A Reactant-Coordinate-Based Wave Packet Method for Full-dimensional State-to-State

Transcription

1 Submitted to J. Chem. Phys. //06 A Reactant-Coordinate-Based Wave Packet Method for Full-dimensional State-to-State Quantum Dynamics of Tetra-Atomic Reactions: Application to Both the Abstraction and Exchange Channels in the H + H O Reaction Bin Zhao, Zhigang Sun, and Hua Guo,* Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 873, USA Center for Theoretical and Computational Chemistry, and State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 603, China * Corresponding author: hguo@unm.edu

2 Abstract An efficient and accurate wave packet method is proposed for the calculation of the stateto-state S-matrix elements in bimolecular reactions involving four atoms. This approach propagates an initial state specific wave packet in reactant Jacobi coordinates. The projection in product channels is carried out on projection planes, which have one less degree of freedom, by transforming both the time-dependent wave packet and final product states into a set of intermediate coordinates. This reactant-coordinate-based (RCB) method is more efficient than product-coordinate-based (PCB) methods because it typically requires a smaller number of basis functions or grid points, and allows the determination of S-matrix elements for multiple product channels from a single propagation. This method is demonstrated in calculating the (J tot =0) stateto-state S-matrix elements for both the abstraction and exchange channels of the H + H O reaction.

3 I. Introduction Thanks to recent advances in quantum reactive scattering theory, accurate characterization of reaction dynamics has entered its post-atom-diatom era., Coupled with new developments in ab initio electronic structure theory and high-fidelity fitting methods of multidimensional global potential energy surfaces (PESs), our understanding of gas phase bimolecular reaction dynamics has reached an unprecedented level. 3, 4 The progress is the most apparent for tetra-atomic (atom-triatom or diatom-diatom) reactive systems, for which full-dimensional quantum scattering calculations can now be performed routinely on accurate PESs. An array of issues concerning the reaction dynamics, such as mode specificity, bond selectivity, and dynamical resonances, can thus be addressed with the full six internal degrees of freedom., 4 The most detailed observable for a reactive scattering problem is the quantum stateresolved differential cross section (DCS), which requires the state-to-state scattering (S) matrix. 5 Despite much progress in quantum reactive scattering involving four atoms, most fulldimensional calculations for tetra-atomic systems so far have been initial state specific, which resolve no final states. 6-4 Although state-to-state probabilities have been reported before, 5-7 only a few studies have computed state-to-state S-matrix elements in reduced 8, 9 and fulldimensionalities In fact, state-to-state DCSs have only been reported for one reaction (H + H O H + H O) and its isotopic analogs, by Zhang and coworkers The lack of state-tostate quantum reactive scattering calculations can be attributed to several intrinsic difficulties. First, the breaking and forming of bonds render it very difficult to design an optimal coordinates for both the reactant and product arrangement channels. This notorious coordinate problem is present in atom-diatom reactions, but becomes more pronounced in tetra-atomic reactions. Second, the number of grid points and/or basis functions needed to converge the calculations 3

4 increases exponentially with dimensionality, making tetra-atomic systems substantially more challenging. There exist several strategies for state-to-state quantum reactive scattering calculations for tetra-atomic reactions. The first is based on the so-called reactant-product decoupling (RPD) 40, 4 method, which was initially proposed by Peng and Zhang, and further developed by Althorpe and coworkers. 4, 43 In this approach, the wave packet is propagated in the reactant Jacobi coordinates and the parts reached a product arrangement channel are collected using an absorbing potential, and further propagated in the product channel in the product Jacobi coordinates. In this way, a column of the S-matrix is obtained. Most of current state-to-state calculations on tetra-atomic reactions were based on the RPD method. However, the RPD approach would encounter problems if the PES possesses a significant post-reaction well, because the resulting recurrence of the wave packet in this well could make the convergence difficult. The second approach is the transition-state wave packet (TSWP) method of Manthe and coworkers. 44, 45 The idea there is to propagate multiple TSWPs, defined as the eigenstates of the thermal flux operator on a dividing surface near the transition state, into both the reactant and product arrangement channels. Because the propagations can be carried out independently in different arrangement channels using the appropriate Jacobi coordinates, the size of the basis and/or grid could be smaller and the full S-matrix elements can be obtained. Unlike the RPD approach, the existence of post-reaction wells has a limited impact on the computational costs as the analysis plane can be placed deep into the asymptotic region without significantly impeding the numerical efficiency. This method has been demonstrated in a number of systems, including reactions involving three, 44, 46 four, and six atoms. 47 It has the added advantage of allowing 4

