Article pubs.acs.org/JPCA
Formation of the SiP Radical through Radiative Association Nikolay V. Golubev,†,‡ Dmitry S. Bezrukov,† Magnus Gustafsson,‡ Gunnar Nyman,‡ and Sergey V. Antipov*,‡,¶ †
Department of Chemistry, M. V. Lomonosov Moscow State University, Moscow 119991, Russia Department of Chemistry and Molecular Biology, University of Gothenburg, 41296 Gothenburg, Sweden
‡
ABSTRACT: Formation of the SiP radical through radiative association of Si(3P) and P(4S) atoms is studied using classical and quantum dynamics. Rate coefficients for formation in the two lowest doublet states and the two lowest quartet states are calculated for T = 10−20 000 K. Breit−Wigner theory is used to properly account for contribution from quantum mechanical resonances.
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same is true for a4Π and b4Σ+. (The ordering, a and b, of these quartet states will be discussed below.) Thus, the radiative association through electronic transitions that are the converse of those in reactions 1 and 2 can be important, namely
INTRODUCTION Formation of molecules is one of the cornerstones of chemistry, and understanding molecular formation in different environments is of central interest. Radiative association is one of several processes that contribute to production of molecules in the interstellar medium.1 In this process, two particles collide to form a new molecule, and the excess energy is taken away by spontaneous emission of a photon. Laboratory measurements of radiative association rate coefficients have so far been carried out only for certain ionic species,2 and thus, results of theoretical studies are of interest in astrochemistry. In the present paper, we investigate formation of the SiP radical through radiative association of Si(3P) and P(4S) atoms, which may take place in the inner envelope of the evolved stars.3 Collisions of Si(3P) and P(4S) can lead to formation of SiP in 36 molecular electronic states, which can be of doublet, quartet, and sextet multiplicities. In a previous theoretical study,3 a semiclassical model1,4 was applied to formation of SiP through the following reactions Si(3P) + P( 4S) → SiP(A2Σ+) → SiP(X2Π) + hν
(1)
Si(3P) + P( 4S) → SiP(a4 Π) → SiP(b4 Σ+) + hν
(2)
(3)
Si(3P) + P( 4S) → SiP(b4 Σ+) → SiP(a4 Π) + hν
(4)
Here, we calculate radiative association rate coefficients for reactions 1−4 over the temperature interval of 10−20 000 K.
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THEORY OF RADIATIVE ASSOCIATION Various methods exist for the calculation of radiative association cross sections based on classical or quantum dynamics for the description of the nuclear motion. Detailed overviews of such methods can be found in refs 1 and 5. Here, we briefly summarize two of them, which are relevant for the present study, the perturbation theory (PT) and the semiclassical (SC) approaches. PT Approach. The quantum mechanical PT treatment of radiative association is based on the semiclassical theory of light−matter interactions and thermodynamic relations for the Einstein A coefficient describing the spontaneous emission of a photon, which yields the following formula for the cross section1
and the rate constants were calculated for temperatures from 300 to 14 100 K. However, as was shown recently,5 the conventional semiclassical theory overestimates the high-energy cross section and, consequently, the high-temperature rate coefficient for radiative association. In this work, we use quantum mechanical and refined semiclassical approaches to calculate the rate coefficient at T < 300 K and to improve the high-temperature data. From ab initio calculations,6−8 it is known that the X2Π and 2 + A Σ states have very similar dissociation energies, and the © 2013 American Chemical Society
Si(3P) + P( 4S) → SiP(X2Π) → SiP(A2Σ+) + hν
σΛS →Λ′ S(E) =
∑ Jυ ′ J ′
64 π 5 PΛS SΛSJ Λ′ SJ ′ |MΛSEJ Λ′ Sυ ′ J ′|2 3 4π ϵ0 κ 2 λE3Λ′ Sυ ′ J ′ (5)
