Ion-Polar Molecule Encounters

Ion-Polar Molecule EncountersAuthor(s): David BatesSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 384, No. 1787 (Dec. 8, 1982), pp. 289-300Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2397224 .

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Proc. R. Soc. Lond. A 384, 289-300 (1982)

Printed in Great Britain

Ion-polar molecule encounters

By SIR DAVID BATES, F.R.S. Department of Applied Mathematics and Theoretical Phystcs,

Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland

(Received 2 June 1982)

It is permissible to assume that the rate coefficient for collisions between ions and polar molecules does not depend on the moment of inertia of the latter because the rotation time is brief compared with the collision time. On taking the moment of inertia to be vanishingly small the classical collision problem can be solved exactly when the angular momentum vector is normal to the orbital plane. Use is made of the adiabatic invariance of f p dq/2r in which p is an appropriate momentum and q is the conjugate coordinate. This adiabatic invariant fixes the change in the rotational energy in moving from an infinite separation to any chosen position. The average dipole orientation is thereby determined, which fixes the force acting. The potential energy function (including due allowance for the rotational energy stored) is now written down and an integral expression for the primitive rate coefficient is thence obtained. The ratio of the primitive rate coefficient to the Langevin rate coefficient depends only on the initial rotational energy and on the dimensionless parameter fli= 2axkT/D2, where a is the polarizability, D is the dipole moment and T is the temperature. Extensive computations have been performed. Tables are presented giving the primitive rate coefficient and also approximations to the thermally averaged rate coefficients for linear and for spherical top molecules.

INTRODUCTION

The theory of ion-polar molecule encounters is of practical interest in connection with momentum transfer cross sections and with the thermal rate coefficients for charge transfer, proton transfer and other reactive collisions as may be seen from the compilations Interactions between ions and molecules (Ausloos 1975), Kinetics of ion-molecule reactions (Ausloos I979) and Gas phase ion chemistry (Bowers 1979). It also forms an integral part of the proper treatment of radiative association of ions and polar molecules (Bates i982) and in order to cover conditions in interstellar clouds it must allow the rotational level of the neutral reactant to be specified.

Moreover the dynamical problem presented is sufficiently fundamental to be of interest in itself because the long-range interaction between an ion of charge e and a molecule of polarizability a and dipole moment D

V(r) -ae2/r4-eD * r/r3 (1)

[ 289 ]



290 Sir David Bates

has the simplest orientation-dependent form occurring naturally, and the internal structure (the changing orientation and rotational energy of the neutral molecule) is also simple. Exact numerical solution of the classical Lagrangian or Hamiltonian equations of motion is possible and several sets of trajectory calculations have been performed (Dugan & Magee I967; Dugan et al. I970; Dugan I97I; Dugan & Can- wright 1972; Dugan & Magee '973; Chesnavich et al. I980). A quite demanding computational effort is required to ensure that each trajectory is accurate and also that the statistical errors in the overall results are acceptably low. Although trajectory analysis has yielded important information it is cumbersome and ill suited to the task of determining the classical coefficient y for hard (orbiting) ion-polar molecule collisions. Consequently much of the theoretical thrust has been towards the development of approximate methods. The best known of these is the average dipole orientation (ADO) method of Su & Bowers (I973) as modified by Bass et al. (I975). With (1) written as

V(r) -ae2/r4 - eD cos O/r2, (2) the ADO method consists of replacing cos 0 by an average <cos 0> obtained from the relations

(cos 0> fP(6) cos 0 dO/ P(l) dO, (3)

P(O) cc (sin 0)/6 (4)

cc sin 0/[E(r, 0)]I, (5)

where E(r, 0) is the instantaneous rotational energy corresponding to the component of the angular momentum normal to the orbital plane calculated on the implicit assumption that E(r, sir) remains invariant along a trajectory. This assumption is incorrect, which leads to the derived rate coefficient y being considerably in error (Bates I98I). Two other approximate methods must be- mentioned. On the supposition that there is complete energy randomization between radial, orbital and rotational motion during the encounter Chesnavich et al. (I980) obtained an expression for y from variational rate theory. At almost the opposite extreme Celli et al. (I980) took the orbital and rotational angular momenta to be conserved separately. Using activated complex theory and averaging over the ion-dipole potential energy in the Boltzmann factor, they introduced a temperature-dependent central potential from which an approximation to y could easily be derived. The results obtained from the two methods are in satisfactory accord with each other, with those obtained from trajectory analysis and with laboratory data. However, neither assumption regarding the rotational energy is valid. It will be shown that this energy is a function of r parametrically dependent on its value at infinite separation.

