Spin-orbit effect in the energy pooling reaction

THE JOURNAL OF CHEMICAL PHYSICS 126, 124304 2007 Spin-orbit effect in the energy pooling reaction O 2 a 1 +O 2 a 1 \O 2 b 1 +O 2 X 3 Rui-Feng Lu and Pei-Yu Zhang Academy of Sciences, Dalian 116023, China and Graduate School of the Chinese Academy of Sciences, Beijing, 10039, China Tian-Shu Chu Academy of Sciences, Dalian 116023, China and Institute for Computational Sciences and Engineering, Qingdao University, Qingdao 266071, China Ting-Xian Xie Academy of Sciences, Dalian 116023, China Ke-Li Han a Academy of Sciences, Dalian 116023, China and Virtual Laboratory for Computational Chemistry, CNIC, CAS, China Received 19 October 2006; accepted 9 February 2007; published online 23 March 2007 Five-dimensional nonadiabatic quantum dynamics studies have been carried out on two new potential energy surfaces of S 2 1 A and T 7 3 A states for the title oxygen molecules collision with coplanar configurations, along with the spin-orbit coupling between them. The ab initio calculations are based on complete active state second-order perturbation theory with the 6-31+G d basis set. The calculated spin-orbit induced transition probability as a function of collision energy is found to be very small for this energy pooling reaction. The rate constant obtained from a uniform J-shifting approach is compared with the existing theoretical and experimental data, and the spin-orbit effect is also discussed in this electronic energy-transfer process. 2007 American Institute of Physics. DOI: 10.1063/1.2713399 I. INTRODUCTION The chemical oxygen-iodine laser COIL gas-phase kinetics is very complicated for various energy transfer processes. With respect to this field, many studies on vibrational-to-vibrational, vibrational-to-translational, vibrational-to-electronic VE, and electronic-to-electronic EE energy transfers have been reported. 1 7 Among these processes, energy pooling reaction O 2 a 1 +O 2 a 1 O 2 b 1 +O 2 X 3 that occurs from a single electronic state to a triple electronic state plays an important role in laser efficiency which depends on concentration of the energy carrier: metastable O 2 a 1. Derwent and Thrush 8 measured the rate constant of the title process to be 2.0±0.5 10 17 cm 3 molecule 1 s 1 at room temperature 295 Heidner et al. 9 employed a temperature-controlled kinetic flow tube equipped with spectroscopic diagnostics to revisit the reaction and found a weak temperature dependency in the range of 259 353 K, with the 295 K rate constant of 5.1 10 17 cm 3 molecule 1 s 1. Using a combined discharge flow/shock tube technique Borrell et al. 10 measured the rate constants at high temperatures. Subsequently, Lilenfeld et al. 11 recommended the room temperature rate coefficient a Author to whom correspondence should be addressed. Electronic mail: klhan@dicp.ac.cn to be 2.7 10 17 cm 3 molecule 1 s 1. Theoretically, in terms of the production channel of ozone and the O 2 2 dimer, the potential energy surfaces PESs of O 2 +O 2 system involving ground state of O 2 have extensively been studied. 12 16 Remarkably, for the O 2 X 3 2 dimer, the intermolecular potential has been well studied to elucidate experimental results. 17,18 To explain VE energy transfer, excited state studies have also been carried out, electronic structure calculations involving the ground and two excited states of O 2 have been carried out by Dayou et al., 5,6 and spin-orbit coupling was proposed to be responsible for the VE energy transfer. To the best of our knowledge, very few theoretical works were carried out for the title reaction involving the EE energy transfer. Bussery and Veyret 19 performed ab initio calculations for the low-lying singlet excited states which dissociated into O 2 a 1 +O 2 a 1 a+a. Recently, ab initio results of O 2 X 3,a 1,b 1 +O 2 X 3,a 1,b 1 have been obtained by Liu and Morokuma, and the mechanism of nonadiabatic process have also been explored qualitatively including the spin-orbit coupling. 20 In their paper, four singlet states denoted as S 1, S 2, S 3, and S 4 correspond to the dissociation limit O 2 a 1 +O 2 a 1 and two triplet states T 6 and T 7 correspond to the dissociation limit O 2 b 1 +O 2 X 3 b+x. 20 No nonadiabatic quantum scattering calculations beyond 0021-9606/2007/126 12 /124304/5/$23.00 126, 124304-1 2007 American Institute of Physics

