PHYSICAL REVIEW AVOLUME 50, NUMBER 2
q-de o med ib on model o dia omic molecnles
AUGUST 1994
R. N. Al a ez, 'Dennis Bona sos, and Yu. F.Smi no '
'Ins i u e o Nuclea Physics, Moscow S a e Uni e si y, 117234 Moscow, Russia
Ins i u e o Nuclea Physics, Na ional Cen e o Scien i c Resea ch Demok i os, GR 153-10Aghia Pa aske i, A iki, G eece
(Recei ed 29 No embe 1993)
Ade o med e sion o he ib on model o dia omic molecules is cons uc ed. Bo h he O(4) and
U(3) dynamical symme ies o he model a e ew i en, using he concep o complemen a y subalgeb as,
in amo e con enien o m, which is subsequen ly de o med. The p esen model uni ies he so a in-
dependen success ul quan um-algeb aic app oaches o o a ional and o ib a ional spec a o dia omic
molecules. In addi ion, he me hod can be used o he cons uc ion o de o med e sions o he U(5)
and O(6) limi s o he in e ac ing boson model o nuclea s uc u e.
PACS numbe (s): 33.10.Cs, 31.15.+q, 02.20.S
I. INTRODUCTION
The ma hema ical s uc u e o quan um algeb as
(quan um g oups) [1—
4] has ecen ly been a ac ing
much a en ion. They a e de o med e sions o he usual
Lie algeb as, o which hey educe when he de o ma ion
pa ame e qis se equal o 1. In pa allel, applica ions o
quan um algeb as in physics ha e begun o de elop in
pa icula in cases in which Lie algeb as a e known o de-
sc ibe app oxima ely he symme ies o aphysical sys-
em. The quan um algeb a su (2) has been success ully
used o desc ibing o a ional spec a o dia omic mole-
cules [5—
7], de o med nuclei [8—
10], and supe de o med
nuclei [11]. Vib a ional spec a o dia omic molecules
ha e been desc ibed in e ms o de o med oscilla o s
[12—
16], as well as in e ms o an SU (1,1) symme y
[17,18]. Po en ials gi ing spec a equi alen o hose o
he de o med oscilla o s jus men ioned ha e been con-
s uc ed [19,20] and ound o be de o med e sions o he
modi ied Poschl- Telle po en ial o , equi alen ly, he
Mo se po en ial.
On he o he hand, he ib on model [21—
23], ha ing
an o e all U(4} symme y, is known o p o ide auni ied
desc ip ion o molecula o a ions and ib a ions h ough
he use o algeb aic echniques, in away simila o he
desc ip ion o collec i e nuclei in e ms o he in e ac ing
boson model (IBM) [24]. The O(4} limi ing symme y o
he ib on model has been ound o be app op ia e o di-
a omic molecules, while he U(3) limi ing symme y has
been used o he desc ip ion o clus e ing e ec s in nu-
clei, as well as o he quasimolecula desc ip ion o
hea y-ion esonances (see [25] o lis s o e e ences).
The ques ion is he e o e c ea ed i ade o med e sion
o he ib on model can accommoda e in auni ied ame-
wo k he imp o ed desc ip ions o o a ional and ib a-
'On lea e o absence om he Ins i u e o Nuclea Physics,
Moscow S a e Uni e si y, 117234Moscow, Russia.
ional molecula spec a ob ained so a in e ms o
sepa a e quan um algeb as. The p oblem o cons uc ing
he de o med e sion o he ib on model (o o he IBM)
is no asimple one since he cons uc ion o he educ ion
chains o Ue(4) and U(6) has no been achie ed ye . I
su ices o be men ioned ha he educ ion o n SU (3)
o SO (3) has been ca ied ou only o ully symme ic
i educible ep esen a ions (i eps} o SU (3) [26]. How-
e e , a ew e o s owa ds cons uc ing de o med e -
sions o he ib on model [27,28] and he IBM [29,30] al-
eady exis .
In his pape ade o med e sion o bo h he O(4) and
U(3) dynamical symme ies o he U(4) ib on model will
be cons uc ed, aking ad an age o he echniques o
complemen a y algeb as, in oduced by Quesne and co-
wo ke s [31—
33],which bypass he di icul ies in he con-
s uc ion o educ ion chains o quan um algeb as. In ad-
di ion o uni ying he exis ing independen quan um-
algeb aic desc ip ions o o a ional and o ib a ional
molecula spec a, he p esen app oach allows, in asim-
ple way, o he in oduc ion o c oss e ms desc ibing
he coupling be ween hese wo exci a ion mechanisms.
