arXiv:2106.13190v1 [physics.chem-ph] 24 Jun 2021
Vibrational effects in the quantum dynamics of the H + D+
2 charge
transfer reaction
O. Ronceroa , V. Andrianarijaonab,c , A. Aguadod and C. Sanz-Sanzd
a
Instituto de Fı́sica Fundamental, IFF-CSIC, c/ Serrano 123, 28006 Madrid, Spain, b
Department of Physics , Pacific Union College, Angwin, California, United States, c Current
address: Department of Physics and Engineering, Southern Adventist University, Collegedale,
Tennessee, United States d Unidad Asociada UAM-CSIC, Departamento de Quı́mica Fı́sica
Aplicada, Facultad de Ciencias M-14, Universidad Autónoma de Madrid, 28049, Madrid,
Spain
ARTICLE HISTORY
Compiled June 25, 2021
This is an accepted manuscript of an article that will be published by
Taylor & Francis in “Molecular Physics” on 2021, will be avilable on line:
http://wwww.tandfonline.com/10.1080/00268976.2021.1948125
ABSTRACT
The H + D+
2 (v=0,1 and 2) charge transfer reaction is studied using an accurate wave
packet method, using recently proposed coupled diabatic potential energy surfaces.
The state-to-state cross section is obtained for three different channels: non-reactive
charge transfer, reactive charge transfer, and exchange reaction. The three processes
proceed via the electronic transition from the first excited to the ground electronic
state. The cross section for the three processes increases with the initial vibrational
excitation. The non-reactive charge transfer process is the dominant channel, whose
branching ratio increases with collision energy, and it compares well with experimental measurements at collision energies around 0.5 eV. For lower energies the
experimental cross section is considerably higher, suggesting that it corresponds to
higher vibrational excitation of D+
2 (v) reactants. Further experimental studies of
this reaction and isotopic variants are needed, where conditions are controlled to
obtain a better analysis of the vibrational effects of the D+
2 reagents.
KEYWORDS
charge transfer, quantum dynamics, non-adiabatic dynamics
1. Introduction
H2 is the most abundant molecule in the interstellar medium. The gas phase routes
to form molecular hydrogen present very slow rate constants, and its formation in
local galaxies is attributed to reactions on cosmic grains and ices[1, 2]. However, in
environments where grains and ice do not exist, like in the Early Universe, one key
process is the formation of H2 in gas phase. One of such processes is the charge transfer
CONTACT O. Roncero. Email: octavio.roncero@csic.es
reaction
+
H + H+
2 → H2 + H .
(1)
This reaction has been studied experimentally at rather high energies, > 1 keV, by
Karpas et al[3] and by McCartney et al.[4]. Andrianarijaona et al[5, 6] measured the H
+
+ D+
2 → H +D2 charge transfer cross section in a broad energy range, from 0.1eV/u10 keV/u, thus including energies more relevant for the astrophysical environment. At
this point it is worth mentioning that the D+
2 reactant is excited vibrationally since
it is generated by photoionization or electronic impact. The main goal of this work
is to compare the simulated cross sections for each D+
2 (v) vibrational state with the
experimental measurements. In this line, the measurements made in the H + D+
2
reaction [5, 6] can then be considered as benchmark for the comparison with the
theoretical simulations, which conclusions can be generalized to other isotopes and to
similar reactions for other cations.
From the theoretical point of view, reaction (1) has been studied by several authors
[7–13], using different and increasingly more accurate coupled diabatic potential energy surfaces (PESs)[14–18]. Last et al. [7] used a quantum method based on negative
imaginary potential combined with a variational quantum method in a L2 basis set,
with the helicity decoupling approximation, and using approximated PESs. Krstic[9]
used a close coupling method based on the infinite order sudden approximation (IOSA)
with Delves hyperspherical coordinates. Gosh et al. [12] and Sanz-Sanz et al. [13] used
accurate quantum wave packet methods, in hyperspherical and Jacobi coordinates
respectively, and used different coupled diabatic PESs based on accurate ab initio calculations. All these calculations showed an important enhancement on a particular
vibrational state of H2 due to a quasi-degeneracy of H+
2 (v=0) and H2 (v’=4). This
effect depends strongly on the long-range behavior of the PESs, very accurately described in the work of Aguado et al[19], and already used by Sanz-Sanz et al. [13]. In
this latter work, the charge transfer reaction was studied for several isotopic variants.
