Laser-Driven Isomerization of HCN â HNC: The Importance of

Article pubs.acs.org/JPCA

Laser-Driven Isomerization of HCN → HNC: The Importance of Rotational Excitation Zhaopeng Sun† and Yujun Zheng*,†,‡ †

School of Physics, Shandong University, Jinan 250100, China Beijing National Laboratory for Molecular Sciences (BNLMS)

‡

ABSTRACT: We report a time-dependent quantum wave packet theory, which is employed to interpret the isomerization dynamics of HCN molecules induced by an intense picosecond infrared laser field. Considering the molecular rotational degrees of the freedom, the wave functions are expanded in terms of molecular rotational bases. Our full-dimensional quantum model includes the full Coriolis coupling in the molecular kinetic energy Hamiltonian and dipole approximation in interaction terms. The numerical results show that the field-induced molecule rotational excitation plays an important role in the isomerization dynamical process. Some phenomena appear such as two-step two-photon absorption and highly oscillatory structure in rotational state distributions. The centrifugal sudden (CS) approximation calculation is also carried out and compared in this work; it is shown that the Coriolis couplings may lead to a significant decrease in the isomerization rate but highly enhanced molecular rotational excitation.

■

Gong et al.22 reported a full-dimensional study of classical dynamics of HCN → HNC isomerization driven by a strong infrared laser field. They pointed out that when analyzing molecular dynamics in the presence of strong fields, neglecting some degrees of freedom of the molecule to simplify the dynamical description is dangerous. Moreover, the rotation of the H atom around the C−N bond provides an important and highly efficient energy absorption mechanism. On the quantum optimal control side,24 with the centrifugal sudden (CS) approximation, the C−N stretching mode was found to be important in the dynamics produced by the optimal field. However, all of the quantum dynamics studies mentioned above adapted some approximation such as using the CS approximation, fixed at J = 0, or confining the molecule in a plane. For one thing, due to its limitation, the approximate calculation only can bring a rough description of the molecular dynamics; some connotative but important phenomenon will be lost. For the other, in the previous fulldimensional Lagrangian dynamical study,22 some new and fascinating features have been revealed. Thus, to better comprehend and clarify the physical and chemical mechanisms of the laser-induced HCN isomerization reaction, the full-dimensional quantum study is needed. Our specific motivations and aims can be attributed to three aspects. First, we provide a generalized wave packet method for describing the dynamics of a triatomic molecule exposed to the intense external fields. Here, we are interested in the method for controlling molecular dynamics in the category of triatomic molecules. Also, we will use this method to study the lasercontrolling triatomic molecular dynamics such as photodissociation

INTRODUCTION The application of the time-dependent wave packet method to solve dynamical problems in molecular physics has been successful for the understanding of basic atomic and molecular phenomena. A variety of chemical problems such as photodissociation1−6 and reactive scattering,7−12 have been investigated using the timedependent quantum wave packet method. The triatomic molecular system is ideally suited for time-dependent quantum calculation because its potential surfaces can be obtained accurately via ab initio calculations. However, when the molecule is exposed to an external field, it will not be a conservative system anymore. In order to obtain more comprehensive information, all six degrees of freedom of the triatomic molecule should be included; hence, description of its dynamical behaviors is a challenging problem. The HCN molecule has received much attention as a prototypical case of a triatomic molecule in studying of optical control of molecular dynamics.13−25 Putting the low-temperature isocyanide matrix through ultraviolet rays13 or acetonitrile in a flowing afterglow of nitrogen,14 the isomerization product CNH can be observed experimentally. After the observation of the HCN ⇌ HNC equilibrium from its infrared spectrum,26 the double-well potential of the ground-state surface of HCN was confirmed by Murrell and Carter.27 The advances on the experimental side attracted much theoretical attention to study its laser-induced isomerization dynamics. In 1996, Dion et al.16 simulated the isomerization dynamics of HCN in the presence of two perpendicular intense laser pulses using a plane-confined model and found that the reaction proceeds mainly via the bending of the molecule. After that, with the fields acting on the pure HCN vibrational states through stimulated Raman adiabatic passage processes, nearly 100% population transfer was realized from HCN to HNC.18,19 On the classical simulation side, © 2015 American Chemical Society

