† Corresponding author. E-mail:

‡ Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11474141and 11274149), the Program for Liaoning Excellent Talents in University, China (Grant No. LJQ2015040), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China (Grant No. 2014-1685), and the Special Fund Based Research New Technology of Methanol Conversion and Coal Instead of Oil and the China Postdoctoral Science Foundation (Grant No. 2014M550158).

Quasi-classical trajectory calculations are performed to study the stereodynamics of the _{2}(1^{2}A′) potential energy surface reported by Li *et al*. [Li Y Q and Varandas A J C 2010 *J. Phys. Chem. A* *k**k**k**k** j*′ of the product H

_{2}is not only aligned, but also oriented along the

*y*axis. The alignment parameter, the disposal of total angular momentum and the reaction mechanism are all analyzed carefully to explain the polarization behavior of the product rotational angular moment.

The reaction H + NH → H_{2} + N has played an important role in the chemistry of nitrogen containing fuels, the catalyst realm and the renewable energy production.^{[1]} Owing to their significant effects, the ground and first excited electronic state of NH_{2} have been studied quite extensively by various experimental^{[2–5]} and theoretical techniques.^{[6–33]} In the previous theoretical studies, most of them mainly forced on the dynamic behaviors of the ground and first excited state NH_{2}. In this case, such studies have often been concentrated on the scalar properties of the H + NH → H_{2} + N reaction. To comprehensively understand the NH_{2}dynamics, it is insufficient to investigate only the scalar properties. Vector correlation can provide the detailed information about the whole atom motion when reaction occurs, which can help experimentalists rationalize the experimental results. So it is important to study not only their scalar properties, but also their vector properties. The most familiar vector correlation between the reagent and product relative velocity (*k**k**σ*/d*ω _{t}* (DCS), and the most important vector correlation is the correlation among three vectors

*k*

*k*

*′ (the product rotational angular momentum). The correlations among the three vectors in the center-of-mass (CM) frame can be characterized by certain interesting double and triple vector correlations.*j

In order to fully comprehend the excited-state property of NH_{2} reaction Li and Varandas^{[6]} reported an adiabatic PES of NH_{2} first excited state 1^{2}A′ based on the double many-body expansion (DMBE) theory. In the present work, the quasi-classical trajectory (QCT) method is used to mainly explore the vector correlations (such as *P*(*θ _{r}*),

*P*(

*ϕ*), and

_{r}*P*(

*θ*

_{r},

*ϕ*

_{r})) based on the first excited state NH

_{2}(1

^{2}A′) DMBE-SEC PES.

^{[6]}We select three different collision energies (10, 20, and 30 kcal/mol) to investigate the influences on stereodynamics with

*v*= 0 and

*j*= 0. In turn, four different initial rotational levels (

*j*= 0, 3, 6, 9) are chosen to further study the different effects originating from rotational excitation under the collision energy 30 kcal/mol with initial vibrational level

*v*= 0. The rest of this paper is organized as follows. In Section 2 we briefly review the theoretical methodologies adopted in the current study. In Section 3 we present and discuss the calculated results. Finally, in Section 4 we draw some conclusions from the present studies.

The accurate first excited state 1^{2}A′ PES constructed by Li and Varandas^{[6]} is adopted in the present calculations. The calculation method of QCT is the same as that in Refs. [7]–[12]. In this work, three different collision energies (10, 20, and 30 kcal/mol) are selected to investigate the influences on stereodynamics with *v* = 0 and *j* = 0. In turn, four different initial rotational levels (*j* = 0, 3, 6, 9) are chosen to further study the different effects originating from rotational excitation under the collision energy 30 kcal/mol with *v* = 0. The classical Hamilton equations are numerically integrated in three dimensions. In our calculations, the integration step is set to be 0.1 fs to ensure a conservation of the results. In order to ensure no interaction between attacking atom H and the center-of-mass (CM) reference frame of the diatomic molecule NH, the distance between them is set to be 15 Å. A total of 3 × 10^{4} trajectories are sampled, and the appropriate value of the maximum impact parameter is selected after preliminary running batches of 3×10^{3} trajectories at each collision energy. The orientation of the diatomic molecule and the phase of the diatomic vibrational motion are randomly sampled by a Monte Carlo procedure. Moreover, the impact parameters are optimized before running the trajectories. That is to say, the initial azimuthal angle and polar angle of the reagent molecule internuclear axis are randomly sampled based on Monte Carlo method, and the range of the angles is from 0° to 180° and from 0° to 360°, respectively.

