Classical Approach to Associative Detachment in

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A Mixed Quantum-Classical Approach to Associative Detachment in Low-Energy H+ H Collisions -

Tasko P Grozdanov, and Ronald McCarroll J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 25 Mar 2019 Downloaded from http://pubs.acs.org on April 1, 2019

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The Journal of Physical Chemistry

A Mixed Quantum-Classical Approach to Associative Detachment in Low-Energy H−+ H Collisions Tasko P. Grozdanov∗,† and Ronald McCarroll∗,‡ Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia, and Sorbonne Université, CNRS(UMR 7614),Laboratoire de Chimie Physique-Matière et Rayonnement, 75231-Paris Cedex 05, France E-mail: [email protected]; ronald.mac [email protected]

∗

To whom correspondence should be addressed University of Belgrade ‡ Sorbonne Université †

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Abstract A theoretical approach, in which the rotational degrees of freedom are treated quantum mechanically, while translational degrees of freedom classically or semiclassically, is applied to calculate the total and final-state specific cross sections for the reaction H− + H → H2 (v, j)+e− in the range of low collision energies E = (1 − 754) meV. Calculated cross sections agree well with existing experimental and theoretical results. Results are also presented for the inverse dissociative attachment process: H2 (v, j)+e− → H− + H.

INTRODUCTION At low relative collision energies of H− and H, associative detachment (AD) is assumed 2 + to proceed via the decay of the unstable collision complex H− 2 ( Σu ). For relative collision

energies less than the electron affinity of H− , E < EA =0.754 eV, the only possible decay process is: 2 + 1 + − H− (1 S) + H(2 S) → H− 2 ( Σu ) → H2 ( Σg ; v, j) + e .

(1)

A detailed understanding of this process and of the inverse process of dissociative attachment (DA) is of importance in various physical environments. In many astrophysical plasmas these processes determine the thermal equilibrium densities of H− and H2 . For example, in the prestellar universe, it is believed that the main cooling mechanism is due to the radiative decay of the highly excited rovibration states of H2 produced in the AD process. 1 In laboratory hydrogen plasmas the main source of H− ions is the DA process. These ions can than be used for formation of neutral beams or injection into thermonuclear fusion devices. The process (1) has been extensively studied theoretically 2–7 and more recently also experimentally. 1,8,9 The earlier theoretical approaches 2–5 were based on the application of 2 + quantum scattering theory and the description of the adiabatic resonant state H− 2 ( Σu ) in

terms of a local complex potential. A more complete approach,but much more computationally demanding, is based on the nonlocal resonant scattering theory 6,7 which was proved to

2


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be in excellent agreement with measurements of total reaction cross sections 1,8 and thermal rate coefficients. 9 In the present work, for the study of reaction (1) we shall apply a modification of the simpler, classical approach, originally developed by W. Miller 10 for the related processes of Penning and associative ionizations. In our approach, as explained in more details in the following section, all angular momenta will be treated as quantized whereas the translational degrees of freedom will be treated classically and semiclassically. This kind of theoretical approach is not uncommon. For example, a related mixed quantum/classical theory of inelastic molecular collisions has been proven to be very useful. 11,12 The application of our method is expected to be successful in describing the total (summed over the final states) cross sections for reaction (1) in the range of relative collision energies from few meV up to 0.76 eV where a new channel, for production of H+H+e− is opened. However, the final-state cross sections are expected to be less accurate when specific quantum effects may become of greater importance. Atomic units will be used throughout the work except when explicitly indicated otherwise.

METHODS Interaction potentials The main characteristics of the process (1) are based on the fact that the stable adiabatic − (Born-Oppenheimer) electronic state 2 Σ+ u of H2 exists only for internuclear separations R >

Rc = 2.96. As seen from Fig.1, at the critical distance Rc this potential curve crosses the potential curve of the stable ground electronic state 1 Σ+ g of H2 and for R < Rc the collision complex H− 2 is represented by an unstable state (resonance) that can decay into fragments H2 + e− . The full quantum mechanical description of the process requires a formulation of the non-local theory based on ”projection operator method”. 6,13 However, in the local 2 + approximation, 6,13 which we also adopt here, the resonance H− 2 ( Σu ) is described by a local