Received: November 17, 2014 Revised: March 5, 2015 Published: March 6, 2015 2982

DOI: 10.1021/jp511440w J. Phys. Chem. A 2015, 119, 2982−2988

Article

The Journal of Physical Chemistry A

Here, J ̂ and j ̂ are the total and diatomic angular momentum operators with Jẑ and jẑ as their projections onto the BF z axis. J+̂ (J−̂ ) and j+̂ (j−̂ ) are the corresponding raising (lowering) operators, which are responsible for the Coriolis coupling. In the CS approximation,28−30 the last two terms in eq 3 are ignored. Under the dipole moment approximation, the coupling between the laser field and molecule reads

and photoionization. Second, with the external field, we present a full-dimensional quantized molecular model to simulate the laser-inducing isomerization dynamics of the title reaction. In consideration of the molecular rotation, some new phenomena are observed and revealed. Third, the feasibility of the CS approximation is discussed for the isomerization reaction of the title system. On the basis of these purposes, we solve the timedependent Schrödinger equation by propagating the rotational coupled wave packets to describe the dynamics of HCN interacting with an intense infrared laser pulse. The isomerization probabilities, rotational state distributions, and state-to-state probabilities are obtained by virtue of the orthogonal projection of the final wave function onto the bound state of the HCN.

Ĥ ′ = −μ(R , r , θ ) ·E(t )

(4)

where μ(R,r,θ) is a permanent dipole moment surface. E(t) =eE(t) is the external laser field in the SF frame (e is the unit vector of the external field). In this paper, the laser field E(t) is assumed as a Gaussian envelope

■

⎡ −4 ln 2·(t − t )2 ⎤ 0 ⎥ cos(ωt ) E(t ) = E0 exp⎢ 2 τ ⎦ ⎣

THEORY In order to conveniently describe the isomerization dynamics of HCN → HNC, the Jacobi coordinate system (R,r,θ) is employed. As depicted in Figure 1, the length of C−N is denoted

(5)

where ω is the laser frequency, E0 indicates the amplitude of the laser field, τ denotes the full width at half-maximum (fwhm) of the laser pulse, and t0 is the turn on time. When the molecular system of HCN is exposed to an external field, the molecule itself will not be a conservative system. This can lead to nonconservation of the total angular momentum. Thus, the wave function should be defined as the sum of a series of J-fixed bound-state eigenfunctions Jmax

|ψ (R , r , θ , ω)⟩ =

J

∑ ∑ ψ J ,K ,p(R , r , θ)|J , K , M , p⟩ J=0 K =λ

(6)

with λ = (1/2)[1 − (−1)J+p]; Jmax is the possible maximum value of J. The parity-adapted rotational basis functions in the above expression are as follows Figure 1. SF to BF coordinates frame used for HCN → HNC isomerization.

| J , K , M , p⟩ =

by r, the distance between H and the center mass of CN is defined by R, and the angle between r and R is θ. The z-axis of the space-fixed (SF) frame is parallel to the direction of the external field. The isomerization dynamics is confined to the bodyfixed (BF) xz plane. In BF coordinate, the molecular field-free Hamiltonian can be expressed as (ℏ = 1)

(7)

where the MR and Mr are the reduced masses for the radical in Jacobi coordinates and V(R,r,θ) is the potential energy surface (PES) of the corresponding electronic state. The reduced masses are defined as MR =

i

For a total angular momentum J, the squared orbital angular momentum operator can be expressed as follows 2

l ̂ = (J ̂ − j ̂)2 2

2

+ −

∂ |ψ ⟩ = (Ĥ 0 + Ĥ ′)|ψ ⟩ ∂t

(8)

Though the Hamiltonian of the title system is time-dependent, it can commute with time. Thus, the time evolution operator can be used to solve the above equation. In this work, the evolution time is discretized in a uniform grid {t0 ≡ 0, t1, t2, ..., tN ≡ t}, with Δtn = tn − tn−1 ≡ Δt (n = 1, 2, ..., N). The time-dependent Hamiltonian Ĥ (t) is supposed to vary smoothly in the time interval Δtn, and the time-dependent Hamiltonian Ĥ (t) is approximated with Ĥ (tn) in the time interval Δtn. On the basis of this assumption, the time evolution can be written as follows

(2)

− +

= J ̂ + j ̂ + 2Jẑ jẑ − J ̂ j ̂ − J ̂ j ̂

2(1 + Δ0, K )

