q-Deformed Vibron model for diatomic molecules

PHYSICAL REVIEW AVOLUME 50, NUMBER 2 q-deformed vibron model for diatomic molecnles AUGUST 1994 R. N. Alvarez, 'Dennis Bonatsos, and Yu. F.Smirnov ' 'Institute ofNuclear Physics, Moscow State University, 117234 Moscow, Russia Institute ofNuclear Physics, National Center for Scientiftc Research Demokritos, GR 153-10Aghia Paraskevi, Attiki, Greece (Received 29 November 1993) Adeformed version of the vibron model for diatomic molecules is constructed. Both the O(4) and U(3) dynamical symmetries of the model are rewritten, using the concept of complementary subalgebras, in amore convenient form, which is subsequently deformed. The present model unifies the so far independent successful quantum-algebraic approaches to rotational and to vibrational spectra of diatomic molecules. In addition, the method can be used for the construction of deformed versions of the U(5) and O(6) limits of the interacting boson model ofnuclear structure. PACS number(s): 33.10.Cs, 31.15.+q, 02.20.Sv I. INTRODUCTION The mathematical structure of quantum algebras (quantum groups) [1— 4] has recently been attracting much attention. They are deformed versions of the usual Lie algebras, to which they reduce when the deformation parameter qis set equal to 1. In parallel, applications of quantum algebras in physics have begun to develop in particular in cases in which Lie algebras are known to describe approximately the symmetries of aphysical system. The quantum algebra su (2) has been successfully used for describing rotational spectra of diatomic molecules [5— 7], deformed nuclei [8— 10], and superdeformed nuclei [11]. Vibrational spectra of diatomic molecules have been described in terms of deformed oscillators [12— 16], as well as in terms of an SU (1,1) symmetry [17,18]. Potentials giving spectra equivalent to those of the deformed oscillators just mentioned have been constructed [19,20] and found to be deformed versions of the modified PoschlTeller potential or, equivalently, the Morse potential. On the other hand, the vibron model [21— 23], having an overall U(4} symmetry, is known to provide aunified description of molecular rotations and vibrations through the use of algebraic techniques, in away similar to the description of collective nuclei in terms of the interacting boson model (IBM) [24]. The O(4} limiting symmetry of the vibron model has been found to be appropriate for diatomic molecules, while the U(3) limiting symmetry has been used for the description of clustering effects in nuclei, as well as for the quasimolecular description of heavy-ion resonances (see [25] for lists of references). The question is therefore created if adeformed version of the vibron model can accommodate in aunified framework the improved descriptions of rotational and vibra- 'On leave of absence from the Institute of Nuclear Physics, Moscow State University, 117234Moscow, Russia. tional molecular spectra obtained so far in terms of separate quantum algebras. The problem of constructing the deformed version of the vibron model (or of the IBM) is not asimple one since the construction of the reduction chains of Ue(4) and U(6) has not been achieved yet. It suffices to be mentioned that the reduction frotn SU (3) to SO (3) has been carried out only for fully symmetric irreducible representations (irreps} of SU (3) [26]. However, afew efforts towards constructing deformed versions of the vibron model [27,28] and the IBM [29,30] already exist. In this paper adeformed version of both the O(4) and U(3) dynamical symmetries of the U(4) vibron