5 the analysis of the transition-state control of reactivity , Nevertheless, the number of TSWPs can increase very rapidly with both energy and the mass of the atoms in the molecule. In this publication, we introduce a reactant-coordinate-based (RCB) method for the calculation of state-to-state S-matrix elements for four atom systems. This approach bears strong resemblance to the RCB method proposed for atom-diatom reactions, 5-53 and differs from the product-coordinate-based (PCB) methods in that the wave packet is propagated in the reactant Jacobi coordinates. The key idea in the RCB approach is to define in the product asymptote a projection plane, in which both the time-dependent wave packet and N- dimensional product states are transformed from the respective reactant and product Jacobi coordinates into a set of intermediate coordinates for the calculation of overlaps. Here, N is the dimensionality of the system, which equals six for tetra-atomic reactions. Considering the extension of grid range into the product asymptotic region, the grid/basis size in this RCB approach is much larger than the calculation of initial state specific total reaction probability, but it is typically smaller than that needed in a PCB calculation and the transformations are sufficiently inexpensive not to impact the overall efficiency of the calculations. Like RPD and PCB, RCB only calculates a column of the S-matrix, thus avoiding the multiple propagations needed in the TSWP approach. Furthermore, internal state analysis of multiple product channels can be performed with a single propagation. In this work, we discuss this RCB method and demonstrate its applicability in both the abstraction and exchange channels of the H + H O reaction for zero total angular momentum. It should be noted that the three hydrogens in this system are indistinguishable, so the exchange reaction is only meaningful with isotopic substations. However, we ignore this issue here because of the focus on method development. This publication is organized as follows. The RCB method is outlined in Sec. II, where the 5

6 transformations between various coordinate systems are defined and details for extracting the S- matrix elements are discussed. In Sec. III, the application of the RCB method to the prototypical tetra-atomic reactive system is demonstrated. Finally, summary and conclusions are given in Sec. IV. II. Theory II-A. Hamiltonian and Wavefunctions For a tetra-atomic reactive system, quantum scattering calculations at the state-to-state level require the definition of both the reactant and product coordinates. As shown in Fig., Jacobi coordinates of the A+BCD, AB+CD, and B+ACD arrangements are denoted as ( R, r, r,,, ), ( R, r, r,,, ), and ( R, r, r,,, ), respectively. The z-axis of the body-fixed (BF) frame for the A+BCD, AB+CD, and B+ACD arrangements are defined to be along the R, R, and R vectors, respectively. The vectors r, r, and r lie in the corresponding x-z planes. It should be noted here that the triangle formed by A, B, and the CD center-of-mass (COM) spans the common plane for the three coordinate systems, and r, r, and r are the same vector. In a reactant-coordinate-based (RCB) treatment of an A+BCD reaction, only the Hamiltonian for the A+BCD arrangement is explicitly required in the propagation ( = hereafter): ( ˆ ˆ ) ˆ ˆ J j ˆ j H h ( r ) V ( R, r, r,,, ) R R r tot k k k, () k k k 6

7 where is the reduced mass between A and the COM of BCD, is the reduced mass between B and the COM of CD, and is the reduced mass of CD. ĵ and ĵ are the rotational angular momentum operators of BCD and CD, respectively, ˆj ˆj ˆ j is the orbital angular momentum operator of B relative to CD, and J ˆtot is the conserved total angular momentum operator of the system. The reference vibrational Hamiltonians h ˆ ( r ) are defined as k k ˆ h ( r ) V ( r ), ( k,), () k k k k k rk where V ( r ) are the one-dimensional (D) reference potentials and the potential term V in Eq. k k () consists of the total PES with the reference potentials removed. The Hamiltonians for the two product arrangements have the same form as Eq. (), but with the operators and variables replaced by the corresponding primed and double primed ones for the AB+CD and B+ACD arrangements, respectively. However, the definitions of operators and variables are different. In the AB+CD arrangement, specifically, and are diatomic reduced masses for AB and CD, is the reduced mass between the COMs of AB and CD. ĵ and ĵ are the rotational angular momentum operators of AB and CD, respectively, and they are coupled to form ĵ. In the B+ACD arrangement, on the other hand, is the reduced mass between B and the COM of ACD, is the reduced mass between A and the COM of CD, and is the reduced mass of CD. ĵ and ĵ are the rotational angular momentum operators of ACD and CD, respectively, and ˆj ˆ ˆ j j is the orbital angular momentum operator of A relative to CD. 7

8 The time-dependent wave packet is expanded in terms of BF bases in the reactant arrangement channel: ( R, r, r, t) F ( t) u ( R) ( r ) ( r ) y ( Rˆ, rˆ, rˆ ). (3) J tot M J, tot M J tot M n jk n jk n,, j, K The wavefunctions in the two product channels can be expanded in a similar fashion, with the appropriate variables in primes and double primes. In Eq. (3), M, which is the projection of J tot in the space-fixed (SF) frame, is normally set to zero because the interaction potential is isotopic in the SF frame. The composite index denotes the two vibrational quantum numbers, and k, and the vibrational eigenfunctions ( ) are the eigenfunctions of the reference vibrational k r k Hamiltonians hˆ ( r ). An L-shape scheme is adopt for the sake of efficiency and the k k, corresponding translational basis u ( R) in the asymptotic and interaction regions are defined as: 58 n u, n n ( R R ) ( N ) ( ) ( R) n ( R R ) ( N ) ( ) sin, asy, asy, R ( R, R3 ), R R NR R sin, asy, asy, R ( R, R ), R R NR R (4) where, R ( R R) / ( NR ) R3 R ( N R ) R separates the R coordinate range ( R, R ) into the asymptotic and interaction regions. 6 Different translational bases are used in the two regions with the same grid points. and asy are chosen to be the number of energetically open asy vibrational channels plus one or two closed channels for the two vibrational degrees of freedom. 8

