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Reactant Coordinate Based State-to-State Reactive Scattering Dynamics Implemented on Graphical Processing Units Pei-Yu Zhang and Ke-Li Han* State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People’s Republic of China ABSTRACT: A parallel code for state-to-state quantum dynamics with propagation of time-dependent wavepacket in reactant coordinates has been developed on graphical processing units (GPUs). The propagation of wavepacket and the transformation of wavepacket from reactant to product Jacobi coordinates are entirely calculated on GPUs. A new interpolation procedure is introduced for coordinate transformation to decrease the five-loop computation to two four-loop computations. This procedure has a negligible consumption of extra GPU memory in comparison with that of the wavepacket and produces a considerable acceleration of the computational speed of the transformation. The code is tested to get differential cross sections of H+HD reaction and state-resolved reaction probabilities of O+HD for total angular momenta J = 0, 10, 20, and 30. The average speedups are 57.0 and 83.5 for the parallel computations on two C2070 and K20m GPUs relative to serial computation on Intel E5620 CPU, respectively.

I. INTRODUCTION During the past few decades, the development of quantum scattering methods and advances in computer technologies have allowed us to tackle state-to-state reactive dynamics involving complicated triatomic reactions or even larger systems. The results of the calculations involve the most detailed observables being comparable with the experiments and help us to investigate the reaction resonance, nonadiabatic transition, and other interesting phenomena in chemical reactions.1,2 A central issue in the use of quantum scattering calculations involves an efficient choice of the coordinate system. The hyperspherical coordinates have been commonly used in the time-independent (TID) coupled-channel method.3,4 In timedependent wavepacket (TDWP) methods, the coordinates can be chosen as reactant, product Jacobi, or hyperspherical coordinates.5−7 Typically, the initial wavepacket is prepared in reactant Jacobi coordinates, and the state-to-state information on product is obtained in the product Jacobi coordinates. The coordinate transformation of the wavepacket is needed and can be implemented once before the propagation, continuously at the end of every time step of propagation, or in the interaction region. The quantum scattering approaches with these three different transformations are called product coordinate based (PCB), 8,9 reactant coordinate based (RCB),6,7 and reactant−product decoupling (RPD) approaches,10,11 respectively. The RPD method was developed in the early state-to-state quantum scattering computation and has been successfully used in reactions such as H + H2 and F + H2.10−12 Through analyzing the effect of the mass combinations in the reaction, the results of Gomez-Carrasco and Roncero indicate that in general the use of reactant Jacobi coordinates is more efficient than the PCB approach.6 Moreover, the RCB © 2014 American Chemical Society

approach is found to work well for the reactions with a deep well or a long-range attractive interaction in both the reactant and product channels.7 In the past three years, parallel computing on graphical processing units (GPUs) has been applied to quantum scattering calculations using both TID and TDWP methods. Baraglia et al. developed a GPU version of the TID quantum reactive scattering ABC code,3,13 and the test shows a 6.98 speedup obtained by three GPUs and three central processing unit (CPU) cores on computational efficiency. Comparing the TID method, the time-dependent method is based on matrix− matrix multiplication. The computational speedups increase almost linearly with the number of processors and have shown high parallel efficiency. Pacifici et al. have found the speedups on the order of 2−20 times for GPU versus CPU in evaluation of the reactive probabilities of two direct reactions.14 In our previous work, we presented an implementation of state-tostate reactive scattering dynamics on GPUs in product Jacobi coordinates.15 The result shows global speedups of more than 22.11 and 44.80 times for one and two GPUs, respectively. In this work we developed a parallel code for state-to-state quantum dynamics on GPUs in the RCB approach as an important complement of the previous PCB code. The major change between the present RCB code and previous PCB one involves the coordinate transformation of the wavepacket at the end of every time step of wavepacket propagation. Special Issue: International Conference on Theoretical and High Performance Computational Chemistry Symposium Received: January 28, 2014 Revised: June 17, 2014 Published: June 18, 2014 8929

