A Symmetry-Adapted Algebraic Approach to Molecular Spectroscopy

arXiv:physics/9611029v1 [physics.chem-ph] 29 Nov 1996 A Symmetry-Adapted Algebraic Approach to Molecular Spectroscopy A. Frank1,2), R. Lemus1), R. Bijker1), F. P´erez-Bernal3) and J.M. Arias3) 1) Instituto de Ciencias Nucleares, U.N.A.M., A.P. 70-543, 04510 M´exico D.F., M´exico 2) Instituto de F´ısica, Laboratorio de Cuernavaca, A.P. 139-B, Cuernavaca, Morelos, M´exico 3) Departamento de F´ısica At´omica, Molecular y Nuclear, Facultad de F´ısica, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Espa˜na 1 Introduction The study of molecular vibrational spectra [1] requires theoretical models in order to analyze and interpret the measurements. These models range from simple parametrizations of the energy levels, such as the Dunham expansion [2], to ab initio calculations, where solutions of the Schr¨odinger equation in different approximations are sought [3, 4, 5, 6]. In general, the latter involve the use of internal coordinates and the evaluation of force field constants associated to derivatives at the potential minima. While this method can be reliably applied to small molecules [7], it quickly becomes a formidable problem in the case of larger molecules, due to the size of their configuration spaces. New calculational tools to describe complex molecules are thus needed. In 1981 an algebraic approach was proposed to describe the roto-vibrational structure of diatomic molecules [8], subsequently extended to linear triand fouratomic molecules [9] and certain non-linear triatomic molecules [10]. Although these were encouraging results, the model could not be extended to polyatomic molecules, due to the impossibility of incorporating the underlying discrete symmetries. This difficulty could be surmounted by treating the vibrational degrees of freedom separately from the rotations. In 1984 Van Roosmalen et al. proposed a U(2) based model to describe the stretching vibrational modes in ABA molecules [11] which was later extended to describe the stretching vibrations of polyatomic molecules such as octahedral and benzene like molecules [12]. Recently the bending modes have also been included in the framework, which was subsequently applied to describe C2v-triatomic molecules [13] and the lower excitations of tetrahedral molecules [14], using a scheme which combines Lie-algebraic and point group methods. In a different approach, it has also been suggested [15] to use a U(k+ 1) model for the k= 3n−3 rotational and vibrational degrees of freedom of a n-atomic molecule. This model has the advantage that it incorporates all rotations and vibrations and takes into account the relevant point group symmetry, but for larger molecules the number of possible interactions and the size of the Hamiltonian matrices increase very rapidly, making it impractical to apply. Although the algebraic formulations have proved useful, several problems remained, most important of which is the absence of a clear connection to traditional methods. On the other hand, a related problem is the lack of a systematic procedure to construct all physically meaningful interactions in the algebraic space. In this paper we show that both these issues can be resolved by considering