Ab ie accoun o he ib on model o dia omic mole-
cules will be gi en in Sec. II, while in Sec. III he model
will be o mula ed in ano he way, using he echniques
o complemen a y algeb as. In Sec. IV he q-de o med
e sion o he complemen a y analogs o bo h he O(4)
and U(3) dynamical symme ies o he ib on model will
be gi en. Sec ion Vwill con ain discussion o he p esen
esul s and plans o u he wo k.
II. THK VIBRON MODEL
FOR DIATOMIC MOLKCULES
In his sec ion he b ie es possible accoun o he ib-
on model [21—
23] is gi en in i s usual o m. In he ib-
on model he o a ions and ib a ions o adia omic mol-
ecule a e desc ibed in e ms o ou bosons: ascala bo-
son o posi i e pa i y and angula momen um 1=0,
deno ed by s+, and he h ee componen s o a ec o bo-
1050-2947y94g5Og)g1O88(8)A%06. 0O 50 Qc 1994 The Ame ican Physical Socie y
50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1089
[T 'T '] '= g(k, uk~ u3~k3 u3)T„'T„',
Q)Q2
(2.1)
one obse es ha he 16 possible bilinea quan i ies
[bi+eh&. ]gene a e he algeb a U(4), which is, he e o e,
he o e all symme y o he ib on model.
The e a e wo chains o subalgeb as o u(4) con aining
he angula -momen um algeb a so(3) as asubalgeb a.
These a e
I: u(4) Do(4) Dso(3) Dso(2),
II: u(4) Ou(3) Dso(3) Dso(2) .
(2.2)
(2.3)
In he case o chain I he basis has he o m ~N ELM ),
whe e he a ious quan um numbe s a e de ined as ol-
lows.
(i) Nis he o al numbe o bosons. I cha ac e izes he
i eps o U(4), which a e ully symme ic, since we a e
dealing wi h asys em o bosons.
(ii) co is he senio i y quan um numbe , cha ac e izing
he i eps o O(4) and ob aining he alues
co=N, N—
2, ...,1o 0.
(iii) Lis he angula momen um quan um numbe , la-
beling he i eps o SO(3) and aking he alues
L=a),co —
1, .. . ,1,0.
(i ) Mdeno es he zcomponen o he angula momen-
um, labeling he i eps o SO(2) and ha ing he alues
—
L+M+L.
When he Hamil onian is cha ac e ized by he dynami-
cal symme y o chain I, i can be w i en in e ms o he
Casimi ope a o s o he algeb as appea ing in his chain:
H =ep+ eC~ (u(4) )+63C3(U(4))
+ACz(o(4) )+BC'(so(3)), (2.4)
whe e Nand Na e ela ed o he i s - and second-o de
Casimi ope a o s o u(4). The eigen alues o he Hamil-
onian in he basis gi en abo e a e hen
E(N, co,L)=ep+e)N+e3N(N+3)
+Ace(co+2)+BL (L +1}.(2.5)
Usually he ib a ional quan um numbe
N—
co
2(2.6)
is in oduced, and he ene gy eigen alues a e ew i en as
E(N, u, L)=op+ e',N+e~N 4A (N +2)(u +—,
—
'}
+4A (u+ ,
'}+BL(L+1), —
(2.7)
whe e E'(), E'),Ep a e ela ed o E'0 E'~ 6'p A. I should be no-
iced ha he ou h and i h e ms on he igh -hand
son o nega i e pa i y and 1=1, deno ed by p„+,
p=0,+1. The co esponding annihila ion ope a o s
ans o ming as sphe ical enso s a e s=sand
P„=(—
1)' "p „.Deno ing hese bosons by b&+„, I=0,1
and —
1&p&I, and bI „=(—
1)' "b~ „,and de6ning he
enso p oduc o wo ope a o s T„and T„as
12
+aC3(u(3))+pC, (so(3)) .(2.8)
The eigen alues o he Hamil onian in he basis gi en
abo e a e
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 al e na i e o mula ion o he ib on model can be
achie ed in e ms o complemen a y algeb as. The no-
ion o complemen a y algeb as was in oduced by
Moshinsky, Quesne, and co-wo ke s [31—
33]. I is espe-
cially ui ul in he case o mul idimensional ha monic
oscilla o s o many-pa icle sys ems o ew kinds o bo-
sons. In he p esen case o ou kinds o bosons (s+,p„+,
@=0,+1) he hos algeb a is sp(S,E). Two chains o
subalgeb as a e
sp(8, E)Du(4) Do(4) Dso(3) Dso(2),
sp(S, E)Dsp(2, R)Du(1) .
