# Tunneling and reflection above the repulsive Coulomb barrier of fullerene dianions

Bruno Concina, Christian Bordas

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Bruno Concina, Christian Bordas. Tunneling and reflection above the repulsive Coulomb barrier of fullerene dianions. Chemical Physics Letters, 2025, 868, pp.142003. 10.1016/j.cplett.2025.142003. hal-05012252

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# Tunneling and reflection above the repulsive Coulomb barrier of fullerene dianions

Bruno Concina*, Christian Bordas

Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, UMR5306, F-69100 Villeurbanne, France

## ARTICLE INFO

### Keywords:

- Multiply charged anion

- Thermionic emission

- Tunneling

- Reflection above the barrier

- Fullerene

## ABSTRACT

Isolated fullerene dianions exhibit a repulsive Coulomb barrier towards electron loss. A kinetic energy release distribution of thermionic emission from $C_{84}^{2-}$ has been measured in a velocity map imaging spectrometer and is compared with a detailed-balance model. A key point is the calculation of the transmission probability through and above the barrier. Previously published, a Wentzel-Kramers-Brillouin (WKB) approach included tunneling but disregarded reflection over the barrier. However, the two processes being at the same fundamental quantum level, reflection must be taken into account. We propose a refined model based on the WKB Kemble formula which includes both.

## 1. Introduction

Due to the peculiar shape of the interaction potential between the remaining anion and the additional electron, the electronic properties of isolated multiply charged anions (MCAs) have attracted much attention [1,2]. In the case of dianions, the outer electron experiences an asymptotic repulsive interaction with the singly charged anion core, which contrasts with the attractive long-range potentials of most molecular systems (neutrals, cations, and singly charged anions). A very specific feature of the MCAs is the repulsive Coulomb barrier (RCB) resulting from the combination of the long-range Coulomb repulsion and the short-range attractive interactions. Fig. 1 depicts such a potential calculated within the framework of the classical model of the dielectric sphere for the fullerene dianion $C_{84}^{2-}$ [3,4]. The RCB leads to unique features in the direct photodetachment of MCAs [2,5,6] due to the trapping of the outer electron inside the barrier. The most obvious example is the cutoff on the low kinetic energy side of photoelectron spectra [2]. It also drastically influences delayed emission processes as we shall see below.

Indeed, after photon absorption by MCAs, direct photodetachment is not the only relaxation process. In the case of complex molecules or clusters [7], it is even far from being dominant, as the internal energy is shared among many degrees of freedom. More specifically in the case of systems with relatively low electronic binding energy, decay can occur by thermionic emission. After a fast excitation process (by collision or laser excitation as in the present case), the electronic excitation may be

rapidly converted into vibrational energy in the electronic ground state, the timescale typically being hundreds of femtoseconds to picoseconds [8], and a delayed electron emission occurs after thermal equilibration. Far fewer studies have been devoted to this indirect process than to the direct photodetachment, mainly because the latter enables electronic or vibrational spectroscopic measurements. However, owing to its relevance to the long timescale dynamics of molecular systems (typically in the nanosecond to the millisecond range), delayed thermionic emission deserves to be studied in detail. Pioneering studies assumed a pure over-the-barrier detachment and consequently disregarded tunneling through the RCB [3,9]. Nevertheless, our previously published measurement of kinetic energy release distribution (KERD) of photoexcited fullerene dianions unambiguously demonstrated that the majority of the detected electrons tunnel through the RCB [10]. The detailed-balance model proposed in this previous contribution included tunneling but disregarded reflection above the barrier: the transmission probability was assumed to be uniformly equal to one above the barrier. It should be emphasized that, as tunneling (through the barrier) and reflection (above the barrier) are at the same fundamental quantum level, it is not consistent to regard one and to neglect the other. Tunneling is of course the only transmission process below the barrier, thus it fundamentally changes the nature of the decay process allowing electrons to be ejected under classically forbidden conditions. Comparatively, reflection above the barrier never affects the overall process more than by (about) a factor of 1/2. Consequently, we initially neglected reflection above the barrier in the description of delayed electron emission [10]. In order to

