rydberg states of atoms and molecules. basic group theoretical...

Physics Reports 341 (2001) 173}264

Symmetry, invariants, topology. III

Rydberg states of atoms and molecules.Basic group theoretical and topological analysis

L. Michel!,1, 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 1751.1. Dynamical symmetry of Rydberg states 178

2. Groups and their actions appropriate for theRydberg problem 1792.1. Structure of O(4) 1792.2. The adjoint representation of O(4); induced

action on R"S2]S

2182

2.3. Action of O(3), SO(3), O (3)?T,SO(3)?T, and SO(3)?T

son R; their

strata, orbits and invariants 1852.4. Invariants of the one-dimensional Lie

subgroups of O (3) acting on R 1882.5. One-dimensional Lie subgroups of

O(3)?T and their invariants 1912.6. Orbits, strata and orbit spaces of the one-

dimensional Lie subgroups of O(3) actingon R 192

2.7. Invariants of "nite subgroups of O(3)acting on R 196

2.8. Orbits, strata and orbit spaces of "nitesubgroups of O (3) acting on R 197

2.9. Orbits, strata and orbit spaces forT-dependent subgroups of O(3)?T 198

3. Construction and analysis of RydbergHamiltonians 2033.1. E!ective Hamiltonians 203

3.2. Qualitative description of e!ectiveHamiltonians invariant with respect tocontinuous subgroups of O(3) 205

3.3. Qualitative description of e!ectiveHamiltonians invariant with respect to"nite subgroups of O(3) 210

4. Manifestation of qualitative e!ects in physicalsystems. Hydrogen atom in magnetic andelectric "eld 2134.1. Di!erent "eld con"gurations and their

symmetry 2134.2. Quadratic Zeeman e!ect in hydrogen

atom 2154.3. Hydrogen atom in parallel electric and

magnetic "elds 2174.4. Hydrogen atom in orthogonal electric and

magnetic "elds 2234.5. Where to look for bifurcations? 227

5. Conclusions and perspectives 228Appendix A. Geometrical representation 231

A.1. O(3) or SO(3) invariant Hamiltonian 231A.2. C

=invariant Hamiltonian 233

A.3. C=v

invariant Hamiltonian 235A.4. C

=hinvariant Hamiltonian 236

A.5. D=

invariant Hamiltonian 238A.6. D

=hinvariant Hamiltonian 239

*Corresponding author.E-mail address: [email protected] (B.I. ZhilinskimH ).1Deceased 30 December 1999.

0370-1573/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 9 0 - 9

Appendix B. Molien functions for point groupinvariants 240B.1. C

1group 241

B.2. C2

point group 242B.3. C

ipoint group 243

B.4. Cspoint group 243

B.5. C2v

point group 244B.6. D

3hgroup 245

B.7. ¹dpoint group 248

Appendix C. Strata and orbits for point groups 250Appendix D. Qualitative description of e!ective

Hamiltonians based on equivariant Morse}Botttheory 255D.1. SO (3) continuous subgroup 255D.2. C

=continuous subgroup 256

References 258

Abstract

Rydberg states of atoms and molecules are studied within the qualitative approach-based primarily ontopological and group theoretical analysis. The correspondence between classical and quantum mechanics isexplored to apply the results of qualitative (topological) approach to classical mechanics developed byPoincareH , Lyapounov and Smale to quantum problems. The study of the action of the symmetry group of theproblems considered on the classical phase space enables us to predict qualitative features of the energy levelpatterns 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

174 L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264

1. Introduction

This chapter is devoted to the qualitative analysis of Rydberg states of atoms and moleculesbased on the extensive use of group actions combined with topological arguments introduced inChapter I. Some results of such applications are published (Sadovskii et al., 1996; Sadovskii andZhilinskii, 1998; Cushman and Sadovskii, 1999) but there are still many open problems to studywithin the formalism developed here. Rydberg states of atoms and molecules are very slightlybound quantum states of an electron and a positively charged ion. In the approximation one canneglect the excitations of the ionic core and all spin e!ects, the spectrum of states has a structuretypical of that of the hydrogen atom: grouping in multiplets of n2 states for the large principalquantum number n. In the following, we shall restrict ourselves to the analysis of the internalstructure 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; Stebbingsand Dunning, 1983). There are many works on Rydberg atoms isolated or in di!erent con"gura-tions of magnetic and electric "elds (Beims and Alber, 1993; Boris et al., 1993; Cacciani et al., 1988,1989, 1992; Fabre et al., 1977, 1984; Flothmann et al., 1994; Frey et al., 1996; Fujii and Morita, 1994;Hulet and Kleppner, 1983; Jacobson et al., 1996; Jones, 1996; Konig et al., 1988; Lahaye andHogervorst, 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 studyof Rydberg molecules is beginning (Bordas et al., 1985, 1991; Bordas and Helm, 1991, 1992; Broyeret al., 1986; Dabrowski et al., 1992; Dabrowski and Sadovskii, 1994; Davies et al., 1990; Dietrichet 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 andItzykson, 1966; Bixon and Jortner, 1996; Braun, 1993; Braun and Solov'ev, 1984; Chiu, 1986; Clarket 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; Panand 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 beobtained by a general qualitative analysis of the relevant problems based on general methods thatwe have introduced in the initial Chapter I of this issue. Further applications to atomic andmolecular 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 smallcompared to the splitting between them. The dynamical system which describes the internalstructure of an individual n multiplet has two degrees of freedom. For large n, the set of n2 quantum

L. Michel, B.I. Zhilinskin& / Physics Reports 341 (2001) 173}264 175

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

O(3) Rydberg atoms with the closed shell coreC

=vRydberg atoms in the presence of a weak electric "eldHeteronuclear diatomic Rydberg molecule AB

C=h

Rydberg atoms in the presence of a weak magnetic "eldC

=Rydberg atoms in the presence of parallel magnetic and electric "elds

D=h

Homonuclear diatomic Rydberg molecules A2

Point group G Polyatomic Rydberg molecules with the point group symmetry GH

3(D

3h), NH

4(¹

d),2

levels is su$ciently large to be well studied by classical analysis, as a Hamiltonian dynamicalsystem de"ned on a four-dimensional phase space with non-trivial topology and symmetry that weshall soon make precise.

Second, the isolated hydrogen atom has a large dynamical symmetry: that of the orthogonalgroup O(4) and time reversal. As we shall recall, that is an exact symmetry for the quantummechanical study, in the non-relativistic approximation, of an electron in the static Coulombpotential of a point-like nucleus; then the energy of bound states depends only on the principalquantum number n; that exceptional energy degeneracy between the states of di!erent orbitalmomenta of value l, 04l4n!1 contains + l (2l#1)"n2 states whose state vectors form thespace 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 exactat the asymptotic limit nPR when there are no external "elds. Their geometric symmetry is thatof the positive ion: it is at most O(3) (case of the isolated hydrogenoid atom) and it is a "nitesubgroup of O(3) in the case of non-aligned molecules. Several important physical examples ofgeometric symmetry are listed in Table 1. Further extension of geometric invariance groups tohigher 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 theShroK dinger equation. In a convenient unit system, the Hamiltonian for a hydrogen atom can bewritten

H"

12k

p2!1r, E

n"!

12n2

. (1)

Due to the speci"c Coulomb interaction there are two vector integrals of motion: J } the angularmomentum vector, and X } the Laplace}Runge}Lenz vector,

X"p]J!rr~1 . (2)

In other words, the Hamiltonian operator H commutes with the vector operators J and X and alltheir functions.

After the energy-dependent scaled transformation

K"X(!2En)~1@2"Xn , (3)


it was shown by Hulthen (1933) and Fock (1936), that J and K form a Lie algebra with de"ningcommutation relations:

[Ja , Jb]"ieabcJc , (4)

[Ja , Kb]"ieabcKc , (5)

[Ka ,Kb]"ieabcJc . (6)

We shall show in Section 2.1 that this Lie algebra is the one of the group O(4) or SO(4) (or severalother groups). In the hydrogen atom the operators J, K satisfy moreover the two importantrelations:

J )K"K ) J"0 , (7)

J2#K2"n2!1 , (8)

which give the value of the two SO(4) Casimir operators for the Hilbert space of vector states withthe energy E

n. Instead of J and K we can introduce the two linear combinations:

J1"(J#K)/2 , (9)

J2"(J!K)/2 . (10)

Using only relations (4) and (5) one veri"es that these operators satisfy

[Jka , Jkb]"ieabcJkc , k"1, 2 , (11)

[J1a ,J2b]"0 . (12)

That shows that the Lie algebra of SO(4) is isomorphic to the Lie algebra of the direct productS;(2)]S;(2) of two groups S;(2). Therefore, an irreducible representation of SO(4) can be labeledby a pair ( j

1, j2); its dimension is (2j

1#1)(2j

2#1). Relations (9) and (10) imply J2

1!J2

2"J )K.

Then (7) is equivalent to

J21"J2

2. (13)

This last relation proves that the relevant SO(4) irreducible representations for the Rydbergproblems are of the form ( j, j) with n"2j#1; hence its dimension is n2.

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 inChapter I and used for molecular problems in Chapter II where we introduce the general approachby using examples from molecular physics which are less complicated than Rydberg state problemfrom the point of view of the mathematical technique and give to the reader the opportunity to getthe 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).


Section 2 will give a more thorough study of the symmetry we have to use. The main result of thissection is the description of the symmetry group action on the phase space, construction of therings of the polynomial invariant functions and the orbifolds for all cases of the continuousinvariant symmetry groups of the Rydberg problem and for "nite point symmetry groups.Section 3 gives the construction and properties of phenomenological Rydberg Hamiltonians basedon the integrity bases introduced in Section 2. Section 4 will study some examples of qualitativee!ects. We restrict ourselves with the application of the technique developed to the problem of thehydrogen atom in external electric and magnetic "elds. In spite of the fact that this problem seemsto 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 featuresand to give di!erent (and we hope useful) interpretations of some qualitative modi"cations of thedynamical behavior. In particular, the simple geometrical description of the collapse phenomenonis given recently by Sadovskii et al. (1996). It is based on the orbifold construction discussed indetail in this chapter. The conclusion summarizes brie#y further steps to apply the techniquedeveloped in this chapter to a much wider class of molecular problems. The appendicesgive more details about the geometrical representation of Rydberg orbifolds, explain moretechnical mathematical constructions like Molien functions, and collect some auxiliary tablesneeded for further application of the qualitative approach to molecular problems with "nite pointsymmetry group.

1.1. Dynamical symmetry of Rydberg states

Now, we can formulate the classical limit construction for the Rydberg problem studied. Generalscheme 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 withintroducing a dynamical algebra g whose generators play the role of the dynamic variables of theproblem. 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 (seeSection 1, Eqs. (4)}(6)). An equivalent way to represent the same algebra is to use two commutingvector operators J

1and J

2. 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 geometricalsigni"cance of this space is clearly seen in the J

1, J

2representation. Di!erent points of the classical

limit phase space are in one-to-one correspondence with orientations of two vectors J1, J

2. This

space is the topological product of two two-dimensional spheres S2: S

2]S

2. 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 classicalphase space for the Rydberg problem). Following steps of the qualitative analysis of the Rydbergproblem may be formulated now as follows:

(i) Study of the action of the invariance group on R, classi"cation of orbits and strata. Finding thecritical orbits (see Section 2).

(ii) Application of Morse theory to the complexity classi"cation of functions de"ned over R in thepresence of the invariance symmetry group (see Section 3).


2The symbol ' is a shorthand for de"ning the groups generated by the groups or group element written on each sideof this symbol.

3The exact sequences (see Appendix A to Chapter I) used here are all of the type 1PAPBPCP1. They mean A isthe invariant subgroup of B and C"B/A. This notation is now currently used even in undergraduate studies. For moredetails see e.g. Michel (1980).

(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 descrip-tion of the corresponding quantum energy level patterns and wave functions (see Sections 3and 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 thetime reversal.2 For other physical systems this group can be used as "rst approximation: onlya subgroup of it is the exact symmetry. In the beginning of this section we build the group O(4)'Tand the phase space R of Rydberg problems. The rest of the section is devoted to the study of theaction 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 commuteswith 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 sameLie algebra, e.g. SO(3)]SO(3), O(3)]O(3), S;(2)C2 (de"ned in Eq. (26)), SO(4) and O(4) as shown inEqs. (22), (23) and (28). There is some ambiguity in choosing between these di!erent groups; buta non-connected one is de"nitely richer and better adapted to the physics, so we choose here O(4) asan 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 doublecovering called Spin(n). For n"3, SO(3) is the group of rotations in the three-dimensional spaceand its spin group Spin(3)"S;(2), the group of 2]2 unitary matrices with determinant 1. Therelation between these two groups is the homomorphism S;(2) pP SO(3). The image of thishomomorphism is the full SO(3) group and its kernel is the center MI

2,!I

2N of S;(2).We denote it

by Z2(!I

2). This whole information can be expressed in one-line way:3

1PZ2(!I

2)PS;(2) pP SO(3)P1 . (14)

This notation is an exact sequence. A less explicit notation is SO(3)"S;(2)/Z2(!I

2).

To understand better the interrelations between di!erent Lie groups having the samesu(2)]su(2) algebra we give below their realization as groups of transformations of the quaternions.

The elements q3Q, the quaternion "eld, can be represented by the 2]2 matrices (e.g. Duval,1964):

q"q4I#iq

1p1#iq

2p2#iq

3p3

, (15)


qa3R, a"1, 2, 3, 4, where the three Hermitian matrices pk"pH

kare the Pauli matrices:

p1"A

0 1

1 0B, p2"A

0 !i

i 0B, p3"A

1 0

0 !1B . (16)

Notice that q is not Hermitian. Indeed qH"q8 with

q84"q

4, q8

1"!q

1, q8

2"!q

2, q8

3"!q

3. (17)

We verify

qqH"qHq"DqD2I with DqD2"4+a/1

q2a (18)

where DqD"DqHD is called the norm of the quaternion q. Notice that DqD'0 when qO0 andDqq@D"DqD Dq@D. That last property can be checked from Eq. (18) or from

DqD2"det q . (19)

Eq. (18) shows that the set of quaternions forms a dimension 4 orthogonal space with the scalarproduct:

(q, q@)"12tr qq@H"

12tr q@qH"

4+a/1

qaq@a . (20)

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 DqD"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.the quaternions of norm 1, is S

3, the unit sphere in the dimension 4 orthogonal space of Q.

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 ofa given norm: indeed the quaternion DqDI is transformed into the arbitrary quaternion q of the samenorm by u"qDqD~1, v"I. This shows the existence of a group homomorphism

S;(2)]S;(2) hP O(4) , (22)

Im h"SO(4), Ker h"Z2(I,!I) . (23)

Indeed, h is continuous so its image is connected; the transitivity property we have just establishedrequires Im h"SO(4). By de"nition Ker h is the set of group elements acting trivially on Q; they areu"v"$I.

The stabilizer of the quaternions of the form q4I is made of the elements v"u; one calls it the

diagonal subgroup S;(2)dLS;(2)]S;(2). Moreover, S;(2)d transforms into itself the three-dimensional subspace of quaternions orthogonal to I (their trace vanishes); that establishes thewell-known homomorphism S;(2)PSO(3), [S;(2)/Z

2"SO(3)].


We shall consider the 4]4 orthogonal matrix

s"A!I

30

0 1B, s2"I, det s"!1 . (24)

It is the inversion through the origin for the subgroup O(3)LO(4) which leaves "xed the coordinateq4; as Eq. (17) shows, it transforms q into qH. Every element of O(4) not in SO(4) is of the form

sh(u, v). Let us study the induced action of these elements on S;2]S;

2from their known action

on every quaternion q:

sh(u, v)s~1 ) q"sh(u, v) ) qH"s ) (uqHvH)"vquH , (25)

it is the permutation of the two factors in S;2]S;

2. So we are led to consider the wreath product

S;2C2 which is de"ned as the semi-direct product:

S;2C2&(S;

2]S;

2) JZ

2(P) , (26)

where Z2(P) denotes the group of permutations of the two factors of S;

2]S;

2. To summarize, we

have established the two exact sequences (see Appendix A to Chapter I)

1PZd2(!I

2,!I

2)PS;

2]S;

2hP SO(4)P1 , (27)

1PZd2(!I

2,!I

2)PS;

2C2 h{P O(4)P1 . (28)

Notice that SO(4) and O(4) have a two-element center generated by the matrix !I4. Eq. (27) shows

that S;(2)]S;(2)"Spin(4). Its elements can be labeled by a pair of indices u1, u

23S;(2). Then

h(u1, u

2) de"nes an orthogonal matrix g3SO(4)

SO(4) U g&h(u1, u

2) . (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,042j3Z; so the irreducible representations of S;(2)]S;(2) are usually labeled by the pair ( j

1, j2).

They have the dimension (2j1#1)(2j

2#1). The representations of S;(2)C2 are labeled by

( j1, j2)=( j

2, j1) (of dimension 2(2j

1#1)(2j

2#1)) when j

1Oj

2. When j

1"j

2this representation

becomes reducible into the direct sum of two reducible representations ( j, j)B which are, respective-ly, symmetric and antisymmetric for the permutation of the two S;(2) factors. Among allrepresentations those with j

1$j

23Z form the set of irreducible representations of O(4). They are

faithful when j1, j2

are both half-integers; when j1, j2

are both integers their image is that of the`adjointa group of O(4), that is the quotient O(4)/Z

2(!I

4) of O(4) by its center. It is easy to see that

it is isomorphic to SO(3)C2. This can be summarized by

1PZ2(!I

4)PO(4) 0

P SO(3)C2P1 . (30)

Eq. (13) shows that the only irreducible representations carried by the n2-dimensional space of statevectors of a Rydberg multiplet have j

1"j

2. As we shall show they are ( j, j)B with n"2j#1. It is

why we say that it is O(4) and not S;(2)C2 which is the symmetry group of the Rydberg problem.


Remark that these representations are faithful only when j is half integer. That is the case of (12, 12)`,

the vector representation (i.e. that of dimension 4) of O(4) that we have studied in this subsection.

2.2. The adjoint representation of O(4); induced action on R"S2]S

2

The adjoint representation of O(4) is (1, 0)=(0, 1); indeed, by de"nition, it is the representation ofO(4) on the six-dimensional vector space of its Lie algebra. We denote this space by <"<

1=<

2direct sum of two three-dimensional vector spaces. The operators J

1, J

2de"ned in Eqs. (9) and (10)

act e!ectively on their respective space <1,<

2and trivially on the other one; that yields the

representation of Eqs. (11) and (12). A vector of the six-dimensional space <1=<

2is naturally

decomposed in a direct sum that we denote by (xy). Moreover, <

1=<

2is an orthogonal space with

the scalar product obtained as the matrix product

(x@ y@)AxA

yAB"x@ )xA#y@ ) yA . (31)

Each three-dimensional subspace is invariant by a SO(3). For the group SO(4), its matrixg&h(u

1, u

2) [see Eq. (29)] is represented by the matrix A(g) (written in 3]3 blocks):

A(g)"Ap(u

1) 0

0 p(u2)B , (32)

where p has been de"ned in Eq. (14). In the preceding section we have shown that s3O(4) (de"nedin Eq. (24)) corresponds to the permutation of the two three-dimensional subspaces (see Eq. (25)):

A(s)"A0 I

3I3

0 B . (33)

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<"<

1=<

2corresponding to the operators J

1, J

2, it is interesting to use the decomposition <"<

!=<

1corresponding to the e!ective action of the axial vector operator J and the polar vector operator K.This can be done by the matrix

S"S12A

I3

I3

I3

!I3B . (34)

The conjugation by S diagonalizes A(s):

SA(s)S~1"A@(s)"AI3

0

0 !I3B , (35)

SA(g)S~1"A@(g)"12A

p(u1)#p(u

2) p(u

1)!p(u

2)

p(u1)!p(u

2) p(u

1)#p(u

2)B . (36)


In this new decomposition we write the component vectors as

Aj

kB"SAx

yB, i.e. j"x#y

J2, k"

x!y

J2. (37)

We denote by gd the matrices of SO(4) which correspond through Eq. (57) to the pairs u"u1"u

2.

The matrices A@(gd) which represent them in the adjoint representation are block diagonal

A@(gd)"Ap(u) 0

0 p(u)B . (38)

They are the matrices of SO(4) which commute with A@(s). They form the subgroup SO(3)d actinge!ectively on the axes 1, 2, 3 of the vector representation of O(4) and leaving the axis 4 "x. Theorthogonal group O(3)d"SO(3)d's is the orthogonal group of the physical space; it transformsj3<

!, k3<

1, respectively, as axial and polar vectors.