124304-2 Lu et al. J. Chem. Phys. 126, 124304 2007 FIG. 1. The Jacobi coordinates for the O 2 +O 2 system. The angle is the out-of-plane torsional angle; in this work, it is restricted to zero. three-dimensional 3D space have been carried out up to date. 21 The title process provides the very prototype for us to perform a multidimensional nonadiabatic quantum dynamics study. In this paper, we focused on studying the two states S 2 and T 7 within C s symmetry 1 A and 3 A, respectively, as shown in Ref. 20, and the spin-orbit coupling matrix elements between them. Ab initio methodology and dynamical method are described in Sec. II, while Sec. III presents some properties of the newly constructed PESs and the spin-orbit coupling; the results of dynamical calculations are also in this section. The final section summarizes our conclusions. II. THEORY A. Ab initio methodology The proper description of two interacting open-shell systems demands the use of multiconfigurational wave functions to take into account of large nondynamical correlation effect. However, exact quantum treatment for the O 2 +O 2 system that consists of 32 electrons is a formidable task, especially for the title reaction on the high excited states. For simplicity, we chose the two states S 2 and T 7 within C s symmetry 1 A and 3 A, respectively to be studied; 20 the structure is shown schematically in the Jacobi coordinates in Fig. 1. A stateaveraged complete active state self-consistent field SA-CASSCF scheme was employed, where the active orbitals of 9a -16a and 1a -4a, and seven states S 1, S 2, S 3, S 4, T 6, and T 7, as well as singlet state S 0 corresponding to X+X manifold were included, and the energies were computed by the CASSCF-based second-order perturbation CASPT2 method. The spin-orbit coupling matrix elements between S 2 and T 7, calculated by the full Breit-Pauli spinorbital Hamiltonian 22 at the CASPT2 level, are found to be the largest ones among the states that correspond to two manifolds a+a,b+x. In all ab initio calculations the MOLPRO 2002.6 suite of quantum chemistry program 23 was used. To assess the theoretical accuracy of potential energies, we listed some properties related to this system in Table I. Two basis sets, 6-31+G d and 6-311+G d, were adopted. The O 2 molecules are optimized with the active space consisting of eight 2p electrons in six 2p orbitals 8e/6o. The binding energies shown in the table for the H geometry of the van der Waals complexes have been corrected for the basis set superposition error, utilizing by the widely used Boys-Bernardi counterpoise method. 24 There are small differences in optimized structures and binding energies between the two basis sets, and generally, 6-311+G d gives better results than 6-31+G d. In preliminary ab initio calculations, tests were performed in terms of the spin-orbit couplings, and some results of previous work 20 were reproduced. Additionally, as pointed out by Liu and Morokuma, 20 the reduced active space did not cause serious problems in the low-energy region. Therefore, the CASPT2/6-31+G d level with the active orbitals described above is realistically employed in order to balance computational cost and accuracy. The planar potential energy surfaces were fitted by Levenberg-Marquardt nonlinear least-squares fit of 5692 ab initio data to Aguado-Paniagua functional forms in manybody expansion method, 25,26 and analytical spin-orbit couplings were obtained in an analogous way. B. Dynamics The theory presented here for calculating the nonadiabatic processes in five dimensions 5D is based on the planar PESs mentioned above. The diatom-diatom Hamiltonian expressed in the Jacobi coordinates shown in Fig. 1 for a given total angular momentum J can be written as H = 1 2 2 R 2 + J j 12 2 2 2 j 2 R 2 + 1 2 2 1 r + j 2 2 1 2 2 r 2 where + V R,r 1,r 2, 1, 2, + h 1 r 1 + h 2 r 2, 1 V = V S 2 V so V so V T7. The definitions of variables in Hamiltonian can be found in Ref. 27. In the present five-dimensional time-dependent wave packet calculations, the out-of-plane torsional angle is TABLE I. Equilibrium distances r e in angstrom of the O 2 molecule, and equilibrium intermolecular distances R e in angstrom along with binding energies E in mev at r=r e for the H geometry of the O 2 2 dimer. r e X r e a r e b R e S 0 E S 0 R e T 7 E T 7 R e S 2 E S 2 6-31+G d 1.226 1.237 1.251 3.12 19.9 3.37 13.1 3.41 12.3 6-311+G d 1.207 1.216 1.228 3.20 18.7 3.41 12.6 3.43 11.2 Dayou et al. a 1.209 1.217 1.230 3.10 19.9 3.42 12.5 Bussery et al. 3.23 b 19.0 b 3.44 c 12.4 c Expt 1.208 1.216 1.227 3.56 d 17.0 d a Values corresponding to Ref. 6. b Values corresponding to Ref. 15. c Values corresponding to method B in Ref. 19. d Values corresponding to Refs. 17 and 18. Absolute uncertainties are estimated as ±0.07 Å on R e and ±0.8 mev on E.