dx.doi.org/10.1021/jp5009779 | J. Phys. Chem. A 2014, 118, 8929−8935

The Journal of Physical Chemistry A

Article

|χi ⟩ = ΨαJMv0εj l0 = G(R α)φv j (rα)|JMj0 l0ε⟩

The typical transformation of the wavepacket is given by the interpolation of wavepackets in reactant Jacobi coordinates to obtain the wavepacket of product rovibrational states.7,10 Gomez-Carrasco and Roncero proposed a procedure using several intermediate coordinates for the transformation.6 The test in their program shows high efficiency of the transformation and the negligible percentage of the total time consumed by the transformation. The procedure constructs and saves the body-fixed transformation matrix and the matrix for the transformation between the intermediate coordinate and product coordinate. The memory for the transformation matrices is smaller than that of one wave packet in the reactions with a relatively small number of rovibrational product states. The size of these matrices increases with the number of product states. Thus, the state-to-state dynamics calculation with intermediate coordinates can require several times the computer memory compared to the calculation with the ordinary transformation approach in the reactions with a large number of product states. This memory consumption is not a big issue on CPU computation. However, it will severely affect the computational capacity on GPU computation, since a GPU card only has a few GB of memory, and there are still drawbacks of communication between GPU and the memory chips on the system board. In this work, a new procedure of the transformation between product Jacobi coordinates and reactant Jacobi coordinates is proposed and embodied in the RCB parallel code for state-tostate quantum dynamics on GPUs. This procedure involves the transformation of the vibrational basis to regular grids and a linear interpolation to finish the transformation between reactant and product coordinates. The paper is organized as follows. The next section outlines theoretical methods for the construction of wavepacket and Hamiltonian, the propagation of wavepacket, the procedure of the transformation, and the calculation of reaction probabilities, differential, and integral cross sections. In Section III the code is applied to H+HD and O(1D)+HD reactions. Section IV concludes the paper.

⎯⇀ ⎯ ⎛ ⇀ Δ⎞ ̂ Δ/2) Ψ JMp(R , r , t + Δ) = exp⎜ −iĤ 0 ⎟exp( −iVrot ⎝ 2⎠ ⎯⇀ ⎯ ⎛ ⎞ ̂ Δ/2)exp⎜ −iĤ 0 Δ ⎟Ψ JMp(R , ⇀ exp(−iV Δ)exp( −iVrot r , t) ⎝ 2⎠

(4)

where ℏ2 ∂ 2 ℏ2 ∂ 2 · 2 − · + Vr(r ) Ĥ 0 = − 2μR ∂R 2μr ∂r 2 ̂ = Vrot

ℏ2 ∂ 2 + Vr(r ) 2μr ∂r 2

(6)

2

⟨J ′K ′j′K ′|L̂ |JKjK ⟩ = ⟨J ′K ′j′K ′|(J ̂ − j ̂)2 |JKjK ⟩ = [J(J + 1) + j(j + 1) − 2K 2]δjj ′δkk ′ − [J(J + 1) − K (K + 1)]1/2 [j(j + 1) − K (K + 1)]1/2 δjj ′δkk ′+ 1 − [J(J + 1) − K (K − 1)]1/2 [j(j + 1) − K (K − 1)]1/2 δjj ′δkk ′− 1 (7) K

Since the Coriolis coupling matrix H is tridiagonal, it is divided into two block-diagonal matrices, A and B, which preserve the exponential of the A and B block-diagonal.

(1)

where μR is the reduced mass between atom A and diatom BC and μr is the reduced mass of BC. J ̂ is the total angular momentum operator, and j ̂ is the rotational angular momentum operator of BC. V(R, r, θ) is defined as Vpes − Vr(r), where Vpes is the potential energy of the triatomic system, and Vr(r) is the diatomic potential energy of BC. h(r) is the diatomic reference Hamiltonian defined as h(r ) = −

2 2 ĵ L̂ + 2μR R2 2μr r 2

(5)

where L = J − j. To obtain state-to-state information, several K(NK) components of wave function for each total angular momentum J are used to get convergent dynamics information. In the body-fixed representation L is not a good quantum number for rotational basis of associated Legendre polynomials.

