Dynamical chains for molecular tops. I. Dynamical chains

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Dynamical chains for molecular tops. I. Dynamical chains

F. Michelot

To cite this version:

F. Michelot. Dynamical chains for molecular tops. I. Dynamical chains. Journal de Physique, 1989,

50 (1), pp.45-62. �10.1051/jphys:0198900500104500�. �jpa-00210899�

Dynamical chains for molecular tops.

I. Dynamical ^chains

F. Michelot

Laboratoire de Spectronomie Moléculaire et Instrumentation Laser, Unité de Recherche associée

au C.N.R.S., 6, ^bd Gabriel, ²¹¹⁰⁰ Dijon, ^France

Résumé.

Des chaînes dynamiques ^sont proposées pour l’étude des toupies moléculaires. Tous les calculs sont explicités dans la chaîne SU(4) ~ SO(4) ~ SU(2) ~ SU(2) qui convient, ^en première approximation, à l’étude des toupies sphériques. ^{Tous les} opérateurs rotationnels qui peuvent intervenir dans l’étude d’une molécule, ^sont construits par ^un produit ^tensoriel

«

étendu

formé à partir des éléments d’une base d’intégrité appropriée.

Abstract.

Dynamical ^{chains are} proposed ^{for the} study of molecular tops. ^{All the} computations

are performed within the chain SU(4) ~ SO(4) ~ SU(2) ~ SU(2) adapted ^{in first} approximation

to the study ^of spherical tops. Rotational operators involved in molecular spectroscopy problems

are obtained as a

streched

tensor product formed from members of an appropriate integrity

basis.

Classification

Physics ^Abstracts

31.00

02.10

1. Introduction.

The study ^{of rotors or} tops ⁱⁿ quantum mechanics is a very old subject (see [1, 2] ^{for a} large bibliography) ^{and the} geometrical invariance group of their Hamiltonian is well-known :

where LO (3 ) is the rotation group of the space-fixed ^frame (SFF) ; the other groups ^are those which leave the inertia ellipsoid invariant and goes from the rotation group MO (3 ) ^{of the top}

fixed frame when all principal ^moments of inertia are equal ^to D2 h ^when they ^are all different.

These various types ^of tops ^are encountered for instance in molecular spectroscopy ^when

the Hamiltonian is expressed in molecular coordinates [3, 4] ; ^{the axes} of the molecule fixed frame (MFF) ^{are chosen so that} they coincide with the principal ^{inertia axes} of the molecule in its equilibrium configuration. ^{In most} cases, the Hamiltonian of the corresponding rigid top may be taken ^as the zero-order rotational Hamiltonian. However to take intramolecular interactions into account, ^one introduces rotational operators ^{which are not} invariant in the groups given ⁱⁿ equation (1) ^{and even in some cases, for} instance when hyperfine interactions

[5] ^or intensities [6, 7] ^are considered, ^{one must} go outside the enveloping algebra ^{built from}

the generators of these groups.

JOURNAL DE

PHYSIQUE. -

50,

1,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500104500

Up ^{to now these} problems have been solved individually ^{for each} type ^of top, ^the computations being mostly made in the Schrôdinger picture ; ^{we also note that} rotational and vibrational degrees of freedom are always ^treated differently.

In a previous paper [8], ^{we described an} algebraic approach ^to the vibrational spectra ^of molecules, here, ^we will extend this method to the case of molecular rotation. For this, ^we

introduce a dynamical algebra ^{for the} tops and realize it in terms of boson operators ;

different subduction chains are associated with the various tops. ^The first consequence is ^a uniform treatment of rotational and vibrational degrees of freedom. In addition all operators involving rotational variables are obtained as a « stretched » tensor product ^of suitably

defined elementary ^factors (or irreducible fundamental tensors) [9, 10, 11], ^{which assures that} only linearly independent ^{tensors are} considered.

Previous group theoretical treatments of tops ^{have been} proposed ^{and a brief review is}

necessary ^to emphasize ^{in which} respects ^our approach ^{differs. Two kinds of} studies have been performed ; ⁱⁿ [12, 13] ^an algebraic ^{treatment of} rigid tops ^is given ^{but it} is restricted to the degeneracy algebra and the construction of tensors in the enveloping algebra ^{is not}

considered.