The CM reference frame is used in the present investigation. The reagent initial relative velocity vector *k**z* axis in the CM frame. Scattering plane *x*–*z* contains *k**k**θ*_{t} is the angle between *k**k** j*′ are

*ϕ*

_{r}and

*θ*

_{r}, respectively.

The distribution function of *P*(*θ*_{r}) describing the *k** j*′ correlation can be expanded in a series of Legendre polynomials as follows:

^{[34–43]}

*k*is odd,

*k*is even, it is called alignment parameter. For

*k*= 2, the coefficient of the expansion is

The dihedral angle distribution function of *P*(*ϕ*_{r}) describing the *k**k** j*′ correlation can be expanded in a Fourier series

^{[34–43]}

The joint probability density function of angles *θ*_{r} and *ϕ*_{r}, which determine the direction of * j*′, can be expressed as

*k*] = (2

*k*+ 1), and the polarization parameter is expressed as

The full three-dimensional angular distribution associated with *k**k**j*′ can be represented by a set of generalized polarization dependent differential cross-sections (PDDCSs) in the CM frame that is described in Refs. [34]–[43]. The fully correlated CM angular distribution is written as

*–*k

*k*

Figure *k**k*_{2} ground state PES.^{[15]} With the increase of the collision energy, both forward scattering and backward scattering become higher, in which the forward scattering has a bigger magnitude of variation. According to the capture model,^{[44]} it may be easy to understand that the forward scattering increases with the increase of collision energy. The influence of collision energy on the *k**k**k**k**v* = 0, *j* = 0, 3, 6, 9) for NH molecular at collision energy of 30 kcal/mol are shown in the plot of Fig.

To obtain a clearer understanding of the reaction mechanism, the variations of internuclear distances of H_{a}–H_{b}, H_{b}–N, and H_{a}–N are presented each as a function of propagation time based on these three collision energies in Fig. _{a} undergoes several collisions with the target molecule H_{b}–N and the complexes have relatively long lifetime before breaking up. The long-term survival of the metastable states could allow the system to perform extensive rotations before breaking up, thereby making the products come out in more random directions. So the indirect reactive trajectories, namely “trapped” trajectories, can be observed. Therefore it is no surprise that the forward scattering and backward scattering are observed as can be seen in Fig. _{a}–H_{b} distance reaches its lowest value before the H_{a}–N does, that is to say, the attacking H_{a} atom “hits” firstly on the H_{b} atom of the target molecule and later migrates to form the H_{a}N product. As a consequence, the reaction takes place by a migration mechanism.^{[45]} Figure _{a}–H_{b}, H_{b}–N, and H_{a}–N each as a function of propagation time based on variation of initial reagent rotational excitations at 30 kcal/mol. It should be noted that the abstraction mechanism could be founded except for *j* = 0. That is to say, the initial reagent rotational excitation transforms the reaction mechanism from insertion to abstraction. And indirect reactive trajectories (i.e. many intermediate complexes appear) of interest are observed only when *j* = 6.

*k*

*j*

*k*

*k*

*j*

In this section, the energy-dependent behaviors of the angular distributions of *P*(*θ*_{r}) and *P*(*ϕ*_{r}) are discussed in detail, with the reactant NH selected at a specific initial state *v* = 0 and *j* = 0. The calculated distributions of *P*(*θ*_{r}) for the product H_{2} molecule from *P*(*θ*_{r}), describing the *k** j*′ correlation, has a maximum at a

*θ*

_{r}angle value of 90° and symmetric with respect to 90°. It can be concluded that

*′ is distributed with cylindrical symmetry in the product scattering frame and the direction of*j

*′ is preferentially perpendicular to the k direction. As we can see, the distribution of*j

*P*(

*θ*

_{r}) becomes weaker when the collision energy increases from 10 kcal/mol to 20 kcal/mol, while it becomes stronger when the collision energy increases from 20 kcal/mol to 30 kcal/mol. The product rotational alignment parameter 〈

*P*

_{2}(

*j*

*)〉 is used to investigate the trend of*k

*P*(

*θ*

_{r}) distribution in depth, which are –0.3486, –0.3413, and –0.3683 at these three collision energies respectively (the closer to 0.5 the 〈

*P*

_{2}(

*j*

*k*

*L*

*j*

*j*

*L*

*L*

*j*

*v*= 0,

*j*= 0))

*L*

*j*

*L*

*L*=

*b*

_{max}(2

*μ E*

_{col})

^{1/2}(although

*b*

_{max}decreases with collision energy, the degree of its reduction is incomparable to that of the collision energy increment). As a result, larger parts of

*L*

*′, which finally induces stronger alignment of*j

*′.*j

The dihedral angle distribution of *P*(*ϕ*_{r}) describing the *k**k** j*′ correlation is shown in Fig.

