Electronically nonadiabatic transitions in a collinear H₂ + H⁺ system: Quantum mechanical understanding and comparison with a trajectory surface hopping method

Abstract

Electronically nonadiabatic transitions in a collinear H₂ + H⁺ system have been studied both quantum mechanically and with classical mechanics using a diatomics-in-molecules type potential energy surface fitted to recent ab initio data. The quantum dynamical calculations are carried out by employing a standard close-coupling method in hyperspherical coordinates and the quasiclassical trajectory calculations were carried out with a basic surface hopping method. Special emphasis is placed on qualitative analysis of the physical mechanisms of the electronically nonadiabatic transitions. To make such an analysis, the basic idea proposed by Nobusada et al. (K. Nobusada, O. I. Tolstikhin and H. Nakamura, J. Chem. Phys., 1998, 108, 8922) in the study of vibrationally nonadiabatic transitions is applied, and also the reaction dynamics is visualized in detail by using classical trajectories. It is found that the electronically nonadiabatic transition occurs in a rather narrow configuration space and its reactivity depends strongly on the initial vibrational state of H₂.

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I Introduction

Accurate theoretical calculations of quantum reaction dynamics have become feasible. A number of calculations for triatomic¹ and also polyatomic^1–3 systems have been carried out. Most of these reactions are usually assumed to occur on the ground adiabatic potential energy surface (PES) only. Although several groups have carried out accurate quantum mechanical calculations of electronically nonadiabatic reactions,^4–13 the detailed quantum dynamics of those reactions have not yet been well analyzed. On the other hand, a quasiclassical trajectory surface hopping (TSH) method has been much more extensively applied to reactions accompanying electronically nonadiabatic transitions.^14–19 The TSH method has the advantage of visualizing reaction dynamics more clearly and is very helpful if the method is used together with a quantum approach to elucidate the reactions accompanying electronically nonadiabatic transitions.

In the present study, we apply quantum and classical mechanical approaches to a reactive collinear system of H₂ + H⁺ accompanying electronically nonadiabatic transitions. It is still demanding to make exact (especially quantum mechanical) calculations of the electronically nonadiabatic transition in a full 3D system. Even though we can accomplish such tough calculations, qualitative understanding of the mechanisms is not sufficient. Therefore, we analyze the present collinear H₃⁺ system in detail and get a deeper insight into electronically nonadiabatic transitions. The quantum dynamical calculations are carried out by using a standard time-independent close-coupling method in hyperspherical coordinates and the classical calculations by the TSH method. In both calculations the new diatomics-in-molecules (DIM)^20,21 type PES is employed. The DIM PES is con structed by fitting a 3 × 3 DIM type matrix to recent ab initio data calculated by Ichihara and Yokoyama.²² In this paper, we place special emphasis on qualitative analysis of the mechanisms of electronically nonadiabatic reactions. From a quantum mechanical viewpoint, we aim to conceptualize electronically nonadiabatic transitions by taking advantage of a basic idea, which was previously established by Nobusada et al. to treat vibrationally nonadiabatic transitions.^23–25 On the other hand, from a classical mechanical viewpoint, we visualize the dynamics more clearly by analyzing the classical trajectories.

This paper is organized as follows. In the next section, we define the present collision system and describe the DIM PES of the system. The methods of the quantum mechanical and classical TSH calculations are given in Section III. In Section IV, the results obtained by the quantum and classical dynamical calculations are discussed in detail. A comparison between the two results is also made in this section. The concluding remarks are given in Section V.