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: bruno.concina@univ-lyon1.fr (B. Concina).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Interaction potential $V(r)$ between the singly charged anion $C_{84}^{-}$ and the electron as a function of the distance $r$ between the anion center and the electron (black line) and the inverted parabolic potential $V_{par}(r)$ used for the electron transmission $T(\epsilon)$ calculation (dashed red line). $V_{par}(r)$ has been adjusted to $V(r)$ by setting its maximum at the same value and location as $V(r)$ and by setting its second derivative at the maximum at the same value as the one of $V(r)$. The horizontal dotted lines ($\epsilon = 0$ and $\epsilon = 1.4$ eV) forms the limits of the energy ranges corresponding to the transmission through the barrier and over the barrier. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

deal properly with all the effects due to the RCB, we need to take it explicitly into account. The present contribution proposes a renewed analysis of the experimental KERD of fullerene dianions (published in Ref. [10]) which includes both tunneling and reflection above the barrier.

Many contributions have dealt with quantum effects in reactions. Tunneling and reflection corrections to the KERD have been calculated for different types of unimolecular reactions [11]. Such issues are at the center of the developments of the thermal reaction rate theory [12,13].

The present article is organized as follows. In Sec. 2, the experimental setup and the detailed-balance model of thermionic emission are briefly described. Section 3 is devoted to the discussion regarding tunneling and reflection above the barrier. The previously published [10] model is discussed and a refined one including reflection above the barrier is introduced. A comparison with the experimental KERD is provided, highlighting the advantages and limitations of our model reformulation. Finally, Sec. 4 is devoted to the conclusion.

## 2. Summary of experiment and detailed-balance model

The experimental setup was previously described [10,14,15]. Briefly, the fullerene dianions are transferred to the gas phase by electrospray ionization (ESI), mass-selected by a quadrupole, and perpendicularly accelerated in a linear time-of-flight mass spectrometer (TOFMS) with a pulsed extraction. A velocity map imaging (VMI) spectrometer is mounted orthogonally to the ion drift tube at the exit of the TOFMS. Proper synchronization allows the photoexcitation of the mass-selected fullerene dianions by a nanosecond laser (frequency doubled diode pumped Nd:YAG laser; 532 nm wavelength; 350 Hz repetition rate). VMI time resolution is achieved by gating the voltages applied on the position-sensitive detector [14]. In the present study, the detection window is delayed by 150 ns with respect to the laser pulse. No contribution from direct photodetachment is measured for such a delay. Consequently, a pure thermionic emission KERD spectrum is recorded for a well-defined time window of 35 ns-width (Full Width at Half Maximum, FWHM) centered on the 150 ns-delay [10]. Such a spectrum

for the fullerene dianion $C_{84}^{-}$ is reported in Fig. 2 (after subtraction of a small contribution due to the thermionic emission of hot $C_{84}$ produced by the detachment of $C_{84}^{-}$) [10]. The RCB height of $C_{84}^{-}$ has been determined by photoelectron spectroscopy in a magnetic bottle by measuring the cutoff at the low kinetic energy side [16]. The value $\epsilon_{C} = 1.4$ eV has been determined [16] and is reported in Fig. 2. By looking at the experimental KERD, it can be established that most of the electrons detected in the chosen time window are emitted with a kinetic energy lower than the RCB height $\epsilon_{C}$ and thus tunnel through the RCB.

