. To "nd now the integrity basis in the A, L representation we can simply transform invariants from J, J to A, L representation. Let us now decrease slightly the symmetry and consider two non-parallel and non-orthogonal "elds. The "nite symmetry group in this case has order 2 and includes one non-trivial operation: product of time reversal and space re#ection in the plane de"ned by two "elds. To specify the action of the symmetry group on basis polynomials we "x the coordinate frame in such a way that the x-axis coincides with the electric "eld vector and the y-axis belongs to the plane of two "elds and has the positive projection of the magnetic "eld on it. Let Tp be the symmetry operation for the generic con"guration of two "elds considered. It ¸ ¸ follows immediately that AV, AW, V, W are invariant with respect to this symmetry operation, ¸ whereas AX, X are pseudo-invariant (change the sign). We see as well, that all basic invariants of the ring of polynomials (167), and (169) are invariant with respect to the Tp operation. At the same L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 225 time between three non-trivial numerator (auxiliary) invariants only, one is invariant with respect to the Tp operation, whereas two others change sign under this operation. This means that the Molien function and the module of invariants on R in the case of a generic con"guration of two "elds has the form 1#j M$ '%"R" , (170) (1!j)

$ '%" " ⅷ P R P[(J)V,(J)W,(J)V,(J)W] (1, (J)X(J)X) . (171) The symmetry group for the case of two orthogonal "elds is higher. It includes four symmetry p p elements E is the identity, T , introduced just above, ) the re#ection in the plane orthogonal to the magnetic "eld B (this plane includes the electric "eld vector), and TC the product of the time reversal and the C rotation around the electric "eld. For the particular case of orthogonal "elds it is useful to change the notation of axes in order to show explicitly their orientation with respect to external "elds. We use below in this section the coordinate frame +e, b, p, with vector e along the electric "eld vector, vector b along the magnetic "eld vector, and vector p chosen to form the right-hand frame. ¸ Symmetry properties of A?, @ and of all basic and auxiliary invariants of module (167) and (171) are summarized in the Table 21. The "rst consequence is the necessity to change the two basic invariants which de"ne the R " " # " ! " sub-manifold . Instead of J J const. we can use J J const. and J J 0. The "rst transformed equation is invariant with respect to the symmetry group. The second is pseudo-invariant. To deal with invariants only on the 6D-space we should "rst change the integrity

Table 21 Transformation properties of dynamic variables for two orthogonal "elds under the action of G group p(C@) p(CN) (C) E T TC ## # # AC ## ! ! A@ #! # ! AN ¸ ## ! ! C ¸ ## # # @ ¸ #! ! # N ## ! ! (J)C (J)C (J)C ## (J)@ (J)@ (J)@ ## ! ! (J)C (J)C (J)C ## (J)@ (J)@ (J)@ #! ! (J)N (J)N (J)N #! ! (J)N (J)N (J)N ## # # (J)N(J)N ## (J) (J) (J) ## (J) (J) (J) 226 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

h h u basis by introducing as basic invariants ?, ?, and as an auxiliary invariant @ and instead of (J)C,(J)@,(J)C,(J)@ their linear combinations which have well-de"ned symmetry properties (irreducible tensors) with respect to the symmetry group h " # h " ! u " ! ? (J J), ? (J J) , @ (J J) , (172) h " ! h " # (J)C (J)C, (J)@ (J)@ , (173) h " # h " ! (J)C (J)C, (J)@ (J)@ , (174) u " ! u " # u " (J)N (J)N, (J)N (J)N, (J)N(J)N . (175)

Now, we have the description of the ring of C invariant polynomial on the 6D-space in terms of the integrity basis which includes invariants and pseudo-invariants of the symmetry group for two ! orthogonal "elds. The Molien function and the ring of invariants P can be written as (1#2j#j)(1#j) M! " , (176) (1!j)(1!j)(1!j)

!" " h h h h h h ⅷ u u u u P P[ , , , , ?, ?] (1, , , )(1, @) . (177) R h h Reduction on should be done now by eliminating two basic invariants, ?, ? and one auxiliary u R invariant @ which corresponds to zero on . Among basic denominator invariants we have two, h h ( , ), which are pseudo-invariants with respect to the total symmetry group of the problem. To insure that all basic numerator polynomials are invariants of the total symmetry group we can h h h "h change the integrity basis by introducing instead of , two new basic invariants ? , h "h u "h u "h u "h h ? and three auxiliary polynomials ? , @ , A . After such a modi"cation we can rewrite the Molien function on R and the module of polynomials on R in terms of an integrity basis including as basic polynomials only invariant polynomials with respect to the total symmetry group, and as auxiliary polynomials both invariants and pseudo-invariants of the total symmetry group:

(1#2j#j)(1#j) M "R" , (178) ! (1!j)(1!j)

!" " h h h h ⅷ u u u u u u P R P[ , , ?, ?] (1, , , )(1, ?, @, A) . (179) The list of polynomials forming the integrity basis is as follows: h " h "¸ h "¸ h " AC, @, ? C, ? A@ , (180) u " u "¸ u " AN, N, (J)N(J)N , (181) u "¸ u " u "¸ ? C, @ A@, A CA@ , (182) u u u u u u u u u u u u ?, @, A, ?, @, A , (183) u u u u u u ?, @, A . (184) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 227

Table 22 R Invariant manifolds for the G symmetry group action on (Hydrogen atom in perpendicular electric and magnetic "elds.)

Stabilizer dim. Topology Equations

¸# " G 1 S @ AC 1 ¸" " ¸! 4 CQ 2 S C A@ 0, N AN 0 ¸" " ¸! 5 T 2 S C A@ 0, N AN 0 ; ¸#¸# # " TQ 2 S S C @ AC A@ 1 ; R # " ) " C 4 S S (L A 1; (L A) 0)

Now, we can simply eliminate all auxiliary polynomials which are not invariant with respect to the total symmetry group for two orthogonal "elds. This gives the following Molien function and the module of invariant on R: (1#2j#j) M) $"R" , (185) (1!j)(1!j)

) $" " ¸ ¸ ⅷ ¸ ¸ P R P[AC, @, C, A@] (1, CA@,(J)N(J)N,(J)N(J)N CA@) . (186) ¸! Remark that we can replace (J)N(J)N by ( N AN). Next step of the qualitative analysis is the strati"cation of the phase space R under the action of the G symmetry group. Invariant manifolds are listed in Table 22. Keeping in mind this information we apply the Morse theory arguments to get restrictions on the number and location of stationary points of the Hamilton function. As long as there are no zero-dimensional strata of the group action, there are no critical orbits. Nevertheless, we can say that there should be at least two stationary orbits on the G invariant subspace and at least one additional stationary TQ invariant orbit formed by two equivalent points. Under the variation of the strength of two "elds these stationary points move but they are obliged to be always on these invariant subspaces. If we compare results of the qualitative analysis of the hydrogen atom in parallel "elds with that for hydrogen atom in orthogonal "elds it becomes clear that for su$ciently low "elds the evolution of the Zeeman multiplet into the Stark multiplet goes through a sequence of bifurcations for parallel "elds, whereas for orthogonal "elds no generic bifurcations are present. Immediately a natural question arises. What can be said about generic "eld con"gurations? Is it reasonable to expect the presence of bifurcations under some variation of relative "elds and their orientation?