model will be constructed, taking advantage of the techniques of complementary algebras, introduced by Quesne and coworkers [31— 33],which bypass the difficulties in the construction of reduction chains ofquantum algebras. In addition to unifying the existing independent quantumalgebraic descriptions of rotational and of vibrational molecular spectra, the present approach allows, in asimple way, for the introduction of cross terms describing the coupling between these two excitation mechanisms. Abrief account of the vibron model for diatomic molecules will be given in Sec. II, while in Sec. III the model will be formulated in another way, using the techniques of complementary algebras. In Sec. IV the q-deformed version of the complementary analogs of both the O(4) and U(3) dynamical symmetries of the vibron model will be given. Section Vwill contain discussion of the present results and plans for further work. II. THK VIBRON MODEL FOR DIATOMIC MOLKCULES In this section the briefest possible account of the vibron model [21— 23] is given in its usual form. In the vibron model the rotations and vibrations of adiatomic molecule are described in terms of four bosons: ascalar boson of positive parity and angular momentum 1=0, denoted by s+, and the three components of avector bo1050-2947y94g5Og)g1O88(8)A%06. 0O 50 Qc 1994 The American Physical Society 50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1089 [T 'T '] '= g(k, uk~ u3~k3 u3)T„'T„', Q)Q2 (2.1) one observes that the 16 possible bilinear quantities [bi+eh&. ]generate the algebra U(4), which is, therefore, the overall symmetry ofthe vibron model. There are two chains of subalgebras of u(4) containing the angular-momentum algebra so(3) as asubalgebra. These are I: u(4) Do(4) Dso(3) Dso(2), II: u(4) Ou(3) Dso(3) Dso(2) . (2.2) (2.3) In the case of chain Ithe basis has the form ~N ELM ), where the various quantum numbers are defined as follows. (i) Nis the total number of bosons. It characterizes the irreps of U(4), which are fully symmetric, since we are dealing with asystem ofbosons. (ii) co is the seniority quantum number, characterizing the irreps of O(4) and obtaining the values co=N, N— 2, ...,1or 0. (iii) Lis the angular momentum quantum number, labeling the irreps of SO(3) and taking the values L=a),co — 1, .. . ,1,0. (iv) Mdenotes the zcomponent of the angular momentum, labeling the irreps of SO(2) and having the values — L+M+L. When the Hamiltonian is characterized by the dynamical symmetry of chain I, it can be written in terms of the Casimir operators ofthe algebras appearing in this chain: Ht =ep+ eC~ (u(4) )+63C3(U(4)) +ACz(o(4) )+BC'(so(3)), (2.4) where Nand Nare related to the firstand second-order Casimir operators of u(4). The eigenvalues of the Hamiltonian in the basis given above are then E(N, co,L)=ep+e)N+e3N(N+3) +Ace(co+2)+BL (L +1}.(2.5) Usually the vibrational quantum number N— co 2(2.6) is introduced, and the energy eigenvalues are rewritten as E(N, u, L)=op+ e',N+e~N 4A (N +2)(u +—, — '} +4A (u+ , '}+BL(L+1), — (2.7) where E'(), E'),Ep are related to E'0 E'~ 6'p A. It should be noticed that the fourth and fifth terms on the right-hand son of negative parity and 1=1, denoted by p„+, p=0,+1. The corresponding annihilation operators transforming as spherical tensors are s=sand P„=(— 1)' "p „.Denoting these bosons by b&+„, I=0,1 and — 1&p&I, and bI „=(— 1)' "b~ „,and de6ning the tensor product oftwo operators T„and T„as 12 +aC3(u(3))+pC, (so(3)) .