Dissociative recombination and vibrational excitation of BF⁺ in low energy electron collisions

Ab initio R-matrix calculations of electron collision with the BF⁺ ion yielding the relevant molecular data were performed by three of us (Chakrabarti and Tennyson 2009, Chakrabarti et al 2011). The R-matrix method is a state-of-the-art quantum scattering technique which can be used to calculate the properties of bound and resonant electronic states of a molecule (Tennyson 2010). The BF⁺ target states were first obtained by performing a configuration interaction (CI) calculation. Subsequently this CI target wave function was used in an R-matrix calculation. The bound states of the BF molecule were obtained by searching for negative energy solutions using the BOUND program available within the R-Matrix code suite (Sarpal et al 1991, Carr et al 2012). This also produces quantum defects of the bound states. Resonances were detected and fitted to a Breit–Wigner profile (Tennyson and Noble 1984) to obtain their energies and widths. Subsequently the electronic couplings ${{V}_{{{d}_{j}},l}}$ were obtained from the resonance widths ${{ \Gamma }_{{{d}_{j}},l}}$ using the relation

$$\begin{eqnarray}&&{{V}_{{{d}_{j}},l}}=\sqrt{\frac{{{ \Gamma }_{{{d}_{j}},l}}}{2\pi}}.\end{eqnarray} \tag{ 8 }$$

The calculations were repeated for 11 internuclear distances in the range $1.5~{{a}_{0}}-3.5~{{a}_{0}}$. While the R-matrix calculations used several partial waves to represent the scattering, one single global (i.e. not l-resolved) resonance width was produced for each symmetry. Consequently, one generic partial wave for each symmetry, corresponding to the highest effective quantum number produced by the R-matrix computation (see figure 3 from Chakrabarti et al (2011)), was considered to take in charge the total strength of the Rydberg-valence interaction in the current calculations. Dissociative curves were constructed using the resonance data above the ion PEC and their continuation as bound states below the ion PEC.

Our earlier studies on molecular systems such as H₂ (Epée Epée et al 2015), HD (Motapon et al 2014), NO (Motapon et al 2006), N₂ (Little et al 2014) and CO (Mezei et al 2015) revealed that accurate and complementary R-matrix and quantum chemistry calculations are needed for a reliable description of the molecular dynamics in electron/molecular cation collisions. In particular, we found that the DR cross section is extremely sensitive to the position of the PECs of the neutral dissociative states with respect to that of the target ion. Indeed, a slight change of the crossing point of the PEC of a neutral dissociative state with that of the ion ground state can lead to a significant change in the predicted DR cross section. In addition, the PECs of the dissociative states must also go to the correct asymptotic limits for large values of the internuclear distance R.

In order to fulfil these accuracy criteria, we smoothly matched the PECs corresponding to R-matrix resonances, situated above the ion PEC, to branches of adiabatic PECs coming from previous quantum chemistry calculations (Magoulas et al 2013). Some of these latter PECs allowed us to complete the relevant diabatic landscape. More specifically, since for the states having ${{}^{1}}{{ \Sigma }^{+}}$ and ${{}^{1}} \Pi$ symmetry the avoided crossings involved always two adiabatic states only, the adiabatic PECs can be a transformed into (quasi-)diabatic PECs using a $2\times 2$ rotation matrix (Roos et al 2009). Meanwhile, far from the avoided crossing, we assumed that the adiabatic and diabatic potential curves are identical. This assumption forces the rotational angle to be a function that varies steeply from 0 to $\pi /2$, chosen in the present case a tangent hyperbolic function. In this way we were able to extract the diabatic PECs for the dissociative states with singlet symmetries relevant for DR.

The electronic couplings provided by the R-matrix theory (Chakrabarti et al 2011) for the triplet states are at least one or even two orders of magnitude smaller then those of the singlet ones, so they are omitted in the present calculation.

