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A new theoretical method for calculating the radiative association cross section of a triatomic molecule: application to N2–H! T. Stoecklin,*a F. Liqueb and M. Hochlafc We present a new theoretical method to treat the atom–diatom radiative association within a time independent approach. This method is an adaptation of the driven equations method developed for photodissociation. The bound state energies and wave functions of the molecule are calculated exactly
Received 1st March 2013, Accepted 21st June 2013
and used to propagate the overlap with the initial scattering wave function. In the second part of this
DOI: 10.1039/c3cp50934f
radiative association cross sections are analysed and the magnitude of the calculated rate coefficient at
paper, this approach is applied to the radiative association of the N2H! anion. The main features of the 10 K is used to discuss the existence of N2H! in the interstellar medium which could be used as a tracer
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of both N2 and H!.
1. Introduction Radiative association (RA) in ion–molecule collisions is considered to be an important step of the synthesis of polyatomic species in interstellar clouds.1,2 The cosmic rays ionise the atoms and the molecules which can then recombine by radiative association to produce new molecules. As a matter of fact the very low density of atoms and molecules in even the densest clouds means that stabilization of any collision complexes must occur by the emission of a photon, rather than by collisions involving a third body. About 14 positive ions have been detected so far in the interstellar medium as well as several carbon chain anions. Because of its abundance, the H3+ ion is the most documented but the radiative association of the most abundant positive ions with H2 are also considered in the astrochemical models. Conversely, the detection of stable negative ions in circumstellar envelopes was a real and recent surprise and they are not yet taken into account in the models. For a long time, the radiative association of diatomic molecules has been the object of many theoretical3 studies while the radiative association of triatomic molecules has up to very recently not received much interest compared to a
Institut des Sciences Mole´culaire, UMR5255-CNRS, Universite´ de Bordeaux, 351 cours de la Libe´ration, 33405 Talence Cedex, France. E-mail:
[email protected] b LOMC - UMR 6294, CNRS-Universite´ du Havre, 25 rue Philippe Lebon, BP 540, 76058, Le Havre, France c Universite´ Paris-Est, Laboratoire Mode´lisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Valle´e, France
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photodissociation which is the reverse process. The main reason for this lack of studies is the experimental difficulty of measuring these cross sections which limited the experimental effort to the systems4 expected to be the most important for the chemistry of interstellar clouds. On the theoretical side for many years all studies were based on statistical approximation and differed very often from the experimental estimates by several orders of magnitude. The most successful of these methods5,6 which includes tunnelling gives rate coefficient values which are expected to be one order of magnitude accurate. The first state to state quantum calculation of the dynamics of the radiative association reaction was performed by Mrugala et al.7 in 2003 for the He–H2+ complex using the CC-BF-diabatic approximation. The first Close Coupling study which is even more recent was performed in 2011 by Ayouz et al.8 for the H3! anion. This later study was motivated by the search for a probe for the presence of H! in the interstellar medium. We present here a new method to treat the atom diatom radiative association within a time independent approach. We use the formal theory of three dimensional photodissociation of a triatomic molecule which is the reverse process and was developed long ago by Band, Freed and Kouri9 and Balint-Kurti and Shapiro.10 We will more specifically present an adaptation of the driven equations method developed at the same time for photodissociation by Heather and Light.11 In the second part of this paper, this approach is applied to the radiative association of the N2H! anion. The bound state energies and wave functions of this anion which we calculated exactly in a recent study12 are used to propagate the overlap
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with the initial scattering wave function. The main features of the radiative association cross sections are analysed and the magnitude of the calculated rate coefficient at 10 K is used to discuss the existence of N2H! in the interstellar medium which could be used as a tracer of both N2 and H!.