9 The composite index j denotes ( j, j, j ), is the parity of the system defined as ˆ ˆ ˆ. ( ) j j l with l being the quantum number for the orbital angular momentum l Jtot j The parity-adapted BF rotational basis J totm y ( ˆ, ˆ ˆ jk R r, r) is given as ˆ J y ( R, rˆ, rˆ) [ D (,, ) Y (,, ) JtotM tot Jtot * jk jk K, M jj 8 ( K 0) ( ) D (,, ) Y (,, )], j j j Jtot Jtot * j K K, M jj (5) where D is the Wigner rotation matrix 59 that rotates the SF frame to BF frame by J tot * KM, (,, ) j j j J three Euler angles (,, ). The restriction, ( ) tot, for K 0 partitions the rotational basis set into even and odd parities. The definition of Y (,, ) is different for the A+BCD, AB+CD and B+ACD j K jj jk arrangement channels. The A+BCD rotational basis Y (,, ) is given by 8 jj j Y (,, ) D (0,, ) j mj 0 j m y (,0), (6) j K j * j j Km jm m 4 j where the Wigner rotation matrix D * Km (0,, ) rotates the BF frame to the molecular-fixed (MF) frame (the z axis of this frame lies along vector r ) by the Euler angles ( 0,, ), m is the projection of j and j on the z-axis of the MF frame, y jm is the spherical harmonics, and j mj 0 j m is the Clebsch-Gordan coefficient. 59 Similarly, the B+ACD rotational basis is given by Eq. (6) with coordinates replaced by the corresponding double primed ones. On the j K other hand, the AB+CD rotational basis Y (,, ) is different and given by 6 jj 9

10 Y (,, ) jm jk m j K y (,0) y (, ), (7) j K j j j m j K m m where m is the projection of j on the z-axis of the BF frame. II-B. Extraction of State-to-state S-matrix Elements In the time-dependent wave packet method, the state-to-state S-matrix element from the initial reactant state i to the final product state f is expressed as the Fourier transform of a timedependent cross-correlation function: 60 ( ) S E e e dt, (8) iet p iht r pf ri ( ) ( ) ( ) 0 f i f E i E where p f and r i are the final product and initial reactant wave packets, respectively, with ( E) and ( E) f i as their corresponding energy normalization factors. However, the initial reactant and final product wave packets are best defined in their corresponding coordinate systems, i.e. the A+BCD, AB+CD and B+ACD Jacobi coordinates for a tetra-atomic reactive system, so that they assume simple product forms. Thus, before carrying out the overlap of these wave functions, they need to be transformed into a common set of coordinates. II-C. Coordinate Transformation In the RCB method, two schemes have been proposed to solve the coordinate problem for triatomic reactive systems, and both can be straightforwardly generalized to tetra-atomic systems. One employs an interpolation scheme to evaluate product states on the grids of reactant coordinates, 5, 60 and the other projects both reactant and product states onto a set of intermediate 0

11 coordinates. 5, 53 The interpolation scheme has the advantage of saving grid points in expressing the product wavefunctions thus saving computer memory, and the intermediate coordinate method is numerically more efficient. In this paper, the intermediate coordinate approach is adopted in our implementation. The calculation of the time-correlation functions, namely the projection, is performed in a set of intermediate coordinates. 5, 53 The intermediate coordinates for a particular product channel are designed to include the corresponding scattering coordinate R or R between the two products. This enables the expression of the product wave packet with a product form, in which the product wave packet in the scattering coordinate is given by a delta function that defines the projection plane. For the A+BCD reaction, the projection planes for the AB+CD and B+ACD product channels can be defined as R R p and R R p in the corresponding Jacobi coordinates, respectively. On such a projection plane, the product wavefunction have one less degree of freedom as R or R is fixed. In our current implementation of the RCB method for tetra-atomic reactive systems, the remaining degrees of freedom of the intermediate coordinates are chosen from reactant Jacobi coordinates in order to minimize the size of the transformation. In general, two options are available: (): ( Rp / p; R, r,,, ) and (): ( Rp / p; r, r,,, ). The choice of such two sets of intermediate coordinates is inspired by noting the existence of a common plane for all the three Jacobi coordinates for the reactive tetra-atomic system. As noted in the previous section, A, B, and the COM of CD form a common plane, and the coordinate transformation becomes the same as in triatomic systems if CD is considered as a single combined atom. With such a consideration, the ( r,, ) coordinates are shared by both the reactant and intermediate coordinates, thus requiring no transformation. Only the ( Rp / p; R, ) or ( Rp / p; r, ) coordinates