*k*

*k*

*ϕ*

_{r}= 180°) and could be used to reflect the strong polarization of product rotational angular momentum directly. In fact, the peaks in the distribution of

*P*(

*ϕ*

_{r}) at

*ϕ*

_{r}angles close to 90° imply a preference for right-handed product rotation while the peaks at

*ϕ*

_{r}angles close to 270° indicate a preference for left-handed product rotation. The peaks of

*P*(

*ϕ*

_{r}) appear at

*ϕ*

_{r}= 90° and

*ϕ*

_{r}= 270° indicate that the rotational angular momentum vectors of the products are mainly aligned along the

*y*axis of the CM frame. It can be seen that the peaks at

*ϕ*

_{r}= 90° are stronger than those at

*ϕ*

_{r}= 270° at these three collision energies, which means that the product rotational angular momentum vector

*′ is oriented along the positive direction of the*j

*y*axis. And the difference in peak between

*ϕ*

_{r}= 90° and

*ϕ*

_{r}= 270° increases with the increase of collision energy. It should be noted that the peaks of

*P*(

*ϕ*

_{r}) distributions at higher collision energies are broader than those at lower collision energies which indicates that the rotation of the product molecule changes from the “in-plane” reaction mechanism into the “outofplane” mechanism. In turn, as depicted in Fig.

*P*(

*ϕ*

_{r}) distributions each display a distinct ‘V’-shaped splitting separately around

*ϕ*

_{r}= 90° and 270° at 10 kcal/mol, which predicts a dynamic enhancement of the degree of orientation along the

*y*axis. A similar splitting was also found in the study of stereodynamics for reaction O + H

_{2}.

^{[46]}The angular momentum polarization in the form of polar plot of the distribution of

*P*(

*θ*

_{r},

*φ*

_{r}) is also depicted in Fig.

*P*(

*ω*

_{t},

*ω*

_{r}). The distributions of the

*P*(

*θ*

_{r},

*ϕ*

_{r}) peaks at (

*π*/2, 3

*π*/2) are in good accordance with the distributions of

*P*(

*θ*

_{r}) and

*P*(

*ϕ*

_{r}), which indicates that the HH products are preferentially polarized perpendicularly to the scattering plane.

*k*

*j*

*k*

*k*

*j*

The effects of the initial rotational excitation of NH on the alignment of product angular momentum * j*′ in the stereodynamics for the reaction H + NH (

*v*= 0,

*j*= 0, 3, 6, 9) are also investigated. The calculated results of

*P*(

*θ*

_{r}) distribution for this reaction at 30 kcal/mol collision energy are shown in Fig.

*P*(

*θ*

_{r}) become broader and lower when the reagent rotations are in the cases of

*j*= 3, 6, 9, indicating that the product rotational alignment effect becomes weaker when the reagent is in the excited rotation state. This is also confirmed by the values of the product rotational alignment parameter 〈

*P*

_{2}(

*j*

*k*

*j*= 0, 3, 6, and 9, respectively. It should be emphasized that the few peaks of

*P*(

*θ*

_{r}) distribution close to

*θ*

_{r}= 0° and 180° appear gradually with the enhancement of the initial rotational excitation of target molecular. It implies that as the initial rotational excitation of target molecular increases, the polarization parallel to the direction of the relative velocity of reactants appears, nothing but the degree of polarization is relatively weak.

In order to examine the influence of initial rotational state of NH on the orientation of product angular momentum * j*′, we show the distributions of

*P*(

*ϕ*

_{r}) in Fig.

