Algebraic description of stretching and bending modes in nonlinear triatomic-molecules

Revista Mexicana de Física 41, No. S (199S) 703-727 Algebraic description of stretching and bending modes in non-linear triatomic molecules J.M. ARIAS Departamento de Física Atómica, Molecular y Nucle01' Facultad de Física, Universidad de Sevilla Apartado 1065, 41080 Sevilla, España A. FRANK*, R. LEMUS Instituto de Ciencias Nucleares UNA M Apartado postal 70~543, 04510 México, D.F., México AND F. PÉIlEZ BERNAL Departamento de Física Atómica, Molecular y Nuclear Facultad de Física, Universidad de Sevilla Apartado 1065, 41080 Sevilla, España Recibido el 3 de marzo de 1995; aceptado el 21 de abril de 1995 AnSTRACT. \Ve extend the U(2) model oCn~collpled anharmonic oscillators, originally proposed to describe stretching vibrations, to include bending modes. The model describes the infrared spectrum of the molecules H2160, I"h 32S, 3251602, and 1603with rms deviations ranging from 13.42 to 1.05 cm-l. In addition, we present a calculatioIl of the dipole transition intensities for the H~60 molecule. RESUME:>. En este trabajo extendemos el modelo algebraico U(2) de osciladores anarmónicos acoplados con el objeto de incluir los modos vibracionales de flexión. rvlediante este modelo se describe el espectro infrarrojo de las moléculas H2l60, H232S, 32S1602 y 1603con desviaciones en las energías \'ibracionales que van de 13.42 a1.05 cm-l. Además se presenta un cálculo de intensidades de transición dipolares para la molécula de H~60. rACS: 03.65.Fd: 33.10.Cs; 33.10.Gx I. ];-.;TRODCCTIO:> With the advent of laser spectroscopy teehniques and their inereased power of resolution, highly excited overtone-combination vibrational speetra of moleeules can now be observed. There is therefore a renewed interest in developing theoretieal deseriptions of the physical proeesses involved. A detailed analysis of the obserwd spectral properties, however, is .Also at Lahoratorio de ClIcrnavaca, Instituto de Física. üNAi\1; Apartado postal 139-ll, CuC'rHayaca. ~lorelos. 11éxico. 704 J.M. AHIAS ET AL. quite eomplieated and different degrees of approximation are used to study the problem, ranging fram the simple Dunham energy expansion appraaeh to attempting the solution of the Sehrodinger equation by ah initio ealculations. In 1981 a novel appraaeh based an a!gebraie methods was proposed: the vibran modcl [1]' whieh was originally intradueed to describe the rato-\'ibrational strueture of diatomie moleeules and subsequently extended to include linear polyatomie moleeules and non-linear triatomie moleeules 12]. An alternative, symmetry-adapted algebraie model for t.riatomic molecules has been proposed by I3ijker et al. [3]. An important. advantage eommon to these algebraie appraaehes is that the powerful methods of graup theory provide relative!y simple solutions. It is well known t.hat. a reasonable pot.ent.ial to describe the vibrations of diatomie moleeules is the Mors!' potentia! [4]. This potentia! is c1ose!y associated to the 0(4) dynamical symmetry of t.he vibran mode! [5]' whieh explains the !atter's success in deseribing molecular vibrational spectra. Although this is a t.hree dimensional result, analogous re!ations hold in one and two dimensions. In a OIle-dimensional system the realization of SU(2) on the sphere can be associated to a ~Iorse potential [6J. The U(2) a!gebraic mode! may thus be considered as the onedimensiona! limit of the vibran mode! and can be generalized to molecu!es with severa! bonds. In 1984 O.S. van Roosma!en et al. ana!yzed the case of two bonds [7] by considering the stretching vibrational modes of specific molecules like H20, S02 and 03. The extension of the U(2) modcl to arbitrary polyatomic mo!ecules was earried out by Iachello and Oss [8]' who studied the vibrational spectra of benzene and severa! octahedral moleeules. Reeently the model has been also applied to infinite systems, name!y to linear and square crystal !attices [9]. Because of the one-dimensionalit.y of t.he modcl, however, the applications to molecular systems were restricted to the description of stretching vibrations. In this article we extend the U(2) model to describe both the stretching and bending modes of molecules by considering the particular case of triatomic mo!ecu!es. Iachello and Oss have proposed an extension 1.0 incorporate bending modes, but using a different technique [lO]. The genera!ization to molecu!es with more atoms will be presented in forthcoming publications [11]. The paper is organized as follows: In the next section we present the modcl of n-coup!ed Morse oscillators, while in Sect. 3 we consider its extension to ineorporate the bending modes. An analysis of the loca! and normal mode baB!'s is also included in this section. Section 4 contains our main results, corresponding to the vibrational spectra of the molecu!es H2160, H 2 32S, 32S1602, and 1603, whi!e the caku!ation of dipole transition intensities in H2160 is presented in Sect. 5. Finally, in Sect. 6 \Ve smnmarize our results and make sorne concluding frmarks. 