+
Among them the H+ D+
2 (v=0) → H + D2 charge transfer reaction was studied and
compared with the experimental values [5, 6], showing good agreement for collision energies around 0.5 eV, but some differences appear at lower energies. These differences
were attributed to vibrational excitation of the D+
2 , not taken into account by SanzSanz et al. [13]. The goal of this work is to analyze the effect of vibrational excitation
of D+
2 , which is expected to present important discrepancies due to the variation of
the energy difference between the initial D+
2 (v) and final D2 (v’) vibrational levels.
The manuscript is distributed as follows. A brief description of the theoretical
method is presented in section 2. In section 3, the quantum results are described
and compared with the experimental results. Finally, in section 4 are outlined some
conclusions.
2. Theoretical method
In this work we use the coupled diabatic PESs developed by Aguado et al. [19] based
on a Diatomics-in-Molecules (DIM) 3×3 matrix description[20, 21], corrected with
diagonal and non-diagonal three body-terms [22] to fit the energies obtained in MultiReference Configuration interaction (MRCI) calculations with complete basis set extrapolation (CBS). Dealing with a system of two electrons, MRCI calculations give
full configuration interactions (FCI) results within the space spanned by the orbital
2
set. To obtain near FCI results cfor complete basis set, a complete basis set extrapolation method was performed. These PESs include the long-range interaction very
accurately. The dominant terms for reactants H + H+
2 channel are the charge-induced
dipole and charge-induced quadrupole dispersion interactions, varying as R−4 and R−6
respectively. For the products H+ + H2 channel, the main long-range terms are the
charge-quadrupole and the charge-induced dipole dispersion energies, which vary as
R−3 and R−4 respectively. The DIM diabatic representation is non diagonal for the
description of any H+
2 + H fragment. Therefore, a transformation to a new diabatic
representation should be done, in which the PESs are diagonal in the reactants channel
while they are non-diagonal in the two product rearrangement channels, as described
by Sanz-Sanz et al. [13]. The features of the potential in the reactant channel, as a
function of Jacobi r variable, are shown in Fig. 1. At long distances, the potentials of
+
D2 and D+
2 cross at r = rc =1.323Å, as shown in Fig. 1.a. The amplitude of D2 (v)
vibrational state is non-zero at this distance, and, in this outer classical turning point
region, the D+
2 (v) levels have a matching overlap with different D2 (v’) vibrational
states. As the two reactants approach each other, the degeneracy between the two
first adiabatic states dissappears: the ground state gets stabilized forming the very
+
+
stable HD+
2 system, which correlates to H +D2 and D +HD adiabatic asymptotes,
while the excited adiabatic state becomes repulsive (see Fig. 1.b).
The reactive and charge transfer collisions have been studied with a quantum wave
packet method using the MADWAVE3 code [23, 24], using the parameters listed in
Table 1 of Ref. [13]. The wave packet is propagated for each total angular momentum J, using a modified Chebyshev propagator[25–29]. The wave packet is represented in reactant Jacobi coordinates and is transformed to product Jacobi coordinates [30] at each iteration to extract the individual state-to-state reaction probabiliJ
ties, PαvjΩ→α
′ v ′ j ′ Ω′ (E), where α denotes the rearrangement channel and the electronic
state, v, j correspond to the vibrational and rotational state of the diatomic fragment,
and Ω is the projection of the total angular momentum in the z-axis of the body-fixed
frame. The integral state-to-state reactive and inelastic cross sections are calculated
using the partial wave expansion as
σαvj→α′ v′ j ′ (E) = qe
−2
πkαvj
(2j + 1)
X
J
(2J + 1)PαvjΩ→α
′ v ′ j ′ Ω′ (E),
JΩΩ′
(2)
where qe = 1/4 is the electronic partition function - note that this factor was not
included in the previous study [13] -. There are four electronic states correlating to
2 +
H(2 S)+D+
2 ( Σg ) asymptote, one singlet and three triplet states. The three triplet
states are not connected to the H+ + H2 (X1 Σ+
g ) channel, and therefore cannot undergo
J
charge transfer reactions. In this study, P (E) are calculated with the MADWAVE3
code for all values with J < 15 and for J = 20, 30, 40, 50, 60, 70 and 80. The rest
of reaction probabilities for all the intermediate J ′ s are obtained by a linear interpolation based on the J-shifting approximation, as described before [31, 32], to save
computational effort. In these calculations, the maximum projection, Ω, considered in
the dynamic calculations is 23. In a previous study[13], the role of the trunction of
Ωmax was checked, giving excellent results for the NCRT channel, and an accuracy
better than 2-3 % for the other two channels.