124304-3 Spin-orbit effect in energy pooling: O 2 a 1 +O 2 a 1 O 2 b 1 +O 2 X 3 J. Chem. Phys. 126, 124304 2007 fixed at zero =0. The time-dependent Schrödinger equation of the diatom-diatom reaction system can be written as i t i t = H i t, 2 where i i=1,2 is the component of the total unitary wave function corresponding to each of the two potential energy surfaces; each is expanded in terms of translational basis U n R, vibrational basis v1 r 1 and v2 r 2, and the body-fixed total angular momentum eigenfunction JM Y jk R,r 1,r 2. 27 The extended split operator scheme utilized to propagate the wave packet is similar to that for the 3D nonadiabatic reaction systems. 21,28,29 The initial-state specified total reaction probabilities are finally obtained by calculating the reaction flux at a fixed surface s=s 0, P J E = 1 Im i E s s 0 s i E, 3 for the collisional energy transfer, i E relates to state T 7, and s=r in the reactant Jacobi coordinates. Because of high cost in dynamical computations, we calculated the rate constant by the uniform J-shifting approach, 30,31 which is r u T = 2 T kt 3Q0 2J +1 e B T J J+1 /kt. 4 J The temperature-dependent shifting constant is determined by kt Q0 B T = ln J J +1 Q J, 5 where k is the Boltzmann constant, T is the temperature, Q 0 is a partitionlike function defined as Q 0 = P 0 E e E/kT de, and Q J is similarly defined as Q J = P J E e E/kT de. Generally, using reaction probabilities evaluated at three values of J can yield accurate rate constants. Of course, the more values of J that are available, the better the rate constant obtained in the uniform J-shifting approach. Thus, one can use Eq. 5 to define more shifting constants, kt QJi B i T = J i+1 J i+1 +1 J i J i ln +1 Q i+1 J i = 1,2,3.... III. RESULTS AND DISCUSSION The contour plots of interaction potentials for states S 2 and T 7 associated with cis-o 2...O 2 structures are indicated in Figs. 2 a and 2 b, respectively. The angles 1 and 2 as shown in Fig. 1 are fixed at 100 and 80, respectively, and 6 7 8 FIG. 2. Contours of potential surfaces as a function of the shortest O 2...O 2 distance R and the internuclear distance r of the O 2 molecule for cis-o 2...O 2 structures, with the angles 1 and 2 as shown in Fig. 1 are fixed at 100 and 80, respectively, and the distance of one diatomic O 2 fixed at 1.237 Å a S 2 1 A ; b T 7 3 A. The dashed line indicates the crossing seam between the two surfaces. The contour lines are drawn in ev. the internuclear distance of one diatomic O 2 is fixed at 1.237 Å. As can be seen, both surfaces are repulsive, with the energy increasing rapidly for the intermolecular O 2...O 2 distance R less than 2.0 Å. The dotted line represents the crossing seam between S 2 and T 7. Obviously, the position of the crossing point is characterized by R values becoming large with the decrease of r, which is the internuclear distance of the other diatomic O 2. The minimum on the crossing seam in Fig. 2 is at about r=1.237 Å and R=2.25 Å, with the energy of about 0.8 ev above the a+a dissociation limit. The plots also show that the S 2 state is only 0.29 ev above the T 7 state in the asymptotic limit with r fixed at 1.237 Å. The spin-orbit couplings between states S 2 and T 7 for cis-o 2...O 2 configurations described above are displayed in Fig. 3. It is clear that the absolute value of the spin-orbit coupling is zero in the asymptotic limit and shows a substantial increase as the two diatoms approach each other. The value of the spin-orbit coupling on the minimum crossing point r=1.237 Å, R=2.25 Å is about 10 cm 1. Note that the crossing point is energetically reachable with moderate collision energy and therefore it plays a significant role in the EE energy transfer from S 2 state to T 7 state. It has also been found that this collision-induced spin-orbit effect is more pronounced at large r values. To clarify the reaction mechanism in this energy pooling process, 5D nonadiabatic quantum dynamical calculations have been performed for several partial waves. In Fig. 4, we depicted the transition probabilities from S 2 to T 7 through spin-orbit coupling for the total angular momentum J = 0, 50, 100, 150, and 200, both O 2 a 1 molecules being in the