The introduction of a dynamical algebra is necessary if ^{one wants to} include within the same

formalism all operators which may be involved in ^a given (molecular) problem ; obviously

these are not confined to those operators ^one may build from the generators ^{of the} degeneracy algebra. ^This point of view is adopted ⁱⁿ [14, 15] ^where dynamical algebras ^are proposed. ^{For a} rigid ^{rotator, as} appropriate ^{for the} study ^{of diatomic} molecules, ^one may ^use

0(3,1) [14] ^{or its} associated compact ^form 0(4) [16]. ^For symmetric tops ^an algebra

E 2(3), contracted form of 0(3, 2), is introduced. Alternatively, ^a systematic search for the invariants of the time-dependent Schrôdinger equation [15] ^{led to} 0(3, 2) ^{for the} rigid ^rotator

and SU(2, 2) ^{for the} symmetric top. These various non-compact algebras possess ^an irreducible representation (IR) ^{which covers the infinite set} of levels of the rigid system ^{and in} each case the matrix elements for the generators ^are given. ^{As noted} by ^{several authors} [14, 16-18] ^the requirements usually imposed ^{on the} dynamical algebra explain ^{that different}

solutions exist for the same physical problem. ^{But our} purpose is ^not just ^to propose ^{a new} subduction chain. Rather we look for a solution that :

a) simultaneously includes the three kinds of tops since there is a natural symmetry breaking ^{when one} goes from the spherical top ^{case to the} asymmetric ^{ones ;}

b) ^{admits an IR} TD, ^such that upon reduction ^to the degeneracy algebra ^{one obtains} only physical states ;

c) ^{allows us to} obtain the zero-order spectrum ^when only the first invariants of each element in the chain are retained ;

d) ^{allows an} easy construction of operators ^{of an} arbitrary degree ^with regard ^{to the} generators.

All these requirements ^{are not met} by previous studies ; ^for instance, ^the spherical top ^case

cannot be obtained from [14], ^{and in} [15], ^the SU(2, 2) algebra ^{includes ladder} generators ^of

half-integer ^{rank, so that the} representation space includes states which are not appropriate ^to

the description ^of top ^levels.

But is it necessary ^to go ^into the complexity ^of non-compact algebras ^{and to} give ^{a full}

treatment of a rigid top ? ^{Even if the} Hamiltonian of such a system is introduced as zero order rotational Hamiltonian in most molecular systems, allowing ^{thus in} principle ^{an infinite set of}

rotational sublevels (or ^J values), it is clear that the number of levels which can actually ^be

observed is finite. With the introduction of a compact dynamical algebra and thus of a finite

dimensional unitary IR, ^we will show in a second paper (II) that in fact we are closer to the

actual physical situation than with the ideal model ; ^{it is to} emphasize ^our point ^{of view that}

we use the expression dynamical ^{chain for a molecular} top.

Also, ^we point ^out that if the methods we use are of rather common practice in atomic and nuclear physics [17, 19-21] they ^have only recently ^been introduced in the field of molecular

physics [22-24]. ^{Yet, as we shall see,} they ^{allow us to recover and} generalize ^{standard results}

without any explicit coordinate representation.

The dynamical ^chains and the IR r D adapted ^{to the} study ^{of the various} types ^of tops ^are obtained in section 2. Sections 3 and 4 are devoted to the construction of the most general

tensor operator which may have ^non-zero matrix elements within fp ^{and to the} computation

of its reduced matrix elements. In those two sections only ^the theory ^{of Lie} algebras iand ^of

their representations is necessary ; if the results obtained depend ^{on the} particular ^{chain and}

IR chosen they ^are independent ^{of the} particular realization we have in mind. However, ^the

notations and phase conventions have been chosen so as to make easier the correspondence

with paper II where the application ^{to molecular tops is} considered.

2. Dynamical algebra for molecular tops.

We shall mainly ^{consider the most} degenerate ^case corresponding ^to spherical tops (Eq. (la)),

other ones being ^deduced through group-subgroup ^reduction.

