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Orbiting collisions between ions and polar molecules: Semiclassical PRS approaches
The perturbed-r&tational-state(PRS) method is used to obtain the c?te coefficiekof the orbiting (capture).collisions betwee& ions and polar m&cules~~ Adiab& &&n&k curves in the- PRS theory &e easily .calculated by Ihe_ Bohr-Soqni&feld (semiclassical) quant&tion rule. which is found to be very usefu! fo; the pr&ent-ca&. Thd r&coe~fi&&ts _& comp&& with th& of Bates, who has used the (classical) adiabatic method but hnd& the assumtitioti that thk an&lar momentum &ct&‘of
the molecule is pkpendicular
to the-intermoleculpr
axis. Further&ore, the p&sent- rektlts are compared with’th&k of the Chesnavich;and a fairly good agreement is obiained. Their~empirica~ formula fdr the rate coefficient is found to_be applicable even at very low-temperatures as in the interstellar clouds;
classical trajeciory (Mdnte Carlo) Study bf Su.&d
_ mechanics
1.Intr&l+ion Ion diffusions ‘in polar gases and ion-polarmolecule reactions are strongly’ influenced by the long-range polarization and dipole interactions, i.e.
V= - $a/r4 + D(cos
8)/2,
0)
where r is the- intermolecular -distance; 8 is--the angle between. the- molecular and intermolecular axes (the molecule being-assumed to be linear); (Y and D are, respectively, the polarizability and the dipole moment of the-:molecule. .We use the atomic units. It is well-known that the capture (orbiting) rate coefficient calculated by the’use of the &tentiaf (1) is of.critical importance in the study of the collision frequency and. the chemical reaction [I]. Since the interaction (1) can -be attract&+nd repulsive depending -onthe molccuhu orientation; it is iot’easy -to .calculate~the &ptu&:rate coeffi$ent. For_e&mple, the classical trajectory,(Monte Carlo) method,- if- emplcyed, require{-avery large number.df.the-Gajectory.calcrilations[2,4]. :Theie: fore,. many authors:proposed effective potential (or ~other..appro+&xi)~a~pproaches [5]‘. -Most.-of- them treated z_’t,he,Lmole&lar: :&tat&n’ -.in>the. classical _._‘, .:_ _...~ ~,~ . . . .,-
with
the intention
that they are applica-
ble at_ the room (or higher) temperatures. In lower-temperature gases, however; the discreteness of the rotational energy levels should .be considered. In-a previous work [6], we used- the perturbed-rotational-state- (PRS) -theory [7] to take account of the quantum nature of-the molecular’ rotation correctly. The PRSmethod is based on the following quaritumTmechanical adiabatic representation:
[BJ2~+~(cose)/rf]X=E(r)
(2)
x.,
where B is the rotational constant of the molecule;
J is the angular momentum -vector (here it being a quantum mechanical operator); x and E eigenfunction and- the eigenvalue (adiabatic tial curve),. respec.tively. The effective.dipole tial- which detei-&nes the relative motion _ collision energiti .is_-giv@i _by
+3J(.r+1,. :- _-‘@? ,. This quantity depends on the angular~momentum quantum. numbers J -+d m (m being. ihe pFoj?tion : quantum $u.tmber onto . the~.,intermolecular axis). Althoughthe method’ using_:the,.effective potential (3) is~reasonably accurate under lo&em-
~03~1~0104/g4/$~3.00 Q Elsevier.Science Publishers B-V; -- .(~~rth-Holland:physids-Publis~gIDivisio~) , ‘.. -_ I, .. .-,_ .-
V,,(r)
are the pqten‘poten:at. i$&
A(i)
‘. -__ _-
_ .~_-
_I ._-
-:
274
K. Sakitno?o /
perature conditions [6], it requires a diagonalization of a large-size matrix to solve eq. (1) at small f. or large J. Recently, Bates [8,9] has improved the averaged-dipole-orientation (ADO) method of Su et al. [lo]. In the ADO method, cos 6 in eq. (1) is replaced by its averaged cos 0. In doing this, Bates has used the classical adiabatic invariance defined in ref. [ll], while Su et al. introduced an assumption that the sum of the rotational energy (BJ’) and the dipole potential D(cos 0)/r’ is conserved. The adiabatic invariance is the classical version of the adiabatic representation of the PRS theory. Bates has calculated an adiabatic invariant (action variable) by assuming that J is perpendicular to the intermolecular axis. Then, he has obtained the rate coefficients, which agree well with the results of the variational [3] and the statistical [12] methods. In the Dresent paper, ‘an adiabatic invariant is calculated with no approximations. i.e. for any orientation of J. Since we are primarily interested in the low-temperature conditions, the discreteness of the rotational energy levels is taken into account. For this purpose, we can use the Bohr-Sommerfeld quantization rule_ We define the effective potential not by the ADO methods but by eq. (3) (see later)_
2. Perturbed rotation (semiclassical theory) if we freeze the relative motion,
the total energy
is given by
can define
D(cos
B)/f2.
