Chemical Physics 106 (1986) 1-9

North-Holland, Amsterdam

THE ALGEBRAIC HAMILTONIAN FOR DIATOMIC MOLECULES IN THE VIBRON MODEL

S.K. KIM ‘, I.L. COOPER ’ and R.D. LEVINE

The Fritz Haber Research Center for Molecular Dynamics, The Hebrew Vnibersity, Jerusalem 91904, Israel

Received 3 February 1986

A quantitative description of the higher-lying vibrotational energy levels of diatomic molecules requires going beyond the lowest-order vibron hamiltonian. A systematic expansion to higher orders which is simple to derive and apply is discussed.

The essential point is that only such generators of U(4) or powers thereof, that transform under rotation as scalars can appear in the hamiltonian. Higher-order scalars are generated as the products of scalars and scalar products of vectors. The resulting

expansion for the rotationally invariant hamiltonian is equivalent to that derived using the spherical tensors formalism. Applications to H, are described.

1. Introduction

The recently introduced algebraic approach proves effective in describing molecuk vibration and rotation energy levels [l-7]. It is therefore of interest to apply the method so as to approach the level of precision available from spectroscopic techniques or from conventional, geometrical (i.e. differential equation) numerical computations. Al- ready at the lowest order, the energy levels pro- vided by the algebraic approach go beyond the familiar harmonic approximation. However, to obtain real quantitative accuracy it proves neces- sary to account for vibration-rotation coupling. This is particularly the case for diatomic mole- cules for which a numerical solution of the radial Schriidinger equation * is readily performed.

In the algebraic approach [8] the starting point is the (closed) set of generators G, of some Lie group G [9,10]. The hamiltonian is expanded as a

’ Permanent address: Department of Chemistry, Temple

University, Philadelphia, Pa 19122, USA. * Permanent address: School of Chemistry, The University,

Newcastle upon Tyne, NE1 7RU, UK.

* Such an approach requires, however, that the potential be

provided as an input. Only for the simpler diatomic mole- cules are such potentials available from ab-initio computa- tions to spectroscopic accuracy.

power series in terms of these generators

H= &;G;+ &xijGiGj+ . . . . (1) , i.j

Group theory enters the computations in two es- sential ways. The first is in providing a convenient basis in which the hamiltonian (1) can be di- agonalized [ll]. The other is in delineating special forms for the expansion of the hamiltonian. This is based on the introduction of Casimir operators of subgroups of G.

For a vibrating-rotating diatomic molecule, the group U(4) has been proposed [2-41 and applica- tions thereof are known as the vibron model. The group U(4) has sixteen generators which can be conveniently realized in terms of boson creation and annihilation operators [9,10,12,13],

(G;} = {ata,; Y, j~=l,..., 4}, (2)

[a”, a;] = 8,,, [a,, a,] = 0, [ut, ut] = 0.

With sixteen generators, the use of a systematic procedure for generating the higher-order terms in the expansion (1) for the hamiltonian is essential. This is particularly so since we are dealing with a system where the angular momentum is a good quantum number so that rotational invariance of the hamiltonian must be ensured. Indeed, a proce-

0301-0104/86/$03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

2 S. K. Kim et al. / Diatomic molecules in the oibron model

dure which achieved both aims and which is based on spherical tensors formalism has been described

and implemented [2,3]. In this paper we achieve the same goals through geometrical considerations which eliminate some of the complexities at the price of having a less powerful but for our pur- pose, sufficient, tool.

In section 2 we consider the transformation properties of operators under rotation, using the Schwinger representation. It will become clear that for our purpose it suffices to consider operators which behave as scalars or as vectors under spatial rotations and reflections. Using these operators it is possible to generate (section 3) in a systematic fashion, scalars which are of increasingly higher order in the boson operators. In section 4 the

hamiltonian is then expanded using these scalars. Numerical applications to vibrational levels of H, are provided in section 5.