Here, ψ is the bound-state wave function of the corresponding electronic state. M and K are the z components of a certainty J J in the SF and BF frames. The Wigner matrix DK,M (ω) is the function of the three Euler angles,31 namely, ω ≡ (α,β,γ), (α = 0 is adopted in our simulation because the H atom is along the z axis in the SF frame initially), which can represent the coordinate transformation between SF and BF. The parity of the bound-state wave function is given by (−1)p, where p = 1 for negative parity states and p = 0 for positive parity states. Propagation Details. When the wave packet is set up in the above basis, the time evolution can be governed by the timedependent Schrödinger equation (ℏ = 1)

(1)

mH(mC + mN) mH + mC + mN mCmN Mr = mC + mN

1

× [DKJ , M (ω) + ( −1) J + K + p D−J K , M (ω)]

2 2 Ĵ 1 ∂2 1 ∂2 l̂ − + + Ĥ 0 = − 2MR ∂R2 2M r ∂r 2 2MR R2 2M rr 2

+ V (R , r , θ )

2J + 1 8π 2

(3) 2983


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The Journal of Physical Chemistry A N

U (t , 0) ≃

In the CS approximation, the Coriolis coupling terms are ignored, which leads the common eigenvalues (M) of Jẑ and jẑ to be good quantum numbers. Thus, the angular kinetic energy operators (KEOs) are diagonal with respect to K and M, which appear only as parameters. Without loss of generality, M = 0 is taken. However, in the Coriolis-coupled (CC) treatment, the coupling between different M channels results in the appearance of nondiagonal elements in the FBR, which make the computation more expensive.37 The use of the FBR simplifies the rotational KEOs but complicates the calculation of the action of the potential energy operators. This can, however, be conveniently transformed to the FBR consisting of the normalized associated Legendre function with the following transformation matrix38,39

∏ e−iĤ (t )Δt n

(9)

n=1

The strategy is then to expand the propagator with the complex Chebyshev polynomial,32,33 and the wave function is, in the time interval Δt, given by ̂

ψ (t + Δt ) = e−iH(t )Δt ψ (t ) ≈ e−iE̅ Δt (2 − δn0) ∑ Jn (ΔE ·Δt )Φn( − iHnorm)ψ (t ) n

(10)

The spectral half-with and mean E̅ = (1/2)(Emax + Emin), ΔE = (1/2)(Emax − Emin) can be obtained from the maximum and minimum energy (Emax = Vmax + Tmax, Emin = Vmin) supported by the grid. Jn(·) is the Bessel function of the first kind, which converged to zero exponentially with n increasing. The complex Chebyshev polynomials are generated by the three-term recurrence relation ϕn + 1 = −2iHnormϕn + ϕn − 1

UjKβ =

(12)

where β denotes the index of the Gauss−Legendre quadrature points for the internal angular coordinate and ωβ is the corresponding weight. Now, we are considering the latter term Ĥ ′ϕ. In general, the dipole moment μ combined with the direction operator e is a vector operator with three Cartesian components μx, μy, and μz. Redefining it in spherical polar coordinates, the dipole moment operator μi can be written as2

(11)

with ϕn = Φn(−iHnorm)ψ(t). The recurrence starts with ϕ0 = ψ(t) and ϕ1 = −iHnormψ(t). The normalized Hamiltonian is given by Hnorm = 2[Ĥ − I(̂ Emax + Emin)/2 ]/(Emax − Emin). The reason for normalizing the Hamiltonian is the uniform character of the Chebyshev polynomials in distributing the errors in the interval [−1,1]. Because the Hamiltonian is changed with time, the normalization procedure needs to repeat in each interval. Actually, our time propagation strategy is similar to the split operator (SPO) method,34 but we do not need to separate T̂ and V̂ . For the SPO method, however, because T̂ and V̂ are not commuted, the error will be proportional to the commutator [T̂ ,V̂ ] and the time interval Δt. For example, for the typical ex̂ ̂ ̂ ̂ pression of the propagator of SPO e−iHΔt ≈ e−(i/2)TΔte−iVΔte−(i/2)TΔt, 34,35 3 the error is iΔt ((1/12)[T̂ ,[V̂ ,T̂ ]] + (1/24)[V̂ ,[V̂ ,T̂ ]])... It is obviously that, for a given system, the values of the commutator are fixed. To reduce the error, the time interval Δt should be small enough. For the long wave-driven pulse, the total Hamiltonian may change more slowly over time, and the larger time interval can effectively ensure the convergency. Thus, the SPO method will be infeasible. In this case, our proposed method can provide a convenient way to solve the time-dependent Schrödinger equation without losing accuracy and efficiency. The major computation task in our approach is the matrix− vector multiplication, Ĥ ϕ = Ĥ 0ϕ + Ĥ ′ϕ. In this work, the direct product discrete variable representation (DVR) is used for the two radial degrees of freedom (R,r), while a finite basis representation (FBR) spreads the angular degrees of freedom (θ).36