The experimental data are analyzed in the framework of a detailed-balance model using the microcanonical temperature defined below [4,10]. Thermionic emission is an indirect decay process in which the electronic energy resulting from the photon absorption is transferred to the vibrations in the electronic ground state. Thermal equilibrium (i.e. equipartition of the energy over all accessible internal modes) is reached on the timescale of a few hundred femtoseconds [8]. Electron emission statistically occurs after a given delay related to the amount of vibrational energy $E$. We use the formalism introduced by Weisskopf [17] based on the detailed-balance principle and level densities to derive both the differential decay rate $k(E, \epsilon)$ (depending on the parent vibrational energy $E$ and the kinetic energy $\epsilon$) and the integrated decay rate $K(E)$ (resulting from the integration of $k(E, \epsilon)$ over $\epsilon$) [4]. In this approach, the microcanonical temperature $T(E)$ of the parent $C_{84}^{-}$ is related to the derivative of the vibrational level density $\rho$ [4,18],

$$ \frac{1}{k_B T(E)} = \frac{d}{dE} \ln \rho(E) \qquad (1) $$

with $k_B$ the Boltzmann constant and $E$ the vibrational energy of $C_{84}^{-}$. The so-called daughter temperature $T_d$, characterizing the excitation of the daughter anion $C_{84}^{-}$ after decay is defined by [4]

$$ T_d = T(E - E_b) \qquad (2) $$

<figure><img src="image_4.png" /><figcaption>Fig. 2. The experimental KERD of the delayed electron emission from $C_{84}^{-}$ is displayed as a black line. The RCB height $\epsilon_{C}$ is marked by a dashed black line forming the limit between the tunneling and the over-the-barrier detachment. The KERD resulting from the simple WKB formula $e^{-2W}$ (dashed red line) is compared to the one resulting from new calculations based on the Kemble WKB formula below the barrier and the transmission through an inverted parabolic potential above the barrier (blue line). In both cases, the daughter temperature $T_d$ is deduced from the detailed balance using the experimental time window (resp. $T_d = 1175$ K and $T_d = 1185$ K). The normalization constant $S$ of both theoretical KERDs (see Eq. (4)) has been used as a free parameter in the fit to the experimental distribution. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
with E the vibrational energy of the parent dianion $C_{84}^{2-}$ and $E_b$ the electron binding energy (i.e. the second electron affinity of $C_{84}$, $EA_2 = 0.75$ eV [16]). The differential decay rate $k(E, \epsilon)$ reads [4]

$$k(E, \epsilon) = \frac{m}{\pi^2 \hbar^3} \frac{g_f}{g_i} \epsilon \sigma(\epsilon) \exp\left(-\frac{\epsilon}{k_B T_d}\left(1 + \frac{\epsilon}{2C_d T_d}\right)\right) \exp\left(-\frac{E_b}{k_B T_e}\right) \quad (3)$$

with m the electron mass, $g_i$ and $g_f$ the electronic degeneracies respectively in the initial and the final states, $C_d$ the heat capacity (temperature derivative of the energy) of the daughter and $\sigma(\epsilon)$ the cross section of the reverse process, namely the electron attachment to the singly charged anion. The emission temperature $T_e$ is defined by: $T_e = T(E - E_b/2)$ [4]. Thus, the KERD of the electrons emitted from the $C_{84}^{2-}$ dianions of vibrational energy E is as follows

$$f(\epsilon) = S\epsilon\sigma(\epsilon)\exp\left(-\frac{\epsilon}{k_B T_d}\left(1 + \frac{\epsilon}{2C_d T_d}\right)\right) \quad (4)$$

with S a normalization constant. In these expressions (Eq. (3) and Eq. (4)), it is necessary to include the second term of the expansion of the level density $\ln\rho(E - E_b - \epsilon)$ in power series of the kinetic energy $\epsilon$ because of the relatively large value of $\epsilon$ ($\epsilon \sim 1$ eV) [4]. For a comparison with the experiment, the daughter temperature $T_d$ has to be determined. It is deduced from the equality between the integrated decay rate $K(E)$ and the inverse of the experimental delay $\tau$ (here $\tau = 150$ ns), giving a good approximation of the most probable vibrational energy E of the parent dianion [4],

$$K(E) = 1/\tau \quad (5)$$

The daughter temperature $T_d$ may also be obtained by fitting the experimental KERD. We will use these two possibilities in the following section.