a symmetryadapted version of the U(2) algebraic model for the analysis of molecular vibrational spectra. In this approach it is possible to construct algebraic operators with well defined physical meaning, in particular interactions fundamental for the description of the degenerate modes present in systems exhibiting high 1 degree of symmetry. The procedure to construct them takes full advantage of the discrete symmetry of the molecule and gives rise to all possible terms in a systematic fashion, providing a clear-cut connection between the algebraic scheme and the traditional analyses based on internal coordinates, which correspond to the harmonic limit of the model [16]. As a test of the symmetry-adapted approach we discuss an application to three D3h-triatomic molecular systems, namely H+ 3, Be3and Na+ 3, and to two tetrahedral molecules, the Be4cluster and the methane molecule. 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. We establish an exact correspondence between configuration space and algebraic interactions by studying the harmonic limit of the U(2) algebra. This general procedure not only allows to derive algebraic interactions from interactions in configuration space, but can also be applied to cases for which no configuration space interactions are available. The D3h-triatomic molecules constitute the simplest systems where degenerate modes appear and where the new interactions in the model become significant. In the case of Be4we present a comparison with ab initio calculations, while for CH4we present a detailed comparison with experiment. The application of these techniques to other molecules, as well as a more complete presentation can be found in references [16, 19, 20, 21]. 2 The U(2) vibron model The model is based on the isomorphism of the U(2) Lie algebra and the one dimensional Morse oscillator H=−¯h2 2µ d2 dx2+D(e−2x/d −2e−x/d),(1) whose eigenstates Ecan be associated with U(2) ⊃SO(2) states [22]. In order to see how this isomorphism comes about, consider the radial equation 1 2−1 r d dr rd dr +σ2 r2+r2φ(r) = (N+ 1)φ(r),(2) which corresponds to a two-dimensional harmonic oscillator (in units where ¯h=µ=e= 1) associated to a U(2) symmetry algebra [23]. By carrying out a change of variable r2= (N+ 1)e−ρ, Eq. (2) transforms into −d2 dρ2+N+ 1 22 (e−2ρ−2e−ρ)φ(ρ) = −σ 22 φ(ρ).(3) This can be identified with Eq. (1) after defining x=ρd and multiplying by ¯h2/2µd2, provided that D=¯h2 8µd2(N+ 1)2,(4) E=−¯h2 2µd2m2,(5) where we have defined m=σ/2. In the framework of the U(2) algebra, the operator ˆ Ncorresponds to the total number of bosons and is fixed by the potential shape according to Eq. (4), while m, the eigenvalue of the SO(2) generator Jz, takes the values m=±N/2, ±(N−2)/2,.... The Morse spectrum is reproduced twice and consequently for these applications the m-values must be restricted to be positive. In terms of the U(2) algebra, it is clear from Eqs. (3-5) that the Morse Hamiltonian has the algebraic realization ˆ H=−¯h2 2µd2ˆ J2 z=−Aˆ J2 z.