2. Method The method of the driven equations as initially developed for photodissociation relates in a single equation the initial and final nuclear wave functions of the colliding system. We will only present the main steps of this method which lead to new formulation because of the differences between radiative association and photodissociation and we will use a greek index for the bound states and latin letters for the scattering states. A first simplification of the application of this approach to radiative association is due to the fact that very often a single electronic PES participates in the radiative association process while at least two different electronic surfaces are involved in photodissociation. We then use the Born–Oppenheimer approximation and write the total wave function of the system as a superposition of the initial and final states: CTotal = zeCi(R, r, g)|0i + zeCaf (R, r, g)|1i
(1)
where ze and |0i,|1i are, respectively, the electronic state of the triatomic system and the photon states. A second simplification appears in this first equation as we use the same space fixed Jacobi coordinates (R, r, g) to describe the initial Ci and final Caf nuclear wave functions whereas Heather and Light used different coordinates for photodissociation. Here the scattering state associated with the collision of N2 and H! is the initial state Ci of the triatomic system whereas it would be the final state in a photodissociation process. We use a dipole approximation to the matter–radiation interaction Hamiltonian. After operating on the left by h0|ze* and h1|ze* and integrating one obtains a system of coupled equations:13 ˆ ! E] Ci(R, r, g) = m(R, r, g) Cf (R, r, g) [H
(2)
ˆ ! E] Cf (R, r, g) = m(R, r, g) Ci(R, r, g) [H
(3)
where m(R, r, g) is the projection of the dipole moment of the lowest electronic state of the triatomic system along the direcˆ is the space-fixed tion of the emitted photon field and H triatomic Hamiltonian in Jacobi coordinates: " ! ! 2 " " 2 !h2 1 @ l^ ^ H ¼ ! R þ 2mH! !N2 R @R2 2mH! !N2 R2 (4) # ! ! "" 2 !h2 1 @ 2 j^ r þ þ V ðR; r; gÞ ! 2mN2 r @r2 2mN2 r2 with ˆj and ˆl = ˆJ ! ˆj being, respectively, the operators associated with the rotational angular momentum of the diatomic molecule and with the relative angular momentum. mN2 and mH! !N2 are, respectively, the relative masses of N2! and of the triatomic system N2H!. As we know the final bound state wave functions we can ignore the second equation and look only for the initial
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state wave function Ci(R, r, g) solution of eqn (1). This is exactly the reverse process of photodissociation which for the initial state is known and the final state is looked for. The computation of the bound state energies efa and wave functions Caf (R, r, g) of the triatomic molecule in space fixed coordinates which is the next step of the calculation was detailed in our first paper dedicated to N2H! 12 and will not be described again. As usual in a Close Coupling calculation the eigenvalues enj and eigenfunctions of the free diatomic Hamiltonian hdiat are first determined: m
m
hdiat jnj ðrÞyj j ð^ rÞ ¼ enj jnj ðrÞyj j ð^ rÞ
(5)
where n, j and mj are, respectively, the quantum numbers associated with the vibration of the diatomic molecule, to its rotational angular momentum ˆj and to its projections along m the z space fixed axis. yj j ð^ rÞ is a spherical harmonic function of the angles ˆr associated with the diatomic vector r. Both the initial and final wave functions are then expended in the basis set jnj(r) describing the vibration of the diatomic molecule N2 and in a coupled space fixed angular basis set describing both the rotation of N2 and the relative movement of N2 towards H!. & # $ X% # $ m l ^ ^ ^ r ¼ R (6) jmj l; ml &jJM iyj j ð^ rÞym YjlJM R; l mj ;ml
where l and J are, respectively, the quantum numbers associated with the angular momentum ˆl and ˆJ defined above while ml and M are those associated with their projections along the z space fixed axis. # $ 1 X JM ^ r^ Ci ðR; r; gÞ ¼ w ðRÞjnj ðrÞYjlJM R; (7) Rr n; j;l n; j;l Caf ðR; r; gÞ ¼
$ 1 X aJ 0 M 0 0 0# ^ ^ o ðRÞjnj ðrÞYjlJ M R; r Rr n; j;l n; j;l
(8)
These two expressions define the functions w JM n, j,l (R) and 0
0
M oaJ n; j;l ðRÞ which are, respectively, associated with the variation