(3.1)
(3.2)
The quan um numbe s N, p3, L,M, labeling he i eps o
he subalgeb as o he i s chain, ha e been desc ibed in
Sec. II. sp(2,R) is isomo phic o su(1,1). The i eps o
su(1,1) and u(1) a e labeled by he quan um numbe s j
and m, espec i ely. Two subalgeb as A& and Az o a
la ge algeb a Aa e complemen a y wi hin ade ini e i -
ep o Ai he e is aone- o-one co espondence be ween
all he i eps o A& and o Az con ained in his i ep o A
[31]. In he example gi en abo e, he only i eps o he
hos algeb a sp(S,E} ha can be ealized in aFock boson
space a e he e en i ep [0], including he ec o s
~NcoLM )wi h Ne en, and he odd i ep [1],including
he ec o s wi h Nodd. I can hen be p o ed ha o(4}
and sp(2,E) [and hus also o(4) and su(1,1)] a e comple-
men a y. The same holds o u(4) and u(1).
Fo con enience le us deno e he ou kinds o bosons
in oduced in Sec. II by b, =1,2,3,4, co esponding o
p+&,p„po,s, espec i ely. To each kind o bosons
co esponds an algeb a sp"(2,E},gene a ed by
K+ =,'bP„, K' =,
'b b„,—ECp=
—,
'(N, +——,
'), (3.3)
side co espond o he spec um o he Mo se po en ial
[34].
In he case o chain II he basis is ~Nn LM ),whe e N
is again he o al numbe o bosons, while he o he quan-
um numbe s a e de ined as ollows.
(i) nis he numbe o pbosons labeling he i eps o
U(3) and ob aining he alues n=0,1, ...,N
(ii) Lis labeling he i eps o SO(3}, ob aining he
alues L=n,n—
2, ...,1o 0.
(iii) Mis labeling he i eps o SO(2), wi h alues
—
L+M+L.
When he Hamil onian is cha ac e ized by he dynami-
cal symme y o chain II, i can be w i en in e ms o he
Casimi ope a o s o he algeb as appea ing in i :
H„=ep+ e,C,(u(4) )+ezCz(u(4) )+eC,(u(3))
R. N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50
whe e N„=bP .These gene a o s sa is y he commu a-
ion ela ions
[K(),K+ ]=+K~, [K+,K" ]= 2—
K() .(3.4)
The sp(2,E}=su( 1, 1}algeb a, men ioned abo e, is eal-
ized in he space o ou kinds o bosons. The e o e we
a e going o use o i he symbol
sp" '(2, E)=su" '(l, l). This algeb a is gene a ed by
=—,
'gbP„, K=—,
'gb„b„, K=—,
'(N+2),
~NcoLM) he quan um numbe s Nco by he quan um
numbe s jm o he complemen a y subalgeb as. Fu he -
mo e, in he Hamil onian o Eq. (2.4) one is en i led o e-
place he second-o de Casimi ope a o o o(4) by he
second-o de Casimi ope a o o su" '(l, l) and he
i s - and second-o de Casimi ope a o s o u(4) [N and
N(N +3)]by he i s - and second-o de Casimi ope a-
o s o u(1) (Ko and Ko ).
In he case o chain II, he u(3) subalgeb a o u(4) in-
ol es only he pbosons. The hos algeb a is hen
sp(6,E),ha ing he wo chains o subalgeb as
(3.5)
whe e N=g„b„b, The. se gene a o s sa is y he com-
mu a ion ela ions
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 Casimi ope a o is
Cz(sp" '(2,E))=—
K+K +K()(K()—
1),
(3.6)
(3.7)
1n—
4
J—N+
22(3.8)
wi h eigen alue j(j+1).I is known ha when o(n) and
su(1, 1) a e complemen a y, he quan um numbe s co and j
cha ac e izing hei i eps a e connec ed by [35]
whe e he supe sc ip (123) means ha only he bosons
b„bz, b3a e in ol ed in he o ma ion o
su" '(1,1}=sp" '(2, E). Fu he de ails on hese chains
a e gi en below [see Eqs. (3.26)—
(3.28}].