2 (J ̂ − j ̂)2 ĵ ℏ2 ∂ 2 + + + h(r ) 2μR ∂R2 2μR R2 2μr r 2

+ V (R , r , θ )

(3)

|JMj0l0ε⟩ represents the SF rotational basis and describes the angular motion with M being the projection of total angular momentum J for the initial state (v0, j0, l0). ϕv0j0(rα) represents the rovibrational eigenfunction of diatomic molecules. Then this wave function can be propagated in BF product Jacobi coordinates by a split-operator scheme

II. THEORY A. Hamiltonian and Propagation of Wavepacket. The Hamiltonian of the A + BC reaction in body-fixed (BF) reactant Jacobi coordinates can be written as16 H=−

00

0

⎡ HK /2 H K ⎤ 12 ⎢ 11 ⎥ ⎢ HK H K /2 ⎥ 22 ⎢ 12 ⎥ K K ⎥ A=⎢ H33/2 H34 ⎢ ⎥ K K ⎢ H34 H44 /2 ⎥ ⎢ ⎥ ⎣ ⋱⎦

(8)

⎡ HK /2 ⎤ ⎢ 11 ⎥ K K ⎢ ⎥ H 22 /2 H 23 ⎢ ⎥ K K ⎢ ⎥ B= H 23 H33/2 ⎢ ⎥ ⎢ K K⎥ H44 /2 H45 ⎢ ⎥ ⎢ K H45 ⋱ ⎥⎦ ⎣

(9)

Then the exponential of HK can be expressed by operator splitting. Using the new expression for operator splitting, every GPU/CPU will communicate with the neighbors rather than all of the other GPUs. Thus, it decreases communication time among GPUs/CPUs and is a good choice in multiple GPU/

(2)

The reactant is prepared in a space-fixed (SF) reactant Jacobi coordinates system using the Gaussian wavepacket in the R direction 8930



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CPU computations. The tests of previous work on two GPUs show 38.80 and 44.80 time speedups without and with this split-operator of the Coriolis coupling matrix, respectively.15 B. Coordinate Transformation. The radial component of the product wavepacket is chosen to be a delta function times the outgoing asymptotic radial functions8 |χf ⟩ = δ(R − R ∞)φv j (rβ)|JMjf l f ⟩

ψ (R ∞ , ri , θj , Kβ) =

Kα

∑ Kα

Cn,jrα is linear interpolated between the two points r (R ∞ , ri , θj)

Cnα, j

(11)

∑ Cn,v ,jun(R α)φv(rα)yjK (θα) α

n,v ,j

(12)

where the un has a form of sine function, and φv is vibrational basis for the diatom BC. yjK is a symmetry-adapted spherical function. Thus, in the two dimensions Rα and θα, the product wavepackets can be derived directly by sine and spherical functions. However, φv has no special functional form. We divide the transformation procedure into two steps. (1) In the vibrational dimension rα, we transform the wavepacket of eq 12 from vibrational FBR to densely regular grids, rv1 (v1 = 1, 2, ..., Nr) ψα(t , R α , rαv1 , θα , Kα) =

∑ Cnv,1jun(R α)yjK (θα) α

n,j

(v1 = 1, 2, ..., N r )

(13)

where the Cv1 is derived by the transformation matrix

Cnv,1j = Tv , v1Cn , v , j

(16)

The number of densely regular grids Nr should be adjusted to get convergent results by this simple interpolation. The detailed tests will be shown in Section III. Equations 13 and 15 are the loops of four coordinates (Rα, θα, ri, θj) and (rv1, Rα, rα, θα), respectively. Thus, this procedure of two steps replaces the original five-loop to two four-loop calculations. Moreover, the main time consumption for the transformation is at the computation vibrational transformation in eq 13 at the first step. This equation is the same for both the product channels B + AC and C + AB. It means the computation of two product channels shares the first step of transformation. Thus, the transformation shows more time speedup when the computation of two product channels is needed. C. Parallel Computing on GPUs. Fortran language CUDA extension included in the PGI compiler is used to implement the GPU version of the quantum dynamics code. The initial wavepacket, the matrix for transformation, and Hamiltonian are calculated on a CPU. After preparation for propagation on a CPU, the data are copied from CPU memory to GPU memory. Then the propagation of the wavepacket and the transformation of the wavepacket entirely run on GPUs. The detailed description of parallel computing can be found in ref 15. Here we concentrate on the parallelism of transformation of the wavepacket between reactant and product coordinates. The time-dependent wavepacket in eq 13 is distributed to NP GPUs, and every GPU saves the components of the wavepacket with a number of specific Kα. Thus, eqs 13 and 15 are implemented individually for the components of a wavepacket in their own GPUs. The last product functions at the dividing surface are the summation of product functions on all of the GPUs. The performance of procedure of transformation will be tested in the next section. There is still a memory limitation problem to be solved before calculation. In the first step of coordinate transformation, the wavepacket on the lhs of eq 13 should be saved in GPU memory and requires a substantial memory. We divide the range of the regular grids rv1 to several sections, in which a loop for the calculation of eqs 13 and 15 is executed to complete the transformation in the entire range. This treatment solves the problem of memory occupation in eq 13. D. State-to-State Information on Products. The timedependent expression for the scattering matrix element in BF representation is obtained17

