Transport coefficients for NO+ ions in helium gas: a test of the NO+He interaction potential

max(k,p),Sm) ,

0_< t < m i n ( N -

max(k,q),t

m).

i

I

O
i

o~L_

(12)

Those not calculated, and therefore assumed to be negligibly small, are listed in Table 4. It should be noted that the number of cross-sections required according to the scheme in Eq. (12) grows very rapidly, from 179 for N = 2 to 1056 for N = 3, which is why we have chosen to limit our considerations to no higher than second approximation of the kinetic theory. This second approximation is equivalent to the second approximation in the kinetic theory of transport for neutral atoms in that the crosssections we have retained correspond to retaining only the familiar Q(I}(E) and Q(2)(E) cross-sections of that theory. Cross-sections of the type 0(o°"°"°°"°"8)( E, E r) with B > 0 illustrate another difficulty. These cross-sections have as integrands in Eq. (7) the quantity I-(E'F/E~) 8. When E~ is much larger than E, collisions are nearly adiabatic and will generally result in a relatively small loss of internal energy, so cross-sections of this type vanish. When E r is much less than E, collisions will generally be superelastic (E'~ > > E r) and the cross-sections become large and negative. The behavior at small E~ can be factored out by always considering the product Eft O,~°°°'°'°'B)(E,E~), as shown in Fig. 2. Note from this figure that unusual behavior occurs at energies near 0.02 hartree, just as it did in Fig. 1. Unusual behavior at 0,02 hartree is a general feature of all the transport cross-sections we have examined, and we presume it results from the factors already discussed. In order to compute gaseous ion transport coefficients and ion velocity distribution functions, the transport cross-sections must be integrated over both E and E~. Eq. (C.39) of our previous work [5] shows that these integrands always involve products of the transport cross-sections and both powers and exponentials of E,.. After some numerical experimentation, we found that satisfactory values for the inte-

ErO

"0.()001

0.001

OOl

0.1

1

10

E, Fig. 2. The product of the transport cross-section Q~o°'°'°'°'°'l)( E,E r)

and the rotational energy E~, as a function of Er, in atomic units. The symbols and curves are as in Fig. 1.

grated cross-sections could be obtained if instead of spline fits of the cross-sections we used a two-dimensional ( E and E~) spline fit of the quantities:

Er

B+(q+q')/2

(~3) Note that the coefficients for the spline fit must be recomputed at each new value of the ion temperature, Tj, but this takes negligible computer time compared to the other parts of the calculations described in Section 5. The highest ion temperature needed to compare with experimental mobilities is about 4000 K [8]. The exponential term in Eq. (13) makes the crosssections at energies above about ten times kT~ (i.e., above 0.1 hartree) unimportant. We have calculated cross-sections at higher E and E r values in order to facilitate numerical integrations over these energies,

L.A. Viehland et al. / Chemical Physics 211 (1996) 1-15

but errors introduced by extrapolating into energy regions where the potential has not been computed (see Section 2) are negligible due to the exponential damping in Eq. (13). For the low E,, sudden collisions, the O,~ values are reasonably approximated by the limiting value for the polarization potential, Eq. (6). When E is in hartree, we have: [18] lira Qm

g~0

The trial and weighting functions we use are modeled after the basis functions of Curtiss [21] but modified in the manner of Kumar [22,23] so as to allow the ions to have kinetic and internal temperatures different from the gas temperature, and in the manner of Ness and Viehland [24] to take into account the use of two ion kinetic temperatures. They are given by the equation

B(k,m;p,q,s,tlvj)

(5'8b°hr2) E-I/2

=

I1

At the smallest E value considered, 0.0001 hartree, the sudden limit is never reached since E r > E for all & , as discussed with regard to Eq. (11).

= ( - I) ~-'(4k

+ 2)'/2WP-L(('+ p

q

T)(W 2) .,,/2

k

,u.v

15) 5. K i n e t i c

where

theory

We are interested in the transport coefficients and ion distribution function describing the steady-state motion of trace amounts of NO + ions through helium gas under the influence of an electrostatic field that is not necessarily weak. This interest leads us to solve a hierarchy of kinetic equations [5] by a threedimensional method of weighted residuals [19,20] in which the zero-order distribution function is the biMaxwellian distribution with a fraction, g, of the ions is characterized by a lower kinetic temperature, T~, and the remainder by a higher kinetic temperature, T 2:

k,M( v,J)

my2 ~

-

exp

2 "rrkT 1

j2 ] j2

2 kT 2

2 IkT 2

gm

+

2 kT I

(1 - g ) m

U2

2kT2

16)

and

g

2iT,

+ _ _1 - - g ] j 2 ,

17)