4.5. Where to look for bifurcations?

To answer this question we represent in Fig. 12 the space of relative con"gurations of two "elds F, G imposing the restriction of the type F#aG"S, where S is supposed to be su$ciently small and the positive parameter a can be chosen in such a way that the splitting of the Rydberg multiplet in Zeeman and Stark limits are approximately the same. If electric and magnetic "elds are non-collinear and non-orthogonal, the only non-trivial symmetry operation is the composition of time reversal and the space re#ection in the plane 228 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Fig. 12. Schematic representation of qualitatively di!erent Morse-type Hamiltonians for a hydrogen atom in two external "elds (electric and magnetic). Two parameter family of Hamiltonians is split into regions corresponding to the simplest and non-simplest Hamiltonians (as a Morse-type functions).

including two "elds. So the symmetry group for such generic orientation is the TQ group introduced earlier. The only non-trivial invariant subspace is the TQ invariant torus. On this torus four stationary points are generically present. The same four stationary points should always be present for the R phase space. The appearance of additional stationary points should be veri"ed "rst of all near the collinear con"guration of two "elds corresponding to the non-simplest Morse-type Hamiltonian. Near this point the Hamiltonian is not of the simplest Morse type as we show below. Let us consider the collinear con"guration of two "elds corresponding to the range of F, G parameters such that additional extrema on R exist. These additional extrema correspond to points on p"0. After a small deformation of the con"guration of "elds ("elds become non-orthogonal after an arbitrarily small perturbation) the symmetry is broken but each stationary orbit leads to one stationary orbit of the lower symmetry group. As soon as for parallel "elds on a TQ invariant torus (with one particular orientation of symmetry plane) there are more than four stationary points, their number should be conserved after the symmetry is broken by small perturbation. So near the collapse region for parallel "elds there should be the region where the Hamiltonian is a non-simplest Morse-type function even for non-parallel "elds. Fig. 12 schematically demonstrates this fact.

5. Conclusions and perspectives

The main idea of this chapter was to demonstrate how general group-theoretical and topological methods work in a particular physical problem, Rydberg states of atoms and molecules. As compared to the similar analysis realized earlier for molecular rotations and vibrations, the mathematical di$culty was overcome in this study. This is the presence of continuous symmetry related with the generalization from standard Morse theory of functions with isolated non- degenerated stationary points to Morse}Bott theory of functions with non-degenerate stationary L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 229 manifolds. At the same time the general scheme of the qualitative analysis follows the way developed earlier for molecular rotations and vibrations (Zhilinskii and Pavlichenkov, 1987, Pavlichenkov and Zhilinskii, 1988; Zhilinskii, 1989a,b; Sadovskii and Zhilinskii, 1993a; Zhilinskii, 1996) and brie#y summarized in Chapters I and II. In contrast to many speci"c theoretical molecular analyses, we do not impose at the beginning the concrete form of the Hamiltonian. This allows us to "nd some features of the energy spectrum (or of the classical dynamical behavior) which are common for a relatively large set of possible Hamiltonian functions. The Hamiltonian which is interesting from the point of view of physical applications is sometimes just a single particular operator from the wide set of objects considered. General statements can be surely applied to one particular example but the information obtained in such way is certainly more restrictive than the result of concrete numerical analysis of the particular problem. This is a disadvantage of the generic qualitative analysis. Otherwise, concep- tually it is extremely useful to understand the presence of features which are independent of the precise form of the Hamiltonian, especially taking into account the fact that any Hamiltonian used for a concrete physical application is always an approximation. For Rydberg problems we have studied very simple case of small splitting of n-shell. It is certainly possible to "nd some physical situations when such a model is rather accurate and reasonable. At the same time it is clear that for real molecular and atomic systems there are many interactions (and additional degrees of freedom) which become important and often can modify considerably even the qualitative behavior. Among the simplest but essential corrections are those due to spin, the motion of the center of mass in the presence of external "elds (Johnson et al., 1983; Farrelly, 1994). Highly excited states of the hydrogen atom occupy rather special place among Rydberg states of atomic systems with one excited electron. The origin of this is the additional dynamical O(4) symmetry. Many di!erent experimental and theoretical analyses of the non-hydrogenic atom Rydberg states were done. The quantum defect theory is the most popular and powerfool tool of the theoretical analysis and interpretation of experimental data. Aymar et al. (1996) reviewed recently this subject. At the same time the systematic qualitative analysis of e!ective n-shell Rydberg Hamiltonians for free non-hydrogenic atoms or atoms in external "elds has yet not been done. Doubly excited Rydberg states of atoms give much more complicated examples with rich qualitative structure which was pointed out initially by Herrick and Sinanoglu (1975) on the basis of comparison of the approximate O(4) dynamical symmetry and extensive numerical calculations. To understand better the qualitative features of electronic excited states and especially their localization properties [see for example papers by Goodson and Watson (1993), Dunn et al. (1994) and references therein] it would be interesting to perform the topological and symmetry analysis of a two-electron problem on the same basis as it is done in the present paper for one-electron Rydberg problem. We have not practically touched in the present review the concrete applications of the qualitative theory to Rydberg states of diatomic or polyatomic molecules except for the general symmetry analysis. It should be noted that the application of a qualitative analysis should begin with the > reanalysis of the excited states of the simplest one-electron diatomic molecule H . In spite of its > apparent simplicity the description of the "ne structure of highly excited states of H still remains a question of interest (Brown and Steiner, 1966; Coulson and Joseph, 1967; Grozdanov and 230 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Solov'ev, 1995). Diatomic molecules are of particular theoretical interest due to the presence of continuous symmetry groups which are di!erent for homonuclear and heteronuclear molecules. Due to the variety of diatomic molecules it is possible to "nd examples of molecules with slight or strong breaking of the interchange symmetry of two nuclei. Moreover, for a number of diatomic molecules experimental or numerical data exist (Bordas et al., 1985; Dabrowski et al., 1992; Dabrowski and Sadovskii, 1994; Davies et al., 1990; Fujii and Morita, 1994; Howard and Wilkerson, 1995; Jacobson et al., 1996; Jungen, 1988; Jungen et al., 1989, 1990; Greene and Jungen, 1985; Kim and Mazur, 1995; Merkt et al., 1995, 1996, Michels and Harris, 1963; Watson, 1994) and experimental data or results of alternative numerical modelization are ready to be compared through the qualitative analysis in order to reveal some universal features of the system of Rydberg states of diatomics. Among polyatomics, the H molecule is surely the most popular as an object to study Rydberg states (Bordas and Helm, 1991; Bordas et al., 1991; Bordas and Helm, 1992; Dodhy et al., 1988; Helm, 1988; Lembo et al., 1989, 1990; Herzberg, 1981; Ketterle et al., 1989; King and Morokuma, 1979; Pan and Lu, 1988; Stephens and Greene, 1995). In fact, this molecule belongs to the class of so-called Rydberg molecules for which the chemical bonding is formed due to the Rydberg electron (Herzberg, 1987). One can imagine the Rydberg molecule as a stable molecular ion plus an electron on a high Rydberg orbit. Typically, Rydberg molecules are bound only in excited electronic states and their predissociation becomes more pronounced under electronic desexcitation. H and NH are typical Rydberg molecules (Herzberg, 1981). More exotic examples of Rydberg dimers like (H) or (NH), etc., are discussed by Boldyrev and Simons (1992a), Boldyrev and Simons (1992b) and Wright (1994). Chemical processes related with Rydberg electrons were studied even for such big objects as fullerenes (Weber et al., 1996) or metal surfaces (Ganesan and Taylor, 1996). Comparison of simple model electronic Rydberg calculations with concrete molecular experiments is naturally much more complicated due to the presence of additional degrees of freedom (vibration and rotation). At the same time this gives possibility for new qualitative e!ects like, for example, the stabilization of unstable rotational axes of an asymmetrical top molecule due to the interaction with a Rydberg electron as proposed by Basov and Pavlichenkov (1994) or core induced stabilization of molecular Rydberg states discussed by Lee et al. (1994). Monitoring of intramolecular dynamics through preparation of special Rydberg electron wave packets is no longer a science "ction but the subject of current interest (Beims and Alber, 1993; Boris et al., 1993; Dietrich et al., 1996; Frey et al., 1996; Jones, 1996; Rabani and Levine, 1996; Remacle and Levine, 1996a,b; Thoss and Domcke, 1995). Naturally, it is impossible to expect that a simple qualitative model based on the n-shell approximation can be used in the region where n is no longer an approximate integral of motion. So a further natural question arises: How to extend the approach presented in this paper to the case of `overlappinga and to the case of more serious `interactionsa of di!erent n-shells? This problem can be attacked from two opposite sides. One possibility is to start with the model of `complete classical chaosa and to study peculiarities of quantum systems through some kind of statistical or some other `chaologicala methods (Bixon and Jortner, 1996; Friedrich and Wintgen, 1989; Hasegawa et al., 1989; Lombardi et al., 1988; Lombardi and Seligman, 1993; von Milczewski et al., 1994, 1996; Zakrzewski et al., 1995). An alternative approach consists in studying qualitative e!ects related to the violation of the individual n-shell approximation. Such an extension of the qualitative approach to the qualitative theory allowing the `couplinga of di!erent n-shells should be extremely L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 231 useful for the Rydberg problem in the intermediate region between quasiregular and completely chaotic motion. The analog of such an extension was proposed earlier in the qualitative analysis of molecular rovibrational structure. Rotational multiplets of di!erent individual vibrational components can be considered as formal analogs of di!erent n-multiplets. In the classical limit the violation of individual components (vibrational components for rovibrational problems or n-shells for Rydberg problems) is associated with the formation of `diabolica points (conical intersection points) between di!erent energy surfaces. As was shown on simple initial example by Pavlov-Verevkin et al. (1988) the new qualitative phenomenon, namely the redistribution of energy levels between di!erent branches in the energy spectrum typically appears in such a case under variation of the integral of the motion. Each elementary qualitative phenomenon can be characterized by a topological invariant which is stable under small deformation of the Hamiltonian. Recent theoretical analysis (Zhilinskii and Brodersen, 1994; Brodersen and Zhilinskii, 1995b; Brodersen and Zhilinskii, 1995a; Zhilinskii, 1996) supports this point of view and makes some interesting relations with rather di!erent physical phenomena like recoupling of angular momenta and quantum Hall e!ect (Avron et al., 1983, 1988; Bellissard, 1989; Leboeuf et al., 1992; Niu et al., 1985; Simon, 1983) or purely mathematical questions like topological obstructions to integrability (Nekhoroshev, 1972; Duistermaat, 1980; Cushman and Bates, 1997). Complete classical analysis of the redistribution phenomenon discussed in Section 6.1 of Chapter II (Sadovskii and Zhilinskii, 1999) has given an interesting mathematical relation with classical monodromy. Further study of this phenomenon (Cushman and Sadovskii, 1999) shows the presence of classical monodromy for the hydrogen atom in orthogonal electric and magnetic "elds. The redistribution of energy levels between di!erent n-shells in quantum Rydberg problems and corresponding analysis of associated classical models will surely become a new subject of further theoretical and experimental study.