The approach adopted here is to confine attention to the limit where the moment of inertia I of the polar molecule is vanishingly small. This limit is of interest because the value of the rate coefficient y when I is zero can be taken to apply to the range of I relevant to polar molecules. The reasoning is that as I is increased



Ion-polar molecule encounters 291

from zero, y must remain unaltered at least until the rotational time period ceases to be brief compared with the orbital time period; the trajectory studies (Dugan & Magee I967; Chesnavich et al. I980) show that y is indeed remarkably insensitive to I: for example in the H+ + NH3 system the statistical error of 10 % makes it uncertain if y changes even when I is raised to 100 times the true value; and the insensitivity has been confirmed by Sakimoto (I98I a) using the perturbed rotational state approach of Takayanagi (I978).

Progress may be made in the limit mentioned because recognition that there is an adiabatic invariant enables the exact solution to the classical dynamical problem to be found easily wh1en the rotational angular momentum vector is normal to the orbital plane. Moreover a good approximation to the rate coefficient y may be obtained from them. Quantal transitions are weak (Sakimoto I98I a, b) and will be ignored. They are most readily treated by the perturbed rotational state approach (Takayanagi I978, i982).

2. THE ORY

2.1. Adiabatic invariance As we let the moment of inertia I - 0 so also T -- where T is the time period of

the rotation or libration of the polar molecule. The orbital motion in interval T

therefore causes only an infinitesimally small change in the field on the dipole. Consequently the rotation and libration about an axis perpendicular to the orbital plane are characterized (cf. Landau & Lifshitz I960) by having as an adiabatic invariant

J p dq/2n (6)

= (21)1 6 -J a (1- cos0) dO/2ir, (7)

where e is the instantaneous rotational energy at the orientation at which the potential energy of the ion-dipole interaction is a minimum.

Denote the temperature by T and introduce the dimensionless quantities

x _ r(kT/eD)i = r/d (say), (8)

g = e/kT, (9) and suppose that

A as x - oo. (10)

If x > (2/C)I, (1 1)

rotation occurs in which event (7) becomes

J = (2I1kT)!j {g-x-2 (1 , cos 0)}i dO/it (13)

= (2/it) (2IkT)l Cl E(l /m), (13)

where m = 1cx2, (14)



292 Sir David Bates

and E is the complete elliptic integral of the second kind. Since

E(?) = 1it E(1) = 1, (15) using (10) and (11) and supposing this last is an equality at (CO, x0) we see that

CIE(1/m) = liAl, x > xo, (16)

with x0= 8ji/tAi, (17)

and = Ii2A. (18)

If x < x0 (19)

condition (11) is of course not satisfied and instead of rotating the polar molecule librates with

- OL < ? < OL' (20) where

COS OL= -CX2. (21) It follows that

J = (2kT)i {- X-2 (1- cos 0)}i dO/it (22)

= (2/it) (2IkT)i ,irnm- {E(m) - (1-rm) K(m)} (23)

where K is the complete elliptic integral of the first kind. On using (10) again we obtain

gim-r {E(rn) - (1-rm) K(m)} = iAi. (24)

With the aid of the polynomial expansions for the elliptic integrals given by Milne-Thomson (I964) we can now easily compute C for any desired A as a function of m from (16) and (24) and thence from (14) as a function of x: that is we can find how the rotational energy of the polar molecule varies with the distance from the ion.

If we know C the calculation of <cos 0> from (3)-(5) is straightforward. We obtain

(cos 0> = 1-2x2 for 0 < x < x0, (25)

and <cos0> = I{1+X(X2C2-2C)i-X2C} for x0 < x. (26)

2.2. Potential energy and rate coefficient It is evident that in the limit r -?0 the effective force between the ion and the

rotating or librating polar molecule is given by the standard expression for the force with cos 0 replaced by <cos 0> and it is evident also that this effective force is conservative. The work done by an external force in bringing the ion and molecule from an infinite separation to a separation x is the sum of the potential energy kT V(x) and the gain kTp(x) in the rotational kinetic energy at angle

0 = arecos <cos 0>, (27) which gain is determined by

p(x) = {-A-( 1-<cos 0>)/X2}. (28)




We hence have that

V(X)= =- x-4-2 <C > dx -p(x) (29)

with ,i 2akT/D2. (30)

The ADO procedure of replacing cos 0 in expression (2) for V(r) by <cos 0> provides in contrast to (29), which is exact, only an approximation.