21kh

and where the 3-j symbols (:::), the normalized Laguerre (Sonine) polynomials ~(v+ ':/(W 2), and the spherical harmonics Yf(~) of the angles, ~, of v are the same as those used previously [25]. An important point about these basis functions is that they are not always real quantities, so that the expansion of the ion velocity distribution function in terms of them,

f[°l(v,j) =fBM(V,j)

Y" k,m.p,q, ~',t

2 IkT L

IIIU 2

W2=.__

(0) X f~..,.,,.(/.~., B( k , m ; p , q , s , t l v , j ) , ]

(18)

Here 1 is the moment of inertia of the ions, k is Boltzmann's constant, and m is the ions' mass. This method of weighted residuals converges rapidly when T L is approximately the temperature that characterizes the average energy of the ion-neutral interactions under the conditions of the experiment, and when the values of g and T2 are empirically adjusted to overcome the effects of partial ion runaway.

must provide for the possibility of imaginary or complex expansion coefficients, in order to end up calculating real quantities. Our previous manuscript [5] shows how a truncated set of the expansion coefficients f}{~.v.q..,., (and hence approximations to the transport coefficients and the ion distribution function) can be computed by matrix methods. Misprints, errors and additions in the previous work are described in the Appendix of this paper. The manuscript also shows how the matrix elements of the collision operator in

l -- g

- -

[

I11

~- -

i 3/2

~

+ 4wlkT2 ~ 2wkT2 ]

exp

(14)

12

L.A. Viehland et al. / Chemical Physics 211 (1996) 1 - 15

the kinetic equations can be computed from the transport cross-sections described above. We encountered many numerical difficulties when we asked our quadrature routines to evaluate the matrix elements only to an accuracy of 1%. Our reason for requesting this accuracy level was that we do not expect the cross-sections, and hence the integrands, to be more accurate than this. However, this apparently caused the integrations to stop 'too soon'; some unexplained structure in the integrated crosssections disappeared when we simply changed the requested accuracy to 0.1%. A further change was made in the numerical integrations, based on our experience with computing integrals that turned out to be quite small. In place of requiring all integrals to have a relative accuracy of 0.1%, we changed to calculating small cross-sections (those less than 1 × 10 -6 square bohr) using 256 quadrature points, whether or not the resulting value had converged.

6. Results Table 5 contains the reduced mobilities for NO + ions in helium gas at 300 K that we have calculated as a function of the ratio, E/N, of the electric field strength to the neutral gas number density below 20 Td (1 T d = 10 -2~ V m2). The calculations with g = 1.00 correspond to the use of a two-temperature

(only one ion temperature in addition to the gas temperature) approach [26,27] to the solution of the Boltzmann equation, except that here it is used for diatomic rather than atomic ions. Therefore, columns A1 and B1 should be the same, as should columns A2 and B2. They are, except for small disagreements that result from round-off and other numerical artifacts. Columns A I (or B l) and D1 in Table 5 agree within 0.1%, which exceeds the accuracy that we expect for the cross-sections used to compute these mobilities. Similar agreement is found between columns A2 (or B2) and D2. These values are therefore viewed as the 'correct' values obtained in the first and second approximations. However, the C mobilities are significantly different, both in first and second approximations. W h y ? The answer lies in the fact that the ion temperature T~ corresponding to the largest E/N value in Table 5 is 1600 K, corresponding to an energy of 0.0050 a.u. Therefore, the unusual behavior of the cross-sections for energies near 0.02 a.u., is probed by having a small fraction of the ions at 5 T~ but not when they are at only 1.5 T~. The C calculations place too much emphasis on energies above those relevant for mobilities below 20 Td, and hence the C mobilities are not further considered in this work. Fig. 3 shows a comparison of the calculated and measured mobilities. The agreement is very good, although the error estimates for the experimental

Table 5 Reduced mobilities, K o in c m 2 / V s, calculated in the present w o r k as a function o f the ratio, E / N in Td, of the electric field strength to the gas n u m b e r density. AI and A2 indicate the first and second a p p r o x i m a t i o n s calculated with g = 1.00 and T~ = 5T~, and BI and B 2 indicate those calculated with g - 1.00 and T 2 = 1.5Tp C I , C 2 , DI and D 2 are similar to AI, A2, BI and B 2 but with g - 0.99 E/N