1 (μ − iμy ) 2 x

μ+1 = − μ0 = μz μ−1 =

1 (μ + iμy ) 2 x

(13)

With this definition, the dipole moment can transform more conveniently between SF and BF frames. The transformation can be expressed as 1

μmSF =

∑

μi BF Di1, m(ω)

(14)

i =−1

BF where μSF m and μi are the components of the dipole moment in the SF and BF frames, respectively. Because isomerization dynamics occurs on a single electronic state, it can be considered as a parallel transition (A′ ← A′), which means that the dipole moment is parallel to the plane of the molecule. This leads the μy in eq 13 to be zero. While considering the direction of the electronic field along the SF z axis, using eq 14, the production of the dipole moment and wave packet can be written as

J′ ⎧ ⎛1 J ⎪ J′ ⎞ ( −1)−K ⎡ ⎛ 1 J J ′ ⎞ ⎢μ ⎜ ∑ ( −1)M ⎜ ⎟ (2J ′ + 1)(2J + 1) × ⎨ ⎟|J ′, K + 1, M , p + 1⟩ ⎪ 2 ⎢⎣ x ⎝ 0 K −K ⎠ ⎝ 0 M −M ⎠ J = 0 J ′= J − 1 ⎩ K =1 ⎤ ⎛1 J J′ ⎞ ⎛1 J J′ ⎞ + ( −1) J + J ′ 1 + δ0, k − 1 μx ⎜ ⎟|J ′, K , M , p + 1⟩⎥ ⎟|J ′, K − 1, M , p + 1⟩ + 2 μz ⎜ ⎥⎦ ⎝1 −K K − 1⎠ ⎝ 0 K −K ⎠ Jmax

μzSF |ψ ⟩ =

ωβ P ̅jK (cos θβ)

J+1

∑ ∑

⎫ ⎛1 J J′ ⎞ ⎛1 J J ′ ⎞ ⎪ ψ J ′ , K , p(R , r , θ) + δλ ,0μx ⎜ ⎟|J ′, 0, M , p + 1⟩⎬ ⎟|J ′, 1, M , p + 1⟩ + μz ⎜ ⎪ ⎝1 0 −1⎠ ⎝ 0 K −K ⎠ ⎭

where (:::) denotes the 3 − j symbol. All of the computations of the bound-state wave functions and energy levels are performed by

(15)

diagonalization of the molecular field-free Hamiltonian H0 with the Lanczos algorithm.40,41 2984


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The Journal of Physical Chemistry A Isomerization Probability. The time dependence of the isomerization probability of HCN to HNC can be calculated using PI(t ) =

∑∫ J ,K

π

dθ

π /2

∫

Table 1. Calculated Low-Lying HCN Vibrational Energy Levels (cm−1), Relative to the HCN (0,0,0) J = 0 State

dR dr |ψ J , K , p(R , r , θ , t )|2

J=0

(16)

This means that when the hydrogen atom appears in the region close to nitrogen, the isomerization reaction occurs. After separating each J channel, the rotational distributions and stateto-state distributions are obtained PJ(t ) =

∑ ∫ dR dr dθ |ψ J ,K ,p(R , r , θ , t )|2 K

(ν1, ν2 , ν3) Pvib (t ) =

(17)

∑ ∫ dR dr dθ⟨ν1, ν2 , ν3|ψ J ,K ,p(R , r , θ , t )⟩ K

(18)

where ν1, ν2, and ν3 are the quantum numbers of stretching, bending, and antisymmetric stretching modes, respectively.