2.2.1. Extension of O(4) by including the time reversalIn 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 ofangular momenta. More generally in physics, T changes the sign of axial vectors (e.g. the magneticinduction) 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-unitaryoperator. T is an exact symmetry of atomic and molecular physics (except for tiny e!ects inducedby weak interaction!) (Sakurai, 1964). We denote by O(4)'T the group generated by O(4) andT; itis 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) (generallycalled P, the parity operator) is de"ned in Eq. (35). Remark that O(4)'T is not the direct productof O(4) by Z

2(T) since T does not commute with the elements of O(4) not in the subgroup

O(3)d]Z(!I4) (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 representationand on the phase space R

We begin by studing the action of SO(4) on <. It is de"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 uniqueinvariant, the vector norm x ) x50 which de"nes the orbit space; the two strata are de"ned byx ) 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)WO(2)"SO(2), which are index2 subgroups of the stabilizers of O(3). Note that the vector representation of SO(3) coincides with itsadjoint representation. The adjoint representation of SO(3)]SO(3) is the direct sum of two adjointrepresentations of SO(3).

We obtain the strata on the space <"<1=<

2by using the general fact (see Chapter I):

In a reducible representation of a compact group, the stabilizer of a vector is the intersection of thestabilizers of its components in the irreducible subspaces.


Table 2Strata of the SO(4) adjoint representation

Stablizer! Invariants dim.str." Topology#

SO(3)]SO(3) x ) x"0"y ) y 0 One pointSO(2)]SO(3) x ) x'0"y ) y 3 S

2SO(3)]SO(2) x ) x"0(y ) y 3 S

2SO(2)]SO(2) x ) x'0(y ) y 6 S

2]S

2

! 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-algebraicdomain x ) x50, y ) y50. Its inside corresponds to the generic stratum. Its boundary is the unionof the three strata x ) x'0"y ) y, x ) x"0(y ) y, and x ) x"0"y ) y. The results are given inTable 2. To obtain the stabilizers in SO(4) one must take the inverse image 0~1 of those inSO(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): itexchanges the vectors x and y. To write the invariants of O(4) (or of its image SO(3)C2) we use thesymmetric 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 theinvariants of the O(4) action on < are i, o2 and the orbit space is de"ned by restrictionsi50, i25o250. The inside of the orbit space (obtained by replacing 5 by ') corresponds tothe generic stratum. The three non-generic strata form the boundary. Their equations are,respectively, i'0, o2"0; i'0, i2"o2'0; and i"0(No2"0). The corresponding orbits inthe image SO(3)C2 are SO(2)C2, SO(3)]SO(2), SO(3)C2.

Since i and o2 are also T invariant, O(4)'T has same orbits and same orbit space as O(4). Thecorresponding stabilizers [for the image (SO(3)C2)'T] are given in Table 3 where T

sis the

product Ts"sT.Taking into account the physical relation in Eq. (13) we have found that the phase space of

Rydberg problem has the topology S2]S

2and we have denoted it by R. In < this phase space is

de"ned by the equations

R&S2]S

2: i"1, o"0 . (42)

It is an orbit of O(4)'T [and even of SO(4)] belonging to the stratum o2"0 of stabilizer(SO(2)C2)'T.


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

Stablizer! Invariants" dim.str.# Topology$

(SO(3)C2)'T i"0"o2 0 One point(SO(2)]SO(3))'T

si2"o2'0 3 2S

2(SO(2)C2)'T i2Oo2"0 3 S

2]S

2(SO(2)]SO(2))'T

si2!o2'0 6 2(S

2]S

2)

! The image of the stabilizer is given. For the O(4) adjoint representation T or Ts

should be omitted." i50 is implicit.# dim.str. means the dimension of the stratum.$ The topological structure of each orbit is given.

2.3. Action of O(3), SO(3), O(3)'T, SO(3)'T, and SO(3)'Ts

on R; their strata,orbits and invariants

In the four-dimensional space, the subgroup SO(3) leaving invariant the fourth coordinate ischaracterized in Eqs. (32) and (36)

r3SO(3)% u1"u

2%SO(3)dLSO(3)]SO(3) . (43)

In Eq. (24) we have de"ned s, the space inversion of O(3)"SO(3)]Z2(s). The restriction of the

adjoint representation of O(4) to O(3) is reducible. The matrix S reduces this representation (thereduction is given explicitly in Section 2.2, see Eqs. (34)}(37)). This reducible representation of O(3)can be denoted by 1`=1~ where 1B is, respectively, the axial and polar vector representation. Inour problem, the axial and polar vectors j, k correspond, respectively, to the angular momentum andthe Runge}Lenz vector. They satisfy the relations in Eqs. (39) and (40) which completely characterizethe 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) thatwe have determined in Table 3. This gives immediately the classi"cation of the strata in the actionof O(3) on R. Indeed, the stabilizers in the three-dimensional subspaces of j and k are, respectively,O(3) for the null vector, C

=h( j) for jO0 and C

=v(k) for kO0. So for the points of R, when both j, k

are non-zero vectors (they satisfy j ) k"0) the stabilizer is Cs( j), the group generated by the

re#ection through the plane orthogonal to j. Table 4 summarizes the data of the three strata whichappear in the action of O(3) on R.

The stabilizers appearing in the action on R of the subgroup SO(3) are obtained by takingthe intersection of SO(3) with the stabilizers of the O(3) action. This shows that on R the orbits arethe same for O(3) and SO(3). However, the SO(3) orbits form only two strata: the same open densestratum with trivial symmetry and one stratum with symmetry C

=formed by two orbits m"$1.

Any SO(3)-invariant function on R is invariant by O(3) (see Table 4).


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

2are critical

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

O(3) C=v

(k) 2 m"!1 1 S2

C=h

( j) 2 m"1 1 S2

Cs( j) 4 !1(m(1 R RP3

SO(3) C=

2 DmD"1 2 S2

1 4 !1(m(1 R RP3

! 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 ofSO(n) (n53) are made from all distinct scalar products and determinants. Its application to theaction of SO(3) on the six-dimensional space < proves that i, m,o form a minimal set of generatorsof the invariant polynomials since they are algebraically independent. As a scalar product of anaxial vector j and a polar vector k, o is a pseudo-scalar for O(3). For this group, i, m,o2 form anintegrity basis. There is no di!erence in R between the PO(3) and PSO(3) invariant polynomial ringssince 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. So on <, 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 ringof invariant polynomials of SO(3) and O(3) are the polynomial rings:

PSO(3)6

"P[i, m,o] , (45)

PO(3)6

"P[i, m,o2] . (46)

Table 4 shows that any function on R, invariant under O(3) or SO(3), depends only on the variablem de"ned in Eq. (44); indeed this parameter labels completely the orbits (common to O(3) andSO(3)). The geometrical representation of the orbifold (which is a 1D-segment) is given in AppendixA. Table 4 also shows that the two orbits de"ned by m"$1 are isolated in their strata; thereforethey are critical: these two orbits are orbits of extrema of any O

3or 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

Aj

kB3R6, +m"2Aj

!kB . (47)


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

2are critical

G Stabilizer Equation

O(3)'T C=v

(k)'T m"!1C

=h( j)'T

s! m"1

Cs( j)'T

s{" !1(m(1

SO(3)'T C=

'T m"!1C

='T

2# m"1

T2$ !1(m(1

SO(3)'Ts

C=

'Ts

DmD"1T

s!1(m(1

! Ts

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 reversalT and the re#ection s@ in plane of j and k.# T

2is the product of time reversal T by C

2rotation around axis orthogonal to vector j.

$ T2

is the product of time reversal T by C2

rotation around k vector.

For a function on R, we have to project +m on the tangent plane to R. Two normal vectors at(jk)3R are obtained by di!erentiating the equations i"1, o"0 de"ning R, i.e.

v1"ARi/RjRi/RkB"A

j

kB, v2"ARo/RjRo/RkB"A

k

jB, vivj"d

ij. (48)

The rank one orthogonal projectors on these two vectors are, respectively:

P1"A

jTS j jTSk

kTS j kTSkB, P2"A

kTSk kTS j

jTSk jTS jB . (49)

Since P1P

2"0"P

2P

1the gradient of m on the tangent plane at the point (j

k)3R is

+Rm"(I6!P

1!P

2)+m"A

(1!m)j

!(1#m)kB . (50)

To end the proof of this lemma, we simply verify that this vector vanishes for m"1 (so k"0) andfor m"!1 (so j"0). Remark that this vector never vanishes on the interval !1(m(1. So theonly other orbits of extrema of f (m) occur for the values of m on this interval, satisfyingf @(m),df/dm"0.

The extension to the case of three di!erent two-dimensional Lie groups including T-dependentsymmetry transformations can be easily done because the space of orbits remains the same and thesame invariant m can be used to distinguish orbits and strata. These groups are SO(3)'T,O(3)'T, and SO(3)'T

s. 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

Ts"sT is represented by !I

6. Hence T

sleaves i, m,o invariant and the orbits and strata of


4We could have chosen for the direction n( the third coordinate axis; then k"j3, l"k

3.

SO(3)'Tsare those of SO(3). Similarly, the orbits and strata of O(3), O(3)'T, and SO(3)'T are

identical. Invariants i, m,o2 characterize orbits unambiguously. So we have

PSO(3)6

"PSO(3)\Ts

6, (51)

PO(3)6

"PO(3)\T

6"PSO(3)\T

6. (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 traditionalnotations in molecular physics are

C=

, C=h

, C=v

, D=

, D=h

. (53)

We denote unit vectors by a hat, e.g. n( , and by C=

(n( ) the group of rotations around the axis de"nedby n( . The stabilizers for the action of these groups on R and on the six-dimensional space < are theintersections 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 R6, the vector space of the O(4) adjoint representa-tion. 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 forC

=; we label them by Greek letters (i, m,o 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.4However, this is not enough for generating the ring PC= of C

=polynomial invariants and for

labeling all the C=

-orbits. Indeed, the Molien function is

MC=

(j)"12pP

2p

0

dh(1!j)2(1!2j cos h#j2)2

"

1#j2

(1!j)2(1!j2)3. (59)

The degrees of j in the "ve factors of the denominator are equal to the degrees of the "vehomogeneous polynomials of variables j

i, k

jof Eqs. (54)}(58). But the numerator requires a sixth


5We use the property that the square of the determinant on m m-component vectors va ,14a4m is the determinant ofthe matrix of elements va .vb .

generator of PC= of degree 2. The square of this invariant should be a polynomial function of sixbasic invariant polynomials. This statement can also be veri"ed geometrically: indeed any set ofvalues satisfying Schwarz inequalities for the scalar products:

o24i2!m2 , (60)

k2412(i#m) , (61)

l2412(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,5

p2"14(i2!m2)#okl!A

14o2#

i2(k2#l2)!

12mk2#

12ml2B50 . (64)

So the module of C=

-invariant functions on<, the six-dimensional vector space of the O(4) adjointrepresentation may be represented as

PC=6

"P[i, m,o,k, l]v(1,p) . (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 <DC

=.

The four other one-dimensional subgroups have C=

as invariant subgroup. We can build theirorbit spaces from the one of C

=. Let G be any of the three groups C

=h, C

=v, D

=. Then G is the

union of two C=

cosets: G"C=

XCXC

=where C

Xdenotes, respectively, the re#ections C

v,C

h,

and C2

a rotation by n around an axis (in the plane h) orthogonal to the rotation axis of C=

. Theaction of the quotient group G/C

=on the C

=orbit space can be obtained by computing the action

of these three operations CX

on the C=

invariants o, k, l, p (the O3-invariants i, m 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 willobtain the set of denominator invariants of G by replacing the pseudo-invariants among o,k, l bytheir squares. The product of any two pseudo-invariants transforming according to the sameone-dimensional representation of the symmetry group is an invariant which will be a numeratorinvariant. We can treat similarly the group D

=hwhich 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,k2, l2,o2. The "rst threelines 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 otherone-dimensional subgroups of O(3), can be compared with their Molien functions. To compute


Table 6Action of C

=and discrete symmetry generators of the group O(3)'T on a set of C

=-invariants. i and m are invariant of

the total group O(3)'T. They are omitted from the table. The ! sign indicates the multiplication by !1

Element o k l p

C=

! # # # #

Cv

! ! # #

Ch

! # ! !

C2

# ! ! !

T ! ! # !

Tv

# # # !

Th

# ! ! #

T2

! # ! #

!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@G

of the C=

cosets not containing1 of the "rst three groups. We obtain

M@C=v

"

12pP

2p

0

dh(1!j2)3

"

1(1!j2)3

, (66)

M@C=h

"

12pP

2p

0

dh(1!j2)((1!j2)2!4j2(cosh)2)

"

1(1!j2)(1!j4)

, (67)

M@D=

"

12pP

2p

0

dh(1#j)2(1!j2)2

"

1(1#j)2(1!j2)2

. (68)

We can express the complete Molien function for the invariants in terms of these M@:

MC=v

"12(M

C=#M@

C=v) , (69)

MC=h

"12(M

C=#M@

C=h) , (70)

MD=

"12(M

C=#M@

D=) , (71)

MD=h

"14(M

C=#M@

C=v#M@

C=h#M@

D=) . (72)

Then we obtain for the expression of the Molien functions:

MC=v

"

1#j2#j3#j5

(1!j)(1!j2)3(1!j4) A"1#j3

(1!j)(1!j2)4B , (73)


MC=h

"

1#2j3#j4

(1!j)(1!j2)3(1!j4), (74)

MD=

"

1#j2#2j3

(1!j2)5, (75)

MD=h

"

1#j3#j4#j5

(1!j2)4(1!j4). (76)

We verify again that Molien functions are rational fractions. However, for Eq. (73) the form wehave used does not correspond to the reduced one (which is given between brackets). The situationis di!erent from the Sloane counter example referred to in Chapter I (Sloane, 1977). The reducedform of the C

=vMolien 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):

PC=v6

"P[i, m,o2,k2, l]v(1,p, ok,pok) , (77)

PC=v6

"P[i, m,p,k2, l]v(1,ok) . (78)

To prove the equivalence of these two modules we replace p by its square in the denominatorinvariants of Eq. (78) and add to the numerator invariants the numerator invariants multiplied byp; the two modules have same basis. Eq. (64) shows that p2 is a function of o2 and of i, m,k2, l,ok;since the latter are other invariants of Eq. (78), we can replace p2 by o2; that ends the transforma-tion of Eq. (78) into Eq. (77). We prefer to use the "rst form because o"0 in one of the equationsde"ning the manifold R and our future qualitative analysis uses basically G-invariant functions onR (de"ned by i"1, o"0).

Table 7 enables us to write explicitly the structure of the "ve modules of G-invariant functionson R:

PC=6

DR"P[m,k, l]v(1,p) , (79)

PC=v6

DR"P[m,k2, l]v(1,p) , (80)

PC=h6

DR"P[m,k, l2]v(1, lp) , (81)

PD=6

DR"P[m,k2, l2]v(1,kl,kp, lp) , (82)

PD=h6

DR"P[m,k2, l2]v(1, lp) . (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 ofChapter I. We summarize here in Table 7 the system of invariants of all 16 one-dimensionalsubgroups of the group O(3)'T.

The restriction of the system of invariant polynomials on R becomes especially simple for foursymmetry groups C

='T

v, C

=v'T, C

=h'T

2, and D

=h'T. For all these groups we have on

R only three denominator invariants:

PC=\Tv

6DR"P[m,k, l] , (84)


Table 7Denominator and numerator invariants in the action on R6 of the 16 one-dimensional Lie subgroups of O(3)'T. h

1"i

and h2"m are present for all subgroups and therefore omitted from the table. Invariants which have a "xed value on

R,S2]S

2(due to o"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 theD

=h'T group with eight connected components

G h3

h4

h5

u1

u2

u3

C=

(o) k l p

C=v

(o2) k2 l p (ok) (okp)C

=h(o2) k l2 lp (ol) (op)

D=

(o) k2 l2 kl kp lpC

='T (o2) k2 l kp (ok) (op)

C=

'Tv

(o) k lC

='T

h(o) k2 l2 p kl (klp)

C=

'T2

(o2) k l2 p (lo) (lop)

D=h

(o2) k2 l2 lp (kop) (klo)C

=v'T (o2) k2 l (ko)

C=h

'T (o2) k2 l2 klp (po) (klo)D

='T (o2) k2 l2 kp (lko) (lpo)

C=v

'Th

(o2) k2 l2 p (lko) (lkpo)C

=h'T

v(o2) k l2 (lo)

D=

'Tv

(o) k2 l2 kl

D=h

'T (o2) k2 l2 (okl)

PC=v\T

6DR"P[m, k2, l] , (85)

PC=h\T2

6DR"P[m,k, l2] , (86)

PD=h\T

6DR"P[m,k2, l2] . (87)

It is curious to note that especially these groups have the most natural physical interpretation.C

=v'T is the symmetry group for the atom in a static electric "eld or for Rydberg states of

a heteronuclear diatomic molecule.C

=h'T

vis the symmetry group of an atom in a constant magnetic "eld (Zeeman e!ect).

C=

'Tv

is the maximal common subgroup of C=v

'T and C=h

'Tv. Thus, it corresponds to

the symmetry of an atom in the simultaneous presence of two parallel magnetic and electric "elds.The D

=h'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) conjugationclasses: those of C

=v(k), C

=h( j), C

s( j). The stabilizers of the "ve groups G of Eq. (53) are their


Table 8Strata 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?)RDG strat. and inequalities orbits orbits

C=

C=

c 0 k2"1, l2"1 4 A,B,C,D crS3

1 g 4 k2(1, l2(1 R3 S1

C=v

C=v

c 0 l2"1 2 C,D crC

=c 0 k2"1 1 AB cr

B3

Cs( j@) 3 k"0, l2(1 R2 S

11 g 4 0(k2(1Nl2(1 R3 2S

1

C=h

C=h

c 0 k2"1 2 A,B crC

=c 0 l2"1 1 CD cr

S3

Cs(n) 2 04

1#m2

"k2(1Nl"0 R S1

Ci

2 m"1, k2(1 R S1

1 g 4 m(1, 0(k2(1Nl2(1 R3 2S1

D=

C=

c 0 k2#l2"1, kl"0 2 AB,CD crSusp. C

2c 1 m2"1, k"l"0 2 C

1,C

2cr

RP2 1 g 4 (m2"1, 0(k2(1, 0(l2(1) R3 2S1

D=h

C=h

c 0 k2"1 1 AB crC

=vc 0 l2"1 1 CD cr

C2h

( j) c 1 m"1, k"l"0 1 C1

crC

2v(k) c 1 m"!1, k"l"0 1 C

2cr

B3

Cs(n) 2 04

1#m2

"k2(1Nl"0 R 2S1

Ci

2 m"1, 0(k2(1 R 2S1

Cs( j@) 3 k"0, l2(1, m(1#l2 R2 2S

1

1 g 4 0(k2(1#m

2(1, l2(

1!m2

(1, R3 4S1

(m!1)2#k2#l2'0

!Column 1: Below the group G is given the topological nature of the orbit space RDG (Susp. is for suspension)."Column 2: Stabilizer of the stratum. C

s( j@)"C

s( j) except when j"0; then it is C

s(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 ofstrata, 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.


First, let us write again all equations on the C=

invariants when they are restricted on themanifold R de"ned by i"1 and o"0. This corresponds to the study of the three-dimensionalorbifold RDC

=. Eqs. (60)}(62) and (64) become

k2412(1#m) , (88)

l2412(1!m) , (89)

!124p41

2(90)

and

044p2"(1!m2)!2(1!m)k2!2(1#m)l241 . (91)

This last equation shows explicitly that p is determined up to a sign by m,k2, l2. From the twoprevious equations we deduce

k"$1Nm"1, l"0"p , (92)

l"$1Nm"!1, k"0"p . (93)

This means that for a given n( there are four points A,B, C,D of R "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 S2]S

2the 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 anextremum on each of these four points of R.

Similarly, we have two circles on R:

C1: k"0, n ) j"0Nm"1, k"l"0 , (98)

C2: j"0, n ) k"0Nm"!1, k"l"0 . (99)

They correspond, on the two equators of S2]S

2, to pairs of identical points, diametrically opposed

points, respectively. Each of these two circles C1, C

2form a critical orbit of D

=or D

=h.

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 theorigin (while any non-trivial polar vector changes of sign), m"1 contains a stratum with stabilizer


6Some 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 geometricalrepresentation (see Appendix A).

Ci(the group generated by the symmetry through the origin) for the groups C

=hand D

=h. When

m"!1 then j"0 and the polar vector k is invariant by the symmetry through any planecontaining it. The symmetry through the plane which contains also n( belongs to a stabilizer ofC

=vand D

=hand this is still true when jO0 and collinear to n(]k, see Table 8.