The data on the PECs are limitted to internuclear distances with $R\leqslant 8$ a₀. We completed the diabatic PECs by adding a long-range tail of $D-{\sum}_{n}{{C}_{n}}/{{R}^{n}}$ type (Stone 1996, Lepers and Dulieu 2011) for larger values of R, where D is the dissociation limit as given in the Kramida et al (2015). For the ground state of BF⁺ , the leading term ${{C}_{3}}/{{R}^{3}}$ of the multipolar expansion comes from the interaction between the charge of B⁺ and the quadrupole moment of fluorine in its ground state $^{2}{{P}^{o}}\left({{M}_{L}}=0\right)$, with M_L the azimuthal quantum number of F with respect to the internuclear axis. The corresponding coefficient C₃  =  Q  =  0.731 a.u. Medved et al (2000) gives rise to a repulsive interaction, which is balanced by the next term ${{C}_{4}}/{{R}^{4}}$ of the multipolar expansion, due to the polarisation of the electronic cloud of F by the ion B⁺ . The coefficient is given by ${{C}_{4}}=-{{\alpha}_{zz}}/2=-1.72$ a.u., where ${{\alpha}_{zz}}=3.43$ a.u. (Medved et al 2000) is the static dipole polarisability of the ground state ${{}^{2}}{{P}^{o}}\left({{M}_{L}}=0\right)$ of fluorine. As for the states of the BF, the C_n coefficients are calculated as fitting parameters to ensure a smooth connection with the existing part of the PECs.

The top panel of figure 1 shows the PEC of the ion ground state—together with its vibrational structure—and the PECs of the DR-relevant dissociative states obtained using the procedure outlined above and taken into account in the present MQDT calculation. The electronic couplings of these latter states with the ionization continua, computed from the R-Matrix-produced autoionization widths followed by Gaussian-type extrapolations to small and large internuclear distances, are shown in the mid panel of the same figure. The bottom panel gives the quantum defects characterising the Rydberg series for the different symmetries. They are small in absolute value and slowly variable with the internuclear distance R, with a notable exception in a small region around 2.5 a₀ in the case of the ${{}^{1}} \Delta$ symmetry, due to local relatively strong Rydberg-valence interactions.

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Using the set of molecular data (PECs, electronic couplings and quantum defects) determined as described in the previous section, we performed a series of MQDT calculations of cross sections for DR and competitive processes, assuming BF⁺ to be initially in its electronic ground state X ${{}^{2}}{{ \Sigma }^{+}}$ and on one of its lowest vibrational levels $v_{\text{i}}^{+}=0,1$ and 2. We have considered that the reactive processes take place via BF-states of total symmetry of ${{}^{1}} \Delta$, ${{}^{1}} \Pi$, ${{}^{1}}{{ \Sigma }^{+}}$—see figure 1—and we neglected rotational and spin-orbit effects.

We consider incident electron energies from 0.01 meV up to 5 eV—the dissociation energy of BF⁺ X ${{}^{2}}{{ \Sigma }^{+}}$ electronic state—and all the 30 vibrational levels of the ion are included in the calculations.

At very low energy, all the excited levels are associated with closed ionisation channels, as defined in section 2, responsible for temporary resonant capture into Rydberg states. As the energy increases, more and more ionisation channels open, which results in autoionization, leading to competitive processes, such as IC and SEC, and decreasing the flux of DR.

The direct electronic couplings between ionisation and dissociation channels—mentioned in paragraph (i) of section 2—were extracted from the autoionization widths (see equation (8)) of the valence states calculated by Chakrabarti and Tennyson (2009). For each dissociation channel available, we considered interaction with the most relevant series of Rydberg states including only one partial wave for each symmetry.

By examining the magnitude of the valence-Rydberg couplings (shown in the middle panel of figure 1) and the positions of the crossing points between the dissociative states correlating to the B(2p¹)  +  F(2p⁵) atomic limits with the ionic ground state—blue and black curves in the first figure of the upper panel of the same figure—at low collision energies (up to 1 eV) one may predict that the major part of the total cross section comes from the ${{}^{1}} \Delta$ symmetry (solid black line in figure 2), while the ${{}^{1}} \Sigma$ and ${{}^{1}} \Pi$ symmetries (solid green and red lines in the same figure) only make a minor contribution to the cross section. The importance of these latter two symmetries is revealed at higher collision energies, when the crossing of the first dissociative state with the molecular ion becomes favorable. At collision energies about 1 eV, the ${{}^{1}}{{ \Sigma }^{+}}$ states give a larger contribution to the DR cross section than the ${{}^{1}} \Delta$ symmetry, which in turn is exceeded by the ${{}^{1}} \Pi$ symmetry at about 2 eV. Moreover, one can notice that at higher collision energies, around 3.5 eV, when all the dissociative states are open, the DR cross section shows a sharp revival and, at 5 eV, the resonance structures disappear since all the ionization channels are open, and consequently the direct process only drives the DR.