There are two appendices. Appendix A estab- lishes the connection with the spherical tensors, previously employed [2-51 in the algebraic ap- proach. Appendix B expresses some useful oper- ators in the present notation.

2. Classification of boson operators

With a given n x n matrix T we can associate an operator ?, bilinear in the boson operators by the definition

?=A+TA. (3)

Here A is a column vector of n annihilation operators and At is the corresponding row vector,

A’= (a!, a; ,...) aI;>. (4)

The Schwinger representation [9,10,12,13] intro- duces a set of matrices T which satisfy the same communication relations as those of an abstract Lie algebra. For example, for the SO(~) subalge- bra * of u(4) which generates the infinitesimal rotations in three dimensions, the three matrices

* We see the convention (e.g. ref. (lo]) that low case letters are used for the algebra of a given Lie group.

are given by

Jr=[i 8 %;i, Jz=(yi i ii,

0 . 0 J,= i

i 1

i’ 0 . (5) 0 0 0

The Schwinger representation of the correspond- ing operators is given by (3) or, explicitly, as

j, = -i(a -aLa,), (6)

where i, j and k are a cyclic permutation of 1, 2, 3.

Given four boson operators a,, Y = 1,. . . ,4 and their adjoints it follows from (6) and (2) that

[j,, a!] = iCcijkai,

[ 4, a,] = ii.,jka,; k

[jl, 41 =o, [ 8, a,] = 0,

(7)

where eJjk is the Levi-Civita symbol *. The J, are the generators of infinitesimal spatial

rotations and hence (7) establishes that a4 and ai

behave as scalars and (a,, a,, a,) and (a!, ai, af) as vectors under rotations. It is convenient to use the notation **

a; = .+

(al, a;;

a4 = 6,

af) = ,+ , (a,, a2r +)=r. (8)

It follows from the transformation properties (7) that wt and rr belong to the same vector space. Hence one can introduce new scalars using the scalar product of vectors. The three scalars which

* For any right-handed vector basis e, X ej = &$,jkek so that c,,~ is non-zero only if i, j, k is a permutation of 1, 2, 3.

**The vector R as used here has three Cartesian components Q, = a,, 4 = a2 and 3= a3. It is closely related to the spherical tensor of rank 1 which was used in refs. [2,3] whose components are s+ Ir n,, n_,. The linear transfor- mation between these two vectors is provided in appendix

A.

S. K. Kim er al. / Diatomic molecules in the vibron model 3

are bilinear in the boson operators are

rrr+=E,+.s+ ) Tr==s*R, t Tl l w. (9)

In terms of rt and rr the generator j of infinitesi-

mal rotations (i.e., the angular momentum) is given by the vector product

,. iJ=r+Xn. (10)

For several purposes an explicit geometrical realization of the operators is useful. In terms of the coordinates of a four-dimensional Cartesian space

.+=x Y “, a, = tl/ax, = ip, (11)

or

a: = (x, - ip,)/Jjl, a, = (xv + ip,)/fi,

where Y= I..., 4. Either realization shows ex- plicitly that a+( = ~1) and (I transform as scalars while ,+ and II transform as vectors with proper parity with respect to spatial rotations * and re-

flections. We have restricted attention in this section to

infinitesimal rotations which is all that is required to ensure rotational invariance (i.e. [ri, .Q = 0) of the hamiltonian. However, the Schwinger repre- sentation proves also to be very convenient in discussing finite transformations. We plan to pursue this application in a separate publication.