### 3. Tunneling and reflection above the repulsive coulomb barrier

Introduced because of the detailed-balance principle, the cross section $\sigma(\epsilon)$ of the electron attachment to $C_{84}$ is the crucial ingredient of the model. It is related to the electron attachment potential for which a simple expression $V(r)$ is proposed based on the classical dielectric sphere model [3,4]. We use the first terms of the classical expression of the energy of a point charge $-e$ at a distance $r$ from the center of a singly negatively charged dielectric sphere. In addition to the Coulomb and standard polarizability $1/r^4$ terms, a $1/r^6$ term is included in order to account for a finite-size sphere,

$$V(r) = \frac{e^2}{r} - \frac{ae^2}{2r^4} - \frac{Ke^2}{r^6} \quad (6)$$

with $K = \frac{\alpha^{2}2r_{+}^{4}+4}{2r_{+}^{2}+3}$ where $\alpha$, $r_0$ and $\epsilon_r$ are respectively the polarizability, the radius and the dielectric constant of the sphere [3,4]. Like in the previous works [4,10], we use $\alpha = 130$ Å$^3$, $r_0 = 6.54$ Å and $\epsilon_r = 3.6$. The potential curve $V(r)$ is displayed in Fig. 1. The RCB maximum is reached for $R_C = 8.1$ Å and is given by $\epsilon_C = 1.4$ eV in agreement with the experiment [16], which supports the choice of the potential [4]. We propose a simple hard-sphere expression for the attachment cross section with a capture radius equal to $R_C$,

$$\sigma(\epsilon) = T(\epsilon)\pi R_C^2 \quad (7)$$

weighted by the transmission probability $T(\epsilon)$ relative to the potential $V(r)$. We propose to calculate the factor $T(\epsilon)$ in the Wentzel-Kramers-Brillouin (WKB) framework which allows to introduce both tunneling through the barrier and reflection above the barrier. Our model describes only the electronic capture by the potential $V(r)$. Once the electron is transmitted through the potential to the well (small values of $r$), it is assumed that it cannot cross the potential barrier again and return to large values of $r$. Actually, the electron excess energy is trans-

ferred to the molecular vibrations to lead to a stable dianionic state. This second step, called intramolecular vibrational redistribution (IVR), is not described in our model.

The central potential $V(r)$ defined by Eq. (6) is only realistic in the range where $V(r) > 0$. At short-range and more specifically when $V(r) < 0$, the complex interaction between the additional electron and the rest of the system (including the other excess electron) is described by an effective central potential $V_{eff}(r)$ that differs significantly from $V(r)$ at short distances. In particular, the definition of $V_{eff}(r)$ for $r < R_C$ must restore the correct value for the second electronic affinity, which is clearly not the case for the potential defined by Eq. (6). However, the short-range correction to the potential has no influence, at least to a first approximation, on the capture cross-section.

In our previous contributions [4,10], we used the simplest approach to include tunneling, namely the well-known WKB formula for the probability of transmission $T(\epsilon)$ of a particle of kinetic energy $\epsilon$ through the potential $V(r)$ [19],

$$T(\epsilon) = e^{-2W(\epsilon)} \quad \text{for } \epsilon \le \epsilon_C \quad (8)$$

with

$$W(\epsilon) = \frac{1}{\hbar} \int_{r_{-}(\epsilon)}^{r_{+}(\epsilon)} \sqrt{2m(V(r) - \epsilon)} dr \quad (9)$$

where m is the electron mass, $r_{\pm}(\epsilon)$ are the turning points (i.e. $V(r_{\pm}) = \epsilon$). Eq. (8) leads to a transmission probability equal to one at the top of the barrier ($\epsilon = \epsilon_C$). By continuity, this factor remains equal to one over the barrier,