Because we are directing our attention to the limit I 0 the rotational angular momentum is vanishingly small and therefore the orbital angular moment is conserved. In this circumstance the radial motion is controlled in the usual way by the sum of the centrifugal and potential energies, that is by kTU(x), where

U(x) = (yb2/d2) X-2-.1l-4- 23' (Cos 0>dx-P(x), (31)

in which y is the incident kinetic energy of relative motion in units of kT and b is the impact parameter.

At the separation x corresponding to the top of the centrifugal barrier two equations are satisfied:

dU(x)/dx = 0 (32)

and = U(x) (33)

- U(x) + ix dU(x)/dx. (34)

The value of y for any chosen x is given by (34) in which the terms involving b cancel; and with y thus determined (32) then gives b and hence rb2, the cross section for hard (or orbiting) collisions.

On substituting from (31) into (34) and (32) we obtain

=4-4 + <COS 6>x-2 <COS 0> X-3dx - p(x) - lx dp(x) +<COSO>X '7 dx (35) and

ird2 ( dp(x) (6 irb2 = (2,X-2+2<coso>>-X3 dx (36)

Multiplication by the velocity of relative motion and integration over a Maxwellian distribution gives y(A), the primitive (that is rotational energy dependent) rate coefficient for hard collisions, the energy associated with the rotation about an axis perpendicular to the orbital plane at infinite separation being as indicated. It is convenient to express y(A) in units of the Langevin rate coefficient

=- 2r(ce2/4et)j, (37)

It being the reduced mass of the colliding pair. We find

Y(A)/YI = (f)je-fe (fX-2 + 2<coso> 6- 3dp(x) dy. (38)



294 Sir David Bates

The quadrature entailed must be done numerically after computing x as a function of y from (35) by inverse interpolation. Where x(g) is multivalued the largest of the values is the one relevant.

If the angular momentum vector is not normal to the orbital plane the angle between it and the line of centres changes during the encounter, which complicates the problem considerably. A simple approximate procedure for averaging y(A)/yL over all orientations of the vector is to take the distribution in A to be that associated with the component of the angular momentum vector in the plane normal to the line of centres at infinite separation.

If the rotational angular momentum vector has magnitude Jh the probability of the square of its projection on the plane perpendicular to the line of centres being in the interval (dK2) h2 around the value K2hi2 is

f(K) dK2 = d(K2)/J2, K < J? (39) =0, K > J

Hence on the approximation indicated the mean value of y(A)/yL for a molecule of rotational energy Ao is

1 AO Y(A)/YL = Ao y(A)/yLdA. (40)

Suppose now that the rotational levels are thermally populated. For a linear molecule of rotational constant B the probability of the rotational angular momentum being within interval dJh around Jh is

g(J) dJ = (2BJ/kT) exp (-BJ2/kT) dJ. (41)

Combined with (39) this leads to the mean

Y(A)/7L =7 (y(A)/yL) E1(A) dA, (42)

where E1(A) is the exponential integral. For a spherical top molecule (41) is replaced by

g(J) dJ = (4/7i) (B/lkT)i J2 exp (- BJ2/kT) dJ (43)

and (42) is therefore replaced by 00

Y(A)/yL = 2 (y(A)/yL) erfc Al dA. (44)

3. CALCULATIONS AND RESULTS

The dimensionless parameter ,8 defined by (30) was chosen to make

f-i = 2, 4,6,8,12, 16,24or 32. (45)

Interpolation of the results to be presented is feasible. The fld- range covered was extended to higher values than are required for laboratory chemistry in order to




meet needs that arise in connection with interstellar clouds, which may be at a temperature as low as 1OK: the needs are concerned with reactive ion-molecule collisions (Sakimoto Ig8I a) and with ion-molecule radiative association (Bates I982).