II Physical system and potential energy surfaces

In the present study, we chose a collinear H₂ + H⁺ reactive collision system accompanying electronically nonadiabatic transitions, i.e., charge transfer. This system has two different electronic channels in the energy range considered:

If we assume that all three hydrogen atoms are distinguishable, there are four different processes in this collision system, exchange and nonexchange without charge transfer and exchange and nonexchange with charge transfer. In the following discussions, we distinguish between these four processes. The present collision system is described by using the DIM PES. The DIM PESs of H₃⁺ have already been reported.^26,27 To reconstruct the DIM PES, however, we fit the 3 × 3 minimal DIM potential to recent ab initio data for the ground and first excited states of H₃⁺ calculated in 734 physically important nuclear configurations.²² The fitting procedure is fulfilled in a framework of the simple many body expansion method²⁸ by adding correcting polynomials described below. The 3 × 3 DIM Hamiltonian matrix is given by

Here,

where i, j, k(i≠j≠k) = 1, 2, 3 and r_i,j,k are internuclear distances of the diatoms. ¹Σ_g⁺ represents the ground electronic state of H₂, and ²Σ_g⁺ and ²Σ_u⁺ represent the ground and first excited electronic states of H₂⁺, respectively. Functional forms of these states are given by

where a = 1.13804 and b = 0.52961, and the other diatomic parameters are listed in Table 1. Three-body terms C_h,g,u in eqn. (4)–(6) decrease exponentially to zero when one of the atoms moves away from the other two. The three-body terms are explicitly given by

Table 1 Diatomic parameters of eqn. (7)–(9)

h	g	u
d	−0.17445	0.10272	0.62871
β	2.09607	0.71935	0.87080
r	1.40104	2.00313	0.34405

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Here P_l is the cubic polynomial which is represented in the following form taking account of the symmetry of H₃⁺:

The fitting parameters c_n(n = 1–14) defined for each l( = h, g, u) are given in Table 2.

Table 2 Fitting parameters of eqn. (10) and (11)

h	g	u
c 1	8.13033	−6.22155	−3.94414
c 2	−3.02555	3.70543	0.57600
c 3	1.65898	−1.98676	0.32953
c 4	0.93108	−1.96016	0.01044
c 5	0.92263	−2.08734	0.03277
c 6	−0.36808	5.76902	−0.06461
c 7	−1.77260	1.77340	−0.10262
c 8	−0.12637	0.17791	−0.00117
c 9	0.11083	0.21520	−0.00541
c 10	−0.34325	−1.03671	0.01565
c 11	−0.30780	−0.01557	0.00594
c 12	0.44247	−0.20215	−0.00982
c 13	0.30672	−0.04772	0.00182
c 14	0.78463	0.93662	0.58787

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We plot contour maps of the H₃⁺ DIM PES for the ground state in Fig. 1 and for the first excited state in Fig. 2 as a function of u₁ and u₂, where u₁ is the mass-scaled distance between H and the center-of-mass of H₂⁺ and u₂ is the mass-scaled H–H internuclear distance. The ground PES has a deep well at u₁≈2.5 a₀ and u₂≈1.4 a₀ corresponding to the stable H₃⁺ triatomic complex. On the other hand, the first excited PES is generally repulsive, at least in the energy region considered.

Fig. 1 Contour map of the ground electronic state of H₃⁺ as a function of u₁ and u₂, where u₁ is the mass-scaled distance between H and the center-of-mass of H₂⁺, and u₂ is the mass-scaled H–H internuclear distance. The thick broken curves indicate the seam lines. The contour lines are drawn in eV relative to the lowest vibrational state of H₂.

Fig. 2 As Fig. 1 but for the first excited state.

III Calculations of reaction dynamics on multi-potential energy surfaces

A Quantum mechanical treatment

The Schrödinger equation taking account of nuclear and electron motions is given by

where T(r) and K(R_el) are the kinetic energy parts with respect to nuclear and electron motions, respectively. Here, r( = r₁, r₂, r₃) and R_el respectively indicate sets of nuclear and electron coordinates. V(r, R_el) is the interaction potential as a function of r and R_el, and E is the total energy measured from the lowest vibrational state of H₂. If we consider a collinear H₃⁺ system, eqn. (12) is easily rewritten by using the hyperspherical coordinates in a usual way:

where μ is the characteristic mass,

is the hyper-radius and ϕ( = tan⁻¹u₂/u₁) is the hyper-angle and H_el = K(R_el) + V(ρ, ϕ, R_el). The wavefunction Ψ is expanded in terms of the wavefunctions with respect to electron and nuclear motion as follows:

where Φ_n(R_el; ρ, ϕ) is the electronic basis set forming the DIM Hamiltonian of eqn. (3) and ψ_n(ρ, ϕ) is the nuclear basis function. After substituting eqn. (14) into eqn. (13), we obtain the following set of coupled differential equations given by

where U_mn( = 〈Φ_m∣H_el(ρ, ϕ, R_el)∣Φ_n〉) is represented by eqn. (3). Without difficulty eqn. (15) is further rewritten into a set of coupled equations with respect to ρ if ψ_n(ρ, ϕ) is expanded in terms of proper basis sets as a function of ϕ. Then, we solve these coupled equations using the R-matrix propagation method²⁹ up to the asymptotic region along ρ and obtain the S-matrix by imposing a proper scattering boundary condition there.

B Trajectory surface hopping approach

We carry out the classical calculations by employing a primitive TSH method described elsewhere.¹⁴ To obtain quantitative results, we should use the improved TSH methods.^15,17,19 However, in this paper we do not intend to make a quantitative comparison between the quantum and classical results. The electronically nonadiabatic transition probability between the two PESs is evaluated by using the Landau–Zener (LZ) formula at every position where the classical trajectory crosses the seam line. The LZ parameters are determined by interpolating the adiabatic PESs in the section normal to the seam line. Whether the surface hop actually occurs or not is determined by comparing the magnitude of the LZ nonadiabatic transition probability with random numbers (i.e., an anteater method).

IV Results

A Quantum mechanical description of electronically nonadiabatic transitions

Fig. 3 shows the vibronic adiabatic potential energy curves as a function of ρ. The vibrational quantum numbers of H₂ and H₂⁺ are indicated on the right-hand edge. The curves are roughly categorized into two types of diabatic curves; (i) those having deep potential wells and (ii) simply repulsive curves. All the potential curves asymptotically correlate to definite H₂ or H₂⁺ vibrational energy levels and are double degenerate. However, the diabatic curves correlating to H₂ split into two at small ρ. This is simply energy splitting, well known in the case of a double well potential. These splittings are mainly responsible for the atom exchange process. In the region of E≳1.8 eV and 7≲ρ/a₀≲10, many avoided crossings are located between the adiabatic potential curves. As will be described below, these avoided crossings play a very important role in electronically nonadiabatic transitions.

Fig. 3 Vibronic adiabatic potential energy curves as a function of ρ. Vibrational quantum numbers of H₂ and H₂⁺ are indicated on the right-hand edge.

In Fig. 4(a)–(d), we show the quantum mechanical probabilities of three inelastic collision processes for H₂([italic v] = 2–5) as a function of the total energy. The three processes are exchange and nonexchange with charge transfer, and exchange without charge transfer. Here, it should be noted that all the probabilities are smoothed out because the original raw data give a huge number of (Feshbach) resonances owing to the deep potential well shown in Fig. 3. All the quantum probabilities reported below are similarly smoothed out. These figures indicate two interesting features: (i) Some representative maximum peaks appear. (ii) The three processes compete with each other strongly, depending on the initial vibrational state of H₂. These characteristic features are qualitatively interpreted in terms of the idea proposed by Nobusada et al. in the study of electronically adiabatic rearrangement reactions.^23–25 In their papers, they demonstrated that the rearrangement reactions were regarded as vibrationally nonadiabatic transitions occurred at some important avoided crossings. Let us first consider the dynamical feature of (i) mentioned above. To relate the electronically nonadiabatic transitions to specific avoided crossings, we pick up several avoided crossings from Fig. 3. Fig. 5 is a magnification of Fig. 3 to show such avoided crossings more clearly, labeled a–d. We can roughly ascribe the main maximum peaks labeled in Fig. 4 to these specific avoided crossings. In fact, each main peak position approximately corresponds to the energy positions of the avoided crossings a–d. To confirm the above explanations more quantitatively, we evaluate the electronically nonadiabatic transition p at these crossings by introducing the a² parameter, which judges the significance of avoided crossings.³⁰ According to the semiclassical analysis by Zhu et al.,³⁰ effective nonadiabatic transitions (0.005≲p≲0.96) occur in the range 0.05≲a²<100. The present a² parameters are 0.402 (crossing a), 0.246 (crossing b), 0.265 (crossing c), and 0.287 (crossing d). These results clearly indicate that electronically nonadiabatic transitions occur effectively at these avoided crossings. The representative maximum peaks in Fig. 4 appear as a result of these effective electronically nonadiabatic transitions.