The "ve orbit spaces RDG, are of dimension three.6 We will "rst make a direct study of theirtopology. We shall "rst prove that topologically RDC

=&S

3. Then, as suggested by the } signs in

the columns of k, l, p of Table 6, the orbit spaces of C=v

, C=h

, D=

are obtained by identifying thepoints symmetrical through, respectively, a three-, two-, one-dimensional linear manifold contain-ing the center of the sphere S

3.

In the four-dimensional space of parameters m,k, l,p, when !1(m(1, Eq. (91) shows that thesection of RDC

=&S

3by a hyper-plane m"constant is an ellipsoid &S

2:

!1(m(1 ,

2p2#(1!m)k2#(1#m)l2"12(1!m2) . (100)

Indeed this is true because each coe$cient (function of m) is strictly positive. Furthermore, thisequation 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$1 ofthe ellipsoids de"ned by Eq. (100). This proves that the orbifold for the C

=action on R is

the 3D-sphere

RDC=&S

3(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`orthogonal coordinatesa p, m,k, l, the orbit space RDC

=has three symmetry hyper-planes p"0,

k"0, l"0, mutually orthogonal.Table 7 shows that RDC

=vis obtained from RDC

=by identifying its points of coordinates

m,p, l,$k. We can represent it by the intersection of RDC=v

with the closed half-space k50. Thisorbit space RDC

=vis therefore topologically equivalent to a hemisphere, i.e. a ball B

3:

RDC=

W(k50)"RDC=v

&B3

. (104)

This is also the topological nature of the projection of RDC=v

on its symmetry hyper-plane k"0;indeed from (1!m)k250 we deduce from Eq. (91) that

2p2#12m2#(1#m)l241

2, (105)


but one must remark that there is a unique point of this projection which does not correspond toa unique point of the orbit space RDC

=v: indeed m"1, p"k"l"0 if the projection of the whole

segment m"1, p"l"0, !14k41. To summarize, Table 8 shows that the orbit space RDC=v

contains four strata represented by

(1) two points C,D de"ned in Eqs. (97) and (98): each corresponds to a one point C=v

-orbit;(2) one point X: m"1"k, p"0"l, representing a two point C

=-orbit, MA,BN (see Eqs. (95) and

(96));(3) the boundary (RB

3&S

2) minus the two points C,D representing the C

sstratum;

(4) the interior of B3

minus the point X representing the generic stratum.

We now prove RDC=h

&S3. Indeed, from Table 7 in the parameter space one obtains RDC

=hfrom

RDC=

by identifying the points symmetrical through the two-plane P(m,k) of the coordinates m,k (itis the intersection of the two symmetry hyper-planes l"0, p"0). Consider the sphere S

3:

m2#k2#l2#p2"1 topologically equivalent to RDC=

. By identi"cation of its points in the pairsB"(m,k,$l,$p (which are not in P(m,k))), one obtains a topological space &RDC

=h. The

projection on the two-plane P(m,k) is the disk B2: m2#k241. Note that RB

2"S

3WP(m,k) while

each point p(m, k) of the interior of B2

is the projection of a circle Cp

of equationl2#p2"1!m2!k2 in the two-plane perpendicular to P(m,k). By the identi"cation of its pointssymmetric through its center the circle C

pis transformed into a circle C@

p; this holds for each p in the

interior of B2. Correspondingly this point identi"cation transforms S

3into S@

3.

We note from Table 7 that one transforms RDC=h

into RDD=h

by identifying the points ofopposite k coordinate. As for the transformation of RDC

=into RDC

=vstudy above, we obtain that

RDD=h

&B3.

We are left with the study of RDD=

. It is obtained from RDC=

by identifying the points in each pairsB"(m,$k,$l,$p), 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.We remark that RDC

=&S

3is topologically equivalent to the double cone of vertices $1, 0, 0, 0

and basis B"(RDC=

)WH&S2; this is also called a suspension of B. The point identi"cation

transforms B into B@&RP2. So:

RDD=&suspension (RP2) . (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 inthis 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 construc-tion of the integrity basis on the six-dimensional space where the action of the symmetry group ofthe problem is linear. The Molien function and the invariants themselves for the six-dimensionalspace x=y or k=j may be found from known expressions for Molien functions and integrity basesfor irreducible representations. Next step includes the restriction of the polynomial algebra on thesub-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 Section 5.2 of Chapter I. Thesub-manifold R is de"ned in the six-dimensional space x=y by the polynomial equationsx2"1

2, y2"1

2. 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 casewe just eliminate them from the integrity basis constructed for the 6D-space and the resultingintegrity 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 thesimilar problem for the proper rotation subgroup and after that take into account the e!ect ofimproper 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 wejust summarize the results for the case of the C

2point group as the invariance group of the

problem.Let us take x

3and y

3to coincide with the C

2-axis. In such a case x

1,x

2, y

1, y

2transform

according to the anti-symmetric representation B and x3, y

3according to the totally symmetric

representation A of the C2

point group. We are interested in the module of invariant functionsde"ned on the sub-manifold R given by two polynomial equations:

x21#x2

2#x2

3"1

2, (107)

y21#y2

2#y2

3"1

2. (108)

After the restriction on R the Molien function for invariants has the form

MC2

DR"1#6j2#j4

(1!j2)2(1!j)2. (109)

The explicit form of the module of invariant functions on R may be easily given as well:

PC2 DR"P[x21, x

3, y2

1, y

3]v(1,u

1, u

2, u

3, u

4, u

5, u

6, u

7) , (110)

where

u1"x

1y1, u

2"x

2y1, u

3"x

1y2, u

4"x

2y2

, (111)

u5"x

1x2, u

6"y

1y2, u

7"x

1x2y1y2

. (112)

More condensed notation for the set of numerator invariants may be used

PC2 DR"P[x21, x

3, y2

1, y

3]v((1,x

1x2)(1, y

1y2), (x

1,x

2)(y

1, y

2)) . (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 thatthe choice of denominator and numerator invariants is not unique and the choice proposed here isjust 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 (k=j which isthe sum of polar and axial vector representations) follow immediately from the well-known results


Table 9Strata and orbits in the action of C

2von R

Stabilizer gc! dim." Number# Nature$

C2v

c 0 2 1C

2c 0 1 2

Cvs

2 R2 2Cd

s2 R2 2

1 g 4 R4 4

!g is for generic stratum; c is for closed stratum."Dimension of the stratum.#Number of orbits in the stratum. Rn 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 forseveral point groups which are the most interesting from the point of view of possible applications.These are examples of molecules with the C

2v, D

3h, and ¹

dpoint symmetry. Similar tables for all

other possible group symmetries are given in Appendix C. Table 9 listed in this subsection show incolumn 1, the stabilizer of the stratum. In column 2 closed and generic strata are indicated. Weremark once more that some strata are neither closed nor generic. In column 3 the dimension of thestratum 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 thenumber of orbits in the stratum is "nite. If the dimension of the stratum is positive the number oforbits in the stratum is in"nite. We denote it as Rn 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 isalways "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 symmetrygroup O(3)'T of the problem. There are naturally many "nite subgroups which can be obtainedin a way similar to our construction of one-dimensional subgroups. Extension of point subgroupsof O(3) to points subgroups of O(3)'T is analogous in some sense to the construction ofantisymmetry (Shubnikov, 1951) or magnetic symmetry groups (color groups) well known insolid-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 groupsincluding time reversal. We have chosen those groups which have simple physical realization assymmetry groups of the hydrogen atom in the presence of di!erent external "elds. This simplequantum 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 ofcorresponding symmetry groups:

f Hydrogen atom without any external "elds. The symmetry group includes the full orthogonalgroup O(3) and the time reversal. We can write the group as the direct product O(3)'T where


Table 10Strata and orbits in the action of D

3hon R


C3h

c 0 1 2C

3vc 0 1 2

C2v

c 0 2 3C

2c 0 1 6

C1h

2 R2 6C

s2 R2 6

1 g 4 R4 12


Table 11Strata and orbits in the action of ¹

don R


C3v

c 0 2 4C

3c 0 1 8

S4

c 0 1 6C

2vc 0 1 6

Cs

2 R2 121 g 4 R4 24


Table 12Invariant manifolds for the C

=h'T

vsymmetry group action on R (Hydrogen atom in magnetic "eld.)

Stabilizer dim. Equations

C=h

'Tv

0 k"$1C

='T

v0 l2"1

[Ch'T

v]! 2 l2"0; m"2k2!1

[Ci'T

v]" 2 l2"0; m"1

Tv

3 p"0T

23 l2"0

C1

4 R

!This group includes four elements: E, ph, T

v, T

2.

"This group includes four elements: E, i } inversion, Tv, T

2.


Table 13Invariant manifolds for the C

='T

vsymmetry group action on R (Hydrogen atom in parallel magnetic and electric

"elds.)


C=

'Tv

0 l2"1; k2"1T

v3 p"0

C1

4 R

the two element group Z2(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 usethis higher group to introduce the approximate quantum number corresponding to an indi-vidual n-shell of the Rydberg atom even in the case of external "elds (assuming that the splittingof the n-shell is small as compared to the inter-n-shell gap). Di!erent external "elds break thissymmetry till various subgroups.

f Hydrogen atom in the presence of constant magnetic "eld. This case corresponds to the Zeemane!ect for the hydrogen atom. C

=hgroup is the invariance group of spatial geometric transforma-

tions. The complete symmetry group including time-reversal operations is C=h

'Tv. It is the

semi-direct product of C=h

and two element group Tv"ME,T

vN including two elements: the

identity E and the product of time reversal and space re#ectionTvin the plane of symmetry axis.

The group C=h

'Tv

includes the following classes of conjugated elements: MEN is the identity,Mp

hN the re#ection in the plane orthogonal to the magnetic "eld axis, MR

(N the rotation on angle

/ around magnetic "eld, MS(N the rotation-re#ection on angle / around magnetic "eld, MT

v(/)N

the time reversal followed by re#ection in a plane passing through the magnetic "eld, MT2(/)N

the product of time reversal and rotation by n around an axis orthogonal to the "eld axis. AnoperationT

2can be equivalently described as a product of T

vand S

(operations. Table 12 gives

invariant manifolds for the C=h

'Tv

group action on R. We give here invariant manifoldsinstead of strata to simplify the de"ning equations.

f Hydrogen atom in the presence of constant electric "eld. This is the case of the Stark e!ect for thehydrogen atom. C

=vgroup 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 C

=v'T. This group includes the

following symmetry operations: E is the identity, T the time reversal, R(

the rotation on angle/ around electric "eld, p

(the re#ection in plane passing through the "eld axis, TR

(the time

reversal followed by rotation, Ts(/) the time reversal followed by re#ection. The system of

invariant manifolds of the C=v

'T action on R is given in Table 14.f 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 completeinvariance group can be written as C

='T

v. Naturally, this group is the subgroup of the

symmetry group appropriate for the case of only electric or only magnetic "eld. The system ofinvariant manifolds of the C

='T

vaction on R is given in Table 13.

f Orthogonal electric and magnetic "elds. The symmetry group is "nite in this case and includesonly four symmetry elements: identity E, re#ection in the plane orthogonal to the magnetic "eld


Table 14Invariant manifolds for C

=v'T symmetry group action on R (Hydrogen atom in electric "eld.)


C=v

'T 0 l"$1C

='T

v0 k2"1

[Cs'T

s{]! 2 k2"0; m"1!2l2

[Cs'T] 2 k2"0; m"!1

Cs

3 k2"0T

v3 p"0

C1

4 R

!This group includes four elements: E is the identity, p the re#ection in the plane including the "eld axis and a vectororthogonal to the vector L, T

s{the time reversal followed by re#ection in the plane formed by "eld axis and the vector L,

T2

the time reversal followed by rotation over p around "eld axis.

ph, product Tp

vof time reversal T and re#ection p

vin plane formed by two orthogonal "elds,

and the symmetry operation TC2

which is the product of the time reversal T and theC

2rotation around the electric "eld direction. We will use the notation G

4for this symmetry

group. This group has three subgroups of order two which will be useful below. The groupT

vincludes as non-trivial symmetry operation the product Tp

vof time reversal T and

re#ection pvin plane formed by two orthogonal "elds. The group C

sincludes the re#ection in the

plane orthogonal to the magnetic "eld (this plane includes the electric "eld vector). The groupT

2includes as a non-trivial symmetry operation, the product of time reversal and the

C2

rotation around the electric "eld direction.f Generic non-orthogonal non-collinear con"guration of two "elds. The symmetry group T

shas

order two and includes one non-trivial symmetry operation Tp which is the product of the timereversal and the re#ection in plane formed by the two "elds. T

sis the subgroup of the symmetry

group G4

for orthogonal "elds and the subgroup of the symmetry group C=v

'Tsfor collinear

"elds. It is the only common subgroup for these two important limiting cases of parallel andorthogonal "elds.

f To complete the list of interesting symmetry groups we add here the symmetry group of thequadratic Zeeman e!ect or of Rydberg states of homonuclear diatomic molecules. This is theD

=h'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 impor-tant cases as the Zeeman e!ect, Stark e!ect, hydrogen atom in parallel "elds or quadratic Zeemane!ect the integrity basis and the geometrical representation of the orbifold. Going from the C

=to

C=

'Ts, from C

=vto C

=v'T, from C

=hto C

=h'T

s, and from D

=hto D

=h'T leads to an

integrity basis consisting of only basic `denominatora invariants. The space of orbits for extendedgroups possesses richer strati"cation (see Tables 12}15) but the geometrical form of the orbifoldbecomes simpler (see Figs. 1}3) as compared to the geometrical form of the orbifold for purelygeometrical point group symmetry studied in Appendix A.


Table 15Invariant manifolds for the D

=h'T symmetry group action on R (Complete symmetry group of the quadratic Zeeman

e!ect.)


C=h

'Ts

0 k2"1C

=v'T 0 l2"1

C2h

'Ts! 1 m"1, k2"l2"0

C2v

'T 1 m"!1, k2"l2"0C

s(n)'T

s" 2 1#m"2k2

Ci'T

s2 m"1

[Cs'T] 2 k2"0; m"!1

[Cs'T

s{]# 2 k2"0; m"1!2l2

Cs( j)'T

2$ 2 k2"0, l2"0

Cs

3 k2"0T

23 p2"0

Ts

3 l2"0C

14 R

!C2h

subgroup includes the symmetry plane orthogonal to the vector L; Tsis the time reversal followed by re#ection in

plane formed by L and magnetic "eld."C

sis the re#ection in plane orthogonal to the C

=axis;T

sis 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 vectororthogonal to the vector L, T

s{is the time reversal followed by re#ection in the plane formed by "eld axis and the vector

L, T2

the time reversal followed by rotation over p around "eld axis.$T

2is the time reversal followed by C

2rotation around K.

Fig. 1. Space of orbits for the C=h

'Ts

action on R. (H atom in the presence of magnetic "eld.)

Fig. 2. Space of orbits for the C=

'Ts

action on R. (H atom in the presence of parallel magnetic and electric "elds.)


Fig. 3. Space of orbits for the C=

'Ts

action on R. (H atom in the presence of electric "eld.)

Fig. 4. Three slices of D=h

'T orbifolds for di!erent k2 values. Di!erent strata are denoted by letters as follows:a!C

2h'T

s; b!C

s( j)'T

2; c!C

2v'T; d![C

s'T]; e!C

=v'T; f![C

s'T

s{]; g!C

s; h!C

i'T

s; i!T

2;

j!Cs(n)'T

s; k!T

s; l!C

1; m!C

=h'T

s. De"ning equation for the strata are given in Table 15.

To see better the strati"cation of the R under the action of D=h

'T symmetry group whichincludes 13 di!erent strata we give in Fig. 4 the schematic representation of the space of orbitsthrough three sections corresponding to k2"0, 0(k2(1, and k2"1.

Similar extension of the geometrical point group symmetry can be done in cases of "nitesymmetry groups; e.g. see the example of the G

4symmetry group for the hydrogen atom in two

orthogonal "elds mentioned a little earlier in this section is analyzed in more details in Section 4.4and 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 subgroupsof the O(4) group and their relation with invariant functions de"ned on the R manifold. We startwith the Hamiltonian for a hydrogen atom which is O(4) invariant. It was introduced in Eq. (1) andrewritten in terms of the angular momentum vector J and the transformed Laplace}Runge}Lenzvector, 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 thepresence of n multiplets typical for the hydrogen atom (this is precisely the situation with Rydbergatoms and molecules in the limit of small splitting of n multiplets). An e!ective Hamiltonian fora given n shell may be written as a phenomenological e!ective Hamiltonian constructed from J andK operators. Relations (7) and (8) are important to reach the correspondence between the quantum


operators J and K and their classical analogs, because the classical variables j, k satisfy thenormalization condition (40), (39), (42) which is independent of the particular realization of thequantum operator for any concrete n shell. As a consequence, the correspondence betweenquantum operators and classical dynamic variables has the form

Ja%Jn2!1ja , (114)

Ka%Jn2!1ka . (115)

Using the integrity basis for the classical representation we can now construct easily the basicpolynomial invariants formed by the quantum operators and to indicate their classical analogs

J2!K2"(n2!1)m , (116)

Jz"Jn2!1k , (117)

Kz"Jn2!1l . (118)

For higher-order invariants one needs to make the anti-symmetrization of the operators which donot commute. For example, the operators representing the auxiliary numerator invariants have theform

MJx, K

yN!MJ

y,K

xN"(n2!1)p , (119)

MKz, (MJ

x, K

yN!MJ

y, K

xN)N"(n2!1)3@2lp , (120)

MJz, K

zN"(n2!1)kl , (121)

MJz, (MJ

x, K

yN!MJ

y, K

xN)N"(n2!1)3@2kp , (122)

where

MA,BN"12(AB#BA) . (123)

Sometimes it is useful to work in the x, y representation (40), (39) instead of j, k. We denote thecorresponding quantum operators by

J1"(J!K)/2 , (124)

J2"(J#K)/2 . (125)

The Hilbert space of wave functions associated with the n multiplet may in such a case be formedby the basis

DJ1"J

2"(n!1)/2; M

1, M

2T , (126)

Mi"!(n!1)/2,2, (n!1)/2 . (127)

The correspondence between quantum operators J1, J

2and classical dynamic variables follows

directly from Eqs. (116)}(122):

J1) J

2"(n2!1)m/4 , (128)


J1z#J

2z"2Jn2!1k , (129)

J1z!J

2z"Jn2!1l , (130)

J1y

J2x

!J1x

J2y"(n2!1)p/2 , (131)

M(J1z!J

2z), (J

1yJ2x

!J1x

J2y

)N"(n2!1)3@2lp , (132)

J21z!J2

2z"(n2!1)kl , (133)

M(J1z#J

2z), (J

1yJ2x

!J1x

J2y

)N"(n2!1)3@2kp . (134)

There are two ways leading to the construction of the e!ective operators. One is purely a phenom-enological approach based on the utilization of all operators allowed by symmetry with theircoe$cients being the adjustable parameters of the model. The other one uses the transformation ofthe initial operator to the e!ective one by applying some kind of perturbation theory or contacttransformation, etc. This second procedure results in the e!ective operator with "xed coe$cientswhich depend on the initial Hamiltonian. Both approaches result in the Hamiltonian which may bewritten in terms of integrity basis. We remind that as soon as invariant polynomials are known, anarbitrary Hamiltonian may be written in the form of expansions (properly symmetrized to take intoaccount the non-commutativity of quantum operators) in terms of denominator invariants h

iand

numerator invariants us

H"

=+

n1 ,n2 ,2,nk

+s

Cn1 ,n2 ,2,nk _s

ushn11

hn22 2hnk

k. (135)

An explicit form of invariants for all continuous subgroups of O(3) was found in Section 2.4 (seeTable 7) and for some point groups in Section 2.7 and in Appendix B. The purely phenomenologi-cal way to represent e!ective Hamiltonians supposes that all the C

n1 ,n2 ,2,nk _scoe$cients are

adjustable parameters, whereas perturbation treatment gives the Cn1 ,n2 ,2,nk _s

coe$cients inEq. (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!ectiveHamiltonians taking into account their symmetry properties. The main idea of our approach is tointroduce along with the phenomenological construction some kind of complexity classi"cation ofe!ective Hamiltonians which is based on the quantitative measure } the number of orbits ofextrema. 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 continuoussubgroups of O(3)

We discuss here the qualitative classi"cation of e!ective Hamiltonians invariant with respect todi!erent symmetry groups. By qualitatively di!erent Hamiltonians we mean those operators whichare characterized by di!erent sets of stationary orbits with each stationary orbit characterized byits 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


number of stationary orbits (compatible with the topological structure and the symmetry groupaction according to the equivariant Morse}Bott theory). We will list below qualitatively di!erentMorse}Bott Hamiltonians of some low level of complexity, whereas in Appendix D more generalresults are outlined.

