(X1Î£+) Reaction on A New ab Initio Potential ... - ACS Publications

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Accurate Study on the Quantum Dynamics of the He + HeH+ (X1Σ+) Reaction on A New ab Initio Potential Energy Surface for the Lowest 11A′ Electronic Singlet State Wenwu Xu and Peiyu Zhang* State key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Science, Dalian 116023, P. R. China ABSTRACT: A time-dependent quantum wave packet method is used to investigate the dynamics of the He + HeH+(X1Σ+) reaction based on a new potential energy surface [Liang et al., J. Chem. Phys. 2012, 136, 094307]. The coupled channel (CC) and centrifugal-sudden (CS) reaction probabilities as well as the total integral cross sections are calculated. A comparison of the results with and without Coriolis coupling revealed that the number of K states NK (K is the projection of the total angular momentum J on the body-fixed z axis) significantly influences the reaction threshold. The effective potential energy profiles of each NK for the He + HeH+ reaction in a collinear geometry indicate that the barrier height gradually decreased with increased NK. The calculated time evolution of CC and CS probability density distribution over the collision energy of 0.27−0.36 eV at total angular momentum J = 50 clearly suggests a lower reaction threshold of CC probabilities. The CC cross sections are larger than the CS results within the entire energy range, demonstrating that the Coriolis coupling effect can effectively promote the He + HeH+ reaction. (TDQM) wave packet approach. Most recently, Liang et al.15 constructed a new PES (LYWZ PES) by using the complete active space self-consistent field method to determine the molecular orbitals and employing the high-theoretical-level multireference configuration interaction with the Davidson correction (MRCI+Q)/d-aug-cc-pV5Z calculations to obtain 15 682 ab initio energy points. No other method apart from quasi-classical trajectory was used by Liang et al.15 to test the effect of the new PES. Thus, quantum dynamics investigations based on the new PES need to be conducted for the title reactive system. A series of quantum dynamics investigations16−38 on ion−molecule reactions demonstrate that the Coriolis coupling effect plays an important role in chemical reactions. Panda et al.14 just investigated the K = 0−3 reaction probabilities and cross section of the He + HeH+ reaction and considered that the agreement between the centrifugal-sudden (CS) and the coupled channel (CC) results becomes poorer with increased J. They indicated that the effect of inclusion of Coriolis coupling in calculations for the He + HeH+ reaction is worth studying. In this paper, both CC and CS calculations by a timedependent wave packet method are carried out on the He + HeH+ reaction using the LYWZ PES. The rest of the paper is organized as follows. Section II describes the characteristics of

I. INTRODUCTION Helium atoms (He) and He-containing species are important in the collision processes of interstellar media, plasmas, planetary ionospheres, and high-energy physics studies and have been predicted to be abundant in various astrophysical objects such as interstellar media,1 planetary nebulae,2 helium-rich white dwarfs,3 or metal-poor stars.4 Although the He-involving He + HeH+ → HeH+ + He proton-transfer reaction under study is too limited because of the difficulties in distinguishing experimentally inelastic and elastic collision from collision involving the exchange of the proton, theoretically the title reaction was investigated early in the 1970s.5−15The potential energy surface (PES) is of great importance in quantum chemistry to study the dynamics of the ion−molecule reaction. However, most of the investigations in the literature focus on the equilibrium geometry and well depths or region around the geometry of the system. The investigations for analytical PES are very limited. In 2003, Panda et al.13 computed a global many-body expansion PES (PS PES) by an ab initio calculation with coupled cluster single and double excitations with perturbative triple excitations [CCSD(T)] using a correlation consistent basis set d-aug-cc-pVTZ. A many-body expansion function was fitted to the ab initio potential energy values and the resulting fit has a root-mean-square (rms) deviation of 10.8 meV (0.25 kcal/mol). Based on this PES, the bound and quasi-bound states13 of He2H+ and He2D+, as well as the reaction probabilities and cross sections14 of He + HeH+ reaction, were calculated by a time-dependent quantum-mechanical © 2013 American Chemical Society

