Theoretical study of radiative electron attachment to CN, C2H, and C4H radicals Nicolas Douguet, S. Fonseca dos Santos, Maurice Raoult, Olivier Dulieu, Ann E. Orel, and Viatcheslav Kokoouline
Citation: J. Chem. Phys. 142, 234309 (2015); doi: 10.1063/1.4922691 View online: http://dx.doi.org/10.1063/1.4922691 View Table of Contents: http://aip.scitation.org/toc/jcp/142/23 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 142, 234309 (2015)
Theoretical study of radiative electron attachment to CN, C2H, and C4H radicals Nicolas Douguet,1 S. Fonseca dos Santos,1 Maurice Raoult,2 Olivier Dulieu,2 Ann E. Orel,1 and Viatcheslav Kokoouline3 1
Department of Chemical Engineering and Materials Science, University of California at Davis, Davis, California 95616, USA 2 Laboratoire Aimé Cotton, CNRS/Université Paris-Sud/ENS Cachan, bât. 505, Campus d’Orsay, 91405 Orsay Cedex, France 3 Department of Physics, University of Central Florida, Orlando, Florida 32816, USA
(Received 19 May 2015; accepted 7 June 2015; published online 19 June 2015) A first-principle theoretical approach to study the process of radiative electron attachment is developed and applied to the negative molecular ions CN−, C4H−, and C2H−. Among these anions, the first two have already been observed in the interstellar space. Cross sections and rate coefficients for formation of these ions by direct radiative electron attachment to the corresponding neutral radicals are calculated. For the CN molecule, we also considered the indirect pathway, in which the electron is initially captured through non-Born-Oppenheimer coupling into a vibrationally resonant excited state of the anion, which then stabilizes by radiative decay. We have shown that the contribution of the indirect pathway to the formation of CN− is negligible in comparison to the direct mechanism. The obtained rate coefficients for the direct mechanism at 30 K are 7 × 10−16 cm3/s for CN−, 7 × 10−17 cm3/s for C2H−, and 2 × 10−16 cm3/s for C4H−. These rates weakly depend on temperature between 10 K and 100 K. The validity of our calculations is verified by comparing the present theoretical results with data from recent photodetachment experiments. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4922691]
I. INTRODUCTION
The present theoretical study of radiative electron attachment (REA) to neutral molecules is motivated by the recent discoveries of molecular anions in the interstellar medium (ISM). Six anions have been detected so far in the ISM: C6H−,1–6 C4H−,7 C8H−,8,9 C3N−,10 C5N−,2,5 and CN−.11 More recently, based on analysis of data from 1986 Giotto mission, it was reported12 that certain anions could also be present in the coma of comet Halley. Limited resolution did not allow unambiguous identification of the species. The possibility for atomic anions, such as H−, to be formed in the ISM by REA was first suggested by McDowell13 in 1961. Later, Dalgarno and McCray14 have discussed the role of negative atomic ions in the formation of neutral molecules in the ISM. The formation of molecular anions in the ISM by REA has been proposed by Herbst,15 who has also developed a theoretical approach3,15 to evaluate rate coefficients for REA. More than 20 yr after his prediction, negative molecular ions were indeed detected in the ISM. Since their discovery in the ISM, the carbon chain anions have been included in chemical modeling of the ISM16–18 and planetary atmospheres. There have been also suggested that several other anions, such as C2H−19 or CH2CN−20,21 could also be present in ISM environment similar to the one where the mentioned anions have been detected. The theoretical approach proposed by Herbst, the phase-space theory (PST), has been used in a number of studies3,15,16,22–25 to calculate the REA rate coefficients and to 0021-9606/2015/142(23)/234309/8/$30.00
model formation of anions in the ISM.16,17 The approach has been successful in interpreting the observed column density of C8H−, C6H−, C5N−, and C3N− ions, while the agreement with observations is not as good for the C4H− and CN− ions. The PST relies on several assumptions. First, it considers REA as a two-step process. As a first step, the incident electron is captured by a neutral target molecule M into an electronic state of M − through non-Born-Oppenheimer coupling, thus forming a vibrational resonance of M − in a highly excited vibrational level. The resonant state can decay back to the M + e− electronic continuum spectrum through the same nonBorn-Oppenheimer coupling, i.e., the electron can autodetach. Another possibility for the resonance, considered as the second PST step, is to emit a photon, thus stabilizing the M − system. In PST, it is assumed that the probability of the first step of the process is unity and, therefore, the cross section for the electron capture is approximated by the unitary limit formula for s-wave scattering3 σc = π/k 2, where k is the wave number of the incident electron with energy Eel = (~k)2/(2me ), and me is the electron mass. In the second PST step, stabilization of the M − resonance is represented as an emission of a photon by a set of harmonic oscillators of the molecule in the normal mode approximation of molecular vibrations. A larger number of available vibrational modes decreases the probability of autodetachment and thus allows for a larger probability in the second PST step. Therefore, in PST, the overall two-step REA cross section σPST grows rapidly with the number of atoms in the molecule and approaches the unitary limit σPST → σc . For example, the REA rate coefficient for formation of CN−
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calculated by PST is about 10−17 cm3/s25 and much larger, 3 × 10−7 cm3/s, for C6H−3 at 10 K. In the case of CN−, the theoretical value is smaller by several orders of magnitude than needed to explain the [CN−]/[CN] abundance ratio obtained from the astrophysical observations.11 The PST assumption about the unitary probability of the first step of the process can hardly be justified for small molecular ions. Nevertheless, there exists, in principle, an analog quantum-mechanical approach based on first principles to describe both steps of the REA mechanism suggested by Herbst. Because this mechanism is a two-step process, it will be referred to as indirect radiative electron attachment (IREA) in the following development. However, there also exists another mechanism for REA, which can be considered quantum-mechanically: the electron incident on a neutral molecule emits a photon and becomes bound without any intermediate step of capture into a vibrationally excited state of the ground electronic state of the negative ion. To distinguish the two mechanisms, we call the latter direct radiative electron attachment (DREA). The DREA and IREA mechanisms are schematically represented in Fig. 1 for the case of CN/CN−. Recently, we have developed a fully quantum theoretical approach for DREA based on first principles only.26 The approach considers the radiative electron attachment of a continuum electron through spontaneous decay to the anion ground state, as described as well in Fig. 1. Note that this process does not include an intermediate vibronic state of M − populated through the non-Born-Oppenheimer coupling, as represented by the two-step REA mechanism in PST. The approach actually relies on wave functions of the continuum spectrum of the M + e− system and transition dipole moments (TDMs) to the bound electronic state of M −, calculated ab initio. Using the developed approach, we have calculated the REA cross section and rate coefficient for the formation of the cyanide ion, CN−. Our results confirmed the previous assessment that the REA rate coefficient for CN− is too small, 8 × 10−16 cm3/s at 10 K,26 to explain the CN− abundance observed in the ISM. In the present study, we extend the theoretical approach26 to larger molecules, C2H and C4H. Because there is no experimental data on the REA process for carbon chain molecules, we use a similar theoretical approach to determine cross sections for the inverse process of DREA, namely, photodetachment (PD), and compare with available data from recent photodetachment experiments.27,28 This comparison allows us to verify and confirm the validity of our results. In addition, we also consider, using a quantum mechanical approach, the IREA for the simplest case of the CN molecule. More precisely, we calculate explicitly the probability of electron capture by CN during the first step of the IREA process using ab initio methods and compute the overall cross section of the IREA process. The indirect process for larger molecules will be considered in a separate study. It requires an additional theoretical development due to a large number of vibrational degrees of freedom in linear polyatomic molecules. In Sec. II, we present our theoretical approach to study DREA and apply it to the CN−, C2H−, and C4H− ions. Section III is devoted to the comparison of our results with data from photodetachment experiments. Section IV presents the
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FIG. 1. Schematic representation of two mechanisms for REA of an electron to the CN radical: (i) DREA: The electron incident on CN in its ground vibronic state spontaneously emits a photon of energy ~ω (green arrow) and forms the CN− ground state. The photon energy is equal to the difference between the initial total energy of the system E tot (horizontal dashed line at near energy of the v − = 19 state) and the energy of CN− ground state. (ii) IREA: As a first step of the process, the incident electron is captured via non-Born-Oppenheimer coupling into the ground electronic state of CN− without emitting a photon. Because the total energy E tot is conserved, the vibrational level v − of CN− in this first step of the process is highly excited: for low incident energies of the electron, it corresponds to v − = 18 or 19. After the first step, the electron captured in an excited vibrational level can either autodetach or stabilize by photon emission. The photon changes the rotational and vibrational states of the CN− molecule. The largest probability for the second step corresponds to a change of v − by one quantum. The dashed red line shows the derivative of the vibrational ground state wave function of CN, which enters in the calculation of the non-Born-Oppenheimer coupling in Eq. (12).
theoretical approach of IREA to the CN molecule. Finally, Sec. V summarizes the important findings of the study.