We start from the well-known results [25, 26] ^{that the} generators associated with

Lo (3) ^x mO(3) span the Lie algebra SU(2) ED SU(2) isomorphic ^to SO(4) and that the Hamiltonian for a spherical top ^can be written as :

where the notation L, f ^{is used to} distinguish ^{when the} angular ^{momentum J is} projected ^on

the SFF axes (Lj,

Ji, ; ^i’

X, Y, Z) ^{or on the MFF axes} (fi = - Ji ; i

^{x, y,}

z), ^{with the standard} commutation relations :

As a result to each energy level is associated ^a unitary irreductible representation ^of SU(2) 0 SU(2) which may be denoted :

whether one uses the standard Cartan [pqj

Ul + j2, jl -7z] ^labels for the irreducible

representations ^of SO(4) ^{or the} SU(2) 0 SU(2) ^labels [17]. ^The degeneracy of each level is

(2 j + 1 )2.

For the present problem, ^a compact dynamical algebra ^is easily ^{found from the} branching

rules G:::> O(n) [27-29] which show that we may take G

U(4) ^{for which we have :}

where [NO ] represents ^a totally symmetric representation ^of U(4) ^and where p ^{takes the}

values N, N - 2 ... 1 or 0.

If we want only integer ^values for j (which ^are appropriate ^{for the rotor} levels), ^{we must}

choose N _{even ;} but the computations ^can be made for arbitrary ^{N : odd values of}

N lead to half-integer representations (k/2, k/2) (kodd) ^of SU(2) ^E9 SU(2) ^{which is}

consistent with the fact that the top eigenfunctions ^are found among the whole ’J) (j,j) ^>

representations ^of SU(2) E9 SU(2). ^This illustrates the main difference between our approach

and that realized in [15] ^for symmetric tops ; in the reduction of the representation space

To

[NO], ^{N even,} only physical ^states appear (or equivalently ^{our ladder} generators ^{will be} of integer ^{rank in} SU(2) (B SU(2)).

Also we note that in this case the hypothesis formulated in [30] ^which suggest ^{to take}

U(p ⁺ 1) ^{as a} dynamical algebra ^{for a} problem with p degrees ^{of freedom,} is verified.

However, ^{for a} rigid ^rotator (2 degrees ^of freedom), U(3) ^is only ^a double-jump dynamical algebra [16] ^and 0(4) ^the appropriate one-jump algebra.

In the reduction U (4) SU (4) ^and 0 (4) 1 SO (4), ^the representations [NO] ^and [pO]

remain irreducible, ^{also the} algebras SU(4) ^and SO(6) ^are isomorphic, ^{so the} dynamical ^chain

we shall use for spherical tops ^{is :}

At this point, several remarks can be made :

(i) ^{If in the} following, ^{we will use} equation (6) with the shorter notation SU (4) ZD ^SO (4 ) ^or

SU (4 )

^SU (2) ^{0 SU} (2), it is written to emphasize that it should not be confused with the chain :

used recently [31] in connection with the beta and gamma vibrations in deformed nuclei.

Also the same algebra SU (2) ^Q SU (2) ^$ U (1 ) ^denoted SO (4) Q) SO (2) ⁱⁿ [17] ^{is the}

maximal compact subalgebra ^{for the} hydrogen ^{atom, and is not} appropriate ^for rigid spherical tops.

(ii) On the other hand, this chain is quite analogous ^to that used in [32] for shell model calculations within the Wigner supermultiplet ^scheme.

The main differences lie in the realization of the Lie algebras (besides ^fermion operators

are used by ^{Hecht et al.} [32]) and in the representations considéred for state vectors as well ^as for operators.

(iii) The chains for other tops ^are obtained with the reduction SU(2) ⁰ SU(2) ::> SU(2) ^(B SO(2) ^for symmetric tops ^{and =3} SU(2) ^Q) D2 ^for asymmetric ^ones.

3. Construction of rotational operators.

3.1 KET AND TENSOR LABELLING.

Within the totally symmetric representation [NO]

(NO) ^of SU(4), ^kets symmetrized in the chain (6) will be denoted

since we have no missing ^label problem ; ^{M and K are the usual} SO(2) ^e SO(2) labels. For the highest weight ^{states in} SO(4) ^the abbreviated notation :

will be used

The possible symmetries ^{for tensor} operators which may have ^non-zero matrix elements within these states, ^are obtained from the standard product rules and given by :

whether one uses the Young partition ^notation [27, 28] ^{or the} Cartan-Weyl labelling [11] ; throughout Ô stands for repeated ^zero.