Received: March 31, 2013 Revised: July 18, 2013 Published: August 19, 2013 8184
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Here, E is the kinetic energy of the colliding particles, λEΛ′Sυ′J′ is the wavelength of the emitted photon, k = (2μE)1/2/ℏ is the channel wavenumber, μ is the reduced mass of the molecule, PΛS is the statistical weight factor for approaching along a particular potential curve VΛS(R), and SΛSJΛ′SJ′ is the Hönl− London factor for the corresponding transition. Notations for all quantum numbers used throughout the paper are listed in Table 1. The statistical weight factor PΛS can be calculated as1 PΛS
(2S + 1)(2 − δ0Λ) = (2LSi + 1)(2SSi + 1)(2L P + 1)(2SP + 1)
SC Approach. Within the SC approach, the radiative association cross section can be calculated as4,11 ⎛ μ ⎞1/2 σΛS →Λ′ S(E) = 4π ⎜ ⎟ PΛS ⎝ 2E ⎠ ×
AΛS →Λ′ S (R ) =
Table 1. Notation for the Quantum Numbers electronic orbital angular momentum of atom X total electronic orbital angular momentum of the diatomic system: L⃗ = L⃗ A + L⃗ B
SX S l J Λ υ
electronic spin of atom X total electronic spin of the diatomic system: S⃗ = S⃗A + S⃗B rotational angular momentum of the nuclei total angular momentum of the diatomic system: J ⃗ = L⃗ + S⃗ + l ⃗ projection of L on the internuclear axis vibrational quantum number
X2Π → A2Σ+
A2Σ+ → X2Π
a4 Π → b4Σ+
b4Σ+ → a4Π
4/36 1
2/36 2
8/36 1
4/36 2
Σ+ → Π
J−1 J J+1
(J + 1)/2 (2J + 1)/2 J/2
J−1 2J + 1 J+2
(8)
(9)
(10)
which should be used in eq 8. The first condition in eq 10 checks that the classically bound region exists on the final effective potential, and the second condition is that the initial kinetic energy of the system does not exceed the optimal energy of the emitted photon, that is, a vertical transition to the classically bound region is possible.
(7)
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RATE COEFFICIENT When the radiative association cross section is obtained, the corresponding rate coefficient at a temperature T can be calculated as kΛS →Λ′ S(T ) 3/2 ⎛ 8 ⎞1/2 ⎛ 1 ⎞ =⎜ ⎟ ⎜ ⎟ ⎝ πμ ⎠ ⎝ kBT ⎠
∫0
∞
EσΛS →Λ′ S(E)e−E / kBT dE
(11)
where kB is the Boltzmann constant. A typical SC cross section is a smooth curve,5,12 and the corresponding rate coefficient is easily obtained by numerical integration using eq 11. Conversely, a quantum mechanical PT cross section typically shows a complex resonance structure with many exceedingly narrow resonances,5,12−15 which arise due to quasi-bound states supported by the effective potential of the initial electronic state. Such features make it practically impossible to evaluate the integral in eq 11 numerically. Moreover, it is known that the PT approach gives nonphysical heights for the resonances with tunneling widths smaller than
Table 3. Parity-Averaged Hönl−London Factors, SΛSJΛ′SJ′9,10 Π → Σ+
2 64 π 4 ⎛ 2 − δ0, Λ+Λ′ ⎞ DΛS Λ′ S(R ) ⎜⎜ ⎟⎟ 3 3 (4π ϵ0)h ⎝ 2 − δ0, Λ ⎠ λΛS →Λ′ S(R )
⎧ Eb2 ⎪ AΛS →Λ′ S (R ) VΛ′ S(R ) + 2 < 0 ∪ ⎪ R AΛEbS →Λ′ S (R ) = ⎨ E < VΛS(R ) − VΛ′ S(R ) ⎪ ⎪ ⎩0 otherwise
where R is the internuclear distance and DΛSΛ′S(R) is the matrix element of the dipole moment operator between the corresponding molecular electronic wave functions. Here, ψΛSEJ(R) is the radial part of the continuum energy-normalized wave function of the initial state and ψΛ′Sυ′J′(R) is the radial part of the final rovibrational wave function, normalized to unity. Equations 5 and 7 are derived assuming that there are no couplings between different electronic states of the system and that J ⃗ − S⃗ ≈ J,⃗ 1 that is, the spin is not accounted for in the dynamics explicitly. Due to the latter ansatz, the Hönl−London factors SΛSJΛ′SJ′ are approximated by those for transitions between singlet states (Table 3). Moreover, the summation over parity is not performed in eq 5, but it is instead taken care of by the factor PΛS. Thus, the parity-averaged Hönl−London factors are used.