(6) 2 In addition, the U(2) algebra includes the raising and lowering operators ˆ J+and ˆ J−, which connect different energy states, while the angular momentum operator is given by ˆ J2=ˆ N(ˆ N+ 2)/4, as can be readily shown. The Morse Hamiltonian of Eq. (6) can be rewritten in the more convenient form ˆ H′=ˆ H+Aˆ N2 4=A 2[( ˆ J+ˆ J−+ˆ J−ˆ J+)−ˆ N],(7) where we have used the relation ˆ J2 z=ˆ J2−(ˆ J+ˆ J−+ˆ J−ˆ J+)/2 and added a constant term Aˆ N2/4 in order to place the ground state at zero energy. The parameters Nand Aare related to the usual harmonic and anharmonic constants ωeand xeωeused in spectroscopy. To obtain this relation it is convenient to introduce the quantum number v=N 2−m , (8) which corresponds to the number of quanta in the oscillator. In terms of v, the corresponding energy expression takes the form E′=−A(m2−N2 4) = −A 2(N+ 1/2) + A(N+ 1)(v+ 1/2) −A(v+ 1/2)2,(9) from which we immediately obtain ωe=A(N+ 1) , xeωe=A . (10) Thus, in a diatomic molecule the parameters Aand Ncan be determined by the spectroscopic constants ωeand xeωe. We now consider the Ui(2) ⊃SUi(2) ⊃SOi(2) algebra, which is generated by the set {ˆ Gi} ≡ {ˆ Ni,ˆ J+,i,ˆ J−,i,ˆ J0,i}, satisfying the commutation relations [ˆ J0,i,ˆ J±,i] = ±ˆ J±,i ,[ˆ J+,i,ˆ J−,i] = 2 ˆ J0,i ,[ˆ Ni,ˆ Jµ,i] = 0 ,(11) with µ=±,0. As mentioned before, for the symmetric irreducible representation [Ni,0] of Ui(2) one can show that the Casimir operator is given by ~ J2 i=ˆ Ni(ˆ Ni+ 2)/4 [23], from which follows the identification ji=Ni/2. The SOi(2) label is denoted by mi. 3 The Be4cluster As a specific example, we consider the Be4cluster, which has a tetrahedral shape. D3hmolecules can be similarly treated. In the Be4case there are six Ui(2) algebras involved (i= 1,...,6). In the present approach each relevant interatomic interaction is associated with a Ui(2) algebra. The operators in the model are expressed in terms of the generators of these algebras, and the symmetry requirements of the tetrahedral group Tdcan be readily imposed [14, 24]. The local operators {ˆ Gi}acting on bond ican be projected to any of the fundamental irreps Γ = A1,Eand F2. Using the ˆ Jµ,i generators we obtain the Tdtensors ˆ TΓ µ,γ = 6 X i=1 αΓ γ,i ˆ Jµ,i ,(12) where µ=±,0 and γdenotes the component of Γ. The explicit expressions are given by ˆ TA1 µ,1=1 √6 6 X i=1 ˆ Jµ,i , 3 ˆ TE µ,1=1 2√3ˆ Jµ,1+ˆ Jµ,2−2ˆ Jµ,3+ˆ Jµ,4−2ˆ Jµ,5+ˆ Jµ,6, ˆ TE µ,2=1 2ˆ Jµ,1−ˆ Jµ,2−ˆ Jµ,4+ˆ Jµ,6, ˆ TF2 µ,1=1 √2ˆ Jµ,1−ˆ Jµ,6, ˆ TF2 µ,2=1 √2ˆ Jµ,2−ˆ Jµ,4, ˆ TF2 µ,3=1 √2ˆ Jµ,3−ˆ Jµ,5.(13) The Hamiltonian operator can be constructed by repeated couplings of these tensors to a total symmetry A1, since it must commute with all operations in Td[14]. All calculations are carried out in a symmetry-adapted basis, which is projected from the local basis U1(2) ⊗ ··· ⊗ U6(2) ⊃SO1(2) ⊗ ··· ⊗ SO6(2) ⊃SO(2) ↓ ↓ ↓ ↓ ↓ |[N1], . . . , [N6] ; v1, . . . , v6;Vi (14) in which each anharmonic oscillator is well defined. By symmetry considerations, Ni=Nfor the six oscillators, vi=Ni/2−midenotes the number of quanta in bond iand V=Piviis the total number of quanta. The local basis states for each oscillator are usually written as |Ni, vii, where vi= (Ni−2mi)/2 = 0,1,...[Ni/2] denotes the number of oscillator quanta in the i-th oscillator. The states with one quantum V= 1 are denoted by |iiwith vi= 1 and vj6=i= 0. Using the same projection technique as for the generators (13), we find the six fundamental modes |1φΓ γi= 6 X i=1 αΓ γ,i |ii.