as a function of the intermolecular coordinate of the initial and final wave functions. We obtain for the function w JM n, j,l (R) associated with the motion of H! relative to N2 the usual Close Coupled equations but with a non-zero right hand side describing the dipolar coupling with the final bound states: ' 2 ( 0 0 0 d l ðl þ 1Þ n0 j0 l0 2 ! þ k ðEÞ ! U ð R Þ wnnjlj l ðRÞ ¼ lanjl ðRÞ (9) nj njl dR2 R2 where we used the usual notation of the matrix elements of the intermolecular potential 0 0 0
njl ðRÞ Unjl
¼ 2mH! !N2
Z
# $ # $ ^ r^ VðR; r; gÞjn 0 j 0 ðrÞY JM ^ ^ ^ nj ðrÞY JM R; dr d^ r dRj jl j 0 l 0 R; r (10)
and of the channel wave vectors a function of total and diatomic energies: E and enj knj2(E) = 2mH!–N2[E ! enj]
Phys. Chem. Chem. Phys., 2013, 15, 13818--13825
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While lanjl ðRÞ ¼ !2mH! !N2
Z
* # # $ ) $ ^ nj ðrÞY JM R; ^ ^r m R;~ ~ r Ca R; ^ r^ dr d^ r dRj jl f (12)
is called the driving term and is a real function while the 0 0 0
wnn;jj;ll ðRÞ functions defined in eqn (7) are complex: & & 0 0 0 0 0 0 0 0 0 & & wnnjlj l ðRÞ ¼ wnnjlj l ðRÞ& þiwnnjlj l ðRÞ& R
I
(13)
The imaginary part of the initial wave function appears then to be a simple inelastic scattering wave function as it follows the usual Close Coupled equations: ' 2 ( & 0 0 0 d l ðl þ 1Þ & n0 j0 l0 2 ! þ k ðEÞ ! U ð R Þ wnnjlj l ðRÞ& ¼ 0 (14) nj njl 2 2 I dR R
While the real part is solution of a system of equations involving the driving term ' 2 ( & d l ðl þ 1Þ & n0 j0 l0 n0 j0 l0 2 ! þ k ðEÞ ! U ð R Þ w ðRÞ & ¼ lanjl ðRÞ (15) nj njl njl R dR2 R2
When compared to ours, the formalism of the Franck– Condon approach chosen by Mrugala et al.7 or Ayouz et al.8 appear to be quite different as they also use different numerical methods. The approach of Mrugala et al. is not strictly equivalent to ours as they use the CC-BF-diabatic approximation, while the Close Coupling approach used by Ayouz et al. is expected to be equivalent to ours as demonstrated by Takatsuka and Gordon.14 In general the dipole moment surface has to be determined ab initio. In our first paper12 dedicated to the N2–H! system, we checked by comparing with ab initio values that a very good approximation of the dipole moment of the complex is to take it to be lying along the intermolecular coordinate: ' ( ' ( # $ mH! !N2 m ~ ) mm ’ R H! !N2 C m R^ ~ (16) m’ R 1 mH mH ˆ where Cm 1 (R) are spherical harmonics in the Racah normalisation. We obtain for the driving term the simple following expression: # $2 X n 0 ; j 0 ;l 0 !2 mH! !N2 0 ;M 0 lanjl ðRÞ ¼ dn;n 0 Gn; j;l Roa;J (17) n 0 ; j 0 ;l 0 ðRÞ mH u0 ; j 0 ;l 0 0
0 0
where Gnn;;j;lj ;l contains the angular part of the integral: 0
0 0
1
0
Gnn;;j;lj ;l ¼ ð!1Þ½2Jþlþl þ1' ½ð2J þ 1Þð2J 0 þ 1Þð2l þ 1Þð2l 0 þ 1Þ'2 dj; j 0
X
u0 ; j 0 ;l 0
l
1 l0
0
0
0
!(
l
J
j
J0
l0
1
)
the simple analytical expression of the radial part of the local scattering wave function G J,M n, j,l(R) in the diagonal basis set of each interval [Rn, Rn+1]. The transformation Tn between the asymptotic initial scattering basis set expressed in the space fixed frame and the local diagonal representation of the wave function is obtained by diagonalising the following matrix in the middle of each interval: '+ , ( l ðl þ 1Þ n0 j0 l0 0 dj; j 0 dl;l 0 ! U Tny knj 2 ðEÞ ! d ð R Þ Tn ¼ xn2 n;n njl R2 for R ¼ Rn þ
hn 2
(19)
In this local diagonal representation an analytical expression of the following integral can be obtained from a Taylor expansion through quadratic terms of the driving term: X Z Rnþ1 ) y *J;M # $ on jal ¼ Gn ðRÞ Tny j;k lak ðRÞ dR (20) Rn
j;k
l; j
where for the sake of simplicity, each of the indices l and j replace the three quantum numbers n, j,l. These contributions are accumulated along the propagation and retransformed in the initial asymptotic basis set giving the arrays yn|ai as written in matrix form in the following expression: yn|ai = Q†n+1[(Gn 0 (Rn+1))!1]†{yn!1|ai + on|ai }
where Qn = T†n!1Tn. The propagation is performed up to a value R% = Rn+1 of the intermolecular coordinate in the asymptotic region where the transition amplitude between the initial and the final state is obtained from: X Mia ðEÞ ¼ Dk;l ðTn!1 Þk;l yn!1 jal (22) k;l
In this expression the D matrix results from applying an incoming wave asymptotic boundary condition to the initial scattering 0 0 0
wave function wnnjlj l ðRÞ for our radiative association problem, while
Heather and Light imposed an outgoing wave asymptotic boundary condition to the final scattering wave function to treat the case of photodissociation. We apply the energy density normalization of the scattering wave function15 and find for D: " #12 ( .' $! 1 # $1 !1 2p ! lp # J;M Dn; j;l;n 0 ; j0 ;l 0 ¼ ei knj R! 2 (23) knj 2 þ i