12 5, 53 need be considered, and their transformations are exactly the same as the triatomic case. It is quite straightforward to express the time-dependent wave packet in the intermediate coordinates from that in the reactant coordinates. In Option (), the wave function in the intermediate coordinates at ( Rp / p; R, ) can be obtained from the wave packet in the reactant coordinates ( Rr,, ) where r can be explicitly expressed as r Rcos R (cos ) ( B R p / p ) A, (9) where the plus and minus signs ( ) correspond to the AB+CD and B+ACD product channels, respectively. In Option (), on the other hand, the wave packet in the intermediate coordinates at ( R ; r, ) p/ p can be similarly obtained from the wave function in the reactant coordinates ( Rr,, ) where R can be explicitly expressed as R A r cos ( Ar ) (cos ) ( B R p p ). (0) / The definition of thea and B constants depends on the product channels: for the AB+CD abstraction channel, A =(m A +m B ) / ma ( mc md ) / ( m B +m C +md ) and B =(m A +m B ) / ma, while for the B+ACD exchange channel, A =(m +m +m ) / m m / ( m +m +m ) and A C D A B B C D B =(m A +m C +m D ) / ma. The transformed wave packets in the intermediate coordinates need to be multiplied by the Jacobian factors, which are explicitly given as g B R p/ p A R (cos ) ( B Rp / p), ()

13 for Option (), and g B R p/ p ( Ar ) (cos ) ( B Rp / p) () for Option (). It should be noted here that our wave packet in Eq. (3) is expanded in basis functions rather than on DVR grids and thus the wave function needs to be converted into the DVR in the ( R, ) or ( r, ) degrees of freedoms in order to carry out the projection in the intermediate coordinates. After expressing the time-dependent wave packet in the intermediate coordinates, the R or r degree of freedom can be conveniently kept on DVR grids without taking the extra effort to convert them back to the basis representation. While the angular degrees of freedom can be represented in either the basis representation or the DVR, as along as both the time-dependent wave packet and final product states are expressed with the same representation along the corresponding degrees of freedom in the intermediate coordinates. In the current work, for the sake of memory, rather than numerical, efficiency, the basis representation was adopted for the angular degrees of freedom for projecting the product states information. The transformation of the product wave packets from the corresponding product Jacobi coordinates into the intermediate coordinates is somewhat more complicated. Since the scattering coordinate R or R is fixed in both the product and intermediate coordinates, the coordinate transformation entails a five-dimensional mapping of the final product states, but this transformation needs only be done once. The details of the transformation have been discussed previously, 30, 33 and only a brief description is given here. As noted above, r, r, and r are the same for all the three arrangement channels, and as a result, transformation is only carried out in 3

14 four dimensions: from ( r,,, ) and ( r,,, ) to either ( R,,, ) in Option (), or ( r,,, ) in Option (). With the split operator propagator implemented in our work, the procedure for the transformation can be carried out in the following steps: First, the product wave packets are interpolated on the reactant coordinate grids, ( R,,, ) or ( r,,, ), with the corresponding basis functions in Eq. (3), and then, product wave packets represented on the reactant grids are transformed to the basis representation by transformation matrices in the angular degrees of freedom while the R or r degree of freedom is similarly kept in grid representation, which is the same as the time-dependent wave packet in the intermediate coordinates. Finally, if non-zero total angular momentum and multiple helicity quantum number Ks are involved, rotation of the BF z-axis from R or R to R is also required to complete the coordinate transformation. 5 II-D. Initial Wave Packet and Final State Analysis In the RCB method, the initial wave packet is defined in the reactant asymptotic region as a product of the reactant internal state wave function and a Gaussian wave packet along the R coordinate: J D (,, ) G( R) ( r, r,,, ), (3) r tot J tot * r i K0, M i 8 ( K0 0) where the parity of is removed for the sake of clarity as it can be easily incorporated by adding the negative K term later. GR ( ) is a Gaussian shaped function, 4 R R0 ( ) GR ( ) exp e ik0r, (4) 4

15 where R 0, and k 0 are the central position, width, and mean translational momentum of the r initial wave packet. i ( r, r,,, ) is the internal state wave function of the BCD triatomic molecule and it can be expressed in the following form, ( r, r,,, ) c ( r ) ( r ) Y (,, ), (5) r K0 J0s0, K0 J0K0 i J0s0 j j j j,, j, j where J 0 is the rotational quantum number of the BCD molecule, and it is the same as j in the A+BCD arrangement, s0 is the index of different vibrational states with total rotational quantum number J 0, K is the projection of J 0 on the BF z-axis of the A+BCD arrangement, which is along the R vector. This internal state can be obtained by diagonalizing the BCD Hamiltonian, Hˆ h ( r ) ˆj V ( R, r, r,,, ) ˆ i BCD i i i ir, (6) i where the terms are the same as in Eq. (), and at sufficiently large value of R, V( R,,,,, ) r r only depends on ( r, r, ), i.e., V ( r, r, ). The final product state wave packets are defined as a product of a delta function in the corresponding scattering coordinate, which defines the projection plane, and the internal wavefunction of the products: J D (,, ) ( R R ) p tot J tot * p f K, M p f 8 ( K0), (7a) J D (,, ) ( R R ) p tot J tot * p f K, M p f 8 ( K0 ). (7b) 5