The building up o he bases ela ed o he wo limi ing
symme ies o he ib on model can hen be achie ed as
ollows. To each kind o boson b„, an sp"(2,E) algeb a
co esponds, as al eady men ioned, gene a ed by he
ope a o s gi en in Eq. (3.3). The s a es wi h N„e en co -
espond o he i ep D,while he s a es wi h N„odd
co espond o he i ep D'.Thus o each boson s a e
co N—
U
J=2= 4(3.9)
In he p esen case su" '(l, l) is complemen a y o o(4),
so ha
1
lN)NzN, N4) =QN)!Nz!N3!N4!
X(bi) '(bz) '(bz) '(b4) '0) (3.17)
Cz(A) )=c)cz(Az)+cz .
In he case o o(n) and su(1,1} his ela ion is
(3.10)
I is also known ha he Casimi ope a o s o wo alge-
b as complemen a y o each o he a e connec ed by a
simple, usually linea , ela ion o he ype
one can co espond ase o ou noncompac "angula
momen a" j„=—
—,
'o —
—,
', =1,2,3,4, he alue o each
angula momen um jdepending on he pa i y o he
co esponding boson numbe N.
We can now p oceed o he ec o coupling o he i s
wo angula momen a j, and jz, using he SU(1,1)
Clebsch-Go dan coe icien s [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 he p esen case o o(4} educes o
C,(su"""(1,1))=-,
'C,(o(4)).(3.12)
1n
m= —X+—
22(3.13)
which in he p esen special case o u(4) and u(1) educes
o
m=—,
'(N+2) .(3.14)
The chain o Eq. (3.1) al eady s udied is o in e es in
he case o he chain Io he ib on model. I implies
ha in s udying chain I, one can eplace in he basis
The u(1) subalgeb a o su" '( l, l) is gene a ed by he
ope a o Eo alone, he eigen alues o which we label by
m. In he gene al case o he complemen a y algeb as
u( n)and u(l), he quan um numbe s Nand mcha ac e -
izing hei i eps a e connec ed by [35]
m&m&
X~j)m))~jzmz) .(3.18)
wi h p=0, +1. The hos algeb a o his space o wo
kinds o bosons is sp(4, E}.The ollowing wo chains o
subalgeb as exis :
sp(4, E)&u(2) Dso(2),
sp(4, E)Dsu' )( l, l)DU(1),
(3.20}
(3.21)
whe e he i eps o u(2) a e labeled by he o al numbe o
bosons N, z=N, +Nz, while he i eps o so(2) a e labeled
by M=N, Nz. SO(2) is —
complemen a y o
su" '( 1,1)=sp" '(2, E), he i eps o which a e labeled by
j,z=—,
'(M —
1), (3.22}
This means ha he in e media e su" '(l, l}algeb a has
been in oduced, gene a ed by
(3.19)
50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1091
m(z =—
'(N(z+1), (3.23)
acco ding o Eq. (3.13).
The nex s ep along his line is o couple j12 wi h j3.
In his case h ee kinds o bosons a e in ol ed, so ha he
hos algeb a is sp(6,R). The ele an chains o his case
ha e been gi en in Eqs. (3.15) and (3.16), so(3) being com-
plemen a y o su" '(1,1},which is gene a ed by
g123 g12 +~3 (3.24}
wi h p=0, +1. The esul ing eigen ec o s a e
IJljz{4lz V3'J123m 123 &
{j(2mlzj3m3IJ123m123 &SU(1, ()
m12m3
xIj(jz.j(zm &z )Ij3m 3)(3.25)
The Casimi ope a o s o so(3} and su" '(l, l) a e con-
nec ed by
acco ding o Eq. (3.8), while u(2) is complemen a y o
u(1), he i eps o which a e labeled by
The hos algeb a in his case is sp(8,R), he ele an
chains ha ing been gi en in Eqs. (3.31) and (3.32). The
basis ec o s, deno ed by Ij,jz(j,z)j3(j,z3)j4.
