One of the main purposes of a theoretical spectroscopist is to propose a model to reproduce the xperimental spectra available in the literature as accurate and with as much physical information as possible. In this case it has been studied the Far infrared an Infrared spectra of non-rigid molecules, introducing in the Vibrational Potential their dependence on the large amplitude variables. The molecule that has been selected to test our proposed Hamiltonian is ethane which is a very well studied molecule. The molecule is a non-rigid molecule and their experimental spectra have been reported in the literature. © 2008, Sociedad Química de MéxicoPeer Reviewe

Hidalgo-Olguín, D.R.

Mora-Delgado, M.A.

Senent, María Luisa

Villa, M.

Journal of the Mexican Chemical Society

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            Available in: http://www.redalyc.org/articulo.oa?id=47552115  Red de Revistas Científicas de América Latina, el Caribe, España y PortugalSistema de Información CientíficaDavid Ricardo Hidalgo Olguín, María Villa, María Luisa Senent, Marco Antonio Mora DelgadoEthane Internal Rotation -Vibrational HamiltonianJournal of the Mexican Chemical Society, vol. 52, núm. 1, 2008, pp. 99-102,Sociedad Química de MéxicoMéxico   How to cite      Complete issue      More information about this article      Journal's homepageJournal of the Mexican Chemical Society,ISSN (Printed Version): 1870-249Xeditor@jmcs.org.mxSociedad Química de MéxicoMéxicowww.redalyc.orgNon-Profit Academic Project, developed under the Open Acces InitiativeJ. Mex. Chem. Soc. 2008, 52(1), 98-102© 2008, Sociedad Química de MéxicoISSN 1870-249XArticleEthane Internal Rotation -Vibrational HamiltonianDavid Ricardo Hidalgo-Olguín,1 María Villa,1* María Luisa Senent,2 and Marco Antonio Mora-Delgado11 Departamento de Química, Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana. Av. San Rafael Atlíxco 186, Col. Vicentina, Iztapalapa, C.P., 09340 México, D.F., México, e-mail: mvv@xanum.uam.mx2 Departamento de Astrofísica Molecular e Infrarroja, Instituto de Estructura de la Materia, C.S.I.C.,Serrano 113 b, Madrid 28006, España. E-mail: mvv@xanum.uam.mxRecibido el 27 de septiembre del 2007; aceptado el 1 de abril de 2008Abstract One of the main purposes of a theoretical spectroscopist is to propose a model to reproduce the xperimental spectra available in the literature as accurate and with as much physical information as possi-ble. In this case it has been studied the Far infrared an Infrared spectra of non-rigid molecules, introducing in the Vibrational Potential their dependence on the large amplitude variables. The molecule that has been selected to test our proposed Hamiltonian is ethane which is a very well studied molecule. The molecule is a non-rigid molecule and their experimental spectra have been reported in the literature.Keywords: Ethane, ab initio, FIR-IR Spectrum.Resumen Uno de los objetivos principales de un espectroscopísta teó-rico es generar un modelo que reproduzca los espectros experimentales con la mayor precisión e información física posible. En este caso se estudió el espectro en el Infrarrojo lejano e Infrarrojo de moléculas no rígidas introduciendo en el potencial vibracional su dependencia de las variables de gran amplitud. La molécula de etano fue seleccionada para comprobar nuestro Hamiltoniano propuesto, ya que es una molécula ampliamente estudiada.; además de ser una molécula no rígida cuyo espectro experimental ha sido reportado en la literatura.Palabras clave: Etano, ab initio, espectro de ir lejano.IntroductionA non-rigid molecule [1,2] is a molecule with internal motions (large amplitude motions) associated to low energy barriers, that interchange various isoenergenetic structures, giving rise to a Far-Infrared spectrum. At the same time these molecules have normal modes of vibration (variations of angles and dis-tances) associated to the vibrational