(4)
Hereafter,
all the variables are regarded as classiLet pe and p+ be the conjugate momenta of the polar coordinate 0 and 9 of the molecular orientation, respectively_ Then, J’ is expressed by
cal quantities_
pi
+
pi/sin%,
(8) where 6, and & are the turning pqints pf the &motion (8, < 8,). In the last form (S), we have introduced the dimensionless quantities, x =
p=cose,
r(
B/D)“2,
(9)
and
=
(P
-
a)(P
-
b)(P
-Cl*
(10)
with u(x) = &(r)/B and u > b > c. According to the Bohr-Sommerfeld quantization rule, we choose n as a non-negative integer (and m as an integer). At x ---, co, the integration (8) is easily performed to give n + 4 = u( co)“2
- Irnl_
(11)
Using the relation u(co) = (J + l/2)‘, we obtain J = n + lnzj, i.e. J = Iml, lnzl+ 1,. . . For J fixed, the rotational energy u( co) is (2 J + 1)-fold degenerate_ When x is not infinite, eq. (8) becomes (hereafter the notation uJm(x) being used instead of u(x)) rx(
J - lnzl+ ;)(a
+mL%.
- c)l”
- c)E(q) d/O
- m2x2 [n(
Pl,
+c)lv
where K(q), E(q) elliptic integrals kinds [13] and 4 = (b - c)/(u
= 2( z+,,,x’ - a)K(q) d/O
-
4 (14
and II( p, q) are the complete of the first, second and third
- c), +c).
Pl=(~--)/(l--)~ 03)
(5)
where the t-axis is chosen along the intermolecular direction. Since $ is the cyclic coordinate in eq. (4), p+ is conserved, i.e. p+ = m( = constant)_ We .,,*,
invariant,
(7)
p2=(c-b)/(l 3’ =
adiabatic
.
I
+2(a E(r) = BJ’+
the following
_
L .- .-..
Orbiting collision ‘between ion and polar molecule.
Since 4 c 1, the elliptic integrals of the first and the second kinds are easily computed by the use of the polynomial approximations [13]. For computations of the-elliptic integral of the third kind, we
-.
Conside; _-
0
0.1
0.2
0.3
0.4
0.5
-0.6
0.7
0.8
0.9
the following
d~m&idnless
potenii&
U(x)=Eb2x-2+x~4+-u~,(x)-(J+~)~,-
: -.
X
(in units of B) as a function of x( = r( B/D)‘j2, calculated by the _semiclassical (solid lines) and quantum mechanical(circles) PRS methods. Fig. 1. The adiabatic potential curves
use the Gauss integration method with the Chebishev‘ polyr?omi& in which the -quadrature points are easy to kvaluate [13]. The relation (12) gives the semiclassical adiabatic potential curve u,,,, as .a function df x. Numerically, the u,~(x) are. calculated by Newton’s method. The results are shown in fig. 1 and
(14 where E is’ the collision energy in units of B; b is the impact parameter in units of (O/B)‘/‘; and K = aB/D’. Th e orbiting impact parameter b, and the energy E satisfy the equation dU(x)/dx-70,
&U(x).
(15)
If x is specified, E and 6, are determined. The cross section (in units of DI/B) is defined by a( J, m) = n-6:. To solve eq. (15) we need the value of du,,,(x)/dx. Differentiating eq. (8) by x, we
1.4 1.2
0.8 0.6
.
E .-versus E $n units of B);cakulated by the semiclassical(solid Fig. 2. The-scaled cross s&i& E’&(J Ll, in) (in knits of D/BID) lines) and qua&m mechanical (dashed lines) PRS the&i&. K( = aB/qs) 7 8.78 X lo- ‘. The arrow represents the calculation by the St&k-eff~tmethod.:
=
. 1 :. _
_. ..-
-I-
..
--
.
