**3.4 Rotational alignment parameter (2) A0**

262 Measurements in Quantum Mechanics

Figure 8 shows the rotational excitation effect on the PDDCSs distributions for the N(2*D*) + H2(v=0, j=0-5) → NH + H reactions at the 5.1 kcal/mol collision energy. As depicted in

results means that the rotational excitation leads to the product angular distribution changing from the backward scattering to the both backward and forward scattering. The PDDCS20 distributions in Fig. 8b show negative values in both of the backward and forward scatterings, but they are close to zero for the sideway scatterings. These results suggest that the *j'* polarizes preferentially along the direction perpendicular to *k,* which is consistent with the product P( ) *θr* distributions in Fig. 3. The PDDCS22+ values are negative for all scattering angles, meaning the alignments of the NH products prefer to be along the *y* axis. As can be seen from Fig. 8c, the rotational excitation diminishes the PDDCS22+ peak, implying that the rotational excitation weakened the product alignment. The PDDCS21- distribution has two largest peaks for the N(2*D*) + H2(v=0, j=0) → NH + H

PDDCS21- distribution for other rotational excited reactions. These characteristics illuminate that the rotational excitation reduce the anisotropy of the product angular

Fig. 8. (a) PDDCS00, (b) PDDCS20, (c) PDDCS22+ and (d) PDDCS21- distributions as a function of scattering angle t *θ* for the N(2*D*) + H2(v=0, j=0, 1, 2, 3, 4, 5) → NH + H reaction at the

= 15° and 140°. However, no distinct large peaks were found in the

= 180° for the

= 0°. These

Fig. 8a, with the increasing rotational excitation, the peak at about *t*

reaction at *t*

distribution.

collision energy of 5.1 kcal/mol.

PDDCS00 distribution becomes weaker, but stronger for the peak at about *t*

A simple way to express the degree of product rotational polarization of *j'* can be through the CM frame alignment parameter A ) <sup>2</sup> 2 ( 3cos 1 <sup>2</sup>*<sup>P</sup> <sup>r</sup>* **j K** (2) <sup>0</sup> , which is also a usual and common parameter measured in vector correlation experiments. [1] In this expression, the brackets denote an average over the distributions of *j'* with respect to *K*. The product rotational alignment parameter (2) A0 has also been calculated at the present work, which is shown in Table 1. It can be seen from the (2) A0 expression that, for *j'* parallel or antiparallel to *K*, ( ) <sup>2</sup> *<sup>P</sup>* **<sup>j</sup> <sup>K</sup>** =1 and then the alignment parameter takes the value (2) A0 = 2, while for *j'* perpendicular to *K*, (2) A0 = -1. The values of the alignment parameter discussed above are limiting cases that represent the maximum possible alignment. In general, these parameters take values of small magnitude, which indicates a distribution that tends toward one of the above limits. If all alignment parameters are zero, the *j'* distribution is isotropic about *K*. All of the (2) A0 values displayed in Table 1 are negative, which means that most of the product rotational angular momentum *j'* tend to be perpendicular to *K*. For the N(2*D*) + H2(v=0, j=0) <sup>→</sup> NH + H reactions at collision energies of 2.0, 3.8, 5.1, 7.0, 9.0, and 11.0 kcal/mol, the (2) A0 value becomes larger with the collision energy increasing. For the N(2*D*) + H2(v=0, j=0-5) → NH + H reactions at the collision energy of 5.1 kcal/mol, the (2) A0 value also becomes larger with the rotational excitation increasing. These trends imply that both of the increasing collision energy and reagent rotational excitation depress the product rotational alignment. This is in good consistent with the the *P*( ) *r* distributions (Fig. 2 and 3).


Table 1. Values of the impact parameters *b*max and product rotational alignment parameters (2) A0 calculated with 100, 000 trajectories for the N(2*<sup>D</sup>*) + H2(v=0, j=0-5) → NH + H reactions under collision energies of 2.0, 3.8, 5.1, 7.0, 9.0, and 11.0 kcal/mol.