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Figure 2 shows the importance of the indirect process for the DR of vibrationally relaxed BF⁺ at low energies.

Within the ${{}^{1}} \Delta$ symmetry, the entrance ionization channel associated with the $v_{\text{i}}^{+}=0$ state is strongly coupled to the dissociation continuum (see figure 1), allowing the direct process to dominate. The indirect process diminishes the cross section— destructive interferences with the direct one—due to stronger coupling between closed channels associated with highly-excited vibrational levels—v⁺   =  12–16—and the dissociative ones.

Within the ${{}^{1}} \Pi$ symmetry one finds the opposite. The direct process gives only a minor contribution to the DR cross section, due to the poor Franck–Condon overlap between the vibrational wave function of the initial state of the ion and the dissociative neutral states. However, the much stronger couplings between some closed ionisation channels—v⁺   =  8–10—and the dissociative states significantly increases the total cross section through the indirect mechanism (Pop et al 2012, Schneider et al 2012) resulting into constructive quantum interferences.

Finally, for the states with ${{}^{1}}{{ \Sigma }^{+}}$ symmetry, the indirect process plays a relatively minor role, the magnitude of the total cross section being given mainly by the direct process. The indirect process is responsible for resonant structures which are a consequences of both constructive and destructive interference.

For a given initial vibrational level of BF⁺ , the total DR cross section is obtained by summing over the partial cross sections of the three symmetries contributing to the process—${{}^{1}} \Delta$, ${{}^{1}} \Pi$ and ${{}^{1}}{{ \Sigma }^{+}}$—given in equation (6). The results of the calculations performed for the lowest three vibrational levels of the ground electronic state of the ion are shown in figure 3.

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Besides the dissociative recombination cross sections presented in figure 3, we use equation (7) to calculate the vibrational excitation (VE) and de-excitation (VdE) cross sections for the same vibrational levels of the target. Since many of the features of the VE and VdE cross sections either are similar to, or may be understood with the same reasoning as those of DR, we do not display them here. Instead, we present in this section their Maxwell rate coefficients for a broad range of electronic temperatures, in comparison with those of the DR. All these data are relevant for cold non-equilibrium plasmas.

Figure 4 shows the total DR and vibrational transition—VE and VdE—rate coefficients for the three lowest vibrational levels of the ion. The DR rate for $v_{\text{i}}^{+}=0$ shows different behaviour to those for the $v_{\text{i}}^{+}=1$ and 2 states. It has a maximum at an electron temperature T_e of about 400 K, while the rates for the other two states are monotonically decreasing with T_e, reaching nearly the same magnitude at higher temperatures. This behaviour for $v_{\text{i}}^{+}=0$ comes from the strong resonant peaks just below 0.01 eV, see figure 2 and black curve in figure 3.

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For other recently studied molecular systems, N₂ (Little et al 2014) and CO (Mezei et al 2015), the DR clearly dominates the electron-ion collisions. However, for BF the calculated VdE rate coefficients are larger than the DR-ones, and VE rate coefficients exceed the DR-ones at high T_e by almost an order of magnitude. Thus, one can conclude that in the case of the BF⁺ containing plasma, the internal vibrational energy is much more important for energy-exchange via VE and VdE, than the kinetic energy release via the exothermic DR.

To facilitate the use of our rate coefficients (figure 4) in modeling calculations, we fit them using modified Arrhenius-type formulas. The calculated DR rate coefficients for v  =  0, 1, 2 are given by :

$$\begin{eqnarray}&&k_{\text{B}{{\text{F}}^{+}},v}^{\text{DR}}\left({{T}_{\text{e}}}\right)={{A}_{v}}{{T}_{\text{e}}}^{{{\alpha}_{v}}}\exp \left[-\underset{i=1}{\overset{7}{\sum}}\,\frac{{{B}_{v}}(i)}{i\centerdot {{T}_{\text{e}}}^{i}}\right]\end{eqnarray} \tag{ 9 }$$