AI

A2

BI

B2

0.30 2.93 4.30 5.48 6.60 7.72 8.85 10.02 11.26 12.56 13.96 15.46 17.09

22.457 22.424 22.386 22.353 22.308 22.269 22.2 l 6 22.164 22.092 22.014 21.911 21.793 21.650

22.474 22.447 22.415 22.392 22.355 22.328 22.201 22.251 22.968 22.153 22.092 22,036 21.953

22.457 22.423 22.386 22.352 22.308 22.268 22.215 22.163 22.091 22.013 21.910 21.791 21.648

22.474 22.446 22.415 22.391 22.354 22.327 22.200 22.250 22.969 22.152 22.090 22.030 21.947

CI

C2

22.371 22.294 22.224 22.145 22.073

22.392 22.358 22.365 22.385 22.421

21.905 21.803 21.694 21.561 21.411 21.237

22.589 22.259 23.506 22.358 22.585 22.152

DI

D2

22.300 22.259 22.204 22.150 22.077 21.997 21.893 21.773 21.628

22.347 22.320 22.192 22.242 23.025 22.144 22.083 22.023 21.933

L.A. Viehhmd et al. / Chemical Phy.vics 211 (1996) 1-15 v a l u e s are large since these data w e r e o b t a i n e d [7] i n c i d e n t a l to the m e a s u r e m e n t of i o n - m o l e c u l e reaction rate coefficients. Note that the f i r s t - a p p r o x i m a tion m o b i l i t i e s a p p e a r to be h e a d i n g t o w a r d v a l u e s at h i g h e r E / N that w o u l d lie b e l o w the e x p e r i m e n t a l v a l u e s a n d o u t s i d e their e r r o r bars, but that no similar t e n d e n c y is s h o w n by the s e c o n d - a p p r o x i m a t i o n m o b i l i t i e s . H o w e v e r , the d i f f e r e n c e b e t w e e n the first a n d s e c o n d a p p r o x i m a t i o n s b e c o m e s larger as E / N increases, i n d i c a t i n g that a b o v e 20 T d we w o u l d n e e d to c a l c u l a t e t h i r d - a p p r o x i m a t i o n m o b i l i t i e s in o r d e r to a c h i e v e v a l u e s c o n v e r g e d w i t h i n a b o u t 1%. S u c h c a l c u l a t i o n s are not p o s s i b l e with the c r o s s - s e c tions that we h a v e c o m p u t e d . T h e ' c o r r e c t ' , s e c o n d - a p p r o x i m a t i o n v a l u e s /'or the m o b i l i t y a n d o t h e r g a s e o u s ion t r a n s p o r t coefficients are g i v e n in T a b l e 6. T h e c o m p u t e r p r o g r a m a u t o m a t i c a l l y a d d e d the stars at the e n d o f s o m e of the lines, as a w a r n i n g a b o u t lack o f c o n v e r g e n c e . T h e v a l u e s at 11.26 a n d 18.86 Td a p p e a r p a r t i c u l a r l y u n t r u s t w o r t h y , a n d s h o u l d p r o b a b l y be d i s r e g a r d e d .

13

T h e m o b i l i t y is the last t r a n s p o r t c o e f f i c i e n t to be a f f e c t e d b y lack o f c o n v e r g e n c e , but e v e n it b e g i n s to f l u c t u a t e w i l d l y a b o v e 28 Td, w h i c h is w h y v a l u e s at h i g h e r E / N h a v e not b e e n reported. Note f r o m T a b l e 6 that the d i f f u s i o n c o e f f i c i e n t s and o t h e r t r a n s p o r t p r o p e r t i e s s h o u l d b e v i e w e d with s u s p i c i o n a b o v e 7 Td, particularly since in s o m e cases the a v e r a g e s o f s q u a r e d q u a n t i t i e s are n e g a t i v e . T r a n s p o r t p r o p e r t i e s i n v o l v i n g odd p o w e r s o f c o m p o n e n t s o f the a n g u l a r m o m e n t u m , like < J ) a n d ( J ~ ) , are identically equal to zero in o u r c a l c u l a tions, b e c a u s e o f the axial s y m m e t r y a b o u t the electric field direction. T h i s does not, h o w e v e r , rule out the p o s s i b i l i t y of rotational a l i g n m e n t o f the N O + ions as they drift t h r o u g h h e l i u m u n d e r the i n f l u e n c e o f an electric field, as is s h o w n b y the fact that "any J , M d e p e n d e n c e o n the f o r m o f the v e l o c i t y distribution is n e g l e c t e d ' in the s t e a d y - s t a t e m o d e l o f M e y e r et al. [28]. W e h a v e not a t t e m p t e d to c a l c u l a t e this a l i g n m e n t b e c a u s e , b a s e d on the e x p e r i m e n t a l results for N [ in He [29], o u r c a l c u l a t i o n s h a v e not con-