2

32 8 2

II I I I_ 0 0.2 0.4 0.6 0.8 1.0

x X0

FIGURE 1. Curves of 7q(x) at A = 1.020. The value of ,8-I is given on each curve.

Figure 1 shows representative y(x) curves. It is seen that (x) falls abruptly at the point xo that is at the boundary between the librating and the rotating regions. This is due to the derivative dp/dx being discontinuous at the boundary. The term

q(x)= x-2 <cOs 0> (46)

which appears in expression (28) for p(x), is responsible for the discontinuity. Thus from (25) and (26) it may be seen that

dq/dx =-2x-3-3', 0 <X ( x (47) and

dq I[2g+XZ'(X2g- 1) _2-3 -_ X ' (48) dx- ~ [3 x2F2x j x0 ,

where F = (x2C2-2C)i, (49)

which from (17) and (18) vanishes on the boundary. The abrupt change in y at xo does not of course imply an abrupt change in the kinetic energy there: thus y is the incident kinetic energy of relative motion that just enables the centrifugal barrier having its maximum at separation x to be traversed.

It is evident from figure 1 that only the librating region contributes for the PJs treated. This may be understood by comparing the absolute separation ro



296 Sir David Bates

3-

32

2- p

I

0 2 4

FIGURE 2. Curves of p(q) at A = 1.020. The value of f-t is given on each curve.

0.8 1 1

0.4

2

0 2 4 n

FIGURE 3. Curves of f(yq) at A = 1.020. The value of ,8- is given on each curve.

corresponding to x0 with the separation rL at the top of the centrifugal barrier in the Langevin case of a pure polarization interaction. On substituting from (17) into (8) we obtain

rO = (18/1t) (eD/AkT)I, (50)

while (cf. McDaniel et al. 1970) we have

rL = (0te2/2yIkT)I, (51)




TABLE 1. RATIO Y(A)/YL

(y(A) is the rate coefficient for ion-polar molecule collisions when AkT is the rotational energy at infinite separation associated with the component of the angular momentum vector in the plane normal to the line of centres. YL iS the Langevin rate coefficient. The parameter ,/ is defined in (30).)

g8-i 2 4 6 8 12 16 24 32 A

1.305-2 2.800 4.795 6.821 8.861 12.96 17.08 25.35 33.64 6.747-2 2.312 3.953 5.645 7.359 10.82 14.30 21.29 28.31 1.603-1 1.948 3.297 4.706 6.139 9.040 11.96 17.84 23.73 2.833-1 1.691 3.813 4.002 5.218 7.684 10.17 15.17 20.19 4.256-1 1.515 2.467 3.493 4.548 6.691 8.856 13.21 17.58 5.744-1 1.401 2.226 3.134 4.072 5.984 7.917 11.81 15.71 7.167-1 1.331 2.063 2.888 3.744 5.493 7.264 10.83 14.40 8.397-1 1.291 1.958 2.725 3.525 5.164 6.825 10.17 13.53 9.325-1 1.271 1.894 2.625 3.390 4.960 6.552 9.761 12.98 9.870-1 1.262 1.862 2.573 3.320 4.854 6.410 9.549 12.70 1.020 1.258 1.844 2.544 3.280 4.794 6.331 9.429 12.54 1.102 1.250 1.804 2.479 3.192 4.659 6.149 9.156 12.18 1.237 1.246 1.751 2.389 3.067 4.468 5.894 8.771 11.66 1.408 1.252 1.702 2.301 2.943 4.276 5.635 8.379 11.14 1.592 1.270 1.667 2.230 2.841 4.115 5.417 8.049 10.70 1.763 1.296 1.646 2.181 2.768 3.997 5.256 7.804 10.37 1.898 1.320 1.635 2.151 2.721 3.921 5.151 7.643 10.15 1.980 1.337 1.632 2.136 2.698 3.881 5.095 7.557 10.04 2.034 1.348 1.630 2.128 2.683 3.856 5.062 7.506 9.967 2.169 1.378 1.629 2.110 2.652 3.802 4.986 7.389 9.810 2.381 1.425 1.633 2.091 2.615 3.735 4.891 7.240 9.608 2.619 1.476 1.646 2.080 2.587 3.678 4.808 7.109 9.430 2.831 1.518 1.663 2.077 2.571 3.640 4.752 7.018 9.306 2.966 1.542 1.675 2.078 2.564 3.621 4.723 6.970 9.241 3.020 1.551 1.681 2.079 2.562 3.615 4.713 6.953 9.217 3.102 1.563 1.689 2.081 2.559 3.606 4.698 6.929 9.183 3.237 1.582 1.705 2.085 2.557 3.593 4.677 6.893 9.134 3.408 1.601 1.726 2.093 2.556 3.581 4.656 6.855 9.080 3.592 1.618 1.749 2.103 2.558 3.571 4.637 6.822 9.033 3.763 1.627 1.773 2.114 2.562 3.565 4.624 6.796 8.996 3.898 1.631 1.792 2.124 2.566 3.562 4.615 6.779 8.971 3.980 1.635 1.804 2.130 2.569 3.561 4.611 6.770 8.957 4.078 1.633 1.818 2.138 2.573 3.560 4.607 6.760 8.943 4.405 1.622 1.868 2.165 2.588 3.561 4.597 6.734 8.903 4.962 1.568 1.957 2.219 2.622 3.572 4.594 6.710 8.862 5.700 1.446 2.080 2.296 2.675 3.599 4.606 6.701 8.838 6.553 1.284 2.226 2.390 2.743 3.639 4.631 6.708 8.833 7.447 1.135 2.380 2.491 2.818 3.687 4.663 6.725 8.840 8.300 1.019 2.524 2.589 2.890 3.734 4.698 6.746 8.854 9.038 0.960 2.628 2.674 2.954 3.776 4.728 6.765 8.867 9.595 0.935 2.686 2.738 3.002 3.807 4.752 6.780 8.878 9.922 0.925 2.709 2.776 3.030 3.826 4.766 6.789 8.884