II. DIRECT MECHANISM OF RADIATIVE ELECTRON ATTACHMENT TO CN, C2H, AND C4H
The cross section for DREA of an electron to a neutral linear molecule M, such as Cn H (n = 2, 4) or Cm N (m = 1, 3), initially in its electronic ground state, vibrational level v, and energy Ei , was given in26 σi =
g f 16 πω3me (v→ v f ) 2 d , gi 3 k 2~2c3 lπ π,Γl−π
(1)
where v f is the vibrational state of the ion M − with total energy E f formed after DREA; ω is the frequency of the emitted photon, ~ω = Ei + Eel − E f ; g f = 1 and gi = 4 are multiplicities of final states and initial states. The above cross section
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takes into account averaging over initial rotational states of the neutral molecule and summation over final possible rotational (v→ v ) states of the final negative ion.26 The quantities d π, Γl λf are matrix elements of the components π = −1, 0, +1 of the dipole moment operator between the initial ΨΓl λ (M + incident electron) and the final Ψ f electronic states, integrated over the initial χ v (⃗q) and final χ v f (⃗q) vibrational wave functions of M and M −, respectively, where q⃗ denotes collectively all internuclear degrees of freedom. Their values are given by (v→ v f ) d π, Γl λ = χ v f (⃗q)⟨Ψ f |d π |ΨΓl λ ⟩ χ v (⃗q)d⃗q (2) (see details in Ref. 26), with l the electronic partial wave angular momentum and λ its projection on the molecular axis in the molecular frame. The matrix elements ⟨Ψ f |d π |ΨΓl λ ⟩ of the dipole moment operator dˆπ are given by the integral N ⟨Ψ f |d π |ΨΓl λ ⟩ = − Ψ∗f (r 1, . . . ,r N )er k π k=1
× ΨΓl λ (r 1, . . . ,r N )d 3r 1 · · · d 3r N ,
(3)
where the function Ψ f represents the N-electron final state of the negative ion (r 1, . . . ,r N are the coordinates of the electrons), ΨΓl λ is the electronic continuum state representing the scattering electron with l and λ angular quantum numbers, and Γ labels the initial neutral electronic target state with N − 1 electrons. Finally, r k π is one of the three cyclic components (π = 0, ±1) of the coordinate of the kth electron z k , if π = 0 √ (4) rk π = ∓(x k ± i yk )/ 2, if π = ±1. The calculations of the electronic wave functions and TDMs of Eq. (3) are performed using the complex Kohn variational method, extensively described in past studies.29,30 The electronic continuum functions ΨΓl λ in our calculations are normalized as described in Ref. 26, namely, at large distances, the incoming part of the total electronic wave functions behaves as lπ i rΨΓl λ ∼ √ e−i(k r − 2 ). (5) 2 k With respect to the energy-normalized continuum function φ E (r), also often used in calculations, ΨΓl λ can be written as 2me φ E (r) = ΨΓl λ . (6) π~2 To avoid any confusion, we stress that in this study, all formulas (except Eq. (9)), which contain explicitly or implicitly continuum wave functions, imply the same normalization as ΨΓl λ . The DREA cross section for the formation of CN−, starting from the ground vibrational level v = 0 of CN, was calculated in Ref. 26 using Eq. (1). The vibrational integral of Eq. (2) was computed explicitly from the geometry-dependent matrix elements ⟨Ψ f |d π |ΨΓl λ ⟩ obtained in the complex Kohn calculations. These matrix elements, as a function of the electron energy and of the internuclear distance, were, respectively, shown in Figs. 3 and 4 of Ref. 26. The calculation of the TDMs is computationally intensive, especially for polyatomic molecules. In Ref. 26, we found that
the TDMs weakly depend on the geometry of the molecule near its equilibrium position. Therefore, it seems reasonable to use the Franck-Condon approximation and simplify the calculation of the vibrational integral in Eq. (2) by using the value of ⟨Ψ f |d π |ΨΓl λ ⟩ at a fixed molecular geometry, e.g., the equilibrium position of the negative ion or the equilibrium of the neutral molecule, which are close to each other. The TDMs then take the simple form (v→ v ) d π, Γl λf ≈ ⟨Ψ f |d π |ΨΓl λ ⟩ Q0 χ∗v f (⃗q) χ v (⃗q)d⃗q, (7) where the subscript Q0 refers to the equilibrium geometry of the negative ion. For molecules such as Cn H and Cm N, for which the potential energy surfaces of the initial electronic state of the target and the final state of the negative ion are quite similar in shape near the equilibrium positions of the ion and the neutral molecule,31 the Franck-Condon integral in Eq. (7) is the largest for transitions with v f = v, for which its value is close to unity. For instance, the Franck-Condon integral in Eq. (7) is about 0.90 for C2H/C2H− and 0.87 for C4H/C4H−.31 For transitions to other vibrational levels, the integral is significantly smaller. Therefore, the DREA cross section is well approximated by σi ≈
2 4 πω3me ⟨Ψ f |d π |ΨΓl λ ⟩ Q0 , 2 2 3 3 k ~ c lπ
(8)
where the transition dipole moment is evaluated at the energy of the M + e− system. The transition dipole moments strongly depend on energy, especially for non-zero partial waves, l > 0. At low energies, their behavior is described quite well by the Wigner threshold law.32 The energy dependence of ⟨Ψ |d |Ψ ⟩ for C H/C H− and C H/C H− is shown in 2 2 4 4 f π Γl λ Q0 Figs. 2 and 3. For C2H and C4H, the DREA cross sections were calculated using the approximate formula of Eq. (8), with the transition dipole moments determined only at the equilibrium geometry Q0 of the negative ion for several energies of the incident electron. For comparison, the DREA cross section for CN− has been calculated using the direct integration over internuclear distances, Eqs. (1) and (2), as well as using Eq. (8). Figure 4 shows the DREA cross sections obtained for CN−, C2H−, and C4H−. As can be seen from the figure, the CN− cross sections obtained in the two ways are almost identical. Therefore, Eq. (8) gives a very good approximation of Eqs. (1) and (2) for CN−. For C2H/C2H− and C4H/C4H−, the approximation is likely less accurate because the vibrational functions of the neutral molecule and the ion are not as similar to each other as for CN−. In a future study, we will evaluate the vibrational integral of Eq. (2) numerically, at least, for some of the vibrational modes. This may result in a certain change of the final REA cross section for large molecules, such as C4H/C4H−. However, preliminary calculations have indicated that the change is not significant. The obtained DREA cross sections have been used to determine the thermal rate coefficients, which are shown in Fig. 5. The largest rate coefficient is obtained for CN/CN−, 7 × 10−16 cm3/s, at 300 K. From the point of view of astrophysical modeling of anion evolution in the ISM, such values of rate coefficients mean that the DREA process is too slow
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√ FIG. 2. Matrix elements of the d x = (d −1 − d +1)/ 2 and d z = d 0 components (Eq. (3)) of the transition dipole moment between the C2H− electronic ground state and the e − + C2H system for several partial waves as a function of the incident electron energy, calculated at the equilibrium geometry of C2H−.
to explain the densities of CN− and C4H− observed in the ISM. For example, two different models have suggested that the REA rate coefficient for the C4H− formation should be 9 × 10−11 cm3/s33 or 2 × 10−9 cm3/s33 in order to explain the observed densities of C4H and C4H−.
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FIG. 4. The cross sections for formation of CN−, C2H−, and C4H− by DREA with the ground vibrational level v = 0 of the neutral molecule as the initial state. The CN− cross section is calculated in two different ways: the dotted line corresponds to the calculation using Eq. (8) and the cross section shown with the solid line, “vib FT” (vibrational frame transformation), is obtained evaluating the vibrational integral of Eq. (2) explicitly.