3. The vibron hamiltonian

In the vibron model, a vibration-rotation state of a diatomic molecule is described using N bo- sons (the, so-called, vibrons) distributed amongst four single particle states. The four creation and annihilation operators {at, uvr Y = 1,. . . ,4} act on these states. Since the hamiltonian (1) contains

terms at least bilinear in the boson operators (cf. (2)) the single-particle energy levels will be split into a finite but in general anharmonic progres- sion of levels. By nature of the model, the hamilto-

* Using (10) one readily verifies that the generator j, used in the transformations (7) is realized as the angular momen-

tum in the spatial dimensions j = x X p.

nian must conserve the total number, A, of vibrons. The number operator is defined, as ex-

pected, by

fi=iija+;,,

A, = a+u, ii,=,+*,. (12)

In the Schwinger representation, the matrix N is the 4 x 4. unit matrix. The wavefunctions of the model must (by definition) be totally symmetric

under permutation of the vibrons. The hamiltonian of the vibron model must then

be hermitean, transform as a scalar under rotation and conserve &. It can thus be written as

H = H(O) + H(l) + H(=) + Hc3’ + . . . , (13)

where H(“) is a linear combination of operators from the set SC”) of linearly independent scalars which are of the n th order with respect to terms bilinear in vibron operators given by

S(O) = {constants},

9’) = { 2,)

Here [X, Y], = XY + YX denotes the symme- trized product of X and Y. To arrive at the terms above we have started with the only elementary scalar’s up to terms bilinear in the boson oper- ators. These are u and ut and A, = utu, A, = rrt l

a, rt2 and w2. The requirement that fi is con- served eliminated (in So’) all except ii, and A, and, for a given N one can replace it, by N - A,. All the number conserving hermitean scalar oper- ators which are linearly independent are given in (14) up to the fourth order. To this order, all other such scalars are simply different in the order of the boson operators and hence linearly dependent on those given in (14) through the boson commu- tation relations. The sets of scalar terms are re-

4 S. K. Kim et al. / Diatomic molecules in the vrbron model

lated by: and that the terms in (13) can be written as

G3’ = ([S:“, qc2)] +; all i, j),

sC4’ A ( [ p, y ] +, [s/p, s/q +;

all i, j, k, I),

. . (15)

In evaluating the symmetrized product we have used

,+%2=A,(A,+ 1) -j2 (16)

and that .?’ commutes with all scalars. For later use we also note that

H”’ = e fi 1 77)

H(2) = e j-j2 + e j2 + e 2 n 3

3 4 3

Hc3’ = e,A: + e,A, J2 + e7 [ A,, D’] +,

Hc4’ = e,A;) + e9( .?2)2 + elo(b2)2 + e,,Aij’

+e,,.f2b2 + e,,[Rf, b’] +, (22)

or, directly in terms of Casimirs, up to third order

H = h, + h,@U3) + h2e2(U3) + h3e2(S03)

+h4e2(S04) + h5~,(U3)~2(U3)

+h,QU3)~z(S03)

rt2u2 + .t27r2 = b2 - A,(& + 1) - A&, + 3), +h7[E1(U3), e2(S04)] +,

(17)

where 6,

i, = r+u + a+7r, (18)

is a “dipole” (i.e. vector-like) operator. We have eliminated in (14) those scalars which are linearly dependent on others. Hence there is no additional term from [Si”, Sj”]+ in Sc4’.

where h, up to h, are linear combinations of the parameters e, up to e, which appear in (22). The reader should have no difficulty in adding the fourth-order terms (and beyond) to the expansion

(23).

4. Spectrum

There are two group chains [2,3]

u(4) 1 so(4) 3 SO(3) 1 so(2)

and

(19a)

u(4) 3 u(3) 3 SO(3) 3 SO(2) (19b)

of U(4). The Casimir operators [9,10] of the sub-

groups are [2,3]

e,(u3) = A,,

e2(u3) = A,(?, + 3),

eZ(so3) = ?,

e*(so4) =b2+.?. (20)

The Casimir operators provide an alternative ex- pansion for the hamiltonian. Using (16)-(18) and

We have seen that the most general vibron model hamiltonian can be written in terms of the Casimir operators alone. However, these are oper- ators belonging to two different group chains. Hence, in general, the spectrum need be computed by a numerical diagonalization, cf. eq. (31) below. There are two special cases where the spectrum can be determined analytically [2,3]. These corre- spond to an expansion in powers of Casimir oper- ators of a particular group chain. For chain I, up to fourth order