16 For the AB+CD product channel, the following wavefunction is used to describe the ro-vibrational degrees of freedom of the products: ( r, r,,, ) ( r) ( r) Y (,, ), (8) p K K j K f j j j j j j j j where the composite indices and j are explicitly expanded, ( j ) and ( j ) denote rovibrational states, j ( r ) and ( r ), of AB and CD diatoms, respectively, j determines the j relative orientation. For the B+ACD channel, the internal wavefunction assumes a similar form as in Eq. (5). It is well established that the Coriolis coupling in the BF frame is relatively long ranged. As a result, it is advantageous to construct the initial state wave packet and perform final asymptotic analysis in the SF frame, where the orbital angular momentum lˆ ( Jˆ ˆj ) is tot diagonal. 43, 55, 60 The parity-adapted BF rotational bases the SF frame by the following relation, y ( Rˆ, rˆ, rˆ ) Jtot M j j j K can be transformed to Jtot j Jtot M J Mj j jl C y, (9) tot lk j j j K Jj where C is the parity-adapted orthogonal transformation matrices between the SF and BF frames, 6 lk l C j Kl0 J K. Jtot j lk K 0 tot Jtot (0) For the two product arrangements, similar expression can be obtained by replacing the quantum numbers with corresponding primed and double primed ones. 6

17 The energy normalizing factors in Eqs. (8), ( ) J0s0l E, ( ) 0 jl E, and Js l ( E), for the initial wave packet and the final states on the two corresponding projection planes are obtained in the SF frame as follows: ( E) H ( k R) G( R) dr, (a) J0s0l0 l0 J0s0l0 kj0s 0l0 ( E) H ( k R ), (b) jl l jl p k jl ( E) H ( k R ), Js l l Js l p kj sl (c) where k ( E E ), k ( E E ) and k ( E E ) with E as J0s0l0 J0s0 jl j J s l J s the total energy, E Js, E 0 0 j, and EJs as the energies of the initial and final internal states, are the asymptotic incoming/outgoing Riccati-Hankel functions. H l The state-to-state S-matrix in the SF frame can now be expressed as follows ( ) S ( E) ( E; R ) ( E; R ), (a) J tot p r jl J0s0l0 * jl p J0s0l0 p jl ( E) J ( ) 0s0l E 0 ( ) S ( E) ( E; R ) ( E; R ), (b) J tot p r Js l J0s0l0 * Js l p J0s0l0 p J sl ( E) J ( ) 0s0l E 0 for the abstraction and exchange channels, respectively. 7

18 Finally, the S-matrix is transformed from the SF frame to the BF helicity representation by a standard transformation, 6 l 0 l S i j Kl J K S J K l 0 J K Jtot l0 l Jtot 0 jkj s0K tot 0 jlj0s0l tot 0 ll J 0 tot Jtot, (3a) l l S i JK l 0 J K S J K l 0 J K Jtot l0 l Jtot 0 Js K J 0s0K tot 0 Js l J 0s0l0 ll J 0 tot Jtot tot 0, (3b) and the state-to-state reaction probabilities P ( E ) j and K J s K P ( ) J s K J0s0K E are given as the 0 modulus squares of the corresponding S-matrix elements. III. Results and discussion III-A. Calculation Details In this section, the RCB method is implemented to calculate state-to-state reaction probabilities of the benchmark tetra-atomic reactive system for both the abstraction (H + H O H H + OH) and exchange (H + H O H + H OH) channels with zero total angular momentum (J tot =0) on the PES of Chen et al. 6 The time propagation of wave packet uses a second-order split operator propagator, 63 although other propagators are also amenable. The radial parts of the wave packet are represented by a sine basis for the R coordinate and potential optimized discrete variable representation (PODVR) bases for r and r, 64 while the angular parts use a pseudo-spectral method to transform the wave packet between rotational bases and grids. 65 An L-shaped wave function expansion for R, r and r is adopted for propagation. The absorbing potentials have the following form for x R, r, r: 8