III. COMPARISON WITH THE RESULTS OF PHOTODETACHMENT EXPERIMENTS
There is no experimental data on radiative electron attachment to the CN, C2H, and C4H radicals. However, the calculated transition dipole moments can be used to determine the photodetachment cross sections, for which experimental data
FIG. 3. Same as Fig. 2 for the e − + C4H system.
FIG. 5. The figure shows the thermally averaged rate coefficients ⟨vσ i (E)⟩ for the formation of the CN−, C2H−, and C4H− molecules in the DREA process obtained from cross sections shown in Fig. 4. The average is performed over the thermal distribution of electron-molecule collisional velocities at a given temperature.
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on absolute values of photodetachment cross sections have recently been obtained for CN−,27 C2H−,28 and C4H−28 anions. Here, we discuss the interpretation of the photodetachment experiments only briefly. A detailed and more elaborated study can be found in Ref. 31. The photodetachment cross section is given by (see, for example, Eq. (2.202) of Ref. 34) σPD =
2 4π 2ω d l,π, E , 3c lπ
(9)
where ω is the photon frequency and d l,π, E is the dipole moment between the initial bound state and the final continuum state with energy E and partial wave l of the photodetached electron. The radial part of the wave function φ E (r) of the initial electronic-continuum state used to calculate the transition dipole moment d l,π, E in Eq. (9) is energy-normalized. ∗ Thus, using functions ΨΓl λ , the PD cross section takes the form σPD =
8me πω (v→ v f ) 2 d . 3~2c lπ π, Γl−π
FIG. 6. Theoretical (lines) and experimental (symbols with uncertainty bars) photodetachment cross sections for CN−, C2H−, and C4H−. The triangle is the experimental result for CN−,27 diamonds—C2H−,28 and squares—C4H−.28
(10)
For convenience of comparison with the results of the DREA calculations, the photon frequency can be expressed in terms of the electron affinity Eea and the energy Eel of the incident (in DREA) or emitted (in PD) electron ~ω = Eel + Eea. This formula assumes that both initial and final vibrational levels are not excited, v = v − = 0, and that the zero-point energies for the neutral molecule and the ion are the same. The experimentally measured affinities are 3.862 ± 0.004 eV for CN,35 2.969 ± 0.001 eV for C2H,36,37 and 3.533 ± 0.001 eV for C4H.38,39 We discuss now the accuracy of the present calculations. The main source of uncertainty for the REA and PD cross sections for all three molecules considered here is the quality of electronic continuum and bound state wave functions. The quality of the electronic bound state wave function could be assessed in part by comparing the obtained theoretical affinity with the experimental one. The affinities corresponding to wave functions used in the present calculations are 3.8 eV, 2.2 eV, and 3.0 eV for CN−, C2H−, and C4H−, respectively. Therefore, the agreement is about 1% for CN− and much poorer, about 30%, for C2H− and C4H−. The quality of electronic continuum wave functions is more difficult to assess. Based on our previous experience with the electron-scattering calculations, we assume here that an additional uncertainty in the calculated transition dipole moments due to the quality of continuum wave functions is about 10% for CN−, C2H−, and C4H−. These considerations give an estimated uncertainty in the DREA and PD cross sections of about 20% for CN− and 40% for C2H− and C4H−. For C2H− and C4H−, there are additional sources of uncertainty; the neglected geometry dependence of the transition dipole moments and the neglected role of rovibrational Feshbach resonances, which could be present at low collisional energies. Figure 6 shows the PD cross sections calculated for CN−, − C2H , and C4H− using Eq. (10) and compares them with the available experimental data, which were estimated in Ref. 28 to have about 25% of uncertainty. The agreement is very good for C2H−, especially for experimental data points near the
photodetachment threshold. The agreement for C4H− is also good for the lowest energy point, but is about a factor 23 lower than the experimental value for the second energy point. Similarly, the only experimental data point for CN− measured at 4.65 eV gives a PD cross section that is twice larger than the theoretical value. The reason for this discrepancy is not clear. Overall, the agreement of the theoretical and experimental results is sufficient to conclude that the theory is reliable for calculations of photodetachment cross sections and, therefore, also for DREA cross sections. Therefore, the data from photodetachment experiments validate the present theoretical approach for the DREA process and the calculated cross sections obtained using the approach. In conclusion, the PD calculations and experiments provide an additional confirmation that the observed abundance of the CN− and C4H− ions in the ISM can hardly be explained by the DREA mechanism.
IV. THE CONTRIBUTION OF THE INDIRECT PROCESS TO THE TOTAL REA CROSS SECTION: THE CN EXAMPLE
We now consider the process of radiative attachment mediated by the non-adiabatic couplings, through the IREA mechanism discussed in the Introduction. In this section, we consider the IREA process for CN, for which the non-adiabatic couplings are evaluated numerically. The probability per unit time of a transition from the initial vibronic state |i⟩ of the e− + CN system described above into the final state of CN− being in a vibrationally excited level v f ∼ 19 (see Fig. 1) is given by the Fermi’s golden rule P=
2π ˆ 2 ⟨ f |Λ|i⟩ ρ(Ec ), ~
(11)
where ρ(Ec ) is the density of final states after the electron capture ρ(Ec ) ∼ 1/∆Erv, with ∆Erv the energy splitting between ˆ is the operator of non-adiabatic rovibrational states, and Λ coupling. Denoting by R the CN internuclear distance, the
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matrix element Λfi between initial and final vibronic states is ~2 ∂ χ v (R)dR, (12) Λfi = χ∗v f (R)Λ f ,Γl λ (R) µCN ∂R where µCN is the reduced mass of CN and Λ f ,Γl λ (R) is the following electronic matrix element: ∂ Λ f ,Γl λ (R) = Ψ∗f (⃗r , R) ΨΓl λ (⃗r , R)d⃗r . (13) ∂R In the above equation, ⃗r denotes collectively all electronic coordinates. Note that we neglected the second term of the non-adiabatic couplings, which is expressed with the second derivative of the electronic wave function with respect to the internuclear distance. While calculating Λfi, we did not account for the integral over rotational coordinates. The latter integral is of the order of unity or smaller. A change of rotational quantum number j during the process of the non-adiabatic electron attachment is unlikely for CN− at low collision energies since the values of the non-adiabatic couplings are by far the largest for s-wave scattering at low energy. This is the reason why we use the vibrational splitting in formula (11) to express the density of states ρ(Ec ) ∼ 1/∆Ev . Typically, non-adiabatic couplings can be calculated using standard ab initio programs. However, such calculations are usually limited to couplings between electronic bound states, whereas in the present case, the initial electronic state belongs to the electronic continuum spectrum of the molecule. One possible way to evaluate couplings with a continuum state using standard bound-states ab initio codes is to use a procedure analog to the Stieltjes imaging method40–42 applied in photoionization and photodetachment. By using a large set of square-integrable (L 2) Slater-type orbitals, including diffuse basis functions, we obtain a discretized representation of the continuous spectrum. As more diffuse functions are added to the basis set, a better description of the asymptotic region is achieved, which leads to the appearance of pseudo-state wave functions with positive asymptotic energy. Such states resemble a scattering state of the electron, as long as enough diffuse functions are included in the basis set. Moreover, at low electron energy, the short-range part of the pseudo-state wave function, which is responsible for the non-Born-Oppenheimer coupling with the electronic ground state, is barely energy dependent, up to a normalization factor. Despite the fact that this method cannot give the detailed energy dependence of the non-Born-Oppenheimer coupling, once the correct normalization per unit energy range is performed, a reasonable approximation of the non-adiabatic couplings should be obtained. Performing numerical calculations, we have verified convergence of results (the non-adiabatic couplings) with respect to the number of added diffuse functions. The calculated electronic pseudo-states Ψnl of CN + e− have discretized energies and are labeled by the index n (in order of increasing energy) and dominant angular momentum l. The states should be rescaled to be energy normalized by multiplying by a factor ∆Eel−1/2, where ∆Eel is the energy difference between pseudo-states. Because ∆Eel is changing from one state to another, it is taken here as the average between two energy differences for three neighboring pseudo-states. The calculated values of Λ f ,Γl λ (R) at several internuclear distances
FIG. 7. The matrix element of the non-adiabatic coupling operator of Eq. (13) evaluated for two different electronic “quasi-continuum” states with energies ∼10 and ∼60 meV above the electronic threshold. The continuum state of the incident electron is normalized in the same way as in Ref. 26, where the current density of electrons in the incident plane wave is equal to the electron velocity (see Eqs. (10), (A2), and (A19) in Ref. 26).