H = h, + h3e2(S03)

b’+j~=A(A+2)-($7+2-cT+~)(TrGr~)

(21)

one verifies that this is an equivalent expansion

+h4e2(S04) + h,[e2(S03)]2

+h,,(~2(S04))2h,2~2(S03)~2(S04). (24)

Here there are no terms of the first or third order (i.e. from the set So’ or Sc3’, cf. (14)). In higher orders, all odd orders will be absent.

The eigenstates of such a hamiltonian can be labelled [3,11] ( NwJM) where N is the vibron number, the SO(4) representation quantum num-

S. L Kim et al. / Diatom? molecules in the vihron model

ber w,

i;(SO4)]NwJM)=u(w+2)]NwJM), (25)9

taking the values u = N, N- 2,. ..,l or 0 for N odd or even and the usual angular momentum quantum number J and its projection M, - 1 G N ( J, characterizing the representations of SO(3) and SO(2), respectively. To make contact with the standard notation for vibrational spectrum it is convenient to define the vibrations quantum number u by u=(N-w)/2. Hence, u=O, l,..., N/ 2 or (N - 1)/2 for N even or odd. For this chain the vibron number N determines both the vibrational quantum numbers of the highest bound state (i.e. N/2 or (N - 1)/2 and, through the condition J Q w = N - 20, the rotational quantum number of the highest bound rotational state of any given vibrational manifold of states.

The eigenvalues of the Casimir operator of SO(4) can be written in terms of the vibrational quantum number Y as

= (N + 2)2(1 - 2x6)’ - 1,

u=u++, x=l/(N+2). (26)

Thus, the energy eigenvalues of the hamiltonian (24) are given by the Dunham-like expansion

E = hb + h,J( J + 1) + h;(l - 2x6)’

+~~~J(J+l~~‘+~~~(l -2x$

+A;,(1 - 2xi$J( J -t 1). (27)

Here the coefficients hi are related to those in (24) in an obvious fashion. In (27) and in higher-order terms (1 - 2x6) appears only in even powers, cf. (24). Hence the leading term in the vibration-ro- tation interaction [14] which depends linearly on E; and on J( J + 1) is inherently absent. The expan- sion (24) is thus, in general, too restrictive.

The eigenstates of the group chain II can be labelled 1 Nn,,JM) where n_= N, N- l,...,O and J=n,,n,-2 ,..., 1 or 0 for n, odd or even. The hamiltonian corresponding to this chain, up

to fourth order, is

H = h, + h,A, + h,ii; + h,j2 + h,A;t

+h,A,j’ + h,fi; + h,( j2j2

+h,&j2

and its eigenvalues are given by

E = h, t h,n,, + h,n; + h, J( J +

cw

1) + h,n;

+h~n~J(J+l)~h~n~+h~~J(J+l)~2

+h,,n;J(J+ I), (29)

where O<n,<N and O<J<n,,--2k with an integer k.

Up to any given order, the expansion of the ha~ltonian using group chain II has more flexi- bility in that it has more parameters but has the strong restriction that J must be even (or odd) when n, is. The identification [3] of the vibra- tional quantum number u by

n,=J+2u (30)

satisfies the restriction and can be interpreted in terms of a totally non-rigid molecule [15]. With the identification (30) the resulting vibrotational spectrum leads, however, to rotational spacings which are far too wide for typical diatomic mole- cules. In general therefore it is necessary to use the most general vibron hamiltonian, e.g. eq. (22), and to find its eigenvalues by numerical diagonaliza- tion. Because most terms in (22) are Casimirs of chain II, the eigenstates of this chain offer a very suitable basis. Indeed, only fi’ is non-diagonal:

rt2u2(N, n,,, J, M)

= {(N-n,)(~- II,-l)(n,+J-t=3)