Appendix A. Geometrical representation

We consider in this appendix the geometrical representation of orbifolds which is the initial step for the geometrical representation of qualitatively di!erent types of the Morse}Bott-type functions de"ned over classical phase space for various invariance groups. As long as we are working with functions which are supposed to be the classical analog for quantum e!ective Hamiltonians, we will use the notion `energy levela for a solution of the equation H"const. In our particular case the R" ; Hamiltonian function is de"ned over a four-dimensional space ( S S), so the energy level is normally a three-dimensional region of the phase space which may be characterized by its topology. Taking into account the action of the symmetry group we can reach more detailed information about the structure of each energy level by indicating the topological structure of orbits from one side and the topological structure of the orbifold section from another side.

A.1. O(3) or SO(3) invariant Hamiltonian

The case of SO(3) invariant operator is not very interesting in our particular problem because R" ; any SO(3) invariant operator is also invariant with respect to the O(3) group on the S S 232 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

Fig. 13. Orbifold for the O(3) group action on R"S;S. Orbits are parametrized by the value of one invariant polynomial, m.

Table A.1 Topological description of energy levels for the simplest Morse}Bott-type Hamiltonian (O(3) invariant). `noa in the Morse index column means absence of stationary points

Morse index Dim of orbit Topological structure Local symmetry of the energy level

; 0(2) 2 e S CT ; no 3 e RP CQ ; 2(0) 2 e S CF

manifold. So if one is restricted to the consideration of the diagonal in `na e!ective operators, there is no di!erence between SO(3) and O(3) symmetry groups. R" ; The orbifold for the O(3) action on the S S manifold is shown in Fig. 13. From the R" & topological point of view it is a 1D-ball ( O(3) B). Orbits are parametrized by the value of one invariant polynomial, m. The detailed description of orbits and strata is given in Table 4 of this paper. The simplest Morse}Bott-type function de"ned on R and which is O(3) invariant possesses two critical manifolds coinciding with two critical orbits of the O(3) group action. Di!erent energy levels of such a simplest function are characterized by the topological structure in Table A.1. There are, in fact, two possible choices of the simplest Morse}Bott function di!ering in inter- changing the position of minimum and maximum critical manifolds. As soon as the dimension of critical orbits is two, the Morse indices for minimal or maximal critical manifolds are equal to 0 or 2 correspondingly. Any function H"H(m) which has RH/RmO0 for !14m41 may be used as an example of the simplest Morse}Bott-type function. If we have an e!ective Hamiltonian which possesses an extremum within the interval !1(m(#1, then the system of di!erent energy levels has more complicated topological properties. For example, the e!ective Hamiltonian of the form

" m# H c( ) (A.1) has "ve topologically di!erent energy levels given in Table A.2. The Morse index for the RP stationary manifold may be either 0 or 1 (because the dimension of the RP stationary manifold is 3). One can easily see that the total number of critical manifolds may be arbitrary but the numbers of critical manifolds with given indices are related among themselves. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 233

Table A.2 Topological description of energy levels for the (A.1) type Hamiltonian (O(3) invariant). Morse indices are indicated for the c'0(c(0) choice of the parameter. The disconnected components of the energy level are represented within the curly brackets

Morse index Topological structure Local symmetry of the energy level

; 0(1) e RP CQ + ; , no 2 e RP CQ + ; ,# 2(0) e S CT + ; , no e RP CQ ; no e RP CQ ; 2(0) e S CF

If the total number of critical manifolds is even there are one maximum and one minimum which are situated on critical orbits (CT and CF) and equal number of minima and maxima on generic (CQ) orbits. If the total number of critical manifolds is odd, there are two maxima (or two minima) # on critical orbits (CT and CF) and odd, 2p 1, (even, 2p) number of minima and even, 2p, # 5 (odd, 2p 1) number of maxima on generic (CQ) orbits (p 0 being nonnegative integer).