(15) The expansion coefficients are the same as in Eq. (13). The states with a higher number of quanta |VφΓ γi can be constructed using the Clebsch-Gordan coefficients of Td[14, 24]. Since all operators are expressed in terms of powers of the Ui(2) generators, their matrix elements can be easily evaluated in closed form. The symmetry-adapted operators of Eq. (13) and symmetry-adapted basis states are the building blocks of the model. We now proceed to expicitly construct the Be4Hamiltonian. For interactions that are at most quadratic in the generators the procedure yields ˆ H0=ω1ˆ HA1+ω2ˆ HE+ω3ˆ HF2+α2ˆ VE+α3ˆ VF2,(16) with ˆ HΓ=1 2NX γˆ TΓ −,γ ˆ TΓ +,γ +ˆ TΓ +,γ ˆ TΓ −,γ, ˆ VΓ=1 NX γ ˆ TΓ 0,γ ˆ TΓ 0,γ .(17) Note that we have not included ˆ VA1in ˆ H0, since the combination X Γˆ HΓ+ˆ VΓ=1 4N 6 X i=1 ˆ Ni(ˆ Ni+ 2) ,(18) is a constant 3(N+ 2)/2 . The five interaction terms in Eq. (16) correspond to linear combinations of the Casimir operators of [14]. However, for a good description of the vibrational energies of Be4it is 4 necessary to include interactions which are related to the vibrational angular momenta associated with the degenerate modes Eand F2. These kind of terms is absent in the former versions of the model [12, 14]. We now proceed to show how they can be obtained in the present formalism. In configuration space the vibrational angular momentum operator for the Emode is given by [25] ˆ lA2=−iqE 1 ∂ ∂qE 2−qE 2 ∂ ∂qE 1,(19) where qE 1and qE 2are the normal coordinates associated to the Emode. This relation can be transformed to the algebraic space by means of the harmonic oscillator operators bΓ† γ=1 √2qΓ γ−∂ ∂qΓ γ, bΓ γ=1 √2qΓ γ+∂ ∂qΓ γ,(20) to obtain ˆ lA2=−ibE† 1bE 2−bE† 2bE 1.(21) Here bE γ=PiαE γ,i bi, with a similar form for bΓ† γ, while the αE γ i can be read from Eqs. (12,13). In order to find the algebraic expression for ˆ lA2we first introduce a scale transformation ¯ b† i≡ˆ J−,i/pNi,¯ bi≡ˆ J+,i/pNi.(22) The relevant commutator can then be expressed as [¯ bi,¯ b† i] = 1 Ni [ˆ J+,i,ˆ J−,i] = 1 Ni 2ˆ J0,i = 1 −2ˆvi Ni ,(23) where ˆvi=ˆ Ni 2−ˆ J0,i .(24) The other two commutation relations of Eq. (11) are not modified by the scale transformation of Eq. (22). In the harmonic limit, which is defined by Ni→ ∞, Eq. (23) reduces to the standard boson commutator [¯ bi,¯ b† i] = 1. This limit corresponds to a contraction of SU(2) to the Weyl algebra and can be used to obtain a geometric interpretation of algebraic 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 b† i→¯ b† i=ˆ J−,i/√Niand bi→¯ bi=ˆ J+,i/√Ni. Applying this method to the vibrational angular momentum we find ˆ lA2=−i Nˆ TE −,1ˆ TE +,2−ˆ TE −,2ˆ TE +,1.(25) For the vibrational angular momentum ˆ lF1 γassociated with the F2mode we find a similar expression. The corresponding interactions are ˆ H1=g22 ˆ lA2ˆ lA2+g33 X γ ˆ lF1 γˆ lF1 γ.