X(n,- J-i- 2))“’

IN, n-+2, J, M)

and

u%r2 (N, n,, J, M)

= {(N-n,+2)(N-n,+i)

x(n,+J+l)(n,-J))“2

IN, n,,- 2,-J, M). (31)

6 S.K. Kim et al. / Diutomic molecules in the vibron model

5. Application

A fit of the hamiltonian for chain I up to second-order terms to the computed [16,17] vibro- tational spectrum of H, has been previously con- sidered [2,3]. A reasonable fit was obtained even at this, lowest, level of approximation and all qualitative aspects of the spectrum were correctly reproduced. There are three parameters in such a

fit (i.e. the frequency, the anharmonicity and the rotation constant). In the notation of eq. (27) the

three parameters are h,, h, and x = l/( N + 2). We shall compare our results with those derived from a Dunham-type expansion

E=CCYjfi’[J(J+l)]‘. (32) i j

For this simplest case there are only three non- vanishing Dunham coefficients, i.e. Ya,, Y,, and Y,,. One readily verifies that [2,3] Y,,/Y,, = -(N + 2).

For H, in its ground electronic state, the highest bound vibrational state is [16,17] u = 14. Since the highest possible value for u is N/2 or (N - 1)/2 for N even or odd it follows that N >, 29. For N = 29, Y,,/Y,, = - 31 versus the “experimental”

ratio [18] of - 37.28. A second objection to N = 29

is that the highest value of J for a given u is N - 2u and the assignment of N = 29 excludes quite a number of possible [16,17] high-lying rota- tional states. By choosing N = 35 we not only include all known rotational states up to the dis- sociation limit but also come closest to the experi- mental value of Y,,/Y,,.

Higher-order corrections are best introduced in two stages. In the first, we retain the restriction to chain I (so that the dependence of the eigenvalues on the quantum numbers u and J remains ana- lytic and explicit) but include higher-order terms in the hamiltonian. There are no third-order terms in chain I so that the first improvement of the first stage is the fourth-order hamiltonian as given by (24). The corresponding eigenvalues are given in (27). The resulting five-term expansion is found to be somewhat inferior (cf. table 1) to the corre- sponding five-term Dunham expansion (in which the sum of the exponents i and j is less than or equal to 2). However, the spectrum (27) was geri- erated from an explicit hamiltonian, given by eq. (24) and the eigenstates are also explicitly known. Hence other properties and relevant matrix ele- ments can be computed. This is not the case for the Dunham expansion which is an empirical fit to the energy spectrum.

Table 1

Expansion coefficients, chain I, dynamical SO(4) symmetry ‘)

G(SD4) J(J + 1) [C*(SO4)]2 lJ(J+r)l* C,(SO4)J(J + 1) C,(SW]J(J + 1)12 lJ(J+1)13 rms deviation

- 29.14 27.31 _ - 0.02262 0.02287 _ _ 0.2227

- 28.58 27.02 - 6.60E - 4 - 0.02241 0.02292 _ _ 0.2067

- 28.81 22.68 -5.93E-4 -0.01191 0.02652 -8.8E-6 _ 0.1914

- 29.64 25.70 _ - 0.03292 0.02139 _ l.O7E- 5 0.1764

- 28.47 30.39 - 6.63E - 4 - 0.03270 0.02143 _ l.O7E-5 0.1559

- 28.78 24.91 -5.68E-4 - 0.03336 0.02619 -1.22E-6 1.22E-5 0.1153

‘) rms deviations are the natural logarithm of the rms deviation of the bound levels (in cm-‘) from LeRoy’s tabulation. The rms

deviation for a five-term Dunham expansion is 0.1852.