‘. ;
K. Sakimoto / Orbiting collision between ion and polar molecule
276
have dr~,,~,(x)/dx-=2x--3[(~-c)E(q)/K(+~].
The’cross sections for J = 1 are shown in fig. 2. It should be noted that the cross section a(J, m) for any J, nt depends only on K and E. The dotted lines indicate the results of the previous calcula(x) is obtained by the exact tions [63 in which zlJJn, quantum mechanical treatment_ The present semiclassical method gives fairly good results except at very small E. As E decreases. the orbiting collision occurs at large _x, where the semiclassical calculation of U,,,,(X) is rather poor. For a simple and accurate calculation of ZJ~~,(S) at very large _Y, we can apply the theory of the second-order Stark effect [Z, 141. We have V -=err B
1
J(J+l)-31~2’ 2J(J+1)(2J-1)(2J+3)
F
( = c,,,/2x’).
(34
1
10
IW
P Fig. 3. The reduced rate coefficient for J = 1 (or u(w) = 2.25) k(J)/k, plotted versus /I( = kT/B) for KB( = akT/D2) = 4.88 x 10eJ and 3.47 x 10e3. A comparison is made among the semiclassical PRS (solid lines). the Stark effect (dashed lines) and the ADO of Bates (circles) calculations.
This gives the cross section us,nrk(J, ,,r ) = 7i(21+‘E
=o
if
K’
it
K’<
)I” =
K -
c,,,,
>
0.
0.
(17)
This Stark-effect model is accurate at E < 1 (fig. 2), and can be used to remedy the defect of the semiclassical method at small E. 4. Rate coefficients Consider the rate coefficient k(J) divided by the Langevin value k, = 2 (a/p)‘/‘. where p is the reduced mass of the collision pair. It is given by k( J)/k,
= (2fl/?r’K)‘/zlwo(
J)e-“q
dn,
Fig. 3 shows k( J)/k, for J = 1. The results are compared with those of Bates [8,9] and the Starkeffect method. Bates has assumed that J is perpendicular to the intermolecular axis, i.e. m = 0. However, fig_ 2 shows that the I?,-dependence of the cross section is small at high energies. Therefore. the results of Bates slightly differ from the present semiclassical values except at low temperatures. For /3> 1, the semiclassical method gives the most accurate values among the three methods. When p <( 1. however. the Stark-effect methods will be rather better. The rate coefficient averaged over the rotational states is given by
(18)
k=
l)k(J)ed---,/P))
\J
where /3=kT/B,
q=E/j3,
(19)
and T is the temperature_ In eq. (18). u(J) is the degeneracy-averaged cross section (in units of D/B), u(J)
(x(23+
= (2J+
I)-‘&(
J, m). nr
(20)
+(2J+l)exP(-EJ/fl))-‘3
I
(21)
where EI = (J + l/2) 2_ We .’Improve the semiclassical calculation by taking a(J) = ustnrk( J) at E < 0.5. Performing the classical trajectory calculations,
su a& che~na~cht(4j.lhave-.f0~~d
ihat:@&&pen& alm&t &ly ;ihe param&~ 2 (Z@)T’e as farm:as_‘@i&B- &&. ;;i:- i&ii_ ‘tT& .1& Tb;e*
rigor$us$-k&&for alI:.&&& :@f,:@/&& & ihe variationak[3], sta~$tk&[iZ$klA~O [8,9]-meihods.) -Furthetior&;;'they .bavti- obtained &I eiiipirical formula on therbasis of: their &ssical.~raject&y klctiltitions. ‘IO within-the 3%. a&&$ they-have found the formula in @e form at_&? > 2 :_ E/k,
=-O-47675 +.0.8i,
‘7.2 5 ,, 2.
:
(22)
.have not performed the caJ$ulationS at 5 > 7. The values obtdtied from eq. (22) are shdtin in table 1 and compared with the present semiclassical results. The‘ttio results agree fairly -welleven at < > 7 (f > 7 with K= 1.0 X 10B3 -.corresponding to /3-z 7.813). Table 2 shows the present semiclassical calculations where values of /? and K- are chosen to -give .$= 32 (or ~j3 = 4.883 X 10m4). -All_ the values of k/k, agree with each other within 15%. We conclude ‘therefore th‘at the empirical formula (22), though simply depending on only one parameter 5, is accurate within the error of-15% even at kT ( B. The ADO resutt of Bates agrees with the present one within 25% at a z=1. The difference at p = 0.4883 is = 40% and is expected to be mainly due to the assumption in the- ADO method of Bates that J is perpendicular to the intermolecular axis. The variatidnal [3] and the statistic%1 [12] methods give an tipper bound of the capture rate coefficient. In fact .their results are larger than the
E/k,
.:‘.