spectrum in the Infrared region.One can use a theoretical model in order to reproduce the experimental spectra reported in the literature. Both motions can be considered independent at a first approach due to its difference in energy, even though in order to have a more accurate reproduction of the experimental spectra one has to consider the interaction between both motions.The purpose of this work is to generate a theoretical and systematic model that will take in consideration the vibrational (Infrared) spectrum dependence with respect to the large amplitude coordinates. Using the model previously mention the final idea is to obtain a theoretical spectrum that can reproduce more precisely the experimental data available and at the same time to contain more physical information of the system.One has to choose a widely studied theoretically and experimentally molecule in order to give support to our study, ethane [3, 4, 5] will be used as good example in this study. The coordinates to be considered are the internal large amplitude rotational angle ´θ´ which defines the rotation of one methyl group with respect to the other one and the normal vibrational C-C stretching coordinate. This work will contribute to the high resolution spectroscopy of ethane. In general the model that is developed can be generalized to study the IR spectra of non rigid molecules.TheoryAs mention in the introduction, the molecular system selected for the present study is ethane.The Hamiltonian to be solved can be associated to three different motions (external rotation, internal rotation and vibrational motion). Considering the difference in energy the hamiltonian at a first approach can be solved separetly: Ĥ = Ĥext-rot + Ĥnt-rot + Ĥvib …… (1)The total Hamiltonian must represent the non-rigid sym-metry group (SG) of the molecule[1] S = [e3I Λ UI ] Λ D3 ~ G36. (2)Each partial Hamiltonian can be written as follows:External rotational Hamiltonian: Ĥext-rot = hAĴ 2 a + hBĴ 2 b + hCĴ 2 c (3)where A B and C are the moments of inertia, h is the Plank Constant and Ĵ 2 a, Ĵ 2 b, Ĵ 2 c the corresponding angular momentum operators, in the principal axes system.Internal rotational Hamiltonian:For this case in order to make the Hamiltonian simpler it will be taken only the H-C-C-H dihedral angle (θ) as internal variable: int rot int rot int -rotˆ ( ) ( )H V? ? ?? ?? ?? ?? ? ?? ? . (4)100   J. Mex. Chem. Soc. 2008, 52(1) D.R. Hidalgo.Olguín et al.For simplicity, the only internal variable considered in this Hamiltonian is the H-C-C-H dihedral angle. 2int rot int rot int -rot2ˆ ( )H V? ??? ??? ? ?? . (5) The int-rot potential is expressed as an expansion in terms of cos(θ) functions and it has to be a symmetry adapted θ func-tion (f (θ) = f (θ+120°)). int rot,3int -rot( ) cos (3 )nnV n?V? ??? . (6)The potential will be truncated up to third order terms:int rot,0 int rot,3 int rot,6int -rotint rot,9( ) cos (3 ) cos (6 )cos (9 )V ? ?V V V?V? ? ? ??? ? ??  (7)Vibrational HamiltonianOnly one normal vibrational mode (C-C stretching) is being considered: 2 22ˆ ( ).2vib vibH V SS?? ? ??? (8).The Hamiltonian is written as a function of the internal displacement coordinate defined in terms of the carbon-carbon distance displacement (∆(C-C)) around the equilibrium posi-tion: ? ?? ?,S C C?? ? ?  (9)Where µ is the reduce mass of two carbon atoms, each one with mass mc. 22ccmm? ? . (10)The vibrational potential energy (Vvib(S)) can be written as a Taylor expansion ( )1( ) !n nvib vibnV S V Sn?? , (11)Where: ( ) ( )nn vibvib neqV SV S?? ??? ? ?? ?? ?. (12)Each term must be of a certain order of magnitude in order for the series to converge, and in this particular case the func-tion will be truncated up to the third order term: '' 3 ''' 301 1( ) 2 6vib vib vibV S V V S V S? ? ? . ( 13 )Up to this point the internal rotation and vibrational Hamiltonians have been considered as independent. In order to considered