Table 2

Expansion coefficients for symmetry breaking with U(3)

c* (SD4) J(J+l) C*(So4)J(J + 1) lJ(J+1)12 ]G(SD4). n,l+ n,lJ(J+t)l lJ(J+r)13 rms deviation

- 29.74 26.99 0.02298 -0.02279 - 0.01104 _ 0.2227

- 32.13 31.82 0.01920 - 0.02265 0.0625 _ _ 0.1778

- 32.46 22.77 0.02204 - 0.02277 0.0692 0.3294 _ 0.1739

- 32.05 31.18 0.01971 - 0.02218 0.0602 _ -3.6E-7 0.1753

S. K. Kim et al. / Diatomic molea& in the vibron model 7

One can go to even higher-order hamiftonians of the chain I and the resulting improvements in the fit to LeRoy’s [17] levels is summarized in table 1. As is evident from table 1, it is possible to have a five-term fit which is somewhat more accu- rate than a five-term Dunham by using terms of the sixth order. By including all fourth-order terms and several sixth-order terms (there are no odd- order terms), the convergence is reasonable.

In the second stage we no longer restrict the ha~ltonian to a given dynamical symmetry. It is then most convenient to diagonalize the hamilto- nian in the U(3) basis where the only non-diago- nal terms are those containing a2. The necessary matrix elements are given in eq. (31). Some results obtained in this way are given in table 2. As is clear, on the level of retaining only second- or third-order terms, breaking the dynamical symme- try is beneficial. Note in particular that the oper- ator [C,(SO4), n,]+ which couples the two chains is particularly useful in improving the fit. The terms which originate purely in chain II such as n, or nn -J( J + 1) contribute far less.

6. Concluding remarks

A simple, systematic derivation of the vibron hamiltonian has been discussed. In applications it is found that the hamiltonian is capable of de- scribing vibrotational levels of simple diatomic molecules like H, to an accuracy comparable to that provided by a Dunham expansion. The ad- dition~ advantage is that the h~lto~~ is ex- plicit and its eigenstates are available.

Acknowledgement

We thank F. Iachello, R. Gilmore, OS. van Roosmalen and C.E. Wulfman for useful com- ments and discussion. SKK and ILC thank the members of the Fritz Haber Research Center for their hospitality extended to them during the course of this work. SKK wishes to thank Temple University for supporting his stay in Israel. The visit of ILC to Israel was under the auspices of the Royal Society-Israel Academy of Science ex-

change agreement. This work was partly sup- ported by the Stiftung Volkswagenwerk. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fti die Forschung, mbH, Munich, FRG.

Appendix A: Spherical tensors

Spherical tensors have played a central role in many earlier discussions of algebraic hamiltoni- ans. It is therefore worthwhile to establish the relation with the pedestrian approach used here.

The 21+ 1 components of an Ith-order irre- ducible spherical tensor, ?‘I, s = I, I,. . . , -I-t- 1, - /, are defined by the communication relation

[I*, T,“‘] = [/(f+l)-11(~il)11’2T::),,

(A-1)

c_ * where j, = J, f i Jy. From the coruscation re- lations (7) we can write down immediately the spherical tensors of the zeroth and first order

.+=a,, t

t s+1= -(aT-!-i6zi)/fi,

vJ=i$,

lri, = (+- iui)/fi. (A-2)

Again, from the definitions (A-l), the spherical tensors constructed from the annihilation oper- ators given by

IT = -(-l)‘a_,, a”=a P (A.31

or

p+1= (a, + i+)/j=2, is,= -as,

ii_, = -(a, - ia,)&?.

In the coordinate realization (lla) t “+* = - (xi + i-&fi,

Z+I = (ipi -P,)/v%

rJ=+,

8 S. K. Kim et ai. / Diatomc moIeades in the oibron modef

So= -ip,,

7rt, = (x, - ix,)/6 ;

71-, = -(ip, +p,)/fi. (A.4)

These are clearly vector operators since their signs change under reflection.