2. ALGEBHAIC MOOEL The model is baB!'d on the isomorphism of the U(2) Lie algebra and ¡he one-dimensiona! Morse oscillator (2.1) ALGE!lIlAIC DESCIlIPTION OF STIlETCIIING AND !lENDING MODES... 705 whose eigenstates may be pul inlo a one lo one correspondence wilh lhe U(2) ::J 0(2) states 11 21. \Ve briefty discuss here how this carnes about. Consider the radial equalion 1 ( 1 d d m 22) , - -- - r - + -2 + r q,(r) = (1\ + l)rp(r), 2 r dr dr r (2.2) which corresponds to a two-dimensional oscillator (in units where f, = l' = e = 1) and thus to a U(2) syrnmetry algebra [61. By carrying out the transformation [61 r 2 = (N + 1) exp( -p), Equation (2.2) transfonns into [ 2 ( )2 ] d N + 1 - d p 2+ ~ (e-2p -2e-P) q,(p) = _171 2 q,(p), (2.3) (2.4) so defininig x = p.d and multiplying by r,2/2¡u¡2 we arrive al (2.1), provided that N+1 = JS/ld 2D/r,2 and E = -¿;;,m 2 Since N = 0,1,2, ... and m = c:!:N/2,c:!:(N -2)/2, ... , we see that the 1,lorse spectrum is reproduced twice and that we need to reslricl the m-vahles to non-negati\'e values. \Ve also note the connection between N and the potential depth. In terms of the usual SU(2) algebra, it is then clear from (2.4) that the Morse Hamillonian has the algcbraic realization - r, 2-2 H=--2/ 2 .1" ¡u (2.5) while Nis related lo lhe SU(2) label jthrough j = N/2 [2,10). \Ve can 'lIso \Vrile (2.5) in the form (2.6) where we have defined 6 20(2) := 4f¡ - ¡i,'2 The parameler Ais t1ms related to the Morse parameters, \Vhile the term _¡i,,2 is introduced in arder to place the gronnd state at null energy. \Ve now cOIlsider a molecular systl'lI1 wherc '1 chemical hOllds are iu\'olvcd [8J. In t.he algebraic model a U'(2) algebra is associated to the i-th hondo Therefore the produet U 1 (2) x ... x U"(2) establishes the dynamical group of the system. which means that every operator may be expanded in terms of generators of the V'(2) groups. In particnlar. the Hamiltonian is given in tenns of the invariant (Casimir) operators of the gronp' involved in the different reduetions of the dynamical algebra into its subalgebras. A possihle (iccompositioll ill\'olv('~ tll(, rcductioll Vi (2) x V 2 (2) x ' .. x ü"(2) ::J 0 1 (2) x ... x 0"(2) ::J 0(2). (2.7) 706 J.:--1. AmAS ET AL. where the eoupling to the fin"l 0(2) group is carried oul Ihrough the different intermediate couplings 0'](2). A seeond chain arises from all the possible eouplings of Ihe Ui(2) groups to ohtain a tOlal U(2) group. ",hieh in turn contains the full 0(2) group. The Ilami!tonian. up lo I",o-body interaelions and restrieted lo terms eonserving Ihe total 0(2) quantum Ilumber, is thcn given in tcrms of '/ contributions h¡ = A1 C 20i(2)l rcpresenting the T¡-One dimensional indepclldent .i\.lorsc oscillators, plus tvm types of bond-bond interactioIls: é 20"(2) and Mij, ",hieh eorrespoud lo the Casimir operators of the Oi](2) groups and the ~[ajoraua operators, respeetively 181. The latter are related to the U'](2) Casimir operators C 2U"(2) hy the relation (2.8) ",here N k eorrespouds lO Ihe numher of hosons associaled to the U k (2) group. The Hamiltonian has thus Ihe general fonn ~ ~ ~ -ji = !lo + L A,é20'(2) + L B,]C20"(2) + L .\']'~i)' i=l I>j I>} (2.9) The simplesl hasis to diagoualize the lIami!toniau (2.9) is Ihe oue associated ",ilh the local-mode chain IiNd, .... IN"I..... vl ....• V,,: V). where \\.'C have defillcd 1.J.¡ and \'. (2.10) ~Yi l" = - - 11l I 2 lo '1 V = LVi. i=l (2.11) iu terms of Ihe '1uaulum lllllnhers 711,. The operalors involved iu Ihe first Iwo sums of the lIamiltoniau (2.9) are diagoual iu the hasis (2.10) (INd, .... [N"I: VI •...• v,,: VIC20'(2)IINd, ... ,IN"I; Vi ....• vo: V) = 4(,,; - Nil',), (2.12a) ([Nd, ... , [N"I: ... , Vi •...• Vj, ... : Flé20,,(2)IiNd ..... [1Y"I:.... V,•...• Vj •... : 1') = 4[(11, + v])2 - (Vi +v])(N i +Nj)), (2.12h) while the ~lajoralla operator .'V1ij has hoth diagonal aIld non-diagonal matrix e1ements - V",(v] + 1)(."j - ,,])(Si - Vi + l)b,' ,. -1 6,' ,. +1' (2.12(') l' , j' J ALGEBRAIC IlESCHIPTION OF STHETCIIING AND !lENDING MOllES... 