over the electron temperature range 100 K  <  T_e  <  3000 K. The parameters A_v, ${{\alpha}_{v}}$, and B_v(i) are listed in table 1. The corresponding formula for the vibrational transitions (VE and VdE) has the form:

$$\begin{eqnarray}&&k_{\text{B}{{\text{F}}^{+}},{{v}^{\prime}}\to {{v}^{\prime \prime}}}^{\text{DR}}\left({{T}_{\text{e}}}\right)={{A}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}{{T}_{\text{e}}}^{{{\alpha}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}}\exp \left[-\underset{i=1}{\overset{7}{\sum}}\,\frac{{{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(i)}{i\centerdot {{T}_{\text{e}}}^{i}}\right]\end{eqnarray} \tag{ 10 }$$

over the electron temperature range T_min  <  T_e  <  3000 K. The parameters T_min, ${{A}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$, ${{\alpha}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$, and ${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(i)$ for $i=1,2,\ldots,7$ are given in tables 2–4.

Table 1. Parameters used in equation (9) to represent the DR rate coefficient of BF⁺.

v	Av	${{\alpha}_{v}}$	Bv(1)	Bv(2)	Bv(3)	Bv(4)	Bv(5)	Bv(6)	Bv(7)
0	0.703 050 632[−5]	−0.108 298 227[1]	−0.363 984 388[2]	0.685 227 53[6]	−0.341 534 791[9]	0.862 623 62[11]	−0.117 919 938[14]	0.821 543 800[15]	−0.229 194 862[17]
1	0.205 402 433[−11]	0.557 133 316[0]	−0.185 580 405[4]	0.110 399 946[7]	−0.316 387 400[9]	0.531 242 790[11]	−0.525 109 108[13]	0.283 203 635[15]	−0.643 843 159[16]
2	0.144 845 145[−5]	−0.806 777 388[0]	0.285 629 778[4]	−0.322 036 339[7]	0.160 926 033[10]	−0.409 338 097[12]	0.558 087 964[14]	−0.388 815 002[16]	0.108 682 270[18]

Note: Powers of 10 are given in square brackets.

Table 2. Parameters used in equation (10) to represent the VE rate coefficient of BF⁺ (v  =  0).

${{v}^{\prime}}\to {{v}^{\prime \prime}}$	Tmin(K)	${{A}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$	${{\alpha}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(1)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(2)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(3)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(4)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(5)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(6)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(7)$
$0\to 1$	100	0.262 533 924[−5]	−0.547 999 875[0]	0.214 096 395[4]	0.639 318 590[6]	−0.338 435 157[9]	0.871 079 196[11]	−0.119 891 991[14]	0.844 485 990[15]	−0.238 785 179[17]
$0\to 2$	250	0.173 038 633[−11]	0.899 303 013[0]	−0.154 506 863[4]	0.103 113 831[8]	−0.802 809 891[10]	0.345 037 537[13]	−0.831 154 448[15]	0.104 995 341[18]	−0.541 063 017[19]
$0\to 3$	350	0.228 322 632[−19]	0.298 753 871[1]	0.362 608 159[4]	−0.187 052 923[8]	0.462 415 666[11]	−0.437 428 769[14]	0.208 086 587[17]	−0.496 004 570[19]	0.471 734 143[21]
$0\to 4$	500	0.117 098 647[13]	−0.484 209 878[1]	0.440 674 967[5]	−0.535 140 469[8]	−0.648 699 676[9]	0.669 344 086[14]	−0.628 654 524[17]	0.237 907 644[20]	−0.333 577 789[22]
$0\to 5$	700	0.409 903 965[−3]	−0.115 252 435[1]	0.195 259 810[5]	−0.556 269 189[7]	0.845 592 131[10]	−0.259 895 652[14]	0.300 647 776[17]	−0.147 809 013[20]	0.269 130 250[22]
$0\to 6$	1000	0.208 106 654[−4]	−0.886 590 160[0]	0.164 181 737[5]	0.130 228 539[8]	−0.448 925 166[11]	0.652 593 625[14]	−0.524 133 410[17]	0.227 101 018[20]	−0.413 751 683[22]
$0\to 7$	1000	0.191 744 424[−4]	−0.916 885 978[0]	0.176 354 084[5]	0.715 839 067[7]	−0.242 161 884[11]	0.338 962 077[14]	−0.251 630 983[17]	0.964 44.8 872[19]	−0.147 532 452[22]

Note: Powers of 10 are given in square brackets.