3.2.1. O(3) invariant HamiltonianAny 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 operatorsgiven by Eq. (116) (or (128)) in the J,K (or J

1,J

2) representation. So any e!ective

Hamiltonian may be written as a power series in m or more generally as an arbitrary function ofm as in Eq. (136).

Any O(3) invariant classical Hamiltonian possesses two critical manifolds due to the presence ofthe two critical orbits. These two critical orbits have very simple physical meaning. C

=hcritical

orbit corresponds on the orbifold to the point m"1 ( j ) j"1, k ) k"0), i.e. the quantum statelocalized near this point has maximal (possible for a given n) value of the orbital momentuml"n!1. Another critical orbit corresponds to the point m"!1& ( j ) j"0, k ) k"1) i.e. for thequantum problem the quantum state localized in the phase space near this point has minimumpossible value of the orbital momentum l"0.

More complicated Hamiltonians can be classi"ed according to the number of stationary orbits(the RP3 manifolds) which belong to the generic C

sstratum. 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 ofthe group O(3). This one regular sequence may be rather complicated in the general case but for thesimplest Morse}Bott-type Hamiltonian (p"0 in Table 16) the sequence should be monotonic. It isreasonable to assume that the number of extrema of the H"H(m) function in a general case ismuch smaller than the number n imposing limit on possible l values, l4n!1. In such a case theenergy spectrum explicitly shows the regular behavior. In contrast, if the number of extrema onH(m) is larger or of the same order as n (assuming n is large), the energy spectrum may seem to beerratic.

Near an extremum, corresponding to the C=h

critical orbit (m"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 theC

=vcritical orbit (m"!1&( j ) j"0, k ) k"1)) there is a regular system of energy levels,

E(l)"const. l(l#1) (138)


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

Level of C=v

C=h

N45! N

45complexity" stratum stratum with index 0 with index 1

0 max min no no0 min max no no1 max max no 11 min min 1 no2 max min 1 12 min max 1 1

2p max min p p2p min max p p

2p#1 max max p p#12p#1 min min p#1 p

!N45

is the number of stationary RP3 manifolds on Cs

stratum."p is a non-negative integer.

which is characterized by l"0, 1, 2,2 . Near an extremum corresponding to generic Csorbit the

system of energy levels may be described as a bent series of the form

E(l)"const.[l(l#1)!l0(l

0#1)]2 (139)

with the l value varying around some generally non-integer l0

value.

3.2.2. C=

invariant HamiltonianThe C

=invariant phenomenological Hamiltonian may be written as

HC=

"+Cn1,n2 ,n3

mn1kn2ln3#+Cpn1 ,n2 ,n3

pmn1kn2ln3 (140)

with m,k, l, p given in Eqs. (116)}(119), or in (128)}(131) with all necessary symmetrization. AnyHamiltonian (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 Hamiltonianpossesses one minimum and one maximum on two critical orbits and two saddle points (withMorse index 2) on the other pairs of critical orbits. The description of qualitatively di!erentHamiltonians 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 fora given set of the qualitatively similar Hamiltonians. To reduce the number of di!erent classeswe 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 thetransformation of the Morse indices of stationary points (k% (4!k)) and of one-dimensionalstationary manifolds (k % (3!k)). For several classes both Hamiltonians H and (!H) aredescribed by the same Morse counting polynomial (we add to the level of complexity the index


Table 17Classi"cation of qualitatively di!erent C

=invariant Hamiltonians

Level of C=

! C1"

complexity# stratum stratum

0B 1#2t2#t4 no

1 1#3t2 t31 1#t2#2t4 t2

2B 1#2t2#t4 t#t22 1#2t2#t4 1#t2B 4t2 1#t32B 2#2t4 t#t22 1#3t4 2t22 2#2t2 t#t3

!Morse counting polynomial is given in this column to characterize the system of stationary orbits."Morse counting polynomial for the S

1stationary orbits is given.

#The superscript $ in the "rst column indicates those classes which are invariant under the sign inversion of theHamiltonian.

$ in such a case, see Table 17). For all other classes the change of the sign of the Hamiltonian willmodify the redistribution of the stationary orbits over Morse indices. For example, there are twoextra classes of the Hamiltonians of the "rst level of complexity and three additional classes for thesecond level of complexity, but we omit them from the table.

Near minimum and maximum the simplest Morse-type C=

-invariant Hamiltonian (zero level ofcomplexity) may be approximated as 2D-isotropic harmonic oscillator. So it is characterized bya qualitative energy level pattern shown in Fig. 5. Lower and upper parts of the multiplet areformed by a sequence of polyads typical for 2D-isotropic harmonic oscillator.

3.2.3. C=v


=vinvariant phenomenological Hamiltonian may be written as

HC=v

"+Cn1,n2 ,n3

mn1 (k2)n2ln3#+Cpn1 ,n2 ,n3

pmn1(k2)n2ln3 (141)

with m, (k2), 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 criticalorbits: one C

=orbit consisting of two stationary points (A,B) and two C

=vorbits 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 C

=vsymmetry 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,k2, l with respect to time reversal and the alternation of thesign of p under the same operation the general form of Hamiltonian (141) becomes simpler. In thecase of Stark e!ect the e!ective Hamiltonian (141) is independent of p.


Fig. 5. Qualitative energy level pattern for e!ective C=

-invariant Hamiltonians of the simplest Morse}Bott type. In theneighborhood 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. C=h


=hinvariant phenomenological Hamiltonian may be written as

HC=h

"+Cn1,n2 ,n3

mn1kn2 (l2)n3#+Clpn1 ,n2 ,n3

(lp)mn1kn2(l2)n3 (142)

with m,k, (l2) as denominator invariants and (lp) as one numerator invariant as given inEqs. (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

C=h

orbits consisting of one stationary point (A,B) each (stationary points A, B,C,D, seeTable 8 and Eq. (101)).

In the case of the Zeeman e!ect after extending the symmetry group from C=h

to C=h

'Ts(see

Section 2.9) the general form of the e!ective Hamiltonian becomes independent of (lp).

3.2.5. D=

invariant HamiltonianThe D

=invariant phenomenological Hamiltonian may be written as

HD=

"+Cn1 ,n2,n3

mn1(k2)n2(l2)n3#+Clpn1 ,n2 ,n3

(lp)mn1(k2)n2(l2)n3

#+Ckpn1 ,n2 ,n3

(kp)mn1 (k2)n2 (l2)n3#+Clkn1 ,n2 ,n3

(lk)mn1(k2)n2 (l2)n3 (143)

with m, (k2), (l2) as denominator invariants and (lp), (kp), and (lk) as three numerator invariantsgiven 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

two C2

orbits (both being S1

stationary manifold C1, C

2) (stationary points A,B,C, D, and

stationary circles C1,C

2see Table 8 and Eqs. (101) and (102)).

3.2.6. D=h

invariant HamiltonianThe D

=hinvariant phenomenological Hamiltonian may be written as

HD=h

"+Cn1 ,n2,n3

mn1(k2)n2 (l2)n3#+Clpn1 ,n2 ,n3

(lp)mn1(k2)n2 (l2)n3 (144)


with m, (k2), (l2) 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: oneC

=horbit consisting of two stationary points (A,B), one C

=vorbit consisting of two stationary

points (C,D), one C2h

orbit consisting of a S1

stationary manifold C1, and one C

2vorbit consisting

of a S1

stationary manifold C2

(stationary points A,B, C,D, and stationary circles C1, C

2are given

in Table 8 and in Eqs. (101) and (102)).

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

We characterize below in Tables 18 and 19 the simplest Morse-type functions for all "nite pointgroup symmetries. For "nite groups, orbits include a "nite number of isolated points and anyMorse-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 ofstationary points are indicated up to trivial change of Morse indices c

k% c

4~k. Stationary points

for each group are split into columns according to their Morse index k. The number of stationarypoints c

kwith 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. Itshould be noted that for many point groups the stationary points are situated on zero-dimensionalstrata only and their positions are "xed (i.e. only critical orbits are present) (see Section 2.8 andAppendix C for the description of all strata for the "nite group action on R). For some lowsymmetry groups (C

1, C

s, S

2,C

i) 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 closed2D-stratum for the C

sand S

2groups 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 therestriction of the complete initial function on the close stratum.

For Ohand >

hgroups, 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 Cnv

and Cnh

criticalorbits.

Near minima or maxima any Hamiltonian may be approximately represented as 2D-harmonicoscillator. Taking into account the presence of several, say k (k is the dimension of the orbit ofstationary points) equivalent by symmetry minima the model problem appropriate for the descrip-tion of internal dynamics near the extrema is the motion of a particle in a 2D-potential withk equivalent extrema (the potential can be anisotropic or isotropic and slightly anharmonic neareach extremum) assuming small tunneling between di!erent extrema. We can introduce threecharacteristic 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 levelpatterns for quantum problems near extrema. Let d!/* be the anisotropy of the model Hamiltoniannear an extremum

d!/*&hDl1!l

2D/(l

1#l

2) , (145)

where li, i"1, 2, are two harmonic frequencies of the model operator, d56/ be the characteristic

energy splitting due to tunneling, and d!/) be the anharmonicity correction.


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

Point group c0

(c4) c

1(c

3) c

2c3

(c1) c

4(c

0)

C1

1(C1) no 2(C

1) no 1(C

1)

Cs

1(Cs) no 2(C

1) no 1(C

s)

1(Cs) no 1(C

s)#1(C

s) no 1(C

s)

S2

1(S2) no 2(C

1) no 1(S

2)

1(S2) no 1(S

2)#1(S

2) no 1(S

2)

S2n

1(S2n

) no 2(Cn) no 1(S

2n)

n52

Cn

1(Cn) no 1(C

n)#1(C

n) no 1(C

n)

n52

Cnh

1(Cnh

) no 2(Cn) no 1(C

nh)

n52

Cnv

1(Cnv

) no 2(Cn) no 1(C

nv)

n52

D2! 2(C

2) 2C

22(C

2)#2(C

2) 2(C

2) 2(C

2)

Dn" 2(C

n) n(C

2) n(C

2)#n(C

2) n(C

2) 2(C

n)

n53 2(Cn) n(C

2) n(C

2)#2(C

n) n(C

2) n(C

2)

n(C2) n(C

2) 2(C

n)#2(C

n) n(C

2) n(C

2)

n"3, 5 only 2(Cn)#2(C

n) n(C

2) n(C

2) n(C

2) n(C

2)

Dnh

(Dnd

) 2(Cnh

) n(C2v

) 2n(C2) n(C

2v) 2(C

nv)

n53, odd

Dnh

(Dnd

) 2(Cnh

) n(C2h

) n(C2v

)#n(C2h

) n(C2v

) 2(Cnv

)n52, even 2(C

nh) n(C

2h) n(C

2v)#n(C

2v) n(C

2h) 2(C

nv)

# 2(Cnh

) n(C2v

) n(C2v

)#n(C2h

) n(C2h

) 2(Cnv

)2(C

nh) n(C

2v) n(C

2h)#n(C

2h) n(C

2v) 2(C

nv)

n"4 only 2(C4h

)#2(C4v

) 4(C2v

) 4(C2h

) 4(C2v

) 4(C2h

)n"4 only 2(C

4h)#2(C

4v) 4(C

2v) 4(C

2h) 4(C

2h) 4(C

2v)

n"4 only 2(C4h

)#2(C4v

) 4(C2v

) 4(C2v

) 4(C2h

) 4(C2h

)n"4 only 2(C

4h)#2(C

4v) 4(C

2h) 4(C

2h) 4(C

2v) 4(C

2v)

n"4 only 2(C4h

)#2(C4v

) 4(C2h

) 4(C2v

) 4(C2v

) 4(C2h

)n"4 only 2(C

4h)#2(C

4v) 4(C

2h) 4(C

2v) 4(C

2h) 4(C

2v)

!There are three di!erent C2

strata for the D2

group. Each stratum includes two orbits. To reach complete descriptionof qualitatively di!erent Morse-type functions it is necessary to specify the stratum for all stationary points."There are two di!erent C

2strata for even n54, whereas there is only one stratum for odd n.

#Further speci"cation of strata is needed. There are three di!erent C2v

and three di!erent C2h

strata for the D2h

(D2d

)group. There are two di!erent C

2vand two di!erent C

2hstrata for the D

nh(D

nd) group for n54, even.


Table 19Simplest Morse-type functions de"ned on the R manifold in the presence of point group symmetry (higher symmetrygroups). To complete the list of Morse functions for O

hand>

hgroups it is necessary to add other sets of stationary points

resulting from di!erent permutations within pairs of Cnv

and Cnh

strata

Point group c0

(c4) c

1(c

3) c

2c3

(c1) c

4(c

0)

¹ 4(C3) 6(C

2) 4(C

3)#4(C

3) 6(C

2) 4(C

3)

¹h

4(C3h

) 6(C2h

) 8(C3) 6(C

2v) 4(C

3h)

¹d

4(C3v

) 6(S4) 8(C

3) 6(C

2v) 4(C

3v)

O 6(C4) 12(C

2) 6(C

4)#8(C

3) 12(C

2) 8(C

3)

6(C4) 12(C

2) 8(C

3)#8(C

3) 12(C

2) 6(C

4)

8(C3) 12(C

2) 6(C

4)#6(C

4) 12(C

2) 8(C

3)

6(C4)#6(C

4) 12(C

2) 8(C

3) 12(C

2) 8(C

3)

Oh

6(C4h

) 12(C2h

) 6(C4v

)#8(C3h

) 12(C2v

) 8(C3v

)(a) 6(C

4h) 12(C

2h) 8(C

3h)#8(C

3v) 12(C

2v) 6(C

4v)

8(C3h

) 12(C2h

) 6(C4h

)#6(C4v

) 12(C2v

) 8(C3v

)6(C

4h)#6(C

4v) 12(C

2h) 8(C

3h) 12(C

2v) 8(C

3v)

> 12(C5) 30(C

2) 12(C

5)#20(C

3) 30(C

2) 20(C

3)

12(C5) 30(C

2) 20(C

3)#20(C

3) 30(C

2) 12(C

4)

20(C3) 30(C

2) 12(C

5)#12(C

5) 30(C

5) 20(C

3)

12(C5)#12(C

5) 30(C

2) 20(C

3) 30(C

2) 20(C

3)

>h

12(C5h

) 30(C2h

) 12(C5v

)#20(C3h

) 30(C2v

) 20(C3v

)12(C

5h) 30(C

2h) 20(C

3h)#20(C

3v) 30(C

2v) 12(C

5v)

20(C3h

) 30(C2h

) 12(C5h

)#12(C5v

) 30(C2v

) 20(C3v

)12(C

5h)#12(C

5v) 30(C

2h) 20(C

3h) 30(C

2v) 20(C

3v)

In some cases the harmonic approximation of the Hamiltonian near an extremal orbit should beisotropic due to symmetry (extremal orbit is a critical one with its stabilizer being a group withhigh symmetry). In such a case we have only the anharmonicity parameter d!/) and the tunnelingsplitting parameter d56/. Typically d!/) is su$ciently small but nevertheless at the same timed!/)'d56/. In such a case the energy spectrum of the quantum problem near the extremum may berepresented as a spectrum of a k-fold 2D-dimensional isotropic (slightly anharmonic) oscillator. Itmeans that each polyad of a 2D-isotropic (slightly anharmonic) harmonic oscillator is replaced bya k-fold cluster of similar polyads.

Typically for the model problem with an anisotropic harmonic oscillator we can neglect theanharmonicity correction and assume equally that d56/(d!/*. In such a case the energy spectrum ofthe quantum problem near the minimum or maximum may be represented as a spectrum of a k-fold2D anisotropic harmonic oscillator. It means that each non-degenerate level of the 2D-anisotropicharmonic oscillator is replaced by a k-fold cluster of energy levels. If both d56/ and d!/) are smalland have the same order of magnitude, then polyads are formed by energy levels of each almostisotropic harmonic oscillator and the total energy level pattern is a system of k-fold clusters ofvibrational polyads. Internal structure of each cluster depends on the relation between d56/ and


d!/) and may be complicated and vary signi"cantly from one polyad to another because the ratiobetween anisotropic and tunneling splittings changes with the vibrational energy increase.

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

Hydrogen atom in magnetic and electric "elds gives us an opportunity to study the dependenceof 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 lowto ensure the small splitting of the n multiplets with respect to the splitting between multiplets. Inthe case of non-zero electric "eld the ionization is always possible and consequently the strictnon-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 ourselveswith 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 anexternal magnetic "eld, for example, we "nd here conditions which enable one to treat n-multipletsseparately. It is su$cient that the diamagnetic shift which roughly varies as G2n4 (G is a magnetic"eld strength in atomic units, 1 a.u."2.35]105T) remains smaller than the energy di!erencebetween two consecutive multiplets (*E"1

2n2!1

2(n#1)2&1/n3), i.e. Gn7@2(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!ectiveHamiltonians 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 thesymmetry of the problem created by the two "elds. To specify the symmetry we de"ne "rstthe 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 absolutecon"gurations which can be mutually transformed by some rotation of the laboratory frameshould 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 tothe classi"cation of orbits of the O(3) group on the six-dimensional space generated by one polarand one axial vector.

Three parameters, F2, G2, and the scalar product (FG) with a natural relation between them(FG)24F2G2, 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 andpoints of the "lled cone in the 3D-space (see Fig. 6). The cone shown in Fig. 6 is the orbifold of theO(3) group action on the space of absolute "eld con"gurations. Qualitatively di!erent relativecon"gurations are characterized by di!erent symmetry groups. There are six di!erent types of "eldcon"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


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

Table 20Symmetry 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

G2"F2"0 Point O O(3)'T No "eldsG2"0, F2'0 Ray OF2 C

=v'T Stark e!ect

G2'0, F2"0 Ray OG2 C=h

'Ts

Zeeman e!ectG2'0, F2'0, (GF)2"G2F2 Conical surface C

='T

sParallel "elds

G2'0, F2'0, (GF)"0 Plane G2OF2 G4

Orthogonal "eldsG2'0, F2'0, Interior of the cone T

sGeneric

(GF)2(G2F2, con"guration(GF)O0

operations: purely geometrical spatial transformations forming one of the standard point sym-metry groups and symmetry operations related to time reversal.

We use the following notation. The group T includes two elements, identity E and time reversal¹. The group T

sincludes also two elements, identity E and the symmetry operation (¹p) which is

the product of the time reversal and the re#ection in a plane including both "elds. GroupG

4includes four symmetry elements: identity E, re#ection in the plane orthogonal to the magnetic

"eld ph, product ¹p of time reversal ¹ and re#ection p in plane formed by two orthogonal "elds,

and the symmetry operation ¹C2

which is the product of the time reversal ¹ and the C2

rotationaround the electric "eld direction. G

4has three subgroups of order two: C

s"(E,p

h), T

s"(E,¹p),

and T2"(E,¹C

2).

We can study now the dependence of the dynamics on the ratio of the "eld strengths assumingthat the total e!ect of both "elds is kept to be more or less the same even if we change the "eldcon"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 wetake into account that the energy correction to the hydrogen atom in the linear Zeeman limit is


*En+$Gn3 and in the linear Stark limit is *E

n+$3Fn4 where n is the principal quantum

number for a perturbed hydrogen atom n+J!1/(2E). Thus for given n it is natural to "xS"J(Gn3)2#(3Fn4)2 and to vary only the relative strength of the two "elds:

Gs"

Gn3

J(Gn3)2#(3Fn4)2, (146)

Fs"

3Fn4

J(Gn3)2#(3Fn4)2. (147)

We consider the case of small S but we allow large variations of the relative strengths of the two"elds 04F

s,G

s41 with the restriction F2

s#G2

s"1.

A very interesting question concerns the qualitative behavior of the n-shell dynamics under thevariation of "eld parameters (assuming S to be small enough). Are e!ective Hamiltonians for then-shell always of the simplest Morse type? Or there are some regions in the space of parameterswhere 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 "eldlimit) 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 representedby points on the surface of cone in Fig. 6) and the case of orthogonal "elds (con"gurationsrepresented by points on the bisectral plane of the cone in Fig. 6). But before looking on these caseswe 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

H"

p2

2!

1r#

G2

8(x2#y2) . (148)

This Hamiltonian is very popular from the point of view of theoretical investigations of di!erentdynamical 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; Sadovskiiet al., 1995; Solov'ev, 1981,1982; Tanner et al., 1996; Uzer, 1990).

The Hamiltonian in Eq. (148) is D=h

'T invariant. Remark that its symmetry is higher than thatof an atom in the presence of magnetic "eld (which is C

=h'T

s) because we neglect the terms linear

in angular momentum. Consequently any e!ective diagonal in n operator can be rewritten in termsof operators corresponding to the D

=hpolynomial invariants, m, k2, l2 (see Eqs. (116)}(118)). The

auxiliary invariant lp does not enter in Hamiltonian (148) due to additional time-reversalinvariance. We replace Hamiltonian (148) by the diagonal in n e!ective operator just by projectingit on the manifold of the non-perturbed hydrogen atom wave functions with a given n quantumnumber. Such procedure is physically meaningful in the low magnetic "eld limit.