707 Thus the local basis owes its name to the faet that in it the individual Morse oseillators are well defined, with Vi the number of quanta in the i-th oscillator [12]. These simple results for the matrix elemeuts allow the diagoualization of il in a straightfOlward way. The Hamiltoniau (2.9), however, is arbitrary and does not iu general satisfy the possible symmetry requirements of the moleeule, whieh are eonsidered in the next seetion. The algebraie model eau also provide transition intensities. In the traditional approaeh, the use of the I3orn-Oppenheimer approximation has as a eonsequenee that the effeetive transition operator is not given direetly by the dipole operator. Instead, the dipole funetion is expanded in tenns of single bond eoordinates 00 d(r) = L a,,(r - Te)". n=O It is also possible to perfonn a Taylor expansion iu powers of the l\lorse variable rather than in powers of (r - Te). It has been shown that the most convenient fonn, whieh has the appropriate limit behavior, is [13] (2.13) In the algebraic approaeh the dipole function is expanded in terms of elements of the dynamieal algebra. It has been sllggested that in the SU(2) model the matrix elements of the transition operator ti, associated to the i-th bond, can be parametrized in the form 18] ([i N] [~' ]. ,/ . ,rlt-I[N] Il\']' 1). • V) - e-~Iv:-v.l 1 "", ~"'r¡ •... 'l' .... ~ 1 41, ...• 1] •••• , l"'" -, • (2.14) The operators ti are thllS associaíe<1 lo the ;-t1l bond in the local pictllre. The molecular dipole transition opcrat.or is t.IH'll given in t,erms of an expansioIl of t}l(' local operators ti. For any molecular systelll t.hp dipole opcrator T has tlJn'(' COlllPOllPlltS, giVCll in terms of local operators. Vp to lillrar t.PrIllS. [01' cxample, (~=x,y,z), (2.15) wlIere th(' relative va1tH's of t.he ('oefficiC'llts ü7 are detcrmillcd according t.o t.he molecular symuH't ry. The transit.ioll illtellsities Ii~f~ frolIl an illitial st.ate i to a final state ¡. are t1lell ('omputed in the usual [01'111 (2.1G) ( 708 J.M. ARIAS ET AL. where the sta tes li) and If) may be expressed in terms of the local basis li) = I>~l.V'..V, i[N¡J, ... , [N~l; VI,' .. ,v~;Vi), {v;} If) = L aL".v, I [NI], ... ,[N~I; VI,' .. ,v~; V f), {v;} (2.17a) (2.17b) with the coefficients a~¡,.. ,v, and at,. __ ,v, provided by the diagonalization of the Hamiltonian. Expressions (2.9) and (2.15) are still quite general and we shall see that symmetry considerations impose certain conditions on the parameters. In the next section we present the particular form of the Hamiltonian and the infrared operators T~ for the molecules we consider in this paper. 3. NON-LINEAR TRIATOMIC MOLECULES In the standard approach the potentials between nuclei are expressed in terms of a set of internal coordinates. For bent triatomic molecules the usual set corresponds to the bond distances rl, r2 and the angle O between them, as shown in Fig. 1. The potential is then expanded in the form V(1'I, 1'2, O) = V o + L ao/h rf r~ O'. o~, (3.1 ) It is equally possible, however, to expand the potential in terms of r3 (the distance between the A atoms in Fig. 1) instead of the angle O V(rlr2r3) = Vo + L bo~, rf r~ rj, o~, (3.2) and the Schrodinger equation can be written in terms of either (3.1) or (3.2). In the algebraic approach the bond coordinates rl and r2 are replaced by U(2) Lie algebraic structures. This procedure essentially corresponds to tbe potential (3.1) witb O constant, wbicb leads to a description of stretrbing modes only. In this papel' we propose to generalize the algebraic description following tbe scbeme (3.2). \Ve ran tben replare the three roordinates r1, r2 and r3 by U(2) algebrair structures, whirh leads to a description of both the stretcbing and bending modes in terms of Morse potentials. \Ve shall show tbat this approach is valid and consistent, and rompute both energy and intensity lits, which turn out to have the same level of accuracy as otber metbods. Tbe alternative description in Fig. 1 bas the additional advantage of providing a natural way to incorporate tbe bending degree of freedom to 1l10lecules like 03, for wbich a successful algebraic description has not been presented. ALGEIlIlAIC DESCRII'TION OF STRETCHlNG AND BENDlNG MODES. . . 