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A quantum chemical study of the N2H+ + e− → N2 + H reaction I: The linear dissociation path
Chemical Physics 332 (2007) 298–303 www.elsevier.com/locate/chemphys
A quantum chemical study of the N2H+ + e ! N2 + H reaction I: The linear dissociation path D. Talbi
*
Groupe de Recherche en Astronomie et Astrophysique du Languedoc UMR 5024, Universite´ de Montpellier II, Place Euge`ne Bataillon, 34090 Montpellier, France Received 19 September 2006; accepted 14 December 2006 Available online 21 December 2006
Abstract A theoretical investigation of the dissociative recombination (DR) of linear N2 Hþ ðX1 Rþ g Þ to give N2 + H has been undertaken because it is of interest for astrochemistry and also because it has been recently studied experimentally. Using state of the art quantum chemical methods, it is shown that the lowest 2R repulsive state of N2H leading to the N2 and H fragments in their ground electronic states does not cross the curve of the ion nor the one of the lowest N2H Rydberg state. This lowest 2R repulsive state is very low in energy. Its curve passes below the 1R N2H+ state and below the lowest bound 2R N2H states. However, it is also shown that there exist higher repulsive 2R and 2D states of N2H (the second and third repulsive states) crossing the ion curve. These states will lead to the formation of 3 + N2 in its 3 Rþ u and Du states. This study, the first of its type, shows that the DR of linear N2H should involve the direct mechanism and that it should lead to N2 in its first excited states. However this process may not be efficient for N2H+ in its ground vibrational state (v = 0), a state in which it exists in the cold environment of the interstellar medium. For the DR to be efficient for N2H+ in its ground v = 0 vibrational state, bent geometries of the ion might have to be considered. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Electronic dissociative recombination; Ab initio calculations; Potential energy surfaces; Excited linear states of N2H
1. Introduction Nitrogen is the fifth most abundant element in the universe. In the interstellar medium it is thought to be mostly molecular (N2). Since N2 has no observable rotational or vibrational transitions, it is traced through its parent ion, N2H+, to which it is related by the following reactions: þ N2 þ Hþ 3 ! N2 H þ H2
N2H+ is detected in a large number of astronomical [1–6] objects. In astrochemical models, it is considered that N2 is almost completely recycled through the electronic dissociative recombination (hereafter DR) of N2H+ N 2 H þ þ e ! N 2 þ H *
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However, the recent experimental measurements of Geppert et al. using the CRYRING ion storage ring apparatus [7] have thrown doubt on the validity of this hypothesis. In this experiment, it was shown that the most exoergic N2H+ + e ! N2 + H channel only accounts for 36% of the total reaction. N2H+ + e ! NH + N channel accounted for the remaining 64%. These unexpected branching ratios, posing the problem of the mechanism involved by the DR of N2H+ as well as of the abundance of N2 in space, have resulted in a strong demand from both the astrochemical and experimental community for a theoretical study of this process to rationalize the experimental results and to understand the mechanism involved in this process. I have undertaken this theoretical study using the most sophisticated ab initio methods of quantum chemistry. In this paper the investigation of the N2H+ + e ! N2 + H linear reaction path is presented; the other path (i.e. N + NH) is in progress and will be presented in a forthcoming paper.
D. Talbi / Chemical Physics 332 (2007) 298–303
Dissociative recombination corresponds to the capture of an electron by a positive molecular ion forming a neutral molecule in an excited electronic state. If this state is repulsive, dissociation may take place and the process is called ‘‘direct’’. If this state is a vibrationally excited electronic state of the neutral molecule, two mechanisms are possible. Either autoionisation takes place, or the temporarily bound state relaxes to a lower energy state, leading, if it is dissociative in nature, to fragmentation of the molecule. In this case the process is called ‘‘indirect’’ and most often involves Rydberg-type excited states because the complex is formed in a state where the electron is far from the nuclei, the core being essentially that of the positive ion ground state. The efficiency of both direct and indirect processes depends upon the existence of at least one repulsive state and also upon its position with respect to the bound excited or ionic states. Understanding such processes requires knowledge of the curves governing the corresponding mechanism, i.e. the ionic, excited (mainly Rydberg) and dissociative potential energy surfaces of the parent neutral molecule. The difficulty in treating such problems resides in, first of all, the description of the excited and dissociative states along dissociative paths. This is not a trivial task because, very often the nature of states changes along such channels and carefully designed wave functions are thus necessary in order to follow these transformations. Another difficulty is that an even-handed treatment is required for all the states involved in the mechanism. The N2H+ ion is a strongly bound linear species and is well known both experimentally and theoretically [8,9]. Its neutral parent N2H, a less familiar molecule, has been the subject of many theoretical investigations. The latest one [10] confirmed that its bent minimum energy structure is unstable with respect to N2 and H but is separated from the dissociation products by a barrier sufficient to support at least a zero point level. Experimentally its observation is more recent [11], probably because of its metastable structure. Very little is known about the excited states of N2H [12] and to my knowledge nothing has been published on the potential energy surfaces of the excited linear states that are of importance for understanding the N2H+ + e! N2H* ! N2 + H process. 1.1. The theoretical approach In the simple case of the closed shell ion N2H+, corresponding electronic configurations can be written as core (2spN)2(rNN)2(rNH)2 (pNN)4. DR involves the capture of an electron, which will lead to the formation of linear excited states of the parent neutral molecule with the additional electron occupying a virtual orbital. If the occupied virtual orbital is the anti-bonding NH* the resulting electronic configuration: core (spN)2(rNN)2(rNH)2 (pNN)4 1 (rNH Þ , for which the NH bond is weakened, is the lowest of those governing the DR of N2H+ leading to the H + N2 products in their ground electronic states. The additional electron can also occupy a diffuse orbital far from the