(26) With this method we obtain an algebraic realization of arbitrary configuration space interactions. As a simple example, a one-dimensional harmonic oscillator Hamiltonian ˆ Hi= (b† ibi+bib† i)/2, transforms into 1 2N(ˆ J−,i ˆ J+,i +ˆ J+,i ˆ J−,i) = 1 N(ˆ J2 i−ˆ J2 0,i) = ˆvi+ 1/2−ˆv2 i N,(27) 5 where in the last step we used Eq. (24). The spectrum of Eq. (27) has an anharmonic correction, analogous to the quadratic term in the Morse potential spectrum. We are thus substituting harmonic oscillators by Morse oscillators. A more interesting application is to use our model to fit the spectroscopic data of several polyatomic molecules. In the case of Be4the energy spectrum was analyzed by ab initio methods in [17], 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 quanta using the analysis in [25]. For the algebraic Hamiltonian we take [16] ˆ H=ω1ˆ HA1+ω2ˆ HE+ω3ˆ HF2+X12 ˆ HA1ˆ HE+X13 ˆ HA1ˆ HF2+X33 ˆ HF22 +g33 X γ ˆ lF1 γˆ lF1 γ+t33 ˆ O33 +t23 ˆ O23 ,(28) The terms ˆ O33 and ˆ O23 represent the algebraic form of the corresponding interactions in [25] which are responsible for the splitting of the vibrational levels in the (ν1, νm 2, νl 3) = (0,00,22) and the (0,11,11) overtones [16]. In Table I we show the the results of a least-square fit to the vibrational energies of Be4with the Hamiltonian of Eq. (28). The r.m.s. deviation obtained is 2.6 cm−1, which can be considered of spectroscopic quality. We point out that in [25, 26] several higher order interactions are present which we have neglected. Since our model can be put into a one to one correspondence with the configuration space calculations, it is in fact possible to improve the accuracy of the fit considerably, but we have used a simpler Hamiltonian than the one of [25, 26]. When no ab initio calculations are available (or feasible) the present approach can be used empirically, achieving increasingly good fits by the inclusion of higher order interactions [16]. We note that the Be4Hamiltonian of Eq. (28) preserves the total number of quanta V. This is a good approximation for this case according to the analysis of [25, 26], but it is known that Fermi resonances can occur for certain molecules when the fundamental mode frequencies are such that (V, V ′) states with V6=V′are close in energy. These interactions can be introduced in the Hamiltonian but the size of the energy matrices grows very rapidly, so the best way to deal with this problem is through perturbation theory. 4D3htriatomic molecules For D3hmolecules we follow a similar procedure, namely, we construct the D3hsymmetry-adapted operators and states analogous to Eq. (13,15) and carry out the building up procedure to construct the Hamiltonian and states with a higher number of quanta with the appropriate projection operators and Clebsch-Gordan coefficients [19]. In Table II we present the fits to the spectra of Be3, Na+ 3and H+ 3up to three quanta. While remarkably accurate descriptions of the first two molecules can be achieved using a four-parameter Hamiltonian, we were forced to include four additional higher order terms in the H+ 3Hamiltonian in order to properly describe this molecule. This is in accordance with the work of Carter and Meyer [18], who were forced to include twice as many terms in the potential energy surface for H+ 3than for the Na+ 3molecule. The H+ 3ion is a very “soft” molecule which, due to the light mass of its atomic constituents carries out large amplitude oscillations from its equilibrium positions [18]. 