Upon reduction in SO(4) ^{of a} general (kok) representation ^of SU(4) ^a given representation (il, j2) ^{may occur several times ; this} multiplicity introduces an additional r label. So a general

tensor operator acting ^{within states} given by equation (7) ^and symmetrized ⁱⁿ SU(2) ⁰ SU(2)

may be denoted :

These operators which will represent rotational operators involved in molecular top Hamiltonians, lie in the enveloping algebra ^of SU(4) ; ^{so as to obtain} only linearly independent operators, ^{one must not} build them arbitrarily ^{from the} generators. ^{Rather one}

must look for a fundamental set of irreducible tensor operators [11], ^{which are not of the} general ^form (Eq. (10)), but such that any ^tensor in the enveloping algebra ^{can be obtained as}

a

streched » tensor product from members of this fundamental set. Besides, ^{since we are} working ^{in a chain, we} may ^use the elementary multiplets ^method [9, 10] ^{which allows us at}

the same time to solve the multiplicity problem.

An elementary factor is written by Sharp ^{et al.} [9] in the form (,k 1 À2 À3 ; 71/2) ^which symbolizes ^the highest weight component ^{of an} irreducible tensor operator :

The appropriate k 1, k 2, À 3, il, j2 values for the chain SU(4):::> SU(2) (D SU(2) ^are given ⁱⁿ [9]. ^{As we are in a} very degenerate ^case, simplifications ^{occur and the} only elementary ^factors

we need have the symmetry :

Thus the highest weight component ^{of a} general ^tensor (,k 1 0 Àg) r(j1’ j2 ) ^is uniquely ^defined by :

with :

where all coefficients a, b,

^are integers (-- 0) and with the condition that e or

f ^are equal ^{to zero. These} results, ^{as well as} the second member of equation (13) ^{are obtained}

with the property that, for powers of elementary factors, representation ^{labels are additive} (« ^streched

products). ^In particular ^{the kets} (Eq. (8)) ^{and their} conjugates ^are respectively represented by :

with with

which is consistent with equation (5). ^{Also, for the} special ^tensor, equation (10), ^with

ml = j1 and m2

j2, ^{we have in addition :}

hence

k (or D) is thus the degree ^with regard ^{to the} SU(4) generators. The reduction in SO(4) ^{of the}

first IRs (f2Of2 ) ^of SU(4) ^is given in table I. At this point, ^we may also ^note that the rotational operators involved in vibration-rotation Hamiltonians are all scalars in SFF ; ^{if we}

set j1

^{0 in} equations (14), (16), ^{we find that} they ^{will be} represented ^{as :}

with 2 b + f = n. For n fixed, f ^{takes the} values n, n - ^{2 ... 1 or} 0, ⁱⁿ _agreement ^{with the}

well-known reduction rule for the symmetrized power of identical fundamental represen- tations of SO(3) [33]. ^{We will see} in paper II how the operators (Eq. (17)) ^{are related to the}

usual tensor operators RD(O,K)(K == f ) ^{used in} spherical top ^studies [5, ^33, 34].

Hermitian operators ^{with a definite} symmetry ^with respect ^to time reversal and normalized

symmetrized ^kets will be constructed in the following ^sections. But, before, ^{we must find an} explicit realization of these elementary ^factors. Moreover, ^{we note} that, ^while defining ^these quantities, phase ^{choices must be made.} Throughout, ^{these have been fixed so that} they ^{be in} agreement ^{with the} phase conventions for rotational tensors and kets used in spherical top studies, ^{but could} be modified without difficulty.

3.2 REALIZATION OF ELEMENTARY FACTORS.

For the cases considered here, ^the SU(4) algebra may be realized with four boson creation ai ^and annihilation _ai operators

(i 1, - , 4 ) ^which satisfy ^{the usual} commutation relations :

The generators ^{of the} SO(4) subalgebra [35] may be expressed ^{in terms of two} commuting

angular ^{momenta, one for each} SU(2) algebra. ^{In view} of further applications these may be

identified with L and f (Eq. (3)), ^{which allows us to fix} immediately the variances of tensor

Table 1.