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complex potential W (R) = V1 (R)−iΓ(R)/2. In this approach, the real part V1 (R) is the sum of the potential of an unperturbed discrete electronic state Vd (R) and a shift ∆(R) caused by the mixing with the continuum states. The resonance width Γ(R), which is non-zero only for R < Rc , is also determined by the mixing with the continuum states. The quantity Γ(R)/¯ h , is interpreted as total probability per unit time of the decay of the resonance by emission of an electron. In determining V1 (R) and Γ(R), which are shown in Fig.1, we have followed the work of ˇíˇzek et al. 6 The real part is given by: C

V1 (R) = Vd (R) + ∆(R),

(2)

where the potential of the unperturbed discrete state 6 of the molecular ion is parameterized as Vd (R) =

    1.74 e−2.37R − 94.4 e−22.5/R /[(R − 2.54)2 + 3.11]2 , R ≤ 10.6    −0.00845R e−0.35R − 2.25/R4 − 97/R6 ,

(3)

R ≥ 10.6,

and ∆(R) is the resonance shift, which, together with the resonance width Γ(R), is determined by the procedure explained in the work of M¨ undel et al 13 with the correct set of necessary parameters as given in the work of Horáˇcek et al. 14 The potential V1 (R) is in a very good agreement with the available ab initio data of Senekowitsch et al 15 for H− 2 , which are 16 represented as full circles in Fig.1. Also shown and labeled as V0 (R) is the 1 Σ+ g potential

of the ground electronic state of H2 . For convenience, as shown in Fig.1, we take the zero of the energy scale to correspond to dissociation limit V1 (∞) = 0. Also indicated in Fig.1 are the initial relative collision energy E,the final rovibrational level E(v, j) and the energy of the escaping electron determined from the conservation of energy in reaction (1):

E = E(v, j) +

4


(4)

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Note that with this choice of energy scale V0 (∞) = EA so that the dissociation energy of the (v, j) level is EA − E(v, j).

Total cross section The expression for the total cross section (summed over all final states) for reaction (1) is given by: AD

Q

π (E) = 2µE

Lmax (E)

X

gL (2L + 1)P (E, L),

(5)

L=0

where µ is the reduced mass in the reactant arrangement, L is the total motional angular momentum and the factor gL accounts for the nuclear spin statistics; it equals 3/4 for even L and 1/4 for odd L. For the calculation of the total reaction probabilities P (E, L) we shall use the expression valid when the relative motion of H and H− is treated classically: 10

P (E, L) =

    1 − exp[−2ξ(Rt (E, L), Rc )] L ≤ Lmax (E)    0

(6)

L > Lmax (E),

where ξ(R1 , R2 ) =

Z

R2

R1

Γ(R) dR, v(R, E, L)

(7)

Rt (E, L) is the turning point, that is the zero of the radial velocity #1/2

"

2 v(R, E, L) = (E − V1 (R, L)) µ

(8)

defined with effective potential

V1 (R, L) = V1 (R) +

L(L + 1) . 2µR2

(9)

The upper limit Lmax (E) in the summation in Eq.(5) is determined as the maximum integer

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L that satisfies : E > maxR {V1 (R, L)} .

(10)

The condition (10) simply states that in order to reach the reaction region R < Rc the classical relative particle has to overcome any existing (centrifugal or other) barrier in the effective radial potential. If this condition is fulfilled, than the probability that the collision complex (resonance) will decay by emission of an electron with formation of H2 while traversing twice the segment [Rt (E, L), Rc ] is given by expression in the first line of Eq.(6). 10 In the case of low collision energies and large widths Γ(R) the exponential term in Eq.(6) is negligible, so that we find P (E, L) = 1 for L ≤ Lmax (E), and the reaction cross section (5) is approximately equal to complex formation cross section:

QAD (E) ≈ Qcf (E) =

π [Lmax (E) + 1][2Lmax (E) + 4gLmax (E) ] 8µE

(11)

A similar expression was obtained by Sakimoto 4 (see Eq.(9) in this reference where, however, instead of 4gLmax (E) assumed to be 1, which is correct if Lmax (E) is odd, while 4gLmax (E) = 3 if Lmax (E) is even.). The last expression in Eq.(11) is simply the result of calculating the weighted sum of all open channels at given collision energy. In the general case, however, that is to say for smaller Γ(R) and/or higher collision energies the general expressions (5)-(10) have to be used.

