On the possibility of using model potentials for collision integral

Chemical Physics 338 (2007) 62–68 www.elsevier.com/locate/chemphys

On the possibility of using model potentials for collision integral calculations of interest for planetary atmospheres M. Capitelli

a,b

, D. Cappelletti c, G. Colonna b, C. Gorse a,b, A. Laricchiuta G. Liuti c, S. Longo a,b, F. Pirani d

b,*

,

a

c

Dip. Chimica, Universita` di Bari, Italy b IMIP CNR, Bari, Italy Dip. Ing. Civile ed Ambientale, Universita` di Perugia, Italy d Dip. Chimica, Universita` di Perugia, Italy Received 4 May 2007; accepted 27 July 2007 Available online 3 August 2007

Abstract The interaction energy in systems (atom–atom, atom–ion and atom–molecule) involving open-shell species, predicted by a phenomenological method, is used for collision integral calculations. The results are compared with those obtained by different authors by using the complete set of quantum mechanical interaction potentials arizing from the electronic configurations of separate partners. A satisfactory agreement is achieved, implying that the effect of deep potential wells, present in some of the chemical potentials, is cancelled by the effect of strong repulsive potentials.  2007 Elsevier B.V. All rights reserved. PACS: 12.39.Pn; 34.20.Gj; 34.50.s; 52.25.Fi Keywords: Potential models; Elastic scattering; Transport processes in plasma

1. Introduction In the last decades a great effort has been devoted to the study of elementary processes occurring in non-equilibrium systems such as plasmas or planetary atmospheres (Earth, Mars, Venus, Jupiter, etc.). In particular, great attention has been dedicated to the calculation of collision integrals of high temperature species. Different types of interactions, controlling neutral–neutral, neutral–ion, electron–neutral and ion–ion processes, can occur in the high temperature environment, triggered by the impact of hypersonic entering vehicle in the relevant atmospheres. The vehicle kinetic energy is in fact transformed in thermal energy, which activates chemical processes, such as dissociation and ioniza-

*

Corresponding author. E-mail address: [email protected] (A. Laricchiuta).

0301-0104/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2007.07.036

tion. For instance, in the Martian atmosphere, which in normal conditions is composed mainly by CO2 and N2, if the temperature, T, is increased many new species can appear, such as CO2, CO, C, O, C+, O+, N2, N, N+, CN, NO, NO+ and free electrons. The heat transferred from the shock layer to the vehicle surface by the interaction with the gas particles is of paramount importance for the safety of the mission. In turn the heat transfer depends on the gas phase thermal conductivity, which can be estimated from the Chapman–Enskog method once the collision integrals (transport cross sections) for all the relevant particle interactions are known. Different collision integral data sets exist in literature based on a variety of methods for the evaluation of the relevant interaction potentials. They include: • high energy beam collisions to obtain information about the repulsive part of the interaction potential;