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N2H+ core. The result is a Rydberg state of the neutral N2H molecule of electronic configuration core (spN)2 (rNN)2(rNH)2 (pNN)4 (Ri)1 where Ri is a diffuse orbital of spherical (Rs) or axial (Rx, Ry, Rz) symmetry. These types of states have been shown in our previous theoretical studies [13,14] to play an important role in DR reactions involving the indirect mechanism. For a quantitative study of the recombination of N2H+ the even handed treatment of both the ionic, the excited, (mainly Rydberg type), and the dissociative states of linear N2H is required. Correlation energies must be treated through the same CI procedure so as not to favor one state over another. This is true for all states, which might be involved in the recombination of the ion and all along the dissociative reaction coordinate. To fulfill these requirements, I have adopted a methodology, already defined in previous studies[13,14], which is based on the use of orbitals with a strong localized character, to build the CI wave functions. The advantage of such a method is that the CI nparticle space is defined in terms of orbitals whose chemical significance is clearly identified. Configurations can be selected according to chemical evidence and the type of arbitrariness associated with energy related selection processes can be avoided. More important, the CI expansion is directly obtained in a configuration space that contains the approximate diabatic representation of the problem (hereafter quasi-diabatic). Indeed, thanks to the use of orbitals with a strong localized character to construct the configuration space, it is possible to identify most configuration state functions with a quasi-diabatic state (repulsive, excited valence or Rydberg) according to the weights of the relevant configurations in the CI wave functions. (Details of the procedure can be found in Talbi et al. [14].) To obtain orbitals of localized character different procedures can be used. I have, after a series of test calculations, chosen to use MCSCF natural orbitals optimized through the MCSCF n-particle space of Table 1, where three electrons have been distributed in set1. (NH and NH* orbitals), two in set2. (NN and NN* orbitals) four in set3. (p and p* orbitals) and 2 in set4. (spNspN*). The NH and NH* orbitals have been verified to correctly dissociate to the desired limits (i.e. the 2s of nitrogen and the 1s of hydrogen) while keeping a localized character. To construct the molecular orbital basis, a 6-311G(d,p) atomic orbital basis set similar to that of Ref. [15] has been used for nitrogen and a 4s3p1d similar to that of Ref. [16] has been used for H. To account for the Rydberg components, the atomic orbital basis set has been augmented by atomic diffuse functions on each atoms and the valence space obtained following the MCSCF procedure, has been extended by molecular Rydberg orbitals of atomic type (see [13,14] for details). The s (0.026, 0.016) and p (0.023, 0.015) diffuse exponent functions of Dunning and Hay [17] have been used for nitrogen. The s (0.015) and p (0.025) [17] diffuse exponent functions used to extend the 4s3p1d atomic orbital basis set of hydrogen have been shown in a previous study [16] to give (when combined with the above 4s3p1d basis set) reliable
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D. Talbi / Chemical Physics 332 (2007) 298–303
Table 1 MCSCF n-particle space designed for the N2H+ + e system in C2v symmetry Symmetry
Electronic distributions Corea
Internal sets Set1
Set2
a1 b2 b1 a2
rNH rNH
rNN rNN
Number of electronsb
3(2)
Set3
Set4 spN spN
pNN pNN pNN pNN 2
4
2
The number of CSF’s generated is 708 for the neutral and 654 for the ion. a Core orbitals: 1sN. b Configuration of the ion in between brackets.
H(n = 2) energy levels. The H(n = 2) states are states to which the lowest linear Rydberg states of N2H will correlate, they need to be correctly described. The resulting set of MO’s is Schmidt-orthogonalized prior to the CI calculations. This way of building a complete set of orbitals has been shown to give reliable positions for the Rydberg states [18]. The CI n-particle space designed to evaluate the correlation energy is shown in Table 2, where the electronic dis-
tribution in brackets is that for the ion calculated with the same CI n-particle space. This CI n-partitioning designed after a series of test calculations should ensure an evenhanded treatment of all states (ion, valence excited, Rydberg and dissociative) involved in the mechanism, at all points of the dissociation pathway. It will be used for the study of the N + NH exit channel to ensure the same correlated treatment.
Table 2 CI n-particle space designed for the N2H+ + e system in C2v symmetry Symmetry
Electronic distributions Corea
Internal sets Set1
a1
1 2 3 4 5 6 7 8 9 10 11 12
Set2
spN rNN rNH rNN rNH
b2 b1 a2 Configuration classesb,
External set
c
Set3 RsRxRsRx
23
pNN pNN pNN pNN
RyRy RzRz
11 11 3
4(4) 3(3) 3(3) 2(2) 2(2) 5(5) 4(4) 4(4) 3(3) 6(6) 4(4) 4(4)
0(0) 1(1) 0(0) 2(2) 1(1) 0(0) 1(1) 0(0) 2(2) 0(0) 2(2) 1(1)
0(0) 0(0) 1(1) 0(0) 1(1) 0(0) 0(0) 1(1) 0(0) 0(0) 0(0) 1(1)
0(0) 0(0) 0(0)
2(2) 2(2) 2(2)
MO’s
Number of electrons 7(6) 7(6) 7(6) 7(6) 7(6) 6(5) 6(5) 6(5) 6(5) 5(4) 5(4) 5(4)
Number of electrons with respect to referencesd 13 14 15
7(6) 6(5) 5(4)
2(2) 3(3) 4(4)
All configurations corresponding to the electronic distributions specified and satisfying the spin and symmetry requirements are considered. Total number of CSF’s: 1,430,834 for neutral doublet states and 827,916 for the ionic singlet state. a Core orbitals: 1sN. b Core orbitals are doubly occupied in all CSF’s. c Electronic distributions for the ions are given in parentheses. d For the double excitations in the external set, the excitations in the internal set are limited to double excitations. For example, the configuration classes 15 corresponds to double excitation in the external set with respect to all singly and doubly excited configurations that are generated by distributing five electrons in set1 (4 for the ion) and four electrons in set2 (4 for the ion).