Formation of H2 (v, j) The general quantum mechanical expression for the partial cross section for the formation of H2 (v, j) in reaction (1) is, according to the derivations of Bieniek, 17 

QAD vj (E) = =

2

 Lj l  π X  P l (E, L) gL (2L + 1)(2j + 1)    vj 2µE Ll 000

π 1 0 1 (E, j) + gj−1 jPvj (E, j − 1) + gj+1 (j + 1)Pvj (E, j + 1) + ... gj (2j + 1)Pvj 2µE

6


(12)

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The origin of the 3j-symbols in Eq.(12) is the quantum mechanical decoupling of the initial ~ into molecular rotational ~j and electronic ~l angular momenta. orbital angular momentum L The first term in the explicit expansion shown in the last line of Eq.(12) corresponds to l = 0 (and the only possible value L = j). The second term corresponds to l = 1 , L = j − 1 and the third corresponds to l = 1, L = J + 1 (the term with l = 1, L = J is missing because the corresponding 3j-symbol is zero). The remaining sum “+ ... “ contains groups l of terms corresponding to l = 2, 3, .... Quantities Pvj (E, L) contain the squares of certain

transition matrix elements between the radial wavefunctions of initial translational and final vibrational motion of nuclei. 17 In the case of reaction (1), however, due to symmetry restrictions (indistinguishability of the two protons and conservation of parity) the orbital angular momentum quantum numbers of the escaping electron l are restricted to odd values. Moreover, as proven in previous theoretical works (see, for example Ref. 18 ) only the p-electrons (l = 1) make appreciable contribution to the decay of the complex, so that to a good approximation, we find from Eq.(12): QAD vj (E) =

h i π 1 1 (E, j − 1) + (j + 1)Pvj (E, j + 1) . gj+1 jPvj 2µE

(13)

In the original work of Miller 10 only the escape of an s-electron (l = 0) was considered (only the first term in expansion (12) was retained) and a (semi)classical expression for the 0 probability Pvj (E, j) was obtained. The generalization of that result to our case of l = 1 is:

1 Pvj (E, j

∂E(v, j) P (R, E, j ± 1) ± 1) = ∂ ∂v | ∂R [V1 (R) − V0 (R)]| R=R

,

(14)

vj (E)

where in the semiclassical approximation the prefactor is π ∂E(v, j) = R , ∂v µ dR{2µ[E(v, j) − V0 (R, j)]}−1/2

(15)

and the classical probability density that the transition occurs at an internuclear distance R

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is P (R, E, j ± 1) =

2Γ(R) cosh[ξ(Rt (E, j ± 1), R)]e−ξ(Rt (E,j±1),Rc ) . v(R, E, j ± 1)

(16)

with ξ(R1 , R2 ) is defined in Eq.(7). The internuclear distance Rvj (E) at which the right-hand side of Eq.(14) is evaluated is determined as a solution of the equation:

= E − E(v, j) = V1 (Rvj (E)) − V0 (Rvj (E)),

(17)

which represents the classical condition for the vertical (”Franck-Condon”) transition in the process of emission of an electron with energy . The total reaction cross section (5) can now be represented in an alternative form: jmax (E) vmax (E,j)

QAD (E) =

X

X

j=0

v=0

QAD vj (E)

(18)

which, after the substitution of (13), yields π QAD (E) = 2µE



Lmax (E)

X

gL L

X

1 PvL−1 (E, L) + (L + 1)

X

1 PvL+1 (E, L) . (19)

v=0

v=0

L=0



vm (E,L+1)

vm (E,L−1)

In Eq.(18), jmax (E) ≤ Lmax (E) + 1 is the highest energetically accessible rotational quantum number and vm (E, j) is the highest energetically accessible vibrational quantum number for given E and j. The appearance of two terms in Eq.(19) is a direct consequence of the quantum mechanical addition of angular momenta (for a p-electron with l = 1 and molecular rotational angular momenta j = L ± 1). Generalizing again the result of Miller 10 valid for the l = 0 case, we find that vm (E,L−1)

X v=0

vm (E,L+1) 1 (E, L) ≈ PvL−1

X

1 (E, L) ≈ P (E, L), PvL+1

(20)

v=0

where P (E, L) is the total reaction probability defined in Eq.(6). Substitution in Eq.(19) then leads to Eq.(5). Approximate equalities appear in Eq.(20) because we deal with summations 8


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over discrete vibrational quantum numbers. The exact equalities would appear if the sums were replaced with integrals over classical vibrational action variables. 10 We also note that the cross section for the process of dissociative attachment (DA) (the inverse of reaction(1)) is given , according to micro reversibility principle, by:

QDA vj () =

=

µE QAD (E) (2j + 1)gj+1 vj

i h π 1 1 (E, j + 1) , (E, j − 1) + (j + 1)Pvj jPvj 2(2j + 1)

(21)

(22)

where is the impact energy of the electron and E is the relative kinetic energy of the dissociating fragments H and H− .

