Quasi-Classical Trajectory Study of the N(2D) + H

Article pubs.acs.org/JPCA

Quasi-Classical Trajectory Study of the N(2D) + H2(X1ΣG+) → NH(X3Σ−) + H(2S) Reaction Based on an Analytical Potential Energy Surface Chuan-Lu Yang,* Li-Zhi Wang, Mei-Shan Wang, and Xiao-Guang Ma School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025, The People’s Republic of China S Supporting Information *

ABSTRACT: Using the multireference configuration interaction method with the Davidson correction and a large orbital basis set (aug-cc-pV5Z), we obtain an energy grid that includes 17 500 points for the construction of a new analytical potential energy surface (APES) for the N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) reaction. The APES, which contains 145 parameters and is represented with a many-body expansion and a new switch function, is fitted from the ab initio energies using an adaptive nonlinear least-squares algorithm. The geometric characteristics of the reported APES in the literature and those of our APES are also compared. On the basis of the APES that we obtained, reaction cross sections are computed by means of quasi-classical trajectory calculations and compared with the experimental and theoretical values available in the literature.

1. INTRODUCTION A potential energy surface (PES) is necessary for all dynamics investigations, whether reaction or nonreaction processes. A general and effective technique involved is construction of analytical potential energy surfaces (APESs) based on PESs from ab initio calculations. Two approaches can be used to avoid APESs in dynamics calculations. The first technique is called direct dynamics1 or ab initio dynamics.2 It obtains potential energies for every structure of the dynamics system through direct ab initio calculations. It can treat large chemical systems, while it is difficult if not impossible to derive analytic potential energy functions. It is very time-consuming for highlevel ab initio calculations because a large number of geometries have to be considered in the dynamics process. The second technique involves expressing PESs by interpolating points in a predetermined range of the energy grid.3 However, interpolation is also time-consuming. It can provide numerical energy derivatives, but these are more time-consuming compared with analytical ones. This technique is also not widely used. Therefore, APES is the least time-consuming form of PES for dynamics calculations, but its construction remains difficult. In the past decades, construction of APESs for ground systems based on a model function including adjusted parameters4−8 and on expressions fitted from ab initio energies9,10 was very successful. However, the same is not the case when some fragments in the excited state are involved, although the reactive system is still at an electronic state, particularly when the same atom has different states in different dissociative asymptotes (DAs); this phenomenon is very common in insertion reactions of molecules with excited atoms.11 For example, in the N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) reaction, the N atom is in the excited2 D state for the N + H2 asymptote, whereas it is in the ground © 2012 American Chemical Society

state for the NH + H asymptote; as a result, the two DAs will have two different values. So far, many reported expressions of APESs4−11 in the literature failed to represent these cases. Therefore, the reported APESs based on these expressions are accurate in the range of small internuclear distances, but poor behavior manifests when the total energy is close to the dissociation limit.12 New expressions are necessary to construct credible global APESs that can provide reliable potential energies for every structure in dynamics investigations. This work suggests a new expression of PESs that can reproduce two different DAs and can construct APES for the N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) reaction. Some quasiclassical trajectory (QCT) calculations are also performed to test the new APES.

2. EXPRESSION OF THE NEW FUNCTION Colliding fragments can generally be regarded as two radicals before and after collision. To simplify the discussion, we consider a three-body system with two similar atoms, as shown in Figure 1. In this case, the DAs for both incident and product channels include a lone atom. The reaction can be represented as follows: X + Y2 → XY + Y

(1)