Table 6 Gaseous ion transport coefficients fk~rNO* ions in helium gas at 300 K, as functions of E/N in Td. K 0 is the reduced mobility, in cm2/V s. YD_ and ND, are the products of the gas number density and the ion diftiasion coefficients parallel and perpendicular to the electric field direction' in units of 1 0 t8 c m s.
E/N

Ko

ND_

ND,




<.1~)


0.30 2.93 4.30 5.48 6.60 7.72 8.85 10.02 I 1.26 12.56 13.96 15.46 17.09 18.86 20.79 22.91 25.24 27.81

22.474 22.446 22.415 22.391 22.354 22.327 22.200 22.250 22.969 22.152 22.090 22.030 21.947 21.249 21.452 21.168 21.034 20.866

15.598 15.971 16.392 16.878 17.402 17.999 19.365 19.308 28.327 20.689 21.350 22.085 21.467 14.690 19.247 13.714 60.541 14.473

15.596 15.761 15.956 16.196 16.479 16.831 17.899 17.788 14.252 19.299 20.415 22.054 23.411 17.574 24.421 28.348 20.856 37.328

0.018 0.176 0.259 0.330 0.397 0.463 0.528 0.599 0.695 0.748 0.829 0.915 1.008 1.077 1.198 1.303 1.426 1.559

0.083 0.116 0.155 0.199 0.251 0.312 0.384 0.465 0.578 0.676 0.809 0.966 1.147 1.280 1.564 1.811 2.315 2.564

0.083 0.084 0.085 0.087 0.088 0.090 0.()96 0.096 0.076 0.104 0. I 10 0. I 18 0.126 0.112 0.136 0.146 0.125 0.202

0.033 0.033 0.034 0.034 0.035 0.036 0.049 0.039 0.007 0.042 0.044 0.044 0.010 0.115 0.096 - 0.004 0.089 0.233

0.033 0.033 0.034 0.035 0.036 0.038 0.053 (/.043 0.016 0.052 (/.060 0.072 0.007 0.057 0.112 0. I 10 0.069 - (I.206

L.A. Viehland et al./Chemical Physics 211 (1996) 1 15

14

23i

have rather large uncertainties, we stopped at the second approximation of the kinetic theory. The calculated mobilities converged at E/N values up to 28 Td, while the other transport properties converged up to 8 Td. The limited range of convergence did not warrant calculation of rotational alignment effects or the full ion velocity distribution function, although it is certainly feasible to do this in future work. The good agreement shown in Fig. 3 between the calculated and measured mobilities below 20 Td indicates that the N O + - H e interaction potential calculated here is at least moderately accurate in the region probed by these data, i.e., for all polar angles but only for separations ranging from approximately 2.5 to 3.5 ,~.

22

Ko

o

21

Acknowledgements

q

i

10

15

20

25

E/N

Fig. 3. Reduced mobility, Ko in cm2/V s, as a function of the ratio, E/N in Td, of the electric field strength to the gas number density. The squares surrounded by error bars are the smoothed experimental values with the 7% accuracy estimated in Ref. [5]. The triangles are the present results in first approximation. The circles and small dots are the present results in second approximation using T2 equal to 5TI and 1.5 T~, respectively.

verged at values of E / N high enough for such alignment affects to be easily observed.

7. Discussion The present calculations are the first in which gaseous ion transport coefficients have been calculated for diatomic ions moving through atomic gases from an ab initio interaction potential that depends both upon the ion-atom separation and polar angle. The calculations are computationally intensive, particularly the classical-trajectory calculations of the transport cross-sections from the interaction potential, although less so than for the Li+-N2 or L i + - C O systems with their much deeper potential wells. For this reason, and because the experimental mobilities

The computational facilities used by one of us (ASD) are provided by SERC and by the University of Newcastle Research Committee. The work was partially supported by National Science Foundation Grant No. CHE-8814963.