Fig. 7. Qualitative description of critical orbits versus energy for e!ective D=h

-invariant Hamiltonian for quadraticZeeman e!ect.

The explicit expressions for the diagonal in n matrix elements of the diamagnetic terms are eitherdiagonal in l or non-diagonal in l with *l"$2 (they are diagonal in m). It is necessary to "nd suchcombinations of diagonal in n operators which give the same matrix elements as the diamagneticperturbation.

The diagonal in n perturbation of the hydrogen atom may be equivalently represented in theform of the e!ective operator identical to that used by Solov'ev (1981), Herrick (1982), Delande andGay (1986) and many others. In terms of invariant polynomials the Hamiltonian for the n shell hasthe following form:

Hn"const.#G2

n2(n2!1)16

[k2!2m!5l2] , (149)

where the const. is n-dependent. This Hamiltonian in Eq. (149) is invariant with respect to theD

=h'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 onthe D

=h'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:

k2!2m!5l2 . (150)

There are four critical orbits and the topological structure of the energy levels varies by passingthrough the critical orbits only. Corresponding energies are (in increased order) E"!3, C

=v'T

} critical orbit; E"!2, C2h

'T } critical orbit (one-dimensional manifold); E"!1, C=h

'T} critical orbit; E"2, C

2v'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. Thelowest 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 minimummay 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 tothe splitting of the energy levels within a polyad: anharmonicity of the isotropic oscillator and thetunneling between two equivalent wells. The splitting of polyad due to anharmonicity e!ects


preserves the C=

symmetry. As a consequence, the degeneracy of energy levels with the same DmD,the projection of the angular momentum is present.

If we neglect the tunneling splitting, for any DmDO0 there are four degenerate energy levels. Thenumerical results for the quantum problem clearly show this patterns. We can estimate the positionof the classical energy minimum for the C

=vorbit from quantum results by assuming the harmonic

oscillator model and taking the energy of the zero level equal to the fundamental frequency (this isthe linear approximation). Thus for example, for n"45 the harmonic frequency (the fundamentaltransition) is 0.09442 and the estimated energy of the classical energy minimum found from theentirely quantum calculations is E

-*/(C

=v)"!2.99797 as compared to (!3) for the purely

classical model.At the other end of the energy multiplet (near C

2vorbit) the energy level pattern is similar to that

of a one-dimensional rotator plus one-dimensional oscillator. The one-dimensional harmonicoscillator describes energy levels with m"0. The harmonic frequency (for n"45) can be againestimated 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 (C

2v) critical orbit:

E-*/

(C2v

)"1.99987 as compared with E"2 in the classical limit. This numerical comparison iscompletely 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 regularsequence of non-degenerate energy levels within the energy interval 25E5!1 and the regularsequence of doublets within the energy interval !25E5!3. This fact well correlates with theenergy of the critical orbit C

2hcharacterizing by the energy E(C

2h)"!2.

It is important to note that the existence of four critical orbits is the consequence of the symmetryof the Hamiltonian and does not depend on the concrete form of the Hamiltonian. At the sametime the relative positions in energy of critical orbits strictly depend on the concrete form of theoperator.

4.3. Hydrogen atom in parallel electric and magnetic xelds

We demonstrate shortly in this section one particular application of the qualitative analysis ofRydberg states by studying the transition from a Zeeman to a Stark structure of a weakly splitRydberg 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 theenergy levels. The use of classical mechanics, topology, and group theory provides detaileddescription of the modi"cations of dynamics due to the variation of the electric "eld. We focus onthe point where the collapse of the Zeeman structure occurs, give the sequence of classicalbifurcations responsible for the transition between di!erent dynamic regimes, and compare it to thequantum energy-level structure.

In fact, Rydberg atoms in parallel magnetic and electric "elds have been extensively studied boththeoretically and experimentally during the last decade. In particular, many studies have focusedon the situation where the "elds are (relatively) weak and the dynamics can be analyzed in terms ofadditional 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


Fig. 8. Collapse of the Zeeman structure for the magnetic "eld G"0.06 due to the increasing electric "eld. Quantumlevels are calculated for the n"10 multiplet of the hydrogen atom. Scaled electric "eld strength is given in units FI

0"GI /3

[see Eq. (153)].

di!erent strengths of the electric (F) and magnetic (G) "elds. These regimes clearly manifestthemselves in the energy level pattern in Fig. 8. At very weak electric "eld, where most of the studieswere done, the energy levels are grouped according to the value of ¸

z, the projection of the angular

momentum on the "eld axis. The internal structure in this region is mainly due to quadraticZeeman e!ect (see Fig. 9 and discussion in the Section 4.2). When F increases this structure quicklydisappears. 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 contraryto 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 magneticG and electric F "elds (along the z-axis) has the form (in atomic units)

H"

p2

2!

1r#

G2¸

z#

G2

8(x2#y2)!Fz (151)

with G and F in units of 2.35]105 T and 5.14]109 V/cm. We restrict ourselves to the low "eldcase where the splitting of an n-shell caused by both "elds is small compared to the splittingbetween neighboring n-shells (see Fig. 10). As is well known, in the absence of electric "eld low-msubmanifolds of the n-shell show characteristic pattern of the second order Zeeman e!ect. Whenthe electric "eld e!ect is of the same order as the quadratic Zeeman e!ect (see Fig. 9), this patterndisappears 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 Starksplitting of the n-shell (J3Fn2) is of the same order as the n-shell splitting due to magnetic "eld


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

Fig. 10. Neighboring Rydberg multiplets n"9, 10, 11 of the hydrogen atom.


(JGn) (Fig. 8), in other words when

F0"G/(3n) . (152)

Eq. (152) gives the collapse condition for shell n. To ensure that this collapse happens when then-shell splitting remains small compared to the gap between n-shells, we take G(1/n4. Under thisassumption we can study an isolated n-shell and can use scaling

FI "Fn4, GI "Gn3, EIn"2n2E

n#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 andRunge}Lenz vector A"p]L!r/r, or, alternatively, in terms of their linear combinationsJ1"(L#A)/2 and J

2"(L!A)/2. For the linear Stark}Zeeman e!ect in parallel "elds the

e!ective Hamiltonian is

H"

12n2

(!1#Gn2¸z#3Fn3A

z) . (154)

If we impose the relation between "eld strengths (152) this Hamiltonian becomes

H"

12n2

(!1#3F0n3(J

1)z) . (155)

The n2 energy levels in the n-shell described by Eq. (155) form n-fold degenerate groups. The levelsin each group are labeled by the same value of (J

1)z

and by di!erent values of (J2)z. Fig. 8 shows

how the Zeeman structure of the n-shell at f"0 transforms into this highly degenerate structure atF"F

0(FI "GI /3). We call this e!ect the collapse of the Zeeman structure caused by electric "eld.

To describe the "ne structure of each (J1)z

manifold of states the second-order e!ects should betaken into account.

To develop the e!ective n-shell Hamiltonian to higher orders we consider n as an integral ofmotion and use the perturbation theory to reduce the initial problem (151) to two degrees offreedom. Naturally, the pair (¸

z,/

Lz) describes one of these degrees; the other degree can be

described by Az

and /Az

(Farrelly et al., 1992). Of course, for Hamiltonian (151) ¸z

is strictlyconserved and the n-shell Hamiltonian does not depend on /

Lz. However, to study the collapse we

should consider the energy level structure of the n-shell as a whole, and therefore, we should keep¸z

as a dynamical variable. Hence our n-shell Hamiltonian is a function of dynamical variables(¸

z, A

z,/

Az) and parameters (n, F,G).

The classical phase space R for e!ective n-shell Hamiltonian is a 4D space with topology S2]S

2.

Its parametrization can be done either using the L, A variables with L2#A2"n2, and L )A"0, orusing the J

1, J

2variables with J2

1"J2

2"n2/4. (In the classical limit n is su$ciently large and

n2+n2!1.)


For the qualitative analysis of the n-shell dynamics we use invariant polynomials

l"Az/n; k"¸

z/n; m"(¸2!A2)/n2 (156)

forming integrity basis which is used both to label the points of the phase space and to expand theHamilton function as explained in Section 5.6 of Chapter I (see also Appendix A). Furthermore, theproper scaling in Eqs. (156) and (153) results in equations which do not depend on n.

The symmetry group G"C=

'Tsof the problem and its action on R 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 ofChapter 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 thephase space. The strati"cation of the R phase space under the action of the symmetry groupC

='T

sis given in Tables 8 and 13.

The Hamilton function can be expressed as a polynomial H"H(l,k, m) of invariant polynomialsl,k, m. Up to quadratic in F and G terms the scaled energy EI has the form

EI "GI k#3FI l!14GI 2m#1

8(9FI 2#GI 2)k2#1

8(3FI 2!5GI 2)l2#1

8(3GI 2!17FI 2) . (157)

To qualitatively characterize classical and quantum dynamics we "nd the system of stationarypoints (manifolds) of the energy function on the phase space. Group theory asserts that four pointsA,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. Morseinequalities con"rm that the simplest Morse-type functions possessing stationary points only onthe four critical orbits really exist on R and have one minimum, one maximum, and two saddlepoints. For more complicated functions any other stationary points can be found by looking forthose energy sections of the orbifold which correspond to the modi"cation of the topology of theenergy section.

Simple geometrical analysis shows that in the linear (in F and G) approximation for F(F0

theenergy function is of the simplest type with minimum in B, maximum in A, and two saddle points inC and D. For F'F

0the 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 theenergy function to another one in the linear model occurs due to the formation of the degeneratestationary manifold at value F"F

0corresponding to Hamiltonian (155). In the linear model

the energy surface touches the orbifold through the whole interval [C,A] or [B, D]. Introductionof the F2 and G2 terms into the energy function removes this degeneracy. The energy surface (157) isthe second-order surface in m,k, l variables. It can touch the orbifold O at some isolated points onthe p"0 surface which are di!erent from the critical orbits A,B,C, D. If this happens, additionalstationary orbits are present. The detailed analysis of a system of stationary points as a function ofF near the collapse value F

0shows how the transformation from the Zeeman-type energy function

(with only four stationary critical orbits having minimum and maximum in B and A) to theStark-type energy function (with only four stationary critical orbits having minimum and max-imum in D and C) occurs. Two sequences of bifurcations are present with two bifurcations in eachsequence. As F increases, one sequence begins with a bifurcation at point B which creates a new


Fig. 11. Bifurcation diagram near the collapse region. Classical (left) versus quantum (right) representation.

stationary S1

orbit of EI on R. The corresponding point on the surface of the orbifold moves fromB to D and disappears at D after the second bifurcation. Another sequence of bifurcations proceedsin the similar way with two bifurcations at C and A and the new additional stationary orbit movingfrom 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 vectorto the energy surface k"(kl , kk , km) are collinear. This geometric view gives us extremely simpleconditions for bifurcations at points A,B,C,D:

A: 4kmkk"k2l!k2k , dA+!GI 2/8!GI 3/16 , (158)

C: 4kmkl"k2l!k2k , dC+2GI /3#GI 2/72 , (159)

B: 4kmkk"k2k!k2l , dB+!GI 2/8#GI 3/16 , (160)

D: 4kmkl"k2k!k2l , dD+!2GI /3#GI 2/72 . (161)

When d varies between dA

and dC

an additional stationary orbit exists on the surface of the orbifoldand moves from the point A to the point C. Similarly, for d between d

Dand d

Banother additional

stationary orbit moves from D to B. The energies of all stationary orbits near the bifurcation pointsand the quantum energy levels are shown in Fig. 11.


Simple quantum mechanical interpretation of the e!ect of the transformation of theZeeman-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 eachextremal state by two average values S¸

zT and SA

zT. From the positions of stationary points on

the orbifold it follows immediately that for the state with maximal energy S¸zT+n for d(d

A,

S¸zT+0 for d'd

C, whereas S¸

zT varies almost linearly with d for d

A(d(d

C. For the same

state SAzT+0 for d(d

A, SA

zT+n for d'd

C, whereas SA

zT varies almost linearly with d for

dA(d(d

C.

We conclude that important qualitative modi"cations of dynamics take place in the collapseregion. This suggests new experimental investigations which can use the detailed information onmany energy-level crossings in the collapse region to obtain quantum states with desired propertiesby "ne tuning of the "eld parameters. Existence of collapsed levels with di!erent projections of theorbital momentum m can be used in the experiment to selectively produce states with any possiblem 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 manyexperimental (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 ofqualitative information can be obtained taking into account only general topology and symmetryinformation.

The hydrogen atom in the presence of two orthogonal "elds gives us an example of a system withthe "nite symmetry group G

4(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 theorbifold in a geometrical way. Nevertheless, very useful topological and group-theoretical informa-tion can be found by studying the invariant subspaces of di!erent symmetry. We remind on thisexample again major steps of the qualitative analysis realized in Chapter II for di!erent rovibra-tional 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 G4

invariantpolynomials on R. The procedure we follow here is formally the same as that used in Appendix Bfor molecular point group symmetry.

The simplest way to write the Molien function is to work in the J1, J

2representation, to consider

the ring of all invariant functions constructed from six dynamic variables and to restrict this ringto 4D classical phase space, R. We start with the situation without any additional symmetry(the symmetry group is trivial, C

1). On the six-dimensional space (J

1)a , (J

2)b , (a, b"x, y, z) the

Molien function for invariants has a trivial form

MC1"

1(1!j)6

. (162)


The ring of the invariant polynomials on the 6D-space PC16

is the ring of all polynomials

PC16"P[(J

1)x, (J

1)y, (J

1)z, (J

2)x, (J

2)y, (J

2)z] . (163)

To restrict the polynomial ring on the sub-manifold

h5"(J

1)2x#(J

1)2y#(J

1)2z"const. , (164)

h6"(J

2)2x#(J

2)2y#(J

2)2z"const. , (165)

we are obliged to introduce two second-order polynomials, h5, h

6given by Eqs. (164) and (165) as

denominator invariants. This may be done by multiplying both the numerator and denominator ofthe Molien function (162) by (1#j)2. The new form of the Molien function

MC1"

1#2j#j2

(1!j)4(1!j2)2(166)

corresponds now to another description of the ring of invariants PC16

which is considered as a freemodule

PC16"P[(J

1)x, (J

1)y, h

5, (J

2)x, (J

2)y, h

6]v(1, (J

1)z)(1, (J

2)z) (167)

with four auxiliary invariants u1"1, u

2"(J

1)z, u

3"(J

2)z, u

4"(J

1)z(J

2)z. We use the nota-

tion (a, b)(c, d)"(ac, ad, bc, bd) to show that four numerator invariants are represented as productsof more simple terms. Having the form (166) of the Molien function and the form (167) of themodule of invariant functions it is easy to make the restriction to the sub-manifold R. We justeliminate two denominator invariants, h

5, h

6, corresponding to the equations de"ning R. The

resulting Molien function and the module of invariants on R are written as follows:

MC1

DR"1#2j#j2

(1!j)4, (168)

PC1 DR"P[(J1)x, (J

1)y, (J

2)x, (J

2)y]v(1, (J

1)z)(1, (J

2)z) . (169)

The choice of basic and auxiliary invariants is ambiguous and the ones proposed here is just anexample. The important point is that the integrity basis may be constructed which includes fournumerator invariants (including 1) and four denominator invariants.

To "nd now the integrity basis in the A, L representation we can simply transform invariantsfrom J

1, J

2to 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 actionof the symmetry group on basis polynomials we "x the coordinate frame in such a way that thex-axis coincides with the electric "eld vector and the y-axis belongs to the plane of two "elds andhas the positive projection of the magnetic "eld on it.

Let Tp be the symmetry operation for the generic con"guration of two "elds considered. Itfollows immediately that A

x,A

y,¸

x,¸

yare invariant with respect to this symmetry operation,

whereas Az,¸

zare 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


Table 21Transformation properties of dynamic variables for two orthogonal "elds under the action of G

4group

E Tp(eb) p(ep) TC(e)2

Ae

# # # #

Ab

# # ! !

Ap

# ! # !

¸e

# # ! !

¸b

# # # #

¸p

# ! ! #

(J1)e

# # !(J2)e

!(J2)e

(J1)b

# # (J2)b

(J2)b

(J2)e

# # !(J1)e

!(J1)e

(J2)b

# # (J1)b

(J1)b

(J1)p

# ! !(J2)p

(J2)p

(J2)p

# ! !(J1)p

(J1)p

(J1)p(J

2)p

# # # #

(J1)2 # # (J

2)2 (J

2)2

(J2)2 # # (J

1)2 (J

1)2

time between three non-trivial numerator (auxiliary) invariants only, one is invariant with respectto the Tp operation, whereas two others change sign under this operation. This means that theMolien function and the module of invariants on R in the case of a generic con"guration of two"elds has the form

M2F '%/DR"1#j2

(1!j)4, (170)

P2F '%/DR"P[(J1)x, (J

1)y, (J

2)x, (J

2)y]v(1, (J

1)z(J

2)z) . (171)

The symmetry group for the case of two orthogonal "elds is higher. It includes four symmetryelements E is the identity, Tp, introduced just above, p

035) Bthe re#ection in the plane orthogonal

to the magnetic "eld B (this plane includes the electric "eld vector), and TC2

the product of thetime reversal and the C

2rotation 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 withrespect to external "elds. We use below in this section the coordinate frame Me, b, pN with vectore along the electric "eld vector, vector b along the magnetic "eld vector, and vector p chosen toform the right-hand frame.

Symmetry properties of Aa ,¸b 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 thesub-manifold R. Instead of J2

1"J2

2"const. we can use J2

1#J2

2"const. and J2

1!J2

2"0. The

"rst transformed equation is invariant with respect to the symmetry group. The second ispseudo-invariant. To deal with invariants only on the 6D-space we should "rst change the integrity


basis by introducing as basic invariants h5a

, h6a

, and as an auxiliary invariant u6b

and instead of(J

1)e, (J

1)b, (J

2)e, (J

2)b

their linear combinations which have well-de"ned symmetry properties(irreducible tensors) with respect to the symmetry group

h5a"(J2

1#J2

2) , h

6a"(J2

1!J2

2)2, u

6b"(J2

1!J2

2) , (172)

h1"(J

1)e!(J

2)e, h

2"(J

1)b#(J

2)b

, (173)

h3"(J

1)e#(J

2)e, h

4"(J

1)b!(J

2)b

, (174)

u1"(J

1)p!(J

2)p, u

2"(J

1)p#(J

2)p, u

3"(J

1)p(J

2)p

. (175)

Now, we have the description of the ring of C1

invariant polynomial on the 6D-space in terms ofthe integrity basis which includes invariants and pseudo-invariants of the symmetry group for twoorthogonal "elds. The Molien function and the ring of invariants PC1

6can be written as

MC16"

(1#2j#j2)(1#j2)(1!j)4(1!j2)(1!j4)

, (176)

PC1 D6"P[h

1, h

2, h

3, h

4, h

5a, h

6a]v(1,u

1,u

2,u

3)(1,u

6b) . (177)

Reduction on R should be done now by eliminating two basic invariants, h5a

, h6a

and one auxiliaryinvariant u

6bwhich corresponds to zero on R. Among basic denominator invariants we have two,

(h3, h

4), 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 canchange the integrity basis by introducing instead of h

3, h

4two new basic invariants h

3a"h2

3,

h4a"h2

4and three auxiliary polynomials u

4a"h

3, u

4b"h

4, u

4c"h

3h4.

After such a modi"cation we can rewrite the Molien function on R and the module ofpolynomials on R in terms of an integrity basis including as basic polynomials only invariantpolynomials with respect to the total symmetry group, and as auxiliary polynomials both invari-ants and pseudo-invariants of the total symmetry group:

MC1

DR"(1#2j#j2)(1#j)2

(1!j)2(1!j2)2, (178)

PC1 DR"P[h1, h

2, h

3a, h

4a]v(1,u

1,u

2, u

3)(1,u

4a, u

4b,u

4c) . (179)

The list of polynomials forming the integrity basis is as follows:

h1"A

e, h

2"¸

b, h

3a"¸2

e, h

4a"A2

b, (180)

u1"A

p, u

2"¸

p, u

3"(J

1)p(J

2)p

, (181)

u4a"¸

e, u

4b"A

b, u

4c"¸

eA

b, (182)

u1u

4a, u

1u4b

, u1u

4c, u

2u

4a, u

2u

4b, u

2u

4c, (183)

u3u

4a, u

3u4b

, u3u

4c. (184)


Table 22Invariant manifolds for the G

4symmetry group action on R (Hydrogen atom in perpendicular electric and magnetic

"elds.)