We usually set the energy of the system as zero when the atoms are completely separated and are in ground states. Therefore, the two lone atoms (X or Y) in eq 1 are in ground states, and their energies are set to zero for both asymptotes. A conventional function that has zero asymptotical value can be Received: September 16, 2012 Revised: October 28, 2012 Published: December 4, 2012 3

dx.doi.org/10.1021/jp309197v | J. Phys. Chem. A 2013, 117, 3−8

The Journal of Physical Chemistry A

Article

Rn1 → zero because n is larger than 1 and exp[−P2(R2 + R3)/Rn1] → 1.0. Obviously, the value of the f(R1,R2,R3) function tends to be zero, which means that the lone atom Y is in the ground state. The function can also be expanded to a more complex case. If the lone atoms in both asymptotes are also in excited state, the values of X + Y2 and XY + Y DA become nonzero, and we can increase a similar term to represent their excited energies. The function is written as g (R1 , R 2 , R3) = A1{1.0 − exp[−A 2 R1(1/R 2m + 1/R3m)]} + P1{1.0 − exp[−P2(R 2 + R3)/R1n]}

Figure 1. Three-body collision system.

Both m and n are numbers larger than 1, whereas A1, A2, P1, and P2 are parameters larger than zero and are determined by fitting. The first term of eq 3 is zero in the reactant asymptote X + Y2, where R1 → constant and R2,R3 → ∞. P1 is still the excited energy of the atom X. For the XY + Y asymptote, R1 → ∞ and R2 or R3 must increase to ∞. If R2 → ∞ and R3 → finite constant, A2R1/Rm2 → 0 but A2R1/Rm3 → ∞ because of R1 → ∞. If R3 → ∞ and R2 → finite constant, A2R1/Rm2 → ∞ but A2R1/ Rm3 → 0. Both cases result in exp[−A2R1(1/Rm2 + 1/Rm3 )] → 0. A1 becomes the asymptotical value of g(R1,R2,R3) because the second term of eq 3 tends to be zero in these cases. Therefore, A1 is the value of DA XY + Y or the excited energy of atom Y. Moreover, g(R1,R2,R3) satisfies the zero value of DA X + Y + Y, where R1, R2, and R3 → ∞. However, the reaction involving two nonzero DAs is not very common. This work focuses on an actual N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) system in the succeeding section. The APES for the three-body system can be represented by manybody expansion as proposed by Murrell and Carter,8 which is as follows:

used to construct the APES. Many APESs have been built in this kind of system, except when the lone X or Y atoms are in excited states. In the excited state, the values of two DAs are different, and PESs cannot be described by the reported functions in the literature, such as many-body expansions, as suggested by Murrell et al.8 or Aguado et al.,9,10 because their expressions have only one value (zero) for DAs in the threebody term. Murrell et al.8 observed this problem and employed a switching function to solve it. Varandas et al.12 presented a modified one for N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S), where two different asymptotes are described with two separate functions and the parameters are determined before fitting. In this work, we present another function that can describe the different DAs on PES. Previous functions for the atomic ground states also remain valid as long as the present function is added to them. In the reaction that involves eq 1, two possible DAs of the reaction of the XY2 system are X + Y2 and XY + Y. We assume that only atom X is in the excited state, with atom Y in the ground state. For the reactant asymptote X + Y2, the three-body term of APES should obviously demonstrate the excited energy of X in addition to the energy of the Y2 molecule. In other words, the three-body term of the APES should provide the excited energy of X. For the product asymptote XY + Y, the three-body term of the APES only provides the energies of the XY molecule because the energies of the ground-state atom Y are supposed to be zero. Hence, the values of the three-body terms of the APES should be nonzero in the X + Y2 asymptote but zero in the XY + Y asymptote. To describe the two DAs, we suggest the following function: f (R1 , R 2 , R3) = P1{1.0 − exp[−P2(R 2 + R3)/R1n]}

(3)

3

Vtot(R1 , R 2 , R3) =

∑

V i(1)

+

i

∑ V n(2)(R n) n=1

(3)

+ V (R1 , R 2 , R3)

(4)

where Vtot is the potential energy of the NH2 reaction system, (i = 1−3) are the atomic energy terms, V(2) V(1) i n (n = 1−3) represent the two-body terms, and V(3) is the three-body term that becomes zero at all dissociation limits. In the present case, the values for V(1) i cannot be set to zero. They are represented by the new suggested function of eq 2 because of the different DAs. Hence