the dependence of the vibrational Hamiltonian (Hvib) in terms of the internal variable θ, the potential function can be written as function of both variables (S and θ): '' 2 ''' 301 1( , ) ( ) ( ) ( )2 6vib vib vibV S V V S V S? ? ? ?? ? ? . ( 14 )Vibrational Spectra can be obtained by calculating the vibrational part in two different ways: a) θ-independent (equa-tion 10) and b) θ-dependent (equation 11)MethodologyOnce that the system and the methodoly hve been selected the first the first thing to be done is to obtain the minimun energy conformation by using the Gaussian03 program, MP2 method, 6-311G** basis set. At the same time the harmonic analysis of the minimum was also done.In Table 1 it is observed that that the method and the basis used in this work reproduce quite reasonably the more pre-cise methods reported in the literature. Other results obtained are the Rotational constants and the frequencies under study (Tables 2 and 3).The method for calculating the β parameter from the opti-mized geometry is well described in the literature [7, 8] and it has been used for the ethane molecule. β =10.76190 cm-1 (15)As it was mentioned before, it will be considered a con-stant and not q dependent.By using the same method the energies at different values of θ (each ∆θ = 10°) were calculated:Table 1. Minimum energy Conformation (θ = 60º)parameters 6-311G** CC-pCVTZ [6] C-C  1.5420 Å 1.5265 Å C-H  1.0954 Å 1.0908 Å H-C-H  107.8368º 107.68° H-C-C  111.0652º 111.21°  θ  60º 60°E(cm-1) 17,428,896.3Ethane Internal Rotation-Vibrational Hamiltonian 101The final result is a θ-dependent vibrational potential, similar to eq. (14), which will be used for obtaining the so called coupled spectra where the vibrational potential includes the θ dependence which is our main purpose: V(S,θ) = 0.2179 S - 0.2644 S Cos3θ + 0.3068 S Cos6θ                     - 0.5569 S Cos9θ + 35.8196 S2 - 0.2074 S2 Cos3θ                     - 0.0223 S2 Cos6θ + 0.0232 S2 Cos9θ - 1.0637 S3                     + 0.0081 S3 Cos3θ + 0.0003 S3 Cos6θ                      - 0.0012 S3 Cos9θ	 (19)SpectraThe energy levels are calculated variationally using a two-dimensional flexible model where the torsional and the vibra-tional modes are considered independent variables and the remaining 3N-8 coordinates are allowed to be relaxed.The program bidim [9] has been used for calculating the spectra.a) With the internal rotation potential and (eq. 16) and the one-dimensional vibrational potentials (eq. 17) a non-cou-pled spectra is calculated for each θ. In Table 5 only the results for θ (0,30,60) are shown:The vibrational energy levels have been classified by the vibrational (nvib) and rotational (nrot) quantum numbers. It is important to note the triple degeneracy of the energy levels, in accordance to the molecular symmetry and the angle depen-dence of the vibrational frequency (ν)b) The coupled spectra is calculated by using the internal rotation potential and the two-dimensional (S, θ ) vibrational potential (eq. 19).Table 2. Rotational constants.A (cm-1) B (cm-1) C (cm-1)2.6680 0.6577 0.6577Table 3. Frequencies.Frequency (cm-1) Normal mode327.5665 CH3 Rotation1012.7524 C-C StretchingTable 4. Potential Energies at different θ angle. 0º 10º 20º  30º 40º 50º 60ºE (cm-1) 1135.07 1057.8 847.399 562.385 280.037 74.897 0Table 5. Uncoupled energy levels (cm-1).nvib  nrot 0° 30° 60°0 0 0.0000 0.0000 0.00000 0 0.0032 0.0032 0.00320 0 0.0032 0.0032 0.00320 1 303.2862 303.2862 303.28620 1 303.2862 303.2862 303.28620 1 303.4232 303.4232 303.42320 2 575.5118 575.5118 575.51180 2 577.9414 577.9414 577.94140 2 577.9414 577.9414 577.94140 3 804.7870 804.7870 804.78700 3 804.7870 804.7870 804.78700 3 827.1955 827.1955 827.19551 1 997.8629 v 1001.2754 v 1003.5018 v1 1 997.8660 v 1001.2785 v 1003.5050 v1 1 997.8660 v 1001.2785 v 1003.5050 vEach value and conformation is considered a partial mini-mum, and the int-rot potential energy