(2.8) The eigenvalues of the Hamiltonian in the basis given above are E(N, n&,L)=Fp+F.,N+ezN(N+3) +en +an (n +2)+PL(L+1) . III. ALTERNATIVE FORMULATION OF THE VIBRON MODEL (2.9) An alternative formulation of the vibron model can be achieved in terms of complementary algebras. The notion of complementary algebras was introduced by Moshinsky, Quesne, and co-workers [31— 33]. It is especially fruitful in the case of multidimensional harmonic oscillators or many-particle systems of few kinds of bosons. In the present case of four kinds of bosons (s+,p„+, @=0,+1) the host algebra is sp(S,E). Two chains of subalgebras are sp(8, E)Du(4) Do(4) Dso(3) Dso(2), sp(S, E)Dsp(2, R)Du(1) . (3.1) (3.2) The quantum numbers N, p3, L,M, labeling the irreps of the subalgebras of the first chain, have been described in Sec. II. sp(2,R) is isomorphic to su(1,1). The irreps of su(1,1) and u(1) are labeled by the quantum numbers j and m, respectively. Two subalgebras A& and Az of a larger algebra Aare complementary within adefinite irrep of Aif there is aone-to-one correspondence between all the irreps of A& and of Az contained in this irrep of A [31]. In the example given above, the only irreps of the host algebra sp(S,E}that can be realized in aFock boson space are the even irrep [0], including the vectors ~NcoLM )with Neven, and the odd irrep [1],including the vectors with Nodd. It can then be proved that o(4} and sp(2,E) [and thus also o(4) and su(1,1)] are complementary. The same holds for u(4) and u(1). For convenience let us denote the four kinds of bosons introduced in Sec. II by b, v=1,2,3,4, corresponding to p+&,p„po,s, respectively. To each kind vof bosons corresponds an algebra sp"(2,E},generated by K+ =,'bP„, K' =, 'b b„,—ECp= —, '(N, +——, '), (3.3) side correspond to the spectrum of the Morse potential [34]. In the case of chain II the basis is ~Nn LM ),where N is again the total number ofbosons, while the other quantum numbers are defined as follows. (i) nis the number ofpbosons labeling the irreps of U(3) and obtaining the values n=0,1, ...,N (ii) Lis labeling the irreps of SO(3}, obtaining the values L=n,n— 2, ...,1or 0. (iii) Mis labeling the irreps of SO(2), with values — L+M+L. When the Hamiltonian is characterized by the dynamical symmetry of chain II, it can be written in terms ofthe Casimir operators ofthe algebras appearing in it: H„=ep+ e,C,(u(4) )+ezCz(u(4) )+eC,(u(3)) R. N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50 where N„=bP .These generators satisfy the commutation relations [K(),K+ ]=+K~, [K+,K" ]= 2— K() .(3.4) The sp(2,E}=su( 1, 1}algebra, mentioned above, is realized in the space of four kinds of bosons. Therefore we are going to use for it the symbol sp" '(2, E)=su" '(l, l). This algebra is generated by =—, 'gbP„, K=—, 'gb„b„, K=—, '(N+2), ~NcoLM) the quantum numbers Nco by the quantum numbers jm of the complementary subalgebras. Furthermore, in the Hamiltonian of Eq. (2.4) one is entitled to replace the second-order Casimir operator of o(4) by the second-order Casimir operator of su" '(l, l) and the firstand second-order Casimir operators of u(4) [N and N(N +3)]by the firstand second-order Casimir operators of u(1) (Ko and Ko ). In the case of chain II, the u(3) subalgebra of u(4) involves only the pbosons. The host algebra is then sp(6,E),having the two chains of subalgebras (3.5) where N=g„b„b, The. se generators satisfy the commutation relations sp(6, E)Du(3) Dso(3) Dso(2), sp(6, E)Dsu" '(l, l)Du(1), (3.15) (3.16) [Kp,K~]=+K~, [K+,K]=— 2K() . The Casimir operator is Cz(sp" '(2,E))=— K+K +K()(K()— 1), (3.6) (3.7) 1n— 4 J—N+ 22(3.8) with eigenvalue j(j+1).It is known that when o(n) and su(1, 1) are complementary, the quantum numbers co