With this equation, the direct transformation of the wavepacket is a five-loop computation, which involves the interpolation of three-dimensional wavepackets in reactant Jacobi coordinates to get the two-dimensional product wavepacket. The timedependent wavepacket in reactant Jacobi coordinates (Rα, rα, θα, Kα) in finite basis representation (FBR) is ψα(t , R α , rα , θα , Kα) =

= (Cnva, j(rα − r va) + Cnvb, j(rα − r vb))/(r va − r vb),

r va < rα < r vb

R ∞ri ψ (R α(R ∞ , ri , θj) R αrα

, rα(R ∞ , ri , θj), θα(R ∞ , ri , θj), Kα)d KJεβ*Kα

(15)

α

R∞ is the location of the dividing surface. It is a fixed radial coordinate at the asymptotic region in product Jacobi coordinates. The basis sets of the rovibrational product correspond to a set of the grid points (R∞, ri, θj, Kβ), Kβ being the projection of total angular momentum J in BF product coordinates. The typical procedure of transformation between wavepackets from reactant to product Jacobi coordinates at the dividing surfaces is the interpolation from the wavepacket in reactant Jacobi coordinates (Rα, rα, θα, Kα) ψ (R ∞ , ri , θj , Kβ) =

n,j

R ∞ri rα(R∞, ri , θj) Cn , j un(R α R αrα

(R ∞ , ri , θj))yjK (θα(R ∞ , ri , θj))d KJεβ*Kα

(10)

f f

∑∑

(14) v1

Tv,v1 is vibrational wavepacket at the grid r . It is evaluated by transformation of vibrational basis to Nr sine functions. Not all of the wavepackets in vibrational dimension should be transformed to that of regular grids. The range of regular grids is chosen to cover the region of product functions in reactant coordinates and is determined by the transformation of the product rovibrational wavepacket from product coordinates to reactant coordinates before wavepacket propagation. Thus, after this step, the wavepacket in reactant coordinates is represented by regular grids in the vibrational dimension rα and basis functions in the translational and angular dimensions. (2) The product functions are given by

Sfi(E) =

+∞

1 2πaαi(E)adf *(E)

∫−∞

̂

dte(i / ℏ)Et ⟨χβf |e(i / ℏ)Ht |χαi ⟩ (17)

The coefficients are given by aαi(E) = ⟨φi|χi+ ⟩ = 8931

i

μR kαi 2π

Rhl(2) (kαiR )|G(R α) α (18)