A.2. C invariant Hamiltonian R Orbits for the C action on are parametrized by values of three denominator invariants (m, k, l) and one numerator invariant, p. So, we can represent the orbifold in m, k, l variables taking into account that for p"0 there is one-to-one correspondence between points of the orbifold and m, k, l values satisfying the equalities in Eq. (16). If p'0 there are two orbits with the same m, k, l values but with the di!erent signs of p p"$ !m ! !m k! #m l [(1 ) (1 ) (1 ) ] . (A.2) We represent an orbifold as consisting of two parts, one corresponding to p50 and other corresponding to p40. Both these parts are shown in Fig. 14. They are identical in 3D-(m, k, l)- space. For m"1 we have l"0 and k41. For m"!1 we have k"0 and l41. For any m(1, putting p"0 we can "nd the geometrical form of the boundary of each part of the orbifold in the 3D-(m, k, l)-space. The equation for the boundary has the form 1 1 1 k# l" . (A.3) 1#m 1!m 2 So any m"const. section of one part of the orbifold is an ellipsoid. The complete orbifold may be constructed by identifying the surfaces of two 3D-bodies corresponding to p50 and p40 parts of the orbifold. To specify the topological structure of the orbifold it is necessary to use some additional information about topology of three-dimensional manifolds (Fomenko, 1983; Thurston, 1969). 234 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

R" ; Fig. 14. Orbifold for the C group action on S S . Each part of the orbifold is parametrized by three invariant polynomials, m, k, l. Two parts corresponding to di!erent signs of the auxiliary polynomial p should be glued together through the identi"cation of respective points on the boundary p"0.

R" ; Fig. 15. Representation of the orbifold for the C group action on S S in non-polynomial variables " m " k#l " k!l e arccos , e arccos( ), e arccos( ). Only one part of the orbifold is shown.

Every three-manifold can be obtained from two handle-bodies (of some genus) by gluing their boundaries together. Such a representation is called a Heegard decomposition. It is not unique and de"ned by number g (the genus of the boundary of two auxiliary manifolds) and by mapping of the boundaries. It is known that the only three-manifold, possessing the Heegard decomposition of genus 0 is the 3D sphere. The above constructed orbifold is given just in the form of the Heegard decomposition of genus R" & 0, thus the topological structure of the orbifold is the three-dimensional sphere S ( C S) with four marked points corresponding to the 0-D stratum. This result agrees perfectly with the result of the analytical treatment made in Section 2.6. Using a non-polynomial transformation of the variables m, k, l to new ones of the type arccos m, arccos(k#l), arccos(k!l) , (A.4) we can represent the orbifold in a more simple geometrical way (as two tetrahedra with identi"ed surfaces) but with the same topological properties (see Fig. 15). These non-polynomial variables may be interpreted as angles in the (x"j#k, y"j!k) representation characterizing angles between x and y and the symmetry axis. For the group C an example of the simplest e!ective invariant operator may be written in the form of the linear combination of the invariant polynomials " m# k# l H c c c . (A.5) In fact, if the auxiliary numerator invariant p does not enter in Hamiltonian (A.5) the complete ' symmetry of this Hamiltonian is higher, namely it is C TQ. To get the generic Hamiltonian it is necessary to choose the coe$cients cG in such a way that any section of the orbifold includes at most L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 235 one critical orbit. Let us choose one particular form H"2k!l . (A.6) We have four sections of the orbifold which go through the critical orbits and three connected components of the generic sections. They are denoted by letters according to their energy: a, E"!2; b, !1'E'!2; c, E"!1; d, 1'E'!1; e, E"1; f, 2'E'1; g, E"2. Four sections a, c, e, g corresponding to energies of critical orbits are shown in the Fig. 16. If we vary the coe$cients in Eq. (A.5) the orientation of the constant energy level planes with respect to the orbifold changes. This means that critical orbits which are stationary for any choice of Hamiltonian can change its stability. One particular simple physical example of a hydrogen atom in parallel electric and magnetic "elds corresponds to Hamiltonian (A.5) in the simplest approximation (see Section 4.3).

A.3. CT invariant Hamiltonian " Orbifolds for other one-dimensional Lie subgroups of O(3), G CT, CF, D, DF, can be constructed from that for the C group. It is su$cient to take into account the action of G/C on invariant polynomials given by Table 6 and to note that the action on orbits is equivalent to the action on invariant polynomials. p The orbifold for the CT subgroup results from that of a C one by noting that the T operation relates two orbits which belong to the same part of the C orbifold characterized by a given sign of the p invariant. So it is su$cient to take the k'0 parts of both 3D-bodies corresponding to p m l k di!erent signs of . To properly represent the CT orbifold we use the , , variables which are m" mO$ the invariant polynomials for the CT group. In such variables each const., 1, section of the orbifold is a parabola. The orbifold is shown in Fig. 17.

" k!l p5 Fig. 16. Representation of energy levels of the Hamiltonian H 2 on the C orbifold. Only one part ( 0) of the orbifold is shown.

R Fig. 17. Orbifold for CT group action on . Two parts corresponding to di!erent signs of the auxiliary polynomial p should be glued together through the identi"cation of respective points on the boundary p"0. 236 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

It should be noted that the identi"cation of points on the surfaces of C orbifold results in the identi"cation of the C points on the surfaces of two parts of the CT orbifold. All points on the surfaces which do not belong to the CQ stratum should be identi"ed by pairs. At the same p5 p4 time the CQ invariant orbits lying on 0 and on 0 parts of the CT orbifold at k"0, pO0 are di!erent. This is due to the fact that they correspond to pO0. We can again use non-linear transformation of variables to reach a more simple geometrical form of the orbifold. In new variables arccos m, arccos((k#l), arccos((k!l) , (A.7) each part of the orbifold is a tetrahedron shown in Fig. 18. These non-polynomial variables may be again interpreted as angles in the (x"j#k, y"j!k) representation characterizing angles between x and y and the symmetry axis. From the topological point of view the orbifold is a 3D-ball, B, with one marked point (C orbit) inside and with the S surface which includes 2D-stratum (CQ) plus two isolated points (CT orbits). The schematic topological structure of the orbifold is shown in Fig. 19. It is useful to note that the sub-manifold formed by both CQ and CT strata is a closed 3D-manifold invariant with respect to the CQ subgroup of the CT invariance group. Its topological structure is the suspension of the 2D-torus (see Section 2.6 for discussion of the topology of this closed sub-manifold). This fact is important for a detailed classi"cation of the CT-invariant Morse}Bott functions.

A.4. CF invariant Hamiltonian p p l For the CF subgroup the F operation relates C orbits with opposite and values. In order to have the one-to-one correspondence between orbits and invariant polynomial values, we must unify at one point of the CF orbifold pairs of orbits of the C orbifold according to the rule (m, k, l)"(m, k,!l) . (A.8)

R " m " Fig. 18. Orbifold for the CT group action on represented in non-polynomial variables k arccos , k ar- "k"#l " "k"!l ccos( ), k arccos( ). Only one part of the orbifold is shown.