RESULTS AND DISCUSSION Total cross section Inspection of Fig.2 shows that the effective potentials appearing in Eq.(10) are characterized by inner and outer potential barriers. The situation is similar to some other reactions, such as for example S+H2 considered in our recent work. 19 The critical classical angular momentum when both barriers are of equal heights Vc = 15.7 meV occurs for Lc = 22.3. This means that for collision energies E < Vc , Lmax (E) is determined by centrifugal barriers at large internuclear distances and for E > Vc inner barriers define Lmax (E). Figure 3 shows the calculated dependence Lmax (E) from Eq.(10) and the two above mentioned energy regions ate clearly distinguished. The above discussion has a consequence on the change of slope in the energy dependence of the total reaction cross section when calculated by using approximate formula (11) as shown in Fig.4. Also shown are the results of quantum mechanical nonlocal scattering calculations, 6,8 some earlier quantum calculations using different potentials 3 and experimental results. 8 The agreement between the present results and quantum non-local theory is very 9



good while the experimental points lie somewhat higher. Our results exhibit the characteristic zig-zag structures due to neglect of tunneling through the potential barriers shown in Fig.2. Results of Bieniek 3 depart significantly from other results at low energies, indicating their use of an incorrect long-range part of the resonance potential. If we include in the calculations the exponential factor in Eq.(6), the results obtained practically coincide with those obtained using Eq.(11). The small differences occur only in the region of the highest energies considered, but they would scarcely be visible on the scale of Fig.4 and are therefore not shown.

Partial cross sections For the calculation of partial cross sections using Eqs.(13)-(17) it is crucial to have a knowledge of the classical probability density (16). This function for the case of head-on collisions (L = 0) at impact energy E = 100 meV is shown in Fig.5. It has an integrable singularity at the turning point R = Rt (E, 0) and goes to zero at R = Rc = 2.96. As seen from Eq.(13) or Eq.(19) the partial wave L = 0 contributes only to population of j = 1 levels with probabil1 (E, 0). The positions Rv1 from Eq.(17) are indicated as initial points at the R-axis ities Pv1

of the vertical lines in Fig.5 and circles at the top of these lines indicate the required values P (Rv1 , E, 0) which appear in Eq.(14). The required values of the rovibrational levels E(v, j) are taken from Refs. 5,20 1 The relative populations of various vibrational levels Pv1 (E, 0)/P (E, 0) (with P (E, 0)

calculated from Eq.(20)) as functions of energy of the detached electrons are shown in Fig.6. They agree well with corresponding results of Gauyacq 21 obtained using a quantum mechanical effective-range model. Figure 6 can also be interpreted as spectrum of emitted electrons in head-on collision. Clearly, a total spectrum of all emitted electron will look much more complicated if all accessible final rovibrational states of H2 (v, j) at given collision energy are included. This is demonstrated in Fig.7 for the case of collision energy E = 129 meV. Although a wider range 10


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of electronic energies is energetically accessible, the spectrum is dominated by low energy electrons indicating that the population of the highly excited rovibrational states of H2 is predominant, which is in accord with previous findings. 6 For the case of E = 129 meV, there are detailed quantum mechanical calculations of Dalgarno and Bieniek 2,3 of partial cross sections for populating various final rovibrational states. Comparisons for four characteristic cases of final j = 0, 5, 10 and 20 states are shown in Fig.8. While in the case of j = 0 agreement is satisfactory, for j = 5 our results are lower. At j = 10 agreement is again good while at j = 20 our results are higher.

Dissociative attachment The differences observed in Fig.8 are probably mostly due to the fact that different real and ˇíˇzek imaginary potentials of the resonant state were used in two calculations. In the work of C et al 6 on the AD process, both, a non-local and local quantum theory based on the same potentials as ours were employed. Generally good agreement between the two approaches was found but no detailed information on the partial cross sections was provided. However, in the work of Horáˇcek et al 14 the same non-local quantum theory was applied in the study of the reverse DA process:

− − 2 + ∗ − 2 1 H2 (1 Σ+ g ; v, j) + e → H2 ( Σu ) → H ( S) + H( S).