24.42 9;786‘ 4.883 2.442
__.
.17_6@0)~‘--
-. -.._ I
::,i7_5(10)
_
~. 17.3 (10)
..
:
--
16.3(S) 1X1(5)
0.9786
_:
I, :
-17-I(6) 16.9 (5)
-0.4883
They
Table
37.86
” ( )I) means that the J = O-n rotational account in the thkmal averaging.
-. states
are taken &to
classical trajectory:. and the present semiclassical results (table i)_
4. Summary
The adiabatic potential curves in- the PRS theory are easily (and accuraiely at small x j obtained by the BohrSomtierfeld quantization rule. .The empirical formula (22) obtained by Su and Chesnavich -gives values very close to the present semiclassical results at all the temperatures considered here: For the calculations of the rate coefficient averaged over.the.translational and the rotational energies, the quantum effect (the discreteness of the rotational energy levels) is not so important even at very low temperatures. The
1 at K( =a13/D2)=l.0X10-3
B( = kT,‘W
Variational
Statistical
Classical
ADO (Bates)
trajectory
_ Present semiclassical
ref. [S] 125 31.25 13.89 7.813 3.472 1.953.
1.739
-1.85
1.827
1.83
1.57
1.58 (20) a)
2.819 4.006 5.230
3.10 4.40 5.70
2.981 4.213 5.474 8.031 10.61
2.97 4.20 5.45 8.00 10.6
2.53 3.48 4.43 6.34 8.25
2.52 (10) 3.48 (10) 4.49 (8) 6.53 (5) 8:44(5)
0.8681 Ok883 a) (n)
rpeans that the J = O-n rotational
.-
.
15.80 21.00
states are taken into account
_
)
15.8 21.0 in.the
thermal
12.1 15.9 averaging.
~
12.5 15.1
(5) (5)
-
278
K
Sokimoto /
Orbiting ‘colkion betwt+n ion and polar molecule
empirical formula (22) will be useful -to estimate the ion-molecule reaction rate coefficient in the interstellar chemistry. The following, however, should be taken into mind: Because of the low density of interstellar clouds, the population of the rotational energy levels of the molecules is not necessarily in the thermal equilibrium [15]. Then, we have to deal with the quantity k(J) rather than & Further, k(J) has a strong J-dependence at small J and at low temperatures [6].
Acknowledgement
The author would like to thank to Professor Itikawa for his reading the manuscript_
Y.
References of ion-molecule reactions P. Ausloos. ed.. Kinetics (Plenum Press, New York, 1979): M.T. Bowers. ed., Gas phase ion chemistry (Academic Press, New York. 1979). J.V. Dugan and J.L. Magee. J. Chem. Phys. 47 (1967) 3103;
J.V. Dugan, Chem. Phys.. getters 21’ (1973): 476,. and references therein. . [3] WJ. Chemavich. T. Su and M-T_ Bowers, J_ Chem.‘Ph$s.. 72 (1980) 2641. [4j T. SU and W.J. Chesnavich, J. &&I. Phys. 76 (1982) 5183.. [S] K. Takayanagi. in: Physics of electronic and atomic coIli;
sions, ed. S. Datz (North-Holland, Amsterdam, 1982) p. 343. [6] K. Sakimoto and K. Takayanagi, J. Phys. Sot. J&n (1980) 2076; K. Sakimoto. Chem. Phys. 63 (1981) 419. [7] K. Takayanagi. J. Phys. Sot. Japan 45 (1978) 976. [8] D.R. Bates. Proc. Roy. Sot. A384 (1982)
48 .
289.