.jm ), o by
Ij(zj3(j(z3 )j4.jm )in he case in which he sho ened e -
sion o Eq. (3.30) is used o he ec o s wi h angula
momen um j12, co espond o he i eps o he
sp' (2,R)=su' (l, l) algeb a, gene a ed by
g1234 g123 +g4(3.32)
I
wi h @=0,+1. He e Ka e he gene a o s o he
su (1,1) algeb a, associa ed wi h he s-bosons. The o al
noncompac angula momen um j, cha ac e izing he i -
eps o su" 4'(l, l), is connec ed o he senio i y quan-
um numbe (0, cha ac e izing he i eps o o(4), by Eq.
(3.8), while he Casimi ope a o s o hese wo comple-
men a y algeb as a e connec ed by Eq. (3.12).
Gi en he abo e, i is clea ha o he chain Io he
ib on model, ins ead o he basis INcoLM), he basis
Ij,zjz{j,z3)j4.jm )can be used. Fu he mo e, in he case
o chain II, ins ead o he basis INn~LM), he basis
Ij,zj3:j,z,m, z, )Ij4m4 )can be used. I is clea ha he
connec ion be ween he wo new bases o he dynamical
symme ies o he ib on model is
Cz(so(3) )=4Cz(su" "(1,1))+—
', ,{3.26) Ij(zjz(j]z3V4:Jm &
acco ding o Eq. (3.11), while he quan um numbe s la-
beling hei i eps, Land j123, espec i ely, a e connec ed
by m&23 m4
&j(z3m(z3j4m4ljm &sU((, (i
j(z3=I'(L (3.27) xIj(zj3 j(z3m]z3 &Ij4m4 &(3.33)
acco ding o Eq. (3.8). The eigen alues o
Cz(su" '(l, l)) in he abo e-men ioned basis a e gi en
by j,z3(j,z3+1). Fu he mo e, in he chains o Eqs.
(3.15) and (3.16), u(3) and u(1) a e complemen a y, he
quan um numbe s n~ and m,23 labeling, espec i ely,
hei i eps being connec ed by
The Hamil onian o chain I, gi en in Eq. (2.4), can be
ew i en using he complemen a i y ela ions desc ibed
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 eigen alues o his Hamil onian a e
Ij„m„&= (b$) '(bz) 'I0&,
QN (!Nz!
wi h
(3.30)
J(z 2(N( Nz —
1), m„=—,
'(N, +N, +1), (3.31)
[which a e in ag eemen wi h Eqs. (3.22) and (3.23)] a e
eigen ec o s o he Casimi ope a o Cz(su"z'(1, 1)),wi h
eigen »ues j»(j»+1). In his way one can a oid he
couphng o j1 and j2 o j». The coupling o j» and j3
canno be a oided howe e . The esul ing ec o s in his
case we deno e by Ij,zj3:
j,z3m (z3 ).
In he las s ep he coupling o j123 o j4 is pe o med.
acco ding o Eq. (3.13).
The coupling o he wo angula momen a j, and jz,
pe o med abo e, can be a oided by no icing ha he
su" '(l, l) can be gene a ed by
E'+ =b,bz, E' =b(bz, Ep =—,
'(N, +Nz+1) .(3.29)
The ec o s
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)
whe e on he igh -hand side he second and hi d e m
co espond o he i s - and second-o de Casimi ope a-
o o he u(1) algeb a o he chain o Eq. (3.2), while he
ou h and i h e ms co espond o he i s - and
second-o de Casimi ope a o s o he u(1) algeb a ap-
pea ing in he chain o Eq. (3.16). The eigen alues o his
Using Eqs. (3.14), (3.9), and (3.27), which connec he
quan um numbe s m, j,j,z3 o he p e ious ones (N, co,L),
i is easily e i ied ha Eq. (3.35) is an al e na i e way o
w i ing Eq. (2.5}.
Simila ly he Hamil onian o chain II, gi en in Eq.
(2.8), can be ew i en, aking in o accoun he com-
plemen a i y ela ions gi en in Eqs. (3.1},(3.2), (3.15), and
(3.16), as
1092 R. N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50
Hamil onian a e
E(m, m, 23 J)23)= eP+ eIm+e2m +e m)23+a'm(23
one can p o e ha su" '(1,1) is gene a ed 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 connec
m, m,23,j,23 o N, n~, L, i is easily e i ied ha Eq. (3.37)
is an al e na i e way o w i ing Eq. (2.9).
In his sec ion we ha e he e o e ew i en he bases
and he Hamil onians co esponding o he wo dynami-
cal symme ies o he ib on model in e ms o comple-
men a y subalgeb as. This o mula ion is use ul because
i can be qde o med in a e y simple way.