and j characterizing their irreps are connected by [35] where the superscript (123) means that only the bosons b„bz, b3are involved in the formation of su" '(1,1}=sp" '(2, E). Further details on these chains are given below [see Eqs. (3.26)— (3.28}]. The building up of the bases related to the two limiting symmetries of the vibron model can then be achieved as follows. To each kind of boson b„, an sp"(2,E) algebra corresponds, as already mentioned, generated by the operators given in Eq. (3.3). The states with N„even correspond to the irrep D,while the states with N„odd correspond to the irrep D'.Thus to each boson state co N— U J=2= 4(3.9) In the present case su" '(l, l) is complementary to o(4), so that 1 lN)NzN, N4) =QN)!Nz!N3!N4! X(bi) '(bz) '(bz) '(b4) '0) (3.17) Cz(A) )=c)cz(Az)+cz . In the case of o(n) and su(1,1}this relation is (3.10) It is also known that the Casimir operators of two algebras complementary to each other are connected by a simple, usually linear, relation ofthe type one can correspond aset of four noncompact "angular momenta" j„=— —, 'or — —, ', v=1,2,3,4, the value of each angular momentum jdepending on the parity of the corresponding boson number N. We can now proceed to the vector coupling of the first two angular momenta j, and jz, using the SU(1,1) Clebsch-Gordan coefficients [36,37] Cz(su(1, 1))=—, 'Cz(o(n ))+n(n — 4) (3.11) lj)jzj)zm)z&= g&J(m)jzmzlj(zmiz}sU(), )) which in the present case ofo(4}reduces to C,(su"""(1,1))=-, 'C,(o(4)).(3.12) 1n m= —X+— 22(3.13) which in the present special case of u(4) and u(1) reduces to m=—, '(N+2) .(3.14) The chain of Eq. (3.1) already studied is of interest in the case of the chain Iof the vibron model. It implies that in studying chain I, one can replace in the basis The u(1) subalgebra of su" '( l, l) is generated by the operator Eo alone, the eigenvalues of which we label by m. In the general case of the complementary algebras u( n)and u(l), the quantum numbers Nand mcharacterizing their irreps are connected by [35] m&m& X~j)m))~jzmz) .(3.18) with p=0, +1. The host algebra of this space of two kinds of bosons is sp(4, E}.The following two chains of subalgebras exist: sp(4, E)&u(2) Dso(2), sp(4, E)Dsu' )( l, l)DU(1), (3.20} (3.21) where the irreps of u(2) are labeled by the total number of bosons N, z=N, +Nz, while the irreps of so(2) are labeled by M=N, Nz. SO(2) is — complementary to su" '( 1,1)=sp" '(2, E),the irreps of which are labeled by j,z=—, '(M — 1), (3.22} This means that the intermediate su" '(l, l}algebra has been introduced, generated by (3.19) 50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1091 m(z =— '(N(z+1), (3.23) according to Eq. (3.13). The next step along this line is to couple j12 with j3. In this case three kinds ofbosons are involved, so that the host algebra is sp(6,R). The relevant chains for this case have been given in Eqs. (3.15) and (3.16), so(3) being complementary to su" '(1,1},which is generated by g123 g12 +~3 (3.24} with p=0, +1. The resulting eigenvectors are IJljz{4lz V3'J123m 123 & {j(2mlzj3m3IJ123m123 &SU(1, () m12m3 xIj(jz.j(zm &z )Ij3m 3)(3.25) The Casimir operators of so(3} and su" '(l, l) are connected by according to Eq. (3.8), while u(2) is complementary to u(1), the irreps of which are labeled by The host algebra in this case is sp(8,R), the relevant chains having been given in Eqs. (3.31) and (3.32). The basis vectors, denoted by Ij,jz(j,z)j3(j,z3)j4. .jm ), or by Ij(zj3(j(z3 )j4.jm )in the case in which the shortened version of Eq. (3.30) is used for the vectors with angular momentum j12, correspond to the irreps of