19 x x s Vabs ( x) ic x e x s n, xs x xe, (4) where x s and x e denote the starting and ending positions of the absorbing potential, respectively, C is the strength parameter and n is the order of absorbing potential. All parameters used in the calculations are listed in Table I. It should be noted here that about 000 coupled rotational basis functions are used to represent the angular degrees of freedom, while about grid points are required to represent the same degrees of freedom in the DVR, underscoring the numerical advantage of the basis representation in projecting the product state resolved attributes. A two-dimensional plot of the interaction region of the PES in the L-shape scheme is displayed in Fig. for both the abstraction (H + H O H H + OH) and exchange (H + H O H + H OH) channels along the reactant Jacobi coordinates R and r. Due to the extended range of r in the RCB method and the skew angle for the abstraction channel, a nearly square grid is used along these two coordinates and only a small grid range along R is used for the asymptotic part of PES (not shown). On the other hand, if only the exchange channel is considered, a much smaller grid is sufficient to cover this channel while the flux into the abstraction channel can be absorbed early. The initial wave packet is prepared in the reactant asymptotic region and propagated into the interaction region. Final product states are analyzed at the two projection planes in the two corresponding product arrangement channels. 9

20 III-B. Convergence The selection of the intermediated coordinates from the following options (): ( R ; R, r,,, ) p/ p and (): ( Rp / p; r, r,,, ), depends on the relative masses of reactants and products. It resembles the case of triatomic reaction systems 5, 53 and a comparison is informative. To this end, it is convenient to define the following parameters: a m / ( m m ) A A B and c ( m m ) / ( m m m ) for the abstraction channel, and a m / ( m m m ) C D B C D A A C D and c m / ( m m m ) for the exchange channel. The abstraction channel with a B B C D and c has similar values to those of the H + HF H + F reaction and both options 5, 53 have comparable performances. For the exchange channel, both a and c have very small values, a and c 0.056, so the use of Option () will result in a very narrow range of because in this channel r R and R r. 5, 53 As shown in Fig., the projection plane of the exchange channel is almost perpendicular to the r axis. If Option () is used, a very dense grid is required for the r coordinate to properly represent the final product states on the projection plane. As a result, Option () was adopted in the current calculation for both channels. As a necessary condition, the sum of all state-to-state reaction probabilities obtained via the RCB method should be the same as the total reaction probabilities calculated using the flux method. As shown in Fig. 3, they do agree very well with each other in both channels, suggesting the accuracy of the RCB method in calculating the state-to-state reaction probabilities. It is well known that one of the OH bonds in H O can be treated as a spectator for the abstraction reaction, for which a small basis is sufficient. On the other hand, a large basis is needed for the nonreactive OH bond in the exchange channel, as it cannot be treated as a spectator. 3 In our 0

21 implementation, the two formally equivalent OH bonds in H O reactant are treated differently. State-to-state reaction probabilities is explicitly calculated only for one OH bond (O-H ), while the other OH bond (O-H ) is treated as non-reactive. Specifically, no state-to-state information is extracted for the O-H bond cleavage and the flux into the product channel is absorbed. This treatment thus focuses on one of the two sets of equivalent reaction channels and the final reaction probabilities are multiplied by a factor of two. To test the size-dependence of the basis/grid, 4 and 6 PODVR points, denoted as S (small) and L (large), respectively, were used for the non-reactive OH bond. As shown in Fig. 3, the total reaction probabilities are essentially the same for the abstraction channel, while the results for the exchange channel are quite different. This confirms the non-spectator nature of the non-reactive OH bond for the exchange channel. Tests indicate that the exchange reactivity converges with 6 PODVR points for this moiety. We note in passing that the exchange channel has a much larger reactivity than the abstraction channel. The non-spectator nature of the non-reactive OH bond originates from a shallow potential well of H 3 O with C 3v symmetry. In this well, the three OH bonds are equivalent with the bond lengths slightly elongated by Å compared to the equilibrium geometry of H O. In fact, this well supports a shape resonance, 6 which manifests as the sharp peak at E c = 0.85 ev in Fig. 3(b). The dissociation of the H 3 O complex to H + H O can occur in three equivalent channels, thus requiring large bases for all O-H coordinates for the exchange reaction. To ascertain the validity of the RCB method, we have also compared in Fig. 4 the J tot =0 final product state distribution of the abstraction channel at E c =.0 ev on the YZCL PES, 66 which have been calculated earlier using the TSWP method. 36 As shown, the agreement is excellent, confirming the accuracy of the RCB method.

22 III-C. State-to-state Reaction Probabilities The calculated final product state distributions for the H + H O(000, J 0 =0) reaction on the PES of Chen et al. 6 are shown in Fig. 5 for both the abstraction and exchange channels at two difference collision energies. Brackets in the figure denote threshold energies for the products in the corresponding vibrational states. The first and second integers in the bracket of Figs. 5(a) and (b) denote the vibrational quantum numbers of H and OH, respectively, while the three integers in the bracket of Figs. 5(c) and (d) denote the quantum numbers of the symmetric stretching, bending, and anti-symmetric stretching modes of H O, respectively. For the abstraction channel shown in Figs. 5(a) and (b), the H and OH products are only found in their ground vibrational states at the two selected collision energies, even though sufficient energy is available to produce either H or OH in the first excited vibrational state at E c =.5 ev. The increase of collision energy from 0.85 to.5 ev only results in hotter rotational distributions of the two products. On the other hand, for the exchange channel shown in Figs. 5(c) and (d), the final H O state distributions are quite interesting. No state-to-state calculation has been done for this channel prior to the current work. The two collision energies are chosen to be the position of the peaks in the total reaction probability in Fig. 3(b). At E c = 0.85 ev (the first sharp peak), H O is produced solely in the ground vibrational state with different rotational states. With E c increased to.5 ev (the second broad peak), H O is still mostly produced in the ground vibrational state but also in the bending excited states, (00) and (00), and even in the stretching excited states, (00) and (00).