Fig. 4 Quantum probabilities of exchange with charge transfer (solid line), nonexchange with charge transfer (dotted line), and exchange without charge transfer (broken line) as a function of the total energy. Initial vibrational states of H₂ are (a) 2, (b) 3, (c) 4 and (d) 5. Some representative maximum peaks are labeled as a–d.

Fig. 5 Magnification of Fig. 3 showing some avoided crossings much more clearly. These avoided crossings are labeled with notation corresponding to those of Fig. 4.

We now explain the reason for the characteristic feature (ii) listed above by drawing sections of the ground and first excited PESs of H₃⁺ in Fig. 6 as a function of ϕ/π at ρ = 6, 8 and 14 a₀. The reagent and product channels are well separated by the potential barrier. The potential barrier of the first excited PES is so high that the exchange reaction cannot occur, at least in the energy range considered. On the other hand, the barrier of the ground PES becomes lower with decreasing ρ and then the exchange reaction seems to occur easily on the ground PES. Combining these sections of the PESs with the vibronic adiabatic potential curves in Fig. 3, we analyze the quantum dynamics of H₃⁺. Let us follow the vibronic adiabatic potential energy curves correlating to H₂([italic v]⩽3) in Fig. 3. These potential curves do not have important avoided crossings at which the electronically nonadiabatic transitions occur effectively. Therefore, the reaction for H₂([italic v]⩽3) + H⁺ occurs mainly on the ground PES corresponding to the solid line in Fig. 6. As mentioned above, the exchange reaction on the ground PES can occur at small ρ. In fact, exchange without charge transfer occurs easily when [italic v]⩽3 as shown in Fig. 4(a) and (b) (see broken lines). However, charge transfer also occurs even when [italic v] = 2 or 3. This is because the vibrational excitation occurs during the collision and then the system has a chance to encounter the important avoided crossings shown in Fig. 5. Nevertheless, this process is minor and thus the probability of charge transfer is much smaller than that of exchange without charge transfer in the case of H₂([italic v]⩽3) + H⁺. Next, we follow the vibronic adiabatic potential energy curves correlating to H₂([italic v]⩽4). Differently from the case of H₂([italic v]⩾3) + H⁺, these curves encounter the important avoided crossings shown in Fig. 5. As mentioned above, electronically nonadiabatic transitions occur efficiently at these crossings. Thus, the sum of the probabilities for the two charge transfer processes (solid line + dotted line) is larger than that for H₂([italic v]⩽3) + H⁺. In particular, Fig. 4(c) and (d) show that nonexchange with charge transfer occurs more efficiently than exchange with charge transfer. This is because the charge transfer occurs effectively in the entrance channel and then the collsion system is repelled backwards on the first excited PES owing to the high potential barrier, as mentioned above.

Fig. 6 Sections of the present PESs of the ground (solid line) and first excited (dotted line) states at ρ = (a) 6, (b) 8 and (c) 14 a₀ as a function of ϕ/π.

B Classical mechanical description of electronically nonadiabatic transitions

Fig. 7 show the classical mechanical probabilities of four collision processes for the initial vibrational states of H₂([italic v] = 2, 4 and 6) as a function of the collision energy E_col . The dynamics of these processes is illustrated in Fig. 8 by showing the corresponding typical classical trajectories. The seam lines are the same as those shown in Fig. 1 and 2. As clearly seen from these figures, the dynamics depends strongly on the initial vibrational state of H₂. The dynamical features are explained qualitatively as follows.