the sp' (2,R)=su' (l, l) algebra, generated by g1234 g123 +g4(3.32) I with @=0,+1. Here Kare the generators of the su (1,1) algebra, associated with the s-bosons. The total noncompact angular momentum j, characterizing the irreps of su" 4'(l, l), is connected to the seniority quantum number (0, characterizing the irreps of o(4), by Eq. (3.8), while the Casimir operators of these two complementary algebras are connected by Eq. (3.12). Given the above, it is clear that for the chain Iof the vibron model, instead of the basis INcoLM), the basis Ij,zjz{j,z3)j4.jm )can be used. Furthermore, in the case of chain II, instead of the basis INn~LM), the basis Ij,zj3:j,z,m, z, )Ij4m4 )can be used. It is clear that the connection between the two new bases for the dynamical symmetries ofthe vibron model is Cz(so(3) )=4Cz(su" "(1,1))+— ', ,{3.26) Ij(zjz(j]z3V4:Jm & according to Eq. (3.11), while the quantum numbers labeling their irreps, Land j123, respectively, are connected by m&23 m4 &j(z3m(z3j4m4ljm &sU((, (i j(z3=I'(L (3.27) xIj(zj3 j(z3m]z3 &Ij4m4 &(3.33) according to Eq. (3.8). The eigenvalues of Cz(su" '(l, l)) in the above-mentioned basis are given by j,z3(j,z3+1). Furthermore, in the chains of Eqs. (3.15) and (3.16), u(3) and u(1) are complementary, the quantum numbers n~ and m,23 labeling, respectively, their irreps being connected by The Hamiltonian of chain I, given in Eq. (2.4), can be rewritten using the complementarity relations described in Eqs. (3.1), (3.2), (3.15), and (3.16},as H) =ep+e(C((u(l))+ezCz(u(1)) +4ACz(su" '(1,1))+4BCz(su" '(1,1)).(3.34) m,z3=—, '(n +— ', ), (3 28) The eigenvalues of this Hamiltonian are Ij„m„&= (b$) '(bz) 'I0&, QN (!Nz! with (3.30) J(z 2(N( Nz — 1), m„=—, '(N, +N, +1), (3.31) [which are in agreement with Eqs. (3.22) and (3.23)] are eigenvectors ofthe Casimir operator Cz(su"z'(1, 1)),with eigenv»ues j»(j»+1). In this way one can avoid the couphng of j1 and j2 to j». The coupling of j» and j3 cannot be avoided however. The resulting vectors in this case we denote by Ij,zj3: j,z3m (z3 ). In the last step the coupling ofj123 to j4 is performed. according to Eq. (3.13). The coupling of the two angular momenta j, and jz, performed above, can be avoided by noticing that the su" '(l, l) can be generated by E'+ =b,bz, E' =b(bz, Ep =—, '(N, +Nz+1) .(3.29) The vectors E(m,jj(z3)=Ep+eIm +ezm '+4Aj(j+1) +4Bj123{j123+1).(3.35) Hn &O+e1m +edam +e'm123+a'm 123 +4P'Cz{su" '(1,1)), (3.36) where on the right-hand side the second and third term correspond to the firstand second-order Casimir operator of the u(1) algebra of the chain of Eq. (3.2), while the fourth and fifth terms correspond to the firstand second-order Casimir operators of the u(1) algebra appearing in the chain of Eq. (3.16). The eigenvalues of this Using Eqs. (3.14), (3.9), and (3.27), which connect the quantum numbers m, j,j,z3 to the previous ones (N, co,L), it is easily verified that Eq. (3.35) is an alternative way of writing Eq. (2.5}. Similarly the Hamiltonian of chain II, given in Eq. (2.8), can be rewritten, taking into account the complementarity relations given in Eqs. (3.1},(3.2), (3.15), and (3.16), as 1092 R. N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50 Hamiltonian are E(m, m, 23 J)23)= eP+ eIm+e2m +e m)23+a'm(23 one can prove that su" '(1,1) is generated by [38— 40] K''=a"a K" '=a aK" '=— '(N, +N2+1), +4&'J123(j123+1).