D. Talbi / Chemical Physics 332 (2007) 298–303
All these calculations have been performed in C2V symmetry using the ALCHEMY-CI II program [19]. 1.2. Potential energy surfaces Provided the crossing we are looking for is located not very far from the equilibrium geometry of the cation, it is known from previous studies [13,14,20] that the best compromise while studying the DR of an HXY+ molecule, is to keep the geometry of the XY bond fixed at its geometry in the ion along the dissociative reaction coordinate H–X. Moreover, the NN bonds being almost identical in N2H+ and N2, I have for the study of the [N2H+ + e] complex along the N–H dissociative coordinate, fixed the NN bond at its geometry in the ion optimized at the present MRCI ˚. level, i.e. NN = 1.12 A 1.2.1. The adiabatic representation The expected reliability of the present CI calculations summarised in Table 2 can be assessed from the energetics deduced from the N2H (2R) and N2H+(1R) energies calculated at the dissociation limits within the super molecule approach. With the present MRCI calculations, the N2H+ to N2 + H+ dissociation energy is calculated to be 5.4 eV (without considering zero point energies). This value is identical to the best calculated one [9]. At the present 3 1 MRCI level, N2 ð3 Rþ u Þ, N2( Du) and N2( Dg), respectively lies at 6.0 eV, 7.3 eV and 9.2 eV above N2 ð1 Rþ g Þ. These ver-
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tical excitation energies (calculated without considering zero point energies) are consistent with the N2 excitation energies reported in the literature [21]; respectively 6.2 eV, 7.4 eV and 8.9 eV. The present calculated ionization energy for hydrogen is 13.4 eV and the H(n = 1) ! H(n = 2) excitation energy 11.3 eV. These numbers are in a satisfying agreement with the experimentally known values, 13.6 eV [22] and 10.2 eV [23], respectively. The above-calculated relative energies are in a good enough agreement with the known data from the literature to consider the present MRCI level of sufficient reliability to describe the N2H+ + e ! N2H* ! N2 + H process. Adiabatic potential energy surfaces for the four lowest 2 A1 states of N2H, states of R and D symmetry in a C1v notation, as well as that of the ion N2 Hþ ðX1 Rþ g Þ are presented in Fig. 1. They are calculated along the NH reaction ˚ . Fig. 1 coordinate keeping the NN bond frozen at 1.12 A 2 shows that the lowest R N2H state, has a repulsive character. It is very low in energy and does not cross the N2H+ ion curve. Its MRCI wave function analysis reveals that its electronic configuration is: core (spN)2 (rNN)2(rNH)2 1 (pNN)4ðrNH Þ . This state is the one correlating to the N2 and H fragments in their ground electronic states. The three next higher excited states show a strongly avoided crossing indicating a change in their character along the N–H reaction coordinate. A close analysis of their respective MRCI wave functions reveals that these states have a Rydberg character at their equilibrium geometry, with an
Fig. 1. N2H four lowest adiabatic potential energy surfaces of linear N2H are reported for the NH dissociative reaction coordinate. These states are of 2A1 symmetry in the present C2V calculations and of 2R and 2D symmetry in a C1v notation. The upper bond state is the X 1 Rþ g ionic state. Energies are in atomic units and distances in Angstroms. These curves have been constructed with a ‘‘spline’’ function using points that have been calculated with steps of ˚ for N–H distances varying from 0.85 A ˚ to 1.5 A ˚ . These steps were increased to 0.25 A ˚ for NH varying between 1.5 A ˚ and 3 A ˚ . Steps of 0.5 A ˚ were 0.1 A ˚. used for NH distances larger than 3 A
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Fig. 2. The N2H five lowest (bond and repulsive) quasi-diabatic potential energy surfaces of 2R and 2D symmetry are reported as well as that of the N2H+ ion of 1R symmetry (the top one) for the NH dissociative reaction coordinate. The two lowest bond curves correspond to the lowest Rydberg state of spherical and axial symmetry. Energies are in atomic units and distances in Angstroms. The curves have been constructed with a ‘‘spline’’ function using points deduced from the adiabatic MRCI energies as explained in the text.
electronic configuration of core (spN)2(rNN)2(rNH)2(pNN)4 (Ri)1 that changes to states of excited valence character with electronic configurations of core (spN)2 (rNN)2(rNH)2 1 (pNNp*NN)4ðrNH Þ . These states correlate to the excited states of N2. The shape of the adiabatic surfaces as well as the analysis of the corresponding MRCI wave functions (outlining the change in character of the higher excited states) argue for the existence of curve crossings, in a quasi-diabatic representation, between bound valence excited states of N2H and repulsive states. 1.2.2. The quasi-diabatic representation Since I am interested in the states involved in the DR of N2 Hþ ðX1 Rþ g ; v ¼ 0Þ, only potential energy surfaces of the ionic, the two lowest Rydberg of spherical and axial symmetry and the three linear lowest repulsive states of N2H have been reported for simplicity. However, even if only the lowest Rydberg states are shown, the reader has to keep in mind that above them exists a series of parallel Rydberg curves converging towards the ionic state. The quasi-diabatic potential energy surfaces governing the DR of N2 Hþ ðX1 Rþ g Þ are displayed in Fig. 2. As has already been outlined from the adiabatic representation, the lowest dissociative state leading to the ground N2 ð1 Rþ g Þ state, does not cross the ion nor the lowest N2H 2 R bond state, i.e. the Rydberg state. However, there is a second low repulsive state, dissociating to excited N2 ð3 Rþ u Þ þ H which crosses the ion not very far from its minimum geometry, i.e. for v = 0. However this crossing
˚ ) is at the turning point of the vibrational (for NH 1.25 A wave function of the ion and of the nuclear wave function of the dissociative state. Because of the resulting week overlap between these wave functions, we do not expect the DR of N2 Hþ ðX1 Rþ g ; v ¼ 0Þ to be efficient in the cold space environment. For the v = 1 level of the ion the overlap between the nuclear wave functions of the ionic and repulsive states should be strong. If so, the DR of N2 Hþ ðX1 Rþ g ; v ¼ 1Þ should efficiently lead to vibrationally excited N2 ð3 Rþ u Þ state. There is a third repulsive state, a 2D state, that crosses the ion for much higher vibrational states of N2H+. If this crossing is of less interest for the present study, it shows that for highly excited vibrational state of N2H+, there exists more pathway to dissociation. 2. Conclusions and discussion The present theoretical study of the N2 Hþ ðX1 Rþ gÞþe , reaction using ab initio methods of quantum chemistry have shown that for the linear channel leading to N2 and H fragments:
1. DR of N2 Hþ ðX 1 Rþ g Þ should involve a direct mechanism, i.e. a direct transition from the ionic state to a dissociative state, thanks to the existence of curve crossing between these two states. This confirms the statement of Geppert et al. who concluded from the absence of resonances that a direct mechanism must be occurring.