IV. qDEFORMATION OF THE VIBRON MODEL
he ele an commu a ion ela ions being
[K(12) K(12) )+K(12)
0~+j—+
[K(12) K(12) ]= (2K()2) ]
The ec o s
~j»m»), =(a)) '(a2) '~0),
V'[N(]q)[N2)q(
(4.10)
(4.11)
The co esponding Hamil onians we e hen w i en in
e ms o he Casimi ope a o s o he new educ ion
chains. An e iden possibili y o qde o ming hese
Hamil onians is o subs i u e he su(1, 1) algeb as o Eq.
(4.1) by hei q-de o med coun e pa s su (1,1) [38—
40],
su" '(1,1)Dsu" '(1,1)Dsu" '(1, 1)ZU (1) .(4.2)
In his sec ion we shall explain how his can be achie ed,
a e gi ing ab ie accoun o he necessa y ma hema ical
de ails.
qnumbe s a e de6ned as
(4.3)
Fo q eal (q =e'wi h eal), hey can be w i en as
sinh x
sinh~ (4.4)
while in he case o qbeing aphase (q =e"wi h eal),
hey ob ain he o m
sin x
qsin7- (4.5)
In he limi ing q~ 1( ~0), qnumbe s educe o usual
numbe s.
q-de o med oscilla o s [41,42] a e in oduced h ough
he ela ions
aalu —
q—
'a 'a =q, [N, a]=a, [N,a]=—
a, (4.6)
whe e aand aa e he q-de o med boson c ea ion and
annihila ion ope a o s and N he ele an numbe ope a-
o . Using Eq. (4.6) one can easily show ha
aa=[N]~, aa =[N+1]~ .(4.7)
q-de o med algeb as can be exp essed in e ms o q-
de o med bosons. In oducing a, as he q-de o med ana-
logs o b; (i =1,2, 3,4), wi h he p ope ies
[a ,a„"]=[a,
a„]=[a„,
a„]=0, ip, (4.8)
In he p eceding sec ion he subalgeb a chains o he
ib on model we e educed o equi alen chains o com-
plemen a y subalgeb as
su" '(1 1)~su" '(1 1)~su" '(1 1)~u(1) .(4.1)
wi h j,2, m)2 s ill gi en by Eq. (3.31), a e eigen ec o s o
he de o med Casimi ope a o
C(su" '(1 1))= K" 'K
—
''+[K" '] [K" '—
1]
(4.12)
(4.13)
whe e Nis he numbe o bosons. These gene a o s
sa is y he commu a ion ela ions
[K(),K+ ]=+K+ [K+ K]= [2K()]2.(4.14)
We a e no going o use he su'(l, l) and su (1,1) alge-
b as explici ly in couplings, since o he su" '( l, l) alge-
b a we al eady ha e he o m gi en in Eq. (4.9), which
a oids he di ec coupling. In o de o be able o couple
su (1,1) and su (1,1) o su" '(l, l), i is use ul o ha e
he same de o ma ion pa ame e in all o hese algeb as,
i.e.,i is use ul o ha e he same de o ma ion pa ame e
in he commu a ion ela ions o Eqs. (4.10) and (4.14). In
o de o achie e ha , we eplace in Eqs. (4.13) and (4.14)
qby q. As a esul , o =3,4, Eq. (4.6) is mean om
now on wi h q eplaced by &q. Then one also has
a~„=[N,]&—, a,,a„"=[N,+I]&- .(4.15)
Equa ion (4.12), gi ing he Casimi ope a o , is he e o e
alid in his case wi h he usual qnumbe s.
The su" '(1,1) algeb a is gene a ed by he ope a o s
g3 g(12)
+(123) +(12) o+~3o
+q+
~(123) ~(12)+~3
00
(4.16)
112+
i.e.,i is as anda d cop oduc o he i eps Dand
D'o he su (1,1) algeb a. The e o e he basis ec-
o s, in analogy o Eq. (3.25), a e o he o m
wi h eigen alues [j)2]~[j)2+1]».