(3.37) Using Eqs. (3.14), (3.28), and (3.27), which connect m, m,23,j,23 to N, n~, L, it is easily verified that Eq. (3.37) is an alternative way of writing Eq. (2.9). In this section we have therefore rewritten the bases and the Hamiltonians corresponding to the two dynamical symmetries of the vibron model in terms of complementary subalgebras. This formulation is useful because it can be qdeformed in avery simple way. IV. qDEFORMATION OF THE VIBRON MODEL the relevant commutation relations being [K(12) K(12) )+K(12) 0~+j—+ [K(12) K(12) ]= (2K()2) ] The vectors ~j»m»), =(a)) '(a2) '~0), V'[N(]q)[N2)q( (4.10) (4.11) The corresponding Hamiltonians were then written in terms of the Casimir operators of the new reduction chains. An evident possibility for qdeforming these Hamiltonians is to substitute the su(1, 1) algebras of Eq. (4.1) by their q-deformed counterparts su (1,1) [38— 40], su" '(1,1)Dsu" '(1,1)Dsu" '(1, 1)ZU (1) .(4.2) In this section we shall explain how this can be achieved, after giving abrief account of the necessary mathematical details. qnumbers are de6ned as (4.3) For qreal (q =e'with rreal), they can be written as sinhrx sinh~ (4.4) while in the case of qbeing aphase (q =e"with rreal), they obtain the form sinrx qsin7- (4.5) In the limiting q~ 1(v~0), qnumbers reduce to usual numbers. q-deformed oscillators [41,42] are introduced through the relations aalu — q— 'a 'a =q, [N, a]=a, [N,a]=— a, (4.6) where aand aare the q-deformed boson creation and annihilation operators and Nthe relevant number operator. Using Eq. (4.6) one can easily show that aa=[N]~, aa =[N+1]~ .(4.7) q-deformed algebras can be expressed in terms of qdeformed bosons. Introducing a, as the q-deformed analogs of b; (i =1,2, 3,4), with the properties [at,a„"]=[a, a„]=[a„, a„]=0, vip, (4.8) In the preceding section the subalgebra chains of the vibron model were reduced to equivalent chains of complementary subalgebras su" '(1 1)~su" '(1 1)~su" '(1 1)~u(1) .(4.1) with j,2, m)2 still given by Eq. (3.31), are eigenvectors of the deformed Casimir operator C(su" '(1 1))= K" 'K — ''+[K" '] [K" '— 1] (4.12) (4.13) where Nis the number of vbosons. These generators satisfy the commutation relations [K(),K+ ]=+K+ [K+ K]= [2K()]2.(4.14) We are not going to use the su'(l, l) and su (1,1) algebras explicitly in couplings, since for the su" '( l, l) algebra we already have the form given in Eq. (4.9), which avoids the direct coupling. In order to be able to couple su (1,1) and su (1,1) to su" '(l, l), it is useful to have the same deformation parameter in all of these algebras, i.e.,it is useful to have the same deformation parameter in the commutation relations of Eqs. (4.10) and (4.14). In order to achieve that, we replace in Eqs. (4.13) and (4.14) qby q. As aresult, for v=3,4, Eq. (4.6) is meant from now on with qreplaced by &q. Then one also has a~„=[N,]&—, a,,a„"=[N,+I]&- .(4.15) Equation (4.12), giving the Casimir operator, is therefore valid in this case with the usual qnumbers. The su" '(1,1) algebra is generated by the operators g3 g(12) +(123) +(12) o+~3o +q+ ~(123) ~(12)+~3 00 (4.16) 112+ i.e.,it is astandard coproduct of the irreps Dand D'of the su (1,1) algebra. Therefore the basis vectors, in analogy to Eq. (3.25), are of the form with eigenvalues [j)2]~[j)2+1]». For the su'(1, 1) algebras one has the boson realization [38] K' =aa, K" =aa, K() ='(N +-'),— 11 50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1093 Ij)2j3:j)23m)23 &q while the vectors analogous to Eq. (3.33) are Ij)2J3(J123 }J4:Jm & &j)23m)23j4m4IJm &sU, )1,1) m123 m4 xIj)2j3.j)23m]23 &ql j4m4&, .