D. Talbi / Chemical Physics 332 (2007) 298–303
2. Since this curve crossing occurs at the turning point of the vibrational v = 0 wavefunction of the ion and of the nuclear wave function of the second dissociative state, the DR is not expected to be very efficient in the dense cloud of the interstellar medium where N2H+ cannot be vibrationally exited because of the low temperature (10 K) prevailing in this environment. 3. In a warmer environment as diffuse could where N2H+ could be vibrationally excited dissociation should be efficient through the v = 1 vibrational state of the ion leading to vibrationally excited N2 3 Rþ u state. From the present study we show that the DR of N2H+, does not occur through the lowest dissociative state of N2H but involves the second repulsive one. However even with this mechanism the DR may be less efficient for the ground vibrational state of the ion, i.e v = 0. This statement opens two questions: The first one is whether N2H+, for which a DR rate constant has been measured to be rather efficient, was really vibrationally relaxed in the experimental apparatus. The second question and it is the most promising one, is how bent geometries could affect the dissociative reaction path. Indeed my preliminary investigations on DR following bent geometries of the N2H+ ion shows that the overlap between the wave function of the lowest vibrational state of the ion and of the nuclear wave function of the second dissociative state might be improved for these bent geometries. Indeed using low level CI calculations it appears that the energy of the ion is almost not affected when bending its angle from 0° to 10°, while for these geometries, the second repulsive state is lowered toward the minimum of the ionic curve (almost halfway between the first vibrational level v = 0 and the electronic minimum energy of the ion). Of course these investigations are much too preliminary for drawing any firm conclusion but they do open a new perspective toward a more efficient dissociation path for the ion in its ground v = 0 vibrational state to give N2 + H. Work will be continued in this direction for this channel. This still leaves open the question of the behavior of the less exothermic exit channel leading to N and NH. Work is in progress for the N–NH system using the same MRCI calculations using orbitals of localized character to build the CI wave functions from the same CI n-particle space.
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Acknowledgements The CNRS national research program PN-PCMI has supported this study. The calculations were performed at the ‘‘Centre Informatique National de l’Enseignement Supe´rieur (CINES)’’. Both are gratefully acknowledged. Prof. P. Hickman is also gratefully acknowledged for very constructive discussions. References [1] B.E. Turner, Astrophys. J. 193 (1974) L83. [2] A. Fuente, J. Martin-Pintado, J. Cernicharo, R. Bachiller, Astron. Astrophys. 276 (1993) 473. [3] B.E. Turner, Astrophys. J. 449 (1995) 635. [4] V.A. Krasnopolsky, J. Geophys. Res. 107 (E12). doi:10.1029/ 2001JE001809. [5] P. Casellly, C.M. Walmsley, A. Zucconi, M. Tafalla, L. Dore´, P.C. Myers, Astrophys. J. 565 (2002) 473. [6] K. Tatematsu, J. Korean Astron. Soc. 38 (2005) 279. [7] W.D. Geppert et al., Astrophys. J. 609 (2004) 459. [8] J.C. Owrutsky, C.S. Grudeman, C.C. Martner, L.M. Tack, N.H. Rosenbaum, R.J. Saykally, J. Chem. Phys. 84 (1986) 605 (and references therein). [9] W.P. Kraemer, A. Komornicki, D.A. Dixon, Chem. Phys. 105 (1986) 87. [10] L.A. Curtiss, D.L. Drapcho, J.A. Pople, Chem. Phys. Lett. 103 (1984) 437 (and reference therein). [11] S.F. Selgren, P.W. McLoughlin, G.I. Gellene, J. Chem. Phys. 90 (1989) 1624. [12] K. Vasudevan, S.D. Peyerimhoff, R.J. Buenker, J. Mol. Struct. 29 (1975) 285. [13] D. Talbi, F. Pauzat, Y. Ellinger, Chem. Phys. 126 (1988) 291. [14] D. Talbi, Y. Ellinger, Chem. Phys. Lett. 288 (1998) 155. [15] D. Talbi, A. Le Padellec, J.B.A. Mitchell, J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 3631. [16] D. Talbi, R. Saxon, J. Chem. Phys. 89 (1988) 2235. [17] T.H. Dunning, P.J. Hay, in: Schaeffer III (Ed.), Modern Theoretical Chemistry, vol. 3, 1977, p. 1. [18] D. Talbi, P.R. Saxon, J. Chem. Phys. 91 (1989) 2376. [19] A.D. MacLean, M. Yoshimine, B.H. Lengsfield, P.S. Bagus, B. Liu, ALCHEMY II in MOTEC-90. [20] A.P. Hickman, R.D. Miles, C. Hayden, D. Talbi, Astron. Astrophys. 438 (2005) 31. [21] G. Herzberg, Electronic Spectra and Electronic Structures of Polyatomic Molecules, Van Nostrand, New York, 1966. [22] G. Herzberg, Atomic Spectra and Atomic Structure, Dover, New York, 1966. [23] T. Sharp, At. Data 2 (1971) 119.
