摘要:

1 Introduction Vibrational predissociation (VP) is the result of an energy transfer between the intramolecular and intermolecular modes in weakly bound complexes. The coupling between the two modes is typically very small and the corresponding VP lifetimes are relatively long. The study of such processes is useful in providing information about the weak intermolecular forces which hold these complexes together. Due to the long lifetimes theoretical calculations on VP are challenging. There have been numerous exact time-independent and time-dependent quantum mechanical studies of the VP of three-atom complexes.1 Full dimensional quantum mechanical studies have also been applied to the VP of four-atom complexes1 and include D2/H2–HF1–5 and D2/H2 –OH.1,6,7 In this study we consider the VP of He2Br2 which has two weak intermolecular bonds which is more dynamically demanding than H2–HF.Previous theoretical studies on Ne2Cl2, He2Cl2, Ne2I2 and He2I2 8,9,10 have involved reduced dimensionality calculations. In particular, Gray and co-workers have used a three-degrees of freedom model with a time-dependent method.8,9 In addition, Roncero et al. used a four-degrees of freedom model again employing a time-dependent methodology which was very computationally expensive.10 Here we use a three-degrees of freedom model developed by Le Quéré and Gray to perform time-dependent wavepacket studies on the He2Br2 complex. The aim is to study the predissociation mechanism and compare with related systems such as He2Cl2.Section 2 outlines the theoretical procedure employed. Section 3 details the results obtained and Section 4 concludes. Time-dependent wavepacket study of the vibrational predissociation of He2Br2 which one intermolecular bond is broken first and then the second bond is broken. Alternatively, a direct mechanism in which both van der Waals bonds are broken simultaneously is possible. With the sequential mechanism the intermediate complex HeBr2 is formed. Due to the complexity of this process a reduced dimensional theoretical approach is used in which the two He atoms are taken to be at right angles to the Br2 bond and the angle between the two He atoms q is fixed.8,9 The Jacobi coordinates for this system are shown in Fig.1. Therefore, only the product vibrational and not rotational states of the Br2 fragment can be obtained from the evolving wavepacket. Fig. 1 Jacobi coordinates for He2Br2 system where R1, R2 are the dissociation coordinates, r is the Br2 bond distance and q is the angle between the two He atoms. Throughout the calculations both R1 and R2 are kept at 90° to the Br2 bond r and the angle q is also held fixed at 54°. The model Hamiltonian obtained by Gray and Le Quéré may be written as8 (2) where p�, 1 �Pand P �2 are the momentum operators associated with the r, R1 and R2 coordinates, respectively. The reduced masses m and m are the reduced masses of Br2 and He2Br2 system, respectively. The potential VBr2(r) is a Morse Paul J.Krause and David C. Clary Department of Chemistry, University College London, London, UK WC1H 0AJ Received 21st January 1999, Accepted 5th February 1999, Published 17th February 1999 A time-dependent quantum mechanical approach is used to study the vibrational predissociation of He2Br2. This system is particularly challenging to study because two intermolecular bonds are broken in the predissociation process and therefore threedegrees of freedom need to be treated explicitly. The vibrational predissociation lifetimes and an estimate of the vibrational product distributions are obtained. Studying colour plots of the evolving wavepacket provides a picture of the dissociation process. It is found that He2Br2 is very similar to He2Cl2 and dissociates via a sequential mechanism.