$$T(\epsilon) = 1 \quad \text{for } \epsilon \ge \epsilon_C \quad (10)$$

Thus, using Eq. (8) for the transmission below the barrier implies neglecting the reflection above the barrier. $T(\epsilon)$ as a function of $\epsilon$ is reported in Fig. 3. The KERD resulting from this WKB formula is displayed in Fig. 2 using the daughter temperature $T_d$ derived from the model ($T_d = 1175$ K). A non-physical angular point appears for $\epsilon = \epsilon_C$ due to the discontinuity of the derivative of $T(\epsilon)$. Agreement is otherwise good with the experimental KERD. For a more realistic comparison, the

<figure><img src="image_5.png" /><figcaption>Fig. 3. Transmission probability $T(\epsilon)$ as a function of the kinetic energy $\epsilon$. The previous calculation given by the simple WKB formula $e^{-2W}$ (black squares and line) is compared to refined calculations based on the Kemple WKB formula (red circles and line) and the transmission through an inverted parabolic potential (blue line). Inset: comparison between the Kemple WKB formula (red circles and line) and the transmission through the inverted parabolic potential (blue line) in logarithmic scale. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
theoretical KERD should be convoluted by the instrumental resolution. Such a curve is reported in Fig. 4 using $T_d$ as a fitting parameter. The angular point has been suppressed due to the experimental broadening, concealing the non-physical irregularity of the transmission probability. The overall shape of the theoretical KERD is in good agreement with the measured KERD. The temperature deduced from the fit ($T_d = 1220 K$) is almost equal to the prediction of the model, reinforcing the good accord with the experiment.

Despite a satisfactory agreement with the measurements, Eq. (8) has been used outside its range of validity. In its semi classical derivation [19], the tunneled flux is assumed to be exponentially small. Thus Eq. (8) is valid only when the transmission probability $T(\epsilon)$ is small, i.e. in practice for energies well below the barrier top.

The WKB Kemble formula, which is based on a rigorous flux conservation, is valid in the whole energy range below the barrier top [20,21]. It reads

$$T_K(\epsilon) = \frac{1}{1 + e^{2W(r)}} \quad \text{for } \epsilon \le \epsilon_C \qquad (11)$$

with $W(\epsilon)$ still defined by Eq. (9). By contrast with Eq. (8), there is no restriction on the magnitude of the transmission probability $T_K(\epsilon)$. We note that the limit of Eq. (11) at small transmission probability is Eq. (8). The transmission probability is equal to $1/2$ at the top of the barrier. Above the barrier ($\epsilon \ge \epsilon_C$), Kemble formula necessitates the determination of complex turning points and an integration in the complex plane [21], which is not a simple calculation. In order to avoid any mathematical complication, we have made the choice to use the analytical results of the inverted parabola for the region above the barrier [19]. We consider the inverted parabolic potential $V_{par}(r)$ that best approximates the potential $V(r)$, i.e. its maximum is $\epsilon_C$ located in $r = R_C$, leading to the expression

$$V_{par}(r) = \epsilon_C - \frac{a}{2}(r - R_C)^2 \qquad (12)$$

Additionally $a$ ($a > 0$) has been set to the opposite of the second derivative of $V(r)$ for $r = R_C$. $V(r)$ and $V_{par}(r)$ are displayed in Fig. 1. The