709 B O AA B O A FIGURE 1. a) Traditional and b) alternative internal coordinates used to describe the vibrational dcgrees of freedom in bent triatomic molccules. H o H H H H FIGURE 2. Geometrical structure 01 a) water-like molecules H,O, H,S, SO" 03(C,,) and b) V3h molecules showing the assignment 01 the U(2) algebraic structures. \Ve now proceed to establish the lIamiltonian for the triatomic molecules H2 0, H25, 502, and 03. In Fig. 2 we display the representative geometrical structure for these lIlolecules, which corrcsponds to a C 2v syullnctry. In tlle samc figure \Ve show fOf COI11pleteness a cOllfigllratioll with V 3h symllletry, which would corrcspond to molccules such as IIj which, howe\'('r, are not consic!"r"c! in this article. 710 J .~l. ARIAS ET AL. \Ve now eonsider moleeules with C 2v symllletry by noting that the subgroup C2 e C 2v is enough to label the vibrational modes, sinee the tluee atoms define aplane. Aeearding to the general proeedure presented in Rcf. [14]' we first establish the isomorphism between the C2and S2 groups: E ~ (e), C 2~ (12){:l). (3.:¡) The S2 invariant Hamiltonian is theJI obtained in a straightforward way. Takiug iuto account up lo t.\\'o-hody interactiolls we fiud it~2" = h o + Al ((:7 201(2) + 6 202(2)) + ..1 3 6 203(2) + B12C2012(2) +B I3 (6 20"(2) + 6 2013(2)) (3 ..1) which is sYlllmetric agaiust the permutatiou of lahels 1 aud 2. as it should he. \Vhile the lIamiltonian (3.4) describes the general featmes of the spectrum, it is usually uuahle to provi<1e results of spectroscopie qua lit y, whicll n'<¡lIire adding thc llext order (quartic) terms to 'H~2"' The lIamiltouian providiug ti", d,'sired accuracy is then giVl'u hy (3.5 ) For C2t, IIlOICCl1leS. 9 parametrr:-; (...\13 is !lot 1It'('(\t.d as we show in 5('('t. 4) plus tlH' 1I11I1liJ('r of bOSOIlS lY¡ and 1\"3' ar(' thus 11('('<\('<\ lo prodll(,(, J¡i~lJ quality fits. The Hallliltonians (3.5) can be diagoualizl'd in the local hasis (2.10). For C 2,. molecul"s .N I = JV 2 amI tite' basis takrs t.ht' fOl"m (:l.G ) Th" diagonalization of the lIallli1tollian couplt's the local oscillatars. This int"rhond coupliug is illduC<'d hy t.he non-diagonal ).,lajoralla operators, givillg r¡se lo a trallsitioll towards tlOflllalmo<ies. The true behavior of a lllo!(,(,1l1ris in general in-hetwl'l'1l the local atld normal .schcJIlPS ami can be reprodllced by choosing tite appropriate paramct('rs in (3.4) alld (3.5). The local limit is obtailled hy takillg th" " parameters e'lual to zero. whil.' ill thp Ilormallimit all the.4 alld n pal'amett'rs shollld IH' llllll. Illtt'rllH'<iiate sitllatioll:-i call1H' gallgt'd hy the local-llorma1 trallsitioll paralllt'tt'l"s ~ illtrodul'l'd hy Child alHllIalOlll'll [i:J]. whirh fol' C 2t, molt'l'llles is ddilll'd a.s [15] ~= ~ tan -1 (_1_1_'_"_1_2_) . 7r A 1 +B 12 (3.7) ALGE!lRAIC DESCRII'TION OF STRETCIIING AND !lENDING MODES.. . 711 Sinee the Majorana parameters ..\ can take positive and negative values, the range of the parameter ~ is -1 ::; ~ ::; 1. Thus I~I = 1 for a purely normal mode behavior and ~ = Oin the local mode limil [13). The factor 11 in (3.10) is introdueed as a normalization. In moleeules near the loeallimit, e.9. H 2 S, the quantum numbers (3.6) provide a natural labeling of states. \Vhen the normal mode behavior is dominant, however, normallabeling is more appropriate. \Ve next present the eonneetion between these bases [12], for whieh it is eonvenient to use angular momentum labels Ij¡,). In terms of these indiees the local mode basis (3.6) takes the form u l (2) x U 2 (2) x U 3 (2) :J 01(2) x0 2 (2) x 0 3 (2) :J 0(2) 111111 1 (3.8) Ijl J2 J3 l' 1 1'2 1'3 l') \\'herc NI f\T 3 JI =J2 = 2' )3 = 2' Jii = m,! and l' = 1'1 +1'2 + Jl3. The Majorana operator .M 12 in (3.4) is diagonal in the basis x :J 0(2) 1 l' 12) , (3.9) \Vhieh means that this wave funetion is a normal basis with respeet to the bonds 1 and 2, and is related to the local basis IjIl1I)lizl'2) by the coupling coefficients [16) \Ve now proceed to establish the relation between the normal labels 1IA, 1IB (symmetrie amI antisymmetrie normal modes) and the angular momentum labels in (3.9). Applying the é 2 rotation [pennutation (12)] to (3.10) we obtain whieh suggcsts the following normal labeling for the antisymmetric mode 11/3 = JI + iz - jl2 = NI - j12. (3.11 ) (3.12) Since j12 = lV1,lV} - 1, ... ,O, we have that V8 =0,1, ... , ¡VII and the wave function chan!!f's si!!Tl fH'('ordiJ)~ f.o t1u'" nrlritv of 1!IL ,qs P"Xnpct.PíL TllP svmnlf'trir flllantlllll Tlnmhf'r 718 J.1>1. ARIAS ET AL. TABLE IV. Comparisoll hctween ealclllated and cxpf'rimcntal energies Cor 1125, and predicted cncrgies up lo 7 quanta. AlI ctlcrgics in cm-l. (11 m:f: v,) (VI V2 V:i) Theor. Exp. Theor. - Expt. (O O+ 1) (O 1 O) 1183.61 1182.60 1.01 (O O+ 2) (O 2 O) 2353.07 2354.00 -093 (O 1 + O) (1 O O) 2615.26 2614.40 0.86 (O 1 - O) (O O 1) 2628.61 2628.50 0.11 (O 1 + 1) (1 1 O) 377812 