3
4
+
D2
D2
D2
+
D2
D2
+
D2
Energy(eV)
3
2
+
1
0
+
D2 (v=0)
D2 (v’=6)
0.5
1
1.5
+
D2 (v=1)
D2 (v’=6)
2
2.5
0.5
1
r(Å)
1.5
D2 (v=2)
D2 (v’=7)
2
2.5
0.5
1
r(Å)
1.5
2
2.5
r(Å)
(a) Potential of D2 and D+
2 at very long distance of H atom, showing the vibrational levels on each. The
energies of D+
2 (v) are 2.056, 2.251 and 2.439 eV, for v=0, 1 and 2, respectively. The energies of D2 (v’) are
1.902, 2.202, 2.487 and 2.758 for v’= 5,6, 7 and 8, respectively.
0.01
1
0
2 A’
Energy (eV)
−0.01
−0.02
2.2
−0.03
2
1
−0.04
1 A’
2
Diabatic coupling (eV)
2.4
4
6
8
R (Å)
10
12
14
−0.05
(b) Adiabatic potential energies of the first 2 singlet
electronic states as a function of R, for r = rc and at
a T-shaped configuration. The diabatic coupling between the two diabatic energies are also shown.
+ + D charge transfer process, where r is
Figure 1. Features of the 3x3 PES’s describing the H+D+
2
2 → H
the D2 internuclear distance and R is the distance between H and the center-of-mass of D2 .
4
In what follows, three processes are distinguished
H + D+
2 (v, j = 0, J) −→
H+ + D2 (v ′ , j ′ )
non − reactive charge transfer(NRCT)
−→ HD+ (v ′ , j ′ ) + D
exchange reaction(ER)
−→ HD(v ′ , j ′ ) + D+
reactive charge transfer(RCT).
Charged diatomic products correspond to the first excited electronic singlet adiabatic
state, 21 A′ , while the product with a charged atomic fragment corresponds to the
ground adiabatic electronic singlet state, 11 A′ .
2
The H(2 S)+ H+
2 ( Σg ) entrance channel also correlates with the lowest triplet state
+ 3 ′
of H3 ( A ). This triplet state presents a shallow well at collinear geometry, and has
access to the exchange products, as described before[19, 33]. Therefore, the triplet
state contributes to the exchange reaction, ER, but does not contribute to the other
two charge transfer processes, NRCT and RCT. In the present calculations dynamical calculations are all performed for the singlet states. Therefore, the cross section
for the RCT channel presented below is not complete, since they should include the
contributions arising from the triplet states.
3. Results
The vibrationally resolved reaction probabilities for J=0 are shown in Fig. 2 for
H+D+
2 (v) collisions, for v=0,1 and 2, in the left, middle and right panels, respectively. The probabilities for the individual final vibrational states, v’, are included for
the three different processes, NRCT (top panels), ER (middle panels) and RCT (bottom panels). The reaction in the excited adiabatic electronic state has an extremely
high barrier. However, as the two fragments get closer, the electronic couplings bring
a large portion of the wave packet to the ground adiabatic state correlating to the
two charge transfer channels, NRCT and RCT. The NRCT reaction probabilities are
very close to those of RCT at low energies, except when the D2 (v’) level is close to
the D+
2 (v) state considered, D2 (v’=6) for v=0 and 1 and D2 (v’=7) for v=2. According
to Fig. 1.a, the final population of D2 (v’) with an energy slightly above than that of
the initial D+
2 (v) is always enhanced. This is due to the interaction with the H atom,
which stabilizes the ground electronic state, connected to H+ + D2 asymptote, while
the excited state becomes repulsive (see Fig. 1.b).
For lower energies, the amplitude transfered to the ground electronic state enters the
deep well of HD+
2 , forming long lived resonances. In this well, the energy is transfered
among all possible modes and the reaction becomes nearly statistical, as reported for
the reaction dynamics in the ground electronic state [29, 34, 35]. This explains why
NRCT and RCT show similar populations, except for near-resonant v’ levels. This also
allows the transition back to the excited electronic state and explains the appearance
of the ER probabilities in the middle panels of Fig. 2, that cannot reach directly the
excited adiabatic electronic state.