<figure><img src="image_6.png" /><figcaption>Fig. 4. The KERD resulting from the simple WKB formula $e^{-2W}$ (dashed red line) is compared to the one resulting from refined calculations based on the Kemble WKB formula below the barrier and the transmission through an inverted parabolic potential above the barrier (blue line). In both cases, the daughter temperature $T_d$ results from a fit to the experimental distribution (black line): resp. $T_d = 1220 K$ and $T_d = 1370 K$ (the normalization constant $S$ has also been used as a free parameter). The theoretical KERDs are convoluted by the instrumental resolution. The experimental KERD is displayed as a black line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

two potentials are close only in the vicinity of the maximum, ensuring the correct behavior just below and above the barrier top. At lower kinetic energy, $V_{par}(r)$ is significantly thinner than $V(r)$. The transmission probability for the inverted parabola $V_{par}(r)$ is given by the Kemble formula (Eq. (11)) replacing $W(\epsilon)$ with (with)

$$W_{par}(\epsilon) = \frac{\pi}{\hbar} \sqrt{\frac{m}{a}(\epsilon_C - \epsilon)} \qquad (13)$$

Note that in Eq. (13), there is no restriction on the value of $\epsilon$, the result being valid both below and above the barrier [19]. The corresponding transmission probability $T_{par}(\epsilon)$ is reported in Fig. 3. Below the barrier, it is much larger than the one derived from Kemble formula with exponent $W(\epsilon)$ calculated using $V(r)$ (see inset of Fig. 3), the parabolic barrier being much thinner than $V(r)$ (see Fig. 1). The parabolic potential is thus a too crude approximation below the barrier. Therefore, we retain the parabolic potential expression in the range above the barrier and combine it with the Kemble formula using the potential $V(r)$ below the barrier. The connection between the two expressions is smooth at $\epsilon = \epsilon_C$ thanks to the definition of the fitted parabolic potential. Above the barrier, the parabolic potential is a reasonable approximation. In this energy range, the transmission probability increases from $1/2$ ($\epsilon = \epsilon_C$) to 1 for both calculations. For energies significantly larger than $\epsilon_C$, they may differ, but the error is expected to be small as the transmission probability is slightly lower than one in both cases.

Note that such an approximation (Kemble formula below the barrier and parabolic potential above the barrier) has been used for instance in the framework of the microcanonical instanton theory devoted to the calculation of chemical reaction rates [22].

These new formulae using the temperature $T_d$ derived from the model lead to the KERD represented as a blue line in Fig. 2. When compared with the previous calculations, we note that the angular point has disappeared, but that the KERD is slightly off-center in the direction of the low energies with respect to the experimental distribution. This shift is due to the fact that the new calculated transmission is significantly lower than the previous one for $\epsilon \ge 1.2$ eV. Note that the daughter temperature $T_d$ derived from the model using the experimental time window depends very little on the expression of the transmission probability $T(\epsilon)$: $T_d = 1175 K$ (previous calculations) vs. $T_d = 1185 K$ (present calculations). The change in transmission probability results in a variation of about 15 % of the integrated rate $K(E)$. Due to the exponential variation of the rate as a function of the temperature in such a statistical process [4], the change in $T_d$ is insignificant. Clearly, the shift of the presently calculated KERD towards low energies indicates an underestimate of the temperature.

In the process of comparing with the experiment, the daughter temperature $T_d$ may also be considered as a free parameter. Fitting of this parameter leads to $T_d = 1220 K$ (in our previous work) and $T_d = 1370 K$ (in the present work) respectively. The corresponding KERDs convoluted by the instrumental resolution are displayed in Fig. 4 together with the experimental one. In both cases, the KERD shape is in good agreement with the experiment, the new distribution being a bit narrower than the previously calculated one. Noticeably, in the more complete calculation presented here, the difference between the temperature deduced from the emission rate $K(E)$ and the temperature fitted to the experimental spectrum is larger than in the case where reflection above the barrier was neglected. This is a direct consequence of the shift of the newly calculated KERD in Fig. 2 (using the temperature $T_d$ determined from the time window). Finally, from the point of view of $T_d$ temperature, the newly calculated distribution leads to a slightly poorer agreement with the experiment. However, from the theoretical side, it is clear that the newly developed model should be preferred as it includes the reflection above the barrier and avoids a non-physical angular point. The two ways of calculating the transmission probability differs by less than a factor two. This is certainly less than the uncertainty on the differential rate $k(E, \epsilon)$ due to the others ingredients of the model. For
instance, the simple hard-sphere expression chosen for the attachment cross section $\sigma(\epsilon)$ is expected to contribute significantly to this uncertainty and the associated error could be larger than a factor two. The slightly better agreement of the previous calculation regarding $T_d$ could be fortuitous and the difference between the two calculations remains within the model uncertainty.