3779.20 -1.08 (O 1 - 1) (O 1 1) 3791.47 3789.30 2.17 (O 1 - 2) (O 2 1) 493974 4939.20 0.54 (O 2 + O) (2 O O) 5145.42 5145.10 032 (O 2 - O) (1 O 1) 5145.22 5147.40 -0.18 (O 2 + 1) (2 1 O) 6287.36 6288.20 -0.84 (O 21) (1 1 1) 6289.16 6289.20 -om (1 1 + 1) (O 1 2) 6385.86 6388.70 -2.84 (O 3 + O) (1 O 2) 7575.87 7576.30 -0.43 (O 3 - O) (2 O 1) 7575.97 7576.31 -0.34 (1 2 + O) (3 O O) 775338 7751.90 1.48 (1 2 - O) (O O 3) 7779.41 7779.20 021 (O 3 - 1) (2 1 1) 8696.82 8697.30 -0.48 (O 4 - O) (3 O 1) 9910.19 9911.10 -0.91 (1 3 - O) (1 O 3) 10194.07 10194.50 -043 (O 41) (3 1 1) 11009.78 11008.80 098 rms deviation (cm) 1.05 TABLE IV. Canto Prt'dicted l'llergies. (11 m:f: v,) (VI v, V3) Theor. (11 m :f: V,) (VI V, V3) Theor. (O O+ 3) (O 3 O) 3512.31 (O 5 + O) (3 O 2) 12149.89 (O 1 + 2) (1 2 O) 4926.40 (5 OO) (2 O 3) 12149.89 (O O+ 4) (O 4 O) 4665.04 (12+ 4) (34 O) 12159.75 (1 1 + O) (O O 2) 524392 (124) (O 4 3) 1218579 (O 0+ 5) (O 5 O) 5814.73 (13+ 2) (4 2 O) 12372.37 (O 1 + 3) (1 3 O) 6064.02 (132) (1 2 3) 12377.44 (O 1 - 3) (O 3 1) G077.37 (2 2 + 2) (O 2 .1) 1247614 (O 0+ 6) (O 6 O) 6964.61 (1 4 + O) (1 O 4) 12524.76 (O 1 + 4) (1 4 O) 7194.70 (1 4 - O) (4 O 1) 12525.11 (O 1 - 4) (O 4 1) 7208.05 (2 3 + O) (5 O O) 12696.69 (O 2 + 2) (2 2 O) 7414.29 (2 3 - O) (O O 5) 12734.68 (O 2 - 2) (1 2 1) 7416.09 (O 4 + 3) (2 3 2) 13165.42 (1 1 + 2) ~O 2 2) 7512.79 (O 4 - 3) (3 3 1) 13165.43 (O O+ 7) (O 7 O) 8117.69 (O 5 + 1) (3 1 2) 13228.06 (O 1 + 5) (1 5 O) 8321.90 (501) (2 1 3) 13228.06 (O 1 - 5) (O 5 1) 8335.25 (1 3 + 3) (4 3 O) 13444.25 (02+3) (2 3 O) 8530.15 (1 3 - 3) (133) 13449.31 ALGEBHAIC DESCB1PTIO:-;OF STHETCHlNG ANO BENDING MonES... 719 TABLEIV. Cont. Predictecl energies. (o m:f:: v,) (VI v, V3) Theor. (oro:f::v,) (VI v, V3) Theor. (O 2 - 3) (1 3 1) 8531.95 (2 2 + 3) (O 3 4) 13548.01 (1 1+ 3) (O 3 2) 8628.65 (14+ 1) (1 I 4) 13602.93 (O 3 + 1) (1 1 2) 8696.73 (141) (41 1) 13603.28 (12+ 1) (3 1 O) 8874.23 (2 3 + 1) (5 1 O) 13774.86 (1 2 - 1) (O I 3) 890026 (2 3 - 1) (O I 5) 13812.85 (O 1 + 6) (1 6 O) 9448.86 (O 5 + 2) (3 2 2) 14290.01 (O 1 - 6) (O 6 1) 9462.21 (5 O2) (2 2 3) 14290.01 (O 2 + 4) (2 4 O) 9638.63 (O 6 + O) (2 O 4) 14295.69 (O 2 - 4) (14 1) %40.44 (6 OO) (3 O 3) 14295.69 (1 1 + 6) (O 4 2) 9737.13 (1 4 + 2) (1 2 4) 14664.88 (O 3 + 2) (1 2 2) 9802.15 (1 4 - 2) (42 1) 14665.23 (O 3 - 2) (2 2 1) 9802.25 (1 5 + O) (2 O 4) 14763.42 (O 4 + O) (2 O 2) 9910.18 (1 5 - O) (5 O 1) 14663.43 (12+ 2) (3 2 O) 9979.66 (2 3 + 2) (5 2 O) 14836.81 (1 2 - 2) (O 2 3) 10005.69 (2 3 - 2) (O 2 5) 14874.80 (1 3 + O) (4 O O) 10189.01 (2 4 + O) (6 O O) 15037.16 (2 2 + O) (O O 4) 10292.77 (2 4 - O) (3 O 3) 15046.55 (O 2 + 5) (2 5 O) 10743.22 (3 3 + O) (O O 6) 15148.11 (O 2 - 5) (1 5 1) 10745.02 (O 6 + 1) (2 1 4) 15352.27 (1 1+ 5) (O 5 2) 10841.72 (601) (3 1 3) 15352.27 (O 3 + 3) (1 3 2) 108%.10 (15+ 1) (2 I 4) 15820.00 (O 3 - 3) (2 3 1) 108%.19 (151) (51 1) 15820.02 (04+ 1) (2 1 2) 11009.78 (2 4 + 1) (6 1 O) 16093.74 (1 2 + 3) (330) 11073.60 (2 4 - 1) (3 I 3) 16103.13 (1 2 - 3) (O 3 3) 1109963 (3 3 + 1) (O 1 6) 16204.70 (1 3 + 1) (4 1 O) 1128860 (7 O+ O) (3 O 4) 16348.22 (131) (1 13) 11293.67 (O 7 - O) (4 O 3) 16348.22 (2 2 + 1) (O 1 4) 11392.37 (1 6 + O) (5 O 2) 16907.97 (O 3 + 4) (1 4 2) 11982.25 (1 6 - O) (6 O 1) 16907.97 (O 3 - 4) (2 4 1) 11982.34 (2 5 + O) (7 O O) 17279.64 (O 4 + 2) (2 2 2) 12093.55 (2 5 - O) (6 O 1) 17280.46 (O 4 - 2) (32 1) 12093.55 (3 4 + O) (7 O O) 17447.08 (3 4 - O) (O O 7) 17496.25 5. DIPOLE 1'11'\:'\51'1'10:'\5 In Sect. 2we presented the genera! form of the dipo!e operator and indicated the way to compute transition intensities in the framework of the algebraic approach. In this section we calcu!at(' the dipole intensities fOl the H20 ITlo)ecule. Although the linear expansion of the transition operator (2.15) is not sufficient to lit the data. we ha,'e found that the dipo)•. operatOls (3.17a-b), which inrlude '1uadratic teflns. 