Electron collisons with cyclobutane

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Electron collisons with cyclobutane

M. H. F. Bettega^I; A. R. Lopes^II; M. A. P. Lima^II; L. G. Ferreira^II

^IDepartamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, 81531-990, Curitiba, Paraná, Brazil

^IIInstituto de Física "Gleb Wataghin", Universidade Estadual de Campinas, 13083-970, Campinas, São Paulo, Brazil

ABSTRACT

We report integral, differential, and momentum transfer cross sections for elastic scattering of low-energy electrons by cyclobutane (c-C₄H₈), which is an isomer of the C₄H₈ molecule and has a closed chain. Our calculations were performed with the Schwinger multichannel method with pseudopotentials at the static-exchange level of approximation, for energies up to 50 eV. We compare the cross sections of cyclobutane with the cross sections of other three isomers of C₄H₈, namely, isobutene, cis-2-butene, and skew-1-butene. These isomers have open chain and were the subject of a previous study by our group [Lopes et al. J. Phys. B 37, 997 (2004)]. We also show previous calculated integral and momentum transfer cross sections for isomers of C₃H₄, C₃H₆ and C₄H₆, which have open and closed chains. For each isomeric group we discuss the isomer effect focusing on the closed chain isomer. The isomer effect is related to differences in the isomers cross sections due to differences in their geometries.

Keywords: Isomer effect; Cyclobutane; Norm-conserving pseudopotentials

I. INTRODUCTION

In the last three years, motivated by some experimental works [1-4], we have been investigating electron collisions with isomers of small hydrocarbons, focusing on the isomer effect. This effect allows one to distinguish the different isomers of a molecule by differences in their cross sections. The main goal of this investigation was to find differences in the electron-collision cross sections due to differences in the (geometrical) structures of the isomeric groups. We calculated cross sections for electron collisions with isomers of C₃H₄ [5], C₄H₆ [6], C₄H₈ and C₄H₁₀ [7], and more recently by isomers of C₃H₆ [8]. In particular, for C₄H₈ we have not included cyclobutane (c-C₄H₈), which has a closed chain. We have noticed, however, that for the isomers of C₄H₆ and C₃H₆, the integral cross sections for the closed chain isomers, namely cyclobutene and cyclopropane, are smaller than for the other isomers. In this case the isomer effect is more evident for the closed chain isomers.

In this paper we report integral, differential and momentum transfer cross sections for cyclobutane, c-C₄H₈, for energies up to 50 eV. To compute the cross sections we employed the Schwinger multichannel method implemented with pseudopotentials. We compare the cross sections of c-C₄H₈ with the cross sections of isobutane, cis-2-butene and skew-1-butene, which are also isomers of C₄H₈ but having open chains. We also present integral and momentum transfer cross sections for isomers of C₃H₄, C₄H₆ and C₃H₆ and discuss the isomer effect on the point of view of the closed chain isomers.

In the following we briefly discuss the theoretical method and the computational procedures used in our calculations. Then we present our results and discussions. We end the paper with a brief summary of our findings.

II. THEORY

The Schwinger multichanel (SMC) method [9-11] and its implementation with pseudopotentials [12] have been described in detail in several publications. We will only discuss the points relevant for the present calculations.

The SMC method is a variational method which results in the following expression for the scattering amplitude

where

and

In the above equations, is a solution of the unperturbed Hamiltonian H₀ and is a product of a target state and a plane wave, V is the interaction potential between the incident electron and the electrons and nuclei of the target, |c_mñ is a set of (N+1)-electron Slater determinants (Configuration State Functions-CSF's) used in the expansion of the trial scattering wave function, = E—H is the total energy of the collision minus the full Hamiltonian of the system, with H = H₀+V, P is a projection operator onto the open-channel space defined by the target eigenfunctions, and is the free-particle Green's function projected on the P-space. The (direct) configuration space is constructed as:

where |F₁ñ is the target ground state wave function, described at the Hartree-Fock level of approximation, |j_iñ is a one-electron function represented by a virtual (unoccupied) orbital, and is the antisymmetrizer.