5 The methane molecule We now turn our attention to the CH4molecule, for which we shall make a detailed description. For methane we have four U(2) algebras corresponding to the C-H interactions and six more representing the 6 H-H couplings. The assignments and the choice of the Cartesian coordinate system are the same as in [14]. The molecular dynamical group is then given by the product G=U1(2) ⊗U2(2) ⊗...⊗U10(2) .(29) The labeling is such that i= 1,...,4 correspond to the C-H couplings while the other values of i are associated with H-H interactions. Consequently there are two different boson numbers, Nsfor the C-H couplings and Nbfor the H-H couplings, which correspond to the stretching and bending modes, respectively. The tetrahedral symmetry of methane is taken into account by projecting the local operators {ˆ Gi}, which act on bond i, on the irreducible representations Γ of the tetrahedral group Td. The explicit expressions for the Tdtensors for the stretching modes are ˆ TA1,s µ,1=1 2 4 X i=1 ˆ Jµ,i , ˆ TF2,s µ,1=1 2ˆ Jµ,1−ˆ Jµ,2+ˆ Jµ,3−ˆ Jµ,4, ˆ TF2,s µ,2=1 2ˆ Jµ,1−ˆ Jµ,2−ˆ Jµ,3+ˆ Jµ,4, ˆ TF2,s µ,3=1 2ˆ Jµ,1+ˆ Jµ,2−ˆ Jµ,3−ˆ Jµ,4,(30) while for the bending modes we have ˆ TA1,b µ,1=1 √6 10 X i=5 ˆ Jµ,i , ˆ TEb µ,1=1 2√3ˆ Jµ,5+ˆ Jµ,6−2ˆ Jµ,7+ˆ Jµ,8−2ˆ Jµ,9+ˆ Jµ,10, ˆ TEb µ,2=1 2ˆ Jµ,5−ˆ Jµ,6−ˆ Jµ,8+ˆ Jµ,10, ˆ TF2,b µ,1=1 √2ˆ Jµ,5−ˆ Jµ,10, ˆ TF2,b µ,2=1 √2ˆ Jµ,6−ˆ Jµ,8, ˆ TF2,b µ,3=1 √2ˆ Jµ,7−ˆ Jµ,9.(31) As before, the algebraic Hamiltonian can be constructed by repeated couplings of these tensors to a total symmetry A1. The methane molecule has nine vibrational degrees of freedom. Four of them correspond to the fundamental stretching modes (A1⊕F2) and the other five to the fundamental bending modes (E⊕F2) [27]. The projected tensors of Eqs. (30) and (31) correspond to ten degrees of freedom, four of which (A1⊕F2) are related to stretching modes and six (A1⊕E⊕F2) to the bendings. Consequently we can identify the tensor ˆ TA1,b µ,1as the operator associated to a spurious mode. This identification makes it possible to eliminate the spurious states exactly. This is achieved by (i) ignoring the ˆ TA1,b µ,1tensor in the construction of the Hamiltonian, and (ii) diagonalizing this Hamiltonian in a symmetry-adapted basis from which the spurious mode has been removed following the procedure of [14]. We note that the condition on the Hamiltonian that was used in [14] to exclude the spurious contributions, does not automatically hold for states with higher number of quanta. According to the above procedure, we now construct the Tdinvariant interactions that are at most quadratic in the generators and conserve the total number of quanta ˆ HΓx=1 2NxX γˆ TΓx −,γ ˆ TΓx +,γ +ˆ TΓx +,γ ˆ TΓx −,γ, 7 ˆ VΓx=1 NxX γ ˆ TΓx 0,γ ˆ TΓx 0,γ .(32) Here Γ = A1,F2for the stretching vibrations x=sand Γ = E,F2for the bending vibrations x=b. In addition there are two stretching-bending interactions ˆ Hsb =1 2√NsNbX γˆ TF2,s −,γ ˆ TF2,b +,γ +ˆ TF2,s +,γ ˆ TF2,b −,γ , ˆ Vsb =1 √NsNbX γ ˆ TF2,s 0,γ ˆ TF2,b 0,γ .(33) The zeroth order vibrational Hamiltonian is now written as ˆ H0=ω1ˆ HA1,s +ω2ˆ HEb+ω3ˆ HF2,s +ω4ˆ HF2,b +ω34 ˆ Hsb +α2ˆ VEb+α3ˆ VF2,s +α4ˆ VF2,b +α34 ˆ Vsb .