The Journal of Physical Chemistry A aβf (E) =

⟨φf |χf+ ⟩

μR kβf

=

2π

Article

Rhl(1) (kβf R )|δ(R β

Clearly all of the DCSs by the RCB TDWP method in Figure 1 are in perfect agreement with time-independent results calculated by the ABC code over the entire energy range. The convergence of linear interpolation in the coordinate transformation for the reaction H + H′D → HH′ + D/HD + H′ is tested by the calculation of DCS with three different regular grids. The total number of regular grids in the whole range (0.5−11.2) of rα is 300, 400, and 500, respectively. The range of rv1 of the product functions is (4.86−7.06a0) in the reactant coordinates, and there are 65, 82, and 106 regular grids in these ranges, respectively. The wavepackets are transformed by eq 12 to the values at these grids. Figures 2a−2d show the DCSs calculated with two different regular grids Nv1 = 65 and 106. The figures demonstrate that the convergence of the linear interpolation of this reaction can be derived by a small number of regular grids. In the three computations with a different number of regular grids, the percentages of time consumption of the coordinate transformation for J = 10 are 13%, 15%, and 17%, respectively. For J = 20, the values are 13%, 14%, and 15%, respectively. Thus, the percentage of time consumption is almost invariant with increasing J. This is considerable because the computational complexity of eq 13 and eq 15 linearly and squarely increases with increasing J, but the main time consumption of the coordinate transformation happens in eq 13 rather than eq 15. This invariance of percentage is an attractive character since the percentage in the computation with the original transformation method increases with increasing J, and the main time consumption for a state-to-state calculation is at the calculation of scattering matrix with large J. Figure 3 shows speedups of parallel computation on GPU relative to the serial calculation on Intel E5620 CPU with the total angular momentum J. The speedups are 27.5 and 41.6 for one C2070 and K20m averaged over the results of J = 10−24, respectively, and 57.0 and 83.5 for two C2070 and K20m averaged over the results of J = 10−30, respectively. The computing performance of K20m is 1.5 times that of C2070. The speedup increases with the increasing J for the computation on two GPUs. At J = 30, the speedups are 82.5 and 122.0, for two C2070 and K20m GPUs, respectively. As mentioned in Section II. C, the wavepackets are transformed in their own GPUs. Thus, the time consumption of communication among GPUs in the transformation procedure is negligible. For both types of GPU cards, the speedup ratios of two GPUs and one GPU are 1.7 at J = 17 and 2.0 at J ≥ 23, which shows the good parallel efficiency of the wavepacket propagation and newly proposed transformation procedure. B. Reaction Probabilities of O(1D) + HD. Reaction O(1D) + H2 → H + OH is important in understanding the photochemistry process in atmospheric reactions. Many potential energy surfaces have been developed to study the reaction dynamics of the O(1D) + H2 system.22−30 The differential cross sections of the O(1D) + HD reaction have been studied by the quasiclassical trajectory method,31,32 and there exists disagreement between theory and experiment in the OD/OH branching ratio. As far as we know, no state-to-state quantum dynamics have been reported except for zero total angular momentum.33 Here we calculate the scattering matrix of this reaction for J = 0, 10, 20, and 30 with all of the helicity quantum numbers considered. The dynamics calculations have been carried out on DK PESs with the numerical grid parameters shown in Table 1.34 The number of states in one product channel is 5500, 9450, and 12400 for J = 10, 20, and

− R ∞) (19)

Finally the scattering matrix is transformed into helicity representation to derive differential and integral cross sections by scattering matrix summing over all relevant total angular momentum quantum numbers J. dσv ′ j ′ , v0j (θ , E) 0

dΩ

=

1 (2j0 + 1) 1 2ik v0j

∑∑ K′

K0

∑ (2J + 1)dKJ ′ K (θ) 0

J

0

2

SvJ′ j ′ , v0j 0 (20)

σv ′ j ′ , v0j = 0

1 (2j0 + 1)k v20j

0

∑ ∑ ∑ (2J + 1)|SvJ′ j ′ ,v j |2 00

K′

K0

J

(21)

III. EXAMPLE CALCULATION Two reactions, H + HD and O(1D) + HD, are chosen to test the convergence of linear interpolation in the coordinate transformation and the efficiency of the GPU version of the RCB quantum scattering code. These two reactions represent the types of direct reaction and the deep well reaction, respectively. Two types of GPU cards, K20m and C2070, are used in the test. A. Differential Cross Sections of the H + HD Reaction. The H3 system and its isotopic variants are the prototype for the reaction with bond breaking and bond formation and have been studied in detail both experimentally and theoretically.18−20 Here the calculations have been performed employing the BKMP2 surface.21 The differential cross sections are calculated by the GPU version of RCB quantum dynamics code with the new procedure of coordinates transformation. In this test the numerical grid parameters are shown in Table 1. Figures 1a−1d show the state-to-state differential cross sections (DCSs) as a function of total energy for two scattering angles θ = 0 and 90 of rovibrational state v = 0, 1 and j = 0, 2, 4, 6. Table 1. Numerical Grid Parameters Used in the Calculationsa Rmin−Rmax NR, NINTR rmin−rmax Nr, Nrb, Nvb Nj R0, δ E0 (eV) Ttotal, dt NPv NPj Nallv1 a

H + HD

O + HD

0.002−16.0 120, 50 0.5−11.2 300, 60, 5 50 8.0, 0.7 0.7 10000, 5 5 20 300/400/500

0.002−20.0 400, 130 0.7−14.0 300, 120, 4 120 12.0, 0.4 0.2 40000, 5 10 55 300/500/600