(2)

where R1 is the interatomic distance between atoms Y and Y, R2 and R3 represent the interatomic distances between atoms X and Y (see Figure 1), and P1, P2, and n are parameters larger than zero, which can be determined by fitting ab initio energies of the PES. The value n can be fixed or fitted according to the different systems, but it should be larger than 2 according to our experience. Can a simple function describe different DAa? In the following, we describe how this works. For reactant asymptote X + Y2, R1 → constant (the equilibrium position of diatomic Y2, for example, the bond length of H2, is 0.714 Å in the insertion H2 reaction) and R2,R3 → ∞, which results in the function exp[−P2(R2 + R3)/R1n] → 0 (assuming P2 > 0); then, f(R1,R2,R3) → P1. Therefore, P1 is the value of the three-body term of the X + Y2 DA, namely, the energy of the excited state of atom X. For the product asymptote XY + Y, R1 and R2 → ∞, R3→ finite constant, or R1 and R3 → ∞ and R2 → finite constant; both cases result in the exponent part −P2(R2 + R3)/

3

∑ Vi(1) = P1{1.0 − exp[−P2(R 2 + R3)/R1n]} i=1

(5)

The AP function suggested by Aguado and Paniagua9,10 is used to represent the two- and the three-body terms. The twobody term is written as V (2)(R i) =

c0 exp( −βi R i) Ri

N

+

∑ ciRi exp(−αiR i) i=1

(6)

where Ri represents the distances between two atoms. In this work, we truncate the polynomial to a degree of 9(N = 9), which results in 12 parameters (α, β, c0, c1, ..., a9) to be determined for H2 and NH molecules with ab initio energies. The three-body term can be written as 4



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Table 1. Spectroscopic Constants of the Two-Body Terms for H2 and NH H2 (AP) H2 (expt.)26 NH (AP) NH (expt.)26

Re/Å

D0/eV

we/cm−1

wexe/cm−1

Be/cm−1

αe/cm−1

0.74192 0.74144 1.03775 1.03621

4.4728 4.4781 3.3893 3.47

4405.65 4401.21 3277.94 3282.27

123.476 121.33 80.8322 78.35

60.780 60.853 16.6514 16.6993

2.835 3.062 0.6485 0.6490

⎧ M ⎪ ∑ cjklλ1jλ 2k λ3l (k = l ) ⎪ j,k ,l=0 ⎪ V (3)(R1 , R 2 , R3) = ⎨ ⎪ M j k l l k ⎪ ∑ cjklλ1 (λ 2 λ3 + λ 2λ3 ) (k ≠ l) ⎪ j,k ,l=0 ⎩

data,26 and they imply that the present ab initio calculations and the fitting process for the two-body terms are reliable. The parameters of two-body terms are fixed in the further fitting process for the parameters of the one-body and three-body terms. A nonlinear least-squares fitting method is used to determine the left parameters of the one-body and three-body terms. A total of 145 parameters (142 linear and 3 nonlinear) are available in the last fitting process. The fitting calculations are performed with an adaptive nonlinear least-squares algorithm.27 As a large number of both fitting parameters and energies are present, fitting becomes difficult to converge if the initial parameters are unsuitable. However, suitable initial parameters cannot be easily determined. To obtain convergent fitting results, we do not use all ab initio energy points at the beginning of the fitting process. Instead, we start from 5000 energy points and gradually increase the points. Finally, 17 500 energies are used to determine 145 parameters. The root-meansquare error is 1.2 kcal/mol. All fitted parameters and the FORTRAN routine of the APES, including the first derivatives, are presented with the Fortran routine in the Supporting Information to provide fast, accurate, and convenient codes for the dynamic calculation. C. Characteristic of the APES. The energy difference (P1 in the new function) of the asymptotes of N(2D) + H2(X1Σg+) and N(X4S) + H2(X1Σg+) is experimentally reported as 55.0 kcal/mol.26 The present 56.21 kcal/mol value of the APES is an improvement compared with the 56.54 kcal/mol from the switching function of Varandas and Poveda,12 which implies that the new switch function can well describe the energies of atomic excited states. However, our new function has the threebody effect because of its dependence on all three internuclear distances of the molecule. The function can also be directly added to any other expressions of the APES and can fit the parameters together with a common fitting procedure. Therefore, the new function is considered suitable for many interaction systems. Figure 2 shows the contour plots of the APES with the use of internal coordinates. Figure 3 shows the APES at perpendicular and linear configurations. To compare the present plots with their counterparts in the literature, we use the same parameters in the plots. The present APESs show very similar characteristics to those of Zhou et al.19 and Varandas et al.12