M. Capitelli et al. / Chemical Physics 338 (2007) 62–68

• low energy crossed beam experiments to obtain information about the potential well and the first part of the repulsive wall; • ab initio calculations of potential energy surfaces; • semi-empirical methods to provide the whole or the relevant part of the potential; • semi-empirical rules to predict collision integrals of A–B interactions from known A–A and B–B collision integrals. Some of these methods have been used by our group [1] and, more recently, by the NASA group [2] for extensive tabulation of collision integrals of high temperature Earth atmosphere and for the estimation of the possible errors in this kind of calculations. On the other hand, it is important to have easily and quickly available physical meaningful methods which allow the calculation of fundamental features of non-covalent neutral–neutral and neutral–ion inter-molecular interactions, i.e. binding energy and equilibrium distance. Such features can be easily predicted, in unknown systems, by a method proposed few years ago by the Pirani group [3– 7], which exploits correlation formulas given in terms of fundamental physical properties of involved interacting partners (polarizability, charge, number of electrons effective in polarization). Very recently a new model potential, considered as an improvement of the Lennard–Jones function, has been also introduced [8]. Such model, involving the use of few potential features as parameters, properly describes the interaction energy in a wide inter-molecular distance range. The full methodology can be in principle applied to any neutral–neutral and neutral–ion interaction, including open-shell atom–atom and atom–ion systems. Actually a multiplicity of states, bound and repulsive, arise when the two colliding partners approach and the total collision integral results as a weighted average of the contributions of each state. In this kind of averaging the repulsive potentials have a major role due to their high statistical weight. The phenomenological Pirani’s potential, simulating the average interaction, could allow the direct evaluation of internally consistent complete sets of collision integrals for different high temperature atmospheres. Therefore, it is useful to compare the results of this approach and the corresponding collision integral values obtained by using accurate quantum mechanical interaction potentials. In this paper some benchmark systems have been considered, including open-shell atoms and ions. The validation of this phenomenological approach, which does not intend to substitute accurate ab initio calculations, could be important to extend the procedure to the study of complex systems, difficult to be characterized in details. 2. Method of calculation For neutral–neutral systems, the phenomenological method represents the binding energy, e0, and the equilib-

63

rium distance, re, in terms of polarizability, a, of the interacting partners by the following correlation formulas [3–5] 1=3

re ¼ 1:767 e0 ¼ 0:72

ai

1=3

þ aj

ðai aj Þ0:095

Cd r6e

ð1Þ ð2Þ

˚ , a in A ˚ 3 and e0 in eV. The Cd constant where re is given in A 6 ˚ (eV A ) is an effective long-range London coefficient ai aj C d ¼ 15:7 hpffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffii ð3Þ ai =N i þ aj =N j where N is the effective number of electrons which contribute to the polarization of the neutral species. Previous formulas have been also extended [6,7] to neutral–ion systems, introducing the parameter q, representative of the relative role of dispersion and induction attraction components in proximity to the equilibrium distance. 1=3

re ¼ 1:767

ai

þ a1=3 n

ðai an ½1 þ 1=qÞ0:095 z 2 an e0 ¼ 5:2 4 ½1 þ q re ai q¼ pffiffiffiffiffi 2 z ½1 þ ð2ai =an Þ2=3  an

ð4Þ ð5Þ ð6Þ

where z is the ion charge. The model potential [8], providing the interaction energy u as combination of repulsive and attractive terms, has the form: "  nðxÞ  m # m 1 nðxÞ 1  ð7Þ u ¼ e0 nðxÞ  m x nðxÞ  m x where x = r/re, n(x) = b + 4x2. For neutral–neutral and neutral–ion cases the parameter m has the value of 6 and 4 respectively. The value of b parameter can be estimated through an empirical formula: b¼6þ

5 ðs1 þ s2 Þ

ð8Þ

The softness s of the ith species, entering in Eq. (8), is defined as the cube root of its polarizability. For open-shell atoms and ions a multiplicative factor, which is the ground state spin multiplicity, should be also considered. The distance r, where attraction and repulsion balance, i.e. u = 0, is another basic potential parameter, and its value scales with re as ð9Þ

r ¼ x0 r e under the constraint mnðx0 Þ

x0



nðx0 Þ ¼0 m

ð10Þ

A review of the whole procedure is given in Ref. [9]. Potential parameters for selected systems are reported in Table 1. Collision integrals, X(‘,s), in the Chapman–Enskog theory result from a threefold integration: on the interparticle

64

M. Capitelli et al. / Chemical Physics 338 (2007) 62–68

our group, that can handle any potential function regardless the number of extrema [11]. A powerful ‘‘fractal’’ integration technique has been used avoiding problems related to the existence of discontinuities (orbiting) and without any preliminary definition of energy ranges as done in literature [12]. Tabulated collision integral values, reported and analysed in the next section, correspond to the dimensional ˚ 2. quantity r2X(‘,s)w in A