Reduction of ^the first (nOn) irreducible representations of SU(4) ⁱⁿ SO(4)

SU(2) ^Q SU(2).

(*) ^The corresponding expressions ^for tensors

obtained with equations (13), (26).

(**) ^{For each} SO(4) ^IR, only

powers

listed.

components ⁱⁿ agreement ^with previous conventions [5, 36]. The covariant components ^{of all} double tensors must satisfy :

In these equations ^the SU(4) ^{labels have been} omitted, ^{and we note} that these relations determine a tensor only ^{within an overall} phase.

The a; span the IR(1Ô) ^of SU(4) ^{and the} ai the IR(Ô1) contragradiente ^to (lÔ). Using ^the branding ^rules SU (4 ) 1 SO (4 ) [29] ^and equation (19) ^we obtain the first two elementary

factors (Eq. (12)) :

where el and y4 ^are phase factors. In a general way, for all the ^{tensors we} will build, only ^the highest weight component ⁱⁿ SU(2) ^ffi SU(2) ^{will be} given ; ^{others are} easily ^{deduced with} equation (19).

To get ^other elementary factors, ^we proceed by coupling ^{the first ones. The} symmetrized

square ^of (lÔ) = ^{(10) and {1110} = (01) gives} {20} = ^{(20) and {2220} = (02) respect-}

ively ; ^while ( lÔ ) ^{x {1110} gives the} adjoint representation {2110} = (101) and the scalar

one {0} = (0). Upon ^{reduction in} SO(4) ^{we obtain :}

The preceding products ^are multiplicity ^{free in} SU(4) ^{and the reduction in} SO(4) ^{is also} multiplicity ^{free. We can thus} perform ^the couplings ⁱⁿ SO(4) only (Appendix), ^which

amounts to fix the corresponding SU(4) > SO(4) ^{isoscalar factors to} unity :

We thus obtain the additional elementary ^{factors :}

Other tensors in equation (21) ^{do not} give elementary factors, ^as they ^are expressible ^as

«

streched

products of the first (Eqs. (20), (13)). ^The complete ^{set of} elementary factors, with their phases fixed, ^is gathered ^{in table} II ; ^{the other} components ^{of the} corresponding

tensors are obtained with equation (19).

.

(al ⁰ a ) (il, i2) Table II.

Elementary factors (o..)

(AIO A3) VI’ i2)

Table Il.

Elementary factors (Ik 10 ’k 3 i il j2)

^E

_{Il j2}

For completeness, ^we give below the full expressions ^{for the} SU(4) generators symmetrized

in SO(4) :

where we set Ni

at ai (i = 1,..., 4 ). The first six operators ^are SO(4) generators ^and up ^to

a phase factor identical to those given ⁱⁿ [12, 13] ; ^{the nine} remaining generators being ^ladder generators between the various IR of SO(4) contained in a given ^{IR of} SU(4).

3.3 COMMUTATION RELATIONS.

For the elementary factors, ^we only ^{have :}

all other commutators are zero. Those for the generators way be written :

Equations (25a-b-c) ^are just ^the symmetrized ^{form of} equation (3) ^since they only ^involve

the generators ^{of the} degeneracy algebra.

3.4 HERMITIAN OPERATORS AND TIME REVERSAL SYMMETRY. - Hamiltonian operators ^are usually built from Hermitian rotational and vibrational operators ^{and must be invariant} upon time reversal.

Using the well-known Wigner properties [37] ^for SU(2) tensors, extended ^to double tensors

[5], ^{we can} build Hermitian rotational tensors whose highest weight components ^are given by :

with e

0 or 1. Their behaviour _{upon time} reversal is determined from the relations :

and given by :

All these properties ^{have been} determined from the highest ^{or lowest} weight components

of various tensors ; since ^other components ^are generated from these with the standard relations (19), ^{this is} sufficient to assure that they satisfy :

°

3.5 SPECIAL CASE OF SUBGROUP SCALARS. - These are obtained from equations (13), (14), (15) with ji = j2

0 and thus of the general ^{form :}

since b

d within the totally symmetric representation ^of SU(4). ^Besides, straightforward computation ^{of the} SO(4) ^Casimir operators ^from equation (23) shows that :

where N is the linear invariant of U(4) ^{which is a constant} within the representation [NO].