Stabilizer dim. Topology Equations

G4

1 S1

¸2b#A2

e"1

Cs

2 S2

¸2e"A2

b"0, ¸2

p!A2

p40

T2

2 S2

¸2e"A2

b"0, ¸2

p!A2

p50

Ts

2 S1]S

1¸2e#¸2

b#A2

e#A2

b"1

C1

4 S2]S2 R (L2#A2"1; (L )A)"0)

Now, we can simply eliminate all auxiliary polynomials which are not invariant with respect to thetotal symmetry group for two orthogonal "elds. This gives the following Molien function and themodule of invariant on R:

M2035) FDR"(1#2j2#j4)

(1!j)2(1!j2)2, (185)

P2035) FDR"P[Ae,¸

b,¸2

e, A2

b]v(1,¸

eA

b, (J

1)p(J

2)p, (J

1)p(J

2)p¸

eA

b) . (186)

Remark that we can replace (J1)p(J

2)p

by (¸2p!A2

p).

Next step of the qualitative analysis is the strati"cation of the phase space R under the action ofthe G

4symmetry 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 locationof stationary points of the Hamilton function. As long as there are no zero-dimensional strata ofthe group action, there are no critical orbits. Nevertheless, we can say that there should be at leasttwo stationary orbits on the G

4invariant subspace and at least one additional stationary

Ts

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 thatfor hydrogen atom in orthogonal "elds it becomes clear that for su$ciently low "elds the evolutionof the Zeeman multiplet into the Stark multiplet goes through a sequence of bifurcations forparallel "elds, whereas for orthogonal "elds no generic bifurcations are present. Immediatelya natural question arises. What can be said about generic "eld con"gurations? Is it reasonable toexpect 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 "eldsF, G imposing the restriction of the type F2#aG2"S2, where S is supposed to be su$cientlysmall and the positive parameter a can be chosen in such a way that the splitting of the Rydbergmultiplet in Zeeman and Stark limits are approximately the same.

If electric and magnetic "elds are non-collinear and non-orthogonal, the only non-trivialsymmetry operation is the composition of time reversal and the space re#ection in the plane


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

including two "elds. So the symmetry group for such generic orientation is the Ts

groupintroduced earlier. The only non-trivial invariant subspace is the T

sinvariant torus. On this torus

four stationary points are generically present. The same four stationary points should always bepresent 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-simplestMorse-type Hamiltonian. Near this point the Hamiltonian is not of the simplest Morse type as weshow below.

Let us consider the collinear con"guration of two "elds corresponding to the range of F,Gparameters such that additional extrema on R exist. These additional extrema correspond to pointson p"0. After a small deformation of the con"guration of "elds ("elds become non-orthogonalafter an arbitrarily small perturbation) the symmetry is broken but each stationary orbit leads toone stationary orbit of the lower symmetry group. As soon as for parallel "elds on a T

sinvariant

torus (with one particular orientation of symmetry plane) there are more than four stationarypoints, 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 isa 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 topologicalmethods work in a particular physical problem, Rydberg states of atoms and molecules. Ascompared to the similar analysis realized earlier for molecular rotations and vibrations, themathematical di$culty was overcome in this study. This is the presence of continuous symmetryrelated 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


manifolds. At the same time the general scheme of the qualitative analysis follows the waydeveloped 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 beginningthe 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 possibleHamiltonian functions. The Hamiltonian which is interesting from the point of view of physicalapplications 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 obtainedin such way is certainly more restrictive than the result of concrete numerical analysis of theparticular 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 theprecise form of the Hamiltonian, especially taking into account the fact that any Hamiltonian usedfor a concrete physical application is always an approximation.

For Rydberg problems we have studied very simple case of small splitting of n-shell. It iscertainly possible to "nd some physical situations when such a model is rather accurate andreasonable. At the same time it is clear that for real molecular and atomic systems there are manyinteractions (and additional degrees of freedom) which become important and often can modifyconsiderably even the qualitative behavior. Among the simplest but essential corrections are thosedue 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 ofatomic 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 atomRydberg states were done. The quantum defect theory is the most popular and powerfool tool ofthe theoretical analysis and interpretation of experimental data. Aymar et al. (1996) reviewedrecently this subject. At the same time the systematic qualitative analysis of e!ective n-shellRydberg Hamiltonians for free non-hydrogenic atoms or atoms in external "elds has yet notbeen done.

Doubly excited Rydberg states of atoms give much more complicated examples with richqualitative structure which was pointed out initially by Herrick and Sinanoglu (1975) on the basisof comparison of the approximate O(4) dynamical symmetry and extensive numerical calculations.To understand better the qualitative features of electronic excited states and especially theirlocalization 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 ofa two-electron problem on the same basis as it is done in the present paper for one-electronRydberg problem.

We have not practically touched in the present review the concrete applications of the qualitativetheory to Rydberg states of diatomic or polyatomic molecules except for the general symmetryanalysis. It should be noted that the application of a qualitative analysis should begin with thereanalysis of the excited states of the simplest one-electron diatomic molecule H`

2. In spite of its

apparent simplicity the description of the "ne structure of highly excited states of H`2

still remainsa question of interest (Brown and Steiner, 1966; Coulson and Joseph, 1967; Grozdanov and


Solov'ev, 1995). Diatomic molecules are of particular theoretical interest due to the presence ofcontinuous 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 orstrong breaking of the interchange symmetry of two nuclei. Moreover, for a number of diatomicmolecules 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 andWilkerson, 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) andexperimental data or results of alternative numerical modelization are ready to be comparedthrough the qualitative analysis in order to reveal some universal features of the system of Rydbergstates of diatomics.

Among polyatomics, the H3

molecule is surely the most popular as an object to study Rydbergstates (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 ofso-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 electronon a high Rydberg orbit. Typically, Rydberg molecules are bound only in excited electronic statesand their predissociation becomes more pronounced under electronic desexcitation. H

3and NH

4are typical Rydberg molecules (Herzberg, 1981). More exotic examples of Rydberg dimers like(H

3)2

or (NH4)2, 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 suchbig objects as fullerenes (Weber et al., 1996) or metal surfaces (Ganesan and Taylor, 1996).

Comparison of simple model electronic Rydberg calculations with concrete molecular experi-ments 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, forexample, the stabilization of unstable rotational axes of an asymmetrical top molecule due to theinteraction with a Rydberg electron as proposed by Basov and Pavlichenkov (1994) or coreinduced stabilization of molecular Rydberg states discussed by Lee et al. (1994). Monitoring ofintramolecular dynamics through preparation of special Rydberg electron wave packets is nolonger 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-shellapproximation 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 caseof `overlappinga and to the case of more serious `interactionsa of di!erent n-shells? This problemcan be attacked from two opposite sides. One possibility is to start with the model of `completeclassical chaosa and to study peculiarities of quantum systems through some kind of statistical orsome 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!ectsrelated to the violation of the individual n-shell approximation. Such an extension of the qualitativeapproach to the qualitative theory allowing the `couplinga of di!erent n-shells should be extremely


useful for the Rydberg problem in the intermediate region between quasiregular and completelychaotic motion. The analog of such an extension was proposed earlier in the qualitative analysis ofmolecular rovibrational structure. Rotational multiplets of di!erent individual vibrational compo-nents can be considered as formal analogs of di!erent n-multiplets. In the classical limit theviolation of individual components (vibrational components for rovibrational problems or n-shellsfor Rydberg problems) is associated with the formation of `diabolica points (conical intersectionpoints) between di!erent energy surfaces.

As was shown on simple initial example by Pavlov-Verevkin et al. (1988) the new qualitativephenomenon, namely the redistribution of energy levels between di!erent branches in the energyspectrum typically appears in such a case under variation of the integral of the motion. Eachelementary qualitative phenomenon can be characterized by a topological invariant which is stableunder small deformation of the Hamiltonian. Recent theoretical analysis (Zhilinskii and Brodersen,1994; Brodersen and Zhilinskii, 1995b; Brodersen and Zhilinskii, 1995a; Zhilinskii, 1996) supportsthis point of view and makes some interesting relations with rather di!erent physical phenomenalike 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 liketopological obstructions to integrability (Nekhoroshev, 1972; Duistermaat, 1980; Cushman andBates, 1997).

Complete classical analysis of the redistribution phenomenon discussed in Section 6.1 of ChapterII (Sadovskii and Zhilinskii, 1999) has given an interesting mathematical relation with classicalmonodromy. Further study of this phenomenon (Cushman and Sadovskii, 1999) shows thepresence 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 andcorresponding analysis of associated classical models will surely become a new subject of furthertheoretical and experimental study.

Appendix A. Geometrical representation

We consider in this appendix the geometrical representation of orbifolds which is the initial stepfor the geometrical representation of qualitatively di!erent types of the Morse}Bott-type functionsde"ned over classical phase space for various invariance groups. As long as we are working withfunctions which are supposed to be the classical analog for quantum e!ective Hamiltonians, we willuse the notion `energy levela for a solution of the equation H"const. In our particular case theHamiltonian function is de"ned over a four-dimensional space (R"S

2]S

2), so the energy level is

normally a three-dimensional region of the phase space which may be characterized by itstopology. Taking into account the action of the symmetry group we can reach more detailedinformation about the structure of each energy level by indicating the topological structure oforbits 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 becauseany SO(3) invariant operator is also invariant with respect to the O(3) group on the R"S

2]S

2


Fig. 13. Orbifold for the O(3) group action on R"S2]S2. Orbits are parametrized by the value of one invariantpolynomial, m.

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

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

0(2) 2 e]S2

C=v

no 3 e]RP3 Cs

2(0) 2 e]S2

C=h

manifold. So if one is restricted to the consideration of the diagonal in `na e!ective operators, thereis no di!erence between SO(3) and O(3) symmetry groups.

The orbifold for the O(3) action on the R"S2]S

2manifold is shown in Fig. 13. From the

topological point of view it is a 1D-ball (RDO(3)&B1). Orbits are parametrized by the value of one

invariant polynomial, m. The detailed description of orbits and strata is given in Table 4 of thispaper. The simplest Morse}Bott-type function de"ned on R and which is O(3) invariant possessestwo critical manifolds coinciding with two critical orbits of the O(3) group action. Di!erent energylevels 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 ofcritical orbits is two, the Morse indices for minimal or maximal critical manifolds are equal to0 or 2 correspondingly.

Any function H"H(m) which has RH/RmO0 for !14m41 may be used as an example of thesimplest 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 topologicalproperties. For example, the e!ective Hamiltonian of the form

H"c(m#12)2 (A.1)

has "ve topologically di!erent energy levels given in Table A.2. The Morse index for the RP3stationary manifold may be either 0 or 1 (because the dimension of the RP3 stationary mani-fold is 3).

One can easily see that the total number of critical manifolds may be arbitrary but the numbersof critical manifolds with given indices are related among themselves.


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

Morse index Topological structure Local symmetryof the energy level

0(1) e]RP3 Cs

no 2Me]RP3N Cs

2(0) Me]S2N# C

=vno Me]RP3N C

s

no e]RP3 Cs

2(0) e]S2

C=h

If the total number of critical manifolds is even there are one maximum and one minimum whichare situated on critical orbits (C

=vand C

=h) and equal number of minima and maxima on generic

(Cs) orbits. If the total number of critical manifolds is odd, there are two maxima (or two minima)

on critical orbits (C=v

and C=h

) and odd, 2p#1, (even, 2p) number of minima and even, 2p,(odd, 2p#1) number of maxima on generic (C

s) orbits (p50 being nonnegative integer).

A.2. C=

invariant Hamiltonian

Orbits for the C=

action on R 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 takinginto account that for p2"0 there is one-to-one correspondence between points of the orbifold andm,k, l values satisfying the equalities in Eq. (16). If p2'0 there are two orbits with the same m,k, lvalues but with the di!erent signs of p

p"$[14(1!m2)!1

2(1!m)k2!1

2(1#m)l2]1@2 . (A.2)

We represent an orbifold as consisting of two parts, one corresponding to p50 and othercorresponding 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 k241. For m"!1 we have k"0 and l241. For anym2(1, putting p"0 we can "nd the geometrical form of the boundary of each part of the orbifoldin the 3D-(m,k, l)-space. The equation for the boundary has the form

11#m

k2#1

1!ml2"

12

. (A.3)

So any m"const. section of one part of the orbifold is an ellipsoid. The complete orbifold may beconstructed by identifying the surfaces of two 3D-bodies corresponding to p50 and p40 parts ofthe orbifold.

To specify the topological structure of the orbifold it is necessary to use some additionalinformation about topology of three-dimensional manifolds (Fomenko, 1983; Thurston, 1969).


Fig. 14. Orbifold for the C=

group action on R"S2]S2. Each part of the orbifold is parametrized by three invariantpolynomials, m, k, l. Two parts corresponding to di!erent signs of the auxiliary polynomial p should be glued togetherthrough the identi"cation of respective points on the boundary p"0.

Fig. 15. Representation of the orbifold for the C=

group action on R"S2]S2 in non-polynomial variablese1"arccosm, e

2"arccos(k#l), e

3"arccos(k!l). Only one part of the orbifold is shown.

Every three-manifold can be obtained from two handle-bodies (of some genus) by gluing theirboundaries together. Such a representation is called a Heegard decomposition. It is not unique andde"ned by number g (the genus of the boundary of two auxiliary manifolds) and by mapping of theboundaries. It is known that the only three-manifold, possessing the Heegard decomposition ofgenus 0 is the 3D sphere.

The above constructed orbifold is given just in the form of the Heegard decomposition of genus0, thus the topological structure of the orbifold is the three-dimensional sphere S

3(RDC

=&S

3)

with four marked points corresponding to the 0-D stratum. This result agrees perfectly with theresult 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"edsurfaces) but with the same topological properties (see Fig. 15). These non-polynomial variablesmay be interpreted as angles in the (x"j#k, y"j!k) representation characterizing anglesbetween x and y and the symmetry axis.

For the group C=

an example of the simplest e!ective invariant operator may be written in theform of the linear combination of the invariant polynomials

H"c1m#c

2k#c

3l . (A.5)

In fact, if the auxiliary numerator invariant p does not enter in Hamiltonian (A.5) the completesymmetry of this Hamiltonian is higher, namely it is C

='T

s. To get the generic Hamiltonian it is

necessary to choose the coe$cients ciin such a way that any section of the orbifold includes at most


Fig. 16. Representation of energy levels of the Hamiltonian H"2k!l on the C=

orbifold. Only one part (p50) of theorbifold is shown.

Fig. 17. Orbifold for C=v

group action on R. Two parts corresponding to di!erent signs of the auxiliary polynomialp should be glued together through the identi"cation of respective points on the boundary p"0.

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 connectedcomponents 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. Foursections 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 withrespect to the orbifold changes. This means that critical orbits which are stationary for any choiceof Hamiltonian can change its stability. One particular simple physical example of a hydrogenatom in parallel electric and magnetic "elds corresponds to Hamiltonian (A.5) in the simplestapproximation (see Section 4.3).

A.3. C=v


Orbifolds for other one-dimensional Lie subgroups of O(3), G"C=v

, C=h

, D=

, D=h

, can beconstructed 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 theaction on invariant polynomials.

The orbifold for the C=v

subgroup results from that of a C=

one by noting that the pvoperation

relates two orbits which belong to the same part of the C=

orbifold characterized by a given sign ofthe p invariant. So it is su$cient to take the k'0 parts of both 3D-bodies corresponding todi!erent signs of p. To properly represent the C

=vorbifold we use the m, l,k2 variables which are

the invariant polynomials for the C=v

group. In such variables each m"const., mO$1, section ofthe orbifold is a parabola. The orbifold is shown in Fig. 17.


Fig. 18. Orbifold for the C=v

group action on R represented in non-polynomial variables k1"arccos m, k

2"ar-

ccos(DkD#l), k3"arccos(DkD!l). Only one part of the orbifold is shown.

Fig. 19. Schematic topological structure of the orbifold for C=v

group action on R.

It should be noted that the identi"cation of points on the surfaces of C=

orbifold results in theidenti"cation of the C

1points on the surfaces of two parts of the C

=vorbifold. All points on

the surfaces which do not belong to the Cs

stratum should be identi"ed by pairs. At the sametime the C

sinvariant orbits lying on p50 and on p40 parts of the C

=vorbifold at

k2"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(Jk2#l), arccos(Jk2!l) , (A.7)

each part of the orbifold is a tetrahedron shown in Fig. 18. These non-polynomial variables may beagain interpreted as angles in the (x"j#k, y"j!k) representation characterizing anglesbetween x and y and the symmetry axis. From the topological point of view the orbifold isa 3D-ball, B

3, with one marked point (C

=orbit) inside and with the S2 surface which includes

2D-stratum (Cs) plus two isolated points (C

=vorbits). The schematic topological structure of the

orbifold is shown in Fig. 19. It is useful to note that the sub-manifold formed by both Cs

andC

=vstrata is a closed 3D-manifold invariant with respect to the C

ssubgroup of the C

=vinvariance

group. Its topological structure is the suspension of the 2D-torus (see Section 2.6 for discussion ofthe topology of this closed sub-manifold). This fact is important for a detailed classi"cation of theC

=v-invariant Morse}Bott functions.

A.4. C=h


For the C=h

subgroup the ph

operation relates C=

orbits with opposite p and l values. In orderto have the one-to-one correspondence between orbits and invariant polynomial values, we mustunify at one point of the C

=horbifold pairs of orbits of the C

=orbifold according to the rule

(m,k, l)"(m,k,!l) . (A.8)


Fig. 20. Orbifold for the C=h

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

Fig. 21. Orbifold for the C=h

group action on R represented in non-polynomial variables t1"arccos m,

t2"arccos(k#DlD), t

3"arccos(k!DlD). Only one part of the orbifold is shown.

Orbits of C=

having l"0, p"0 are invariant with respect to the C=h

group. Thus, the orbifoldfor the C

=hgroup may be constructed from a C

=one by taking only l50 parts of two bodies with

the identi"cation of all corresponding points on the surfaces. This orbifold is shown in Fig. 20.As soon as invariant polynomial for C

=hhas the form l2 rather than l, it is more meaningful to

change the variables and to give the orbifold in m,k, l2 variables. In these variables each m"const.,mO0, section has the form of a parabola

l2"!

1!m1#m

k2#1!m

2. (A.9)

One should remark that the geometrical form of the C=h

orbifold in the m, k, l2 variables is thesame as the geometrical form of the C

=vorbifold in the m, k2, l 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#Jl2), arccos(k!Jl2) (A.10)

to reach the simpler geometrical form of the orbifold. It is shown in Fig. 21. These non-polynomialvariables are again the angles characterizing the mutual positions of x, y and the symmetry axis inthe (x, y) representation.

From the topological point of view this orbifold is given in the form of the Heegard decomposi-tion of genus 0. So the C

=horbifold is a S

3manifold with one S

1marked circle and three marked

points: one isolated (C=

orbit) point and two (C=h

orbits) lying on the S1

marked circle (formed byC

s, C

iand C

=hstrata).

We remark again that there are two closed sub-manifolds formed each by two di!erent strata.One is formed by C

sand C

=hstrata and another by C

iand C

=hstrata. These both manifolds are

S2

spheres from the topological point of view.


Fig. 22. Splitting of the space of the C=

invariant polynomial variables m,k, l into parts with particular signs ofD

=auxiliary (numerator) invariants.

A.5. D=


In the case of the D=

symmetry the action of the G/C=

group on the C=

orbifold form largerorbits by relating orbits (m,k, l) and (m,!k,!l) with opposite p values. To construct the orbifoldfor the D

=group we "rst subdivide the C

=orbifold into parts with semide"nite signs of the

D=

group numerator invariants (lp,kp,kl). Such a splitting of the m,k, l space of the C=

invariantpolynomials into parts with speci"c signs of the D

=numerator invariants is shown in Fig. 22.

Taking into account the action of the C2

operation on C=

orbits, the D=

orbifold may berepresented as four 3D-bodies shown in Fig. 23 in D

=invariant polynomial variables m, k2, l2 with

the following identi"cation of faces, edges and vortexes:

A1B1C

1D

1"a

1b1c1d1

, (A.11)

A4B4C

4D

4"a

4b4c4d4

,

A1B1C

1"A

4B4C

4, A

1C

1D

1"a

4c4d4

,

A4C

4D

4"a

1c1d1, a

1b1c1"a

4b4c4

, (A.12)

A1C

1"a

1c1"A

4C

4"a

4c4

,

A1B1"A

4B4"a

1b1"a

4b4

, (A.13)

B1C

1"B

4C

4"b

1c1"b

4c4

,

A1D

1"a

4d4"A

4D

4"a

1d1

,

C1D

1"c

4d4"C

4D

4"c

1d1

,

A1"A

4"a

1"a

4, B

1"B

4"b

1"b

4,

C1"C

4"c

1"c

4, D

1"D

4"d

1"d

4. (A.14)


Fig. 23. Orbifold for the D=

group action on R represented in polynomial variables m, k2, l2. 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 consistsof 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)pap"4!5#6!4"1 . (A.15)

Here ap is the number or p-dimensional simplexes. As it is shown in Section 2.6, the orbifold is thesuspension (RP

2).