Received: December 8, 2012 Revised: January 23, 2013 Published: January 24, 2013 1406

dx.doi.org/10.1021/jp312084r | J. Phys. Chem. A 2013, 117, 1406−1412

The Journal of Physical Chemistry A

Article

where V(r) is the diatomic reference potential. The initial wave packet is expanded in terms of the bodyfixed (BF) translational−vibrational−rotational basis uvn(R)̂ 30,31 where n and v are the indices labeling the ϕv(r)YJMε jK (R,r̂), translational and vibrational eigenfunctions, respectively. M and K are the projection quantum numbers of J on the space-fixed and BF z axis, respectively, and ε is the parity of the system. Accordingly, the element of the centrifugal term or the CC matrix is expressed as

the PES and the basic theory method. Section III comprises the results and discussion, and section IV provides the conclusions.

II. METHOD A. Potential Energy Surface. This study uses the lowest 11A′ adiabatic surface for the He + HeH+ reaction.15 The surface is a nonlinear least-squares fitting of multireference configuration interaction calculations with a large basis set. The root-mean-square error of the surface is 17.1 cm−1, which is much smaller than that found in previous works. A 0.577-eVdeep well exists on the reactive path of the exchange reaction of He + HeH+ which corresponds to the global minimum of the system. The equilibrium bond length at global minimum is 0.925 Å. Figure 1 shows the plot of the potential energy

ℏ2 JMε ⟨YjK |(J ̂ − j ̂)2 |YjJMKε ⟩ 2 ′ ′ 2μR R =

ℏ2 δ {[J(J + 1) + j(j + 1) − 2K 2]δKK 2 jj ′ ′ 2μR R

+ + − − − λJK λjK (1 + δK 0)1/2 δK + 1, K − λJK λjK (1 + δK1)1/2 ′ δK − 1, K } (3) ′ 1/2 ± where λ AB = [A(A+1) − B(B ± 1)] . In the CS approximation, the off-diagonal elements are neglected, thus leading to a simplified expression of eq 3

ℏ2 JMε ⟨YjK |(J ̂ − j ̂)2 |YjJMKε ⟩ 2 ′ ′ 2μR R =

ℏ2 δ [J(J + 1) + j(j + 1) − 2K 2]δKK 2 jj ′ ′ 2μR R

(4)

31−38

This method mainly includes three steps. First is constructing the initial Gaussian wave packet that can be written as the product of the motion wave packet and initial vibration eigenfunction. Second is solving the time-dependent Schrödinger equation (iℏ((∂ψ)/(∂t)) = Hψ̂ ) using the reactant Jacobi coordinates by the split-operator scheme as follows

Figure 1. Potential energy contour diagram for the approach of a He atom toward HeH+ in its equilibrium geometry on the LYWZ surface. Successive contours differ by 0.1 eV.

̂

̂

̂

ψ (R , r , t + Δ) = e−iH0Δ /2e−iU Δe−iH0Δ /2ψ (R , r , t )

contour at rHeH = 1.4634a0 in xy coordinates, where x = R cos γ and y = R sin γ (R is the He−HeH+ distance and γ is the Jacobi angle). The reaction path continues to the deep He2H+ bound state well as the He atom approaches HeH+. A comparison between Figure 1 in present paper and Figure 3a in ref 13 reveals that the well of LYWZ PES is slightly wider than that of the PS PES. B. CC Quantum Dynamics Method. Quantum calculations are carried out using the time-dependent wave packet method developed by Zhang et al.30,31 In the reactant Jacobi coordinates, the Hamiltonian of the He2H+ reactive system can be expressed as

(5)

+

H=−

where the reference Hamiltonian H0 is defined as ℏ2 ∂ 2 ℏ2 ∂ 2 − + V (r ) Ĥ 0 = − 2μR ∂R2 2μr ∂r 2

and the effective potential operator U in eq 5 is defined as Û =

(1)

2μR R2

2

+

ĵ

2μr r 2

+ V (R̂ , r )̂ (7)

PvJ0j k 0(E) =

⎤ ℏ ⎡ ∂ Im⎢ ψj(E) δ(s − s0) ψj(E) ⎥ μr ⎣ ∂s ⎦

(8)

σv0j k 0(E) =

π k2

(9)