[9] D.R Bates. Chem. Phys. Letters 97 (1983) 19. [lo] T. Su and M-T. Bowers, J. Chem. Phys. 58 (1973) 30271 T. Su. E.C.F. Su and M.T. Bowers, J. Chem. Phys. 69 (1978) 2243. [ll] L.D. Landau and EM. Lifshitz, Mechanics (Pergamon Press. Oxford, 1960). (121 F. Cell. G. Weddle and D.P. Ridge, J. Chem. Phys. 73
(1980) 801. [13] M. Abramowitz and LA. Stegun, Handbook of mathematical functions with formulas. graphs, and mathematical tables, Applied Mathematics Series 55 (Natl. Bur. Std., Washington. 1965). [14] D. Hyatt and L. Stanton. Proc. Roy. Sot. A318 (1970) 107. [15] L. Spitzer. Physical processes in the interstellar medium (Wiley. New York. 1978).
the molecule is pkpendicular
to the-intermoleculpr
axis. Further&ore, the p&sent- rektlts are compared with’th&k of the Chesnavich;and a fairly good agreement is obiained. Their~empirica~ formula fdr the rate coefficient is found to_be applicable even at very low-temperatures as in the interstellar clouds;
classical trajeciory (Mdnte Carlo) Study bf Su.&d
_ mechanics
1.Intr&l+ion Ion diffusions ‘in polar gases and ion-polarmolecule reactions are strongly’ influenced by the long-range polarization and dipole interactions, i.e.
V= - $a/r4 + D(cos
8)/2,
0)
where r is the- intermolecular -distance; 8 is--the angle between. the- molecular and intermolecular axes (the molecule being-assumed to be linear); (Y and D are, respectively, the polarizability and the dipole moment of the-:molecule. .We use the atomic units. It is well-known that the capture (orbiting) rate coefficient calculated by the’use of the &tentiaf (1) is of.critical importance in the study of the collision frequency and. the chemical reaction [I]. Since the interaction (1) can -be attract&+nd repulsive depending -onthe molccuhu orientation; it is iot’easy -to .calculate~the &ptu&:rate coeffi$ent. For_e&mple, the classical trajectory,(Monte Carlo) method,- if- emplcyed, require{-avery large number.df.the-Gajectory.calcrilations[2,4]. :Theie: fore,. many authors:proposed effective potential (or ~other..appro+&xi)~a~pproaches [5]‘. -Most.-of- them treated z_’t,he,Lmole&lar: :&tat&n’ -.in>the. classical _._‘, .:_ _...~ ~,~ . . . .,-
with
the intention
that they are applica-
ble at_ the room (or higher) temperatures. In lower-temperature gases, however; the discreteness of the rotational energy levels should .be considered. In-a previous work [6], we used- the perturbed-rotational-state- (PRS) -theory [7] to take account of the quantum nature of-the molecular’ rotation correctly. The PRSmethod is based on the following quaritumTmechanical adiabatic representation:
[BJ2~+~(cose)/rf]X=E(r)
(2)
x.,
where B is the rotational constant of the molecule;
J is the angular momentum -vector (here it being a quantum mechanical operator); x and E eigenfunction and- the eigenvalue (adiabatic tial curve),. respec.tively. The effective.dipole tial- which detei-&nes the relative motion _ collision energiti .is_-giv@i _by
+3J(.r+1,. :- _-‘@? ,. This quantity depends on the angular~momentum quantum. numbers J -+d m (m being. ihe pFoj?tion : quantum $u.tmber onto . the~.,intermolecular axis). Althoughthe method’ using_:the,.effective potential (3) is~reasonably accurate under lo&em-
~03~1~0104/g4/$~3.00 Q Elsevier.Science Publishers B-V; -- .(~~rth-Holland:physids-Publis~gIDivisio~) , ‘.. -_ I, .. .-,_ .-
V,,(r)
are the pqten‘poten:at. i$&
A(i)
‘. -__ _-
_ .~_-
_I ._-
-:
274
K. Sakitno?o /
perature conditions [6], it requires a diagonalization of a large-size matrix to solve eq. (1) at small f. or large J. Recently, Bates [8,9] has improved the averaged-dipole-orientation (ADO) method of Su et al. [lo]. In the ADO method, cos 6 in eq. (1) is replaced by its averaged cos 0. In doing this, Bates has used the classical adiabatic invariance defined in ref. [ll], while Su et al. introduced an assumption that the sum of the rotational energy (BJ’) and the dipole potential D(cos 0)/r’ is conserved. The adiabatic invariance is the classical version of the adiabatic representation of the PRS theory. Bates has calculated an adiabatic invariant (action variable) by assuming that J is perpendicular to the intermolecular axis. Then, he has obtained the rate coefficients, which agree well with the results of the variational [3] and the statistical [12] methods. In the Dresent paper, ‘an adiabatic invariant is calculated with no approximations. i.e. for any orientation of J. Since we are primarily interested in the low-temperature conditions, the discreteness of the rotational energy levels is taken into account. For this purpose, we can use the Bohr-Sommerfeld quantization rule_ We define the effective potential not by the ADO methods but by eq. (3) (see later)_
2. Perturbed rotation (semiclassical theory) if we freeze the relative motion,
the total energy
is given by
can define
D(cos
B)/f2.