A quantum chemical study of the N2H+ + e ! N2 + H reaction I: The linear dissociation path D. Talbi
*
Groupe de Recherche en Astronomie et Astrophysique du Languedoc UMR 5024, Universite´ de Montpellier II, Place Euge`ne Bataillon, 34090 Montpellier, France Received 19 September 2006; accepted 14 December 2006 Available online 21 December 2006
Abstract A theoretical investigation of the dissociative recombination (DR) of linear N2 Hþ ðX1 Rþ g Þ to give N2 + H has been undertaken because it is of interest for astrochemistry and also because it has been recently studied experimentally. Using state of the art quantum chemical methods, it is shown that the lowest 2R repulsive state of N2H leading to the N2 and H fragments in their ground electronic states does not cross the curve of the ion nor the one of the lowest N2H Rydberg state. This lowest 2R repulsive state is very low in energy. Its curve passes below the 1R N2H+ state and below the lowest bound 2R N2H states. However, it is also shown that there exist higher repulsive 2R and 2D states of N2H (the second and third repulsive states) crossing the ion curve. These states will lead to the formation of 3 + N2 in its 3 Rþ u and Du states. This study, the first of its type, shows that the DR of linear N2H should involve the direct mechanism and that it should lead to N2 in its first excited states. However this process may not be efficient for N2H+ in its ground vibrational state (v = 0), a state in which it exists in the cold environment of the interstellar medium. For the DR to be efficient for N2H+ in its ground v = 0 vibrational state, bent geometries of the ion might have to be considered. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Electronic dissociative recombination; Ab initio calculations; Potential energy surfaces; Excited linear states of N2H
1. Introduction Nitrogen is the fifth most abundant element in the universe. In the interstellar medium it is thought to be mostly molecular (N2). Since N2 has no observable rotational or vibrational transitions, it is traced through its parent ion, N2H+, to which it is related by the following reactions: þ N2 þ Hþ 3 ! N2 H þ H2
N2H+ is detected in a large number of astronomical [1–6] objects. In astrochemical models, it is considered that N2 is almost completely recycled through the electronic dissociative recombination (hereafter DR) of N2H+ N 2 H þ þ e ! N 2 þ H *
Tel.: +33 4 67 14 41 10; fax +33 4 67 14 45 35. E-mail address: [email protected]
0301-0104/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.12.009
However, the recent experimental measurements of Geppert et al. using the CRYRING ion storage ring apparatus [7] have thrown doubt on the validity of this hypothesis. In this experiment, it was shown that the most exoergic N2H+ + e ! N2 + H channel only accounts for 36% of the total reaction. N2H+ + e ! NH + N channel accounted for the remaining 64%. These unexpected branching ratios, posing the problem of the mechanism involved by the DR of N2H+ as well as of the abundance of N2 in space, have resulted in a strong demand from both the astrochemical and experimental community for a theoretical study of this process to rationalize the experimental results and to understand the mechanism involved in this process. I have undertaken this theoretical study using the most sophisticated ab initio methods of quantum chemistry. In this paper the investigation of the N2H+ + e ! N2 + H linear reaction path is presented; the other path (i.e. N + NH) is in progress and will be presented in a forthcoming paper.
D. Talbi / Chemical Physics 332 (2007) 298–303
Dissociative recombination corresponds to the capture of an electron by a positive molecular ion forming a neutral molecule in an excited electronic state. If this state is repulsive, dissociation may take place and the process is called ‘‘direct’’. If this state is a vibrationally excited electronic state of the neutral molecule, two mechanisms are possible. Either autoionisation takes place, or the temporarily bound state relaxes to a lower energy state, leading, if it is dissociative in nature, to fragmentation of the molecule. In this case the process is called ‘‘indirect’’ and most often involves Rydberg-type excited states because the complex is formed in a state where the electron is far from the nuclei, the core being essentially that of the positive ion ground state. The efficiency of both direct and indirect processes depends upon the existence of at least one repulsive state and also upon its position with respect to the bound excited or ionic states. Understanding such processes requires knowledge of the curves governing the corresponding mechanism, i.e. the ionic, excited (mainly Rydberg) and dissociative potential energy surfaces of the parent neutral molecule. The difficulty in treating such problems resides in, first of all, the description of the excited and dissociative states along dissociative paths. This is not a trivial task because, very often the nature of states changes along such channels and carefully designed wave functions are thus necessary in order to follow these transformations. Another difficulty is that an even-handed treatment is required for all the states involved in the mechanism. The N2H+ ion is a strongly bound linear species and is well known both experimentally and theoretically [8,9]. Its neutral parent N2H, a less familiar molecule, has been the subject of many theoretical investigations. The latest one [10] confirmed that its bent minimum energy structure is unstable with respect to N2 and H but is separated from the dissociation products by a barrier sufficient to support at least a zero point level. Experimentally its observation is more recent [11], probably because of its metastable structure. Very little is known about the excited states of N2H [12] and to my knowledge nothing has been published on the potential energy surfaces of the excited linear states that are of importance for understanding the N2H+ + e! N2H* ! N2 + H process. 1.1. The theoretical approach In the simple case of the closed shell ion N2H+, corresponding electronic configurations can be written as core (2spN)2(rNN)2(rNH)2 (pNN)4. DR involves the capture of an electron, which will lead to the formation of linear excited states of the parent neutral molecule with the additional electron occupying a virtual orbital. If the occupied virtual orbital is the anti-bonding NH* the resulting electronic configuration: core (spN)2(rNN)2(rNH)2 (pNN)4 1 (rNH Þ , for which the NH bond is weakened, is the lowest of those governing the DR of N2H+ leading to the H + N2 products in their ground electronic states. The additional electron can also occupy a diffuse orbital far from the