(23)

Although in that work an additional approximation was adopted by neglecting the angular momentum recoil (that is, L = j was assumed), nevertheless it makes sense to compare their results for DA with ours obtained using Eq.(22). To make these comparisons we had to extend the calculations typically up to values of E = 1.7 eV (here E = + E(v, j) is the kinetic energy of dissociated fragments), assuming that the DA process is still dominated by 2 + the H− 2 ( Σu ) resonance.

We first discuss the transitions from spherically symmetric initial states with j = 0. 11



According to Eq.(22) they correspond in our approach to j = 0 → L = 1 transitions, while as mentioned above, in the approach of Horáˇcek et al 14 they correspond to j = 0 → L = 0. Figure 9 shows DA cross sections for initial vibrational v = 0 and v = 1 states. It can be seen that our results are respectively larger by factors of 7 and 5. This is not surprising, since it is well known that in general, local quantum theories overestimates DA cross sections for low-lying rovibrational states. 13 In addition, in Fig. 9(a) a narrow singularity shows up in our result. This is the consequence of the breakdown of the classical approach when the distance of vertical transition R0,0 (E) approaches the turning point Rt (E, 1) so that the value of density P (R0,0 (E), E, 1) in Eq.(14) becomes infinite (see the analogous singularity in Fig.5 for P (R, E, 0)). In quantum mechanical calculations one would expect instead of this singularity to see a maximum followed by exponential decay in classically forbidden region. In the result of Horáˇcek et al 14 in Fig.9(a) this structure is absent, possibly because they use non local quantum theory and also consider j = 0 → L = 0 transition instead of j = 0 → L = 1. With the increase of the initial vibrational quantum number the agreement between the two theories improves, as demonstrated in Fig. 10(a) for the case of v = 7. At still higher vibrational excitations v = 9, 10 and 11, as shown in Fig. 10, our results are somewhat higher but similar to those of Horáˇcek et al. 14 For initial states that are both vibrationally and rotationally excited the agreement between the two approaches is very good. This is shown in Fig.11 for the case of j = 12 and v = 6. According to Eq.(22) our result is composed from two contributions originated from j = 12 → L = 11 and j = 12 → L = 13 transitions, which when summed up agree well with the j = 12 → L = 12 approximation of the non local theory of Horáˇcek et al. 14 Finally, we consider rotationally highly excited initial states. Figure 12(a) shows cross sections for the case of j = 20, v = 0. Our result is again the sum of two components. The first one, corresponding to j = 20 → L = 19 transition, almost coincide with the result of Horáˇcek et al 14 for j = 20 → L = 20 transition. Our additional component corresponding

12


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to j = 20 → L = 21 transition is classically allowed in the finite range of electron impact energies limited by the two singularities shown in Fig.12(a). The transition is classically allowed if the point of vertical transition R0,20 (E) is larger than the turning point Rt (E, 21). Both points are (independent) functions of energy and in the case under consideration there is a finite range of energy where this condition is fulfilled. In the case of j = 25, v = 0, shown in Fig. 12(b) the only classically allowed transition corresponds to j = 25 → L = 24, but at energies close to the threshold, incidentally, R0,25 (E) approaches Rt (E, 24), hence our result shows a singular behavior in that region. The result of Horáˇcek et al 14 for j = 25 → L = 25 transition shows the expected behavior in a quantum mechanical treatment: the maximum in the region of classical singularity and exponential decay in the classically forbidden region.

CONCLUSIONS We have shown that a combination of the quantum treatment of rotational degrees of freedom and classical and semiclassical treatment of translational degrees of freedom leads to a relatively simple description of the associative detachment process. In calculations of the total reaction cross sections, the classical approximation for reaction probability for given (quantized) orbital angular momenta has been employed taking proper account of quantum mechanical symmetries (parity, permutation of identical particles). Calculated total cross sections exhibit satisfactory agreement with more sophisticated quantum treatments and experimental results. It is the classical component in our approach that brings new and more transparent physical insight into reaction mechanisms. This is particularly obvious, for example, in the explanation of the experimentally confirmed (in Fig.4) change of the slope in the energy dependence of the total reaction cross section. Our classical-quantum analysis (in Fig.3) relates it to the appearance of the inner potential barriers (in Fig.2).

13



In calculations of final state distributions, the quantum mechanical addition of angular momenta of the emitted electron and the molecule is combined with the classical expression for the transition probability and semiclassical treatment of molecular vibrational states. Additional new physical insight is provided by our classical assumption in Eq.(17) that electronic transitions occur ”vertically” in accord with Franc-Condon principle, which reproduces reasonably well partial cross sections (in Fig.8) for dominantly populated excited states, obtained in full quantum treatments. Results presented for the inverse process of dissociative attachment show reasonable agreement with existing quantum mechanical treatments, although certain limitations represented by localized singularities persist due to classical treatment.