Atomic units are used unless otherwise specified. 8932



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Figure 1. Differential cross sections as a function of total energy of the rovibrational state v′ = 0 and j′ = 0, 2, 4, 6, for two scattering angles θ = 180° and 90°. The results of two product channels HD + H and HH′ + D are shown in the upper and lower panels, respectively. The lines and symbols represent results from our work and ABC codes, respectively.3

Figure 2. Differential cross sections as a function of total energy for two scattering angles θ = 180° and 90°. The results of two product channels HD + H and HH′ + D and are shown in the upper and lower panels, respectively. The symbols and lines represent results with the number of regular grids Nv1 = 65 and 106, respectively.

= 10. The computation with 134 grids can get converged results in the D + OH channel and not converged in the H + OD channel. The results of computation with 222 and 266 grids are overlapped with each other, which means the simple linear interpolation in the coordinate transformation with 222 regular grids can get converged differential cross sections in the O + HD reaction. In the parallel computing on two GPUs, the percentages of the time consumption of the coordinate transformation for J = 10 are 19%, 23%, and 25%, respectively.

30, respectively. Thus, the computation is challenging, and the typical coordinate transformation method is difficult to handle in this situation. Three total regular grids, 300, 500 and 600, in the whole range (0.7−14.0) of rα are chosen to test the convergence of linear interpolation. The range of rv1 of the product functions is (4.08−9.89 a0) in the reactant coordinates, and there are 134, 222, and 266 regular grids in these ranges, respectively. Figures 4a and 4b show convergence of the number of regular grids in the computation of reaction probabilities of the product vibrational state v′ = 0 and 5 at J 8933



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Figure 5. OD/OH branching ratios as a function of energy at J = 0, 10, 20, and 30, respectively. Figure 3. Speedups of parallel computation on GPU relative to the serial calculation on CPU versus total angular momentum J. The lines with squares, circles, and up-pointing and down-pointing triangles show the speedups of the parallel computation on two K20m, one K20m, two C2070, and one C2070, respectively.

differential cross sections and explore quantum effects on OD/ OH branching ratios.

IV. CONCLUSIONS In this paper, the reactant Jacobi coordinate based state-to-state quantum dynamics has been implemented on GPUs. A new procedure of the coordinate transformation has been introduced to accelerate the calculation without increasing the memory consumption. The propagation of wavepacket and the coordinate transformation between reactant and product coordinates are entirely calculated on GPUs. Thus, after preparation of initial wavepacket, the CPUs of the computing node can still be used to do other calculations. The tests on two GPUs show 57.0 and 83.5 time speedups for C2070 and K20m relative to Intel E5620 CPU, respectively, and the highest speedup ratio in the test is 122.0 at J = 30. Thus, the GPU parallel computing provides 2 orders of magnitude of increased performance with only two GPUs for the quantum dynamics of complex reactions with large helicity quantum number. Using the simple linear interpolation in the procedure of coordinates transformation can get the converged results in direct reaction H + HD by 62 regular grids and in deep well reaction O(1D)+ HD by 222 regular grids. The percentage of the time consumption of the coordinates transformation is 13−23% for these reactions. Thus, a relatively small percentage of additional time consumption is needed for the computation of DCSs compared with that of total reaction probabilities and cross sections. It should be noted that the approach with intermediate coordinate is a good choice for many reactions, and the procedure proposed in this work can be used in the reaction including a large number of product states or the situation involving the memory limitation.

Figure 4. Reaction probabilities as a function of energy of product vibrational state v′ = 0 and 5 at J = 10. The results of two product channels OH + D and OD + H are shown in the upper and lower panels, respectively.

■

The OD/OH branching ratios for J = 0, 10, 20, and 30 are shown in Figure 5. The average branching ratios are smaller than 2 at J = 0 and 30 and bigger than 2 at J = 10 and 20. The centrifugal potential suppresses the branching ratio at small collision energy. At Ec = 0.196 eV, the branching ratios are 2.24, 2.40, and 1.68 for J = 10, 20, and 30, respectively. These values are consistent with the previous quasiclassical trajectory study of Aoiz et al.32 In the future, the state-to-state quantum dynamics for dozens of J will be performed to get converged

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-0411-8437-9293. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by NSFC (11304310) and the National Basic Research Program of China (2013CB834604). 8934



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