720 J.M. ARIAS ET AL. TABLE V. Comparison between ealculated and experimental energies for SO" and predieted energies up to 7 quanta. AH energies in crn1 (n m io v,) (VI v, V3) Theor. Exp. Theor. - Expt. (O 0+ 1) (O 1 O) 518.61 517.87 0.74 (O 0+ 2) (O 2 O) 1034.46 1035.13 -0.67 (O 1 + O) (1 O O) 1150.95 1151.71 -0.76 (O 1 - O) (O O 1) 1358.12 1362.06 -3.94 (O 0+ 3) (O 3 O) 1549.25 1551.75 -2.50 (O 1 + 1) (1 1 O) 1667.40 1666.33 1.07 (O 1 - 1) (O 1 1) 1874.57 1875.79 -1.22 (O 0+ 4) (O 4 O) 2064.55 2066.87 -2.32 (O 1 + 2) (1 2 O) 2180.97 2179.51 1.46 (O 2 + O) (2 O O) 2293.92 2295.80 -1.88 (O 1 - 2) (O 2 1) 2388.14 2388.92 -0.78 (O 2 - O) (1 O 1) 2498.78 2499.87 -1.09 (O 0+ 5) (O 5 O) 2581.77 2582.30 -0.53 (O 1 + 3) (1 3 O) 2693.36 2693.63 -0.27 (1 1 + O) (O O 2) 2706.91 2713.38 -6.47 (O 2 + 1) (2 1 O) 2808.21 2807.19 1.02 (O 2 - 1) (1 1 1) 3013.06 3010.32 2.74 (1 1 + 1) (O 1 2) 3221.19 3222.25 -1.06 (12+ O) (3 O O) 3428.94 3431.19 -2.25 (O 3 - O) (2 O 1) 3631.56 3629.61 1.95 (1 1 + 2) (O 2 2) 3732.46 3730.90 1.56 (O 3 + O) (1 02) 3837.40 3837.06 0.34 (1 2 + 1) (3 1 O) 3941.04 3939.90 1.14 (O 3 - O) (2 O 1) 3631.56 3629.61 1.95 (1 1+ 2) (O 2 2) 3732.46 3730.90 1.56 (O 3 + O) (1 O 2) 3837.40 3837.06 0.34 (1 2 + 1) (3 1 O) 3941.04 3939.90 1.14 (O 2 - 3) (1 3 1) 4034.31 4029.39 4.92 (1 2 - O) (O O 3) 4046.38 4054.00 -7.62 (1 1 + 3) (O 3 2) 4242.44 4241.50 0.94 (O 2 + 4) (2 4 O) 4339.69 4342.70 -3.01 (1 2 + 2) (3 2 O) 4450.01 4446.90 3.11 (121) (O 1 3) 4558.48 4560.10 -1.62 (O 4 - O) (3 O 1) 4754.48 4751.23 5.25 (O 3 + 2) (1 2 2) 4858.47 4848.14 10.33 (1 2 + 3) (3 3 O) 4957.56 4958.00 -0.44 (1 3 - O) (1 03) 5166.80 5163.62 3.18 (1 4 - O) (4 O 1) 5873.53 5872.10 1.43 (1 4 + O) (1 04) 6487.02 6489.20 -2.18 (2 3 - O) (O O 5) 6697.43 6689.40 8.03 rms deviation (cm) 3.71 ALGEIlHA'C OESCHII'TION OF STHETCII'''G ANO BENDlNG MOOES... 721 TABLE V. Cont. Prcdictcd cllcrgics. (n m:t: V2) (VI V2 V3) Theor. (n m :t: V2) (VI V2 V3) Theor. (O 1 - 3) (O 3 1) 2aOO.52 (1 3 - 4) (O 4 3) 6082.69 (O 0+ 6) (O 6 O) 3102.22 (23+ 1) (5 1 O) 6182.85 (O 1 + 4) (i 4 O) 3206.12 (O 4 - 3) (33 1) 6278.16 (O 2 + 2) (2 2 O) 331a.48 (O 5 - O) (2 O 3) 627a.53 (O 1 - 4) (O 4 1) 3413.29 (1 3 + 2) (O 2 4) 63a3.12 (O 2 - 2) (12 1) 3524.34 (O 4 + 3) (2 3 2) 6481.78 (O 0+ 7) (O 7 O) 3627.05 (O 5 + 1) (3 1 2) 6582.72 (O 1 + 5) (1 5 O) 3720.70 (2 3 + 2) (520) 6687.1 a (O 2 + 3) (2 3 O) 382a.45 (1 3 - 3) (1 3 3) 6688.4a (O 1 - 5) (O 5 1) 3a27.87 (24+ O) (6 O O) 6786.36 (O 3 - 1) (2 1 1) 4143.66 (O 5 - 1) (2 1 3) 6787.23 (O 1 + 6) (1 6 O) 4238.38 (1 4 - 2) (421) 6885.58 (O 3 + 1) (1 1 2) 434a.50 (1 3 + 3) (O 3 4) 68a8.24 (O 1 - 6) (O 6 1) 4445.54 (i 5 - O) (5 O 1) 6a82.76 (O 2 - 4) (1 4 1) 4544.54 (1 4 + 1) (1 1 4) 6aa4.72 (1 3 + O) (4 O O) 4556.00 (O 5 + 2) (3 2 2) 7087.06 (O 3 - 2) (2 2 1) 4652.64 (O 6 + O) (4 O 2) 7182.18 (i 1 + 4) (O 4 2) 4752.67 (2 3 - 1) (O 1 5) 7205.14 (1 3 + O) (4 O O) 4556.00 (O 5 + 2) (3 2 2) 7087.06 (O 3 - 2) (2 2 1) 4652.64 (O 6 + O) (4 O 2) 7182.18 (1 1 + 4) (O 4 2) 4752.67 (2 3 - 1) (O 1 5) 7205.14 (O 2 + 5) (2 5 O) 4851.60 (O 5 - 2) (2 2 3) 72a1.58 (O 4 + O) (2 O 2) 4%0.10 (2 4 + 1) (6 1 O) 72a1.85 (O 2 - 5) (151) 5056.46 (O 6 - O) (3 O 3) 7384.56 (i 3 + 1) (4 1 O) 5065 al (1 5 - 1) (5 1 1) 7488.25 (O 3 - 3) (2 3 1) 5160.19 (1 4 + 2) (i 24) 74aa06 (O 4 - 1) (3 1 1) 5266.38 (O 6 + O) (2 O 4) 758a.87 (O 3 + 3) (1 3 2) 5366.02 (2 3 - 2) (O 2 5) 770a.48 (1 3 + O) (O O 4) 5376.55 (1 5 - O) (i 05) 77a8.04 (O 4 + 1) (2 1 2) 5<17000 (34+ O) (7 O O) 788a.68 (1 3 + 2) (420) 5572.58 (O 6 - 1) (3 1 3) 78aO.06 (123) (O 3 3) 5575.00 (24+ O) (O O 6) 800a.04 (O 3 - 4) (2 4 1) 5667.87 (25O) (6 O 1) 8084.16 (2 3 + O) (5 O O) 5675.14 (O 6 + 1) (2 1 4) 80a5.36 (1 3 - 1) (1 1 3) 5676.71 (1 6 + O) (5 O 2) 8281.5a (O 4 - 2) (32 1) 577305 (i 51) (1 1 5) 8303.53 (O 3 + 4) (142) 5873.71 (O 7 - O) (4 O 3) 8481.a3 (13+ 1) (O 1 4) 5886.46 (2 4 + 1) (O 1 6) 8514.53 (O 4 + 2) (2 2 2) 5976.67 (O 7 + O) (3 O 4) 8685.12 (O 5 + O) (3 O 2) 6075.02 (1 6 - O) (2 O 5) 8891.12 (i 3 + 3) (4 3 O) 60776a (25+ O) (106) aoaa.8a (34O) (O O 7) a311.3a 722 J.M. ARIAS ET AL. TABLE VI. Comparison between calculated and experimental energies for 0 3, and predicted energies up to 7 quanta. AH energies in cm-l. (n m:!: v,) (VI V, V3) Theor. Exp. Theor. - Expt. (O 0+ 1) (O 1 O) 720.34 701.00 19.34 (O 1 + O) (1 O O) 1033.13 1042.00 -8.87 (O 1 - O) (O O 1) 1109.89 1102.00 7.89 (O 1 + 1) (1 1 O) 1727.32 1726.00 1.32 (O 1 - 1) (O 1 1) 1804.08 1796.00 8.08 (O 2 + O) (2 O O) 2049.66 2058.00 -8.34 (O 2 - O) (1 O 1) 2102.08 2110.00 -7.92 (1 1 + O) (O O 2) 2212.40 2201.00 11.40 (O 2 + 1) (2 1 O) 2717.70 2726.00 -8.30 (O 2 - 1) (1 1 1) 2770.12 2785.00 -14.88 (1 1 + 1) (O 1 2) 2880.44 2886.00 -5.56 (O 3 + O) (3 O O) 3043.63 3046.00 -2.37 (O 3 - O) (2 O 1) 3071.56 3084.00 -12.44 (1 2 + O) (1 O 2) 3180.24 3185.00 -4.76 (1 2 - O) (O O 3) 3302.99 3289.00 13.99 (O 4 + O) (4 O O) 4003.02 4000.00 3.02 (O 4 - O) (3 O 1) 4012.74 4009.00 3.74 (1 3 + O) (2 O 2) 4141.35 4139.00 2.35 (1 3 - O) (1 03) 4241.74 4238.00 3.74 (2 2 + O) (O O 4) 4382.12 4371.00 11.12 (O 5 - O) (4 O 1) 4919.44 4922.00 -2.56 (1 4 - O) (2 O 3) 5165.71 5170.00 -4.29 (2 3 - O) (O O 5) 5449.15 5443.00 6.15 (O 6 + O) (4 O 2) 5786.92 5767.00 19.92 (1 5 + O) (6 O O) 6038.55 5997.00 41.55 (1 5 + O) (2 04) 