(7)

where cjkl denotes linear coefficients. The constraints j + k + l ≠ j ≠ k ≠ l, j = k + l ≤ M, and k ≤ l are used to ensure that the three-body term is zero at all dissociation limits. The two expressions of eq 7 are used to satisfy the permutation symmetry of two H atoms. M equals 10 in this work, which results in 140 linear coefficients to be determined in the expression. The λi term is expressed as follows: λi = R i exp( −ωiR i)

(8)

where ωi is the nonlinear parameter and Ri is the distance between two atoms. Three nonlinear parameters are included in eq 7. However, the two ωNH values should be the same because of the commutative symmetry of the two H atoms. Therefore, only two nonlinear parameters (i.e., ωNH, ωHH) must be determined in the present case.

3. COMPUTATIONAL METHOD We use the function to construct the APESs for the insertion reactions of N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) because many studies have been devoted to modeling PESs12−20 for the reaction. Almost all reported PESs use the reproducing kernel Hilbert space interpolation method or three-dimensional cubic spline interpolation to interpolate the calculated ab initio points. Only Varandas and Poveda12,20 built APESs for a reaction that uses a double many-body expansion and the switching function. A. Ab Initio Calculations. The multireference configuration interaction (MRCI) method21,22 is used to obtain the potential energy curves for NH and H2 and the energy grids for the ground state of NH2. The reference wave functions of MRCI include all single and double excitations that are built on complete active space involving 7 electrons and 10 active orbitals. The 1s orbital of the N atom is kept inactive in the calculations. Dunning’s correlation-consistent basis set, aug-ccpV5Z,23,24 is used in the calculations after the accuracy and the CPU time are identified. All calculations are performed with the MOLPRO set of programs.25 Finally, a total of 17 500 ab initio energy points are used to determine 145 parameters in the APES. B. Fitting of the Potential Energy Surface. Using the function expression of eq 6, we determine the parameters for the two-body terms of NH and H2 with ab initio energies. The parameters are presented in Supporting Information. We test the quality of the fitting process by computing the spectroscopic parameters for H2 and NH molecules with the APES of two-body terms. The results in Table 1 show that the present spectroscopic parameters conform to the experimental

4. QUASI-CLASSICAL TRAJECTORY CALCULATIONS To test the applicability of the new APES, we perform QCT calculations and compare the results with those in the literature. We calculate the total integral cross sections (TICSs) for the N(2D) + D2(X1Σg+) → ND(X3Σ−) + D(2S) reaction as these have been experimentally measured and theoretically calculated. For every run, a batch of 500 000 trajectories is performed, and the integration step size in the trajectories is selected to be 0.1 fs, a value that guarantees conservation of the total energy and the total angular momentum. The trajectories start at an initial distance of 10.0 Å between the N atom and the center of the H2 mass. The collisional parameters (bmax) are from 1.43 to 2.10 Å 5



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Figure 3. Contour plots of NH2 APES: (a) contour for C2v insertion of an N atom into H2 in Jacobi coordinates and (b) contour in collinear N−H−H configurations in internal coordinates. The energy zero is defined at the N(2D) + H2 asymptote. The energy is in eV.

