A symmetry adapted approach to molecular spectroscopy: the anharmonic oscillator symmetry model

Revista Mexicana de Física 42, Suplemento 1 (1996) 73-83 A symmetry adapted approach to molecular spectroscopy: the anharmonic oscillator symmetry model A. FRANK', R. LEMUS, R. BIJKER Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México Apartado postal 70-543, 04510 México, D.F., México F. PÉREZ-BERNAL AND J.M. ARIAS Departamento de Física Atómica, Molecular y Nuclear Facultad de Física, Universidad de Sevilla Apartado 1065, 41080 Sevilla, España ABSTRACT. \Ve apply the anharmonic oscillator symmetry model to the description of vibrational excitations in V 3h and Td molecules. A systematic procedure can be used lO establish the relation between the algebraic and configuration space formulations, by means of which new interactions are found in the algebraic model, leading to reHable spectroscopic predictions. \Ve illustrate the method for the case of triatomic D 3h molecules and the T d ne,-cluster. RESUMEN. Utilizamos el modelo de simetría de osciladores anarmónicos para describir las excitaciones vibracionales en moléculas con simetría V 3h yTd. Un procedimiento sistemático permite establecer la reladón entre la formulación algebraica y la de! espado de configuración. Mediante esta conexión se encuentran nuevas interacciones en el modelo algebraico que dan lugar a predicciones espectroscópicas confiables. Ilustramos el método para el caso de moleculas triatómicas V3h y para el cúmulo de berilio Be4 con simetría tctraédrica. PACS: 33.20.Tp; 33.15.Mt; 03.65.Fd The study of molecular vibrational spectra [1] requires theoretical models in order to anaIyze and interpret the measurements [2]. These models range from simple parametrizations of the energy levels, such as the Dunham expansion [21, to ab initio calculations, where solutions of the Schriidinger equation in different approximations are sought [3-5). In general, the latter involve the use of internal coordinates and the evaluation of force field constants assodated to derivatives at the potential minima. While this method can be reliably applied to small molecules [6]' it quickly becomes a formidable problem in the case of larger Illolecules, due to the size of their configuration spaces. New calculational tools to describe cOlllplex molecules are thus needed. In 1981 an algebraic approach was proposed to describe the roto-vibrationaI structure of diatomic molecules [7]' subsequently exten<!cd to linear triand fOllr-atomic molecules [81 • Also at Instituto de Física, Laboratorio de Cucrnavaca, Apartado postal 139-D. Cucrlla\'aca, More!os, México. 73 74 A. FRANK ET AL. and eertain non-linear triatomie moleeules [9]. Although these were eneouraging results, the model eould not be extended to polyatomie moleeules, due to the impossibility of ineorporating the underlying diserete symmetries. This diffieulty eould be surmounted by treating the vibrational degrees of freedom separately from the rotations. In 1984 Van Roosmalen et al. proposed a U(2) ba.sed model to describe the stretehing vibrational modes in ABA moleeules [10], later extended to describe the stretehing vibrations of polyatomie moleeules sueh as oetahedral and benzene like moleeules [11]. Reeently the bending modes have a!so been included in the framework, whieh was subsequently applied to describe C2v-triatomie moleeules [12J and the lower excitations of tetrahedral moleeules [13], using a seheme whieh combines Lie-algebraie and point group methods. In a different approach, it has also been suggested to use a U(k +1) model for the k = 3n - 3 rotational and vibrational degrees of freedom of a n-atomie moleeule. This model has the advantage that it ineorporates all rotations and vibrations and takes into aeeount the relevant point group symmetry [14J, but for larger moleeules the number of possible interaetions and the size of the Hami1tonian matrices inerease very