A.6. D=h


To go from the D=

orbifold to a D=h

one, it is su$cient to consider the action of the phoperation

on the D=

orbifold. As soon as the ph

operation changes, simultaneously signs of l and p it relatesorbits characterized by (lp'0, kp'0, kl'0) and (lp'0, kp(0, kl(0) and in a similar wayorbits characterized by (lp(0, kp(0, kl'0) and (lp(0, kp'0, kl(0). As a consequence,for the D

=horbifold we have only one body instead of each pair of bodies with the same

geometrical form. The D=h

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 Csstratum) must not be identi"ed. From the topological point of view


Fig. 24. Orbifold for the D=h

group action on R represented in polynomial variables m, k2, l2. Two parts correspondingto di!erent signs of the auxiliary polynomial lp should be glued together through the identi"cation of respective pointson the boundary lp"0.

Fig. 25. Schematic topological representation of the orbifold for D=h

group action on R.

the D=h

orbifold is equivalent to a 3D-ball. The boundary is formed by the Cs, C

2v, C

2h, C

=vstrata. Inside the ball there are C

i, C

s, C

=h, and the generic C

1strata. The schematic topological

view of the D=h

orbifold is given in Fig. 25. Several closed subspaces formed by di!erent strata areclearly seen in Fig. 25. It is important to verify that the Morse}Bott inequalities are satis"ed on allthese subspaces.

Appendix B. Molien functions for point group invariants

This appendix deals with a technical question: How to describe the system of invariantpolynomials on the R manifold in the presence of a non-linear action of the symmetry group. Theschematic answer to this question was initially formulated in Section 4 of Chapter I. Someapplications have been discussed in Chapter II and in Section 2.7 of the present chapter. Particular


example of the symmetry group G4

for the hydrogen atom on orthogonal electric and magnetic"elds was treated in more details in Section 4.4. The group G

4has an interesting physical meaning

because it includes symmetry transformations which contain both spatial and time reversaloperations but it is Abelian and this simpli"es signi"cantly the construction of the Molien functionfor invariants. Below we explain on several examples of point group symmetry with increasingcomplexity the procedure of the Molien function construction.

The construction of invariant functions on the R manifold is based on the preliminary construc-tion of the integrity basis on the six-dimensional space where the action of the symmetry group ofthe problem is linear. The Molien function and the invariants themselves for the six-dimensionalspace x=y or k=j may be found from known expressions for Molien functions and integrity basesfor irreducible representations. Next step includes the restriction of the polynomial algebra on thesub-manifold R of the 6D-space. The general procedure of the restriction of the polynomial ringde"ned on the manifold to the sub-manifold was outlined in Chapter I and realized in severalexamples in Section 2.7. The sub-manifold R is de"ned in the six-dimensional space x=y by thepolynomial equations x2"1

2, y2"1

2. If the point group symmetry does not include improper

rotations (inversion or re#ections) these polynomials may always be considered as denominatorinvariants. In such a case we just eliminate them from the integrity basis constructed for the6D-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 thesimilar problem for the proper rotation subgroup and after that take into account the e!ect ofimproper symmetry elements working directly on the 4D-sub-manifold R.

B.1. C1

group

This trivial case is useful to demonstrate how to take into account the restriction of thepolynomial ring on the sub-manifold.

We work in x=y representation. On the 6D-space there are six basic invariants xi, y

i(i"1, 2, 3)

and the Molien function for invariants has a trivial form

MC1"

1(1!j)6

. (B.1)

The ring of the invariant polynomials on the 6D-space PC16

is the ring of all polynomials

PC16"P[x

1, x

2, x

3, y

1, y

2, y

3] . (B.2)

To restrict the polynomial ring on the sub-manifold

h5"x2

1#x2

2#x2

3"1

2, (B.3)

h6"y2

1#y2

2#y2

3"1

2, (B.4)

we are obliged to introduce two second-order polynomials, h5, h

6given by Eqs. (B.3) and (B.4) as

denominator invariants. This may be done by multiplying both the numerator and denominator ofthe Molien function in Eq. (B.1) by (1#j)2. The new form of the Molien function

MC1"

1#2j#j2

(1!j)4(1!j2)2(B.5)


corresponds now to another description of the ring of invariants PC16

which is considered as a freemodule:

PC16"P[x

1, x

2,h

5, y

1, y

2,h

6]v(1,x

3)(1, y

3) (B.6)

with four auxiliary invariants u1"1, u

2"x

3, u

3"y

3, u

4"x

3y3. We use the notation

(a, b)(c, d)"(ac, ad, bc, bd) to show that four numerator invariants are represented as product ofmore simple terms. Having the form (B.5) of the Molien function and the form (B.6) of the moduleof invariant functions it is easy to make the restriction to the sub-manifold R. We just eliminate twodenominator invariants, h

5, h

6, corresponding to the equation de"ning R. The resulting Molien

function and the module of invariants on R are written as follows:

MC1

DR"1#2j#j2

(1!j)4, (B.7)

PC1 DR"P[x1, x

2, y

1, y

2]v(1,x

3)(1, y

3) . (B.8)

The choice of basic and auxiliary invariants is ambiguous and the one proposed here is just anexample. 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 j=k representation we can simply transform invariantsfrom x, y to j, k representation.

B.2. C2

point group

Let us take x3

and y3

to coincide with the C2-axis. In such a case x

1,x

2, y

1, y

2transform

according to the B representation and x3, y

3according to the A representation of the C

2point

group. The Molien function for invariants constructed from 6D reducible representation 4B#2Ahas the form

MC2"

1#6j2#j4

(1!j2)4(1!j)2. (B.9)

The explicit form of the module of invariant functions on 6D-space may be easily given as well:

PC26"P[x2

1, x2

2, x

3, y2

1, y2

2, y

3]v((1,x

1x2)(1, y

1y2), (x

1, x

2)(y

1, y

2)) . (B.10)

There are six denominator invariants and eight numerator invariants. We have in Eq. (B.10) seconddegree denominator invariants which may be replaced by invariants de"ning the sub-manifold R inEqs. (B.3) and (B.4). After such a substitution, to make the restriction on R it is su$cient to omittwo denominator invariants corresponding to equations de"ning R. The resulting Molien functionfor invariant polynomials on R and the description of the ring of invariant function as a module areas follows:

MC2

DR"1#6j2#j4

(1!j2)2(1!j)2, (B.11)

PC2 DR"P[x21, x

3, y2

1, y

3]v((1,x

1x2)(1, y

1y2), (x

1,x

2)(y

1, y

2)) . (B.12)


To "nd the integrity basis in the j=k representation we can use again the transformation from x, yto j, k representation.

B.3. Ci

point group

For point groups which are not the subgroups of SO(3) the procedure of constructing the Molienfunction for invariants on R and the integrity basis is slightly di!erent. We start with theconsideration of the similar problem for the proper rotation subgroup of the point group. There aretwo groups C

i(or in other notation S

2) and C

swhich have no non-trivial rotational subgroups.

So we use them to illustrate the construction of invariant functions for these cases. The Cigroup

includes only one non-trivial symmetry operation, inversion. Its action on x, y variables corres-ponds to interchange x%y. Taking this action into account it is easy to make the transformation ofthe C

1invariant denominator and numerator polynomials (forming the module of invariant

polynomials on R) into form with irreducible transformation properties with respect to theC

igroup:

PCi DR"P[x1#y

1, x

1!y

1, x

2#y

2,x

2!y

2]v((1,x

3#y

3)(1, x

3!y

3)) . (B.13)

This form of the module of the C1

invariant function on R is well adapted to transformation to themodule of C

iinvariant functions. First, we change the denominator invariants (x

1!y

1) and

(x2!y

2) into (x

1!y

1)2&x

1y1

and (x2!y

2)2&x

2y2

which are both C1

and Ciinvariants. This

may be achieved by multiplying the numerator and denominator of the Molien function by(1#j)2. The module of the C

1invariant function on R in such a case will include 16 auxiliary

(numerator) invariants but among them only 8 are Ciinvariants. The resulting Molien function

and the module of the Ciinvariants on R are written as follows:

MCi

DR"1#j#4j2#j3#j4

(1!j2)2(1!j)2, (B.14)

PCi DR"P[x1#y

1, x

2#y

2, x

1y1, x

2y2]v(1,u

1,u

2, u

3,u

4, u

5,u

6, u

7) ,

u1"x

3#y

3, u

2"x

3y3, u

3"(x

1!y

1)(x

2!y

2) , (B.15)

u4"u

1u

3, u

5"u

2u

3, u

6"(x

3!y

3)(x

1!y

1) , (B.16)

u7"(x

3!y

3)(x

2!y

2) . (B.17)

B.4. Cs

point group

The construction of the module of the Csinvariant functions on R is very similar to the realized

above construction for the Ciinvariants. The only di!erence is that the action of the C

snon-trivial

operation (re#ection in the symmetry plane) is now di!erent from the action of the inversion for theC

igroup. Let us suppose the symmetry plane to be orthogonal to x

3(y

3) axes. In such a case

(x1!y

1) and (x

2!y

2) are symmetrical with respect to the C

sgroup whereas (x

1#y

1) and


(x2#y

2) are anti-symmetrical. Taking this into account the Molien function and the module of

Cs

invariants on R are written immediately as follows:

MCs

DR"1#j#4j2#j3#j4

(1!j2)2(1!j)2, (B.18)

PCs DR"P[x1!y

1, x

2!y

2,x

1y1,x

2y2]v(1,u

1, u

2, u

3, u

4, u

5, u

6, u

7) , (B.19)

u1"x

3#y

3, u

2"x

3y3, u

3"(x

1#y

1)(x

2#y

2) , (B.20)

u4"u

1u

3, u

5"u

2u

3, u

6"(x

3!y

3)(x

1#y

1) , (B.21)

u7"(x

3!y

3)(x

2#y

2) . (B.22)

B.5. C2v

point group

For the point group C2v

which is not a subgroup of SO(3), we start again with the considerationof the similar problem for the proper rotation subgroup C

2. We can take the Molien function for

the C2

invariant and integrity basis for the C2

group as the initial point. All invariants of C2

spaninvariants and pseudo-invariants of C

2v(A

1=A

2representations). So we "rst make the linear

transformation of numerator and denominator invariants for the C2

group resulting in a new set ofC

2invariants which are at the same time either of A

1or of A

2type with respect to the C

2vgroup.

This linear transformation does not change the form of the Molien function. It is of the form (B.11)found for the C

2invariants on R. However, this transformation changes the basis of the module of

C2

invariant functions on R.To "nd proper linear combinations which accordingly transform irreducible representation of

the C2v

group we take into account the fact that two symmetry operations (p13

, p23

) which belongto the C

2vgroup but do not belong to the C

2group may be represented as products of inversion

and the C2

rotation around the axis orthogonal to the re#ection plane. We note again that theaction of the inversion results in the interchange of x

iwith y

iand the action of p

13, p

23on x

i, y

imay be represented as follows:

p13

xj%!y

j, ( j"1, 3); p

13x2

% y2

, (B.23)

p23

xj%!y

j, ( j"2, 3); p

23x1

% y1

. (B.24)

So, instead of four denominator invariants (x3, y

3, x2

1, y2

1) we should take two linear combinations

h1s"(x2

1#y2

1) and h

2s"(x

3!y

3) which are invariant with respect to C

2vand two combinations

h1a"(x2

1!y2

1) and h

2a"(x

3#y

3) which are pseudo-invariants of type A

2with respect to C

2v.

In a similar way, we form new numerator invariants. Six numerator invariants are of type A1

with respect to C2v

: 1, u1s"x

1x2!y

1y2, u

2s"x

1y1, u

3s"x

2y2, u

4x"x

1y2!x

2y1,

u5s"x

1x2y1y2. There are equally two numerator C

2invariants which are pseudo-invariants of

the type A2

with respect to C2v

: u1a"x

1y2#x

2y1, u

2a"x

1x2#y

1y2. The module of

C2

invariant functions of R is represented now as

PC2 DR"P[h1s

, h2s

, h1a

, h2a

]v(1,u2s

, u3s

,u4s

,u5s

,u6s

,u1a

, u2a

) . (B.25)

Now, we can transform the basis of the module of the C2

invariant functions in such a way that newdenominator invariants become C

2vinvariants rather than pseudo-invariants. To do that it is


su$cient to multiply the numerator and denominator of the Molien function by (1#j)(1#j2).This action corresponds simply to the substitution of two C

2denominator invariants h

1aand

h2a

by their squares which are C2v

invariants. The new representation of the module of theC

2invariants on R takes the form

PC2 DR"P[h1s

, h2s

, (h1a

)2, (h2a

)2]v(1,u2s

, u3s

,u4s

, u5s

, u6s

, u1a

,u2a

)(1, h1a

)(1, h2a

) . (B.26)

To make the restriction to C2v

invariant functions it is su$cient now to take only those numeratorinvariants which are C

2vinvariants. The representation of the module of C

2vinvariant functions on

R takes the form

PC2v DR"P[h1s

, h2s

, h21a

, h22a

]v((1, h1a

h2a

)(1,+uis), (h

1a, h

2a)(+u

ia)) (B.27)

including 16 numerator invariants. The corresponding Molien function has the form (B.28)

MC2v

DR"1#4j2#3j4#3j3#4j5#j7

(1!j)(1!j2)2(1!j4)(B.28)

"

1#3j2#3j3#j5

(1!j)(1!j2)3, (B.29)

which turns out to be reduced to a more simpler one in Eq. (B.29) which includes only eightnumerator invariants. Construction of the integrity basis corresponding to the reduced form ofthe Molien function (B.29) requires additional analysis because we should "nd three second-order algebraically independent invariants. At the same time the straightforward procedurerealized above gives one of the possible basis of the module of invariant functions although nota minimal one.

B.6. D3h

group

We begin by constructing the Molien function of D3

invariants over the 6D space x=y. This 6Dvector space is the six-dimensional reducible representation of the form (E=A

2)=(E=A

2). The

corresponding Molien function is

MD3

"

1#2j4#4j3#10j4#4j5#2j6#j8

(1!j2)4(1!j3)2. (B.30)

Its restriction on the R subspace has the form

MD3

DR"1#2j4#4j3#10j4#4j5#2j6#j8

(1!j2)2(1!j3)2. (B.31)

To go now to the D3h

group we "rst rewrite the Molien function for the D3

invariants on R usingtwo auxiliary variables j

s, j

ainstead of one j. We use subscript to distinguish the behavior of

D3

invariants with respect to re#ection.


First, the Molien function for the D3

invariants on R with two parameters has the form

MD3

DR"1#2j2

s#2j3

s#6j4

s#2j5

s#2j6

s#j8

s#(2j3

a#4j4

a#2j5

a)

(1!j2s)(1!j2

a)(1!j3

s)(1!j3

a)

. (B.32)

We transform it to new denominator invariants which are at the same time the D3h

invariants:

MD3

DR"(1#2j2

s#2j3

s#6j4

s#2j5

s#2j6

s#j8

s)#(2j3

a#4j4

a#2j5

a)

(1!j2s)(1!(j2

a)2)(1!j3

s)(1!(j3

a)2)

(1#j2a)(1#j3

a) .

(B.33)

To pass to D3h

invariants one must take only those numerator D3

invariants which areD

3hinvariants. The Molien function for D

3hinvariants is written in the form with the only

parameter j

MD3h

DR"(1#j5)[1#2j2#2j3#6j4#2j5#2j6#j8]

(1!j2)(1!j4)(1!j3)(1!j6)

#

(j2#j3)[2j3#4j4#2j5](1!j2)(1!j4)(1!j3)(1!j6)

, (B.34)

which is equivalent to

MD3h

DR"1#2j2#2j3#6j4#5j5#8j6#8j7#5j8#6j9#2j10#2j11#j13

(1!j2)(1!j3)(1!j4)(1!j6).

(B.35)

There are four D3h

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 befactorized and the (1#j2) factor may be simultaneously eliminated from numerator and denomin-ator resulting in the reduced form of the Molien function

MD3h

DR"1#j2#2j3#5j4#3j5#3j6#5j7#2j8#j9#j11

(1!j2)2(1!j3)(1!j6). (B.36)

Although the question is open how to construct integrity basis corresponding to this simpli"edMolien function in Eq. (B.36), the construction of the non-minimal integrity basis corresponding tothe form in Eq. (B.35) of the D

3hMolien function is straightforward.

To give the explicit form of the integrity basis over the six-dimensional space we need the basisfor all invariants and covariants over three-dimensional space. Instead of x

iand y

iwe can use

irreducible tensors with respect to the D3

group on each 3D-subspace to construct the integritybasis on R of functions invariant with respect to the D

3group action on the 6D-space.

First of all we give the explicit form of D3

invariants and covariants on three-dimensional spacewhich form the basis of the module of polynomials. There are three basic (denominator) invariants:

x21#x2

2, x2

3, c

3(x)"x3

1!3x

1x22

(B.37)


and one auxiliary (numerator) invariant

x3p3(x), with p

3(x)"3x2

1x2!x3

2. (B.38)

There are two auxiliary covariants of type A2

(degree 1 and 3), namely x3

and p3(x), and four pairs

of auxiliary covariants of type E (degree 1, 2, 2 and 3)

Ax1

x2B, A

x21!x2

2!2x

1x2B, A

x2x3

!x1x3B, A

2x1x2x3

(x21!x2

2)x

3B . (B.39)

Taking into account the form of these invariants and covariants for 3D x and 3D y spaces and theexpression for the Molien function in Eq. (B.31) for D

3invariants on R we can write the following

representation for the module of D3

invariant functions of R:

PD3 DR"P[x23, c

3(x), y2

3, c

3(y)]v(1,u

1,2,u

23) . (B.40)

There are 24 (including 1) numerator invariants. They are listed below in the form which showsclearly that they are simply invariants produced by coupling x and y covariants:

u1"x

3p3(x), u

2"y

3p3(y), u

3"x

3y3p3(x)p

3(y) , (B.41)

u4"x

3y3, u

5"x

3y2p(y), u

6"y

3x2p(x) , (B.42)

u7"x

2p(x)y

2p(y) , (B.43)

u8"A

x1

x2BA

y1

y2B, u

9"A

x1

x2BA

y21!y2

2!2y

1y2B , (B.44)

u10

"Ax1

x2BA

y2y3

!y1y3B, u

11"A

x1

x2BA

2y1y2y3

(y21!y2

2)y

3B , (B.45)

u12

"Ax21!x2

2!2x

1x2BA

y1

y2B, u

13"A

x21!x2

2!2x

1x2BA

y21!y2

2!2y

1y2B (B.46)

u14

"Ax21!x2

2!2x

1x2BA

y2y3

!y1y3B, u

15"A

x21!x2

2!2x

1x2BA

2y1y2y3

(y21!y2

2)y

3B , (B.47)

u16

"Ax2x3

!x1x3BA

y1

y2B, u

17"A

x2x3

!x1x3BA

y21!y2

2!2y

1y2B (B.48)

u18

"Ax2x3

!x1x3BA

y2y3

!y1y3B, u

19"A

x2x3

!x1x3BA

2y1y2y3

(y21!y2

2)y

3B (B.49)

u20

"A2x

1x2x3

(x21!x2

2)x

3BA

y1

y2B, u

21"A

2x1x2x3

(x21!x2

2)x

3BA

y21!y2

2!2y

1y2B , (B.50)


u22

"A2x

1x2x3

(x21!x2

2)x

3BA

y2y3

!y1y3B , (B.51)

u23

"A2x

1x2x3

(x21!x2

2)x

3BA

2y1y2y3

(y21!y2

2)y

3B . (B.52)

We change the basis of the module of D3

invariant functions in such a way that all invariantsbecome invariants or pseudo-invariants for the D

3hgroup. To do this it is su$cient to form simple

linear combinations of denominator and numerator invariants used in Eq. (B.40). We remark thatthe action of the p

hoperation which should be added to go from D

3group to a D

3hone is given by

phxj%!y

j, ( j"1, 2) and p

hx3

% y3. The new representation of the module of D

3invariant

functions corresponding to the Molien function in Eq. (B.32) with two parameters is as follows:

PD3 DR"P[h1s

, h2s

, h1a

, h2a

]v(1,u1s

,2, u15s

, u1a

,2,u8a

) , (B.53)

h1s"x2

3#y2

3, h

2s"c

3(x)!c

3(y) , (B.54)

h1a"x2

3!y2

3, h

2a"c

3(x)#c

3(y) , (B.55)

u1s"u

1!u

2, u

2s"u

5#u

6, u

3s"u

9!u

12, (B.56)

u4s"u

10!u

16, u

5s"u

11#u

20, u

6s"u

14#u

17, (B.57)

u7s"u

15!u

21, u

8s"u

19!u

22, u

9s"u

3, u

10s"u

4, (B.58)

u11s

"u7, u

12s"u

8, u

13s"u

13, u

14s"u

18, u

15s"u

23, (B.59)

u1a"u

1#u

2, u

2a"u

5!u

6, u

3a"u

9#u

12, (B.60)

u4a"u

10#u

16, u

5a"u

11!u

20, u

6a"u

14!u

17, (B.61)

u7a"u

15#u

21, u

8a"u

19#u

22. (B.62)

We change now two denominator invariants h1a

, h2a

into (h1a

)2, (h2a

)2 which are D3h

invariants.This results in increasing the number of D

3numerator invariants four times. But we are interested

only in those numerator invariants which are D3h

invariants as well. There are 48 such invariantswhich correspond to the form (B.35) of the Molien function. The structure of the module ofD

3hinvariant functions on R may be now given explicitly

PD3 DR"P[h1s

, h2s

, (h1a

)2, (h2a

)2]v((1, h1a

h2a

)(1,u1s

,2,u15s

), (h1a

, h2a

)(u1a

,2,u8a

)) . (B.63)

There are four denominator invariants and 48 numerator invariants which may be reconstructedfrom Eqs. (B.41)}(B.52), (B.53)}(B.62). Apparently, the smaller integrity basis with only 24 numer-ator invariants may be found as the form (B.36) of the Molien function indicates.