(34) The interaction ˆ VA1,s has not been included since, in analogy to Eq. (18), the combination X Γˆ HΓs+ˆ VΓs=1 4Ns 4 X i=1 ˆ Ni(ˆ Ni+ 2) ,(35) corresponds to a constant Ns+ 2. A similar relation holds for the bending interactions, but in this case the interaction ˆ VA1,b has already been excluded in order to remove the spurious A1bending mode. The subscripts of the parameters correspond to the (ν1, νl2 2, νl3 3, νl4 4) labeling of a set of basis states for the vibrational levels of CH4. Here ν1,ν2,ν3and ν4denote the number of quanta in the A1,s,Eb,F2,s and F2,b modes, respectively. The labels liare related to the vibrational angular momentum associated with degenerate vibrations. The allowed values are li=νi, νi−2,...,1 or 0 for νiodd or even [27]. In the harmonic limit the interactions of Eqs. (32) and (33) again attain a particularly simple form, which can be directly related to configuration space interactions. This limit is obtained, as before, by rescaling ˆ J+,i and ˆ J−,i by √Niand taking Ni→ ∞, so that lim Ni→∞ ˆ J+,i √Ni =bi, lim Ni→∞ ˆ J−,i √Ni =b† i, lim Ni→∞ 1 Ni [ˆ J+,i,ˆ J−,i] = lim Ni→∞ 2ˆ J0,i Ni = 1 .(36) where the operators biand b† jsatisfy the standard boson commutation relation [bi, b† j] = δij. Applying the harmonic limit to the interactions of Eqs. (32) and (33) we obtain lim Nx→∞ ˆ HΓx=1 2X γbΓx† γbΓx γ+bΓx γbΓx† γ, lim Nx→∞ ˆ VΓx= 0 , lim Ns,Nb→∞ ˆ Hsb =1 2X γbF2,s † γbF2,b γ+bF2,s γbF2,b † γ, lim Ns,Nb→∞ ˆ Vsb = 0 .(37) 8 Here the operators bΓx† γare given in terms of the local boson operators b† ithrough the coefficients αΓx γ,i given in Eqs. (30,31) bΓx† γ= 10 X i=1 αΓx γ,i b† i,(38) with a similar relation for the annihilation operators. From Eq. (37) the physical interpretation of the interactions is immediate. The ˆ HΓxterms represent the anharmonic counterpart of the harmonic interactions, while the ˆ VΓxterms are purely anharmonic contributions which vanish in the harmonic limit. In an application to the ozone molecule it was found that these terms can account for the strong anharmonicities and incorporate the effect of Darling-Dennison type couplings [20]. The zeroth order Hamiltonian of Eq. (34) is not sufficient to obtain a high-quality fit of the vibrations of methane. Several physically meaningful interaction terms that are essential for such a fit are not present in Eq. (34). They arise in our model as higher order interactions. It is an advantage of the symmetry-adapted model that the various interaction terms have a direct physical interpretation and a specific action on the various modes [16]. Hence the addition of higher order terms and anharmonicities can be done in a systematic way. For the study of the vibrational excitations of methane we use the Td invariant Hamiltonian [21] ˆ H=ω1ˆ HA1,s +ω2ˆ HEb+ω3ˆ HF2,s +ω4ˆ HF2,b +α3ˆ VF2,s +X11 ˆ HA1,s 2 +X22 ˆ HEb2 +X33 ˆ HF2,s 2 +X44 ˆ HF2,b 2 +X12 ˆ HA1,s ˆ HEb+X14 ˆ HA1,s ˆ HF2,b  +X23 ˆ HEbˆ HF2,s +X24 ˆ HEbˆ HF2,b +X34 ˆ HF2,s ˆ HF2,b  +g22 ˆ lA22 +g33 X γ ˆ lF1 s,γ ˆ lF1 s,γ +g44 X γ ˆ lF1 b,γ ˆ lF1 b,γ +g34 X γ ˆ lF1 s,γ ˆ lF1 b,γ +t33 ˆ Oss +t44 ˆ Obb +t34 ˆ Osb +t23 ˆ O2s+t24 ˆ O2b.