R Fig. 19. Schematic topological structure of the orbifold for CT group action on . L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 237

R lp Fig. 20. Orbifold for the CF group action on . Two parts corresponding to di!erent signs of auxiliary polynomial should be glued together through the identi"cation of respective points on the boundary lp"0.

R " m Fig. 21. Orbifold for the CF group action on represented in non-polynomial variables t arccos , " k#"l" " k!"l" t arccos( ), t arccos( ). Only one part of the orbifold is shown.

l" p" Orbits of C having 0, 0 are invariant with respect to the CF group. Thus, the orbifold l5 for the CF group may be constructed from a C one by taking only 0 parts of two bodies with the identi"cation of all corresponding points on the surfaces. This orbifold is shown in Fig. 20. l l As soon as invariant polynomial for CF has the form rather than , it is more meaningful to change the variables and to give the orbifold in m, k, l variables. In these variables each m"const., mO0, section has the form of a parabola 1!m 1!m l"! k# . (A.9) 1#m 2 m k l One should remark that the geometrical form of the CF orbifold in the , , variables is the m k l same as the geometrical form of the CT orbifold in the , , variables (whereas the system of strata is completely di!erent in the two cases). We can use more complicated non-polynomial variables

arccos m, arccos(k#(l), arccos(k!(l) (A.10) to reach the simpler geometrical form of the orbifold. It is shown in Fig. 21. These non-polynomial variables are again the angles characterizing the mutual positions of x, y and the symmetry axis in the (x, y) representation. From the topological point of view this orbifold is given in the form of the Heegard decomposition of genus 0. So the CF orbifold is a S manifold with one S marked circle and three marked points: one isolated (C orbit) point and two (CF orbits) lying on the S marked circle (formed by CQ, CG and CF strata). We remark again that there are two closed sub-manifolds formed each by two di!erent strata. One is formed by CQ and CF strata and another by CG and CF strata. These both manifolds are S spheres from the topological point of view. 238 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

A.5. D invariant Hamiltonian

In the case of the D symmetry the action of the G/C group on the C orbifold form larger orbits by relating orbits (m, k, l) and (m,!k,!l) with opposite p values. To construct the orbifold for the D group we "rst subdivide the C orbifold into parts with semide"nite signs of the lp kp kl m k l D group numerator invariants ( , , ). Such a splitting of the , , space of the C invariant polynomials into parts with speci"c signs of the D numerator invariants is shown in Fig. 22. Taking into account the action of the C operation on C orbits, the D orbifold may be m k l represented as four 3D-bodies shown in Fig. 23 in D invariant polynomial variables , , with the following identi"cation of faces, edges and vortexes: " ABCD abcd , (A.11) " ABCD abcd , " " ABC ABC, ACD acd , " " ACD acd, abc abc , (A.12) " " " AC ac AC ac , " " " AB AB ab ab , (A.13) " " " BC BC bc bc , " " " AD ad AD ad , " " " CD cd CD cd , " " " " " " A A a a, B B b b , " " " " " " C C c c, D D d d . (A.14)

m k l Fig. 22. Splitting of the space of the C invariant polynomial variables , , into parts with particular signs of D auxiliary (numerator) invariants. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 239

R m k l Fig. 23. Orbifold for the D group action on represented in polynomial variables , , . The identi"cation of faces, edges, and vortexes of four bodies is given in Eqs. (A.11)}(A.14).

In fact, we have constructed the representation of the orbifold as a simplicial complex. It consists of four 3D-simplexes, six 2D-simplexes, "ve 1D-simplexes and four 0D-simplexes. The Euler- PoincareH characteristics of this complex di!ers from zero + (!1)NaN"4!5#6!4"1 . (A.15) Here aN is the number or p-dimensional simplexes. As it is shown in Section 2.6, the orbifold is the suspension (RP).

A.6. DF invariant Hamiltonian p To go from the D orbifold to a DF one, it is su$cient to consider the action of the F operation p l p on the D orbifold. As soon as the F operation changes, simultaneously signs of and it relates orbits characterized by (lp'0, kp'0, kl'0) and (lp'0, kp(0, kl(0) and in a similar way orbits characterized by (lp(0, kp(0, kl'0) and (lp(0, kp'0, kl(0). As a consequence, for the DF orbifold we have only one body instead of each pair of bodies with the same geometrical form. The DF orbifold is shown in Fig. 24. It consists of two bodies. The points on two pairs of faces must be identi"ed whereas the third pair of faces (formed by the CQ stratum) must not be identi"ed. From the topological point of view 240 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

R m k l Fig. 24. Orbifold for the DF group action on represented in polynomial variables , , . Two parts corresponding to di!erent signs of the auxiliary polynomial lp should be glued together through the identi"cation of respective points on the boundary lp"0.

R Fig. 25. Schematic topological representation of the orbifold for DF group action on .

the DF orbifold is equivalent to a 3D-ball. The boundary is formed by the CQ, CT, CF, CT strata. Inside the ball there are CG, CQ, CF, and the generic C strata. The schematic topological view of the DF orbifold is given in Fig. 25. Several closed subspaces formed by di!erent strata are clearly seen in Fig. 25. It is important to verify that the Morse}Bott inequalities are satis"ed on all these subspaces.

Appendix B. Molien functions for point group invariants

This appendix deals with a technical question: How to describe the system of invariant polynomials on the R manifold in the presence of a non-linear action of the symmetry group. The schematic answer to this question was initially formulated in Section 4 of Chapter I. Some applications have been discussed in Chapter II and in Section 2.7 of the present chapter. Particular L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 241 example of the symmetry group G for the hydrogen atom on orthogonal electric and magnetic "elds was treated in more details in Section 4.4. The group G has an interesting physical meaning because it includes symmetry transformations which contain both spatial and time reversal operations but it is Abelian and this simpli"es signi"cantly the construction of the Molien function for invariants. Below we explain on several examples of point group symmetry with increasing complexity the procedure of the Molien function construction. The construction of invariant functions on the R manifold is based on the preliminary construction of the integrity basis on the six-dimensional space where the action of the symmetry group of the problem is linear. The Molien function and the invariants themselves for the six-dimensional space xy or kj may be found from known expressions for Molien functions and integrity bases for irreducible representations. Next step includes the restriction of the polynomial algebra on the sub-manifold R of the 6D-space. The general procedure of the restriction of the polynomial ring de"ned on the manifold to the sub-manifold was outlined in Chapter I and realized in several examples in Section 2.7. The sub-manifold R is de"ned in the six-dimensional space xy by the " " polynomial equations x , y . If the point group symmetry does not include improper rotations (inversion or re#ections) these polynomials may always be considered as denominator invariants. In such a case we just eliminate them from the integrity basis constructed for the 6D-space and the resulting integrity basis gives the basis for the 4D sub-manifold. For point groups which are not the subgroups of SO(3) we start with the consideration of the similar problem for the proper rotation subgroup and after that take into account the e!ect of improper symmetry elements working directly on the 4D-sub-manifold R.