*k*

*k*

*′ correlation shows that*j

*P*(

*ϕ*

_{r}) tends to be asymmetric about

*ϕ*

_{r}=

*π*, reflecting the strong polarization of angular momentum directly. The product rotational angular momentum is not only aligned along the

*y*axis (i.e. the distributions have peaks at

*ϕ*

_{r}=

*π*/2 and 3

*π*/2 respectively), but is also oriented along the positive direction of the

*y*axis (i.e., the peaks at

*ϕ*

_{r}=

*π*/2 are larger than those at

*ϕ*

_{r}= 3

*π*/2). The

*P*(

*ϕ*

_{r}) distribution peaking at

*ϕ*

_{r}angle close to

*π*/2 implies a preference for right-handed HH rotation in planes that are parallel to the scattering plane. It is interesting to find that the reaction is symmetric around the relative velocity vector, while the distribution of

*P*(

*ϕ*

_{r}) is asymmetric. The various angular distributions of the product reflect the vector correlation containing rich information about the angular momentum polarization. In order to validate the information about the product molecular axis polarization, the

*P*(

*θ*

_{r},

*φ*

_{r}) distributions averaged over all scattering angles for the reactions

*P*(

*θ*

_{r},

*ϕ*

_{r}) show peaks at

*θ*

_{r}= 90°,

*ϕ*

_{r}= 90° and

*θ*

_{r}= 90°,

*ϕ*

_{r}= 270°, which are found to be in good accordance with the scenarios for the distributions of

*P*(

*θ*

_{r}) and

*P*(

*ϕ*

_{r}) of the HH products. The distributions of

*P*(

*θ*

_{r},

*ϕ*

_{r}) indicate that the products of the reactions are mainly rotated in planes that are parallel to the scattering plane and the products are preferentially polarized perpendicularly to the scattering plane.

In the present work, the energy and rotation-dependent stereodynamics of the reaction _{2} using QCT method. A total of 3 × 10^{4} trajectories are run under each initial condition to investigate the effects of collision energy and rotational state of NH radical on the product rotational polarization. With the increase of the collision energy, both forward and backward scatterings become stronger, in which the forward scattering has a bigger magnitude of variation. The influence of collision energy on the *k**k*

The results of vector correlation show that the product rotational angular momentum * j*′ is not only aligned, but also oriented along the direction perpendicular to the scattering plane. It is found that the three angle distributions are sensitive to initially collision energy and rotational excitation.

**Reference**

1 | |

2 | J. Chem. Phys. 110 8857 |

3 | J. Chem. Phys. 94 4301 |

4 | J. Chem. Phys. 104 9640 |

5 | J. Chem. Phys. 106 4985 |

6 | J. Phys. Chem. A 114 9644 |

7 | J. Chem. Phys. 136 194705 |

8 | Chem. Phys. Lett. 172 180 |

9 | J. Phys. Chem. A 109 2050 |

10 | J. Phys. Chem. A 110 1666 |

11 | Int. Rev. Phys. Chem. 25 201 |

12 | Phys. Chem. Chem. Phys. 10 2431 |

13 | J. Chem. Phys. 122 244322 |

14 | J. Phys. Chem. A 101 5696 |

15 | Can. J. Chem. 57 3182 |

16 | J. Chem. Soc. Faraday Trans. 92 1311 |

17 | |

18 | J. Phys. Chem. A 104 2301 |

19 | Mol. Phys. 70 443 |

20 | J. Chem. Phys. 139 154305 |

21 | Chem. Phys. Lett. 516 17 |

22 | J. Chem. Soc. Faraday Trans. 89 995 |

23 | Mol. Phys. 43 987 |

24 | J. Comput. Chem. 34 1686 |

25 | J. Chem. Phys. 110 9091 |

26 | |

27 | J. Chem. Phys. 104 5558 |

28 | J. Chem. Phys. 128 224316 |

29 | Theor. Chem. Acc. 116 404 |

30 | J. Chem. Phys. 67 5173 |

31 | Phys. Chem. Chem. Phys. 11 10438 |

32 | Phys. Chem. Chem. Phys. 12 7942 |

33 | J. Chem. Phys. 126 034304 |

34 | |

35 | J. Chem. Phys. 109 5446 |

36 | Chem. Phys. 283 463 |

37 | J. Chem. Phys. 118 4463 |

38 | Int. J. Quanttum. Chem. 106 1815 |

39 | J. Chem. Phys. 105 8699 |

40 | |

41 | Chin. Phys. Lett. 27 123103 |

42 | Comput. Theor. Chem. 963 306 |

43 | Chin. Phys. B 22 083402 |

44 | Phys. Chem. Chem. Phys. 1 1141 |

45 | J. Chem. Phys. 136 144309 |

46 | Phys. Chem. Chem. Phys. 2 571 |