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N2H+ core. The result is a Rydberg state of the neutral N2H molecule of electronic configuration core (spN)2 (rNN)2(rNH)2 (pNN)4 (Ri)1 where Ri is a diffuse orbital of spherical (Rs) or axial (Rx, Ry, Rz) symmetry. These types of states have been shown in our previous theoretical studies [13,14] to play an important role in DR reactions involving the indirect mechanism. For a quantitative study of the recombination of N2H+ the even handed treatment of both the ionic, the excited, (mainly Rydberg type), and the dissociative states of linear N2H is required. Correlation energies must be treated through the same CI procedure so as not to favor one state over another. This is true for all states, which might be involved in the recombination of the ion and all along the dissociative reaction coordinate. To fulfill these requirements, I have adopted a methodology, already defined in previous studies[13,14], which is based on the use of orbitals with a strong localized character, to build the CI wave functions. The advantage of such a method is that the CI nparticle space is defined in terms of orbitals whose chemical significance is clearly identified. Configurations can be selected according to chemical evidence and the type of arbitrariness associated with energy related selection processes can be avoided. More important, the CI expansion is directly obtained in a configuration space that contains the approximate diabatic representation of the problem (hereafter quasi-diabatic). Indeed, thanks to the use of orbitals with a strong localized character to construct the configuration space, it is possible to identify most configuration state functions with a quasi-diabatic state (repulsive, excited valence or Rydberg) according to the weights of the relevant configurations in the CI wave functions. (Details of the procedure can be found in Talbi et al. [14].) To obtain orbitals of localized character different procedures can be used. I have, after a series of test calculations, chosen to use MCSCF natural orbitals optimized through the MCSCF n-particle space of Table 1, where three electrons have been distributed in set1. (NH and NH* orbitals), two in set2. (NN and NN* orbitals) four in set3. (p and p* orbitals) and 2 in set4. (spNspN*). The NH and NH* orbitals have been verified to correctly dissociate to the desired limits (i.e. the 2s of nitrogen and the 1s of hydrogen) while keeping a localized character. To construct the molecular orbital basis, a 6-311G(d,p) atomic orbital basis set similar to that of Ref. [15] has been used for nitrogen and a 4s3p1d similar to that of Ref. [16] has been used for H. To account for the Rydberg components, the atomic orbital basis set has been augmented by atomic diffuse functions on each atoms and the valence space obtained following the MCSCF procedure, has been extended by molecular Rydberg orbitals of atomic type (see [13,14] for details). The s (0.026, 0.016) and p (0.023, 0.015) diffuse exponent functions of Dunning and Hay [17] have been used for nitrogen. The s (0.015) and p (0.025) [17] diffuse exponent functions used to extend the 4s3p1d atomic orbital basis set of hydrogen have been shown in a previous study [16] to give (when combined with the above 4s3p1d basis set) reliable
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Table 1 MCSCF n-particle space designed for the N2H+ + e system in C2v symmetry Symmetry
Electronic distributions Corea
Internal sets Set1
Set2
a1 b2 b1 a2
rNH rNH
rNN rNN
Number of electronsb
3(2)
Set3
Set4 spN spN
pNN pNN pNN pNN 2
4
2
The number of CSF’s generated is 708 for the neutral and 654 for the ion. a Core orbitals: 1sN. b Configuration of the ion in between brackets.
H(n = 2) energy levels. The H(n = 2) states are states to which the lowest linear Rydberg states of N2H will correlate, they need to be correctly described. The resulting set of MO’s is Schmidt-orthogonalized prior to the CI calculations. This way of building a complete set of orbitals has been shown to give reliable positions for the Rydberg states [18]. The CI n-particle space designed to evaluate the correlation energy is shown in Table 2, where the electronic dis-
tribution in brackets is that for the ion calculated with the same CI n-particle space. This CI n-partitioning designed after a series of test calculations should ensure an evenhanded treatment of all states (ion, valence excited, Rydberg and dissociative) involved in the mechanism, at all points of the dissociation pathway. It will be used for the study of the N + NH exit channel to ensure the same correlated treatment.
Table 2 CI n-particle space designed for the N2H+ + e system in C2v symmetry Symmetry
Electronic distributions Corea
Internal sets Set1
a1
1 2 3 4 5 6 7 8 9 10 11 12
Set2
spN rNN rNH rNN rNH
b2 b1 a2 Configuration classesb,
External set
c
Set3 RsRxRsRx
23
pNN pNN pNN pNN
RyRy RzRz
11 11 3
4(4) 3(3) 3(3) 2(2) 2(2) 5(5) 4(4) 4(4) 3(3) 6(6) 4(4) 4(4)
0(0) 1(1) 0(0) 2(2) 1(1) 0(0) 1(1) 0(0) 2(2) 0(0) 2(2) 1(1)
0(0) 0(0) 1(1) 0(0) 1(1) 0(0) 0(0) 1(1) 0(0) 0(0) 0(0) 1(1)
0(0) 0(0) 0(0)
2(2) 2(2) 2(2)
MO’s
Number of electrons 7(6) 7(6) 7(6) 7(6) 7(6) 6(5) 6(5) 6(5) 6(5) 5(4) 5(4) 5(4)
Number of electrons with respect to referencesd 13 14 15
7(6) 6(5) 5(4)
2(2) 3(3) 4(4)
All configurations corresponding to the electronic distributions specified and satisfying the spin and symmetry requirements are considered. Total number of CSF’s: 1,430,834 for neutral doublet states and 827,916 for the ionic singlet state. a Core orbitals: 1sN. b Core orbitals are doubly occupied in all CSF’s. c Electronic distributions for the ions are given in parentheses. d For the double excitations in the external set, the excitations in the internal set are limited to double excitations. For example, the configuration classes 15 corresponds to double excitation in the external set with respect to all singly and doubly excited configurations that are generated by distributing five electrons in set1 (4 for the ion) and four electrons in set2 (4 for the ion).