Figure 2. Contour plots of NH2 APES in internal coordinates: (a) contour with the enclosed bond angle at 102.6° and (b) contour with RNH2 =1.923 a0. The energy zero is defined at the N(2D) + H2 asymptote. The energy is in eV.

according to different collisional energy values. As shown in Figure 4, the results of the TICS based on the new APES reveal the same tendency as those from previous PESs, along with an increase in collisional energy. The results from both QCT and the statistical quantum mechanical (SQM) model28 are also presented in Figure 4. Our results are slightly larger than the QCT results and the experimental data in the literature12 in the low-energy range, whereas they are slightly smaller than the QCT results in the high-energy range. However, the present TICSs are almost the same as the experimental data in the medium-energy range. Two factors are responsible for the difference between the present QCT cross sections and previous results.28 The first factor is the different energy from that of the present complete analytical expression of the PES and its interpolated PES and the derivatives from the numerical form in the previous PES and our analytical expression for the same geometry. The second factor is the parameters for the QCT calculations. For example, our trajectories for every run are 5 × 105, but they are 2 × 105 in ref 28. The maximum impact parameter bmax is evaluated for every run in our calculations, but it is estimated with a formula in ref 27. The initial distance between the N atom and the center of H2 is 10 Å in our calculations but 8 Å in theirs. Fortunately, neither

Figure 4. The total integrated cross sections for the N(2D) + D2(X1Σg+) → ND(X3Σ−) + D(2S) reaction.

factor produces very large differences for the cross sections. Obvious differences are observed between the theoretical and experimental values at large collision energy. However, these 6



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Notes

differences are attributed not to the deficiency of the PES expression or QCT calculation but to the effect of the lowest excited state 1A′ or the experiment itself. For brevity, a detailed analysis is not presented in this paper (refer to Bañares et al.28 and Liu29). We also calculate the total cross sections for the N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) (n = 0, j = 0−3) reaction at 0.165 eV collision energy to perform further tests for the new PES. The results are presented in Table 2. The present cross

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation of China under Grants NSFC-10974078 and NSFC-11174117.

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Table 2. Total Cross Sections (in Å2) for the N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S)(n = 0, j = 0−3) Reaction Calculated on Several PESs at 0.165 eV Collision Energy present PES

PES by Pederson et al.17

PES by Ho et al.16

j

QCT

SM30

QM30

QCT30

QCT16

QCT16

0 1 2 3

6.797 6.805 6.812 6.820

6.88 6.85 6.79 7.01

6.68 6.69 6.71

5.81 5.99 6.24 6.85

5.85 5.97 6.27 6.45

5.37 6.33 6.50 6.30

sections closely conform to those in the work of Balucani et al.30 with a rigorous statistical method (SM)31 and the PES constructed by Pederson et al.,17 although they are a little larger than those of quantum mechanical (QM) scattering calculations.30 The cross sections of QCT by Balucani et al. show a more obvious change along with j. However, the QM results for j = 0−2 exhibit only a small change. This result means that the special treatment may overstate the rotational effect. The changes of cross sections for SM and our QCT for j = 0−3 with no special treatment are similar to that of QM. In summary, the present APESs provide reliable dynamics characteristics of the reactive system with high calculation efficiency because they are implemented with complete analytical energy as well as first derivatives. Therefore, additional dynamics calculations based on the new APESs are expected to be performed.

5. CONCLUSIONS We suggested a switch function that can describe two different DAs of the PES for a three-body interaction system. By combining the new function and the traditional expressions of APESs, we obtained highly efficient APESs for the N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) reaction by fitting the parameters from ab initio energies at the MRCI/aug-cc-pV5Z level. The topological characteristic and the TICSs with QCT based on the new APESs agree with those in the literature. The new function is expected to be used to construct additional APESs for similar reaction systems.

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ASSOCIATED CONTENT

S Supporting Information *

Tables for the parameters and the FORTRAN subroutine of the analytical potential energy surface and first derivatives with respect to the internuclear distances for the N(2D) + H2(X1Σg+) → NH(X3Σ−) + H(2S) reaction. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

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*Tel: +86 535 6672870. Fax: +86 535 6672870. E-mail: [email protected]. 7



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Quasi-Classical Trajectory Study of the N(2D) + H

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