Fo he su'(1, 1) algeb as one has he boson ealiza ion
[38]
K' =aa, K" =aa, K() ='(N +-'),—
11

50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1093
Ij)2j3:j)23m)23 &q
while he ec o s analogous o Eq. (3.33) a e
Ij)2J3(J123 }J4:Jm &
&j)23m)23j4m4IJm &sU, )1,1)
m123 m4
xIj)2j3.j)23m]23 &ql j4m4&, .(4.19)
These ec o s a e he qanalogs o he eigen ec o s o he
dynamical symme y Io he ib on model. Simila ly he
ec o s
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„„„a e Clebsch-Go dan
coe icien s o he enso p oduc o wo su (1,1) i eps.
Explici analy ical o mulas o hese coe icien s, as well
as o he ele an su (2) coe icien s, can be ound in
[43—
47].
The su"234'(1, 1)algeb a is gene a ed by he ope a o s
~4 g(123)
~(1234) ~(123) o+~4 o(4.18)
ha e been used. Using Eq. (2.6), Eq. (4.23) can be w i en
as
E(N, , L)=6p+E') [N +2]~—+e2[N +2]~—
+A —
— —
1——
22
+B'[L]~ [L—
+1]~
—, (4.26)
which educes o Eq. (2.7) in he limi q~l, up o a
ede ini ion o E'0
In he case o he dynamical symme y II he Hamil-
onian can be w i en as
H)) =~P+~)[m]q+~2[m]q'+~[m)23)q
+a[m)23]q+PC2(suq) '( l, l)), (4.27)
which is he qanalog o Eq. (3.36) (wi h he p imes o he
coe icien s d opped). The eigen alues o his Hamil oni-
an a e
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 he limi q~1,Eq. (3.37) is ob ained. Assuming ha
m, m,23,j,23 a e connec ed o N, n~, L h ough Eqs. (3.14),
(3.28), and (3.27), Eq. (4.28) can be ew i en in away
esembling i s classical coun e pa , Eq. (2.9), as
a e he qanalogs o he eigen ec o s o he dynamical
symme y II o he ib on model. The e o e Eq. (4.19)
connec s he eigen ec o s o he wo dynamical sym-
me ies, as Eq. (3.33) does in he classical case.
In he case o dynamical symme y I he Hamil onian
eads
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 esul s ob ained in his sec ion call o he ollow-
ing commen s.
(i) Ro a ional- ib a ional spec a o dia omic molecules
a e desc ibed empi ically by he Dunha n expansion [48]
which is he qanalog o Eq. (3.34) (wi h he p imes o he
coe icien s d opped). The eigen alues o his Hamil oni-
an a e 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 he limi q~ 1, Eq. (3.35) is ob ained. Assuming ha
m, j,j123 a e s ill connec ed o quan um numbe s N, co,L
h ough Eqs. (3.14}, (3.9), and (3.27}, Eq. (4.22) can be
ew i en in away esembling i s classical coun e pa ,
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 p oducing Eq. (4.23), iden i ies such as
H) =a)+DC2(su" '(1 1))C2(su" '(1 1)) (4.31)
Then Eq. (4.26}is modi ied as
whe e is he ib a ional quan um numbe , L he angu-
la momen um, and Y;k he Dunham coeScien s, i ed o
expe imen . I is clea ha he Dunham expansion con-
ains powe s o ( +—,
'), powe s o L(L +1), as well as
c oss e ms. Equa ions (4.23) and (4.29) con ain no c oss
e ms. This is due o he ac ha in he Hamil onians o
Eqs. (4.21) and (4.27), only e ms up o quad a ic in he
gene a o s a e included, as in he case o he classical ib-
on model. C oss e ms can be aken in o accoun in he
dynamical symme y I, o example, by modi ying Eq.
(4.21) as ollows:
q
—
[](1/2+ —
1/2) —
1(4.24) E'(N, , L)=E(N, u, L)+D' —— N
—
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) Ro a ional spec a in bo h dynamical symme ies
[Eqs. (4.26) and (4.29)] a e desc ibed by he e m
[L]& [L—+I]&—. This is known o be he Casimi
ope a o o su&—
(2). The su (2) model has been ex en-
q
si ely used o he desc ip ion o o a ional spec a o di-
a omic molecules [5—
7] and de o med [8—
10] and supe -
de o med [11]nuclei. I has been ound [9] ha his e m
is equi alen o an expansion in e ms o powe s o
L(L +1),
[L]q[L+1] =(jp( )L(L+1) j i( '){L(L+1)j +~ j2( )jL(L+1)]3
l
jjp«)]' —
—
', j( )jL (L +1)] +,', j~—( )jL (L +1)] —.), (4.33)
whe e j„( )a e he sphe ical Bessel unc ions o he i s
kind and q=e". This expansion is simila o he one
con ained in he Dunham expansion. In he case o
su (2), howe e , all he expansion coe icien s a e ela ed
o powe s o ~, hus esul ing in economy o pa ame e s.