The amplitude of the D+
2 (v) vibrational state extends over larger radial distances as
v increases, becoming larger at the region of the crossing between the two electronic
states. This can explain why all the probabilities, NRCT, ER and RCT, increase with
v.
As the total angular momentum J increases, the situation gradually changes. The
5
(3)
+
+
NRCT: H+D2(v=0) -> H +D2(v’)
+
+
NRCT: H+D2(v=1) -> H +D2(v’)
+
+
NRCT: H+D2(v=2) -> H +D2(v’)
v’=0
v’=1
v’=2
v’=3
v’=4
v’=5
v’=6
v’=7
v’=8
0.2
0.1
0.1
ER: H+D+2(v=0) -> HD+(v’)+D
ER: H+D+2(v=1) -> HD+(v’)+D
ER: H+D+2(v=2) -> HD+(v’)+D
Probability
v’=0
v’=1
v’=2
v’=3
v’=4
0
+
+
RCT: H+D2(v=0) -> HD(v’)+D
+
+
RCT: H+D2(v=1) -> HD(v’)+D
+
+
RCT: H+D2(v=2) -> HD(v’)+D
v’=0
v’=1
v’=2
v’=3
v’=4
v’=5
v’=6
v’=7
0.2
0.1
0
0
0.4
0.8
0.4
0.8
Collision energy (eV)
0.4
0.8
Figure 2. Vibrationally resolved reaction probability for J=0 H+D+
(left panels),
2 (v=0,j=0)
+
H+D+
(v=1,j=0)
(middle
panels),
and
H
+D
(v=2)
(right
panels),
towards
the
inelastic
charge trans2
2
fer, NRCT (H+ +D2 , in the top panels), the exchange reactive channel, ER (H+HD+ , in the middle panels)
and reactive charge transfer, RCT, channels (D+ +HD, in the bottom panels).
6
reaction probabilites for the three vibrational states studied are shown in Fig. 3 for
the 3 different rearrangent channels. As the rotational barrier increases, the reaction
probabilities shift towards higher energy, as expected. This shift, however, depends on
the mechanism. Thus, the two reactive processes, ER and RCT, have always larger
shifts or effective barriers. As the barrier increases with J, the reaction probabilities
decreases.
This situation is somehow different for the NRCT channel, which is essentially
inelastic and has lower effective barriers. As shown in the top panel of Fig. 3, the
effective barrier decreases progressively when going from v=0 to v=2. The reason is
that the energy difference between the initial (v) and final (v’) vibrational levels also
decreases. Thus, the electronic coupling becomes more effective at longer distances for
higher v’, because lower couplings can induce the transition.
The resonant structure appearing at low energy progressively dissapears as J increases because the rotational barrier increases and the wave packet can not reach the
deep well in the ground electronic state. Finally, as the reactive processes decrease, the
NRCT proccess increases in intensity with J, becoming dominant. The lower effective
barriers observed for the NCRT channel makes necessary to include more partial waves
J to converge the corresponding cross section in Eq. (2), up to J=140, and they were
extrapolated using the J-shifting approximation from the calculated J=80.
The cross section for the NRCT, ER, and RCT channels are shown in Fig. 4. Clearly
the ER channel is one order of magnitude lower than the other two. NRCT and RCT
are of the same order at low energies, but as collision energy increases, RCT decreases
while NRCT is either constant or increases for v=1 and 2. At higher energy, it is
expected that this difference increases, becoming dominant the NRCT channel. It
should be noted that in the case of H+H+
2 the two charge transfer processes, RCT and
NRCT, are indistinguishible [13].
The present theoretical results for each initial v are compared with the experimental
cross sections in the top panel of Fig. 4. In the experimental work by Andrianarijaona
+
et al[5, 6] on the H + D+
2 collisions, the H products were detected, i.e. they provide
a direct evidence on the NRCT cross section. The theoretical NRCT cross section
increases as the initial D+
2 (v) vibrational state increases, and, for v=2 at collision
energies around 0.5 eV, it matches the corresponding experimental measurements.