are plotted in Fig. 7 for two s-wave pseudo-states. These states, corresponding to n = 2 and 3, and have, respectively, ∼10 and ∼60 meV asymptotic electron energies above the electronic threshold. Notice that the non-adiabatic couplings for the two states are not much different and decrease with the internuclear distance. Because the p-wave scattering states cannot effectively penetrate the centrifugal barrier at low energy, we have found that their values are negligible in comparison with the couplings from s-waves scattering states. Finally, although the non-adiabatic couplings were obtained from an approximative treatment, their values should represent an estimation accurate enough for the purpose of this study. The electron capture cross section is then obtained by dividing the probability by the current density jcd in the incident wave. If the incident wave is normalized as in Eqs. (A2) and (A19) of Ref. 26, the current density is then simply equal to the velocity of the incident electron jcd = vcd. It gives the following estimation for the cross section: σc =
π 1 ~4 P ≈ jcd Eel ∆Ev µ2CN 2 ∂ × χ∗v f (R)Λ f ,Γl λ (R) χ v (R)dR . ∂R
(14)
In the above equation, we assumed that the ratio of the number of final rotational levels of the formed anion CN− to the number of initial rotational levels is approximately unity. Moreover, we consider that the value of Λ f ,Γl λ (R) is almost constant in the energy range under study. This represents a relatively good approximation, at least between 10 and 100 meV, on the view of the weak energy dependence of Λ f ,Γl λ (R), as seen in Fig. 7. We have calculated the vibrational integral numerically with the final vibrational level of CN− (v f = 19), which has an energy close to the energy of the initial vibrational level of CN (vi = 0) with a negligible asymptotic energy of the incident
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electron. We obtained the value ∂ χ∗19(R)Λ f , Γl λ (R) χ0(R)dR = −1.5 × 10−5 a.u., (15) ∂R expressed in a.u. (atomic units). The vibrational energy spacing ∆Ev is about 0.01 hartree (H) and the reduced mass µCN is 1.1 × 104 me (a.u.). Therefore, the cross section to capture an electron into a vibrational level is approximately 6 × 10−16 2 a0 . (16) Eel Estimate (16) is an upper bound for the IREA cross section because of the autodetachment process: the actual cross section is reduced because the formed CN− ion in the excited vibrational level can decay back to the CN molecule and a free electron. The overall cross section of the indirect REA is then Γsp σc ≤ σc , (17) σ I RE A = Γtot where Γtot is the total width for the decay of the unstable vibrational state of CN− formed during the collision. For the case of CN/CN−, the total width is the sum Γtot = Γsp + Γad of the widths towards autodetachment Γad and towards spontaneous emission into all possible vibrational levels. The rate coefficient Γad can be calculated using the Fermi’s golden rule and the same matrix element of the non-adiabatic coupling of Eq. (13). The density of final states is calculated differently since it will correspond to an outgoing electronic state. In this case, the density of states is unity because the radial part of the wave function is energy-normalized. Therefore, the probability of autodetachment per unit time is roughly σc ≈
2π |Λfi|2 ∼ 10−17 H/~ = 0.41 s−1, (18) ~ if, at the end of the process, the CN molecule is again in the ground vibrational level. The rate Γsp of spontaneous emission can also be estimated using the standard formula (see Eq. (2.189) of Ref. 34) Γad ∼
Γsp ∼
3 ωsp (v −→ v ′−) 2 d π , ~c3
(19)
π
′−
where d π(v → v ) is the vibrational matrix element of the permanent dipole moment of the anion d π (R) for vibrational transition v − → v ′−, − ′− χ∗v ′−(R)d π (R) χ v −(R)dR. d π(v → v ) = (20) −
In the above equation, v ′− is the vibrational level of CN− after emission of a photon, which should be smaller than v − in order to stabilize the anion. The vibrational matrix element is the largest for ∆v − = 1. For such a transition, the vibrational dipole matrix element is roughly equal to the derivative of d π (R) with respect to R. The order of magnitude of the derivative is about unity in a.u. and, therefore, the vibrational matrix element of the permanent dipole moment is about ea0. For transitions with ∆v − > 1, the matrix elements are significantly smaller. Thus, we only account for the v ′− = v − − 1 transition, for which ωsp = ∆Ev /~, and obtain Γsp ∼ 1.3 × 10−17 H/~ = 0.54 s−1. The rate coefficients Γad and Γsp are thus comparable to each other, meaning that σIREA ≈ σc for the CN molecule.
V. CONCLUSION
We have extended the theoretical approach to study the process of direct radiative electron attachment, developed in our previous study,43 to larger molecules. Using the approach, we have calculated DREA cross sections for the three negative molecular ions CN−, C2H−, and C4H−. The obtained DREA rate coefficients evaluated at temperature 30 K are 7 × 10−16 cm3/s for CN−, 7 × 10−17 cm3/s for C2H−, 2 × 10−16 cm3/s for C4H−. The coefficients depend weakly on temperature between 10 K and 100 K and increase relatively fast with temperature above 200 K. The validity of the obtained results is verified by comparing the present theoretical results with experimental data from recent photodetachment experiments. For formation of CN− by REA, we have also considered the indirect pathway, in which the incident electron is initially captured through non-Born-Oppenheimer coupling into a vibrationally excited state of the anion, forming a resonance, which can then stabilize the anion with respect to autodetachment by photon emission. The contribution of the indirect pathway was found to be negligible compared to the direct mechanism for CN. Theoretical description of the indirect pathway in polyatomic molecules is more complicated due to two reasons. One is that there are more vibrational degrees of freedom, which could be involved in the non-adiabatic capture of the electron. The second reason why the indirect process requires an additional theoretical development is a possible role of low-energy rovibrational resonances of the negative ion, which could exist near electronic detachment thresholds. For example, if a given anion M − has a weakly bound electronic state, say, M −∗ (molecules M with large permanent dipole moments may have one or several excited weekly bound electronic states), rovibrationally excited levels of M −∗ might actually belong to the electronic continuum of the e− + M system and, thus in principle, could play a role in the indirect REA cross section. The contribution of such resonances and low-energy threshold effects for molecules with large permanent dipole moments to the total REA cross section, as well as the role of multiple vibrational degrees of freedom in the non-adiabatic electron capture, will be discussed in future works. In this context, it is appropriate to mention that the currently accepted model (the PST model) of anion formation in the ISM implies an indirect REA mechanism,3,44 as discussed in the Introduction of this study. Our opinion is that this model should be used with a caution, at least, until the unitary-limit approximation for electron capture is justified for large molecules. As it was shown in this study, the approximation is totally unreasonable for CN/CN−. It is commonly believed that carbon-chain anions, Cn H− (n = 2, 4, 6, 8) or Cm N− (m = 3, 5), observed in the interstellar medium, are formed by the REA process. The DREA rate constants obtained in this study are too small to explain the observed abundance of the anions in the ISM. For example, for C4H−, the magnitude of the rate coefficient needed to explain the observed abundance should be of the order of 10−10 cm3/s.3 Although the indirect REA pathway was not considered for C2H and C4H in this study, our preliminary calculations have
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shown that the IREA cross sections for these molecules are very small compared to the DREA contribution. For other polyatomic molecules with relatively large permanent dipole moments, such as C6H or C3N, a more detailed study is needed and is under way. Thus, the present results may suggest that in the ISM, either dipole resonant states or non-local threshold effects increase drastically the REA cross section (and the rate of anion formation), or these anions are formed through a different process than REA. However, REA seems to be excluded as a possible formation mechanism for CN−. For the purpose of a quick estimation of the DREA cross section for molecules other than discussed in this study, one can use Eq. (8). In this equation, the transition dipole moment can be roughly estimated using any quantum chemistry code capable to calculate transition dipole moments between the ground state of the anion and quasi-continuum states, similarly as it is made above for vibronic coupling in CN/CN−. Such an estimate would give a good approximation for the DREA cross section for large molecules. ACKNOWLEDGMENTS
This work is supported by the National Science Foundation, Grant Nos. PHY-11-60611 and PHY-10-68785. Part of the material presented in this manuscript is based on work conducted while A. Orel was serving at NSF. 1M.