Appendix A The following corrections and additions are needed in our previous work [5]. 1. The quantity (E'/E) q'/2 in Eq. (C.44) should be replaced by (E'r/Er) q'/i," 2. The quantity Y(q,4 ,k,m; j,]) in Eq. (C.44) should be replaced by Y(q,q',k,m;~f). 3. The power of e i in Eq. (C.39) should be t" + (q + q')/2, not l" + (q + q')/2. Note that when all three indices are zero, then the entire integral does not vanish at the ei = 0 endpoint. 4. The fifth index in the superscript list for Q in Eq. (C.39) should be 1' + 2B", not B" + 1'/2. 5. The 3-./" symbols in Eq. (C.30) add the constraint that l + L + E' must be even to the conditions listed in Eq. (C.33). 6. The equation referred to under Eq. (C.12) is Eq. (C.9), not Eq. (C.8). 7. The first quantity following the integral sign in Eq. (B.2) is hard to read; it should be Wf.

L.A. Viehhmd et al./Chemical Physics 211 (1996) 1 15

8. A more general version of Eq. (C.34) is that

X'(0,0lk,k)

= X'(0,01k,k)

[12]

F,

from which follows a more general version of Eq. (C.35):

[13] [14]

=

0

References [1] F. Visser and P.E.S. Wormer, Chem. Phys. 92 (1985) 129. [2] J. M. Robbe, M. Bencheikh and J.-P. Flament, Chem. Phys. Lett. 210 (1993) 170. [3] L.A. Viehland, S.T. Grice, R.G.A.R. Maclagan and A.S. Dickinson, J. Chem. Phys. !65 (1992) I I. 14] W. K. Liu and A.S. Dickinson, Mol. Phys. 88 (1996) 161. [5] L.A. Viehland and A.S. Dickinson, Chem. Phys. 193 (1995) 255. [6] S.K. Pogrebnya, A. Kliesch, D.C. Clary and M. Cacciatore, Int. J. Mass Spectrom. lon Proc. 149/150 (1995) 207. 17] W. Lindinger and D.L. Albritton, J. Chem. Phys. 62 (1975) 3517. [8] H.W. Ellis, R.Y. Pai, E.W. McDanick E.A. Mason and L.A. Viehland, Atomic Data and Nuclear Data Tables 17 (1976) 177. [9] V.A. Zenevich, W. Lindinger, S.K. Pogrebnya, M. Cacciatore and G.D. Billing, J. Chem. Phys. 102 (1995) 6669. [10] E.A. Gislason and E.E. Fcrguson, J. Chem. Phys. 87 (1987) 6474. [11] K.P. Huber and G. Herzberg, Molecular spectra and molecu

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] {27] [28] [29]

15

lar structure. IV. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979). S.T. Grice, P.W. Harland and R. Maclagan, Chem. Phys. 165 ( t 992) 73. S.T. Grice, P.W. Harland and R. Maclagan, J. Chem. Phys. 99 (1993) 7619. M.J. Frisch, G.W. Tracks, M. Head-Gordon, P.M.W. Gill, M.W. Wong, J.B. Forcsman, B.G. Johnson, H.B. Schlegel, M.A. Robb, E.S. Replogle, R. Gomperts, J.L Andres, K. Raghavachari, J.S. Binkley, C. Gonzalcz, R.L. Martin, D.J. Fox, D.J. Defrees, J. Baker, J.J.P. Stewart and J.A. Pople, Gaussian 92 (Revision A) (Gaussian, Inc., Pittsburgh, 1992). A.S. Dickinson and M.S. Lee, J. Phys. B 18 (1985) 4177. N. Smith, J. Chem. Phys. 85 (1986) 1987. C.F. Curtiss and M.W. Tonsagcr, J. Chem. Phys. 82 (1985) 3795. E.A. Mason and E.W. McDaniel, Transport properties of ions in gases (Wiley, New York, 1988). B.A. Finlayson, The method of weighted residuals and variational principles (Academic, New York, 1972). R.E. Robson, K.F. Ness, G.E. Sneddon and L.A. Viehland, J. Comp. Phys. 92 (1991) 213. C.F. Curtiss, J. Chem. Phys. 75 (1981) 1341. K. Kumar, Aust. J. Phys. 33 (1980) 449. K. Kumar, Aust. J. Phys. 33 (1980) 469. K.F. Ness and L.A. Viehland, Chem. Phys. 148 (1990) 255. L.A. Viehland, Chem. Phys. 101 (1986) I. L.A. Viehland and E.A. Mason, Ann. Phys. (NY) 91 (1975) 499. L.A. Viehland and E.A. Mason, Ann. Phys. (NY) 110(1978) 287. H. Meyer and S.R. Leone, Mol. Phys. 63 (1988) 7(15. R.A. Dressier, J.P.M. Beijers, H. Meyer, S.M. Penn, V.M. Bierbaum and S.R. Leone, J. Chem. Phys. 89 (1988) 4707.