rapidly, making it impraetieal to apply. A1though the algebraie formulations have proved useful, several problems remained, most important of whieh is the absenee of a clear eonneetion to traditional methods. On the other hand, a related problem is the laek of a systematie proeedure to eonstruet all physically meaningful interaetions in the algebraie spaee. In this paper we show that both these issues can be resolved by means of a general model for the analysis of molecular vibrational speetra, the anharmonie oscillator symmetry model (AOSl\l). In this approaeh it is possible to constrnet algebraie operators with well defined physieal meaning, in particular interaetions fundamental for the deseription of the degenerate modes present in systems exhibiting high degree of symmetry. The proeedure to construet them takes full advantage of the diserete symmetry of the moleeule and gives rise to all possible terms in a systematie fashion. The harmonic limit of the model provides a clear-eut connection between the algebraic scheme and the traditional analyses based on internal coordinates. As a test for this approach we apply the AOSM to the Be4 cluster [15J and to three D 3h triatomic molecular systems, namely Hj, I3e3 and Naj [16]. Since small molecules can in general be well described by means of ab initio calculations [17,18], we emphasize the basic purpose of this work. \Ve have established an exact correspondence between configuration space and algebraic interactions by stndying the harmonic limit of the U(2) algebra. This general procedure not only allows to derive the interactions in the AOSM from interactions in eonfiguration space, but can also be applied to cases for which no configuration space interaetions are available. The 'D3h-triatomie moleeules constitnte the simplest systems where degenerate modes appear and where the new interaetions in the model become significant. In the case of I3e4, a direct comparison with ab initio ealculations will be presented. The application of these techuiqnes to more complex systems, such as the methane moleenles, is presently under investigation [19]. The model is based on the isomorphism of the U(2) Lie algebra and the one dimensional Morse oscillator f¡ 2 d 2 'h r 11 = --- +D(ed - 2e-;1), (1) 2" dx 2 whose eigenstates [ can be a.,"o<:iated with U(2) :> SO(2) states [20]. In order to see how A SYMMETRY ADAPTED APPROACH TO MOLECULAR.. . 75 this isomorphism comes about, consider the radial equation I ( I d d a 22) - ---d rd + 2" + r <,i>(r) = (N + I)<,i>(r), 2 rr r r (2) which corresponds to a two-dimensional harmonic oscillator (in units where h = J1, = e = 1) associated to a U(2) symmetry algebra [21]. By carrying out a change of variable Eq. (2) transforms into [d2 (N+I)2 ] (a)2 - d p 2 + -2- (e2p - 2eP) <,i>(p) = - "2 <,i>(p), (3) which can be identified with (1) after defining x = pd and multiplying by h 2 /2J1,d 2, provided that (4) [= h 2 2 - 2J1,d 2 m , (5) where we have defined m = a /2. In the framework of the U(2) algebra, the operator IV corresponds to the total number of bosons and is fixed by the potential shape according to (4), while m, the eigenvalue of the 50(2) generator i" takes the values m = :f:N/2, :f:(N - 2)/2, .... The Morse spectrum is reproduced twice and consequently for these applications the rn-values must be restricted to be positive. In terms of the U(2) algebra, it is elear from (3-5) that the Morse Hamiltonian has the algebraic realization (6) In addition, the U(2) algebra ineludes the raising and lowering operators .i+, .i_, which connect different energy stat.es in (3), while the angular momentum operator is given by j2 = Ñ(Ñ + 2)/4, as can be readily shown. The Morse Hamiltonian (6) can be rewritten in tlle more convenient fonn (i) wllere we have used the relation .i¡ = .i 2- (';+.L +';_';+)/2 and add"d a constant term AÑ2/4 in order to place the ground state at zero energy. The parameters Nand A