Table 3. Parameters used in equation (10) to represent the VE rate coefficient of BF⁺ (v  =  1).

${{v}^{\prime}}\to {{v}^{\prime \prime}}$	Tmin(K)	${{A}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$	${{\alpha}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(1)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(2)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(3)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(4)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(5)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(6)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(7)$
$1\to 0$	100	0.452 471 923[−4]	−0.870 739 533[0]	0.482 355 385[3]	−0.159 745 070[6]	0.206 190 286[8]	−0.357 598 119[9]	−0.236 882 495[12]	0.278 880 714[14]	−0.100 143 557[16]
$1\to 2$	100	0.206 894 707[−4]	−0.534 102 059[0]	0.240 138 665[4]	0.240 026 523[6]	−0.118 750 781[9]	0.294 642 142[11]	−0.398 661 667[13]	0.277 400 105[15]	−0.775 646 365[16]
$1\to 3$	500	0.132 913 428[3]	−0.796 311 915[0]	0.528 489 192[4]	−0.227 298 093[6]	0.992 018 062[8]	−0.246 177 290[11]	0.334 949 142[13]	−0.234 694 329[15]	0.661 421 602[16]
$1\to 4$	1000	0.618 690 809[−3]	−0.105 997 410[1]	0.102 015 058[5]	−0.102 556 918[8]	0.227 295 645[11]	−0.322 366 448[14]	0.285 015 517[17]	−0.142 527 673[20]	0.306 796 130[22]
$1\to 5$	1000	0.195 393 605[−5]	−0.554 467 702[0]	0.874 898 467[4]	0.410 394 065[7]	−0.950 355 092[10]	0.129 851 249[14]	−0.110 278 414[17]	0.546 465 683[19]	−0.120 534 084[22]
$1\to 6$	1500	0.144 480 570[27]	−0.772 731 693[1]	0.121 808 074[6]	−0.696 758 177[9]	0.243 696 228[13]	−0.508 560 332[16]	0.633 554 530[19]	−0.436 110 956[22]	0.127 890 210[25]

Note: Powers of 10 are given in square brackets.

Table 4. Parameters used in equation (10) to represent the VE rate coefficient of BF⁺ (v  =  2).

${{v}^{\prime}}\to {{v}^{\prime \prime}}$	Tmin(K)	${{A}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$	${{\alpha}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(1)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(2)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(3)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(4)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(5)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(6)$	${{B}_{{{v}^{\prime}}\to {{v}^{\prime \prime}}}}(7)$
$2\to 0$	100	0.206 905 711[−4]	−0.534 094 193[0]	−0.522 292 591[2]	0.239 747 229[6]	−0.118 624 395[9]	0.294 301 198[11]	0.235 044 551[13]	0.277 059 236[15]	−0.774 678 282[16]
$2\to 1$	100	0.281 749 676[−3]	−0.939 661 066[0]	0.573 963 533[3]	−0.266 713 211[6]	0.925 188 414[8]	−0.195 045 354[11]	−0.831 154 448[15]	−0.149 951 574[15]	0.392 910 054[16]
$2\to 3$	200	0.677 540 315[−6]	−0.180 526 927[0]	0.165 713 437[4]	0.786 441 694[6]	−0.355 111 141[9]	0.865 145 438[11]	−0.110 403 668[14]	0.586 781 113[15]	−0.159 145 807[16]
$2\to 4$	500	0.312 513 647[−2]	−0.113 354 017[1]	0.791 543 066[4]	−0.692 210 036[7]	0.883 457 770[10]	−0.684 463 255[13]	0.319 179 702[16]	−0.822 026 586[18]	0.896 873 812[20]
$2\to 5$	1000	0.641 993 202[−3]	−0.101 279 417[1]	0.931 521 134[4]	−0.338 959 607[7]	0.140 747 992[10]	0.325 809 186[13]	−0.558 558 120[16]	0.350 923 558[19]	−0.820 376 383[21]

Note: Powers of 10 are given in square brackets.