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#### **4. Conclusion**

In this chapter, we presented a detailed vector correlation study on the N(2*D*) + H2(v=0, j=0-5) → NH + H reaction at collision energies of 2.0, 3.8, 5.1, 7.0, 9.0, and 11.0 kcal/mol. Three angular distributions *P*( ) *r* , *P*( ) *<sup>r</sup>* , *P*(,) *<sup>r</sup> <sup>r</sup>* , and four PDDCSs were computed by the QCT method with a six-order symplectic integration. Under an accurate 12A state PES, batches of 100, 000 trajectories were running for the investigated reactions. It was found that the *P*( ) *r* distribution has a large peak at about *r* = 90°, meaning that the product angular momentum *j'* is aligned perpendicular to *k*. With the collision energy increasing, for the N(2*D*) + H2(v=0, j=0) → NH + H reactions, the product rotational alignment becomes weaker due to the reduction of the *P*( ) *r* peak at about *r* = 90°. Similarly, the rotational excitation also demonstrated a reducing behaviour for the product rotational alignment in the reactions of N(2*D*) + H2(v=0, j=0-5) → NH + H at 5.1 kcal/mol. However, the increasing collision energy enhanced the peak at about *r* = 270°, demonstrating that the product rotational orientation was enhanced by the increasing collision energy. For the N(2*D*) + H2(v=0, j=0-5) → NH + H reactions at 5.1 kcal/mol, the rotational excitation reduced the *P*( ) *r* peak at about *r* = 270°, and therefore depressed the product rotational orientation. The ( , ) *<sup>P</sup> <sup>r</sup> <sup>r</sup>* distributions was in consistent with the *P*( ) *r* and *P*( ) *r* distributions. These *P*( ) *r* , *P*( ) *<sup>r</sup>* and *P*(,) *<sup>r</sup> <sup>r</sup>* distributions had been interpreted by an "Impulsive model". At the low collision energy of 2.0 kcal/mol, the PDDCS00 shows a large peak at about *θt* = 180° and a very tiny peak at about *θt* = 0° for the N(2*D*) + H2(v=0, j=0) → NH + H reaction, which indicates that the product angular distribution is the backward scattering. As the collision energy increasing, the product angular distribution changes from the backward scattering to the both backward and forward scatterings. Similar behavior was found for the rotational excitation which also leads to the product angular distribution changing from the backward scattering to the both backward and forward scatterings. The PDDCS20 shows an opposite distribution to the PDDCS00. The PDDCS22+ values in the investigated reactions are negative except for the N(2*D*) + H2(v=0, j=0) → NH + H reaction at 2.0 kcal/mol, meaning that most of the product angular momentum *j'* prefer to align along the *y* axis. This alignment was reduced by both of the increasing collision energy and rotational excitation. For the PDDCS21- distribution, the collision energy increasing and reagent rotational excitation decreased the largest peaks, which indicate that the product rotational polarization is anisotropic. This anisotropic distribution was weakened by the rotational excitation and increasing collision energy. The calculated (2) A0 values reflect that the product rotational alignment is perpendicular to *k*, which is consistent with the aforementioned *P*( ) *r* distributions.

### **5. Acknowledgment**

This work is supported by the National Natural Science Foundation of China (No. 21003062) and the Young Funding of Jining University (No. 2009QNKJ02).

#### **6. References**

Alexander, A. J. et al. (2000). Product rotational angular momentum polarization in the reaction O(1D2)*+* H2 *→* OH*+* H. *Physical Chemistry Chemical Physics,* Vol.2, No.4, (February 2000), pp. 571-580, ISSN 1463-9076

264 Measurements in Quantum Mechanics

In this chapter, we presented a detailed vector correlation study on the N(2*D*) + H2(v=0, j=0-5) → NH + H reaction at collision energies of 2.0, 3.8, 5.1, 7.0, 9.0, and 11.0 kcal/mol. Three

method with a six-order symplectic integration. Under an accurate 12A state PES, batches of 100, 000 trajectories were running for the investigated reactions. It was found that the *P*( )

momentum *j'* is aligned perpendicular to *k*. With the collision energy increasing, for the N(2*D*) + H2(v=0, j=0) → NH + H reactions, the product rotational alignment becomes weaker

also demonstrated a reducing behaviour for the product rotational alignment in the reactions of N(2*D*) + H2(v=0, j=0-5) → NH + H at 5.1 kcal/mol. However, the increasing

rotational orientation was enhanced by the increasing collision energy. For the N(2*D*) + H2(v=0, j=0-5) → NH + H reactions at 5.1 kcal/mol, the rotational excitation reduced the

the low collision energy of 2.0 kcal/mol, the PDDCS00 shows a large peak at about *θt* = 180° and a very tiny peak at about *θt* = 0° for the N(2*D*) + H2(v=0, j=0) → NH + H reaction, which indicates that the product angular distribution is the backward scattering. As the collision energy increasing, the product angular distribution changes from the backward scattering to the both backward and forward scatterings. Similar behavior was found for the rotational excitation which also leads to the product angular distribution changing from the backward scattering to the both backward and forward scatterings. The PDDCS20 shows an opposite distribution to the PDDCS00. The PDDCS22+ values in the investigated reactions are negative except for the N(2*D*) + H2(v=0, j=0) → NH + H reaction at 2.0 kcal/mol, meaning that most of the product angular momentum *j'* prefer to align along the *y* axis. This alignment was reduced by both of the increasing collision energy and rotational excitation. For the PDDCS21- distribution, the collision energy increasing and reagent rotational excitation decreased the largest peaks, which indicate that the product rotational polarization is anisotropic. This anisotropic distribution was weakened by the rotational excitation and increasing collision energy. The calculated (2) A0 values reflect that the product rotational alignment is perpendicular to *k*, which is consistent with the aforementioned *P*( )