appearing in (i) are related to the usual harmonic and anharmonic constants We and XeWe 76 A. FRANK ET AL. used in spectroscopy [7). To obtain this relation it is convenient to introduce the quantum number N V=--ln 2 ' (8) which corresponds to the number of quanta in the oscillator [211. In terms of u, the corresponding energy expression takes the form (N2) A E' = -A m24 = -2(N +1/2) + A(N +l)(u +1/2) - A(u +1/2)2, from which we immediately obtain W e = A(N +1), XeWe=A. (9) (10) Thus, in a diatomic molecule the parameters A and N can be determined by the spectrnscopic constants W e and XeWeWe now consider the U,(2) :J 5U,(2) :J 50,(2) algebra, which is generateo by the set {G;} == {Ñ" j+,;, L", jo,;}, satisfying the commutation relations [Ñ i, j~,d = O, (11) with J1 = :l:, O. As mentioned before, for the symmetric irreducible representation [N;, O] of U;(2) one can show that the Casimir operator is given by [2111;2 = Ñ,UV, + 2)/4, from which follows the identification j, = N;j2. The 50,(2) label is denoted by m,. In the algebraic approach eaeh relevant interatomic interaction is associated with a Ui(2) algebra [11]. As a specific example, we eonsider the ne. cluster, whieh has a tetrahedral shape. 1)3. molecules can be similarly treated. In the ne. case there are six Ui(2) algebras involved (i = 1, ... ,6). The operators in the model are expressed in terms of the generators of these algebras, and the symmetry requirements of the tetrahedral group T dcan be readily imposed [13,22]. The local operators {G;} aeting on bond i can be projected to any of the fundamental irreps r = Al, E and F2. Using the j~" generators (11) we obtain the T,¡ tensors (12) where JI = :l:, O and "1 denotes the component uf r. The explicit. expressions are given by 1 6 • tAl '" J ~,1 = .j6 L. ~," 1=1 t;'1 = 2~ (j",l + j",2 - 2j,.,3 + .J",. - 2.J~,5 + .J" ,6) , A SYMMETRY ADAPTED APPROACH TO MOLECULAR •.. 77 (13) The Hamiltonian operator can be constructed by repeated couplings of these tensors to a total symmetry Al, since it must commute with aH operations in T.t. This is accomplished by means of the T.t-Clebsch-Gordan coefficients [13,22,231. AH calculations are carried out in a symmetry-adapted basis, which is projected from the local basis U1(2) (9 ... (9 U6(2) ::> 501(2) (9 ... (9 506(2) ::> 50(2) ! !! !! IINd, ... , [N6]; VI, ... , V6; V) (14) in which each anharmonic osciHator is weH defined. I3y symmetry considerations, Ni = N for the six oscillators, Vi = N;J2 - mi denotes the phonon number in bond i and V = Li Vi is the total number of phonons [13,21]. The one-phonon states V = 1 are denoted by I i) with Vi = 1 and Vj¡ti = O. Using the same projection technique as for the generators (13), we find the six fundamental modes 6 14>; = L ,';';1 i). i=1 (15) The expansion coefficients are the same as in (13). The higher phonon states v 4>; can also be constructed using the Clebsch-Gordan coefficients of T.t [13,22]. 5ince aH operators are expressed in terms of powers of the U,(2) generators, th,eir matrix elements can be easily evaluated in closed formo The symmetry-adapted operators (13) and states (15) are the building blocks of the model. l\'ole that for more complex molecules, the method aHows the exact elimination of spurious states [1U]. \Ve now proceed to expicitly conslruct the I3e4 Hamiltonian. For interactions lhat are at most quadratic in the generalors the procedure yields (16) with (17) 78 A. FRANK ET AL. Note that we have not included V Al in (16), since the combination ¿(itr+Vr) = NI ¿Ñ;(Ñ i +2), r 4 ; (18) is a constant 3(N +2)/4. The five interaction terms in Eq. (16) correspond to linear combinations of the ones obtained in lowest order in Refs. [11,131. However, it is necessary to include interactions which are related to the vibrational angular momenta associated with the degenerate modes E and F 2• These kind of terms is absent in the former versions of the model [11,131. We now proceed