B.1. C group

This trivial case is useful to demonstrate how to take into account the restriction of the polynomial ring on the sub-manifold. " We work in x y representation. On the 6D-space there are six basic invariants xG, yG (i 1, 2, 3) and the Molien function for invariants has a trivial form 1 M " . (B.1) ! (1!j)

! The ring of the invariant polynomials on the 6D-space P is the ring of all polynomials !" P P[x, x, x, y, y, y] . (B.2) To restrict the polynomial ring on the sub-manifold h " # # " x x x , (B.3) h " # # " y y y , (B.4) h h we are obliged to introduce two second-order polynomials, , given by Eqs. (B.3) and (B.4) as denominator invariants. This may be done by multiplying both the numerator and denominator of the Molien function in Eq. (B.1) by (1#j). The new form of the Molien function 1#2j#j M " (B.5) ! (1!j)(1!j) 242 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

! corresponds now to another description of the ring of invariants P which is considered as a free module:

!" h h ⅷ P P[x, x, , y, y, ] (1, x)(1, y) (B.6) u " u " u " u " with four auxiliary invariants 1, x, y, xy. We use the notation (a, b)(c, d)"(ac, ad, bc, bd) to show that four numerator invariants are represented as product of more simple terms. Having the form (B.5) of the Molien function and the form (B.6) of the module of invariant functions it is easy to make the restriction to the sub-manifold R. We just eliminate two h h R denominator invariants, , , corresponding to the equation de"ning . The resulting Molien function and the module of invariants on R are written as follows: 1#2j#j M "R" , (B.7) ! (1!j)

!" " ⅷ P R P[x, x, y, y] (1, x)(1, y) . (B.8) The choice of basic and auxiliary invariants is ambiguous and the one proposed here is just an example. The important point is that the integrity basis which includes four numerator invariants (including 1) and four denominator invariants may be constructed. To "nd now the integrity basis in the jk representation we can simply transform invariants from x, y to j, k representation.

B.2. C point group

Let us take x and y to coincide with the C-axis. In such a case x, x, y, y transform according to the B representation and x, y according to the A representation of the C point group. The Molien function for invariants constructed from 6D reducible representation 4B#2A has the form 1#6j#j M " . (B.9) ! (1!j)(1!j) The explicit form of the module of invariant functions on 6D-space may be easily given as well:

!" ⅷ P P[x, x, x, y, y, y] ((1, xx)(1, yy), (x, x)(y, y)) . (B.10) There are six denominator invariants and eight numerator invariants. We have in Eq. (B.10) second degree denominator invariants which may be replaced by invariants de"ning the sub-manifold R in Eqs. (B.3) and (B.4). After such a substitution, to make the restriction on R it is su$cient to omit two denominator invariants corresponding to equations de"ning R. The resulting Molien function for invariant polynomials on R and the description of the ring of invariant function as a module are as follows: 1#6j#j M "R" , (B.11) ! (1!j)(1!j)

!" " ⅷ P R P[x, x, y, y] ((1, xx)(1, yy), (x, x)(y, y)) . (B.12) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 243

To "nd the integrity basis in the jk representation we can use again the transformation from x, y to j, k representation.

B.3. CG point group

For point groups which are not the subgroups of SO(3) the procedure of constructing the Molien function for invariants on R and the integrity basis is slightly di!erent. We start with the consideration of the similar problem for the proper rotation subgroup of the point group. There are two groups CG (or in other notation S) and CQ which have no non-trivial rotational subgroups. So we use them to illustrate the construction of invariant functions for these cases. The CG group includes only one non-trivial symmetry operation, inversion. Its action on x, y variables corresponds to interchange x%y. Taking this action into account it is easy to make the transformation of the C invariant denominator and numerator polynomials (forming the module of invariant polynomials on R) into form with irreducible transformation properties with respect to the CG group:

!G " " # ! # ! ⅷ # ! P R P[x y, x y, x y, x y] ((1, x y)(1, x y)) . (B.13) R This form of the module of the C invariant function on is well adapted to transformation to the ! module of CG invariant functions. First, we change the denominator invariants (x y) and ! ! & ! & (x y) into (x y) xy and (x y) xy which are both C and CG invariants. This may be achieved by multiplying the numerator and denominator of the Molien function by #j R (1 ) . The module of the C invariant function on in such a case will include 16 auxiliary (numerator) invariants but among them only 8 are CG invariants. The resulting Molien function R and the module of the CG invariants on are written as follows: 1#j#4j#j#j M "R" , (B.14) !G (1!j)(1!j)

!G " " # # ⅷ u u u u u u u P R P[x y, x y, xy, xy] (1, , , , , , , ), u " # u " u " ! ! x y, xy, (x y)(x y) , (B.15) u "u u u "u u u " ! ! , , (x y)(x y) , (B.16) u " ! ! (x y)(x y) . (B.17)

B.4. CQ point group R The construction of the module of the CQ invariant functions on is very similar to the realized above construction for the CG invariants. The only di!erence is that the action of the CQ non-trivial operation (re#ection in the symmetry plane) is now di!erent from the action of the inversion for the CG group. Let us suppose the symmetry plane to be orthogonal to x (y) axes. In such a case ! ! # (x y) and (x y) are symmetrical with respect to the CQ group whereas (x y) and 244 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

# (x y) are anti-symmetrical. Taking this into account the Molien function and the module of R CQ invariants on are written immediately as follows: 1#j#4j#j#j M "R" , (B.18) !Q (1!j)(1!j)

!Q " " ! ! ⅷ u u u u u u u P R P[x y, x y, xy, xy] (1, , , , , , , ) , (B.19) u " # u " u " # # x y, xy, (x y)(x y) , (B.20) u "u u u "u u u " ! # , , (x y)(x y) , (B.21) u " ! # (x y)(x y) . (B.22)

B.5. CT point group

For the point group CT which is not a subgroup of SO(3), we start again with the consideration of the similar problem for the proper rotation subgroup C. We can take the Molien function for the C invariant and integrity basis for the C group as the initial point. All invariants of C span invariants and pseudo-invariants of CT (A A representations). So we "rst make the linear transformation of numerator and denominator invariants for the C group resulting in a new set of C invariants which are at the same time either of A or of A type with respect to the CT group. This linear transformation does not change the form of the Molien function. It is of the form (B.11) R found for the C invariants on . However, this transformation changes the basis of the module of R C invariant functions on . To "nd proper linear combinations which accordingly transform irreducible representation of p p the CT group we take into account the fact that two symmetry operations ( , ) which belong to the CT group but do not belong to the C group may be represented as products of inversion and the C rotation around the axis orthogonal to the re#ection plane. We note again that the p p action of the inversion results in the interchange of xG with yG and the action of , on xG, yG may be represented as follows: p % ! " p % xH yH,(j 1, 3); x y , (B.23) p % ! " p % xH yH,(j 2, 3); x y . (B.24) So, instead of four denominator invariants (x, y, x, y) we should take two linear combinations h " # h " ! Q (x y) and Q (x y) which are invariant with respect to CT and two combinations h " ! h " # ? (x y) and ? (x y) which are pseudo-invariants of type A with respect to CT. In a similar way, we form new numerator invariants. Six numerator invariants are of type A u " ! u " u " u " ! with respect to CT:1, Q xx yy, Q xy, Q xy, V xy xy, u " Q xxyy. There are equally two numerator C invariants which are pseudo-invariants of u " # u " # the type A with respect to CT: ? xy xy, ? xx yy. The module of R C invariant functions of is represented now as