This work is supported by the National Natural Science Foundation of China (No. 21003062)

Alexander, A. J. et al. (2000). Product rotational angular momentum polarization in the

reaction O(1D2)*+* H2 *→* OH*+* H. *Physical Chemistry Chemical Physics,* Vol.2, No.4,

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*r* = 270°, and therefore depressed the product rotational orientation.

distributions had been interpreted by an "Impulsive model". At

*r* and *P*( ) 

, and four PDDCSs were computed by the QCT

*r* = 90°, meaning that the product angular

*r* = 90°. Similarly, the rotational excitation

*r* = 270°, demonstrating that the product

*r* distributions. These

*r*

*r*

**4. Conclusion** 

*P*( ) 

*P*( ) 

The ( , ) *<sup>P</sup> <sup>r</sup> <sup>r</sup>* 

distributions.

**6. References** 

**5. Acknowledgment** 

*r* , *P*( ) 

angular distributions *P*( )

due to the reduction of the *P*( )

*r* peak at about

distribution has a large peak at about

collision energy enhanced the peak at about

*<sup>r</sup>* and *P*(,) *<sup>r</sup> <sup>r</sup>* 

*r* , *P*( ) 

*<sup>r</sup>* , *P*(,) *<sup>r</sup> <sup>r</sup>* 

*r* peak at about

distributions was in consistent with the *P*( )


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**11** 

*China* 

**Schrödinger Transform of Image:** 

*1School of Science, Hubei Province Key Laboratory of Intelligent Robot,* 

*2Department of Computer Science and Application, ZhengZhou Institute of* 

Liantang Lou1, Hua Zeng1\*, Jipeng Xiong1, Lingling Li2 and Wenliang Gao1

Image segmentation is the process of separating or grouping an image into different parts . These parts normally correspond to something that human beings can easily separate and view as individual objects. Computers have no means of intelligently for recognizing objects, and a large number of different methods have been developed in order to segment images, ranging from the simple thresholding method to advanced graph-cut methods. The segmentation process is based on various features found in the image. Those features might be histograms information, information about the pixels that indicate edges or boundaries or

Approaches of Image processing and analysis based on partial differential equation, such as deformable models or snakes (Terzopoulos et al., 1987; Kass, et al., 1987), balloon models (Cohen, L. D., 1991; Cohen, L. D. & Cohen, I., 1993), geometric models (Caselles et al., 1993), discrete dynamic contour models (Lobergt & Viegever, 1995), geodetic active contours (Caselles et al., 1995) and topology adaptive deformable model (McInerney & Terzopoulos, 1999), whose physical background is principle of minimum action or force equilibrium in classical mechanics, are being extensively applied to image segmentation, image smooth, image inpainting, extraction of boundary and so on. Xu and Prince analyzed the reason why snake methods have poor convergence to boundaries with large curvatures and replaced the gradient field with the gradient vector field (GVF), which has a larger capture region and slowly changes away from the boundaries (Xu & Prince, 1998). Consequently, the dependence on initial positions is decreased but the field can attract the moving contour to the right position. Parametric deformable models have high computational efficiency and can easily incorporate a priori knowledge. However, these models cannot naturally handle topological changes and are sensitive to initial conditions. Geometric deformable models are based on the level set method (Osher & Sethian, 1988), which was initially proposed to handle topological changes during the curve evolution. Geometric deformable models have the advantage of naturally handling the topological

**1. Introduction** 

 \*

Corresponding Author

texture information and so on.

**A New Tool for Image Analysis** 

*Wuhan Institute of Technology, Wuhan* 

*Aeronautical Industry Management, Zhengzhou* 

Zhang, X. & Han, K. L. (2006). High-order Symplectic Integration in Quasi-classical Trajectory Simulation: Case Study for O(1D) + H2. *International Journal of Quantum Chemistry,* Vol.106, No.8, (July 2006), pp. 1815–1819, ISSN 0020-7608