Table 1 Parameters of the phenomenological potential (Eq. (7)) for the considered systems ˚) Interaction b e0 (meV) re (A N(4S)–N(4S) N(4S)–N+(3P) O(3P)–O+(4S) Oð3 PÞ–O2 ðX3 R gÞ C(3P)–O(3P) C(3P)–N(4S)

6.605 6.759 6.926 7.264 6.780 6.645

6.43 94.58 93.45 7.98 6.12 6.88

3.58 2.96 2.69 3.62 3.65 3.72

3. Results: analysis of benchmark systems Our investigation starts with the analysis of N(4S)–N(4S) system. According to the Withmer–Wigner rules the interaction occurs along four different potential curves corresponding to 1R, 3R, 5R, 7R electronic terms. Following the pair valency theory we can razionalize the increase of the unbound character of the state with spin multiplicity. So while the singlet state is characterized by a strong chemical bond, the septet exhibits a repulsive potential, as can be appreciated in Fig. 1a, where relevant potential energy curves are reported. In the same figure the curve hui is shown, resulting from the statistical average of the four potentials P wi ui hui ¼ Pi ð15Þ i wi

distance r, on the impact parameter b and on the reduced collision energy c2 = E/kT [10]. Z

1 2



b2 uðrÞ 1 2  E r

1=2 ð11Þ

drr Hðb; EÞ ¼ p  2b rc Z 1 dbbð1  cos‘ HÞ Qð‘Þ ðEÞ ¼ 2p 0 sffiffiffiffiffiffiffiffi Z 1 kT 2 Xð‘;sÞ ðT Þ ¼ dcc2sþ3 Qð‘Þ ec 2pl 0

ð12Þ ð13Þ

H is the classical deflection angle for u(r) interaction potential, rc the distance of closest approach and Q‘ represent the transport cross section, corresponding to the ‘th moment. Suitable quantities for transport properties calculation are the reduced collision integrals X(‘,s)w, displaying the deviation from the rigid sphere model Z 1 4ð‘ þ 1Þ 2 dcc2sþ3 Qð‘Þ ec Xð‘;sÞH ¼ ‘ ðs þ 1Þ!½2‘ þ 1  ð1Þ pr2 0 ð14Þ

where wi and ui represent respectively the statistical weight and the interaction potential associated to the system in the ith electronic state. The statistical weight is the spin multiplicity for R states and two times the spin multiplicity for different symmetries (P, D, etc.). This kind of averaging emphasizes the role of the repulsive states in smoothing the attractive parts of chemical bonds. The curve hui for the N2 system is compared, in Fig. 1b, with the one obtained with the phenomenological procedure summarized in the previous section (potential parameters are given in Table 1). The comparison shows

X(‘,s)w values have been obtained in this paper for the phenomenological potential of Eq. (7) by using a general code for the classical collision integral evaluation developed in

0.01

50

N2

N2 0.005 POTENTIAL ENERGY (eV)

POTENTIAL ENERGY (eV)

40 30 7

20

5 3

10 0 -10 0.5

Σg

Σg

‹ϕ›

Σg

0

-0.01

Σg

1

1.0

1.5

2.0

2.5

INTERNUCLEAR DISTANCE (Å)

3.0

PHENOMENOLOGICAL POTENTIAL

-0.005

-0.015 2.0

‹ϕ› 3.0

4.0

5.0

6.0

7.0

8.0

INTERNUCLEAR DISTANCE (Å)

Fig. 1. The interaction potential energy in the N2 system. (a) Potential energy curves for the electronic states correlating with N(4S)–N(4S); (b) detail of averaged potential hui (full line) and of phenomenological potential (Eq. (7)) (dotted line).