(4)
Hereafter,
all the variables are regarded as classiLet pe and p+ be the conjugate momenta of the polar coordinate 0 and 9 of the molecular orientation, respectively_ Then, J’ is expressed by
cal quantities_
pi
+
pi/sin%,
(8) where 6, and & are the turning pqints pf the &motion (8, < 8,). In the last form (S), we have introduced the dimensionless quantities, x =
p=cose,
r(
B/D)“2,
(9)
and
=
(P
-
a)(P
-
b)(P
-Cl*
(10)
with u(x) = &(r)/B and u > b > c. According to the Bohr-Sommerfeld quantization rule, we choose n as a non-negative integer (and m as an integer). At x ---, co, the integration (8) is easily performed to give n + 4 = u( co)“2
- Irnl_
(11)
Using the relation u(co) = (J + l/2)‘, we obtain J = n + lnzj, i.e. J = Iml, lnzl+ 1,. . . For J fixed, the rotational energy u( co) is (2 J + 1)-fold degenerate_ When x is not infinite, eq. (8) becomes (hereafter the notation uJm(x) being used instead of u(x)) rx(
J - lnzl+ ;)(a
+mL%.
- c)l”
- c)E(q) d/O
- m2x2 [n(
Pl,
+c)lv
where K(q), E(q) elliptic integrals kinds [13] and 4 = (b - c)/(u
= 2( z+,,,x’ - a)K(q) d/O
-
4 (14
and II( p, q) are the complete of the first, second and third
- c), +c).
Pl=(~--)/(l--)~ 03)
(5)
where the t-axis is chosen along the intermolecular direction. Since $ is the cyclic coordinate in eq. (4), p+ is conserved, i.e. p+ = m( = constant)_ We .,,*,
invariant,
(7)
p2=(c-b)/(l 3’ =
adiabatic
.
I
+2(a E(r) = BJ’+
the following
_
L .- .-..
Orbiting collision ‘between ion and polar molecule.
Since 4 c 1, the elliptic integrals of the first and the second kinds are easily computed by the use of the polynomial approximations [13]. For computations of the-elliptic integral of the third kind, we
-.
Conside; _-
0
0.1
0.2
0.3
0.4
0.5
-0.6
0.7
0.8
0.9
the following
d~m&idnless
potenii&
U(x)=Eb2x-2+x~4+-u~,(x)-(J+~)~,-
: -.
X
(in units of B) as a function of x( = r( B/D)‘j2, calculated by the _semiclassical (solid lines) and quantum mechanical(circles) PRS methods. Fig. 1. The adiabatic potential curves
use the Gauss integration method with the Chebishev‘ polyr?omi& in which the -quadrature points are easy to kvaluate [13]. The relation (12) gives the semiclassical adiabatic potential curve u,,,, as .a function df x. Numerically, the u,~(x) are. calculated by Newton’s method. The results are shown in fig. 1 and
(14 where E is’ the collision energy in units of B; b is the impact parameter in units of (O/B)‘/‘; and K = aB/D’. Th e orbiting impact parameter b, and the energy E satisfy the equation dU(x)/dx-70,
&U(x).
(15)
If x is specified, E and 6, are determined. The cross section (in units of DI/B) is defined by a( J, m) = n-6:. To solve eq. (15) we need the value of du,,,(x)/dx. Differentiating eq. (8) by x, we
1.4 1.2
0.8 0.6
.
E .-versus E $n units of B);cakulated by the semiclassical(solid Fig. 2. The-scaled cross s&i& E’&(J Ll, in) (in knits of D/BID) lines) and qua&m mechanical (dashed lines) PRS the&i&. K( = aB/qs) 7 8.78 X lo- ‘. The arrow represents the calculation by the St&k-eff~tmethod.:
=
. 1 :. _
_. ..-
-I-
..
--
.
‘. ;
K. Sakimoto / Orbiting collision between ion and polar molecule
276
have dr~,,~,(x)/dx-=2x--3[(~-c)E(q)/K(+~].