This seems to indicate that measurements made at lower temperature could be due
+
to more vibrationally excited initial D+
2 reactants. In the experiments, the D2 were
produced by a CEA/Grenoble all-permanent magnet Electron Cyclotron Resonance
ion source (ECR) [5, 6], and their vibrational distributions depend on the ECR ion
source parameters, which are not coupled with the ion beam energy. Unfortunately,
these experiments did not include any direct nor indirect measurement of the vibrational distributions of these D+
2 ions. It is expected, though, that slight changes in
the ion source conditions affect the vibrational distributions of the ions as observed in
the case of molecular ions, including D+
2 , produced by Duoplasmatron sources [36]. A
more detailed experimental work needs to be done with controlled conditions on the
vibrational excitation of the D+
2 reactant ions, specially at low collision energies, before
a factual conclusion could be drawn. The 3-D imaging technique similar to the one
used by Urbain et al. [37] was proven to be efficient to measure vibrational distribu+
tions of H+
2 issued from the primary reaction H +H2 . In this technique, two position
sensitive detectors detect the positions and time of flights of fragments, giving access
to the kinetic energy release, which definitely contains information on the vibrational
state of the molecular ions. Given the similarity between Urbain et al..’s molecular
ions and the ones in Andrianarijaona et al,’s experiment [5, 6], the 3-D imagine tech7
+
+
+
+
+
+
NRCT: H+D2(v=0) -> H +D2
NRCT: H+D2(v=1) -> H +D2
NRCT: H+D2(v=2) -> H +D2
ER: H+D+2(v=0) -> HD++D
ER: H+D+2(v=1) -> HD++D
ER: H+D+2(v=2) -> HD++D
RCT: H+D+2(v=0) -> HD+D+
RCT: H+D+2(v=1) -> HD+D+
RCT: H+D+2(v=2) -> HD+D+
0.6
0.4
0.2
Probability
0.1
0
J=0
J=10
J=20
J=30
J=40
J=50
J=60
J=70
J=80
0.6
0.4
0.2
0
0
0.4
0.8
0.4
0.8
0.4
0.8
Collision energy (eV)
+
Figure 3. Reaction probability for different J values in H+D+
2 (v=0) (left panels), H+D2 (v=1) (middle
+
+
panels), and H +D2 (v=2) (right panels), towards the inelastic charge transfer, NRCT (H +D2 , in the top
panels), the exchange reactive channel, ER (H+HD+ , in the middle panels) and the reactive charge transfer,
RCT, channels (D+ +HD, in the bottom panels).
8
100
NRCT: H+D+2(v) −> H++D2
Exp.
10
v=2
v=1
v=0
ER: H+D+2(v) −> D+HD+
Cross section (Å2)
10
v=2
v=1
0.1
v=0
0.01
100
RCT: H+D+2(v) −> D++HD
10
v=2
v=1
v=0
0.1
0
0.4
Collision energy (eV)
0.8
Figure 4. Reaction cross section for H+D+
2 (v=0, 1 and 2) towards the inelastic charge transfer, NRCT
(H+ +D2 , in the top panel), the exchange reactive channel, ER (H+HD+ , in the middle panel) and the reactive
charge transfer, RCT, channels (D+ +HD, in the bottom panel). The experimental values are taken from
Ref.[5, 6].
9
nique would be an appropriate fit to provide data that will accomplish our goal. The
theoretical simulations presented in this work are expected to provide a good guidance
to interpret such measurements. Moreover, this should also be extended to the H+
2 +
H reaction due to its astrophysical relevance, especially in Early Universe models.
In all the mechanisms, the cross section increases with the vibrational excitation.
This is explained, as in the reaction probabilities, by the increase of the overlap between
the initial D+
2 (v) and dominant D2 (v’) (see Fig. 1.a), which acts as a doorway for the
three mechanisms. The amplitude of the initial vibrational D+
2 (v) wave function around
the electronic curve crossing region increases, thus favoring the electronic transition.
The vibrationally resolved state-to-state cross sections for the NRCT and RCT
channels are shown in Fig. 5. There is a clear difference between the two mechanisms.
For RCT, all cross sections decrease with energy, which could be explained by the
near statistical mechanisms, due to the presence of many resonances associated to
the deep HD+
2 well that mediate the reactivity. For NRCT, however, there are several
cases: there is always a dominant v’: v’=6 and 7 for the cases of v=0,1 and v=2,
respectively. This inelastic channel is formed by an electronic transition taking place
at rather long distances. The rest of v’ in the NRCT channel show a progresively
decreasing cross section with decreasing v’, probably due to transitions from vd ’.