124304-4 Lu et al. J. Chem. Phys. 126, 124304 2007 FIG. 3. Contour map of spin-orbit coupling matrix element in cm 1 between S 2 and T 7 for the same structures as in Fig. 2. The dashed line indicates the crossing seam between the two surfaces. ground rovibrational state v=0, j=0. The probability as a function of collision energy is small, thus demonstrating that the energy pooling reaction O 2 a 1 +O 2 a 1 O 2 b 1 +O 2 X 3 is not the prominent process in the COIL system. However, it could be enhanced with the increase of collision energy. The larger the value of J is, the smaller the transition probability is, and the transition probability is almost zero at the value of J = 200. In addition, careful inspection reveals that there are wiggling structures in the curves, which perhaps results from the van der Waals well to form O 2 2 dimers. Based on such small transition probability, the thermal rate constant can be estimated to be very small, which is consistent with the conclusions of experimental measurements 9 11 and the theoretical estimate using a simple Landau- Zener model by Liu and Morokuma. 20 In Fig. 5, the initial state selected rate constant for the ground rovibrational state obtained by the uniform J-shifting approach 30,31 is plotted together with the results of the previous experiments. 9 11 In this work, five partial waves J=0, 50, 100, 150, and 200 as shown in Fig. 4 and additional three partial waves J=15, 30, and 80 not shown have been used. On the whole, the present theoretical values are smaller than experimental values. The difference between computation and experiment is more significant at low temperatures than at high temperatures. It should be noted that the rovibrational excitations of FIG. 4. Calculated transition probabilities as a function of collision energy for energy pooling reaction O 2 a 1,v 1 = j 1 =0 +O 2 a 1,v 2 = j 2 =0 O 2 b 1 +O 2 X 3. Solid, dashed, dotted, dash-dotted, and short-dotted lines correspond to J=0, 50, 100, 150, and 200, respectively. FIG. 5. The initial state selected rate coefficient as a function of inverse temperature for energy pooling reaction O 2 a 1,v 1 = j 1 =0 +O 2 a 1,v 2 = j 2 =0 O 2 b 1 +O 2 X 3 is compared with the experiment measurements. O 2 a 1 molecules are actually not considered here: they should be important at high temperatures. Also it should be kept in mind that since a coplanar treatment as well as a J-shifting approximation is employed, more efforts are needed on this interesting process to get more reasonable results at low temperatures. In practical applications of Eq. 4, it is desirable to calculate the probabilities of partial waves for as many J values as possible in order to obtain accurate rate constants. However, tests show that the calculated rate constants change slightly by varying from five values of J not shown here to eight values of J, and therefore the differences between theory and experiment may arise mainly from the coplanar treatment instead of that from the J-shifting approximation. In spite of these differences, there is an overall agreement of the general trend in rate coefficient over all temperatures considered between the present result and the experimental measurements. IV. CONCLUSIONS In summary, two planar potential energy surfaces have been constructed for the energy pooling reaction O 2 a 1 +O 2 a 1 O 2 b 1 +O 2 X 3, and the spin-orbit coupling matrix element between the two surfaces showing a general collision-induced trend. A five-dimensional nonadiabatic quantum dynamics study was also carried out to elucidate the spin-orbit effect in this reaction. To our best knowledge, this is the first multidimensional nonadiabatic quantum dynamics study beyond three dimensionality. The calculated thermal rate constant based on the transition probability is in agreement with the results of experiment measurements; the existing differences are proposed to most likely result from the coplanar treatment. Naturally, it is worthwhile to extend this kind of calculation to full-dimensional nonadiabatic quantum treatment of four-atom system. This will be deferred to subsequent work with the advances of constructing potential energy surfaces and the developments of computational method.

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