6204.06 6204.00 0.06 (33+ O) (O O 6) 6504.07 6506.00 -1.93 (1 6 - O) (6 01) 6950.14 6987.00 -36.86 (2 5 - O) (4 O 3) 7218.25 7227.00 -8.75 (3 4 - O) (O O 7) 7546.72 7555.00 -8.28 rms deviation (cm) 13.42 lead to a satisfaetory deseription fOl' the intensities in this moleeule. Using the parameter values al = 15.8, a2 = 450.0, a3 = -35.0, a4 = 55.0, a5 = 235.0, {3¡ = 1.1 and (33 = 2.5 we obtain the result given in Table VIII. Although these parameters were not obtained by a least squares fit proeedure, but rather varied to seareh for the best fit, this is not diffieult sinee the effeet of each parameter on the intensities is very specific. A fit to these intensities was carried out previously by Iachello and Oss [151, within the U(4) algebraic method. Although in that case energy fits are quite straightforward, the fit to intensities is very difficult due to the model's sensitivity to parallleter values, in contrast to the case of ALGEBHAIC DESCHII'TION OF STRETCHING ANO BENOING MOOES... 723 TABLE VI. Con!. Predieted energies. (n mor V2) (VI V2 V3) Theor. (n m or V2) (VI v2 V3) Theor. (O 0+ 2) (O 2 O) 1482.61 (1 3 + 2) (2 2 2) 5414.76 (O 0+ 3) (O 3 O) 2286.80 (O 5 + 1) (3 1 2) 5506.92 (O 1 + 2) (1 2 O) 2463.44 (O 5 - 1) (4 1 1) 5509.04 (O 1 - 2) (O 2 1) 2540.20 (1 3 - 2) (1 2 3) 5515.15 (O 0+ 4) (O 4 O) 3132.92 (2 2 + 2) (O 2 4) 5655.53 (O 1 + 3) (1 3 O) 3241.48 (2 3 + 1) (5 1 O) 5688.03 (O 1 - 3) (O 3 1) 3:Jl8.24 (1 4 - 1) (2 1 3) 5755.30 (O 2 + 2) (2 2 O) 3127.66 (O 6 - O) (3 O 3) 5787.25 (O 2 - 2) (1 2 1) 3480.08 (1 4 + 1) (1 1 4) 5886.36 (1 1 + 2) (O 2 2) 3590.41 (O 2 + 5) (2 5 O) 5809.12 (O 3 + 1) (3 1 O) 3685.52 (O 1 + 6) (1 6 O) 5827.16 (O 3 - 1) (2 1 1) 3713.45 (O 2 - 5) (1 5 1) 5861.54 (1 2 + 1) (1 1 2) 3822.13 (O 3 + 4) (34 O) 5862.75 (1 2 - 1) (O 1 3) :19.14.89 (O :14) (2 4 1) 5890.67 (O 0+ 5) (O 5 O) .1020.% (O 1 - 6) (O 6 1) 5903.92 (01+.1) (1 4 O) 4061.45 (O O + 7) (O 7 O) 5922.83 (O 1 - 4) (O 4 1) 41:38.21 (1 1 + 5) (O 5 2) 5971.86 (O 2 + 3) (230) 4179.56 (01 + 3) (4 3 O) 5976.02 (O 2 - 3) (1 3 1) .12:Jl.98 (O 4 - 3) (3 3 1) 5985.74 (1 1 + 3) (O 3 2) 4342.30 (1 2 + 4) (1 4 2) 5999.35 (O 3 + 2) (320) 4369.34 (231) (O 1 5) 6038.74 (O 3 - 2) (2 2 1) 4397.27 (1 5 - O) (5 O 1) 6071.10 (1 2 + 2) (1 2 2) 4505.95 (2 4 - O) (1 O 5) 6339.18 (O 4 + 1) (4 1 O) 4618.76 (1 3 + 3) (2 3 2) 6114.35 (O 4 - 1) (3 1 1) 4628.48 (1 2 - 4) (O 4 3) 6122.11 (1 2 - 2) (O 2 3) 4628.70 (O 5 + 2) (3 2 2) 6138.44 (13+ 1) (2 1 2) 4757.09 (O 5 - 2) (4 2 1) 6140.55 (1 3 - 1) (1 1 3) 4857.48 (1 3 - 3) (1 33) 6214.74 (O 5 + O) (3 O 2) 4917.:1:1 (2 3 + 2) (520) 6319.55 (O 1 + 5) (1 5 O) 492.1.34 (O 6 + 1) (4 1 2) 6350.36 (O 0+ 6) (O 6 O) 4950.93 (O 6 - 1) (3 1 3) 6350.69 (O 2 + 4) (2 4 O) 4973.37 (2 2 + 3) (O 3 4) 6355.12 (2 2 + 1) (O 1 4) 1997.86 (1 4 - 2) (2 2 3) 6386.82 (O 1 - 5) (O 5 1) 5000.10 (1 4 + 2) (1 24) 6517.87 (O 24) (14 1) 5025.79 (1 5 + 1) (6 1 O) 6601.99 (O 3 + 3) (330) 509;).08 (O 7 + O) (5 O 2) 6614.30 (2 3 + O) (.\ O O) 5098.44 (O 7 - O) (4 O 3) 6614.34 (O 3 - 3) (2 :1 1) 512:1.01 (151) (5 1 1) 66:1-1..\4 (1 1 + 4) (O .1 :1) 51:16.12 (2 3 - 2) (O 2 5) 6670.25 (1 2 + 3) (1 :1 2) S:2:31.fi9 (1 5 + 1) (2 1 4) 6767.50 (0.1 + 2) (1 2 O) ,'/276.43 (241) (1 1 ,\) 6902.62 (O .12) (321) 5286. ].\ (1 6 + O) (7 O O) 6940.37 (11 + O) (1 O .1) fJ296.77 (33+ 1) (O I 6) iOG7Ji! (1 2 - 3) (O 3 :1) .'):354.'1.1 (:1 4 + O) (7 O O) 7113 ..1.1 (2 5 + O) (1 O 6) 7371.3;) 724 J .M. ARIAS ET AL. TABLEVII. Values af the ~ parameter far the maleeule, analyzed in Table II. Maleeule ~ 1120 -033 112S -0.08 502-0.82 0 3 -0.96 the U(2) made!. In Table VIII \Vealsa present the results af Reí. [151, as \Vellas previaus calculatians using simple dipole fUllctians. From Table VIII \Ve eonelude that traditional calculations fail campletely ta describe the observed intensities. EVCll though the aceuracy of our results is af the same arder of magnitude than the one of Hef. [15]' \Vehave used a mueh simpler procedure \Vith nearly half the number of parameters. These results emphasize the usefulness of the U(2) model \Vhen only vibrational degrees of freedom are illvolved. Our fitting programs are available on request. 