III. COMPUTATIONAL PROCEDURES

The cross sections were obtained at the static-exchange level of approximation. The molecular geometry was optimized using the package Gaussian [13] at the MP2 (full) level of approximation with the 6-311++G(2d,p) basis set. In the remaining bound state and in the scattering calculations, the 1s core electrons of the carbons were replaced by the pseudopotentials of Bachelet, Hamann and Schlüter [14]. For these calculations, the basis set for the carbon atom included 6 uncontracted s-type functions (with exponents 12.49408, 2.470291, 0.614027, 0.184029, 0.036799, and 0.013682), 5 uncontracted p-type functions (with exponents 5.228869, 1.592058, 0.568612, 0.210326, and 0.072250), and 2 uncontracted d-type functions (with exponents 0.603592, and 0.156753), and was generated according to Ref. [15]. The basis set for the hydrogen atom was taken from Ref. [16] and is shown in Table I. We have not included in our calculations the 3s symmetric combination of d-type functions, namely [( x² + y² + z² )exp(—ar²)], in order to eliminate any possible linear dependence in the basis set.

IV. RESULTS AND DISCUSSION

Figure 1 shows our calculated integral and momentum transfer cross sections for cyclobutane. Unfortunately there are no experimental data nor theoretical results available for this molecule for comparison. The most prominent feature seen in these cross sections is the broad structure located around 12 eV. In Fig 2. we show the symmetry decomposition of the integral cross section according to the D_2d group. We chose to present the two components of the E symmetry separately, in order to show that they are equivalent. This asserts that our results are converged with respect to the numerical quadratures employed in our calculations. The main contribution to the structure seen in Fig 1. comes from the B₂ ad E symmetries. We also found another interesting feature that was hidden in the cross sections shown in Fig 1: a shape resonance located around 10 eV and belonging to the A₂ symmetry. In their study concerning electron collisions with the isomers of C₃H₆, Winstead et al. [17] found a shape resonance in the cross section of cyclopropane belonging to the symmetry. They commented that cyclobutane could present a shape resonance analogous to that of cyclopropane.

Differential cross sections (DCSs) at selected energies are shown in Fig. 3. A very pronounced minimum is seen in the DCSs from 5 to 10 eV at around 75º. Another minimum is visible at around 150º. These features disappear for energies above 10 eV.

Figure 4 compares the integral and momentum transfer cross sections of cyclobutane with the cross sections of other three isomers of C₄H₈, isobutane, cis-2-butene, and skew-1-butene. From this comparison one may notice that the cross sections for cyclobutane lie below the cross sections of the other isomers. The cross sections match at 50 eV; the isomer effect being more evident for cyclobutane, which is the closed chain isomer.

The integral and momentum transfer cross sections of all isomers present a broad maximum located at the same energy (at around 12 eV). However, the cross sections of cyclobutane do not show the low-energy sharp structure seen in the cross sections of the other three isomers. This sharp structure is a p* shape resonance and is a characteristic of hydrocarbons with double carbon-carbon bonds; cyclobutane has only single carbon-carbon bonds. Fig. 5 shows the cross sections for the resonant symmetries of isobutane (C_2v group), cis-2-butene (C_2v group), and skew-1-butene (C_s group).

Comparison of the DCSs at 5, 7, 10, 15, 20, 30, 40 and 50 eV for cyclobutane, isobutane, cis-2-butene, and skew-1-butene are shown in Figs. 6 and 7. The DCSs for cyclobutane differ most from the DCSs of the other three isomers for energies below 15 eV. At 15 eV the differences in the DCSs become smaller, but at 15 eV and above the DCSs of cyclobutane are still distinguishable from the DCSs of the other isomers by the minima located at around 30º and 120º.

In Figs. 8, 9, and ¹⁰ we show the integral and momentum transfer cross section for the isomers of C₃H₄, C₃H₆ and C₄H₆. Except for C₃H₄, the isomers of C₃H₆ and C₄H₆ present the same behavior as that shown by the isomers of C₄H₈. The integral and momentum transfer cross sections of the isomers with closed chain, namely cyclopropane, of C₃H₆, and cyclobutene, of C₄H₆, also lie below the cross sections of the open chain isomers. Cyclopropane has only single carbon-carbon bonds, and its cross sections do not present the low-energy sharp structure, which is seen in the cross section of propane, which is an open chain isomer with a carbon-carbon double bond. The cross sections of cyclobutene also present a low-energy sharp structure. This isomer has a carbon-carbon double bond.

