A general algebraic model for molecular vibrational spectroscopy

arXiv:physics/9612006v1 [physics.chem-ph] 10 Dec 1996 A general algebraic model for vibrational molecular spectroscopy A. Frank1,2R. Lemus1F. P´erez-Bernal3R. Bijker1and J.M. Arias3 1Instituto de Ciencias Nucleares, U.N.A.M., A.P. 70-543, 04510 M´exico D.F., M´exico 2Instituto de F´ısica, Laboratorio de Cuernavaca, A.P. 139-B, Cuernavaca, Morelos, M´exico 3Departamento de F´ısica At´omica, Molecular y Nuclear, Facultad de F´ısica, Apdo. 1065, 41080 Sevilla, Espa˜na 1 Algebraic Model We present a symmetry-adapted version of the vibron model [1]. The model exploits the isomorphism of the U(2) Lie Algebra and the one dimensional Morse oscillator [4]. AU(2) algebra is assigned to each relevant interatomic interaction. The operators in the model are expressed in terms of the generators of the molecular dynamical group [3], which in the case of triatomic molecules is given by the product U1(2) ⊗U2(2) ⊗U3(2) .(1) A simple realization for those generators is given in terms of angular momentum ˆ Jν,i and number ˆ Nioperators {ˆ Ni,ˆ Jx,i,ˆ Jy,i,ˆ Jz,i}, i = 1,2,3.(2) Instead of working with the generators in Eq. (2) we introduce a new set of generators with well-defined tensorial properties under the point group [2]. The choice of cartesian coordinate system, irreducible representations and Clebsch-Gordan coefficients is given in Ref. [3]. The use of the symmetry-adapted generators allows the connection between the algebraic and the configuration space calculations clarifying the geometrical content of the algebraic approach [2]. The relevant symmetry projected generators for D3htriatomic molecules are ˆ TA1 µ,1=1 √3ˆ Jµ,1+ˆ Jµ,2+ˆ Jµ,3, ˆ TE µ,1=1 √62ˆ Jµ,1−ˆ Jµ,2−ˆ Jµ,3,ˆ TE µ,2=1 √2ˆ Jµ,2−ˆ Jµ,3,(3) with µ= +,−,0. 1 The algebraic Hamiltonian is constructed by repeated couplings of these tensors to a total symmetry A1. Terms quadratic in the generators and its products can be expressed in terms of Casimir operators used in former algebraic approaches [4,5]. However, other couplings, which are physically relevant, like ˆ l2written below cannot be expressed in terms of Casimir operators and consequently are not in previous algebraic approaches. According to this we obtain the Hamiltonian ˆ H=αˆ HA1+βˆ HE+γˆ VA1+δˆ l2+α[2] ˆ H2 A1+β[2] ˆ H2 E+ξ[2] ˆ HA1E+ǫ(ˆ T3 ++T3 −),(4) where ˆ HΓx=1 2NxX γˆ TΓx −,γ ˆ TΓx +,γ +ˆ TΓx +,γ ˆ TΓx −,γ,ˆ VΓx=1 NxX γ ˆ TΓx 0,γ ˆ TΓx 0,γ ,(5) and ˆ l=−i√21 Nb [ˆ TE −׈ TE +]A2,ˆ T±=ˆ TE 1±iˆ TE 2, ˆ HA1E≡(ˆ HA1ˆ HE+ˆ HEˆ HA1) 2.(6) The Hamiltonian diagonalization and parameter fitting procedures are enhanced using a symmetry-adapted basis [3]. 2 Application to H+ 3, Be3and Na+ 3 The three D3hsymmetric chosen molecules exhibit a wide range of behaviours, from the very anharmonic spectrum of H+ 3to the almost harmonic of Na+ 3. We present in Table I a least square fit calculation to the ab initio calculated vibrational spectrum for these molecules, using an optimal set of parameters of the Hamiltonian. In the H+ 3case, due to its anharmonicity, we had to include the full set of interactions in Eq. (4) to reproduce the spectrum. The relationship between operators and anharmonicity is clear once the harmonic limit of the AOSM model is explored [2] and the link to configuration space calculations is analyzed. In addition to the 8 parameters in Eq. (4), the value of the boson number N[4] has to be fixed and it was taken to be NNa+ 3=NBe3=NH+ 3= 30. This work suggests that the AOSM represents a systematic, simple and accurate alternative to configuration space methods when the integro-differential approach becomes too complex to be applied. Support by the European Community, contract nr. CI1∗-CT94-0072, DGAPAUNAM, project IN105194, CONACyT-M´exico, project 400340-5-3401E and Spanish DGCYT, project PB95-0533, is acknwoledged. Table I: Least square energy fit to ab initio calculations [6] for Na+ 3, Be3and H+ 3using Hamiltonian (4). We show the energy differences ∆E=Eabinit −Ealg. All energies are given in cm−1. Na+ 3Be3H+ 3 (vA1vl E) Γ ∆E ∆E ∆E (011) e 0.93 0.51 -1.55 (100) a11.95 0.02 0.42 (020) a10.37 -0.74 7.48 (022) e 0.84 0.17 -5.69 (111) e 1.68 0.82 -0.61 (200) a11.26 -0.04 -0.11 (031) e -1.19 -2.05 -4.46 (033) a1-0.34 -1.23 3.18 (033) a2-0.33 0.61 2.44 (120) a1-0.01 1.90 0.66 (122) e 0.34 -1.36 -5.00 (211) e -0.19 0.79 4.07 (300) a1-2.06 -1.66 -1.23 rms 1.33 1.24 5.84 α142.40 458.91 3193.60 α[2] -14.86 Parameters β100.32 396.27 2507.16 β[2] -27.75 γ21.31 209.74 2807.83 ξ[2] -28.04 δ-0.19 -0.95 -13.44 ǫ-0.90 References [1] F. Iachello and R.D. Levine, ‘Algebraic Theory of Molecules’, (Oxford University Press, 1995). [2] F. P´erez-Bernal, R. Bijker, A. Frank, R. Lemus and J.M. Arias, Chem. Phys. Lett. 258, 301-306 (1996). [3] A. Frank, R. Lemus, R. Bijker, F. P´erez-Bernal and J.M. Arias, Ann. of Phys. Nov. (1996) In press. [4] A. Frank and P. van Isacker, ‘Algebraic Methods in Molecular & Nuclear Structure Physics’, (Wiley Interscience, 1994). [5] O.S. Van Roosmalen, I. Benjamin and R.D. Levine J. Chem. Phys. 81, 5986- 5997 (1984). [6] A.P. Rendell, T.J. Lee and P.R. Taylor, J. Chem. Phys. 92, 7050-7056 (1990). S. Carter and W. Meyer, J. Chem. Phys. 93, 8902-8914 (1990).