These relations show first that we have no subgroup SU(2) ^x SU(2) scalars, besides the Casimir operators and their powers, in the SU(4) enveloping algebra. This is in agreement with the general ^theorem [11, 38] ^{which states that the number of} functionally independent subgroup ^{H scalar} operators, ^{in the} enveloping algebra ^{of a} group G, ^is twice the number of

missing labels in the reduction G 1 _{H ;} for the present ^{case, we have no} missing ^label.

Secondly, they ^{show that for} subgroup ^scalars built from Casimir operators, ^{we have three} possibilities :

which _{may be} related through equation (31) ^{and :}

But only ^the operator (Eq. (32a)) ^{is an} irreducible tensor in SU(4) ; ^{the third} possibility (Eq. (32c)) is that used in molecular studies [25, 33].

4. Symmetry adapted kets and reduced matrix elements.

4.1 ROTATIONAL BASIS KETS.

We start from the well-known realization for the totally

symmetric representation [NO] ^of SU(4) :

where JY’ _ (n 1 ! n2 ! n3 ! n4 ! )-1/2, ni ⁺ n2 ⁺ n3 ⁺ n4 = N and 10 > ^{is the vacuum state defined} by :

With the same phase conventions as those used in constructing ^tensor operators (Eq. (19))

covariant kets in the chain SU(4) =3 SU(2) ^(p SU(2) ^must satisfy :

From equations (5), (23), (35), ^{it is} easily ^{seen that the} highest ^{covariant states in} SU(4) ^for

fixed N are characterized by :

for

we thus have :

More generally for fixed N and J values (J : N/2 N/2-1

^{1/2 or} 0) ^the highest ^covariant

states in SU(2) ^Q+ SU(2) ^are determined by (Eqs. (23), (35)) :

With the conditions nl ⁺ n2 ⁺ n3 ⁺ n4 = N, n , ^0, it may be shown that the corresponding

kets are necessarily of the form (Eq. (8)) :

where we set p = N12 - J. The coefficients in equation (38) ^are obtained with either conditions :

and then normalized. The resulting ^states may be written :

Both equations ^are useful for further computations ; ^also equation (40.b) ^{shows that, in the}

present ^case, orthonormal state vectors may be built within the ^same scheme as that used for operators (Eq. (15), ^Tab. II). ^{From the} highest weight ^states (Eq. (40)), ^an arbitrary

normalized ket is obtained with repeated ^{use of ladder} operators (Eq. (35)) :

4.2 MATRIX ELEMENTS.

All computations might be made within SU(4) but this would

mean greater difficulties since we are considering general ^{tensors of} symmetry :

within the totally symmetric representation (NO]- {N0} ^of SU(4). This group is ^not simply

reducible and products {N0} ^x {2kkkO} ^{are not} multiplicity free. Some isoscalar factors and

6 - j symbols ^for SU(4) ^{have been} computed [32, 39] ^but mostly ^for special ^{cases or with}

subduction chains which differ from ours. For the time being, it is much simpler ^to perform ^all

the computations within the SO(4) subalgebra ^{which is} simply reducible and for which all the necessary symbols ^{are wellknown} (Appendix).

To compute all the matrix elements of a general ^tensor operator :

we only need the reduced matrix elements of elementary ^factors (Table II). ^{In fact,} general

formulas may be obtained for any powers of these elementary factors, ^which simplify ^the

determination of the reduced matrix elements of a general ^tensor operator.

Besides the realization which we choose for the elementary factors leads to :

So we are left with only four basic reduced matrix elements. The computations being

tedious but rather straightforward, ^we only ^state the results :

The expression ^for the reduced matrix elements of an arbitrary ^tensor (Eq. (26)) ^is given ⁱⁿ

the Appendix. ^As they ^{are of} particular importance, ^we give below the matrix elements of ladder generators appearing ⁱⁿ equation (23) ^{and of} subgroup ^{scalars :}

The usual expressions for the matrix elements of the SO(4) generators L("’) (’, 0) and

(ioi) ^{(0, 1)} are easily recovered with equations (23a-f), (35). ^For subgroup ^scalars (Eq. (32)),

we find : :

where R" (0, ⁰⁾ is given by :

5. Conclusion.

In paper II ^we will show how these results can be used in molecular spectroscopy ^{and how} they correlate with previous treatments (algebraic ^or not) of rotational dynamics. Obviously,

there will be a one-to-one correspondence ^{if one} restricts the analysis ^{to the} degeneracy algebra. The differences appear when one considers the ladder generators since these lead to a compact dynamical algebra ^{and hence to a} finite dimensional representation space.