No ice ha he dec easing o he coe icien s o inc easing
powe s o L(L +1),as well as he al e na ing signs o he
e ms, ac s ha a e known empi ically o hold, occu in
Eq. (4.33) au oma ically, since is known [5—
11] o ob-
ain small posi i e alues. Fu he mo e, i has been
p o ed [9] ha he su~(2) model is equi alen o he a i-
I
able momen o ine ia (VMI) model, which desc ibes o-
a ional s e ching e ec s. The qpa ame e has been
ound [9] o co espond o he so ness pa ame e o he
VMI model. The implica ions o he su (2) model on he
elec omagne ic ansi ion p obabili ies connec ing he
o a ional le els o nuclei ha e been conside ed [10].
(iii) The ou h e m in Eq. (4.26) co esponds o he
Casimi ope a o o su (1,1), al eady used [17] o he
desc ip ion o ib a ional spec a o dia omic molecules.
I has been p o ed [17] ha his e m, o q=e",can be
expanded as
N
U2u—
1——=j—,
'(cos( ) —
cosj (N+2)]) —
sinj (N+2)](u+ —,
')
sin( )
+Hcosj (N+2) j( +—,
') +—
2 sinj (N+2)}(u+ —,
')
—
—,
' cosj (N+2)]( + —,
') +(4.34)
We ema k ha ase ies o powe s o ( +—,
')is ob ained,
simila o he one con ained in he Dunham expansion.
In he p esen case, howe e , he expansion coeScien s
a e all ela ed o (and N, which in he ib on model is a
cons an o agi en molecule), hus esul ing in economy
in pa ame e s.
(i ) The anha monici y cons an (i.e., he a io
Y2p /Yip )in he classical case [Eq. (2.7)] is ixed o
—
1/(N+2). In he de o med case o Eq. (4.26), howe -
e , i is equal o —
/ an[ (N+2)], as i is easily seen
om he expansion o Eq. (4.34). The ex a eedom
gained his way has been ound [17] o imp o e he i s o
ib a ional molecula spec a.
( ) Since Nis ixed o agi en molecule ( ela ed o he
maximum numbe o bound s a es below he dissocia ion
limi ), he i s h ee e ms in Eqs. (4.26) and (4.29) ha e
no inhuence on he spec um.
( i) In Eq. (4.26) i is clea ha he de o ma ion pa am-
e e o he ib a ional pa o he spec um is ~, while
o he o a ional pa i is /2. The e o e a ela ion is
implied be ween he o a ional s e ching and he anha -
monici y co ec ions. Ca e ul empi ical i s a e needed in
o de o decide i his is a es ic ion o an ad an age o
he p esen model. The e is no ap io i eason, howe e ,
ha hese wo physically di e en mechanisms be de-
sc ibed by he same pa ame e . Amo e gene al e sion
o he model, allowing o hese wo de o ma ion pa ame-
e s o be independen o each o he , migh gi e be e e-
sul s.
V. DISCUSSION
In his pape ade o med e sion o he O(4) and U(3)
dynamical symme ies o he ib on model o dia omic
molecules has been cons uc ed. This has been achie ed
by i s ew i ing, h ough he use o he concep o com-
plemen a y subalgeb as, he model in amo e con enien
o m, which is subsequen ly de o med. The p esen ap-
p oach uni ies in o acommon amewo k he so a
sepa a e algeb aic app oaches o o a ional and o ib a-
ional spec a o dia omic molecules.
Fo he O(4) limi o he p esen model, i ings o ex-
pe imen al da a o dia omic molecules a e equi ed. I s
U(3) limi can be used o he desc ip ion o clus e ing
phenomena in nuclei [49], as well as o he quasimolecu-
la desc ip ion o hea y-ion esonances [50]. The p esen
wo k can be ex ended o he s udy o ia omic molecules
[23]. The me hod o complemen a y subalgeb as can also
be used in cons uc ing [30] he de o med e sions o he
U(5) and O(6) dynamical symme ies o he in e ac ing
boson model [24] o nuclea s uc u e.
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