TABLE 2. LOW ROTATIONAL ENERGY LIMIT TO Y(A)/YL

,4 2 4 6 8 12 16 24 32

Y(O)/YL 3.257 5.514 7.770 10.03 14.54 19.05 28.08 37.11

I I VOl. 384 A



298 Sir David Bates

so that rL/rO = -;n (flA2/4y)1. (52)

The greatest value of /8 we have treated being 0.25, this ratio must be less than unity within our range except when A2/4y is quite large in which circumstance the rotation region contributes.

8

6-

4 -

p

2-

\ X \ 2.034 1.020 4.078

8.300 -2 l l l l l | I I

0 0.2 0.4 0.6 0.8 1.0 x

FIGURE 4. Curves of p(x) in libration region for /34 = 8. The value of A is given on each curve.

The p(y) curves associated with the (x) curves of figure 1 are displayed in figure 2. When f-I is 32 or 8 there are obvious discontinuities and when f-i is 2 there is also a discontinuity although it is imperceptible on the scale used. Sudden changes in the rotational kinetic energy at angle arccos (cos O> are not implied by these discontinuities: thus none occur in the p(x) curves. The discontinuities follow from the form of the I(x) curves (figure 1) coupled with the greatest x consistent with a particular y being always chosen. Figure 2 shows that p becomes a quite considerable energy. The derivative dp/dx is negative (in the libration region). This has the effect of increasing the rate coefficient.

Figure 3 gives the integrand in formula (38) for y(A)/yL expressed as a function of y:

e-8 {fX-2 + 2<coso> -x3 dp(x)/dx} f f(y). (53) Until y attains the value at xo (see figure 1) f(y) decreases as exp (- y). After a slight discontinuity, owing to the jump to a smaller x (figure 1), it rises to a maximum and then decreases more slowly than initially.




Table 1 contains our main results. To facilitate the use of the y(A)/YL values in integrations the As are the Gauss pivots specified as follows:

,>1 X~ ~ ~ ~~2 X3 f1dA, 10 pivots; fdA, 8 pivots; f dA, 6 pivots;

r4 10 f:dA, 8 pivots; f dA, lOpivots. (54)

TABLE 3. THERMALLY AVERAGED RATIOS Y(A)/YL

linear molecule I > spherical top

variational activated adiabatic molecule adiabatic D/(2cakT)i methodt complex theory: invariance? invariance?