M. Capitelli et al. / Chemical Physics 338 (2007) 62–68

that the wells are quite similar, with a depth three orders of magnitude lower than that of the ground singlet state, and located approximately in the same internuclear distance range. In Table 2 diffusion-type collision integrals, obtained integrating the classical deflection angle on the averaged and phenomenological potentials, are reported. In the same table a comparison with results from literature [1,13], obtained with the standard procedure, i.e. adiabatically averaging the contributions coming from the four different states, is also performed. In particular, collision integrals by Capitelli et al. [1] result from a Morse fitting of experimental potential curves for the bound states and an exponential-repulsive function reproducing an Heitler– London calculation of septet state [14], while in the low temperature region (T < 1000 K) a Lennard–Jones potential has been used. Levin et al. [13] results are derived on the base of accurate ab initio calculations. We note a substantial agreement between data sets in literature (Refs. [1,13]). On the other hand a Morse-fitting of

Table 2 ˚ 2), as a function of temperDiffusion-type collision integrals, r2X(1,1)w (A ature for N(4S)–N(4S) interaction obtained with different approaches, compared with accurate theoretical results form Refs. [1,13] T (K)

This work

hui

[1]

[13]

500 1000 2000 4000 5000 6000 8000 10,000 15,000 20,000

7.34 6.30 5.42 4.64 4.40 4.21 3.93 3.72 3.36 3.13

5.54 4.82 4.25 3.74 3.58 3.45 3.26 3.11 2.84 2.66

7.76 6.79 5.25 4.50 4.27 4.09 3.79 3.55 3.12 2.82

7.03 5.96 5.15 4.39 4.14 3.94 3.61 3.37 2.92 2.62

20 DOUBLET QUARTET SESTET

POTENTIAL ENERGY (eV)

15

65

hui potential (whose basic parameters [1] are C = 6.216, ˚ , e0 = 10 meV, re = 2.823 A ˚ ) yields collision r = 2.540 A integrals which differ up to 30% at T = 500 K with the corresponding results of Capitelli et al. [1], the discrepancy being reduced significantly as the temperature increases, due to a better description of the short-range repulsive interaction. A better agreement is found when data in literature are compared with the collision integrals obtained by using the Pirani’s potential. In this case the differences increase with temperature, not exceeding 20% below 15,000 K. Such behaviour indicates that the phenomenological approach describes accurately the potential well which play the major role in the low temperature region. As a second case we consider the interaction in the N(4S)–N+(3P) system along the 12 related electronic states 2,4,6 Rg,u, 2,4,6Pg,u. The relevant potential energy curves (from Ref. [1]) have been reported in Fig. 2 together with the statistical average, hui, and the phenomenological potential (see Table 1 for parameters). hui, fitted with an exponential function (hui[eV] = 287.8 exp (2.4457r)), is dominated by the repulsion which obscures the presence of the well in the intermediate internuclear distance range. On the contrary the Pirani’s potential presents a minimum of about 0.1 eV as a result of the interplay between attractive (polarizability) and repulsive part of the potential. In Table 3 the viscosity-type collision integrals, not affected by the charge transfer process occurring in atom–parent ion collisions, calculated with hui and the phenomenological potential are reported together with data in Refs. [1,16]. Results of Capitelli et al. [1] and Stallcop et al. [16] are in satisfactory agreement with values obtained using hui, the percentage error not exceeding 20%. Collision integrals calculated according to the Pirani’s potential shows a reasonable agreement with Stallcop et al. [16] and the Capitelli et al. [1] results, specially in the temperature range (5000– 20,000 K) in which N and N+ are major species. The behavior, in the considered temperature range, of the absolute error of data obtained with the phenomenological approach with respect to the accurate calculation, based on ab initio potentials for each interaction channel,

10 5

Table 3 ˚ 2), as a function of temperViscosity-type collision integrals, r2X(2,2)w (A ature for the N(4S)–N+(3P) interaction obtained with different approaches, compared with accurate theoretical results from Refs. [1,16]

0 -5

+

N2 -10 0.5

1.0

1.5

2.0

2.5

3.0

INTERNUCLEAR DISTANCE (Å)

Fig. 2. Potential energy curves describing the N(4S)–N+(3P) interaction in the different electronic states of the Nþ 2 molecular ion: (closed circle)-Rg, (open circle)-Ru, (closed diamond)-Pg, (open diamond)-Pu. Averaged potential (full line) and phenomenological potential (Eq. (7)) (dotted line) for Nþ 2 are also displayed.