The reason why the NRCT channels v’=6 and 7 for D+
2 (v=0,1) and v’=7 for
+
D2 (v=2) are dominant at high energy can be easily interpreted in terms of the reaction probabilities at high energies and high angular momentum. As J increases, the
rotational barrier also increases. This barrier avoids the wave packet to reach short
distances between the two reactants, and therefore to enter in the deep insertion well of
the ground electronic state, where the reaction takes place. However, the charge transfer between the initial D+
2 (v) and nearly resonant D2 (v’) can takes places at rather
long distances, at which small electronic couplings can induce transitions between close
lying electronic states. Under these circumstances, as reaction probabilities decreases,
the inelastic NRCT becomes dominant towards a particular final D2 (v’) state.
In order to check this model, we have performed approximate time-independent close
coupling (TICC) calculations, using reactant Jacobi coordinates, including only one
vibrational channel in the initial excited electronic state, D+
2 (v=2), and one vibrational
state in the final ground electronic state, D2 (v’=7), with 20 rotational channels in
each, even j values from 0 to 40. The differential equations are integrated using the
renormalized Numerov method[38, 39] for distances between 2 and 20 bohr, with 300
grid points. Also, the centrifugal sudden approximation is assumed, keeping only one
helicity, Ω=0, since the Coriolis coupling are expected to play a negligible role at the
long distances for which the electronic transition takes place at high total angular
momentum. These TICC calculations include the NRCT only, and moreover, only
consider a possible CT vibrational product, D2 (v’=7). However, the WP presented
above include all rearrangment channels, ER and RCT, and all accessible vibrational
states of the products, becoming much more demanding computationally.
The NRCT probabilities obtained in these model calculations are compared in Fig. 6
with the accurate wave packet calculations described before for J=50, 60 and 70
(denotd by WP). Clearly, as J increases, the agreement between the two methods
improves. This demonstrate the validity of the model. The oscillations obtained as a
function of collision energy are due to the interference between the entrance and final
channels energy difference[40, 41].
This behavior obtained at high J values shows that NRCT is the dominant reaction
channel at high energy. This is valid not only for the reaction studied in this work but
also for all the other isotopic variants, since it is attributed to the crossing between
10
10
+
+
NRCT:H+D2(v=2)−>H +D2(v’)
+
+
+
+
RCT: H+D2(v=2)−>HD(v’)+ D
1
v’=0
v’=1
v’=2
v’=3
v’=4
v’=5
v’=6
v’=7
0.1
10
Cross section (Å2)
+
+
NRCT:H+D2(v=1)−>H +D2(v’)
RCT: H+D2(v=1)−>HD(v’)+ D
NRCT:H+D+2(v=0)−>H++D2(v’)
RCT: H+D+2(v=0)−>HD(v’)+ D+
1
0.1
10
1
0.1
0
0.4
0.8
0
0.4
0.8
Collision energy (eV)
Figure 5. Vibrationally resolved state-to-state cross section for H+D+
2 (v=0, 1 and 2) towards the inelastic
charge transfer, NRCT (H+ +D2 (v’), in the left panels), and the reactive charge transfer, RCT, channels (D+
+HD(v’), in the right panels).
11
1
TICC
J=70
WP
0.5
0
1
TICC
J=60
Probability
WP
0.5
0
1
TICC
J=50
WP
0.5
0
0
0.4
0.8
Collision energy (eV)
Figure 6. NRCT probabilities obtained in H + D+
2 (v=2) collisions at several J values using the “exact”
wave packet results (WP) and the approximated time independent close coupling (TICC) results, as described
in the text.
12
WP
only v’=7 WP
TICC
10
2
Cross section (Å )
Exp.
1
1
10
Collision energy (eV)
Figure 7. NRCT cross section obtained in H + D+
2 (v=2) collisions comparing the total NRCT “exact” wave
packet results (WP), the state-to-state wave packet cross section for D2 (v’=7), and the approximated time
independent close coupling (TICC) results. The experimental values are taken from Ref.[5, 6].
the neutral and cationic diatomic fragments, which takes place at very long distances
between the reactants.
The TICC model has been extended to calculate the NRCT cross section in a wider
energy range, by solving the close-coupling equations for J= 0 to 300. The simulated
NRCT cross sections are compared with the experimental results of Andrianarijaona
et al.[5, 6]. The TICC cross section is very close to the wave packet results for D2 (v’=7)
below 1 eV, but is lower than the total WP NRCT cross section, since it does not take
into account the other v’ levels. As discussed above, the quasi degeneracy between
initial D+
2 (v=2) and final D2 (v’=7) produces a very efficient electronic transition giving rise to the charge transfer. At low energy, for which relatively low total angular
momentum J are dominant, the rotational barriers are low and the dynamics in the
ground electronic state can access to the deep well of H+
3 . Within this well there is
an effective energy transfer towards the vibrational levels of D2 (v’) and also access to
HD products in other rearrangment channels. This explains why TICC results are so
different to the accurate WP results below 1eV, since TICC neglects these processes.