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All these calculations have been performed in C2V symmetry using the ALCHEMY-CI II program [19]. 1.2. Potential energy surfaces Provided the crossing we are looking for is located not very far from the equilibrium geometry of the cation, it is known from previous studies [13,14,20] that the best compromise while studying the DR of an HXY+ molecule, is to keep the geometry of the XY bond fixed at its geometry in the ion along the dissociative reaction coordinate H–X. Moreover, the NN bonds being almost identical in N2H+ and N2, I have for the study of the [N2H+ + e] complex along the N–H dissociative coordinate, fixed the NN bond at its geometry in the ion optimized at the present MRCI ˚. level, i.e. NN = 1.12 A 1.2.1. The adiabatic representation The expected reliability of the present CI calculations summarised in Table 2 can be assessed from the energetics deduced from the N2H (2R) and N2H+(1R) energies calculated at the dissociation limits within the super molecule approach. With the present MRCI calculations, the N2H+ to N2 + H+ dissociation energy is calculated to be 5.4 eV (without considering zero point energies). This value is identical to the best calculated one [9]. At the present 3 1 MRCI level, N2 ð3 Rþ u Þ, N2( Du) and N2( Dg), respectively lies at 6.0 eV, 7.3 eV and 9.2 eV above N2 ð1 Rþ g Þ. These ver-
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tical excitation energies (calculated without considering zero point energies) are consistent with the N2 excitation energies reported in the literature [21]; respectively 6.2 eV, 7.4 eV and 8.9 eV. The present calculated ionization energy for hydrogen is 13.4 eV and the H(n = 1) ! H(n = 2) excitation energy 11.3 eV. These numbers are in a satisfying agreement with the experimentally known values, 13.6 eV [22] and 10.2 eV [23], respectively. The above-calculated relative energies are in a good enough agreement with the known data from the literature to consider the present MRCI level of sufficient reliability to describe the N2H+ + e ! N2H* ! N2 + H process. Adiabatic potential energy surfaces for the four lowest 2 A1 states of N2H, states of R and D symmetry in a C1v notation, as well as that of the ion N2 Hþ ðX1 Rþ g Þ are presented in Fig. 1. They are calculated along the NH reaction ˚ . Fig. 1 coordinate keeping the NN bond frozen at 1.12 A 2 shows that the lowest R N2H state, has a repulsive character. It is very low in energy and does not cross the N2H+ ion curve. Its MRCI wave function analysis reveals that its electronic configuration is: core (spN)2 (rNN)2(rNH)2 1 (pNN)4ðrNH Þ . This state is the one correlating to the N2 and H fragments in their ground electronic states. The three next higher excited states show a strongly avoided crossing indicating a change in their character along the N–H reaction coordinate. A close analysis of their respective MRCI wave functions reveals that these states have a Rydberg character at their equilibrium geometry, with an
Fig. 1. N2H four lowest adiabatic potential energy surfaces of linear N2H are reported for the NH dissociative reaction coordinate. These states are of 2A1 symmetry in the present C2V calculations and of 2R and 2D symmetry in a C1v notation. The upper bond state is the X 1 Rþ g ionic state. Energies are in atomic units and distances in Angstroms. These curves have been constructed with a ‘‘spline’’ function using points that have been calculated with steps of ˚ for N–H distances varying from 0.85 A ˚ to 1.5 A ˚ . These steps were increased to 0.25 A ˚ for NH varying between 1.5 A ˚ and 3 A ˚ . Steps of 0.5 A ˚ were 0.1 A ˚. used for NH distances larger than 3 A
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Fig. 2. The N2H five lowest (bond and repulsive) quasi-diabatic potential energy surfaces of 2R and 2D symmetry are reported as well as that of the N2H+ ion of 1R symmetry (the top one) for the NH dissociative reaction coordinate. The two lowest bond curves correspond to the lowest Rydberg state of spherical and axial symmetry. Energies are in atomic units and distances in Angstroms. The curves have been constructed with a ‘‘spline’’ function using points deduced from the adiabatic MRCI energies as explained in the text.
electronic configuration of core (spN)2(rNN)2(rNH)2(pNN)4 (Ri)1 that changes to states of excited valence character with electronic configurations of core (spN)2 (rNN)2(rNH)2 1 (pNNp*NN)4ðrNH Þ . These states correlate to the excited states of N2. The shape of the adiabatic surfaces as well as the analysis of the corresponding MRCI wave functions (outlining the change in character of the higher excited states) argue for the existence of curve crossings, in a quasi-diabatic representation, between bound valence excited states of N2H and repulsive states. 1.2.2. The quasi-diabatic representation Since I am interested in the states involved in the DR of N2 Hþ ðX1 Rþ g ; v ¼ 0Þ, only potential energy surfaces of the ionic, the two lowest Rydberg of spherical and axial symmetry and the three linear lowest repulsive states of N2H have been reported for simplicity. However, even if only the lowest Rydberg states are shown, the reader has to keep in mind that above them exists a series of parallel Rydberg curves converging towards the ionic state. The quasi-diabatic potential energy surfaces governing the DR of N2 Hþ ðX1 Rþ g Þ are displayed in Fig. 2. As has already been outlined from the adiabatic representation, the lowest dissociative state leading to the ground N2 ð1 Rþ g Þ state, does not cross the ion nor the lowest N2H 2 R bond state, i.e. the Rydberg state. However, there is a second low repulsive state, dissociating to excited N2 ð3 Rþ u Þ þ H which crosses the ion not very far from its minimum geometry, i.e. for v = 0. However this crossing
˚ ) is at the turning point of the vibrational (for NH 1.25 A wave function of the ion and of the nuclear wave function of the dissociative state. Because of the resulting week overlap between these wave functions, we do not expect the DR of N2 Hþ ðX1 Rþ g ; v ¼ 0Þ to be efficient in the cold space environment. For the v = 1 level of the ion the overlap between the nuclear wave functions of the ionic and repulsive states should be strong. If so, the DR of N2 Hþ ðX1 Rþ g ; v ¼ 1Þ should efficiently lead to vibrationally excited N2 ð3 Rþ u Þ state. There is a third repulsive state, a 2D state, that crosses the ion for much higher vibrational states of N2H+. If this crossing is of less interest for the present study, it shows that for highly excited vibrational state of N2H+, there exists more pathway to dissociation. 2. Conclusions and discussion The present theoretical study of the N2 Hþ ðX1 Rþ gÞþe , reaction using ab initio methods of quantum chemistry have shown that for the linear channel leading to N2 and H fragments:
1. DR of N2 Hþ ðX 1 Rþ g Þ should involve a direct mechanism, i.e. a direct transition from the ionic state to a dissociative state, thanks to the existence of curve crossing between these two states. This confirms the statement of Geppert et al. who concluded from the absence of resonances that a direct mechanism must be occurring.