T (K)

This work

hui

[1]

[16]

500 1000 2000 4000 5000 6000 8000 10,000 15,000 20,000

18.54 11.65 7.88 6.09 5.72 5.45 5.09 4.83 4.41 4.13

13.44 11.41 9.54 7.84 7.33 6.92 6.30 5.84 5.04 4.51

13.25 11.32 9.55 7.85 7.32 6.90 6.26 5.79 4.99 4.46

16.41 13.27 10.50 8.33 7.74 7.26 6.48 5.84 4.66 3.87

66

M. Capitelli et al. / Chemical Physics 338 (2007) 62–68 20

POTENTIAL ENERGY (eV)

16

DOUBLET QUARTET SESTET

12 8 4 0

+

-4

O2

-8 0.5

1.0

1.5

2.0

2.5

3.0

INTERNUCLEAR DISTANCE (Å)

Fig. 3. Potential energy curves describing O(3P)–O+(4S) interaction in the different electronic states of the Oþ 2 molecular ion: (closed circle)-Rg, (open circle)-Ru, (closed diamond)-Pg, (open diamond)-Pu. Averaged potential (full line) and phenomenological potential (Eq. (7)) (dotted line) for Oþ 2 are also displayed.

Table 4 ˚ 2), as a function of temperViscosity-type collision integrals, r2X(2,2)w (A ature for the O(3P)–O+(4S) interaction obtained with different approaches, compared with accurate theoretical results from Refs. [1,16] T (K)

This work

hui

[1]

[16]

500 1000 2000 4000 5000 6000 8000 10,000 15,000 20,000

15.22 9.58 6.50 5.05 4.74 4.53 4.23 4.02 3.67 3.45

8.07 5.62 4.33 3.60 3.41 3.27 3.06 2.90 2.62 2.43

10.19 8.73 7.40 6.09 5.65 5.29 4.73 4.31 3.61 3.18

14.78 11.14 8.72 6.94 6.39 5.95 5.26 4.75 3.92 3.41

is the same displayed by Levin et al. in Fig. 1 of Ref. [17]. Levin used the effective potential in the Tang–Toennies form, which is actually a more complex function than the Pirani’s potential. However, it should be noted that in Ref. [17] the binding energy and the equilibrium distance, the two basic potential parameters, have been obtained using the methodology outlined in Section 2. Same considerations apply to the case of O(3P)–O+(4S) collision. Atom and parent ion can interact originating 2,4,6 Rg,u, 2,4,6Pg,u electronic molecular states (Fig. 3). The averaged potential hui is characterized by a small well located in the range of low values for internuclear distances ˚ , e0 = 51 meV), while the phenomenological (re = 2.5 A approach predicts an attractive potential two times larger ˚ , e0 = 93 meV). and a minimum slightly shifted (re = 2.7 A 2 (2,2)w The corresponding r X , calculated by direct integration of different interaction potentials, are presented in Table 4 and compared with data in literature. In this case large differences are found between accurate results from Refs. [1,16] and collision integrals obtained using hui, while the phenomenological approach still gives satisfactory results, especially when compared with data in Refs. [1,16] in the temperature range of interest (5000– 20,000 K). This result confirms that an effective potential, not directly connected with details of the interacting system in different electronic states, can be used for transport cross section prediction. Let us now consider the Oð3 PÞ–O2 ðX3 R g Þ system. Atom–diatom dynamics develops on multi-dimensional surfaces and the interaction potential should take into account all the involved channels. Stallcop et al. [18] performed ab initio multireference configuration–interaction calculations and derived an effective potential for the estimation of transport cross sections. The potential parame˚, ters suggested by this accurate procedure (r = 3.205 A e0/k = 80.7 K), which are relevant to the potential well and affect the collision integrals in the low temperature

0.01

O-O2

0.005

VISCOSITY-TYPE 2 COLLISION INTEGRAL [Å ]