HNC HNC HNC HNC HNC HNC J=1 J=5 J = 10 J = 15 J = 20

■

RESULTS AND DISCUSSIONS In this work, the VQZANO+ PES and dipole surface42 are used for isomerization dynamical calculation. Also, we do not consider the molecular dissociation process because we are interested in the isomerization dynamics. Thus, a small number of basis grids should be enough to cover the double well part of the HCN PES. Equidistant grids with NR = 34 and Nr = 64 points are used for the two radial coordinates in the ranges of R ∈ [1.5,4.0] au and r ∈ [1.6,3.5] au, respectively. The fast Fourier transform is used to calculate the action of the radical KEOs on the propagating wave packet.43 For the Jacobi angle, 46 G-Legendre quadrature points are taken between θ = 0 and π. Both potential and the angular KEOs are truncated at 0.6 hartree to minimize the spectral range. In order to ensure the accuracy of the construction of the initial wave packet, we calculate the corresponding vibrational energy levels using the Lanczos algorithm. As a comparison, the VQZANO+ data42 are shown in Table 1. As the table shows, all of the rotational−vibrational frequencies are in excellent agreement with the VQZANO+ results, which further confirms the accuracy of our calculation. With the purpose of showing the effect of the variation of the initial state, in our calculations, the initial wave packets are set on the J = 0 pure vibrational states (0,0,0) and (0,2,0). The driven Gaussian pulse starts at 0 ps with the fwhm of 1.0 ps and centers at 1.4 ps. The wavelength of the laser pulse is 1386.45 cm−1, which is equal to the energy level difference between the (0,2,0) and (0,4,0) states. The peak intensity is 5.0 × 1012W/cm2, which is lower than the dissociation limit (∼1013W/cm2). Because it is located at the infrared wavelength region, the time step Δt = 0.4 fs can ensure the convergence in the time evolution. Figure 2 displays the time dependence of isomerization probabilities of HCN → HNC for different initial states. The effect of the variation of the initial state is obvious. Because the state (0,2,0) is closer to the top of the barrier, the isomerization probability is much higher than that of the ground state (0,0,0). With the same driven Gaussian pulse, the former appears to be still a step ahead; this means that if the dynamics starts from a high vibrational excited state, the isomerization reaction occurs easily. The CS approximation results are also shown in the figure. (It should be noted that we do not use a damping function in our numerical calculations for the wave packet propagating; the wave packet is rebounded at the grid edges, which causes the

ν1

ν2

ν3

ref 42

this work

0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 2 0 4 0 2 6 0 2 4 0 8 2 4 0 2 4 0 6 2 0 0 0 0 0

0 0 1 0 0 1 0 2 0 1 1 0 2 0 0 0 0 1 0 1 0 0 0 0 0

3481.46 1414.92 2100.58 2801.46 3307.75 3510.99 4176.24 4181.45 4686.29 4891.76 5394.43 5537.76 5586.5 6033.72 5185.64 941.917 1903.10 2024.95 2834.81 2955.05 2.959 44.39 162.72 354.94 620.93

3481.043 1414.81 2100.08 2801.25 3308.18 3510.36 4175.94 4180.61 4686.56 4891.03 5396.69 5537.38 5585.45 6034.06 5185.67 941.85 1902.98 2024.37 2834.63 2954.41 2.958 44.37 162.64 354.78 620.64

Figure 2. Time dependence isomerization probabilities for HCN → HNC. The black and blue solid lines are isomerization probabilities initially from (0,0,0) and (0,2,0) states, while the dotted lines are their corresponding CS approximation results, respectively.

isomerization probabilities to be periodically oscillatory in the long time region.) As shown in the figure, the results of isomerization probabilities under the CS approximation from different initial states are much bigger than the results of full CC calculations, which indicate that the Coriolis coupling effect has significant impact on the title system. On the other hand, the CS approximation should be cautiously used in the quantum wave packet calculation for this isomerization dynamics because the molecule is always located at the long-lived complex, which will cause the invalidity of the CS approximation. More alarming, the CS approximation may seriously influences the molecular rotational excitation. Figure 3 displays the time-dependent distributions for different rotational states of HCN with the same 2985