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Citation: J. Chem. Phys. 142, 234309 (2015); doi: 10.1063/1.4922691 View online: http://dx.doi.org/10.1063/1.4922691 View Table of Contents: http://aip.scitation.org/toc/jcp/142/23 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 142, 234309 (2015)
Theoretical study of radiative electron attachment to CN, C2H, and C4H radicals Nicolas Douguet,1 S. Fonseca dos Santos,1 Maurice Raoult,2 Olivier Dulieu,2 Ann E. Orel,1 and Viatcheslav Kokoouline3 1
Department of Chemical Engineering and Materials Science, University of California at Davis, Davis, California 95616, USA 2 Laboratoire Aimé Cotton, CNRS/Université Paris-Sud/ENS Cachan, bât. 505, Campus d’Orsay, 91405 Orsay Cedex, France 3 Department of Physics, University of Central Florida, Orlando, Florida 32816, USA
(Received 19 May 2015; accepted 7 June 2015; published online 19 June 2015) A first-principle theoretical approach to study the process of radiative electron attachment is developed and applied to the negative molecular ions CN−, C4H−, and C2H−. Among these anions, the first two have already been observed in the interstellar space. Cross sections and rate coefficients for formation of these ions by direct radiative electron attachment to the corresponding neutral radicals are calculated. For the CN molecule, we also considered the indirect pathway, in which the electron is initially captured through non-Born-Oppenheimer coupling into a vibrationally resonant excited state of the anion, which then stabilizes by radiative decay. We have shown that the contribution of the indirect pathway to the formation of CN− is negligible in comparison to the direct mechanism. The obtained rate coefficients for the direct mechanism at 30 K are 7 × 10−16 cm3/s for CN−, 7 × 10−17 cm3/s for C2H−, and 2 × 10−16 cm3/s for C4H−. These rates weakly depend on temperature between 10 K and 100 K. The validity of our calculations is verified by comparing the present theoretical results with data from recent photodetachment experiments. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4922691]
I. INTRODUCTION
The present theoretical study of radiative electron attachment (REA) to neutral molecules is motivated by the recent discoveries of molecular anions in the interstellar medium (ISM). Six anions have been detected so far in the ISM: C6H−,1–6 C4H−,7 C8H−,8,9 C3N−,10 C5N−,2,5 and CN−.11 More recently, based on analysis of data from 1986 Giotto mission, it was reported12 that certain anions could also be present in the coma of comet Halley. Limited resolution did not allow unambiguous identification of the species. The possibility for atomic anions, such as H−, to be formed in the ISM by REA was first suggested by McDowell13 in 1961. Later, Dalgarno and McCray14 have discussed the role of negative atomic ions in the formation of neutral molecules in the ISM. The formation of molecular anions in the ISM by REA has been proposed by Herbst,15 who has also developed a theoretical approach3,15 to evaluate rate coefficients for REA. More than 20 yr after his prediction, negative molecular ions were indeed detected in the ISM. Since their discovery in the ISM, the carbon chain anions have been included in chemical modeling of the ISM16–18 and planetary atmospheres. There have been also suggested that several other anions, such as C2H−19 or CH2CN−20,21 could also be present in ISM environment similar to the one where the mentioned anions have been detected. The theoretical approach proposed by Herbst, the phase-space theory (PST), has been used in a number of studies3,15,16,22–25 to calculate the REA rate coefficients and to 0021-9606/2015/142(23)/234309/8/$30.00
model formation of anions in the ISM.16,17 The approach has been successful in interpreting the observed column density of C8H−, C6H−, C5N−, and C3N− ions, while the agreement with observations is not as good for the C4H− and CN− ions. The PST relies on several assumptions. First, it considers REA as a two-step process. As a first step, the incident electron is captured by a neutral target molecule M into an electronic state of M − through non-Born-Oppenheimer coupling, thus forming a vibrational resonance of M − in a highly excited vibrational level. The resonant state can decay back to the M + e− electronic continuum spectrum through the same nonBorn-Oppenheimer coupling, i.e., the electron can autodetach. Another possibility for the resonance, considered as the second PST step, is to emit a photon, thus stabilizing the M − system. In PST, it is assumed that the probability of the first step of the process is unity and, therefore, the cross section for the electron capture is approximated by the unitary limit formula for s-wave scattering3 σc = π/k 2, where k is the wave number of the incident electron with energy Eel = (~k)2/(2me ), and me is the electron mass. In the second PST step, stabilization of the M − resonance is represented as an emission of a photon by a set of harmonic oscillators of the molecule in the normal mode approximation of molecular vibrations. A larger number of available vibrational modes decreases the probability of autodetachment and thus allows for a larger probability in the second PST step. Therefore, in PST, the overall two-step REA cross section σPST grows rapidly with the number of atoms in the molecule and approaches the unitary limit σPST → σc . For example, the REA rate coefficient for formation of CN−
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calculated by PST is about 10−17 cm3/s25 and much larger, 3 × 10−7 cm3/s, for C6H−3 at 10 K. In the case of CN−, the theoretical value is smaller by several orders of magnitude than needed to explain the [CN−]/[CN] abundance ratio obtained from the astrophysical observations.11 The PST assumption about the unitary probability of the first step of the process can hardly be justified for small molecular ions. Nevertheless, there exists, in principle, an analog quantum-mechanical approach based on first principles to describe both steps of the REA mechanism suggested by Herbst. Because this mechanism is a two-step process, it will be referred to as indirect radiative electron attachment (IREA) in the following development. However, there also exists another mechanism for REA, which can be considered quantum-mechanically: the electron incident on a neutral molecule emits a photon and becomes bound without any intermediate step of capture into a vibrationally excited state of the ground electronic state of the negative ion. To distinguish the two mechanisms, we call the latter direct radiative electron attachment (DREA). The DREA and IREA mechanisms are schematically represented in Fig. 1 for the case of CN/CN−. Recently, we have developed a fully quantum theoretical approach for DREA based on first principles only.26 The approach considers the radiative electron attachment of a continuum electron through spontaneous decay to the anion ground state, as described as well in Fig. 1. Note that this process does not include an intermediate vibronic state of M − populated through the non-Born-Oppenheimer coupling, as represented by the two-step REA mechanism in PST. The approach actually relies on wave functions of the continuum spectrum of the M + e− system and transition dipole moments (TDMs) to the bound electronic state of M −, calculated ab initio. Using the developed approach, we have calculated the REA cross section and rate coefficient for the formation of the cyanide ion, CN−. Our results confirmed the previous assessment that the REA rate coefficient for CN− is too small, 8 × 10−16 cm3/s at 10 K,26 to explain the CN− abundance observed in the ISM. In the present study, we extend the theoretical approach26 to larger molecules, C2H and C4H. Because there is no experimental data on the REA process for carbon chain molecules, we use a similar theoretical approach to determine cross sections for the inverse process of DREA, namely, photodetachment (PD), and compare with available data from recent photodetachment experiments.27,28 This comparison allows us to verify and confirm the validity of our results. In addition, we also consider, using a quantum mechanical approach, the IREA for the simplest case of the CN molecule. More precisely, we calculate explicitly the probability of electron capture by CN during the first step of the IREA process using ab initio methods and compute the overall cross section of the IREA process. The indirect process for larger molecules will be considered in a separate study. It requires an additional theoretical development due to a large number of vibrational degrees of freedom in linear polyatomic molecules. In Sec. II, we present our theoretical approach to study DREA and apply it to the CN−, C2H−, and C4H− ions. Section III is devoted to the comparison of our results with data from photodetachment experiments. Section IV presents the
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FIG. 1. Schematic representation of two mechanisms for REA of an electron to the CN radical: (i) DREA: The electron incident on CN in its ground vibronic state spontaneously emits a photon of energy ~ω (green arrow) and forms the CN− ground state. The photon energy is equal to the difference between the initial total energy of the system E tot (horizontal dashed line at near energy of the v − = 19 state) and the energy of CN− ground state. (ii) IREA: As a first step of the process, the incident electron is captured via non-Born-Oppenheimer coupling into the ground electronic state of CN− without emitting a photon. Because the total energy E tot is conserved, the vibrational level v − of CN− in this first step of the process is highly excited: for low incident energies of the electron, it corresponds to v − = 18 or 19. After the first step, the electron captured in an excited vibrational level can either autodetach or stabilize by photon emission. The photon changes the rotational and vibrational states of the CN− molecule. The largest probability for the second step corresponds to a change of v − by one quantum. The dashed red line shows the derivative of the vibrational ground state wave function of CN, which enters in the calculation of the non-Born-Oppenheimer coupling in Eq. (12).
theoretical approach of IREA to the CN molecule. Finally, Sec. V summarizes the important findings of the study.