J′
c
AΛS →Λ′ S (R ) dR db (1 − VΛS(R )/E − b2 /R2)1/2
and values of the factor including the Kronecker delta are given in Table 2. As was discussed in ref 5, the transition rate defined in eq 9 describes radiative transitions to both bound states, which corresponds to radiative association, and continuum states, which corresponds to radiative quenching. In order to isolate the radiative association contribution, additional restrictions should be applied on AΛS→Λ′S(R), yielding the restricted transition rate
∞
ψΛSEJ(R )DΛS Λ′ S (R )ψΛ′ Sυ ′ J ′(R ) dR
∞
⎛ V (R ) − VΛ′ S(R ) ⎞ = max⎜0, ΛS ⎟ ⎝ ⎠ λΛS →Λ′ S(R ) hc
The transition dipole matrix element is defined as
∫0
∫R
1
Table 2. Statistical Weights
MΛSEJ Λ′ Sυ ′ J ′ =
b
where λΛS→Λ′S(R) is the optimal wavelength of the emitted photon
where for Si(3P) and P(4S) atoms, we have LSi = 1, SSi = 1, LP = 0, and SP = 3/2; S = 1/2 for the doublet or 3/2 for the quartet states, and finally, Λ = 0 for the Σ+ and Λ = 1 for the Π molecular states. The values of PΛS are given in Table 2.
PΛS (2 − δ0,Λ+Λ′)/ (2 −δ0Λ)
∞
where AΛS→Λ′S(R) is the transition rate and Rc is the distance of the closest approach for a given value of the impact parameter b. The transition rate is defined as
(6)
LX L
∫0
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or comparable to radiative widths.13,14 Thus, some other approach is required. The established method to compute the resonance contribution to the radiative association rate coefficient is Breit−Wigner theory.16,17 The general idea is to divide the total rate coefficient into a sum of two terms, the direct term dir kΛS→Λ′S (T), which comes from the baseline of the PT cross section, and the resonance term kres ΛS→Λ′S(T). According to the Breit−Wigner theory, the resonance contribution to the rate coefficient is given by kΛresS →Λ′ S(T ) ⎛ 2π ⎞3/2 (2J + 1)e−EΛSυJ / kBT = ℏ2PΛS⎜ ⎟ ∑ tun rad ⎝ μkBT ⎠ υJ 1/ ΓΛSυJ + 1/ Γ ΛSυJ Λ′ S
(12) Figure 1. Potential energy curves for the two lowest doublet and two lowest quartet electronic states of the SiP radical. The inset shows the corresponding transition dipole moments.
where the summation runs over all quasi-bound states supported by the initial effective potential, EΛSυJ is the energy tun of the quasi-bound state, ΓΛSυJ is the tunneling width, and rad ΓΛSυJΛ′S is the width due to the radiative decay to all bound levels of the final electronic state.
In order to verify our results, additional ab initio calculations utilizing various basis sets (aug-cc-pVNZ, where N = T,Q,5,6, and aug-cc-pVMZ-DK, where M = T,Q,5) were performed. The standard contraction scheme was used for each basis set, and the extrapolation to the complete basis set (CBS) limit was done in the same manner as that in ref 8. The results are summarized in Table 4. From the table, it is clear that the relative position of the 4Π and 4Σ+ states does not depend on the basis set size. The same holds true when averaging over only the quartet states is used in the CASSCF calculations. Moreover, the order does not change when the vibrational ground-state energy on each of the potentials is included. Thus, we suggest that the correct spectroscopic labeling for the quartet states of the SiP radical should be a4Π and b4Σ+. Dynamical Calculations. In calculating the PT cross sections, the bound state wave functions ψΛ′Sυ′J′ are obtained using the discrete variable representation method (DVR) with a uniform grid.19 The continuum wave functions ψΛSEJ are computed with the Numerov (shooting) method.20 The Romberg integration is used to evaluate the integral over R in eq 8 in order to handle the behavior of the integrand when R → R+c . The resonance parameters in eq 12 are calculated using the LEVEL program.21 The energies of quasi-bound states (positions of resonances) EΛSυJ and the tunneling widths Γtun ΛSυJ are given by the program directly, while the radiative widths Γrad ΛSυJΛ′S have been found as