POTENTIAL ENERGY (eV)

O-O2

0

-0.005

-0.01 2.0

3.0

4.0

5.0

6.0

7.0

INTERNUCLEAR DISTANCE (Å)

8.0

10.0

1.0 2 10

10

3

10

4

TEMPERATURE [K]

Fig. 4. (a) Phenomenological potential energy curve for Oð3 PÞ–O2 ðX3 R g Þ interaction. (b) Corresponding viscosity-type collision integrals (full line) compared with results of Stallcop et al. [18] (dotted line), Capitelli et al. [1] (open diamond) and Aubreton et al. [21] (closed diamond).

M. Capitelli et al. / Chemical Physics 338 (2007) 62–68

region, are in satisfactory agreement with the phenomeno˚ , e 0/ logical Pirani’s potential, shown in Fig. 4a (r = 3.21 A k = 92.5 K) and also a reasonable agreement with experimental results [19,20] is found. This aspect is confirmed by the merging of viscosity-type collision integrals curves obtained from the phenomenological potential and from Ref. [18] for T < 10,000 K, shown in Fig. 4b. The high temperature region is dominated by the repulsive branch of the

Table 5 ˚ 2), as a function of temperature for Collision integrals, r2X(‘,‘)w (A Oð3 PÞ–O2 ðX3 R Þ interaction g T (K)

r2X(1,1)w

r2X(2,2)w

r2X(3,3)w

100 150 200 300 400 600 800 1000 1500 2000 3000 4000 5000 6000 8000 10,000 20,000

13.959 11.660 10.458 9.201 8.522 7.748 7.282 6.951 6.397 6.031 5.544 5.217 4.973 4.778 4.483 4.262 3.627

15.597 12.949 11.598 10.238 9.529 8.738 8.263 7.924 7.349 6.963 6.442 6.085 5.816 5.601 5.271 5.022 4.299

12.809 10.911 9.949 8.948 8.394 7.737 7.322 7.019 6.495 6.140 5.661 5.334 5.089 4.893 4.594 4.370 3.722

Table 6 ˚ 2), as a function of temperature for C(3P)– Collision integrals, r2X(‘,‘)w (A O(3P) interaction

67

interaction potential and as can be appreciated in Fig. 7 of Ref. [18] a great uncertainty exists on the behaviour of the potential for small interparticle distances. The potential proposed by Stallcop et al. [18] is, in this region, less repulsive than the Pirani’s potential, explaining the increase of discrepancy between corresponding calculated collision integrals with temperature. In the same figure results from Capitelli et al. [1], obtained for T < 1000 K using a Lennard–Jones interaction potential and for T > 2000 K an experimental exponential repulsive curve (two data sets have been smoothly joined in the interval of 1000– 2000 K), and from Aubreton [21] have been reported, finding in general a satisfactory agreement. A good agreement is found between the present results and the corresponding ones from Ref. [18], while the agreement becomes less satisfactory when compared with data of Ref. [1] for T > 2000 K. In Table 5 collision integrals of different orders, calculated with the present phenomenological potential are reported for Oð3 PÞ–O2 ðX3 R g Þ interaction. To conclude this section, results for two systems of interest for Martian atmosphere, C(3P)–O(3P) and C(3P)– N(4S), based on the phenomenological approach, are presented in Tables 6 and 7, respectively. Comparisons with available data in literature [22,23], obtained with the accurate potential averaging procedure, are displayed in Fig. 5a and b. A satisfactory agreement is found in both cases, specially in the range (2000–10,000 K) of simultaneous existence of carbon–oxygen or carbon–nitrogen atoms.