Fig. 7 Classical probabilities of four processes of exchange with charge transfer (○), nonexchange with charge transfer (□), exchange without charge transfer (△), nonexchange without charge transfer ( × ) as a function of the collision energy. Initial vibrational states of H₂ are (a) 2, (b) 4 and (c) 6.

Fig. 8 Typical classical trajectories for some representative initial conditions of (a) H₂([italic v] = 2); E_col = 1.1 eV, (b) H₂([italic v] = 4); E_col = 1.5 eV and (c) H₂([italic v] = 6); E_col = 1 eV as a function of u₁ and u₂. The thick broken curves represent the seam lines.

The dynamics is fully diabatic in the asymptotically far region of the entrance channel. Thus, if the vibration amplitude is so large, i.e., H₂([italic v]>4), that the trajectories can cross the seam line, the electronically nonadiabatic transition probability is equal to unity. As the atom and the diatom are close to each other, the dynamics transforms from diabatic to adiabatic and then the electronically nonadiabatic transition probability tends to be zero. Since the split between the ground and first excited PESs along the seam line becomes narrower exponentially, the transformation from diabatic to adiabatic occurs in a rather limited region of the valley at u₁∽8 a₀. This small region plays a very important role because it determines whether or not the effective electronically nonadiabatic transitions occur. In the following discussions, we further analyze the details of the electronically nonadiabatic transitions by showing the three typical cases for H₂([italic v] = 2), then H₂([italic v] = 6), and finally H₂([italic v] = 4).

B.1 H₂([italic v] = 2) + H⁺. Since the vibration amplitude is small in the entrance channel, as shown in Fig. 8(a), the trajectories cannot cross the seam line. This means that the dynamics proceeds only on the ground PES and thus the electronically nonadiabatic transitions do not occur in the entrance channel. The trajectories on the ground PES reach the deep potential well at u₁≲6 a₀ and then most of the trajectories enter the product channel out of the potential well. Since the present reactive system is symmetric, there exists also the same small region in the product channel where the effective electronically nonadiabatic transitions occur. The collision system in the deep potential well has a chance to be vibrationally excited and as a result the vibration amplitude becomes large. If the outgoing trajectory passes the above small region in the product channel and crosses the seam line, the electronically nonadiabatic transition can occur. This process ends in the exchange with charge transfer. The trajectory corresponding to this process is clearly illustrated in Fig. 8(a). Fig. 8(a) shows the two typical outgoing trajectories starting from the initial condition of H₂([italic v] = 2) and E_col = 1.1 eV. Exchange with charge transfer is a rather minor process and exchange without charge transfer occurs much more frequently.

B.2 H₂([italic v] = 6) + H⁺. In these collisions the initial vibration amplitude is so large that the trajectories can cross the seam line in the entrance channel, as shown in Fig. 8(c). Thus, the dynamics proceeds fully diabatically up to u₁∽8 a₀ where the electronically nonadiabatic transition probability changes sharply from unity to zero. As frequently mentioned in this paper, this rather narrow region determines whether the processes occur on the ground or the excited PES. If the system remains on the ground PES, the subsequent process is similar to the previous case of H₂([italic v] = 2) + H⁺. However, if the electronically nonadiabatic transition occurs in this narrow region, the collision system is repelled backwards by the high repulsive potential barrier on the excited PES. This dynamics is one of the typical processes in the H₂([italic v] = 6) + H⁺ collision system. In fact, the probability for nonexchange with charge transfer increases much more than that of H₂([italic v] = 2) + H⁺. The trajectories repelled backwards have another chance of electronically nonadiabatic transition. If the system is electronically de-excited to the ground state from the upper state, the dynamical process ends in the nonexchange without charge transfer.