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MOLECULAR DATA AND COMPUTATIONAL DETAILS Ab Initio Calculations. The potential energy curves VΛS(R) and transition dipole moments D(R) used in the present study were calculated according to the following scheme. First, the molecular orbitals were constructed using the CASSCF method with an active space consisting of nine electrons in eight orbitals, which at the dissociation limit correspond to the 3s and 3p orbitals of the separate atoms. In that calculation, all core orbitals were optimized while being kept fully occupied, and the averaging (with equal weights) was done over all states (2Σ+, 4Σ+, 6Σ+, 2Π, 4Π, 6Π), correlating with the lowest dissociation limit of the system, Si(3P) + P(4S). Then, the potential energy curves for the doublet and quartet states and the corresponding transition dipole moments were calculated with the internally contracted MRCI+Q method using the CASSCF molecular orbitals as a reference. The scalar relativistic correction has been accounted for by the secondorder Douglas−Kroll (DK) Hamiltonian. The augmented correlation-consistent aug-cc-pV5Z-DK basis set was employed, optimized for the DK relativisric correction. All calculations were carried out in the C2v symmetry group. The MOLPRO 2010.118 package was used. The calculated potential energy curves and transition dipole moments are shown in Figure 1. It should be noted that our ab initio calculations predict the reverse order of the 4Σ+ and 4Π states compared to the order reported by dos Santos and Ornellas.7 According to their calculation, the minimum of the 4 + Σ state lies 45 cm−1 below the 4Π minimum, while in our calculation, 4Π is the lowest quartet state in the system, with the minimum energy being ∼400 cm−1 lower than that of the 4 + Σ state. Because the minimum energies of the 4Σ+ and 4Π states are very close to each other, determination of the relative position of those states is a difficult task. To our knowledge, no experimental data are available for the quartet states of the SiP radical, and there are only two theoretical studies by dos Santos and Ornellas7 and by Shi and co-workers.8 In the latter study, the 4Π state was not calculated, and 4Σ+ was considered to be the lowest quartet state based on the results of ref 7.
Γ rad ΛSυJ Λ′ S = ℏ ∑ AΛSυJ Λ′ Sυ ′ J ′ υ′J′
(13)
Here, AΛSυJΛ′Sυ′J′ are the Einstein A coefficients for spontaneous emission from the quasi-bound level υJ on the initial electronic state ΛS to the lower-lying bound level υ′J′ of the final electronic state Λ′S.
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RESULTS Cross sections for the formation of SiP through reactions 1−4 have been calculated using the SC and PT approaches, and the results are shown in Figure 2. All PT cross sections demonstrate complex resonance structure with baselines showing a monotonic decrease up to certain threshold energies above which the cross sections decay fast. The threshold energies correspond to the maximum energy at which a classical 8186
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Table 4. Equilibrium Distances Re and Dissociation Energies De of the 4Π and 4Σ+ States of the SiP Radicala basis set
Re(4Σ+)
Re(4Π)
De(4Σ+)
De(4Π)
ΔDe(4Π − 4Σ+)
aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV5Z aug-cc-pV6Z aug-cc-pV(CBS)Z aug-cc-pVTZ-DK aug-cc-pVQZ-DK aug-cc-pV5Z-DK aug-cc-pV(CBS)Z-DKb ref 7c
4.19 4.16 4.15 4.14 4.14 4.19 4.16 4.15 4.14 4.141
4.47 4.44 4.43 4.43 4.42 4.48 4.44 4.43 4.42 4.440
12514 13460 13846 13956 14107 12428 13368 13753 14013 13500 ± 100
12889 13835 14217 14336 14498 12834 13781 14161 14442 13500 ± 100
375 375 371 380 391 407 413 409 429 −45
Re is in bohrs, and De is in cm−1. bThe scalar relativistic correction is calculated with the aug-cc-pV5Z-DK basis set. cNo actual values of De are given in the paper. The estimated values are obtained from the corresponding figure. However, the absolute energies at the minima are presented, which allows for precise determination of ΔDe. a
Figure 2. Semiclassical (SC) and quantum mechanical (PT) cross sections for formation of the SiP radical in (a) doublet and (b) quartet states.