Acknowledgement T. P. G. acknowledges the support by the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project No. 171020.

References ˇíˇzek, M.; Glover, S. C. O.; Miller, K. A.; Urbain, X.; (1) Kreckel, H.; Bruhns, H.; C Savin, D. W. Experimental rezults for H2 formation from H− and H and implications for first star formation. Science 2010, 329, 69–71. (2) Bieniek, R. J.; Dalgarno, A. Associative detachment in collisions of H and H− . ApJ 1979, 228, 635–639. (3) Bieniek, R. J. A source of errors in cross sections of curve-crossing processes. J. Phys. B: At. Mol. Phys. 1980, 13, 4405–4416. (4) Sakimoto, K. Ion-molecule reactions at extremely low enegies: H− + H→ H2 +e. Chem. Phys. Lett. 1989, 164, 294–298.

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(5) Launay, J. M.; Dournef, M. L.; Zeippen, C. J. The H+H− * ) H2 (v, j)+e− reaction: a consistent description of the associative detachment and dissociative attachment processes using the resonant scattering theory. Astron. Astrophys. 1991, 252, 842–852. ˇíˇzek, M.; Horáˇcek, J.; Domcke, W. Nuclear dynamics of the H− (6) C 2 collision complex beyond the local approximation: associative detachment and dissociative attachment to rotationally and vibrationally excited molecules. J. Phys. B: At. Mol. Opt. Phys. 1998, 31, 2571–2583. ˇíˇzek, M.; Kreckel, H.; Urbain, X.; Savin, D. W. (7) Miller, K. A.; Bruhns, H.; Elíaˇsek, J.; C Associative detachment of H− +H→H2 +e− . Phys. Rev. A 2011, 84, 052709. ˇíˇzek, M.; Elíaˇsek, J.; Cabrera-Trullilo, R.; Kreckel, H.; (8) Miller, K. A.; Bruhns, H.; C O’Connor, A. P.; Urbain, X.; Savin, D. W. Isotope effect for associative detachment: H(D)− +H(D)→H2 (D2 )+e− . Phys. Rev. A 2012, 86, 032714. ˇ Rouˇcka,; Zimak, I.; Plaˇsil, R.; Glosík, J. Ion trap studies of (9) Gerlich, D.; Jusko, P.; S. H− + H→ H2 +e− between 10 and 135 K. ApJ 2012, 749:22, 6pp. (10) Miller, W. H. Theory of Penning ionization. I. Atoms. J. Chem. Phys. 1970, 52, 3563– 3572. (11) Babikov, D.; Semenov, A. Recent advances in development and applications of the mixed quantum/classical theory for inelastic scattering. J. Phys. Chem. A 2016, 120, 319–331. (12) Semenov, A.; Babikov, D. Mixed quantum/classical theory: application to rotational excitations in H2 +H2 . J. Phys. Chem. A 2016, 120, 3861–3865. (13) M¨ undel, C.; Berman, M.; Domcke, W. Nuclear dynamics in resonant electron-molecule scattering beyond the local approximation: Vibrational excitation and dissociative attachment in H2 and D2 . Phys. Rev. A 1985, 32, 181–193. 15



ˇíˇzek, K. H.; Kolorenˇc, P.; Domcke, W. Dissociative electron attach(14) Horáˇcek, J.; M. C ment and vibrational excitation of H2 by low-energy electrons: Calculations based on an improved nonlocal resonance model. Phys. Rev. A 2004, 70, 052712. (15) Senekowitsch, J.; Rosmus, P.; Domcke, W.; Werner, H.-J. An accurate potential energy function of the H− 2 ion at large internuclear distances. Chem. Phys. Lett. 1984, 111, 211–213. 3 + 1 (16) Kolos, W.; Wolniewicz, L. Potential-energy curves for the X 1 Σ+ g , b Σu , and C Πu