0

where R is the distance from the He atom to the center-of-mass of HeH+, r is the HeH+ bond length, μR is the reduced mass of He with respect to HeH+, μr is the reduced mass of HeH+, J is the total angular momentum, j is the rotational angular momentum number of HeH+, V(R̂ ,r̂) is the interaction potential excluding the diatomic potential of HeH+, and ĥ(r) is the diatomic reference Hamiltonian ℏ2 ∂ 2 + V (r ) 2μr ∂r 2

(J ̂ − j ̂)2

Third is extracting the dynamic information from the final wave packet. The total reaction probability, the individual and total reaction cross sections can be calculated by

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

h(̂ r ) = −

(6)

0

σv0j (E) = 0

∑ (2J + 1)PvJ j k (E) 00 0

J

1 2j0 + 1

∑ σv j k (E) 00 0

k0

(10)

where ψ(E) is the corresponding time-independent part of the final wave function and k is the wavenumber corresponding to the initial state at a fixed collision energy E.

(2) 1407



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III. RESULTS AND DISCUSSION Table 1 shows the details of the initial wave packet in the 11A′ state, and other parameters used in the quantum calculations.

promote the wave packet to the product channel at low collision energy. In Figure 3, we investigate the NK-dependent convergence for the reaction probabilities of the He + HeH+ (vi = 0, ji = 0)

Table 1. Parameters for the Quantum Calculations (All Quantities Are Given in a.u., Unless Otherwise Indicated) center of initial wave packet on the scattering coordinate width parameter of the wave packet average translational energy (eV) scattering coordinate (R) range number of translational basis functions number of vibrational basis functions jmax for the rotational basis functions internal coordinate (r) range propagation time time step

21.0 0.32 0.197 0.1−30 240 120 110 0.5−18 90 000 10

To obtain the converged results in wave function expansion, 320 sinusoidal translational basis functions are used for the R coordinate within the range of 0.1−30.0 a0, 120 vibrational basis functions for the r coordinate of 0.5−18.0 a0, and jmax = 110 for the rotational basis functions. A propagation time of 90 000 au is used for the wave packet initially on the lowest 11A′ state. The width and average translation energy of initial wave packet are 0.32 a0 and 0.197 eV, respectively. The wave packet is dampened using damping function starting at 26 a0 along the R and at 15 a0 along the r. The J = 0 reaction probabilities of the He + HeH+(vi = 0, ji = 0) reaction and the integral cross sections of the He + HeH+(vi = 0, and 1, ji = 0) reaction on the LYWZ PES are presented in Figure 2 together with the quantum mechanical results obtained by Bhattacharya and Panda using the PS surface.13 Both calculations are conducted using the CS approximation. The overall behavior of the reaction probabilities and cross sections on the two surfaces are found to fit well with each other. The dense oscillation in two sets of results are found, which can be attributed to the deep potential well of the PES of the He + HeH+ reaction. The two theoretical results are somewhat different. For example, the reaction probabilities and cross sections on the LYWZ PES are slightly larger/-smaller at low/-high energies than that on the PS PES. The enhancement of CS reaction probabilities and cross sections at low energies results from the broader potential well of the LYWZ PES.15 A broader potential well is conducive to trap the wave packet and

Figure 3. NK-dependent reaction probabilities of the He + HeH+ (vi = 0, ji = 0) reaction on the LYWZ surface at J = 45, 50, and 55. (black dotted line) NK = 5, (red dashed line) NK = 6, and (blue solid line) NK = 7.

reaction at larger J values of 40, 45, and 50. NK denotes the number of K states used in the CC calculation. We find that at larger total angular momentum there is little difference in probabilities between NK = 6 and NK = 7 probabilities, indicating that it is no necessary to calculate the probabilities at NK > 7. Therefore, the converged cross sections can be obtained using NK = 6. To conserve computational time and resources without sacrificing computational accuracy, NK = 6 is chosen as the optimal value for performing accurate quantum scattering calculations. After determining the CC parameter NK, accurate quantum dynamics calculations on the He + HeH+ reaction can be performed using a new LYWZ surface. Figure 4 shows the CC and CS probabilities for the initial quantum numbers vi = 0 and ji = 0 at J = 10, 20, 30, 40, 50, and 60. We find that the reaction threshold in CC calculations is lower than that in CS