Franck−Condon transition22 to bound states is still possible, and their values can be obtained from the difference between the electronic potentials participating in the process, that is, VΛS(R) − VΛ′S(R). The semiclassical calculations reproduce the quantum mechanical baselines for all studied transitions. The small differences between SC and PT cross sections around the threshold energies are due to non-Franck−Condon transitions, which are not included in the SC treatment.23 Figure 3 shows the calculated radiative association rate coefficients for formation of SiP through transitions involving doublet electronic states. Breit−Wigner theory was used to
compute the resonance contributions, and the direct components of the rate coefficients were obtained from the SC cross sections. The discussed differences between SC and PT cross sections above the threshold energy have a negligible effect on the rate coefficients.5 The rate constant for the X2Π → A2Σ+ transition shows little variation in the temperature interval of 10−2000 K with a maximum value of 1.01 × 10−19 cm3/s at about 800 K. The resonance contribution to the X2Π → A2Σ+ rate coefficient is as big as the direct one at 10 K and decreases with increasing temperature; it contributes with less than 10% above 1000 K. The rate constant for the A2Σ+ → X2Π transition increases steadily up to a maximum of 2.30 × 10−20 cm3/s at around 1700 K. In this case, the resonance contribution varies slowly with temperature and amounts to 10−40% of the total rate constant. Both total rate constants decrease fast for temperatures above ∼2000 K, which is a consequence of the drop in the cross sections. The results of the semiclassical calculation by Andreazza et al.3 for A2Σ+ → X2Π are also shown in Figure 3. Comparison reveals a strong disagreement with the results of the present study. We have no explanation for the difference but are confident in our results. The direct rate coefficient presented here is based on the SC cross section, which agrees well with the baseline of the PT cross section. Note that the PT and SC cross sections are obtained by independent approaches. The radiative association rate coefficients for formation of SiP in the quartet spin states (Figure 4) show the same features as those for doublets. The rate constants for transitions between quartet states are generally lower by a factor of 2−3
Figure 3. Rate coefficients for formation of the SiP radical in doublet electronic states. The results of ref 3 are shown by the black curve. 8187
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REFERENCES
(1) Babb, J. F.; Kirby, K. P. In The Molecular Astrophysics of Stars and Galaxies; Hartquist, T. W., Williams, D. A., Eds.; Clarendon Press: Oxford, U.K., 1998; pp 11−34. (2) Gerlich, D.; Horning, S. Experimental Investigation of Radiative Association Processes as Related to Interstellar Chemistry. Chem. Rev. 1992, 92, 1509−1539. (3) Andreazza, C. M.; Marinho, E. P.; Singh, P. D. Radiative Association of C and P, and Si and P Atoms. Mon. Not. R. Astron. Soc. 2006, 372, 1653−1656. (4) Bates, D. R. Rate of Formation of Molecules by Radiative Association. Mon. Not. R. Astron. Soc. 1951, 111, 303−314. (5) Gustafsson, M.; Antipov, S. V.; Franz, J.; Nyman, G. Refined Theoretical Study of Radiative Association: Cross Sections and Rates for the Formation of SiN. J. Chem. Phys. 2012, 137, 104301. (6) Ornellas, F. R.; Andreazza, C. M.; de Almeida, A. A. A Theoretical Study of the A2Σ+−X2Π System of the SiP Molecule. Astrophys. J. 2000, 538, 675−683. (7) dos Santos, L. G.; Ornellas, F. R. Excited Doublet and Quartet States of SiP: A High Level Theoretical Investigation. Chem. Phys. 2003, 295, 195−203. (8) Shi, D.; Xing, W.; Liu, H.; Sun, J.; Zhu, Z. MRCI Study on Electronic Spectrum of 13 Electronic States of SiP Molecule. Spectrochim. Acta, Part A 2012, 97, 536−545. (9) Hansson, A.; Watson, J. K. G. A Comment on Hönl−London Factors. J. Mol. Spectrosc. 2005, 233, 169−173. (10) Watson, J. K. G. Hö nl−London Factors for Multiplet Transitions in Hund’s Case a or b. J. Mol. Spectrosc. 