## 4. Conclusion

The KERD of thermionic emission of the fullerene dianion $C_{84}^{2-}$ has been measured for a delay of 150 ns after laser excitation. It was compared with the results of two detailed-balance models that differ in the calculation of the transmission probability through and above the repulsive Coulomb barrier. Both calculations are carried out within the WKB approximation, the former neglecting reflection above the barrier, the latter being more refined and including both tunneling and reflection above the barrier in a relatively simple way. We find a good agreement with the experiment for both models, the difference between the two approaches remaining within the model uncertainty.

The WKB formula of the first model applied in our previous work [10] was suffering from two major limitations. Firstly, it was used outside its validity range (it requires small transmission probabilities) and secondly the reflection above the barrier was not taken into account. The Kemble formula introduced in the present model is valid regardless of the magnitude of the transmission probability. Tunneling is described using the potential expression from the classical dielectric model. To avoid any mathematical complication, the parabolic approximation is used for the reflection above the barrier. This approach treats more correctly the energy range close to the top of the barrier and the transition between bound and open electron motion.

As mentioned above, the accuracy of the emission rate calculation depends directly on the precision with which the attachment cross section is represented. This gives some directions in which we can further improve our model in the future. Firstly, we should improve the simple hard-sphere expression.

This relatively crude expression considers only s-wave process. Nevertheless, we should in principle take into account the effect of angular momentum $\ell$ resulting from the separation of motion into radial and angular coordinates. It modifies the potential $V(r)$ by the addition of the centrifugal term $\ell(\ell+1)h^2/2mr^2$. This term significantly increases the RCB height $\epsilon_C$ (1.4 eV for $\ell=0$). The effective RCB height rapidly increases as a function of $\ell$. For example, for the lower values of $\ell$, $\epsilon_C(\ell=1) = 1.52$ eV, $\epsilon_C(\ell=2) = 1.78$ eV and $\epsilon_C(\ell=3) = 2.23$ eV. Consequently, only the lower values of the angular momentum contribute significantly to the capture cross-section in the energy range of the KERD (between approximately 0.5 and 1.8 eV). The present approximation is therefore acceptable. At higher energies, angular momentum would clearly have to be taken into account. It is therefore likely that the present assumption slightly overestimates the capture cross-section, but given the various assumptions included in the present model, it seems reasonable to remain within this approximation.

Secondly, we can go beyond the classical dielectric sphere model for the interaction potential between the outer electron and the remaining anion. This expression should be used for tunneling, and also for the reflection above the barrier, instead of the parabolic approximation.

Finally, another avenue for improving our model lies at the heart of the Kemble formula. Indeed, in this description, the transmission probability is equal to $1/2$ at the top of the barrier whatever the barrier (since, at the top of the barrier, $W(\epsilon) = 0$). This is a relatively conservative assumption, however it is well known that it is not the case for some reference potentials like Eckart [23] and square potentials [12,13]. It cannot be ruled out that this is the case for the potential $V(r)$ presented in this contribution. This will be the subject of a forthcoming work.

## CRediT authorship contribution statement

**Bruno Concina:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Conceptualization. **Christian Bordas:** Writing – review & editing, Project administration, Funding acquisition.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

Financial support of the French "Agence Nationale de la Recherche" (ANR) through Project N° ANR-06-BLAN-0041 is acknowledged.

## Data availability

Data will be made available on request.

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