(1) 2 Theory 2.1 The He2Br2 system The VP of He2Br2 is given by The metastable state m is defined as a particular vibrational excitation level in Br2 with zero-point energy in the intermolecular bending and stretching modes of He2Br2. The VP process may proceed via a sequential mechanism in PhysChemComm, 1999, 2Table 1 Potential parameters for He2Br2 used in the calculations Atomic pair (i) Di/cm –1 4292.57 17.0 7.61 Br–Bra He–Brb He–Hec 1.036 0.8200 1.1250 a From ref. 14. b From ref. 15. c From ref. 8. potential which describes the interaction between the Br atoms. The potential, V(r, R1, R2, q) in eqn. (2) may be written as (3) Each of these potentials are taken to be a sum of atom– atom interactions such that (4) where i=1, 2 and (5) The distances RHeBr and RHeHe are given by (6) and (7) Then VHeBr(RHeBr) and VHeHe(RHeHe) are given by Morse potentials where the parameters are shown in Table 1.The potential of He2Br2 can be seen in Fig. 2. This scheme was first used for Ne2Cl2 and comparison with the experimental data showed that it was realistic.8 Fig. 2 Potential energy contour map for He2Br2 in which the Br2 bond distance r is held fixed at its equilibrium position. The contours correspond to V=–60, –50, –40, ... cm–1 such that the separated atoms are of zero energy. Ri/a0 ai /a0–1 5.0597 7.41 5.5993 2.2 Time-dependent theory The time-dependent Schrödinger equation is given by ( =1) (8) The wavepacket, y(r, R1, R2; t)(º y(t)).can be expressed as (9) u where c are the vibrational states of Br2 and the initial condition is that of the metastable state. If we now insert eqn. (2) into the time-dependent Schrödinger equation, left multiply by cu and finally integrate over the respective coordinates we obtain (10) (11) where (12) u and E corresponds to the energy levels of Br2. In the wavepacket calculations the vibrational states u, u – 1 and u – 2 of the Br2 diatom are only considered, where u is the initial vibrational level in the metastable complex. The symplectic integrator propagation scheme is used with the m=6, n=4 coefficients of Gray and Manolopoulos.11 The lifetimes are then extracted from the autocorrelation function using Prony's method.12,13 Vibrational product distributions of the Br2 fragment are obtained using the method of Le Quéré and Gray.8 3 Results and discussion If we consider the lifetimes obtained from the extraction of the evolving wavepacket for He2Br2 ( u =7, 8, 9, 10, 11, 12), which are shown in Table 2 we may compare them with the lifetimes of Le Quéré and Gray for He2Cl2.It can be seen that as the vibrational level is increased the VP lifetime for He2Br2 decreases, which was also observed for He2Cl2. Both sets of lifetimes are in fact very close to one another. For the lowest vibrational level the lifetime of He2Br2 is slightly greater than that of He2Cl2. However, when the highest vibrational level is considered the lifetime of He2Br2 is slightly less than that of He2Cl2.From the calculations we can also obtain estimates of the vibrational product distributions of the Br2 fragment and these can be found in Table 3. For comparison the vibrational product distributions for He2Cl2 of Le QuéreTable 2 Lifetimes of He2Cl2a and He2Br2 in picoseconds Vibrational level He2Cl2 52.6 42.5 33.8 28.9 24.8 20.1 78910 11 12 a From ref. 8. Table 3 Vibrational product distributions of He2Cl2a and He2Br2 Vibrational level He2Cl2 P u–1 0.04 0.90 0.03 0.04 0.91 0.03 0.04 0.91 0.04 0.04 0.90 0.05 0.04 0.90 0.08 0.04 0.89 0.08 78910 11 12 a From ref. 8. and Gray are shown also. On first inspection we see that for He2Br2 the main Br2 vibrational product is that of u – 2.This was also observed for He2Cl2 where the major vibrational product u – 2 constitutes 90% of the products. In both cases the probability of producing u – 1 and u – 3 products of the diatom is extremely small. This vibrational propensity derives from the symmetry of the potential, which is close to harmonic. In the He2Cl2 calculations it was shown that the complex dissociates via a sequential mechanism such that one of the He–Cl bonds is broken first and then the other.8 The similarity between the lifetimes and the vibrational product distris of the two complexes seems to imply that He2Br2 may also undergo a sequential mechanism. The study of the evolving wavepacket can give more detailed information about the dissociation process. To enable us to consider the dynamics we construct the vibrational state density given by8 (13) Ideally it would have been informative