(4.19) These vectors are the qanalogs of the eigenvectors of the dynamical symmetry Iof the vibron model. Similarly the vectors IJ,2J3.j)23m)23 &, Ij4m4 &, (4.20) =g&j)2m)2J3m3IJ)23m)23&SU (),)) m12m 3 XIj)2m )2 &,Ij3m3 &, ,(4.17) &j,m,j,m, Ijm &s„„„are Clebsch-Gordan coefficients for the tensor product of two su (1,1) irreps. Explicit analytical formulas for these coefficients, as well as for the relevant su (2) coefficients, can be found in [43— 47]. The su"234'(1, 1)algebra is generated by the operators ~4 g(123) ~(1234) ~(123) o+~4 o(4.18) have been used. Using Eq. (2.6), Eq. (4.23) can be written as E(N, v, L)=6p+E') [N +2]~—+e2[N +2]~— +A v— —v— 1—— 22 +B'[L]~ [L— +1]~ —, (4.26) which reduces to Eq. (2.7) in the limit q~l, up to a redefinition of E'0 In the case of the dynamical symmetry II the Hamiltonian can be written as H)) =~P+~)[m]q+~2[m]q'+~[m)23)q +a[m)23]q+PC2(suq) '( l, l)), (4.27) which is the qanalog of Eq. (3.36) (with the primes of the coefficients dropped). The eigenvalues of this Hamiltonian are E(m, m)23 L)=Ep+E')[m]q+e2[m]q+E'[m)23]q +a[m)23], +P[j)23]q[j)23+1]q .(4.28) In the limit q~1,Eq. (3.37) is obtained. Assuming that m, m,23,j,23 are connected to N, n~, Lthrough Eqs. (3.14), (3.28), and (3.27), Eq. (4.28) can be rewritten in away resembling its classical counterpart, Eq. (2.9), as are the qanalogs of the eigenvectors of the dynamical symmetry II of the vibron model. Therefore Eq. (4.19) connects the eigenvectors of the two dynamical symmetries, as Eq. (3.33) does in the classical case. In the case of dynamical symmetry Ithe Hamiltonian reads E(N, n&,L)=Ep+ E,[N +2]&- +ez[N +2]&— +e'[n +— ', ]g- +a'[n~+ —' ,]g-, +P'[L]~[L +l]g- .(4.29} H) =6p+6)[m]q +'e2[m ]q +4AC2(suq '( 1,1)) +4BC2(su"2'(1,1)), (4.21) The results obtained in this section call for the following comments. (i) Rotational-vibrational spectra of diatomic molecules are described empirically by the Dunharn expansion [48] which is the qanalog of Eq. (3.34) (with the primes of the coefficients dropped). The eigenvalues of this Hamiltonian are E(u,L)=g Yk(u+ —, ')'[L(L+1)]", ik (4.30) E(m,j,j)23)=op+@)[m],+e2[m] +4A [j] [j+1] +[»»)q[»»+ )q (422} In the limit q~ 1, Eq. (3.35) is obtained. Assuming that m, j,j123 are still connected to quantum numbers N, co,L through Eqs. (3.14}, (3.9), and (3.27}, Eq. (4.22) can be rewritten in away resembling its classical counterpart, Eq. (2.5), as E(N, a),L)=ep+eI [N+2]~—+e2[N+2]~— +A'[co)~— [p)+2]~—+B'[L]~ [L +1]~ —.— (4.23) In producing Eq. (4.23), identities such as H) =a)+DC2(su" '(1 1))C2(su" '(1 1)) (4.31) Then Eq. (4.26}is modified as where vis the vibrational quantum number, Lthe angular momentum, and Y;k the Dunham coeScients, fitted to experiment. It is clear that the Dunham expansion contains powers of (v+—, '), powers of L(L +1), as well as cross terms. Equations (4.23) and (4.29) contain no cross terms. This is due to the fact that in the Hamiltonians of Eqs. (4.21) and (4.27), only terms up to quadratic in the generators are included, as in the case ofthe classical vibron model. Cross terms can be taken into account in the dynamical symmetry I, for example, by modifying Eq. (4.21) as follows: q — [](1/2+ — 1/2) — 1(4.24) E'(N, v, L)=E(N, u, L)+D' v—— N v— 1—— q2q 2)q [L +— ']q =[L]q[L +1]q [—, ']q [— ', ]q (4.25) X[L]~ [L +1]~ —— (4.32) 1094 R..N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50 (ii) Rotational spectra in both dynamical symmetries [Eqs. (4.26) and (4.29)] are described by the term [L]& [L—+I]&—. This is known to be the Casimir operator of su&— (2). The su (2) model has been extenq sively used for the description of rotational spectra of diatomic molecules [5— 7] and deformed [8— 10] and superdeformed [11]nuclei. It has been found [9]that this term is equivalent to an expansion in terms of