However, at high angular momenta, which dominate at high collision energy, the ro+
tational barrier does not allow the H and D+
2 or H +D2 reactants to approach each
other at distances where the vibrational energy transfer or the exchange reaction can
take place. This explains why the TICC approximation yields results in rather good
agreement with the experimental results for collision energies between 3 and 9 eV.
4. Conclusions
The reactive cross sections for the H + D+
2 charge transfer reaction increase with the
initial vibrational excitation of the D+
(v)
reactant,
due to the increase of the amplitude
2
+
of the vibrational wavefunctions of D2 and D2 in the curve crossing and their mutual
overlaps. The dynamics is dominated by the electronic transition from the excited
13
to the ground electronic state, where the reaction takes place in the deep insertion
well of H+
3 . The charge transfer electronic transition is dominated by the crossing of
the two D2 /D+
2 occurring at rather large distance between reactants, where it can
take place between close lying vibrational states. For this reason, the reaction shows a
rather marked dependency on the initial vibrational excitation. Moreover, this feature
determines that the non-reactive charge transfer becomes the dominant channel as the
collision energy increases, for which the high total angular momentum introduces a
barrier avoiding the access to the insertion minimum.
The non-reactive charge transfer cross section is compared to the experimental
measurements in the 0.01-1 eV collision energy range studied theoretically. A good
agreement is found around 0.5 eV. At lower collision energies, the experimental crosssection are larger than the experimental ones. This is attributted to a possible higher
vibrational excitation in the generation of D+
2 reagents. Also, a TICC model, including
only the initial D+
(v
=
2)
and
the
near
resonant
D2 (v’=7) vibrational states, shows
2
good agreement with the experimental cross-section in the 3-9 eV collision energy
range. It is concluded that a more detailed experimental study, focusing on the control
of experimental conditions used to generate D+
2 , are needed to further analyze the
vibrational effects of reagents on the charge transfer reaction, specially at low energies.
For these studies, the theoretical simulations can be of important help.
Acknowledgement(s)
We acknowledge computing time at Cibeles (UAM) under RES computational grant
AECT-2021-1-0011.
Funding
The research leading to these results has received fundings from Ministerio de Ciencia,
Investigación y Universidades (MICIU) (Spain) under grant FIS2017-83473-C2. V.
Andrianarijaona is supported by the National Science Foundation through Grant No.
PHY - 1530944
References
[1] S.C.O. Glover, Astrophys. J. 584, 331 (2003).
[2] V. Wakelam, E. Bron, S. Cazaux, F. Dulieu, C. Gry, P. Guillard, E. Habart, L. Hornekær,
S. Morisset, G. Nyman, V. Pirronello, S.D. Price, V. Valdivia, G. Vidali and N. Watanabe,
Molecular Astrophysics 9, 1 – 36 (2017).
[3] Z. Karpas, V. Anicich and W.T. Huntress, J. Chem. Phys. 70, 2877 (1979).
[4] P.C.E. McCartney, C. McGrath, J.W. McConkey, M.B. Shah and J. Geddes, J. Phys. B:
At. Mol. Opt. Phys. 32, 5103 (1999).
[5] V.M. Andrianarijaona, J.J. Rada, R. Rejoub and C.C. Havener, J. Phys.: Conf. Ser. 194,
012043 (2009).
[6] V.M. Andrianarijaona, L.M. Wegley, A.Z. Watson, M. Andrianarijaona, C.P. DeGuzman,
K. Kim, E.J. Nuss, J.J. Taylor, R.L. Wilson, R.T. Zhang, D.G. Seely and C.C. Havener,
AIP Conference Proceedings 2160, 070005 (2019).
[7] I. Last, M. Gilibert and M. Baer, J. Chem. Phys. 107, 1451 (1997).
[8] H. Kamisaka, W. Bian, K. Nobusada and H. Nakamura, J. Chem. Phys. 116, 654 (2002).
14
[9] P.S. Krstı́c, Phys. Rev. A 66, 042717 (2002).
[10] P.S. Krstic and R.K. Janev, Phys. Rev. A 67, 022708 (2003).