Figure 2. (a) Dependence of the J = 0 reaction probabilities on the collision energy of the He + HeH+(vi = 0, ji = 0) reaction on the LYWZ and PS surfaces; (b) dependence of the CS integral reaction cross sections on the collision energy of the He + HeH+ (vi = 0 and 1, ji = 0) reaction on the LYWZ and PS surfaces. (red solid line) the LYWZ surface and (black dashed line) the PS surface.14 1408



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effective potential barrier height at different J. Figure 6 shows the CC (NK = 6) and CS (NK = 1) effective potential energy profiles for the He + HeH+ (vi = 0, ji = 0) reaction in a collinear geometry on the LYWZ surface at J = 10, 30, and 50. No significant discrepancy is observed between the CC and CS effective potential energy profiles at J = 10. With the increase of J, the discrepancy becomes larger. From the above discussion, we find that the centrifugal potential barrier plays an important role in He+ HeH+ reaction. In order to deeply understand the reaction mechanism, we print the time evolution of CC and CS probability density distribution in logarithmic scale at J = 50. The probability density distribution is computed over the collision energy of 0.27−0.36 eV, which corresponds to the CC and CS reaction thresholds, respectively. The same parameters as Table 1 are used except the width and average translation energy of initial wave packet are 3.5 a0 and 0.295 eV, respectively, which makes 99% of the initial wave packet is at collision energy of 0.27− 0.36 eV. Figure 7 shows the time evolution of CS (NK = 1) probability density distribution. At propagation time T = 2500 and 5000 au, we can see that the distribution gradually approaches potential well. The shape of the distribution is very regular at this propagation time. The substantial distribution between R = 4 and 6 a0 at T = 7500 au indicates that there exists a centrifugal potential barrier at around R = 4 a0, and a large number of distribution, which cannot enter into the potential well, has been completely blocked by the centrifugal potential barrier. When propagation time T = 10 000, 12 500, and 15 000 au, the blocked distribution is rebounded by the centrifugal potential barrier. We finally find that there is no distribution in the region of product channel, which is consistent with that the CS reaction probabilities at J = 50 is almost zero at collision energy of 0.27−0.36 eV. We also present the time evolution of CC (NK = 6) probability density distribution in Figure 8. It is observed that there exists a centrifugal potential barrier at around R = 5 a0. From Figure 5 we know that the height of CC centrifugal potential barrier is much lower than that of CS centrifugal potential barrier. Because of the lower barrier height, part of distribution can pass over the centrifugal potential barrier to arrive at the region of product channel, while the rest is rebounded by the centrifugal potential barrier. At T = 12500 au, about ten percent of the total wave function has passed by the centrifugal barrier into the potential well. This result is consistent with the CC reaction probabilities over the collision energy of 0.27−0.36 eV at J = 50. From Figures 7 and 8, we can clearly obtain the reason why the reaction threshold in CC calculations is lower than that in CS calculations. Thus it is a good way to qualitatively analyze CC effect on barrierless reaction by the difference centrifugal potential barrier between CC and CS approximation. If the CC centrifugal barrier is obviously smaller than the CS one, it is safe to say that the CC cross sections will larger than the CS cross sections. And this condition will not exist in the reactions which have high reaction barrier or the centrifugal potential barrier being in (or close to) the potential well rather than before the potential well. Figure 9 presents the J-dependent partial wave contributions (weighted over a 2J + 1 factor) to the integral cross section of the He+ HeH+ (vi = 0, ji = 0) reaction at two collision energies of 0.2 and 0.4 eV. The involvement of a large number of partial waves clearly indicates a complex-forming mechanism of the reaction. A comparison of the CC and CS results reveals that

Figure 4. Dependence of the reaction probabilities on the collision energy of the He + HeH+ (vi = 0, ji = 0) reaction on the LYWZ surface at J = 10, 20, 30, 40, 50, and 60. (red solid line) CC and (black dashed line) represents CS.

calculations. The shifting of the reaction thresholds is entirely due to the centrifugal barrier. We plot the effective potential energy profiles of each NK for the He + HeH+ (vi = 0, ji = 0) reaction in a collinear geometry on the LYWZ surface at J = 50 in Figure 5. The effective potential energy profile is obtained by