The realization (A.4) points us to the generali- zation of (A.2) and (A.3) to the Ith order

T+“) = Y (L$, fri ai) la i.P 3 3 (A.51

,;f’)= (-l)‘+‘Y,._,(a,, u2, a,),

where the Y,r(~l, x2, x3) are the solid harmonics in the carte&n coordinates x,, x2, x3.

The k th tensor constructed as a product of two tensor operators T’” and 7”“” is denoted by [T”) x 7’(“‘]$k) where [19,20]

The coefficients in (A.6) are the Clebsch-Gordan ones. In particular for the scalar product one has

(7-C’). p) s ( I I)‘(21 + I)““[ T”’ x pqp

= z: ( - l)“zpS’_‘~. (A.71 I”

The spherical tensors corresponding to the angular momentum 1 introduced in (6) and to the dipole operator b defined by (18) are written in this notation as

k;“=[,+Xa+a+Xii]:“,

while for the scalars

(A.81

n, = [ .+ x a$‘,

n, = J7[ ?7+ x #‘. (A.9)

Replacing the scalars and vectors in (22) by their spherical tensor components will lead to the form of the hamiltonian as discussed in [2,3].

Appendix B

For the sake of completeness, one may write down the transition moments in the present nota- tion. The vector transition operator, which is responsible for emission and adsorption of the dipole radiation, is given by, up to the two-body terms,

where t,, t2 and t, are free parameters. For the quadrupole transition operator we have the fol- lowing symmetric traceless tensor

Q = $( n+lr + w+) - 4(2ci, + I)l, (B-2)

written in the dyadic notation. In [3], the expan- sion for T has a fourth term which can be written as [Q, D], which, however, is linearly dependent on the three terms in (B.l).

References

[l] R.D. Levine and C.E. Wulfman, Chem. Phys. Letters 60 (1979) 372.

[i] E. Iachello, Chem. Phys. Letters 78 (1981) 78. ]3] F. Iachelfo and R.D. Levine, J. Chem. Phys. 77 (1982)

3046. [4] O.S. van Roosmafen, Ph.D. Thesis, Groningen, The

Netherlands, (1982). [5] OS. van Roosmalen, A.E.L. Dieperink and F. Iachello,

Chem. Phys. Letters 85 (1982) 32. [6] OS. van Roosmalen, F. Iachello, RD. Levine and A.E.L.

Dieperink, J. Chem. Phys. 79 (1983) 2515. [7] OS. van Roosmalen, I. Benjamin and R.D. Levine, J.

Chem. Phys. 81 (1984) 5986. [8] A. Arima and F. Iachello, Advan. Nucl. Phys. 13 (1984)

139. [9] B.C. Wyboume, Classical groups for physicists (Wiley,

New York, 1974). IlO] R. Giimore, Lie groups, Lie algebras and some of their

applications (Wiley, New York, 1974). [ll] A. Frank and R. Lemus, .I. Chem. Phys., to be published. [12] J. Schwinger, in: Quantum theory of angular momentum,

eds. L.C. Biedenharn and H. van Dam (Academic Press, New York, 1965).

1131 L.C. Biedenham and J.D. Louck, Angular momentum in quantum mechanics (Ad~son-Wiley, Reading, MA., 1981).

[14] G. Herzberg, Spectra of diatomic molecules (Van Nostrand, Princeton, 1950).

S. K. Kim et al. / Diatomic mokdes in the vibron model 9

[15] M.E. Kellman and R.J. Berry, Chem. Phys. Letters 42 (1976) 327.

1161 T.G. Waech and R.B. Bernstein, J. Chem. Phys. 46 (1967) 4905.

[17] R.J. LeRoy, Ph.D. thesis, Wisconsin, USA (1969). 1181 K.P. Huber and G. Henberg, Molecular spectra and

molecular structure, Vol. 4. Constants of diatomic mole- cules (Van Nostrand-Reinhold, New York, 1979).

[19] A. de Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963).

[20] F. Iachello, in: Nuclear spectroscopy, eds. C. Bertsch and D. Kurath (Springer, Berlin, 1980).