[11] L.F. Errea, A. Macias, L. Méndez, I. Rabadán and A. Riera, Nuclear Instruments and
Methods Phys. Research B 235, 362 (2005).
[12] S. Ghosh, T. Sahoo, M. Baer and S. Adhikari, J. Phys. Chem. A 125, 731 (2021).
[13] C. Sanz-Sanz, A. Aguado and O. Roncero, J. Chem. Phys. 154, 104104 (2021).
[14] R. Preston and J. Tully, J. Chem. Phys. 54, 4297 (1971).
[15] A. Ichihara, J. Chem. Phys. 103, 2109 (1995).
[16] V.G. Ushakov, K. Nobusada and V.I. Osherov, Phys. Chem. Chem. Phys. 3, 63 (2001).
[17] S. Mukherjee, D. Mukhopadhyay and S. Adhikari, The Journal of Chemical Physics 141
(20), 204306 (2014).
[18] B. Mukherjee, K. Naskar, S. Mukherjee, S. Ghosh, T. Sahoo and S. Adhikari, International
Reviews in Physical Chemistry 38, 287 (2019).
[19] A. Aguado, O. Roncero and C. Sanz-Sanz, PCCP 23, 7735 (2021).
[20] F.O. Ellison, N.T. Huff and J.C. Patel, J. Am. Chem. Soc. 85, 3544 (1963).
[21] J.C. Tully, Adv. Chem. Phys. 42, 63 (1980).
[22] L.P. Viegas, A. Alijah and A.J.C. Varandas, J Chem Phys 126 (7), 074309 (2007).
[23] A. Zanchet, O. Roncero, T. González-Lezana, A. Rodrı́guez-López, A. Aguado, C. SanzSanz and S. Gómez-Carrasco, J. Phys. Chem. A 113, 14488 (2009).
[24] O. Roncero, https://github.com/octavioroncero/madwave3 (2021).
[25] V.A. Mandelshtam and H.S. Taylor, J. Chem. Phys. 102, 7390 (1995).
[26] G.J. Kroes and D. Neuhauser, J. Chem. Phys. 105, 8690 (1996).
[27] R. Chen and H. Guo, J. Chem. Phys. 105, 3569 (1996).
[28] S.K. Gray and G.G. Balint-Kurti, J. Chem. Phys. 108, 950 (1998).
[29] T. González-Lezana, A. Aguado, M. Paniagua and O. Roncero, J. Chem. Phys. 123,
194309 (2005).
[30] S. Gómez-Carrasco and O. Roncero, J. Chem. Phys. 125, 054102 (2006).
[31] E. Aslan, N. Bulut, J.F. Castillo, L. Bañares, F.J. Aoiz and O. Roncero, Astrophys. J.
739, 31 (2012).
[32] A. Zanchet, B. Godard, N. Bulut, O. Roncero, P. Halvick and J. Cernicharo, ApJ 766,
80 (2013).
[33] C. Sanz, O. Roncero, C. Tablero, A. Aguado and M. Paniagua, J. Chem. Phys. 114, 2182
(2001).
[34] T. González-Lezana, O. Roncero, P. Honvault, J.M. Launay, N. Bulut, F.J. Aoiz and L.
Bañares, J. Chem. Phys. 125, 094314 (2006).
[35] T. González-Lezana and P. Honvault, Int. Rev. Phys. Chem 33, 371 (2014).
[36] V.M. Andrianarijaona, PhD Thesis, Université Catholique de Louvain (2002).
[37] X. Urbain, N. de Ruette, V.M. Andrianarijaona, M.F. Martin, L.F. Menchero, L. Errea,
L. Méndez, I. Rabadan and B. Pons, Phys. Rev. Lett. 111, 203201 (2013).
[38] F.X. Gadéa, H. Berriche, O. Roncero, P. Villarreal and G. Delgado-Barrio, J. Chem. Phys.
107, 10515 (1997).
[39] O. Roncero, A. Aguado and S. Gómez-Carrasco, in Gas-phase chemistry in space: from
elementary particles to complex organic molecules, edited by Lique, F and Faure, A,
AAS-IOP Astronomy (IOP publishing ltd, Dirac house, Temple Back, Bristol BS1 6BE,
England, 2019).
[40] G.A. Parker, A. Laganà, S. Crocchianti and R.T. Pack, J. Chem. Phys. 102, 1238 (1995).
[41] T. P. Tsien and R. T Pack, Chem. Phys. Lett. 6 (1970).
15