Figure 5. NK-dependent effective potential energy profiles of the He + HeH+ (vi = 0, ji = 0) reaction in collinear geometry on the LYWZ surface at J = 50. The horizontal and vertical axes denote R (the distance of He−HeH+) and potential energy, respectively. Actually, the horizontal axis corresponds to the collinear minimum energy path. In order to specify the position of the centrifugal barrier, we use R instead of collinear minimum energy path.

the formula E(R,r) = F/2 μRR2 + V(R̂ ,r̂), where F is the smallest eigenvalue obtained through diagonalization of the matrix of the centrifugal term in eq 3. We find from Figure 5 that there exists a centrifugal potential barrier before the potential well, and the barrier height of the effective potential energy profile gradually decreases with the increase of NK, suggesting a lower reaction threshold of CC probabilities. We can also find from Figure 4 that for low J value, such as 10 and 20, the differences between the CC and CS probabilities are small. However, the remarkable differences are observed with increased J. The difference between the CS and CC reaction thresholds becomes increasingly obvious with the increase of J. This phenomenon can be well explained by the 1409



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Figure 6. Same as Figure 5. CC (NK = 6) and CS (NK = 1) effective potential energy profiles of the He + HeH+ (vi = 0, ji = 0) reaction in collinear geometry on LYWZ surface at J = 10, 30, and 50. (red solid line) CC and (black dashed line) CS.

Figure 7. Time evolution of CS (NK = 1) probability density distribution in logarithmic scale. The horizontal and vertical axes denote R (the distance of He−HeH+) and r (the bond length of HeH+), respectively.

IV. CONCLUSIONS A time-dependent wave packet study on the He + HeH+ reaction is carried out using the new LYWZ PES of Liang et al. The CC and CS reaction probabilities as well as the total cross sections are calculated. The reaction threshold in CC calculations is lower than that in CS calculations. The effective potential energy profiles of different NK values in collinear geometry at J = 50 further prove the significant influence of NK on the effective potential barrier height, which gradually becomes low with the increase of NK. The time evolution of CC and CS probability density distribution in logarithmic scales over the collision energy of 0.27−0.36 eV at J = 50 clearly suggests a lower reaction threshold of CC probabilities. A comparison of CC and CS cross sections reveals that the CC cross sections are larger than the CS results within the entire energy range. Therefore, the lower reaction thresholds of the CC calculations are the main cause of the enhancement of the CC integral cross sections, and that the Coriolis coupling effect can effectively promote the He + HeH+ reaction.

the omission of CC has a negligible effect on this contribution at low J values, but has an increased effect with increased J. This results is supported by a comparison of the integral cross sections of the He + HeH+ (vi = 0, and 1, ji = 0) reaction in Figure 10. The wave packet propagation is performed for J = 0−60 and 0−65 for vi = 0 and 1, respectively. All the J’s partial waves are calculated to get the cross sections. An obvious oscillatory characteristic of the CC and CS cross sections is observed. The oscillation weakens when the initial vibrational quantum number vi increases to one. Figure 10 shows that the CC cross sections are larger than the CS results within the entire energy range considered for both vi = 0 and 1. The lower reaction thresholds of the CC calculations are the main cause of the enhanced CC integral cross sections. Therefore, the Coriolis coupling effect can effectively promote the He + HeH+ reaction, which demonstrates that the Coriolis coupling effect is significant in this reaction. 1410



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Figure 8. Same as Figure 7. The time evolution of CC (NK = 6) probability density distribution in logarithmic scale.

to understand the importance of the centrifugal potential barrier. A comparison of the results between CC and CS shows the pronounced Coriolis coupling effects, and indicates that high-accuracy quantum dynamics calculations for the title reaction require the inclusion of Coriolis coupling effect.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +86-411-84379352. Notes

The authors declare no competing financial interest.

■

Figure 9. Weighed partial wave contributions to the integral cross sections of the He + HeH+ (vi = 0, ji = 0) reaction on the LYWZ surface as a function of the total angular momentum J at the two collision energies 0.2 and 0.4 eV. (red −●−) CC and (black −■−) CS.

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Figure 10. Dependence of the integral reaction cross sections on the collision energy of the He + HeH+ (vi = 0 and 1, ji = 0) reaction on the LYWZ surface. (red solid line) CC and (black dashed line) CS.

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