23 The vibrational state distributions of the two product channels can be rationalized by the Sudden Vector Projection (SVP) model, which attributes the product energy disposal to the coupling between the product vibrational modes and the reaction coordinate at the transition state leading to the product asymptote. 67, 68 Such coupling is approximately quantified in the sudden limit by the projection of the product normal mode vectors onto the reaction coordinate at the (exit) transition state. In Table II, the SVP values are listed for the two arrangement channels. It is clear from the table that the OH product in the abstraction channel is not expected to be excited as its vibration is essentially decoupled from the reaction coordinate, while the H coproduct could be excited thanks to its large SVP value. On the other hand, none of the water vibrational modes has large SVP values, thus unlikely to couple strongly with the reaction coordinate at the transition state. This is easy to understand, as the geometry of the H O moiety changes little from its equilibrium geometry in either the H 3 O well or the transition state. As a result, the dissociation of the H 3 O complex into the product channel is not expected to increase the vibrational excitation of the H O product. These SVP predictions are consistent with the quantum results at low collision energies. The final H O vibrational-state-resolved and rotational-state-summed state-to-state reaction probabilities are shown in Fig. 6. The first sharp peak is solely from H O product in ground vibration state, which is consistent with the observation in Fig. 5(c). With E c increased to.0 ev, H O product starts to be produced in the first bending excited state and at around E c =.5 ev, also the (00), (00) and (00) vibrational states are produced. At E c below.3 ev, those five vibrational states constitute all the possible vibrational states of the H O product. It is obvious that the second broad peak in Fig. 6 is mostly from the H O product in the first excited bending state (00), which is consistent with the observation in Fig. 5(d), where many rotational 3

24 states of the first excited bending state (00) are also produced with noticeable populations. The vibrational excited H O product at high energies is most likely due to the population of vibrationally excited H 3 O species, rather than the forces exerted by the exit transition state. Note that the exchange pathway has both an entrance and exit barrier flanking a shallow H 3 O well, and as a result the product state distribution is much harder to predict with the SVP model. The final H and OH vibrational-state-resolved and rotational-state-summed state-to-state reaction probabilities for the H + H O H H + OH abstraction channel are shown in Fig. 7. At E c below.4 ev, both the H and OH products are formed in their ground vibrational states. At E c larger than.4 ev, the H product starts to show populations in the first excited vibrational state while OH is still in the ground vibrational state, even though OH has a smaller vibrational frequency energy than H. The absence of OH excitation is in accordance with the spectator character of OH bond in the abstraction channel. 39 Finally, rotational state distributions of final products in their ground vibrational states are displayed in Fig. 8 at three different collision energies. In Figs. 8(a) and (b), it is shown that rotational state distributions of the H and OH products in the abstraction channel become hotter with increasing collision energy. At each collision energy, H product prefers odd rotational states while OH product prefers even rotational states. In Fig. 8(c), the rotational state distributions of H O product in the exchange channel peak at J = 3 for all the three collision energies. Particularly, the peak at E c =.0 ev is rather narrow and distribution at E c =.5 ev has two peaks at J=3 and 5. IV. Conclusions 4

25 In this publication, we present a new state-to-state quantum reactive scattering method for four-atom reactive systems. This RCB method propagates the wave packet in the reactant Jacobi coordinates, and projects out the state-to-state S-matrix elements in multiple product channels using intermediate coordinates. These intermediate coordinates are designed to minimize the transformation from the reactant and product coordinates, thus making the projection numerically efficient. This new method is demonstrated here for the title reaction with zero total angular momentum. Future work will report results for J tot >0 and the cross sections for this and other tetra-atomic reactions. Much work is needed in the future to test the relative efficiency amongst the existing quantum state-to-state methods for tetra-atomic reactions. Acknowledgements: This work was supported by the US Department of Energy (Grant No. DE- FG0-05ER5694 to H.G.) and by the National Natural Science Foundation of China (Grant Nos. 308, and to Z.S.). The calculations were performed at the National Energy Research Scientific Computing (NERSC) Center and at the Center for Advance Research Computing at University of New Mexico. 5