The’cross sections for J = 1 are shown in fig. 2. It should be noted that the cross section a(J, m) for any J, nt depends only on K and E. The dotted lines indicate the results of the previous calcula(x) is obtained by the exact tions [63 in which zlJJn, quantum mechanical treatment_ The present semiclassical method gives fairly good results except at very small E. As E decreases. the orbiting collision occurs at large _x, where the semiclassical calculation of U,,,,(X) is rather poor. For a simple and accurate calculation of ZJ~~,(S) at very large _Y, we can apply the theory of the second-order Stark effect [Z, 141. We have V -=err B
1
J(J+l)-31~2’ 2J(J+1)(2J-1)(2J+3)
F
( = c,,,/2x’).
(34
1
10
IW
P Fig. 3. The reduced rate coefficient for J = 1 (or u(w) = 2.25) k(J)/k, plotted versus /I( = kT/B) for KB( = akT/D2) = 4.88 x 10eJ and 3.47 x 10e3. A comparison is made among the semiclassical PRS (solid lines). the Stark effect (dashed lines) and the ADO of Bates (circles) calculations.
This gives the cross section us,nrk(J, ,,r ) = 7i(21+‘E
=o
if
K’
it
K’<
)I” =
K -
c,,,,
>
0.
0.
(17)
This Stark-effect model is accurate at E < 1 (fig. 2), and can be used to remedy the defect of the semiclassical method at small E. 4. Rate coefficients Consider the rate coefficient k(J) divided by the Langevin value k, = 2 (a/p)‘/‘. where p is the reduced mass of the collision pair. It is given by k( J)/k,
= (2fl/?r’K)‘/zlwo(
J)e-“q
dn,
Fig. 3 shows k( J)/k, for J = 1. The results are compared with those of Bates [8,9] and the Starkeffect method. Bates has assumed that J is perpendicular to the intermolecular axis, i.e. m = 0. However, fig_ 2 shows that the I?,-dependence of the cross section is small at high energies. Therefore. the results of Bates slightly differ from the present semiclassical values except at low temperatures. For /3> 1, the semiclassical method gives the most accurate values among the three methods. When p <( 1. however. the Stark-effect methods will be rather better. The rate coefficient averaged over the rotational states is given by
(18)
k=
l)k(J)ed---,/P))
\J
where /3=kT/B,
q=E/j3,
(19)
and T is the temperature_ In eq. (18). u(J) is the degeneracy-averaged cross section (in units of D/B), u(J)
(x(23+
= (2J+
I)-‘&(
J, m). nr
(20)
+(2J+l)exP(-EJ/fl))-‘3
I
(21)
where EI = (J + l/2) 2_ We .’Improve the semiclassical calculation by taking a(J) = ustnrk( J) at E < 0.5. Performing the classical trajectory calculations,
su a& che~na~cht(4j.lhave-.f0~~d
ihat:@&&pen& alm&t &ly ;ihe param&~ 2 (Z@)T’e as farm:as_‘@i&B- &&. ;;i:- i&ii_ ‘tT& .1& Tb;e*
rigor$us$-k&&for alI:.&&& :@f,:@/&& & ihe variationak[3], sta~$tk&[iZ$klA~O [8,9]-meihods.) -Furthetior&;;'they .bavti- obtained &I eiiipirical formula on therbasis of: their &ssical.~raject&y klctiltitions. ‘IO within-the 3%. a&&$ they-have found the formula in @e form at_&? > 2 :_ E/k,
=-O-47675 +.0.8i,
‘7.2 5 ,, 2.
:
(22)
.have not performed the caJ$ulationS at 5 > 7. The values obtdtied from eq. (22) are shdtin in table 1 and compared with the present semiclassical results. The‘ttio results agree fairly -welleven at < > 7 (f > 7 with K= 1.0 X 10B3 -.corresponding to /3-z 7.813). Table 2 shows the present semiclassical calculations where values of /? and K- are chosen to -give .$= 32 (or ~j3 = 4.883 X 10m4). -All_ the values of k/k, agree with each other within 15%. We conclude ‘therefore th‘at the empirical formula (22), though simply depending on only one parameter 5, is accurate within the error of-15% even at kT ( B. The ADO resutt of Bates agrees with the present one within 25% at a z=1. The difference at p = 0.4883 is = 40% and is expected to be mainly due to the assumption in the- ADO method of Bates that J is perpendicular to the intermolecular axis. The variatidnal [3] and the statistic%1 [12] methods give an tipper bound of the capture rate coefficient. In fact .their results are larger than the
E/k,
.:‘.