Appendix.

In this section, ^we gather ^all the formulas which are used for tensor coupling and for the

computation ^{of matrix elements. When not necessary} SU(4) ^{labels are omitted.}

1. COUPLING COEFFICIENTS.

a) SO(4) coupling coefficients ^{in a} standard basis.

Most published SO(4) coupling

coefficients [17, ^25, 26] ^{are symmetry} adapted in the chain SO(4) :3 SO(3) ^{and thus differ}

from the ones we need which are first coefficients for the chain SO(4) -- SU(2) % SU(2) =3 SO(2) Q) SO(2). Keeping previous conventions [5, 36], ^the coupling ⁱⁿ SO(4) of covariant double-tensors is given by :

Where the F coefficients are a product ^{of two} SU(2) Clebsch-Gordan coefficients ; ⁱⁿ equation (A.1) ^tensor notation is used, which allows the Einstein summation convention to be

applied.

b) Symmetry adapted SO(4) coupling symbols.

^For problems encountered in molecular spectroscopy, ^as considered in part II, ^one needs double tensors and kets symmetrized ⁱⁿ SO(3) ^x SO(3) m SO(3) ^x G, ^{where G is a} point group.

These are obtained from the preceding (since SU(2) = SO(3)), through ^a unitary

transformation :

where p ^{is a} triple index p

(n, C, o- ) which labels SO(3) irreducible tensors and ket components in the chain SO(3) :D ^{G. The} missing ^label problem ^is solved, ^{i.e. the} unitary

transformation (k)G is determined, by ^the diagonalization ^{of the} appropriate subgroup ^scalar

built from the integrity basis for G scalars in the chain SO(3) zD ^G.

We then have :

where the coefficients :

are symmetry adapted Clebsch-Gordan coefficients.

Similar relations apply ^{within a chain} 0(3) ^x 0(3) :D 0(3) ^{x G} by introducing parity ^labels

in a consistent way. Such computations ^are commonly ^{used for} spherical tops [5, 33, 36, 40].

All phase conventions and appropriate 1 - j symbols ^can be found in these references.

2. WIGNER-ECKART THEOREM, REDUCED MATRIX ELEMENTS.

a) ^General formulas.

^{In the chain} SU(4) ::> SO(4) = SU(2) p SU(2), ^{we use the} Wigner-

Eckart theorem in the form :

Different phase conventions are taken for the two SU(2) algebras ⁱⁿ agreement ^with previous ^studies [5, 36]. ^{Also, in} equation (A.5), the notation for the SU(4) ^{labels is} simplified and abbreviated by ^a greek ^letter.

Likewise for a chain SO(3) ^x SO(3) :D SO(3) ^{x G, with} symmetry adapted ^kets (Eq.

(A.2)), ^{we have :}

where the F coefficients are symmetry adapted 3 j symbols.

For cases considered in this paper, ^we only need the reduced matrix element formula for ^a tensor product ^of operators defined upon the ^same space ; these ^are given by (Eq. (A.1)) :

b) ^{Reduced matrix elements} of rotational operators.

^{For these} operators, ^we necessarily

have [p ]

(f2 0 f2 ) ^{and we set :}

The choice of intermediate couplings ^is irrelevant since only « streched » products ^are

considered and the relations between the various indices is unambiguously ^defined through equations (15), (16).

The reduced matrix elements of these operators are non-zero only ^{if :}

and obtained with equation (A.7) ^{and the} expressions for the reduced matrix elements of powers of elementary ^factors given ⁱⁿ equations (42), (43). Taking ^{into account the condition}

that e or f equals ^{zero, we thus obtain :} - when e = f ^{= 0}

- when e = 0, f :0 0

- when e:0 0, f ^{= 0.}

The reduced matrix elements

are obtained from equation (A.11) multiplied by (-1 )e ^{and with the} correspondences :

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