As shown in Fig. 7(c), the three inelastic processes of exchange and nonexchange with charge transfer, and exchange without charge transfer have the same energetic threshold (∽0.1 eV). This threshold is caused by the low potential barrier (∽0.1 eV) at u₁∽9 a₀ in the entrance channel (see Fig. 1). A similar energetic threshold will be also seen in the case of H₂([italic v] = 4) + H⁺.

B.3 H₂([italic v] = 4) + H⁺. This collision process is intermediate between the two described above. In the far asymptotic region in the entrance channel, the trajectories cannot cross the seam line, as shown in Fig. 8(b), but have a chance to cross the line as the line gradually curves. If the electronically nonadiabatic transition occurs before the exchange, the trajectories are repelled backwards owing to the high potential barrier of the excited PES. On the other hand, if the system enters the deep potential well on the ground PES, the subsequent processes are similar to the previous cases. Reflecting the present intermediate collision process, exchange without charge transfer is still a major process as for the case of H₂([italic v] = 2) + H⁺ but the other three processes gradually occur.

B.4 Comparison with the quantum mechanical results

We end this section by comparing the classical results with the quantum ones. Fig. 9 shows the quantum and classical probabilities for H₂([italic v] = 2, 4 and 6) + H⁺ as a function of the collision energy. The processes of exchange and nonexchange with charge transfer, and exchange without charge transfer are plotted. As was expected, the classical method reproduces the quantum results on average. In particular, however, the classical mechanical approach breaks down at small [italic v] and near the threshold energy, and of course does not reproduce the resonance structures.

Fig. 9 Comparison of the quantum and classical probabilities for exchange and nonexchange with charge transfer, and exchange without charge transfer. Initial vibrational states of H₂ are (a) 2, (b) 4 and (c) 6.

V Concluding remarks

We have analyzed quantum and classical reaction dynamics accompanying electronically nonadiabatic transitions in a col linear H₂ + H⁺ collision system. The quantum mechanical calculations are based on the standard close-coupling method in hyperspherical coordinates and the classical mechanical ones have been carried out by using a TSH method. We have also reconstructed the DIM potential energy surface of H₃⁺ by using recent ab initio data. Special emphasis is placed on a qualitative understanding of the mechanisms of the electronically nonadiabatic transitions.

In the quantum mechanical studies, we made use of a basic idea previously proposed by Nobusada et al. in analyzing electronically adiabatic rearrangement reactions. This basic idea has been found to work well, even in the reactive system accompanying electronically nonadiabatic transitions. The electronically nonadiabatic transitions occur at some specific avoided crossings located in a rather narrow configuration space. As a result, the charge transfer probabilities have representative maximum peaks, and the reactivity of the processes depends strongly on the initial vibrational state of H₂. We have also found that the electronically nonadiabatic transition and the exchange process occur at different configuration space. The present analysis will be very helpful in the 3D H₃⁺ system when we obtain the adiabatic potential energy curves of the 3D system.

The TSH method successfully visualizes the reaction dynamics clearly by illustrating the trajectories and thus gives helpful insights into the electronically nonadiabatic transition. Similarly to the quantum results, the classical analysis also reveals that the nonadiabatic transitions occur in a rather narrow region and the reactivity depends on the initial condition. The present TSH method is not suitable to obtain quantitative results for the following reasons: (i) The electronically nonadiabatic transitions are allowed only at predefined crossing seam lines. This restriction would be less accurate in the 3D system. (ii) The electronically nonadiabatic transition probability is estimated by the L/Z formula which is invalid in the energetically threshold and classically forbidden region. These problems have been quite well resolved by improving the TSH method.^15,17,19 We are currently improving the primitive TSH method in the classically forbidden region by using the Zhu–Nakamura formula.³⁰

Acknowledgements

The authors thank Professor H. Nakamura for his useful comments. This work was partially supported by a Research Grant 10440179 and by a Grant-in-Aid for Scientific Research on Priority Area “Molecular Physical Chemistry” from The Ministry of Education, Science, Culture and Sports of Japan.

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Footnotes

† Present address: Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia.

‡ Present address: Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan.

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