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be observed when the angular momentum quantum number changes by 1 (ΔJ = ±1). This selection arises from a first-order perturbation theory approximation of the time-dependent Schrödinger equation. According to this treatment, we display the energy scheme of such a two-photon excitation conversion process in Figure 4B. In the first step, the molecule transits from the ground state [(0,0,0), J] to the first excited state [(0,2,0), J + 1] by absorbing a photon. Ignoring the influence of spontaneous emission, it will absorb another photon and jump to the higher excited state [(0,4,0), J] in the second step. (Note that there is a certain probability on the transition that the molecular can also jump to the [(0,4,0), J + 2] state.) As a result, the molecular vibrational excitation will absorb two photons for a certain rotational state. Figure 5 displays the final rotational state distributions initially from the (0,0,0) and (0,2,0) states. As influenced by the intense

Figure 3. Time-dependent distributions of the rotational state of HCN (HNC). The initial state is the pure (J = 0) vibrational state (0,0,0).

starting initial state of [J = 0, (0,0,0)]. Both CS and CC results show that the HCN (HNC) molecule is rotationally excited by the intense laser pulse, but the excitation of CC tends to be more obvious. Regardless of the J = 0 state, finally, the J = 1 state becomes dominant in the CS simulation, but J = 2 dominates in the CC results. A plausible reason is that as the Coriolis terms are cut to zero, the motion of each atom as well as the external field is restricted to a plane, which leads to partial energy transformation from the contribution of the rotational excitation to that enhancing the isomerization rate. Another obvious feature shown in the figure is that the curves of the rotational state distribution show highly oscillatory structures. This reflects the period of the driving pulse and is common in the intense field dynamical process. The final state-to-state distributions for several J numbers are shown in Figure 4A. We can see that the molecule is highly vibrationally excited after interacting with the infrared external field. For a certain rotational state, the energy difference between the two involved excited states is equal to the sum of the energy of the two photons. It shows intriguing characteristics of twophoton absorption (TPA).44 However, such a phenomenon is not the traditional nonlinear optical process. It is still a linear resonant absorption process and is caused by molecular rotational excitation. Typically, rotational transitions can only

Figure 5. Final rotational state distributions initially from pure vibrational states (0,0,0) and (0,2,0), respectively.

laser pulse, finally, the HCN (HNC) molecule is highly rotationally excited. Thus, the constraint of the system to a fixed J is yet to be agree upon. For (0,0,0) results, the rotational states are distributed mainly in the J = 0−5 region, but for the (0,2,0) state, only 50% of rotational states occupy this region. For this reason, more J numbers are needed to ensure the convergency. While using the population of the Jmax partial wave to

Figure 4. (A) The histogram above is the final state-to-state vibrational distributions for few numbers of rotational states. For a certain J, the molecular vibrational excitation can be considered as the absorption of two photons of identical frequency. (B) The schematic diagram shows the process of the TPA; due to the selection rule, the molecule first transits from the ground state [(0,0,0), J] to the upper state [(0,2,0), J + 1] through absorbing a photon, and then, it has the probabilities to jump to a higher state [(0,4,0), J] by absorbing another photon. 2986


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estimate the convergence errors, in order to reach the errors of 0.5%, for the initial states (0,0,0) and (0,2,0), we need to expand the Jmax to 11 and 21, respectively.

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CONLUSION We have introduced the time-dependent quantum wave packet method to investigate a laser interacting with a triatomic molecular system. By expanding the CC wave packet with a number of series of J-fixed bases, we can describe the laser-driven chemical dynamics in the triatomic molecule system in fulldimensional scale. In the case of HCN, our numerical results show that the field-induced molecule rotational excitation plays an important role in the isomerization dynamical process and brings on the phenomenon of the TPA. The CS results are also compared with the CC results in this work; it indicates that the CS approximation is not suitable for the title system because the Coriolis terms can sightly decrease the isomerization rate but highly enhance molecular rotational excitation. The effect of the variation of the initial state is also illustrated in this paper. When the molecular system is initially in high vibrational states, the isomerization can more easily occur. However, if the molecule is highly rotationally excited, more J numbers are needed to ensure the convergency numerically. Because our proposed theoretical frame provides an overall description of the laser interacting with the triatomic molecule, ideally, it can be suitable for investigating control of chemical dynamics such as photodissociation or reactive scattering. This is the subject of a further study.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation of China (Grant No. 11374191) and partially supported by BNLMS fund project (20140101).

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REFERENCES

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