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28 Table I. Parameters used in the calculations (Atomic units are used if not otherwise stated) Initial wave packet R k E with E 0 =.0 ev k 0 0 Grid range and size R R (0.4,5.0), r N R =, r (0.7,4.0), =80, int N R =0 asy =8 r r (0.7,4.0), int =6&4, asy =4 j j (0,00) j (0,0) j Absorption Potential C x s x e n R r r Total time/time step: 0000/0 Projection planes: R p=.0, R p =.0 flux Flux plane: r =7.0 8

29 Table II. SVP values for the H H + OH abstraction channel and the H + H OH exchange channel in the reaction of H + H O H H + OH H + H OH Mode SVP Mode SVP H 0.36 s 0.06 OH a 0.08 Trans. 0.9 b 0.09 Trans

30 Figure captions: Fig. The reactant A+BCD Jacobi coordinates ( R, r, r,,, ) and the Jacobi coordinates for the AB+CD abstraction channels ( R, r, r,,, ) and the B+ACD exchange channel ( R, r, r,,, ) in tetra-atomic reactive systems. and denote the angles between the BF z-axes in the abstraction and exchange reactions, respectively. Fig.. Two-dimensional contour plot of the PES for both the H + H O H H + OH abstraction (Abs.) and H + H O H + H OH exchange (Exc.) channels in the reactant Jacobi coordinates R and r. Other degrees of freedom are optimized. The initial wave packet is prepared in the reaction asymptotic region and two projection planes are located in the asymptotic regions of corresponding product channels. Fig. 3. Convergence of the initial state specific total reaction probability of (a) the H H + OH abstraction and (b) H + H OH exchange channels for the title reaction H + H O with H O in the ground ro-vibrational state. The results calculated from the flux and state-to-state (SS) methods are compared. Two PODVR bases are used, denoted as the larger (L) and small (S) bases, for the non-reactive OH moiety. The large (L) basis use 6 PODVR basis, while the small (S) one has only 4. The details of the bases are given in Table I. Fig. 4 Comparison of the final product state distribution in the abstraction channel between results obtained using the RCB method and previous result using the TSWP method at E c =.0 ev on the YZCL PES. Fig. 5. Final product state distributions for the reaction H + H O (000) in both product channels at two difference collision energies: (a) and (b) for the H H + OH abstraction channel, (c) and (d) for the H + H OH exchange channel. The rotational states with same j and j but different j are summed over. The H and OH ro-vibrational energies are defined relative to the energy of ground ro-vibrational states of H and OH. The H O ro-vibrational energy is defined relative to the energy of ground ro-vibrational energy of H O. Threshold energies to form corresponding vibrational states are marked with black arrows, along with purple arrows denoting the maximum energy available at the indicated collision energy. The first and second integers in the bracket of (a) and (b) denote the vibrational quantum numbers of H and OH, respectively, while the three integers in the bracket of (c) and (d) denote the quantum numbers of symmetric stretching, bending and anti-symmetric stretching modes of H O. Fig. 6. Final H O vibrational-state-resolved and rotational-state-summed state-to-state reaction probabilities for the exchange channel H + H O(000) H + H OH as a function of collision energy. Fig. 7. Final H and OH vibrational-state-resolved and rotational-state-summed state-to-state reaction probabilities for the abstraction channel H + H O(000) H H + OH as a function of collision energy. 30

31 Fig. 8. Rotational state distributions of the final H (a) and OH (b) products in the abstraction channels and the H O (c) product in the exchange channel at three different collision energies. All the products are in their ground vibrational state. The rotational state populations of H O with same total rotational quantum number but different orientations are summed over. 3

32 Fig. 3

33 Fig. 33

34 Fig. 3 34

35 Fig. 4 35

36 Fig. 5 36

37 Fig. 6 37

38 Fig. 7 38

39 Fig. 8 39

40 (a) Abstraction Channel C C D r (r ') R θ' θ ' R' '' θ θ (b) Exchange Channel r B ' r A ϕ' ϕ A r (r '') r '' θ'' θ '' '' R ϕ'' D θ θ r R'' B ϕ

41 H + H OH H H + OH Exc. projection plane r (a.u.) Abs. projection plane Initial wave packet H + H O R (a.u.)

42 flux (S) sum of SS (S) flux (L) sum of SS (L) (a) Abstraction Reaction Probability (b) Exchange E c (ev)

43 Reaction Probability (a) RCB Max. Energy (b) TSWP Max. Energy H O ro-vibrational energy (ev)

44 Reaction Probability (a) 0.85 ev Max. Energy (0) 0.0 (b) ev Max. Energy (0) H and OH ro-vibrational energy (ev) (c) 0.85 ev (00) (00) (d) ev (00) (00) Max. Energy Max. Energy H O ro-vibrational energy (ev)

45 Reaction Probability sum of all SS sum of the five vib (000) (00) (00) (00) (00) E c (ev)

46 Reaction Probability 3 x 0 6 ν H =0 4 ν H = ν OH =0 ν OH = E c (ev)

47 (a) H E c =0.85 E c =.00 E c = Reaction Probability (b) OH (c) H O Rotational Quantum Number