24.42 9;786‘ 4.883 2.442
__.
.17_6@0)~‘--
-. -.._ I
::,i7_5(10)
_
~. 17.3 (10)
..
:
--
16.3(S) 1X1(5)
0.9786
_:
I, :
-17-I(6) 16.9 (5)
-0.4883
They
Table
37.86
” ( )I) means that the J = O-n rotational account in the thkmal averaging.
-. states
are taken &to
classical trajectory:. and the present semiclassical results (table i)_
4. Summary
The adiabatic potential curves in- the PRS theory are easily (and accuraiely at small x j obtained by the BohrSomtierfeld quantization rule. .The empirical formula (22) obtained by Su and Chesnavich -gives values very close to the present semiclassical results at all the temperatures considered here: For the calculations of the rate coefficient averaged over.the.translational and the rotational energies, the quantum effect (the discreteness of the rotational energy levels) is not so important even at very low temperatures. The
1 at K( =a13/D2)=l.0X10-3
B( = kT,‘W
Variational
Statistical
Classical
ADO (Bates)
trajectory
_ Present semiclassical
ref. [S] 125 31.25 13.89 7.813 3.472 1.953.
1.739
-1.85
1.827
1.83
1.57
1.58 (20) a)
2.819 4.006 5.230
3.10 4.40 5.70
2.981 4.213 5.474 8.031 10.61
2.97 4.20 5.45 8.00 10.6
2.53 3.48 4.43 6.34 8.25
2.52 (10) 3.48 (10) 4.49 (8) 6.53 (5) 8:44(5)
0.8681 Ok883 a) (n)
rpeans that the J = O-n rotational
.-
.
15.80 21.00
states are taken into account
_
)
15.8 21.0 in.the
thermal
12.1 15.9 averaging.
~
12.5 15.1
(5) (5)
-
278
K
Sokimoto /
Orbiting ‘colkion betwt+n ion and polar molecule
empirical formula (22) will be useful -to estimate the ion-molecule reaction rate coefficient in the interstellar chemistry. The following, however, should be taken into mind: Because of the low density of interstellar clouds, the population of the rotational energy levels of the molecules is not necessarily in the thermal equilibrium [15]. Then, we have to deal with the quantity k(J) rather than & Further, k(J) has a strong J-dependence at small J and at low temperatures [6].
Acknowledgement
The author would like to thank to Professor Itikawa for his reading the manuscript_
Y.
References of ion-molecule reactions P. Ausloos. ed.. Kinetics (Plenum Press, New York, 1979): M.T. Bowers. ed., Gas phase ion chemistry (Academic Press, New York. 1979). J.V. Dugan and J.L. Magee. J. Chem. Phys. 47 (1967) 3103;
J.V. Dugan, Chem. Phys.. getters 21’ (1973): 476,. and references therein. . [3] WJ. Chemavich. T. Su and M-T_ Bowers, J_ Chem.‘Ph$s.. 72 (1980) 2641. [4j T. SU and W.J. Chesnavich, J. &&I. Phys. 76 (1982) 5183.. [S] K. Takayanagi. in: Physics of electronic and atomic coIli;
sions, ed. S. Datz (North-Holland, Amsterdam, 1982) p. 343. [6] K. Sakimoto and K. Takayanagi, J. Phys. Sot. J&n (1980) 2076; K. Sakimoto. Chem. Phys. 63 (1981) 419. [7] K. Takayanagi. J. Phys. Sot. Japan 45 (1978) 976. [8] D.R. Bates. Proc. Roy. Sot. A384 (1982)
48 .
289.
[9] D.R Bates. Chem. Phys. Letters 97 (1983) 19. [lo] T. Su and M-T. Bowers, J. Chem. Phys. 58 (1973) 30271 T. Su. E.C.F. Su and M.T. Bowers, J. Chem. Phys. 69 (1978) 2243. [ll] L.D. Landau and EM. Lifshitz, Mechanics (Pergamon Press. Oxford, 1960). (121 F. Cell. G. Weddle and D.P. Ridge, J. Chem. Phys. 73
(1980) 801. [13] M. Abramowitz and LA. Stegun, Handbook of mathematical functions with formulas. graphs, and mathematical tables, Applied Mathematics Series 55 (Natl. Bur. Std., Washington. 1965). [14] D. Hyatt and L. Stanton. Proc. Roy. Sot. A318 (1970) 107. [15] L. Spitzer. Physical processes in the interstellar medium (Wiley. New York. 1978).