(39) The interpretation of the ωiand α3terms follows from Eq. (37). The Xij terms are quadratic in the operators ˆ HΓxand hence represent anharmonic vibrational interactions. The gij terms are related to the vibrational angular momenta associated with the degenerate vibrations. As mentioned before, these interactions, which are fundamental to describe molecular systems with a high degree of symmetry, are absent in previous versions of the vibron model in which the interaction terms are expressed in terms of Casimir operators and products thereof [12, 14]. They give rise to a splitting of vibrational levels with the same values of (ν1, ν2, ν3, ν4) but with different l2,l3and/or l4. Their algebraic realization is given by ˆ lA2=−i√21 Nb [ˆ TEb −׈ TEb +]A2, ˆ lF1 x,γ = +i√21 Nx [ˆ TF2,x −׈ TF2,x +]F1 γ.(40) The square brackets in Eq. (40) denote the tensor coupling under the point group Td [ˆ TΓ1׈ TΓ2]Γ γ=X γ1,γ2 C(Γ1,Γ2,Γ; γ1, γ2, γ)ˆ TΓ1 γ1 ˆ TΓ2 γ2,(41) where the expansion coefficients are the Clebsch-Gordan coefficients for Td[14, 24]. In the harmonic limit the expectation value of the diagonal terms in Eq. (39) leads to the familiar Dunham expansion [27] X i ωi(vi+di 2) + X j≥iX i Xij (vi+di 2)(vj+dj 2) + X j≥iX i gij lilj.(42) 9 Table IV: Fit to vibrational excitations of CH4. The values of the parameters are given in the second column of TableIII. Here ∆E=Ecal −Eexp. The experimental energies are taken from [28]. The wave numbers are given in cm−1. Γ (ν1, ν2, ν3, ν4)Ecal Eexp ∆EΓ (ν1, ν2, ν3, ν4)Ecal Eexp ∆E A1(1000) 2916.32 2916.48 –0.16 (0111) 5844.98 E(0100) 1533.46 1533.33 0.13 (1200) 5974.81 F2(0001) 1309.86 1310.76 –0.90 (1011) 7147.49 (0010) 3018.09 3019.49 –1.40 (0021) 7303.38 (2100) 7315.60 A1(0002) 2587.77 2587.04 0.73 (0120) 7479.48 (0200) 3063.66 3063.65 0.01 (0120) 7557.17 (0011) 4323.81 4322.72 1.09 (1020) 8833.05 (2000) 5790.13 5790 0.13 F1(0003) 3920.46 3920.50 –0.04 (0020) 5966.57 5968.1 –1.53 (0102) 4128.38 4128.57 –0.19 E(0002) 2624.14 2624.62 –0.48 (0201) 4364.39 4363.31 1.08 (0200) 3065.22 3065.14 0.08 (0012) 5620.08 (0011) 4323.09 4322.15 0.94 (0012) 5630.76 (1100) 4446.41 4446.41 0.00 (1101) 5755.58 (0020) 6045.03 6043.8 1.23 (0111) 5829.79 F1(0101) 2845.35 2846.08 –0.73 (0111) 5848.94 (0011) 4323.15 4322.58 0.57 (0210) 6061.57 (0110) 4537.57 4537.57 0.00 (1011) 7147.53 F2(0002) 2612.93 2614.26 –1.33 (0021) 7303.29 (0101) 2830.61 2830.32 0.29 (0021) 7343.21 (1001) 4223.46 4223.46 0.00 (1110) 7361.79 (0011) 4321.02 4319.21 1.81 (0120) 7518.70 (0110) 4543.76 4543.76 0.00 (0030) 8947.65 8947.95 –0.30 (1010) 5845.53 F2(0003) 3871.29 3870.49 0.80 (0020) 6003.65 6004.65 –1.00 (0003) 3931.36 3930.92 0.44 (0102) 4143.09 4142.86 0.23 A1(0003) 3909.20 3909.18 0.02 (0201) 4349.01 4348.77 0.24 (0102) 4131.92 4132.99 –1.07 (0201) 4378.38 4379.10 –0.72 (0300) 4595.26 4595.55 –0.29 (1002) 5523.80 (1002) 5498.66 (0012) 5594.92 5597.14 –2.22 (0012) 5617.16 (0012) 5620.68 (0111) 5836.11 (0012) 5632.36 (1200) 5973.26 (1101) 5740.86 (1011) 7147.56 (0111) 5830.28 (0021) 7300.85 (0111) 5848.46 (0120) 7562.91 (0210) 6054.58 (3000) 8583.81 (0210) 6067.03 (1020) 8727.97 (2001) 7094.16 (0030) 8975.64 8975.34 0.30 (1011) 7145.84 A2(0102) 4161.52 4161.87 –0.35 (0021) 7266.11 (0300) 4595.28 4595.32 –0.04 (0021) 7303.38 (0111) 5844.61 (0021) 7344.87 (0120) 7550.53 (1110) 7365.83 E(0102) 4105.22 4105.15 0.07 (0120) 7514.67 (0102) 4152.15 4151.22 0.93 (2010) 8594.90 (0300) 4592.13 4592.03 0.10 (1020) 8786.05 (1002) 5535.04 (0030) 8907.91 8906.78 1.13 (0012) 5620.36 (0030) 9045.36 9045.92 –0.56 (0111) 5836.45 16