is obtained by fitting the energies to the symmetry adapted function expressed in eq. (7): Vint-rot (θ) = 565.0016 cm-1 +567.4783 cm-1 cos3θ + 2.5786 cm-1 cos6θ + 0.0576 cm-1 cos9θ. (16)For each partial minimum conformation the C-C distance is varied from -0.04 Å up to 0.04 Å in 0.005 Å increments and keeping constant the other molecular parameters. The results are fitted to the Vvib function (eq. 14) for each θ angle. This vibrational potentials are the ones called non θ-dependent, because θ is a parameter and not a function.Vibrational potentials for each	θ	angle   Vvib (S, 0º) = 35.6086 S2 -1.0556 S3 Vvib (S, 10º) = 35.6303 S2 -1.0576 S3  Vvib (S, 20º) = 35.7150 S2 -1.0587 S13  Vvib (S, 30º) = 35.8320 S2 -1.0634 S13 (17)  Vvib (S, 40º) = 35.9457 S2 -1.0687 S13  Vvib (S, 50º) = 36.0297 S2 -1.0729 S13  Vvib (S, 60º) = 35.9539 S2 -1.0688 S13This potentials will be useful to obtained the so called non coupled spectra because the vibrational potential is not θ-dependent.In order to introduce the angle dependence into the V vibrational potential each of the parameters V (S, θ)n vib in equation (17) are fitted to a symmetry function of the θ angle:V (θ)nvib = V n0 + V n3 Cos3θ + V n6 Cos6θ + V n3 Cos9θ (18)102   J. Mex. Chem. Soc. 2008, 52(1) D.R. Hidalgo.Olguín et al.ResultsThe energy barrier ( E(0) – E(60) = 1113.07 cm-1) does not reproduce quite precisely the experimental value (1008.009 cm-1 [5]) due to the short basis set 6-311G** and the method used in the calculation. Nevertheless our main purpose is not to reproduce the rotational barrier but to obtain a more precise spectra.With this method the rotational frequency is improved with respect to the one calculated by the Gaussian03 program package.The vibrational frequencies calculated by including the torsional dependence of the vibrational potential and the anharmonic term are closer to the experimental results than the Gaussian calculation reported in the literature:DiscussionIn this work it has been studied the internal rotation and vibra-tional spectra in the case of ethane which is a simple and well known non-rigid molecule.A systematic method for introducing in the vibrational Hamiltonian the anharmonic correction term and their depen-dence on the internal variables has been presented.The method is mainly based on the difference in energies of both motions under study, and the results obtained repro-duce quite reasonably the experimental results reported in the literature, and improves the theoretical values in available in the literature.The study performed in this paper can be applied to other non-rigid molecules.References 1. Smeyers, Y. G. Teoría de Grupos para Moléculas no rígidas. El Grupo Local. Las aplicaciones; in Memorias de la Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid. Tomo XIII. CSIC, Ed. 1987. 2. Bunker, P. R. Molecular Symmetry and Spectroscopy, Academic Press, 1979. 3. Fernández, J.M.; Montero, S. J. Chem. Phys. 2003, 118, 2657-2671. 4. Roberto-Neto, O; Chakravorty, S.; Machado, F.B.C. J. Mol. Struct. (THEOCHEM) 2002, 586, 29-34. 5. Moazzen-Ahmadi, N. J. Mol. Spectrosc. 2002, 214, 144-151. 6. Marshal, P. J. Phys. Chem. 1999, A103, 4560-4563. 7. Harthcock, M. A.; Laane, J. J.Phys. Chem. 1985, 89, 4231-4240. 8. Senent, M. L. Chem. Phys. Lett. 1998, 296, 299-306. 9. Senent, M. L.; Domínguez-Gómez, R. Chem. Phys. Lett., 2002, 351, 251-258. 10. Bauschlicher Jr., C.W.; Partridge, H. Chem. Phys. Lett., 1995, 239, 246-251.Table 6. Coupled energy levels.Nvib Nrot Energy (cm-1)0 0 0.00000 0 0.00320 0 0.00310 1 303.07510 1 303.07510 1 303.21411 0 1003.28111 0 1003.28441 0 1003.2844Table 7. Rotational frequency (cm-1).Experimental[6] MP2/6-311G** [4,10] This work278 333 303.2861Table 8. Vibrational frequencies C-C.Experimental [6,8] MP2/6-311G** [4,10 ] These work945 1034 1003.2844

Ethane internal rotation-Vibrational Hamiltonian

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