II. DIRECT MECHANISM OF RADIATIVE ELECTRON ATTACHMENT TO CN, C2H, AND C4H
The cross section for DREA of an electron to a neutral linear molecule M, such as Cn H (n = 2, 4) or Cm N (m = 1, 3), initially in its electronic ground state, vibrational level v, and energy Ei , was given in26 σi =
g f 16 πω3me (v→ v f ) 2 d , gi 3 k 2~2c3 lπ π,Γl−π
(1)
where v f is the vibrational state of the ion M − with total energy E f formed after DREA; ω is the frequency of the emitted photon, ~ω = Ei + Eel − E f ; g f = 1 and gi = 4 are multiplicities of final states and initial states. The above cross section
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takes into account averaging over initial rotational states of the neutral molecule and summation over final possible rotational (v→ v ) states of the final negative ion.26 The quantities d π, Γl λf are matrix elements of the components π = −1, 0, +1 of the dipole moment operator between the initial ΨΓl λ (M + incident electron) and the final Ψ f electronic states, integrated over the initial χ v (⃗q) and final χ v f (⃗q) vibrational wave functions of M and M −, respectively, where q⃗ denotes collectively all internuclear degrees of freedom. Their values are given by (v→ v f ) d π, Γl λ = χ v f (⃗q)⟨Ψ f |d π |ΨΓl λ ⟩ χ v (⃗q)d⃗q (2) (see details in Ref. 26), with l the electronic partial wave angular momentum and λ its projection on the molecular axis in the molecular frame. The matrix elements ⟨Ψ f |d π |ΨΓl λ ⟩ of the dipole moment operator dˆπ are given by the integral N ⟨Ψ f |d π |ΨΓl λ ⟩ = − Ψ∗f (r 1, . . . ,r N )er k π k=1
× ΨΓl λ (r 1, . . . ,r N )d 3r 1 · · · d 3r N ,
(3)
where the function Ψ f represents the N-electron final state of the negative ion (r 1, . . . ,r N are the coordinates of the electrons), ΨΓl λ is the electronic continuum state representing the scattering electron with l and λ angular quantum numbers, and Γ labels the initial neutral electronic target state with N − 1 electrons. Finally, r k π is one of the three cyclic components (π = 0, ±1) of the coordinate of the kth electron z k , if π = 0 √ (4) rk π = ∓(x k ± i yk )/ 2, if π = ±1. The calculations of the electronic wave functions and TDMs of Eq. (3) are performed using the complex Kohn variational method, extensively described in past studies.29,30 The electronic continuum functions ΨΓl λ in our calculations are normalized as described in Ref. 26, namely, at large distances, the incoming part of the total electronic wave functions behaves as lπ i rΨΓl λ ∼ √ e−i(k r − 2 ). (5) 2 k With respect to the energy-normalized continuum function φ E (r), also often used in calculations, ΨΓl λ can be written as 2me φ E (r) = ΨΓl λ . (6) π~2 To avoid any confusion, we stress that in this study, all formulas (except Eq. (9)), which contain explicitly or implicitly continuum wave functions, imply the same normalization as ΨΓl λ . The DREA cross section for the formation of CN−, starting from the ground vibrational level v = 0 of CN, was calculated in Ref. 26 using Eq. (1). The vibrational integral of Eq. (2) was computed explicitly from the geometry-dependent matrix elements ⟨Ψ f |d π |ΨΓl λ ⟩ obtained in the complex Kohn calculations. These matrix elements, as a function of the electron energy and of the internuclear distance, were, respectively, shown in Figs. 3 and 4 of Ref. 26. The calculation of the TDMs is computationally intensive, especially for polyatomic molecules. In Ref. 26, we found that
the TDMs weakly depend on the geometry of the molecule near its equilibrium position. Therefore, it seems reasonable to use the Franck-Condon approximation and simplify the calculation of the vibrational integral in Eq. (2) by using the value of ⟨Ψ f |d π |ΨΓl λ ⟩ at a fixed molecular geometry, e.g., the equilibrium position of the negative ion or the equilibrium of the neutral molecule, which are close to each other. The TDMs then take the simple form (v→ v ) d π, Γl λf ≈ ⟨Ψ f |d π |ΨΓl λ ⟩ Q0 χ∗v f (⃗q) χ v (⃗q)d⃗q, (7) where the subscript Q0 refers to the equilibrium geometry of the negative ion. For molecules such as Cn H and Cm N, for which the potential energy surfaces of the initial electronic state of the target and the final state of the negative ion are quite similar in shape near the equilibrium positions of the ion and the neutral molecule,31 the Franck-Condon integral in Eq. (7) is the largest for transitions with v f = v, for which its value is close to unity. For instance, the Franck-Condon integral in Eq. (7) is about 0.90 for C2H/C2H− and 0.87 for C4H/C4H−.31 For transitions to other vibrational levels, the integral is significantly smaller. Therefore, the DREA cross section is well approximated by σi ≈
2 4 πω3me ⟨Ψ f |d π |ΨΓl λ ⟩ Q0 , 2 2 3 3 k ~ c lπ
(8)
where the transition dipole moment is evaluated at the energy of the M + e− system. The transition dipole moments strongly depend on energy, especially for non-zero partial waves, l > 0. At low energies, their behavior is described quite well by the Wigner threshold law.32 The energy dependence of ⟨Ψ |d |Ψ ⟩ for C H/C H− and C H/C H− is shown in 2 2 4 4 f π Γl λ Q0 Figs. 2 and 3. For C2H and C4H, the DREA cross sections were calculated using the approximate formula of Eq. (8), with the transition dipole moments determined only at the equilibrium geometry Q0 of the negative ion for several energies of the incident electron. For comparison, the DREA cross section for CN− has been calculated using the direct integration over internuclear distances, Eqs. (1) and (2), as well as using Eq. (8). Figure 4 shows the DREA cross sections obtained for CN−, C2H−, and C4H−. As can be seen from the figure, the CN− cross sections obtained in the two ways are almost identical. Therefore, Eq. (8) gives a very good approximation of Eqs. (1) and (2) for CN−. For C2H/C2H− and C4H/C4H−, the approximation is likely less accurate because the vibrational functions of the neutral molecule and the ion are not as similar to each other as for CN−. In a future study, we will evaluate the vibrational integral of Eq. (2) numerically, at least, for some of the vibrational modes. This may result in a certain change of the final REA cross section for large molecules, such as C4H/C4H−. However, preliminary calculations have indicated that the change is not significant. The obtained DREA cross sections have been used to determine the thermal rate coefficients, which are shown in Fig. 5. The largest rate coefficient is obtained for CN/CN−, 7 × 10−16 cm3/s, at 300 K. From the point of view of astrophysical modeling of anion evolution in the ISM, such values of rate coefficients mean that the DREA process is too slow
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√ FIG. 2. Matrix elements of the d x = (d −1 − d +1)/ 2 and d z = d 0 components (Eq. (3)) of the transition dipole moment between the C2H− electronic ground state and the e − + C2H system for several partial waves as a function of the incident electron energy, calculated at the equilibrium geometry of C2H−.
to explain the densities of CN− and C4H− observed in the ISM. For example, two different models have suggested that the REA rate coefficient for the C4H− formation should be 9 × 10−11 cm3/s33 or 2 × 10−9 cm3/s33 in order to explain the observed densities of C4H and C4H−.
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FIG. 4. The cross sections for formation of CN−, C2H−, and C4H− by DREA with the ground vibrational level v = 0 of the neutral molecule as the initial state. The CN− cross section is calculated in two different ways: the dotted line corresponds to the calculation using Eq. (8) and the cross section shown with the solid line, “vib FT” (vibrational frame transformation), is obtained evaluating the vibrational integral of Eq. (2) explicitly.