!" " h h h h ⅷ u u u u u u u P R P[ Q, Q, ?, ?] (1, Q, Q, Q, Q, Q, ?, ?) . (B.25) Now, we can transform the basis of the module of the C invariant functions in such a way that new denominator invariants become CT invariants rather than pseudo-invariants. To do that it is L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 245 su$cient to multiply the numerator and denominator of the Molien function by (1#j)(1#j). h This action corresponds simply to the substitution of two C denominator invariants ? and h ? by their squares which are CT invariants. The new representation of the module of the R C invariants on takes the form

!" " h h h h ⅷ u u u u u u u h h P R P[ Q, Q,( ?) ,( ?) ] (1, Q, Q, Q, Q, Q, ?, ?)(1, ?)(1, ?) . (B.26)

To make the restriction to CT invariant functions it is su$cient now to take only those numerator invariants which are CT invariants. The representation of the module of CT invariant functions on R takes the form

!T" " h h h h ⅷ h h +u h h +u P R P[ Q, Q, ?, ?] ((1, ? ?)(1, GQ), ( ?, ?)( G?)) (B.27) including 16 numerator invariants. The corresponding Molien function has the form (B.28)

1#4j#3j#3j#4j#j M "R" (B.28) !T (1!j)(1!j)(1!j)

1#3j#3j#j " , (B.29) (1!j)(1!j) which turns out to be reduced to a more simpler one in Eq. (B.29) which includes only eight numerator invariants. Construction of the integrity basis corresponding to the reduced form of the Molien function (B.29) requires additional analysis because we should "nd three second- order algebraically independent invariants. At the same time the straightforward procedure realized above gives one of the possible basis of the module of invariant functions although not a minimal one.

B.6. DF group We begin by constructing the Molien function of D invariants over the 6D space x y. This 6D vector space is the six-dimensional reducible representation of the form (E A) (E A). The corresponding Molien function is

1#2j#4j#10j#4j#2j#j M " . (B.30) " (1!j)(1!j)

Its restriction on the R subspace has the form

1#2j#4j#10j#4j#2j#j M "R " . (B.31) " (1!j)(1!j) R To go now to the DF group we "rst rewrite the Molien function for the D invariants on using j j j two auxiliary variables Q, ? instead of one . We use subscript to distinguish the behavior of D invariants with respect to re#ection. 246 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

R First, the Molien function for the D invariants on with two parameters has the form # j# j# j# j# j#j# j# j# j 1 2 Q 2 Q 6 Q 2 Q 2 Q Q (2 ? 4 ? 2 ?) M "R " . (B.32) " !j !j !j !j (1 Q )(1 ?)(1 Q )(1 ?)

We transform it to new denominator invariants which are at the same time the DF invariants: # j# j# j# j# j#j # j# j# j (1 2 Q 2 Q 6 Q 2 Q 2 Q Q ) (2 ? 4 ? 2 ?) M "R " (1#j)(1#j). " !j ! j !j ! j ? ? (1 Q )(1 ( ?) )(1 Q )(1 ( ?) ) (B.33)

To pass to DF invariants one must take only those numerator D invariants which are DF invariants. The Molien function for DF invariants is written in the form with the only parameter j

(1#j)[1#2j#2j#6j#2j#2j#j] M "R " "F (1!j)(1!j)(1!j)(1!j)

(j#j)[2j#4j#2j] # , (B.34) (1!j)(1!j)(1!j)(1!j) which is equivalent to

1#2j#2j#6j#5j#8j#8j#5j#6j#2j#2j#j M "R " . "F (1!j)(1!j)(1!j)(1!j) (B.35)

There are four DF denominator invariants (degree 2, 3, 4 and 6) and 48 numerator invariants. Remark that this is not the simplest form of the Molien function because the numerator may be factorized and the (1#j) factor may be simultaneously eliminated from numerator and denominator resulting in the reduced form of the Molien function

1#j#2j#5j#3j#3j#5j#2j#j#j M "R " . (B.36) "F (1!j)(1!j)(1!j)

Although the question is open how to construct integrity basis corresponding to this simpli"ed Molien function in Eq. (B.36), the construction of the non-minimal integrity basis corresponding to the form in Eq. (B.35) of the DF Molien function is straightforward. To give the explicit form of the integrity basis over the six-dimensional space we need the basis for all invariants and covariants over three-dimensional space. Instead of xG and yG we can use irreducible tensors with respect to the D group on each 3D-subspace to construct the integrity R basis on of functions invariant with respect to the D group action on the 6D-space. First of all we give the explicit form of D invariants and covariants on three-dimensional space which form the basis of the module of polynomials. There are three basic (denominator) invariants: # c " ! x x, x, (x) x 3xx (B.37) L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 247 and one auxiliary (numerator) invariant p p " ! x (x), with (x) 3xx x . (B.38) p There are two auxiliary covariants of type A (degree 1 and 3), namely x and (x), and four pairs of auxiliary covariants of type E (degree 1, 2, 2 and 3)

x x!x x x 2x x x , , , . (B.39) A B A! B A! B A ! B x 2xx xx (x x)x Taking into account the form of these invariants and covariants for 3D x and 3D y spaces and the R expression for the Molien function in Eq. (B.31) for D invariants on we can write the following R representation for the module of D invariant functions of :

" " " c c ⅷ u u P R P[x, (x), y, (y)] (1, ,2, ) . (B.40) There are 24 (including 1) numerator invariants. They are listed below in the form which shows clearly that they are simply invariants produced by coupling x and y covariants: u " p u " p u " p p x (x), y (y), xy (x) (y) , (B.41) u " u " p u " p xy, xy (y), yx (x) , (B.42) u " p p x (x)y (y) , (B.43) x y x y!y u " , u " , (B.44) A BA B A BA! B x y x 2yy x y y x 2y y y u " , u " , (B.45) A BA! B A BA ! B x yy x (y y)y x!x y x!x y!y u " , u " (B.46) A! BA B A! BA! B 2xx y 2xx 2yy x!x y y x!x 2y y y u " , u " , (B.47) A! BA! B A! BA ! B 2xx yy 2xx (y y)y x x y x x y!y u " , u " (B.48) A! BA B A! BA! B xx y xx 2yy x x y y x x 2y y y u " , u " (B.49) A! BA! B A! BA ! B xx yy xx (y y)y 2x x x y 2x x x y!y u " , u " , (B.50) A ! BA B A ! BA! B (x x)x y (x x)x 2yy 248 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

2x x x y y u " , (B.51) A ! BA! B (x x)x yy 2x x x 2y y y u " . (B.52) A ! BA ! B (x x)x (y y)y We change the basis of the module of D invariant functions in such a way that all invariants become invariants or pseudo-invariants for the DF group. To do this it is su$cient to form simple linear combinations of denominator and numerator invariants used in Eq. (B.40). We remark that p the action of the F operation which should be added to go from D group to a DF one is given by p %! " p % FxH yH,(j 1, 2) and Fx y. The new representation of the module of D invariant functions corresponding to the Molien function in Eq. (B.32) with two parameters is as follows:

" " " h h h h ⅷ u u u u P R P[ Q, Q, ?, ?] (1, Q, 2, Q, ?, 2, ?) , (B.53) h " # h "c !c Q x y, Q (x) (y) , (B.54) h " ! h "c #c ? x y, ? (x) (y) , (B.55) u "u !u u "u #u u "u !u Q , Q , Q , (B.56) u "u !u u "u #u u "u #u Q , Q , Q , (B.57) u "u !u u "u !u u "u u "u Q , Q , Q , Q , (B.58) u "u u "u u "u u "u u "u Q , Q , Q , Q , Q , (B.59) u "u #u u "u !u u "u #u ? , ? , ? , (B.60) u "u #u u "u !u u "u !u ? , ? , ? , (B.61) u "u #u u "u #u ? , ? . (B.62) h h h h We change now two denominator invariants ?, ? into ( ?) ,( ?) which are DF invariants. This results in increasing the number of D numerator invariants four times. But we are interested only in those numerator invariants which are DF invariants as well. There are 48 such invariants which correspond to the form (B.35) of the Molien function. The structure of the module of R DF invariant functions on may be now given explicitly

" " " h h h h ⅷ h h u u h h u u P R P[ Q, Q,( ?) ,( ?) ] ((1, ? ?)(1, Q ,2, Q), ( ?, ?)( ?, 2, ?)) . (B.63) There are four denominator invariants and 48 numerator invariants which may be reconstructed from Eqs. (B.41)}(B.52), (B.53)}(B.62). Apparently, the smaller integrity basis with only 24 numerator invariants may be found as the form (B.36) of the Molien function indicates.

¹ B.7. B point group ¹ R To construct the Molien function for the B group invariants on we use exactly the same ¹ procedure as for the DF group but we start now from the group in (x y) representation. The L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 249

Molien function for the invariants of the ¹ group on R has the form

(1#j)#2(j#j)#(j#j#2j#j#j) M "R" . (B.64) 2 (1!j)(1!j)

More detailed form of the Molien function with two parameters may be given after the transforma- ¹ ¹ tion of the invariants into the form irreducible with respect to B (A or A symmetry types with ¹ j ¹ ¹ respect to B). We use below Q for those -invariants which are at the same time the B invariants j ¹ ¹ and ? for those -invariants which are pseudo-invariants (of the A type) of the B group. #j#j# j# j# j# j# j#j#j#j " "(1 Q Q 5 Q 3 Q 8 Q 3 Q 5 Q Q Q Q ) M2 R !j !j !j !j (1 Q )(1 ?)(1 Q )(1 ?) j# j# j# j# j# j#j # ( ? 2 ? 3 ? 6 ? 3 ? 2 ? ?) !j !j !j !j . (B.65) (1 Q )(1 ?)(1 Q )(1 ?) ¹ R Now to form the Molien function for the B invariant on we multiply both numerator #j #j and denominator by (1 ?)(1 ?) and take in the numerator only those terms which are ¹ B invariant # jj #j#j# j# j# j# j# j#j#j#j (1 ( ? ?))(1 Q Q 5 Q 3 Q 8 Q 3 Q 5 Q Q Q Q ) M "R " 2B !j ! j !j ! j (1 Q )(1 ( ?) )(1 Q )(1 ( ?) ) j#j j# j# j# j# j# j#j # ( ? ?)( ? 2 ? 3 ? 6 ? 3 ? 2 ? ?) !j ! j !j ! j . (B.66) (1 Q )(1 ( ?) )(1 Q )(1 ( ?) ) The total number of numerator invariants now is 96. If we use only one auxiliary parameter j we ¹ can simplify considerably the Molien function for B invariants. Formula (B.66) becomes (1#j)(1#j)(1#j)(1#4j#2j#4j#j) M "R " (B.67) 2B (1!j)(1!j)(1!j)(1!j)

1#4j#2j#4j#j " . (B.68) (1!j)(1!j)(1!j) ¹ R The last simpli"ed form in Eq. (B.68) of the Molien function for the B invariants on seems to be rather reasonable. Probably, it gives the minimal basis of denominator and numerator invariants on R. If the integrity basis corresponding to this simpli"ed form (B.68) exists it may be used in some applications. Number (12) of auxiliary invariants is not enormous. The initial form (B.67) of the integrity basis including 96 numerator invariants is not encouraging at all. To conclude this appendix we give the generating Molien functions for icosahedral symmetry > and >F: N(>) N(> ) g(>)" , g(> )" F , (B.69) (1!x)(1!x) F (1!x)(1!x) 250 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 where N(>)"1#x#x#6x#6x#6x#6x#15x # 16x#15x#16x#15x#32x#15x # 16x#15x#16x#15x#6x#6x #6x#6x#x#x#x (B.70) and " # # # # # # N(>F) 1 x x 3x 3x 3x 3x # 7x#8x#7x#8x#7x#16x#7x # 8x#7x#8x#7x#3x #3x#3x#3x#x#x#x . (B.71)

Appendix C. Strata and orbits for point groups

R ¹ We gave in Section 2.8, strata and orbits in action on of three point groups CT, DF and B. This choice is due to the fact that among polyatomic Rydberg molecules studied experimentally or theoritically molecules with such symmetry groups are most typical. We can cite examples of H (DF symmetry) (Bordas and Helm, 1991, 1992; Dodhy et al., 1988; Helm, 1988; Lembo et al., 1989; Herzberg, 1981, 1987; Ketterle et al., 1989; King and Morokuma, 1979; Lembo et al., 1990; Pan and Lu, 1988; Stephens and Greene, 1995), Na (DF symmetry) (Broyer et al., 1986), HO ¹ (CT symmetry) (Petfalakis et al., 1995), HF(CT symmetry) (Bordas et al., 1985) and NH ( B symmetry) (Herzberg, 1981, 1987; Herzberg and Hougen, 1983; Watson, 1984). At the same time many other molecules are potentially interesting from the point of view of their excited Rydberg states or even Rydberg character of the ground state (Basov and Pavlichenkov, 1994; Boldyrev and Simons, 1992a,b; Chiu, 1986; Mayer and Grant, 1995; Wang and Boyd, 1994; Weber et al., 1996; Wright, 1994). Their symmetry groups vary from very simple CQ for HCO (Mayer and Grant, 1995) till the highest icosahedral symmetry I) for C (Weber et al., 1996). That is why we give in this appendix orbits and strata for all possible "nite symmetry groups. The list of critical orbits for each symmetry group enable us to give the description of simplest Morse-type functions (see Tables 18 and 19 in Section 3.3). Tables C.1}C.8 listed in this appendix show in column 1 the stabilizer of the stratum. In column 2 closed and generic strata are indicated. We remark once more that some strata are neither closed nor generic. In column 3 the dimension of the stratum is given. For the "nite group action on R, the dimension of the generic stratum is always 4. Column 4 gives the number of orbits in the stratum. If the dimension of the stratum is zero the number of orbits in the stratum is "nite. If the dimension of the stratum is positive the number of orbits in the stratum is in"nite. We denote it RL with n equal to the dimension of the stratum. Column 5 shows the number of points in each orbit. For "nite group actions on R this number is always "nite. L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 251

Table C.1 5 R ` a Strata and orbits in the action of low symmetry groups C, CL, CQ, CLF, S, SL (n 2) on . Column gc indicates generic (g) and closed (c) strata