to have calculated the current density as opposed to the probability density, but that would have involved a great deal more computation.The vibrational state density ru(t=0) which corresponds to that of the initial metastable state can be seen in Fig. 3. Here the contour map of the potential energy surface is shown also. Examination of this plot shows that the density is Gaussian-like in shape with a maximum at the position of the well in the potential energy surface. Examination of the vibrational channel u which corresponds to the vibrational level of the Br2 in the initial metastable state, shows that the general form of the density does not change as time evolves.However, the probability density in the channel decreases. The reason for this is that density is flowing into the lower vibrational channels u – 1 and u – 2. He2Br2 62.8 43.5 31.3 22.1 16.0 12.6 He2Br2 P u–3 P u–2 P u–1 P u–3 P u–2 0.00 0.97 0.02 0.00 0.97 0.02 0.00 0.96 0.03 0.00 0.96 0.03 0.00 0.95 0.04 0.00 0.95 0.04 Fig. 3 The contour plots of the initial probability density, ru (t=0), for He2Br2 ( u=11). The contour map of the potential energy surface can be seen underneath the initial wavepacket and the contours correspond to those of Fig. 2. The vibrational state densities ru-1 (t) and ru-2 (t) for t » 3 ps can be seen in Fig.4. Considering the u – 1 channel we observe that the density stretches from the interaction region, which corresponds to He2Br2, to a region in which He + HeBr2 is formed. It can be seen that there is very little density in the 2He + Br2 region. However, if we consider the u – 2 channel we observe that there is now an area of density in the 2He + Br2 region. It is of particular interest to note that the density in that region seems to be flowing from the He + HeBr2 regions. This would therefore suggest that a sequential mechanism is in operation. A similar dynamical picture was observed for He2Cl2.8 Therefore, the VP process can be expressed as(14) 4 Conclusions A reduced dimensionality time-dependent wavepacket approach has been applied to the study of the vibrational predissociation of He2Br2 in which the Br2 is initially vibrationally excited.A symplectic integrator propagation scheme was used to solve the time-dependent Schrödinger equation. We found that the dissociation of He2Br2 is comparable to that of He2Cl2. In particular, by examining colour plots of the evolving wavepackets, we have found that the complex of He2Br2 undergoes a sequential dissociation mechanism. We hope that these findings will stimulate an experimental study of this problem. Paper 9/00605B Fig. 4 The probability densities at t » 3 ps of the evolution of the wavepacket for the He2Br2 ( u =11) propagation in the (a) u – 1 and (b) u – 2 vibrational channels.References 1 P. J. Krause, PhD thesis, Department of Chemistry, University College London, 1999. 2 D. C. Clary, J. Chem. Phys., 1992, 96, 90. 3 D. H. Zhang, J. Z. H. Zhang and Z. Bacic, J. Chem. Phys., 1992, 97, 927. 4 D. H. Zhang, J. Z. H. Zhang and Z. Bacic, J. Chem. Phys., 1992, 97, 3149. 5 P. J. Krause and D. C. Clary, Mol. Phys., 1997, 93, 619. 6 P. J. Krause, D. C. Clary, D. T. Anderson, M. W. Todd, R. L. Schwartz and M. I. Lester, Chem. Phys. Lett., 1998, 294, 518. 7 M. D. Wheeler, D. T. Anderson, M. W. Todd, M. I. Lester, P. J. Krause and D. C. Clary, Mol. Phys., in the press. 8 F. Le Quéré and S. K. Gray, J. Chem. Phys., 1993, 98, 5396. 9 S. K. Gray, Faraday Discuss., 1994, 97, 143. 10 O. Roncero, G. Delgado-Barrio, M. I. Hernández, J. Campos-Martinez and P. Villarreal, Chem. Phys. Lett., 1995, 246, 187. 11 S. K. Gray and D. E. Manolopoulos, J. Chem. Phys., 1996, 104, 7099. 12 S. Marple, Jr, Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, NJ, USA, 1987. 13 S. K. Gray, J. Chem. Phys., 1992, 96, 6543. 14 R. F. Barrow, T. C. Clark, J. A. Coxon and K. K. Yee, J. Mol. Spec., 1974, 51, 428. 15 T. González-Lezana, M. I. Hernández, G. Delgado- Barrio, A. A. Buchachenko and P. Villarreal, J. Chem. Phys., 1996, 105, 7454. PhysChemComm © The Royal Society of Chemistry 1999

ISSN:1460-2733    

DOI:10.1039/a900605b

出版商:RSC    

年代:1999

数据来源: RSC