states of the hydrogen molecule. J. Chem. Phys. 1968, 43, 2429–2441. (17) Bieniek, R. J. Complex potential and electron spectrnm in atomic collisions involving fast electronic transitions: Penning and associative ionization. Phys. Rev. A 1978, 18, 392–413. (18) Berman, M.; M¨ undel, C.; Domcke, W. Projection-operator calculations for molecular shape resonances: The 2 Σ+ u resonance in electron-hydrogen scattering. Phys. Rev. A 1985, 31, 641–651. (19) Grozdanov, T. P.; McCarroll, R. An empirical dynamical barrier for statistical theory of low-energy reactive S(1 D) + HD(j = 0), H2 (j = 0) collisions. J. Phys. Chem. A 2017, 121, 40–44. (20) LeRoy, R. J.; Schwartz, C. Nonadiabatic Eigenvalues and Adiabatic Matrix Elements for all Isotopes of Diatomic Hydrogen; University of Waterloo: Waterloo, 1987. (21) Gauyacq, H. P. Dissociative attachment in e− -H2 collisions. J. Phys. B: At. Mol. Phys. 1985, 18, 1859–1872.

16


Page 16 of 29

Page 17 of 29

4

Γ 2

H + H E

V ,Γ[e V ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

ε H V -2

E ( v ,j)

V

-

+ H

1

0

-4 0

2

4

6

R

8

1 0

[a .u .]

Figure 1: Potential function V0 (R) of H2 (1 Σ+ g ) in its ground electronic state (dashed curve) and real part V1 (R) and width Γ(R) of the local complex potential W (R) = V1 (R) − iΓ(R)/2 2 + 15 of the H− for H− 2 ( Σu ) (full curves). Ab initio data of Senekowitsch et al 2 are plotted as full circles. Vertical dashed line indicates the position of the crossing point Rc = 2.96 of the two potentials. Schematically indicated also are the initial relative collision energy E, final rovibrational level E(v, j) of H2 and energy of the detached electron

17



2 4

0 .1

(R ,L ) [e V ]

0 .0 1 5 7 0 .0

1

2 2

V

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 29

-0 .1 2 0 1 5

L = 0

-0 .2 1

2

3

4

5

6

7

8

9 1 0

2 0

R [a .u .]

Figure 2: Effective potential V1 (R) + L(L+1) for L = 0, 15, 20, 21, 22, 23, 24. The vertical 2µR2 line at Rc = 2.96 gives the position where potential V1 (R) crosses potential V0 (R). For collision energies E < Vc = 0.0157 eV Lmax (E) is determined by centrifugal barriers at large internuclear distances and for E > Vc inner barriers define Lmax (E).

18


Page 19 of 29

3 0

2 5

1 5

m a x

(E )

2 0

L

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 0

5

0 1 E -6

1 E -5

1 E -4

1 E -3

0 .0 1

0 .1

1

E [e V ]

Figure 3: Maximum relative orbital angular momentum quantum number Lmax as function of relative collision energy E. Vertical dotted line indicates critical energy E = Vc = 0.0157 eV.

19



1 0 0 0 -

H 2

+ e

1 0 0

A D

[1 0

-1 6

c m

2

]

H + H

E x p P re T h e T h e

Q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 29

1 0

1 E -3

.[8 s e o r o r

] n t .[8 ] .[2 ]

0 .0 1

0 .1

E [e V ]

Figure 4: Total cross section for associative detachment as function of relative collision energy. Full line- present result from Eq.(11); dashed line - result of quantum nonlocal scattering theory; 8 stars - quantum calculations using different potential; 3 full circles - experimental results. 8 Vertical dotted line E = Vc = 0.0157 eV indicates critical energy where the change of the slope of the cross section dependence on E occurs.

20


Page 21 of 29

7

E = 1 0 0 m e V L = 0 , j= 1

8 6

1 .2

P (R ,E ,L ) [a .u .]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

v = 9 0 .6

4

3 R t

0 .0 1 .2

0

1

2 R 1 .8

2 .4

R

7 ,1

R c 3 .0

[a .u .]

Figure 5: Classical probability density P (R, E, L) as defined in Eq.(16) for E = 100 meV and L = 0. The full circles indiate values of P (Rv,1 , E, 0) for v = 0, 1...9 with Rv,1 determined from Eq.(17).