to show how they can be obtained in the AOSM. In configuration space the vibrational angular momentum operator for the Emode is given by [24] i A, _ . (E a E a ) - -1 q¡ aqf - q2 aqf ' (19) where qf and qf are the normal coordinates associated to the Emode. This relation can be transformed to the algebraic space by means of the harmonic oscillator operators to obtain rt _ 1 (r a) b~ - J2 q~ - aq~ ' r l(r a) b~ = J2 q~ + aq~ ' (20) (21) Here bf = L; uf,; b i, with a similar form for b~t, while the uf; can be read from (13). In order to find the algebraic expression for i A, we first introduce a scale transformation in (11) (22) The relevant commutator can be expressed as where • Ñ i• Vi = 2Jo,¡o (23) (24) The other two commutators in (11) are not lIlodified by (22). In the harlIlonic limit, which is defined by Ni ~ 00, Eq. (23) reduces to the standard boson commutator [b i, bl) = 1. This lilIlit corresponds to a contractioll of SU(2) to the Weyl algebra and can be used to A SYMMETRY ADAPTED APPROACH TO MOLECULAR... 79 obtain a geometric interpretation of AOSM operators in terms of those in configuration space. In the opposite sense, Eq. (22) provides a procedure to construct the anharmonic representation of harmonic operators through the correspondence b1 -. ti1 = j -,;/..,fliTi and b. -. ti. = j+.;/..,fliTi. Applying this method to the vibrational angular momentum (21) we find (25) For the vibrational angular momentum if' associated with the F2 mode we find a similar expression. The AOSM form of the corresponding Hamiltonian interactions is NI = 922 i A, i A, + 933 L if' if'. 1 (26) \Vith this method we obtain an algebraic realization of arbitrary configuration space interactions. As a simple example, a one-dimensional harmonic oscillator Hamiltonian Ni = (b1b. + b i b1)/2, transforms into (27) where in the !ast step we used relation (24). The spectrum of (27) has an anharmonic correction, analogous to the quadratic term in the Morse potential spectrum. \Ve are thus substituting harmonic oscillators by Morse oscillators in the AOSM. A more interesting application is to use our model to fit the spectroscopic data of several polyatomic molecules. In the case of I3e4the energy spectrum was analyzed by ab initio methods in [171, where force-field constants corresponding to an expansion of the potential up to fourth order in the normal coordinates and momenta were evaluated. We have generated the ab initio spectrum up to three phonons using the analysis in [24). For the algebraic Hamiltonian we take [15] N = Wl HA, + W2 HE + W3 HF, + X33 (HF,) 2 + X 12 (HA, HE) +X I3 (HA, HF,) + 933 L if' if' + t33 0 33 + t13 0 13. (28) 1 The terms 0 33 and 0 23 represent the algebraic form of the corresponding interactions in [24) which are responsible for the splitting of the vibrational levels in the (VI, v2' v!) = (0,0°,2 2) and the (0,1 1 ,1 1) overtones [15). In Table I we show the fit to Be4 using the Hamiltonian (28). The fit includes all levels up to V = 4 phonon sta tes and gives a r.m.s. deviation of 2.6 cm1, which can be considered of spectroscopic quality. In Table 1 we only show the results for V :5 3 levels. We point out that in [17,241 several higher order interactions are present which we have neglected. Since our mode! can be put into a one to one correspondence with the 80 A. FRANKET AL. TABLE I. Vibrational excitations of Be, using the algebraic Hamiltonian with parameters given in the texto The ab initio (N - 00) spectrum is generated with the parameters from [17J. The energies are given in cm-l. V (VI, v2", v!) r Ab initio Present V (Vh v;', v~) r Ab initio Present N-oo N =44 N-oo N =44 1 (1,0°,0°) A, 638.6 637.0 3(1,0°,2°) A, 2106.8 2105.6 (0,1',0°) E 453.6 455.0 (1,0°,2') E 2000.1 1999.8 (0,0°,1') F, 681.9 678.2 F, 2056.8 2052.8 2 (2,0°,0°) A, 1271.0 1269.2 (0,3',0°) E 1341.3 1343.7 (1,1',0°) E 1087.1 1087.0 (0,33,0°) A, 1355.5 1352.5 (1,0°,1') F, 1312.6 1308.3 A, 1355.5 1354.4 (0,2°,0°) Al 898.3 901.4 (0,2°.',1 1) F, 1565.5 1565.7 (0,2',0°) E 905.4 906.1 F, 1584.4 1583.1 (0,1',1') F, 1126.7 1125.1 (0,2',1') F, 1578.5 1578.0 F, 1135.5 1134.1 (O,1',2°.') E 1821.4 1821.6 (0,0°,2°) A, 1484.0 1483.0 E 1929.5 1929.0 (0,0°,2') E 1377.3 1373.9 (O,1',2') A, 1813.3 1813.1 F, 1434.1 1429.6 A, 1830.8 1831.7 3 (3,0°,0°) A, 1897.0 1896.7 F, 1874.4 1873.2 (2,1',0°) E 1714.3 1714.3 F, 1883.2 1883.0 (2,0°,1') F, 1937.0 1933.7 (0,0°,3,,3) F, 2136.5 2134.2 (1,2°,0°) A, 1526.6 1529.2 F, 2327.3 2326.9 (1,2',0°) E 1533.7 1532.8 (0,0° ,3') F, 2199.8 2197.1 (1,1',1') F, 1752.2 1749.7 A, 2256.5 2254.4 F, 1761.0 1759.8 configuration space calclllations, it is in fact possible to improve the accllracy of the fit considerably, but we have used a simpler Hamiltonian than the one of [17,24]. \Vhen no ab initio ca1clllations are available (or feasible) the AOS~I approach can be used empirically, achieving increasingly good fits by the inclusion of higher order interactions [191. The Be, Hamiltonian (28) preserws the total phonon-number V. This is a good approximation for this case accOlding to the analysis of 117,24], but it is known that Fermi resonances can occur for certain molecu!es when the fundamental mode frequencies are such that (V, V') states with V i V' are close in energy. These interactions can be introduced in the lIamiltonian but the size of the energy matrices grows very rapidly, so the best way to deal with this problem is through perturbation theOlY. For D3h molecules we follow an analogous proced ure, nillnely, we construct the D3h symmetry-adapted operators and states cOlresponding to (13) and (15) and carry out the building-up procedure to construct the Hamiltonian amI higher phonon states [16]. Here we omit the details fOl lack ofspace and only present the fit to the energy spectrum [16,25]. A SYMMETRYADAPTEDAPPROACHTO MOLECULAR... 81 TABLE11.Least-square energy lit for the vibrational excitations of Hj, Be3 and Naj. The energy differences I:::t..E = E th - E exp are given in cm-l. H+ Be3 Naj 3 V (v"v~) r tJ.E tJ.E tJ.E 1 (O, 1') E -1.55 0.51 0.93 (1,0°) A, 0.42 0.02 1.95 2(0,2°) A, 7.48 -0.74 0.37 (0,2 2) E -5.69 0.17 0.84 (U') E -0.61 0.82 1.68 (2,0°) A, -0.11 -0.04 1.26 3 (0,3' ) E -4.46 -2.05 -1.19 (0,33) A, 3.18 -1.23 -0.34 (0,33) A 2 2.44 0.61 -0.33 (1,2°) Al 0.66 1.90 -0.01 (1,2 2) E -5.00 -1.36 0.34 (2, 1') E 4.07 0.79 -0.19 (3,0°) A, -1.23 -1.66 -2.06 r.m.s. 5.84 1.35 1.33 Parameters 84 4 In Table II we present AOSM lits to the speetra of 13e3,Nat and Ht up to three phonons. While remarkably aeeurate deseriptions of the first two moleeules can be aehieved using a four-parameter Hamiltonian, we had to inelude four additional higher order terms in the Ht Hamiltonian in order to properly describe this moleeule. This is in aeeordanee with the work of Carter and Meyer [18], who were foreed to inelude twiee as many terms in the potential energy surfa'Ce for Ht than for the Nat moleeule. The Ht ion is a very "soft" moleeule whieh, due to the light mass of its atomie eonstituents earries out large amplitude oseillations from its equilibrium positions [18]. In another test of the model we studied the vibrational energies of two ozone isotopes, 16 0 3 and 18 0 3 [26]. A least-square fit to all published experimental levels (up to ten quanta) yields a r.m.S. deviation of 2.5 and 1.0 em-', respeetively. A still finer test for the model is to use the wave funetions to evaluate infrared and Raman transitions. The algebraie realization of the transition operators can be obtained from their expression in eonliguration spaee using the large N eonneetion, or purely algebraieally by their tensorial properties under the relevant point group [19]. \Ve remark that the rnadel can also be extended to include tite rotational degrees of freerIom, by coupling the vibrational wave funetions to rotational states properly symmetrized to carry the point group representations [24].