Figure 11 compares the integral and momentum transfer cross sections for the closed chain isomers of C₄H₈ (cyclobutane), C₄H₆ (cyclobutene) and C₃H₆ (cyclopropane). The magnitude of the cross sections follows the molecular size, except by the p* shape resonance that appears in the cross sections of cyclobutene.

V. SUMMARY

We presented integral, differential and momentum transfer cross sections for cyclobutane, which is one of the isomers of C₄H₈. We found a shape resonance located around 10 eV and belonging to the A₂ symmetry of the D_2d group. Through the comparison of the integral and momentum transfer cross sections of cyclobutane with those of isobutane, cis-2-butene and skew-1-butene, we concluded that the isomer effect is more evident for cyclobutane, which has a closed chain. The isomer effect is also more evident for closed chain isomers of other molecules. In particular, for cyclopropane, an isomer of C₃H₆, and cyclobutene, an isomer of C₄H₆.

Acknowledgments

MHFB, MAPL and LGF acknowledge support from Brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). MHFB acknowledges support from the Paraná state agengy Fundação Araucária and from FINEP (under project CT-Infra 1). ARL acknowledges scholarship from Fundação de Amparo à Pesquisa de São Paulo (FAPESP). The authors acknowledge computational support from Professor Carlos M. de Carvalho at DF—UFPR and from CENAPAD—SP.

Received on 22 January, 2006

• [1] Cz. Szmytkowski and S. Kwitnewski, J. Phys. B 35, 3781 (2002).
• [2] Y. Nakano, M. Hoshino, M. Kitajima, H. Tanaka, and M. Kimura, Phys. Rev. A 66, 032714 (2002).
• [3] C. Makochekanwa, H. Kawate, O. Sueoka, M. Kimura, M. Kitajima, M. Hoshino, and H. Tanaka, Chem. Phys. Lett. 368, 82 (2003).
• [4] C. Szmytkowski and S. Kwitnewski, J. Phys. B 36, 2129 (2003).
• [5] A. R. Lopes and M. H. F. Bettega, Phys. Rev. A 67, 32711 (2003).
• [6] A. R. Lopes, M. A. P. Lima, L. G. Ferreira, and M. H. F. Bettega, Phys. Rev. A 69, 014702 (2004).
• [7] A. R. Lopes, M. H. F. Bettega, M. A. P. Lima, and L. G. Ferreira, J. Phys. B 37, 997 (2004).
• [8] C. Makochekanwa, H. Kato, M. Hoshino, H. Tanaka, H. Kubo, M. H. F. Bettega, A. R. Lopes, M. A. P. Lima, and L. G. Ferreira, J. Chem. Phys. 124, 024323 (2006).
• [9] K. Takatsuka and V. McKoy, Phys. Rev. A 24, 2473 (1981).
• [10] K. Takatsuka and V. McKoy, Phys. Rev. A 30, 1734 (1984).
• [11] M. A. P. Lima, L. M. Brescansin, A. J. R. da Silva, C. Winstead, and V. McKoy, Phys. Rev. A 41, 327 (1990).
• [12] M. H. F. Bettega, L. G. Ferreira, and M. A. P. Lima, Phys. Rev. A 47, 1111 (1993).
• [13] Gaussian 98, Revision A.11.2, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, N. Rega, P. Salvador, J. J. Dannenberg, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, and J. A. Pople, Gaussian, Inc., Pittsburgh PA, 2001.
• [14] G. B. Bachelet, D. R. Hamann, and M. Schlüter, Phys. Rev. B 26, 4199 (1982).
• [15] M. H. F. Bettega, A. P. P. Natalense, M. A. P. Lima, and L. G. Ferreira, Int. J. Quantum Chem. 60, 821 (1996).
• [16] T. H. Dunning Jr., J. Chem. Phys. 53, 2823 (1970).
• [17] C. Winstead, Q. Sun, and V. McKoy, J. Chem. Phys. 96, 4246 (1992).

Publication Dates

• Publication in this collection
  31 July 2006
• Date of issue
  June 2006