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2. Since this curve crossing occurs at the turning point of the vibrational v = 0 wavefunction of the ion and of the nuclear wave function of the second dissociative state, the DR is not expected to be very efficient in the dense cloud of the interstellar medium where N2H+ cannot be vibrationally exited because of the low temperature (10 K) prevailing in this environment. 3. In a warmer environment as diffuse could where N2H+ could be vibrationally excited dissociation should be efficient through the v = 1 vibrational state of the ion leading to vibrationally excited N2 3 Rþ u state. From the present study we show that the DR of N2H+, does not occur through the lowest dissociative state of N2H but involves the second repulsive one. However even with this mechanism the DR may be less efficient for the ground vibrational state of the ion, i.e v = 0. This statement opens two questions: The first one is whether N2H+, for which a DR rate constant has been measured to be rather efficient, was really vibrationally relaxed in the experimental apparatus. The second question and it is the most promising one, is how bent geometries could affect the dissociative reaction path. Indeed my preliminary investigations on DR following bent geometries of the N2H+ ion shows that the overlap between the wave function of the lowest vibrational state of the ion and of the nuclear wave function of the second dissociative state might be improved for these bent geometries. Indeed using low level CI calculations it appears that the energy of the ion is almost not affected when bending its angle from 0° to 10°, while for these geometries, the second repulsive state is lowered toward the minimum of the ionic curve (almost halfway between the first vibrational level v = 0 and the electronic minimum energy of the ion). Of course these investigations are much too preliminary for drawing any firm conclusion but they do open a new perspective toward a more efficient dissociation path for the ion in its ground v = 0 vibrational state to give N2 + H. Work will be continued in this direction for this channel. This still leaves open the question of the behavior of the less exothermic exit channel leading to N and NH. Work is in progress for the N–NH system using the same MRCI calculations using orbitals of localized character to build the CI wave functions from the same CI n-particle space.
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Acknowledgements The CNRS national research program PN-PCMI has supported this study. The calculations were performed at the ‘‘Centre Informatique National de l’Enseignement Supe´rieur (CINES)’’. Both are gratefully acknowledged. Prof. P. Hickman is also gratefully acknowledged for very constructive discussions. References [1] B.E. Turner, Astrophys. J. 193 (1974) L83. [2] A. Fuente, J. Martin-Pintado, J. Cernicharo, R. Bachiller, Astron. Astrophys. 276 (1993) 473. [3] B.E. Turner, Astrophys. J. 449 (1995) 635. [4] V.A. Krasnopolsky, J. Geophys. Res. 107 (E12). doi:10.1029/ 2001JE001809. [5] P. Casellly, C.M. Walmsley, A. Zucconi, M. Tafalla, L. Dore´, P.C. Myers, Astrophys. J. 565 (2002) 473. [6] K. Tatematsu, J. Korean Astron. Soc. 38 (2005) 279. [7] W.D. Geppert et al., Astrophys. J. 609 (2004) 459. [8] J.C. Owrutsky, C.S. Grudeman, C.C. Martner, L.M. Tack, N.H. Rosenbaum, R.J. Saykally, J. Chem. Phys. 84 (1986) 605 (and references therein). [9] W.P. Kraemer, A. Komornicki, D.A. Dixon, Chem. Phys. 105 (1986) 87. [10] L.A. Curtiss, D.L. Drapcho, J.A. Pople, Chem. Phys. Lett. 103 (1984) 437 (and reference therein). [11] S.F. Selgren, P.W. McLoughlin, G.I. Gellene, J. Chem. Phys. 90 (1989) 1624. [12] K. Vasudevan, S.D. Peyerimhoff, R.J. Buenker, J. Mol. Struct. 29 (1975) 285. [13] D. Talbi, F. Pauzat, Y. Ellinger, Chem. Phys. 126 (1988) 291. [14] D. Talbi, Y. Ellinger, Chem. Phys. Lett. 288 (1998) 155. [15] D. Talbi, A. Le Padellec, J.B.A. Mitchell, J. Phys. B: At. Mol. Opt. Phys. 33 (2000) 3631. [16] D. Talbi, R. Saxon, J. Chem. Phys. 89 (1988) 2235. [17] T.H. Dunning, P.J. Hay, in: Schaeffer III (Ed.), Modern Theoretical Chemistry, vol. 3, 1977, p. 1. [18] D. Talbi, P.R. Saxon, J. Chem. Phys. 91 (1989) 2376. [19] A.D. MacLean, M. Yoshimine, B.H. Lengsfield, P.S. Bagus, B. Liu, ALCHEMY II in MOTEC-90. [20] A.P. Hickman, R.D. Miles, C. Hayden, D. Talbi, Astron. Astrophys. 438 (2005) 31. [21] G. Herzberg, Electronic Spectra and Electronic Structures of Polyatomic Molecules, Van Nostrand, New York, 1966. [22] G. Herzberg, Atomic Spectra and Atomic Structure, Dover, New York, 1966. [23] T. Sharp, At. Data 2 (1971) 119.