powers of L(L +1), [L]q[L+1] =(jp(r)L(L+1) rj i(v'){L(L+1)j +~r j2(r)jL(L+1)]3 l jjp«)]' — — ', rj(r)jL (L +1)] +,', rj~—(r)jL (L +1)] —.), (4.33) where j„(r)are the spherical Bessel functions of the first kind and q=e". This expansion is similar to the one contained in the Dunham expansion. In the case of su (2), however, all the expansion coefficients are related to powers of ~, thus resulting in economy of parameters. Notice that the decreasing of the coefficients ofincreasing powers of L(L +1),as well as the alternating signs of the terms, facts that are known empirically to hold, occur in Eq. (4.33) automatically, since ris known [5— 11] to obtain small positive values. Furthermore, it has been proved [9]that the su~(2) model is equivalent to the variI able moment of inertia (VMI) model, which describes rotational stretching effects. The qparameter has been found [9] to correspond to the softness parameter of the VMI model. The implications of the su (2) model on the electromagnetic transition probabilities connecting the rotational levels of nuclei have been considered [10]. (iii) The fourth term in Eq. (4.26) corresponds to the Casimir operator of su (1,1), already used [17] for the description of vibrational spectra of diatomic molecules. It has been proved [17]that this term, for q=e",can be expanded as N U2u— 1——=j—, '(cos(r) — cosjr(N+2)]) — rsinjr(N+2)](u+ —, ') sin(r) +Hcosjr(N+2) j(v+—, ') +— 2r sinjr(N+2)}(u+ —, ') — —, 'r cosjr(N+2)](v+ —, ') +(4.34) We remark that aseries of powers of (v +—, ')is obtained, similar to the one contained in the Dunham expansion. In the present case, however, the expansion coeScients are all related to r(and N, which in the vibron model is a constant for agiven molecule), thus resulting in economy in parameters. (iv) The anharmonicity constant (i.e.,the ratio Y2p /Yip )in the classical case [Eq. (2.7)] is fixed to — 1/(N+2). In the deformed case of Eq. (4.26), however, it is equal to — r/tan[r(N+2)], as it is easily seen from the expansion of Eq. (4.34). The extra freedom gained this way has been found [17]to improve the fits of vibrational molecular spectra. (v) Since Nis fixed for agiven molecule (related to the maximum number of bound states below the dissociation limit), the first three terms in Eqs. (4.26) and (4.29) have no inhuence on the spectrum. (vi) In Eq. (4.26) it is clear that the deformation parameter for the vibrational part of the spectrum is ~, while for the rotational part it is v/2. Therefore arelation is implied between the rotational stretching and the anharmonicity corrections. Careful empirical fits are needed in order to decide if this is arestriction or an advantage of the present model. There is no apriori reason, however, that these two physically different mechanisms be described by the same parameter. Amore general version ofthe model, allowing for these two deformation parameters to be independent of each other, might give better results. V. DISCUSSION In this paper adeformed version of the O(4) and U(3) dynamical symmetries of the vibron model for diatomic molecules has been constructed. This has been achieved by first rewriting, through the use of the concept of complementary subalgebras, the model in amore convenient form, which is subsequently deformed. The present approach unifies into acommon framework the so far separate algebraic approaches to rotational and to vibrational spectra ofdiatomic molecules. For the O(4) limit of the present model, fittings to experimental data for diatomic molecules are required. 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