6. CONCLUSIONS \Ve have presented all extellsioll of lhe U(2) model \Vhieh ineorporates lhe deseription of bending modes in triatomic molecules. In additioll to the energy fits \Ve have ealculated dipole intensities for the 112 0 maleeule. The rms deviations obtaill"d for the best fits are af the order of a fe\V cm1 ar less (\Vith the exception of 0 3), This result, together \Vith the simplicity of the model (no coupling coefficiellts are illvalved in computing the matrix elements) makes it particularly attractive far the study of overtones and combinations \Vith high number of quanta. This is in contrast \Vith th" tradilional approaeh, based on integrodifferential techniques, \Vh"re the potential is model"d in terms of the force ficld eonstants through eomplex ealculations [271. Using th" parameters obtain"d in th" quadratic fits \Ve also eomputed the para meter ewhieh giv"s a measure of Ih" local-normal behavior of the molecules. In addition. our analysis sho\Vs that the definition of qnantum numbers proposed for the normal states are ver)" elose to the exact quantnm nnmber for molecnles \Vith normal behavior. Our relativel)" simple dipole intensity calculatian in H 2 0 gives a reasanabl)" gaod description af the experimental observations. \Vith similar quality ta more invalved methods. The methad can be improved in several \Vays. The interactions ineluded in the I1amiltanian (3.5) assum" the conservation of the total number of quanta V. This restriction means that onl)" ",me physically ul('aningful interactians. sueh as the DarlingDcnnison interactiolls. han) becll taken ¡nto aCCOl1nt. while othefs 1 like titase leading lo Fermi resonanees. have Bol bl'l'l1 included in th(' Hamiltolliau. Phonoll Iloll-conservation, hawever. can be readily ineluded in the modelthrough the other generators af the 5U(2) graup. In particular, the raising and lowering operators. j+ aud j_. ar their hermitean sum jx = !ti+ + j_). mix the multiplets in precisely lhe required farm for Fermi-like illteractions. ALGEBRAIC DESCRIPTION OF STRETCIIING AND BENDING MODES ... 725 TABLE VIII. Calculated intensilies (0,0,0) - (VI v, V3) in H,O. Lawlon & VI V2 V3 Obs. [23] Present work Iachello Oss Child [24J O 1 O 1040.0 973 1040 100 49.5 55.3 51 49.5 O 01 720.0 670.4 732 74.1 020 7.6 6.5 4.7 1 1 O 3.7 1.8 5.7 O 1 1 80.4 82.4 82.8 200 4.6 5.1 7.7 0.037 1 O 164.3 74.2 28 0.054 002 0.58 1.0 2.2 0.01 030 0.04 0.044 0.02 120 0.31 0.012 0.08 02 1 5.1 0.55 1.1 2 1 O 0.042 0.16 0.03 1 1 1 4.95 9.13 3.9 O 1 2 0.16 0.034 0.001 300 0.62 1.61 2 O 17.96 4.69 1 O 2 5(-6) 0.059 0.16 003 0.27 0.12 040 0.002 0.0003 0.0004 1 3 O 0.023 0.00008 0.0008 03 1 0.089 0.0037 0.011 Number oC parameters 712 Slamard el al. [25J 1040.0 43.0 241.9 0.043 0.051 26.7 0.004 0.004 0.0 0.0014 0.06 0.05 0.0013 0.0036 0.29 0.0039 0.0024 0.058 Carney el al. [26J 1040 35.6 493.0 5.4 1.3 162.5 4.3 22.2 0.15 0.02 0.08 1.7 The matrix elements of jx in the local basis take the form ([N)v + IljxJ[N]v) = h/(N - v)(v + 1), ([N]v - IljxJ[N)v) = h/v(N - V+ 1). The algebraie model can thus naturally ineorporate sueh terms in a simple fashion. On the other hand, the study of vibrational isotopie e/feets in moleeules is of major importanee due to the faet that lo a very high degree of approximation these moleeules correspond lo the same energy potenlials. The di/ferenees in vibrational frequencies are due mainly to the presenee of di/ferent masses. From these eonsiderations it is possible to establish additional equations that determine the force field eonstanls. In lhe algebraic approaeh, however, there is no explicit distinetion between lhe kinetie and the potential energy terms. AII struetural information is eontained in the Hamiltonian parameters, so 726 J .~1. ARIAS ET AL. we cannot reproduce the traditional studies in a straightforward way. Jt is possible, nevertheless, to study any isotopie moleeule by means of the same algebraic seh('me considering the change to lower symmetry, if present. Once we have at our disposal tits to a set of isotopie moleeules we can analyze the sealing properties of the Hamiltonian parameters as a funetion of their masses. whieh allows the predietion of the vibrational s¡wetrum of other isotopic species [151.It is also possible to correlate the Hamiltonian parameters with the force tield eonst ants by means of dosed energy expressions for the fundamentals and tirst overtones, as explained in Ref. [15] for the case of 1120. \Ve remark that alt.llOugh the generalization of the U(2) mode! h,~, only been presented for the case of triatomie moleeules, it is possible to extend our considerations to polyatomie moleeules, although a earcful analysis of spurious degrees of freedom has to be made [10,llj. The \Ilodel seems to represent a v'ery promising framework for the description of infrared s¡",etroseopie properties of eomplex moleeules. AC\(:-;OI\'LEDGE~I E:>:TS \Ve are indebted to 1'. Van Isacker for very helpful (,()\Il\llents. 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