2008, 252, 5−8. (11) Zygelman, B.; Dalgarno, A. Radiative Quenching of He(21S) Induced by Collisions with Ground-State Helium Atoms. Phys. Rev. A 1988, 38, 1877−1884. (12) Antipov, S. V.; Sjölander, T.; Nyman, G.; Gustafsson, M. Rate Coefficient of CN Formation Through Radiative Association: A Theoretical Study of Quantum Effects. J. Chem. Phys. 2009, 131, 074302. (13) (a) Bennett, O. J.; Dickinson, A. S.; Leininger, T.; Gadéa, F. X. Radiative Association in Li+H Revised: The Role of Quasi-Bound States. Mon. Not. R. Astron. Soc. 2003, 341, 361−368. (b) Bennett, O. J.; Dickinson, A. S.; Leininger, T.; Gadéa, F. X. Erratum: Radiative Association in Li+H Revised: The Role of Quasi-Bound States. Mon. Not. R. Astron. Soc. 2008, 384, 1743. (14) Mrugała, F.; Špirko, V.; Kraemer, W. P. Radiative Association of HeH+2 . J. Chem. Phys. 2003, 118, 10547−10560. (15) Barinovs, Ǧ .; van Hemert, M. C. CH+ Radiative Association. Astrophys. J. 2005, 636, 923−926. (16) Breit, G.; Wigner, E. Capture of Slow Neutrons. Phys. Rev. 1936, 49, 519−531. (17) Bain, R. A.; Bardsley, J. N. Shape Resonances in Atom−Atom Collisions I. Radiative Association. J. Phys. B 1972, 5, 277−285. (18) Werner, H.-J.; et al. MOLPRO, version 2010.1, a Package of Ab Initio Programs. http://www.molpro.net (2010). (19) Colbert, D. T.; Miller, W. H. A Novel Discrete Variable Representation for Quantum Mechanical Reactive Scattering via SMatrix Kohn Method. J. Chem. Phys. 1992, 86, 1982−1991. (20) Korn, G. A.; Korn, T. M. Mathematical Handbook for Scientists and Engineers, 2nd ed.; McGraw-Hill Book Company: New York, 1968. (21) Le Roy, R. J. LEVEL 8.0: A Computer Program for Solving the Radial Schrödinger Equation for Bound and Quasibound Levels; University of Waterloo Chemical Physics Research Report CP-663; 2007. (22) Noda, C.; Zare, R. N. Relation between Classical and Quantum Formulations of the Franck−Condon Principle: The Generalized rCentroid Approximation. J. Mol. Spectrosc. 1982, 95, 254−207. (23) Julienne, P. S. Theory of Rare Gas-Group VI 1S−1D CollisionInduced Transitions. J. Chem. Phys. 1978, 68, 32−41. (24) Wakelam, V.; et al. A Kinetic Database for Astrochemistry (KIDA). Astrophys. J., Suppl. Ser. 2012, 199, 21.
Figure 4. The same as in Figure 3 but for the quartet electronic states.
than those for transitions between doublet states of the same symmetry. The reason for this is the lower energy of the emitted photon and smaller transition dipole moment for the quartet transitions. The rate constant for the a4Π → b4Σ+ transition reaches a maximum of 4.37 × 10−20 cm3/s at 250 K, and for the b4Σ+ → a4Π transition, the maximum is 7.64 × 10−21 cm3/s at about 1200 K. Comparison of the direct rate coefficient for the a4Π → b4Σ+ transition with the semiclassical results of Andreazza et al.3 shows reasonable agreement for the temperature interval of 300−2000 K, and the variations are due to the use of different ab initio data. The discrepancy at higher temperatures arises because the unrestricted transition rate was used for calculations in ref 3. The radiative association rate coefficients are of interest for modeling of chemistry in the interstellar medium, and the present results will be made available in the database KIDA.24
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CONCLUSIONS We have presented radiative association cross sections and rate coefficients for formation of SiP through radiative association of Si(3P) and P(4S) atoms. A quantum mechanical approach was employed and the resonance contribution to the rate coefficient was included via Breit−Wigner theory. Radiative association rate constants were calculated in a wide temperature range of 10−20 000 K. The X2Π → A2Σ+ transition was shown to be the main channel for formation of SiP. However, the other considered transitions have comparable rate coefficients (within an order of magnitude), and their contribution to radiative association of SiP should be considered if high accuracy is desired.
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Article
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest. ¶ Researcher ID: A-3907-2012.
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ACKNOWLEDGMENTS Financial support from the Swedish Research Council is gratefully acknowledged. N.V.G. and D.S.B. acknowledge the Russian Foundation for Basic Research under the Project No.11-03-00081 for support and the Supercomputing Center of M. V. Lomonosov Moscow State University for provided computational resources. 8188
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