B.7. ¹d

point group

To construct the Molien function for the ¹d

group invariants on R we use exactly the sameprocedure as for the D

3hgroup but we start now from the ¹ group in (x=y) representation. The


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

MTDR"

(1#j6)2#2(j2#j4)2#(j#j2#2j3#j4#j5)2(1!j3)2(1!j4)2

. (B.64)

More detailed form of the Molien function with two parameters may be given after the transforma-tion of the ¹ invariants into the form irreducible with respect to ¹

d(A

1or A

2symmetry types with

respect to ¹d). We use below j

sfor those ¹-invariants which are at the same time the ¹

dinvariants

and ja

for those ¹-invariants which are pseudo-invariants (of the A2

type) of the ¹d

group.

MTDR"

(1#j2s#j3

s#5j4

s#3j5

s#8j6

s#3j7

s#5j8

s#j9

s#j10

s#j12

s)

(1!j3s)(1!j3

a)(1!j4

s)(1!j4

a)

#

(j3a#2j4

a#3j5

a#6j6

a#3j7

a#2j8

a#j9

a)

(1!j3s)(1!j3

a)(1!j4

s)(1!j4

a)

. (B.65)

Now to form the Molien function for the ¹d

invariant on R we multiply both numeratorand denominator by (1#j3

a) (1#j4

a) and take in the numerator only those terms which are

¹d

invariant

MTd

DR"(1#(j3

aj4a))(1#j2

s#j3

s#5j4

s#3j5

s#8j6

s#3j7

s#5j8

s#j9

s#j10

s#j12

s)

(1!j3s)(1!(j3

a)2)(1!j4

s)(1!(j4

a)2)

#

(j3a#j4

a)(j3

a#2j4

a#3j5

a#6j6

a#3j7

a#2j8

a#j9

a)

(1!j3s)(1!(j3

a)2)(1!j4

s)(1!(j4

a)2)

. (B.66)

The total number of numerator invariants now is 96. If we use only one auxiliary parameter j wecan simplify considerably the Molien function for ¹

dinvariants. Formula (B.66) becomes

MTd

DR"(1#j2)(1#j3)(1#j4)(1#4j4#2j5#4j6#j10)

(1!j3)(1!j6)(1!j4)(1!j8)(B.67)

"

1#4j4#2j5#4j6#j10

(1!j3)2(1!j2)(1!j4). (B.68)

The last simpli"ed form in Eq. (B.68) of the Molien function for the ¹dinvariants on R seems to be

rather reasonable. Probably, it gives the minimal basis of denominator and numerator invariantson R. If the integrity basis corresponding to this simpli"ed form (B.68) exists it may be used in someapplications. Number (12) of auxiliary invariants is not enormous. The initial form (B.67) of theintegrity basis including 96 numerator invariants is not encouraging at all.

To conclude this appendix we give the generating Molien functions for icosahedral symmetry> and >

h:

g(>)"N(>)

(1!x6)2(1!x10)2, g(>

h)"

N(>h)

(1!x6)2(1!x10)2, (B.69)


where

N(>)"1#x2#x4#6x6#6x7#6x8#6x9#15x10

#16x11#15x12#16x13#15x14#32x15#15x16

#16x17#15x18#16x19#15x20#6x21#6x22

#6x23#6x24#x26#x28#x30 (B.70)

and

N(>h)"1#x2#x4#3x6#3x7#3x8#3x9

# 7x10#8x11#7x12#8x13#7x14#16x15#7x16

# 8x17#7x18#8x19#7x20#3x21

#3x22#3x23#3x24#x26#x28#x30 . (B.71)

Appendix C. Strata and orbits for point groups

We gave in Section 2.8, strata and orbits in action on R of three point groups C2v

, D3h

and ¹d.

This choice is due to the fact that among polyatomic Rydberg molecules studied experimentally ortheoritically molecules with such symmetry groups are most typical. We can cite examples of H

3(D

3hsymmetry) (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; Panand Lu, 1988; Stephens and Greene, 1995), Na

3(D

3hsymmetry) (Broyer et al., 1986), H

2O

(C2v

symmetry) (Petfalakis et al., 1995), H2F (C

2vsymmetry) (Bordas et al., 1985) and NH

4(¹

dsymmetry) (Herzberg, 1981, 1987; Herzberg and Hougen, 1983; Watson, 1984). At the same timemany other molecules are potentially interesting from the point of view of their excited Rydbergstates or even Rydberg character of the ground state (Basov and Pavlichenkov, 1994; Boldyrev andSimons, 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 C

sfor HCO (Mayer and Grant, 1995)

till the highest icosahedral symmetry I)

for C60

(Weber et al., 1996). That is why we give in thisappendix orbits and strata for all possible "nite symmetry groups. The list of critical orbits for eachsymmetry group enable us to give the description of simplest Morse-type functions (see Tables 18and 19 in Section 3.3).

Tables C.1}C.8 listed in this appendix show in column 1 the stabilizer of the stratum. In column2 closed and generic strata are indicated. We remark once more that some strata are neither closednor generic. In column 3 the dimension of the stratum is given. For the "nite group action on R, thedimension of the generic stratum is always 4. Column 4 gives the number of orbits in the stratum. Ifthe dimension of the stratum is zero the number of orbits in the stratum is "nite. If the dimension ofthe stratum is positive the number of orbits in the stratum is in"nite. We denote it Rn with n equalto the dimension of the stratum. Column 5 shows the number of points in each orbit. For "nitegroup actions on R this number is always "nite.


Table C.1Strata and orbits in the action of low symmetry groups C

1, C

n, C

s, C

nh, S

2, S

2n(n52) on R. Column `gca indicates

generic (g) and closed (c) strata

Group Stabilizer gc dim. Number Nature

C1

1 g 4 R4 1

Cn

Cn

c 0 4 11 g 4 R4 n

Cs

Cs

c 2 R2 11 g 4 R4 2

Cnh

Cnh

c 0 2 1C

nc 0 1 2

Cs

2 R2 n1 g 4 R4 2n

S2

S2

c 2 R2 11 g 4 R4 2

S2n

S2n

c 0 2 1C

nc 0 1 2

S2

2 R2 n1 g 4 R4 2n

Table C.2Strata and orbits in the action of C

nvon R. Strati"cation is di!erent for n even and n odd. Column `gca indicates generic

(g) and closed (c) strata


Cnv

n even Cnv

c 0 2 1C

nc 0 1 2

Cvs

2 R2 nCd

s2 R2 n

1 g 4 R4 2n

Cnv

n odd Cnv

c 0 2 1C

nc 0 1 2

Cs

2 R2 n1 g 4 R4 2n

Along with tables of strata and orbits given in this appendix we give below strata equations forcritical strata for some point groups. The form of the equation de"ning strata depends on thechoice of variables (representation). At the same time the information concerning strata and orbitslisted in Tables in this appendix does not depend on the representation. Throughout this paper weuse either x=y or j=k representation of the R manifold.


Table C.3Strata and orbits in the action of D

non R. Strati"cation is di!erent for n even and n odd. Column `gca indicates generic

(g) and closed (c) strata


Dn

n even Cn

c 0 2 2C

2c 0 2 n

C@2

c 0 2 n1 g 4 R4 2n

Dn

n odd Cn

c 0 2 2C

2c 0 4 n

1 g 4 R4 2n


nhon R. Column `gca indicates generic (g) and closed (c) strata


Dnh

n odd Cnh

c 0 1 2C

nvc 0 1 2

C2v

c 0 2 nC

2c 0 1 2n

C1h

2 R2 2nC

s2 R2 2n

1 g 4 R4 4n

Dnh

n even Cnh

c 0 1 2C

nvc 0 1 2

C2v

c 0 1 nC

2vc 0 1 n

C2h

c 0 1 nC

2vc 0 1 n

C1h

2 R2 2nC

s2 R2 2n

C@s

2 R2 2n1 g 4 R4 4n

For the Cngroup action on R there is one closed zero-dimensional stratum with the stabilizer C

n.

Equations de"ning this stratum in x, y representation are

x23"1

2, y2

3"1

2. (C.1)

In the j, k representation the same Eqs. (C.1) are

j23"1, k2

3"1 . (C.2)



nd(n52, even) on R. Column `gca indicates generic (g) and closed (c) strata


Dnh

n even S2n

c 0 1 2C

nvc 0 1 2

C2v

c 0 1 nC

2vc 0 1 n

C2h

c 0 1 nC

2vc 0 1 n

C1h

2 R2 2nC

s2 R2 2n

C@s

2 R2 2n1 g 4 R4 4n

Dnh

n odd S2n

c 0 1 2C

nvc 0 1 2

C2h

c 0 2 nC

2c 0 1 2n

C1h

2 R2 2nC

s2 R2 2n

1 g 4 R4 4n

Table C.6Strata and orbits in the action of ¹, ¹

hon R. Column `gca indicates generic (g) and closed (c) strata


¹ C3

c 0 4 4C

2c 0 2 6

1 g 4 R4 12

¹h

C3h

c 0 2 4C

3c 0 1 8

C2h

c 0 1 6C

2vc 0 1 6

Chs

2 R2 12Cv

s2 R2 12

1 g 4 R4 24

For the Cs

group action on R the only closed stratum with the stabilizer Cs

has dimension two.In x, y and j, k representations the same stratum is given, respectively, by

x1"!y

1, x

2"!y

2, x

3"!y

3, (C.3)

j1"j

2"0, k

3"0 . (C.4)


Table C.7Strata and orbits in the action of O and O



O C4

c 0 2 6C

3c 0 2 8

C2

c 0 2 121 g 4 R4 24

Oh

C4v

c 0 1 6C

4hc 0 1 6

C3v

c 0 1 8C

3hc 0 1 8

C2v

c 0 1 12C

2hc 0 1 12

Cs

2 R2 24Cd

s2 R2 24

1 g 4 R4 48

Table C.8Strata and orbits in the action of > and >



> C5

c 0 2 12C

3c 0 2 20

C2

c 0 2 301 g 4 R4 60

>h

C5v

c 0 1 12C

5hc 0 1 12

C3v

c 0 1 20C

3hc 0 1 20

C2v

c 0 1 30C

2hc 0 1 30

Cs

2 R2 601 g 4 R4 120

The Cnh

group action on R produces two closed strata. The Cnh

stratum is de"ned in x, y, ( j, k)representation by

x3"y

3"$1

2, (C.5)

j23"1 . (C.6)

The Cn

stratum is de"ned in x, y, ( j, k) representation by

x3"!y

3"$1

2, k2

3"1 . (C.7)


For the S2

group action on R the S2

closed stratum is given by the following equations in x, y andj, k (C.8) representations:

x1"y

1, x

2"y

2, x

3"y

3, k2"0 . (C.8)

Appendix D. Qualitative description of e4ective Hamiltonians based on equivariantMorse}Bott theory

D.1. SO(3) continuous subgroup

To "nd possible Morse}Bott-type functions we use here the equivariant Morse inequalities.Strata and orbits of the SO(3) group action on R are listed in Table 4. There are two Morsecounting polynomials. One for critical orbits (which are S

2manifolds), and another for generic

orbits (PR3

manifolds with the SO(3) group acting freely on this manifold).We can write the Morse inequalities on the p manifold in the form

M/0/#3*5(t)#(1#t2)M#3*5(t)

1!t4!

1#2t2#t41!t4

"(1#t)Q(t), Q(t)50 . (D.1)

We remind that Q(t)50 in Eq. (D.1) means that all coe$cients of the Q(t) polynomial are notnegative. Here the (1!t4)~1 stands for the Poincare polynomial for the universal classifying spaceof the SO(3) group (see Appendix B of Chapter I)

PB(SO(3))

(t)"1

1!t4. (D.2)

The (1#2t2#t4) is the ordinary Poincare polynomial for R, and the (1#t2) the ordinaryPoincareH polynomial for the S

2sphere. The form of two Morse counting polynomials is

M/0/#3*5(t)"n0#n

1t , (D.3)

M#3*5(t)"c0#c

1t#c

2t2, c

i50, c

0#c

1#c

2"2 . (D.4)

In fact, it is easy to see that c1"0, otherwise the left side of the Morse inequalities is not a "nite

polynomial. To simplify the analysis we can just look for three di!erent cases corresponding tothree di!erent polynomials

M#3*5(t)"1#t2 , (D.5)

M#3*5(t)"2t2 , (D.6)

M#3*5(t)"2 . (D.7)

Three associated Morse inequalities are written in the form

n0#n

1t"(1#t)Q(t) , (D.8)

n0#1#n

1t"(1#t)Q(t) , (D.9)

n0!1#n

1t"(1#t)Q(t) . (D.10)


So, we have an obvious answer. When two critical orbits are minimum and maximum, the numberof generic maxima is equal to the number of generic minima. When both critical orbits are minima,the number of generic maxima is larger by one than the number of generic minima. When bothcritical orbits are maxima, the number of generic maxima is smaller by one than the number ofgeneric minima.

D.2. C=

continuous subgroup

There are four isolated one point orbits with C=

stabilizer and a generic stratum of C1

orbits.So any C

=invariant function on R possesses at least four stationary points coinciding with

critical orbits and probably some extra S1

critical manifolds corresponding to generic C1

orbits.We give below the complete list of qualitatively di!erent C

=invariant functions. This means

that we characterize each function by several numbers giving the numbers of critical orbits of eachMorse index over each stratum. As long as for this concrete example four C

=orbits are critical

and they should have even Morse index, there are 15 possible classes of functions which maybe distinguished by their behavior on the C

=stratum. Within each such class further classi"cation

of generic functions should take into account the number of S1

critical manifolds of each Morseindex.

We will use Morse counting polynomials separately for C=

and C1

strata. For the C=

stratumthis polynomial has the form

MC= (t)"n0#n

2t2#n

4t4, n

0#n

2#n

4"4, n

i50 . (D.11)

It simply indicates that there are n0

C=

-orbits with Morse index 0, n2

C=

-orbits with Morse index2, and n

4C

=-orbits with Morse index 4. It may be veri"ed that there are 15 di!erent possibilities to

choose n0, n

2, n

4to be non-negative integers and to satisfy n

0#n

2#n

4"4.

A similar counting polynomial for critical C1

orbits is a polynomial of the third degree

MC1(t)"k0#k

1t#k

2t2#k

3t3 , (D.12)

because any C1

orbit is a S1

manifold. It is clear that kishould be non-negative integers but further

restrictions on kishould follow from the Morse theory.

If we apply the approach based on ordinary homology, the Morse inequalities take the form

(1#t)MC1 (t)#MC=(t)!(1#2t2#t4)"(1#t)Q(t), Q(t)50 . (D.13)

Unfortunately, this form of Morse inequalities is insu$cient in several cases. For example, it givesno restrictions on the number and type of C

1stationary orbits for the class of C

=invariant

functions characterized by

MC= (t)"1#2t2#t4 . (D.14)

Better results can be obtained by using the equivariant version of Morse inequalities. In such a casecontributions from C

=and C

1stationary orbits count di!erently and the equivariant Morse

inequalities take the form

MC1(t)#MC= (t)1!t2

!

1#2t2#t41!t2

"(1#t)Q(t), Q(t)50 . (D.15)


Table D.1Classi"cation of qualitatively di!erent C


MC=(t) MC1(t) Coe$cients Simplest MC1(t)

1#2t2#t4 a#(a#b)t#(b#c)t2#ct3 a, b, c50 0

3t2#t4 (a#1)#(a#b)t#(b#c)t2#ct3 a, b, c50 12#t2#t4 a#(a#1#b)t#(b#c)t2#ct3 a, b, c50 t1#t2#2t4 a#(a#b)t#(b#c#1)t2#ct3 a, b, c50 t21#3t2 a#(a#b)t#(b#c)t2#(c#1)t3 a, b, c50 t3

3#t4 a#(a#2#b)t#(b#c)t2#ct3 a, b, c50 2t4t2 (a#1)#(a#b)t#(b#c)t2#(c#1)t3 a, b, c50 1#t32#2t2 a#(a#1#b)t#(b#c)t2#(c#1)t3 a, b, c50 t#t32#2t4 a#(a#1#b)t#(b#c#1)t2#ct3 a, b, c50 t#t22t2#2t4 (a#1)#(a#b)t#(b#c#1)t2#ct3 a, b, c50 1#t21#3t4 a#(a#b)t#(b#c#2)t2#ct3 a, b, c50 2t2

3#t2 a#(a#b#2)t#(b#c)t2#(c#1)t3 a, b, c50 2t#t3t2#3t4 (a#1)#(a#b)t#(b#c#2)t2#ct3 a, b, c50 1#2t2

4 a#(a#3#b)t#(b#c)t2#(c#1)t3 a, b, c50 3t#t34t4 (a#1)#(a#b)t#(b#c#3)t2#ct3 a, b, c50 1#3t2

Now, we can even forget our previous statement about the existence of 15 classes of functions withrespect to their behavior on C

=stratum. These 15 classes may be rediscovered from the equivariant

Morse inequalities. The simple requirement is that the left side of the last equation should bea polynomial. Further step is to "nd restrictions on MC1(t) for each of the 15 di!erent classes offunctions. These restrictions follow from the fact that the left part should be a polynomial of degreethree and it should be divisible by (1#t). So, the MC1(t) being the polynomial of degree threeshould depend for each class on three integer numbers.

We summarize results in Table D.1.There is only one type of simplest C

=invariant Hamiltonian. It is characterized by the absence

of critical manifolds of non-zero dimension. The next level of complexity includes Hamiltonianswith one critical manifold on the C

1stratum. There are four such Hamiltonians which belong to

di!erent classes with respect to their behavior on the C=

stratum. To give the list of qualitativelydi!erent Hamiltonians of the next level of complexity (those possessing two C

1stationary orbits) it

should be noted that within each class all Hamiltonians have the same parity of the number ofC

1stationary orbits. This is due to the fact that increasing any of the a, b, c coe$cients by 1 results

in the increase of the number of stationary C1

orbits by 2. So, the set of Hamiltonians of the secondlevel of complexity includes three qualitatively di!erent Hamiltonians from the class 1#2t2#t4and six other Hamiltonians (each from di!erent class). The number of qualitatively di!erentHamiltonians increases rapidly with the increase of complexity. (See Table D.2 showing numbers ofqualitatively di!erent Hamiltonians for several low level of complexity.)

It is interesting to note that among qualitatively di!erent functions constructed above there are7 which are perfect in the equivariant sense (i.e. Q(t) is identically zero in the equivariant Morse


Table D.2Classi"cation of qualitatively di!erent C

=invariant Hamiltonians by their complexity

Level of Numbers of qualitatively di!erentcomplexity C


0 11 42 93 144 265 246 527 42

inequalities). All these perfect in the equivariant sense functions belong to di!erent classes withrespect to their behavior on the C

=stratum. It is evident that they are the most simple functions

within their class but they can have several C1

critical orbits. There is one perfect function with zerolevel of complexity (it belongs to class (1#2t2#t4)), two perfect functions characterized by the"rst level of complexity (they are from classes (3t2#t4) and (1#t2#2t4) correspondingly), twoperfect functions characterized by the second level of complexity (from classes (2t2#2t4) and(1#3t4)), one perfect function with the third level of complexity (class (t2#3t4)), and one perfectfunction with the fourth level of complexity (class (4t4)).

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