III. COMPARISON WITH THE RESULTS OF PHOTODETACHMENT EXPERIMENTS
There is no experimental data on radiative electron attachment to the CN, C2H, and C4H radicals. However, the calculated transition dipole moments can be used to determine the photodetachment cross sections, for which experimental data
FIG. 3. Same as Fig. 2 for the e − + C4H system.
FIG. 5. The figure shows the thermally averaged rate coefficients ⟨vσ i (E)⟩ for the formation of the CN−, C2H−, and C4H− molecules in the DREA process obtained from cross sections shown in Fig. 4. The average is performed over the thermal distribution of electron-molecule collisional velocities at a given temperature.
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on absolute values of photodetachment cross sections have recently been obtained for CN−,27 C2H−,28 and C4H−28 anions. Here, we discuss the interpretation of the photodetachment experiments only briefly. A detailed and more elaborated study can be found in Ref. 31. The photodetachment cross section is given by (see, for example, Eq. (2.202) of Ref. 34) σPD =
2 4π 2ω d l,π, E , 3c lπ
(9)
where ω is the photon frequency and d l,π, E is the dipole moment between the initial bound state and the final continuum state with energy E and partial wave l of the photodetached electron. The radial part of the wave function φ E (r) of the initial electronic-continuum state used to calculate the transition dipole moment d l,π, E in Eq. (9) is energy-normalized. ∗ Thus, using functions ΨΓl λ , the PD cross section takes the form σPD =
8me πω (v→ v f ) 2 d . 3~2c lπ π, Γl−π
FIG. 6. Theoretical (lines) and experimental (symbols with uncertainty bars) photodetachment cross sections for CN−, C2H−, and C4H−. The triangle is the experimental result for CN−,27 diamonds—C2H−,28 and squares—C4H−.28
(10)
For convenience of comparison with the results of the DREA calculations, the photon frequency can be expressed in terms of the electron affinity Eea and the energy Eel of the incident (in DREA) or emitted (in PD) electron ~ω = Eel + Eea. This formula assumes that both initial and final vibrational levels are not excited, v = v − = 0, and that the zero-point energies for the neutral molecule and the ion are the same. The experimentally measured affinities are 3.862 ± 0.004 eV for CN,35 2.969 ± 0.001 eV for C2H,36,37 and 3.533 ± 0.001 eV for C4H.38,39 We discuss now the accuracy of the present calculations. The main source of uncertainty for the REA and PD cross sections for all three molecules considered here is the quality of electronic continuum and bound state wave functions. The quality of the electronic bound state wave function could be assessed in part by comparing the obtained theoretical affinity with the experimental one. The affinities corresponding to wave functions used in the present calculations are 3.8 eV, 2.2 eV, and 3.0 eV for CN−, C2H−, and C4H−, respectively. Therefore, the agreement is about 1% for CN− and much poorer, about 30%, for C2H− and C4H−. The quality of electronic continuum wave functions is more difficult to assess. Based on our previous experience with the electron-scattering calculations, we assume here that an additional uncertainty in the calculated transition dipole moments due to the quality of continuum wave functions is about 10% for CN−, C2H−, and C4H−. These considerations give an estimated uncertainty in the DREA and PD cross sections of about 20% for CN− and 40% for C2H− and C4H−. For C2H− and C4H−, there are additional sources of uncertainty; the neglected geometry dependence of the transition dipole moments and the neglected role of rovibrational Feshbach resonances, which could be present at low collisional energies. Figure 6 shows the PD cross sections calculated for CN−, − C2H , and C4H− using Eq. (10) and compares them with the available experimental data, which were estimated in Ref. 28 to have about 25% of uncertainty. The agreement is very good for C2H−, especially for experimental data points near the
photodetachment threshold. The agreement for C4H− is also good for the lowest energy point, but is about a factor 23 lower than the experimental value for the second energy point. Similarly, the only experimental data point for CN− measured at 4.65 eV gives a PD cross section that is twice larger than the theoretical value. The reason for this discrepancy is not clear. Overall, the agreement of the theoretical and experimental results is sufficient to conclude that the theory is reliable for calculations of photodetachment cross sections and, therefore, also for DREA cross sections. Therefore, the data from photodetachment experiments validate the present theoretical approach for the DREA process and the calculated cross sections obtained using the approach. In conclusion, the PD calculations and experiments provide an additional confirmation that the observed abundance of the CN− and C4H− ions in the ISM can hardly be explained by the DREA mechanism.
IV. THE CONTRIBUTION OF THE INDIRECT PROCESS TO THE TOTAL REA CROSS SECTION: THE CN EXAMPLE
We now consider the process of radiative attachment mediated by the non-adiabatic couplings, through the IREA mechanism discussed in the Introduction. In this section, we consider the IREA process for CN, for which the non-adiabatic couplings are evaluated numerically. The probability per unit time of a transition from the initial vibronic state |i⟩ of the e− + CN system described above into the final state of CN− being in a vibrationally excited level v f ∼ 19 (see Fig. 1) is given by the Fermi’s golden rule P=
2π ˆ 2 ⟨ f |Λ|i⟩ ρ(Ec ), ~
(11)
where ρ(Ec ) is the density of final states after the electron capture ρ(Ec ) ∼ 1/∆Erv, with ∆Erv the energy splitting between ˆ is the operator of non-adiabatic rovibrational states, and Λ coupling. Denoting by R the CN internuclear distance, the
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matrix element Λfi between initial and final vibronic states is ~2 ∂ χ v (R)dR, (12) Λfi = χ∗v f (R)Λ f ,Γl λ (R) µCN ∂R where µCN is the reduced mass of CN and Λ f ,Γl λ (R) is the following electronic matrix element: ∂ Λ f ,Γl λ (R) = Ψ∗f (⃗r , R) ΨΓl λ (⃗r , R)d⃗r . (13) ∂R In the above equation, ⃗r denotes collectively all electronic coordinates. Note that we neglected the second term of the non-adiabatic couplings, which is expressed with the second derivative of the electronic wave function with respect to the internuclear distance. While calculating Λfi, we did not account for the integral over rotational coordinates. The latter integral is of the order of unity or smaller. A change of rotational quantum number j during the process of the non-adiabatic electron attachment is unlikely for CN− at low collision energies since the values of the non-adiabatic couplings are by far the largest for s-wave scattering at low energy. This is the reason why we use the vibrational splitting in formula (11) to express the density of states ρ(Ec ) ∼ 1/∆Ev . Typically, non-adiabatic couplings can be calculated using standard ab initio programs. However, such calculations are usually limited to couplings between electronic bound states, whereas in the present case, the initial electronic state belongs to the electronic continuum spectrum of the molecule. One possible way to evaluate couplings with a continuum state using standard bound-states ab initio codes is to use a procedure analog to the Stieltjes imaging method40–42 applied in photoionization and photodetachment. By using a large set of square-integrable (L 2) Slater-type orbitals, including diffuse basis functions, we obtain a discretized representation of the continuous spectrum. As more diffuse functions are added to the basis set, a better description of the asymptotic region is achieved, which leads to the appearance of pseudo-state wave functions with positive asymptotic energy. Such states resemble a scattering state of the electron, as long as enough diffuse functions are included in the basis set. Moreover, at low electron energy, the short-range part of the pseudo-state wave function, which is responsible for the non-Born-Oppenheimer coupling with the electronic ground state, is barely energy dependent, up to a normalization factor. Despite the fact that this method cannot give the detailed energy dependence of the non-Born-Oppenheimer coupling, once the correct normalization per unit energy range is performed, a reasonable approximation of the non-adiabatic couplings should be obtained. Performing numerical calculations, we have verified convergence of results (the non-adiabatic couplings) with respect to the number of added diffuse functions. The calculated electronic pseudo-states Ψnl of CN + e− have discretized energies and are labeled by the index n (in order of increasing energy) and dominant angular momentum l. The states should be rescaled to be energy normalized by multiplying by a factor ∆Eel−1/2, where ∆Eel is the energy difference between pseudo-states. Because ∆Eel is changing from one state to another, it is taken here as the average between two energy differences for three neighboring pseudo-states. The calculated values of Λ f ,Γl λ (R) at several internuclear distances
FIG. 7. The matrix element of the non-adiabatic coupling operator of Eq. (13) evaluated for two different electronic “quasi-continuum” states with energies ∼10 and ∼60 meV above the electronic threshold. The continuum state of the incident electron is normalized in the same way as in Ref. 26, where the current density of electrons in the incident plane wave is equal to the electron velocity (see Eqs. (10), (A2), and (A19) in Ref. 26).