Table 7 ˚ 2), as a function of temperature for C(3P)– Collision integrals, r2X(‘,‘)w (A N(4S) interaction

T (K)

r2X(1,1)w

r2X(2,2)w

r2X(3,3)w

r2X(4,4)w

T (K)

r2X(1,1)w

r2X(2,2)w

r2X(3,3)w

r2X(4,4)w

100 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 12,000 14,000 15,000 16,000 18,000 20,000

12.504 9.643 8.592 8.001 7.599 7.298 7.057 6.858 6.688 6.540 5.638 5.156 4.832 4.591 4.400 4.242 4.109 3.994 3.894 3.724 3.585 3.524 3.467 3.366 3.278

13.928 10.741 9.636 9.029 8.619 8.310 8.064 7.858 7.682 7.527 6.569 6.044 5.685 5.415 5.200 5.022 4.871 4.740 4.625 4.430 4.270 4.199 4.134 4.017 3.914

11.601 9.305 8.454 7.956 7.605 7.335 7.115 6.930 6.770 6.630 5.752 5.273 4.947 4.703 4.509 4.349 4.214 4.097 3.995 3.821 3.679 3.617 3.559 3.456 3.365

12.330 9.972 9.118 8.616 8.260 7.984 7.757 7.566 7.400 7.253 6.328 5.815 5.465 5.201 4.991 4.817 4.670 4.542 4.430 4.240 4.085 4.016 3.953 3.840 3.740

100 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 12,000 14,000 15,000 16,000 18,000 20,000

13.632 10.332 9.138 8.478 8.036 7.708 7.448 7.233 7.051 6.892 5.934 5.425 5.083 4.827 4.625 4.459 4.319 4.197 4.091 3.911 3.764 3.700 3.640 3.533 3.440

15.227 11.505 10.235 9.552 9.100 8.765 8.498 8.277 8.089 7.924 6.911 6.357 5.980 5.695 5.468 5.281 5.121 4.983 4.862 4.656 4.487 4.412 4.343 4.220 4.111

12.562 9.902 8.944 8.396 8.017 7.727 7.492 7.295 7.125 6.976 6.050 5.545 5.201 4.944 4.739 4.571 4.428 4.305 4.196 4.013 3.863 3.797 3.736 3.627 3.531

13.359 10.599 9.635 9.084 8.700 8.404 8.163 7.960 7.785 7.630 6.656 6.116 5.746 5.468 5.247 5.064 4.908 4.774 4.655 4.455 4.291 4.219 4.152 4.032 3.927

68

M. Capitelli et al. / Chemical Physics 338 (2007) 62–68

CO

CN 10.0 2

2

COLLISION INTEGRAL [Å ]

COLLISION INTEGRAL [Å ]

10.0

1.0 2 10

10

3

10

4

TEMPERATURE [K]

1.0 2 10

10

3

10

4

TEMPERATURE [K]

Fig. 5. Comparison of diffusion (dotted line) and viscosity (full line) type collision integrals obtained with the phenomenological Pirani’s potential with data in literature. (a) C(3P)–O(3P) interaction (open and closed diamonds – r2X(1,1)w and r2X(2,2)w results from Ref. [22]). (b) C(3P)–N(4S) interaction (open and closed diamonds – r2X(1,1)w and r2X(2,2)w results from Ref. [23]).

4. Conclusions The present results emphasize the possibility of using the phenomenological Pirani’s potential for calculating the collision integrals of atom–atom and atom–ion open-shell systems. This potential, which presents a very smooth minimum and a repulsive wall, well reproduces the global behavior of multiple potential interactions due to the negative interference of the attractive potentials with strong chemical bonds with the corresponding repulsive (antibonding) potentials, being the uncertainties comparable with the corresponding ones by quantum mechanical approaches. The use of the proposed method can be recommended not only because of its simplicity and generality but also because it can produce internally complete sets of collision integrals for different high temperatures atmospheres containing atomic and molecular species difficult to be characterized experimentally and theoretically. Acknowledgements The present work has been partially supported by MIUR PRIN 2005 (Project No. 2005033911) and MIUR FIRB (Project No. RBAU01H8FW_003). References [1] M. Capitelli, C. Gorse, S. Longo, D. Giordano, J. Thermophys. Heat Transfer 14 (2000) 259.

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