21



0 .3 0

7

E = 1 0 0 m e V L = 0 , j= 1

8 0 .2 5

6

P v 1 (E ,0 )/P (E ,0 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 29

P re s e n t R e f.1 7

0 .2 0

5

0 .1 5

0 .1 0

v = 9 4

0 .0 5

3 2

1

0

0 .0 0 0

1

2

3

4

ε= E - E ( v , 1 ) [ e V ] Figure 6: Relative populations Pv1 (E, 0)/P (E, 0) of H2 (v, 1) states in head-on (L = 0) collisions at E = 100 meV as function of the energy of the detached electron . Full circles are present results and full squares are results of quantum mechanical effective-range model of Gauyacq. 21

22


Page 23 of 29

1 .5

E = 1 2 9 m e V

-1 6

c m

2

]

1 .0

Q v j [1 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 .5

0 .0 0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

3 .5

4 .0

ε= E - E ( v , j ) [ e V ] Figure 7: Total spectrum of emitted electrons at collision energy E = 129 meV.

23



(a )

0 .1

5 1

(b )

j= 5 9

j= 2 0 5

0 .0 1

v = 9

0

c m -1 6

c m 1 E -4

0 .0 1

0

Q v j [1 0

-1 6

v = 8

]

j= 0

2

1 E -3

2

]

0 .1

Q v j [1 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 29

0 1 E -5

j= 1 0 1 E -3

E = 1 2 9 m e V

1 E -6

v = 0

E = 1 2 9 m e V v = 0 1 E -4

1 E -7 0

1

2

3

4

0

ε= E - E ( v , j ) [ e V ]

1

2

3

ε= E - E ( v , j ) [ e V ]

Figure 8: Partial cross sections for production of H2 (v, j) at collision energy E = 129 meV as functions of energy of detached electrons. Full circles are present results and full squares are results of quantum mechanical calculations of Dalgarno and Bieniek. 2 (a) Cases j = 0, v = 0, 1...9 and j = 5, v = 0, 1...9 are shown. (b) Cases j = 10, v = 0, 1...8 and j = 20, v = 0, 1...5 are shown.

24


Page 25 of 29

0 .0 0 5

(a )

(b ) j= 0 , v = 0

0 .0 0 0 3

j= 0 , v = 1

]

]

0 .0 0 4

[1 0

-1 6

c m

2

0 .0 0 3

Q

x 6 .9

0 .0 0 2

v j

D A

v j

D A

[1 0

-1 6

c m

2

0 .0 0 0 2

Q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


x 5

0 .0 0 0 1

0 .0 0 1

0 .0 0 0

0 .0 0 0 0 3 .5

4 .0

4 .5

5 .0

3 .0

ε[e V ]

3 .5

4 .0

4 .5

5 .0

ε[e V ]

Figure 9: Initial state specific dissociative attachment cross sections as functions of electron impact energy. Full lines are present results and dashed lines are results of Horáˇcek et al. 14 Initial quantum numbers: (a) j = 0, v = 0 and (b) j = 0, v = 1.

25



(a )

j= 0

(b )

8

j= 0

6

v = 9 6

] 2

c m v j

D A

[1 0

[1 0

v = 7 4

v = 1 1

Q

v j

D A

4

-1 6

-1 6

c m

2

]

v = 1 0

Q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 29

2

2

0

0 0 .0

0 .5

1 .0

1 .5

2 .0

0 .0

ε[e V ]

0 .5

1 .0

1 .5

ε[e V ]

Figure 10: Same as Fig. 9 but for: (a) j = 0, v = 7 and v = 9 and (b) j = 0, v = 10 and v = 11.

26


Page 27 of 29

1 0

v = 6 , j= 1 2 8

4

v j

D A

[1 0

-1 6

c m

2

]

6

Q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

0 0 .0

0 .5

1 .0

1 .5

2 .0

ε[e V ] Figure 11: Same as Fig. 9 but for j = 12, v = 6. Dotted lines are the two contributions to present result, according to Eq.(22) .

27



4

0 .3 0

(a ) 0 .2 5

(b ) v = 0 , j= 2 0 3

0 .2 0

] 2

c m

c m

2

]

v = 0 , j= 2 5

[1 0

-1 6

2

Q

0 .1 0

v j

D A

v j

D A

[1 0

-1 6

0 .1 5

Q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 29

1 0 .0 5

0

0 .0 0 1 .0

1 .5

2 .0

2 .5

3 .0

0 .5

ε[e V ]

1 .0

1 .5

2 .0

ε[e V ]

Figure 12: Same as Fig. 9 but for: (a) j = 20, v = 0, dotted lines are the two contributions to present result, according to Eq.(22) and (b) j = 25, v = 0)

28


Page 29 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


e

-

H

e

-

+

H H

H + H

-

e H

H

H

2

29

H

H

-

+

2


+ e

-

Classical Approach to Associative Detachment in

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