2 1.739 1.85 1.827 1.663 4 2.819 3.10 2.981 2.624 6 4.006 4.40 4.213 3.679 8 5.230 5.70 5.474 4.768

12 8.031 6.988 16 10.61 9.231 24 15.80 13.74 32 21.00 18.27

t Chesnavich et al. (I980). I Celli et al. (I980): read from graph. ? Present calculations.

As A is increased from zero y(A)/yL initially falls off steeply from the value

Y(O)/YL = 1 + 2/(it/)l (55)

(table 2) given by the locked dipole approximation of Moran & Hamill (I963). Instead of falling monotonically towards unity the ratio exhibits a complicated pattern because of its being controlled by several competing factors. The derivative dp/dx is important. It becomes more negative in the libration region as A is increased (cf. figure 4). Reference to (38) shows that y(A) YL tends to be enhanced by this. The effect is largely responsible for y(A)/yL rising to a maximum after its initial fall. As mentioned immediately after (52) the rotation region contributes in certain circumstances. For i-4 = 2 and large A the rotation region indeed gives the main contribution. In this region the mean ion-dipole interaction is repulsive, more time being spent in repulsive orientations than in attractive orientations; moreover while p is negative it is less in magnitude than the x dp/dx term of (35) and since the derivative dp/dx is positive the overall effect of the rotation is equivalent to a repulsive potential. In consequence Y(A)/YL appears to be anomalously low (table 1, bottom left).

Finally we carried out integrations (42) and (44) to get y(A)/yL for linear molecules and for spherical top molecules respectively. The results, which should straddle those for other molecular structures, are presented in table 3.

II-2



300 Sir David Bates

For linear molecules comparison may be made with the y(A)/yL values obtained by Chesnavich et al. (I980) using the variational method and by Celli et al. (1980) using activated complex theory (? 1). As may be seen from table 3, these results lie on opposite sides of, and are gratifyingly close to, the results of the present work. Sakimoto (I98I a) gives his perturbed rotational state results in terms of cross sections so comparison is not straightforward.

I thank Mrs Norah Scott for her skill in writing the computer program.

R EF RENCaES

Ausloos, P. (ed.) I975 Interactions between ions and molecules. New York: Plenum Press. Ausloos, P. (ed.) I979 Kinetics of ion-molecule reactions. New York: Plenum Press. Bass, L., Su, T., Chesnavich, W. J. & Bowers, M. T. I975 Chem. Phys. Lett. 34, 119. Bates, D. R. I98I Chem. Phys. Lett. 82, 396. Bates, D. R. I982 Astrophys J. (submitted). Bowers, M. I. I979 Gas phase ion chemistry. New York: Academic Press. Celli, F., Weddle, G. & Ridge, D. P. I980 J. chem. Phys. 73, 801. Chesnavich, W. J., Su, T. & Bowers, M. I. I980 J. chem. Phys. 72, 2641. Dugan, J. V. I97I Chem. Phys. Lett. 8, 198. Dugan, J. V. & Canwright, R. B. I972 J. chem. Phys. 56, 3623. Dugan, J. V. & Magee, J. L. I967 J. chem. Phys. 47, 3103. Dugan, J. V. & Magee, J. L. 1973 J. chem. Phys. 58, 5816. Dugan, J. V., Palmer, R. W. & Magee, J. V. I970 Chem. Phys. Lett. 6, 158. Landau, L. D. & Lifshitz, E. M. I960 Mechanics, ?49. Oxford: Pergamon Press. McDaniel, E. W., Cermak, V., Dalgarno, A., Ferguson, E. E. & Friedman, L. 1970 Ion-

molecule reactions, p. 201. New York: Wiley Interscience. Milne-Thomson, L. M. I964 Handbook of mathematical functions (ed. M. Abramowitz &

I. A. Stegun), p. 587. Washington, D.C.: National Bureau of Standards. Moran, T. F. & Hamill, W. H. I963 J. chem. Phys. 39, 1413. Sakimoto, K. I98Ia Chem. Phys. 63, 419. Sakimoto, K. I98I b J. phys. Soc. Japan 50, 1668. Su, T. & Bowers, M. I. - I973 J. chem. Phys. 58, 3027. Takayanagi, K. I978 J. phys. Soc. Japan 45, 976. Takayanagi, K. I982 In Physics of electronic and atomic collisions (ed. S. Datz), p. 343.

Amsterdam: North Holland.



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Ion-Polar Molecule Encounters

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