are plotted in Fig. 7 for two s-wave pseudo-states. These states, corresponding to n = 2 and 3, and have, respectively, ∼10 and ∼60 meV asymptotic electron energies above the electronic threshold. Notice that the non-adiabatic couplings for the two states are not much different and decrease with the internuclear distance. Because the p-wave scattering states cannot effectively penetrate the centrifugal barrier at low energy, we have found that their values are negligible in comparison with the couplings from s-waves scattering states. Finally, although the non-adiabatic couplings were obtained from an approximative treatment, their values should represent an estimation accurate enough for the purpose of this study. The electron capture cross section is then obtained by dividing the probability by the current density jcd in the incident wave. If the incident wave is normalized as in Eqs. (A2) and (A19) of Ref. 26, the current density is then simply equal to the velocity of the incident electron jcd = vcd. It gives the following estimation for the cross section: σc =
π 1 ~4 P ≈ jcd Eel ∆Ev µ2CN 2 ∂ × χ∗v f (R)Λ f ,Γl λ (R) χ v (R)dR . ∂R
(14)
In the above equation, we assumed that the ratio of the number of final rotational levels of the formed anion CN− to the number of initial rotational levels is approximately unity. Moreover, we consider that the value of Λ f ,Γl λ (R) is almost constant in the energy range under study. This represents a relatively good approximation, at least between 10 and 100 meV, on the view of the weak energy dependence of Λ f ,Γl λ (R), as seen in Fig. 7. We have calculated the vibrational integral numerically with the final vibrational level of CN− (v f = 19), which has an energy close to the energy of the initial vibrational level of CN (vi = 0) with a negligible asymptotic energy of the incident
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electron. We obtained the value ∂ χ∗19(R)Λ f , Γl λ (R) χ0(R)dR = −1.5 × 10−5 a.u., (15) ∂R expressed in a.u. (atomic units). The vibrational energy spacing ∆Ev is about 0.01 hartree (H) and the reduced mass µCN is 1.1 × 104 me (a.u.). Therefore, the cross section to capture an electron into a vibrational level is approximately 6 × 10−16 2 a0 . (16) Eel Estimate (16) is an upper bound for the IREA cross section because of the autodetachment process: the actual cross section is reduced because the formed CN− ion in the excited vibrational level can decay back to the CN molecule and a free electron. The overall cross section of the indirect REA is then Γsp σc ≤ σc , (17) σ I RE A = Γtot where Γtot is the total width for the decay of the unstable vibrational state of CN− formed during the collision. For the case of CN/CN−, the total width is the sum Γtot = Γsp + Γad of the widths towards autodetachment Γad and towards spontaneous emission into all possible vibrational levels. The rate coefficient Γad can be calculated using the Fermi’s golden rule and the same matrix element of the non-adiabatic coupling of Eq. (13). The density of final states is calculated differently since it will correspond to an outgoing electronic state. In this case, the density of states is unity because the radial part of the wave function is energy-normalized. Therefore, the probability of autodetachment per unit time is roughly σc ≈
2π |Λfi|2 ∼ 10−17 H/~ = 0.41 s−1, (18) ~ if, at the end of the process, the CN molecule is again in the ground vibrational level. The rate Γsp of spontaneous emission can also be estimated using the standard formula (see Eq. (2.189) of Ref. 34) Γad ∼
Γsp ∼
3 ωsp (v −→ v ′−) 2 d π , ~c3
(19)
π
′−
where d π(v → v ) is the vibrational matrix element of the permanent dipole moment of the anion d π (R) for vibrational transition v − → v ′−, − ′− χ∗v ′−(R)d π (R) χ v −(R)dR. d π(v → v ) = (20) −
In the above equation, v ′− is the vibrational level of CN− after emission of a photon, which should be smaller than v − in order to stabilize the anion. The vibrational matrix element is the largest for ∆v − = 1. For such a transition, the vibrational dipole matrix element is roughly equal to the derivative of d π (R) with respect to R. The order of magnitude of the derivative is about unity in a.u. and, therefore, the vibrational matrix element of the permanent dipole moment is about ea0. For transitions with ∆v − > 1, the matrix elements are significantly smaller. Thus, we only account for the v ′− = v − − 1 transition, for which ωsp = ∆Ev /~, and obtain Γsp ∼ 1.3 × 10−17 H/~ = 0.54 s−1. The rate coefficients Γad and Γsp are thus comparable to each other, meaning that σIREA ≈ σc for the CN molecule.
V. CONCLUSION
We have extended the theoretical approach to study the process of direct radiative electron attachment, developed in our previous study,43 to larger molecules. Using the approach, we have calculated DREA cross sections for the three negative molecular ions CN−, C2H−, and C4H−. The obtained DREA rate coefficients evaluated at temperature 30 K are 7 × 10−16 cm3/s for CN−, 7 × 10−17 cm3/s for C2H−, 2 × 10−16 cm3/s for C4H−. The coefficients depend weakly on temperature between 10 K and 100 K and increase relatively fast with temperature above 200 K. The validity of the obtained results is verified by comparing the present theoretical results with experimental data from recent photodetachment experiments. For formation of CN− by REA, we have also considered the indirect pathway, in which the incident electron is initially captured through non-Born-Oppenheimer coupling into a vibrationally excited state of the anion, forming a resonance, which can then stabilize the anion with respect to autodetachment by photon emission. The contribution of the indirect pathway was found to be negligible compared to the direct mechanism for CN. Theoretical description of the indirect pathway in polyatomic molecules is more complicated due to two reasons. One is that there are more vibrational degrees of freedom, which could be involved in the non-adiabatic capture of the electron. The second reason why the indirect process requires an additional theoretical development is a possible role of low-energy rovibrational resonances of the negative ion, which could exist near electronic detachment thresholds. For example, if a given anion M − has a weakly bound electronic state, say, M −∗ (molecules M with large permanent dipole moments may have one or several excited weekly bound electronic states), rovibrationally excited levels of M −∗ might actually belong to the electronic continuum of the e− + M system and, thus in principle, could play a role in the indirect REA cross section. The contribution of such resonances and low-energy threshold effects for molecules with large permanent dipole moments to the total REA cross section, as well as the role of multiple vibrational degrees of freedom in the non-adiabatic electron capture, will be discussed in future works. In this context, it is appropriate to mention that the currently accepted model (the PST model) of anion formation in the ISM implies an indirect REA mechanism,3,44 as discussed in the Introduction of this study. Our opinion is that this model should be used with a caution, at least, until the unitary-limit approximation for electron capture is justified for large molecules. As it was shown in this study, the approximation is totally unreasonable for CN/CN−. It is commonly believed that carbon-chain anions, Cn H− (n = 2, 4, 6, 8) or Cm N− (m = 3, 5), observed in the interstellar medium, are formed by the REA process. The DREA rate constants obtained in this study are too small to explain the observed abundance of the anions in the ISM. For example, for C4H−, the magnitude of the rate coefficient needed to explain the observed abundance should be of the order of 10−10 cm3/s.3 Although the indirect REA pathway was not considered for C2H and C4H in this study, our preliminary calculations have
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shown that the IREA cross sections for these molecules are very small compared to the DREA contribution. For other polyatomic molecules with relatively large permanent dipole moments, such as C6H or C3N, a more detailed study is needed and is under way. Thus, the present results may suggest that in the ISM, either dipole resonant states or non-local threshold effects increase drastically the REA cross section (and the rate of anion formation), or these anions are formed through a different process than REA. However, REA seems to be excluded as a possible formation mechanism for CN−. For the purpose of a quick estimation of the DREA cross section for molecules other than discussed in this study, one can use Eq. (8). In this equation, the transition dipole moment can be roughly estimated using any quantum chemistry code capable to calculate transition dipole moments between the ground state of the anion and quasi-continuum states, similarly as it is made above for vibronic coupling in CN/CN−. Such an estimate would give a good approximation for the DREA cross section for large molecules. ACKNOWLEDGMENTS
This work is supported by the National Science Foundation, Grant Nos. PHY-11-60611 and PHY-10-68785. Part of the material presented in this manuscript is based on work conducted while A. Orel was serving at NSF. 1M.
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