Download Diameters of rotationally and vibrationally excited diatomic molecules

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Physica A 247 (1997) 108-120
Diameters of rotationally and vibrationally excited
diatomic molecules
Y u r i y E. G o r b a c h e v a, F r a n c i s c o J. G o r d i l l o - V ~ i z q u e z
J o s e p h A. K u n c c,,
b, 1,
a Theoretical Department, A.F.Ioffe Physical-Technical Institute, St.Petersburg 194021, Russia
b Departamento de Fisica Atdmica, Molecular y Nuclear, Universidad de Sevilla, 41080 Sevilla,
Apdo. Correos 1065, Spain
c Department of Aerospace Engineering and Physics, University of Southern California, Los Angeles,
CA 90089-1191, USA
Received 18 March 1997
Abstract
Analytical expressions for kinetic diameters of rotationally and vibrationally excited diatomic
molecules are derived using the Hamilton-Jacoby theory and the Bohr-Sommerfeld quantization rule. The diameters are obtained assuming the rotating Morse and Tietz-Hua oscillators as
dynamical models of the molecules.
PACS: 05.20.Dd; 33.20.Vq
1. Introduction
Studies of kinetic properties of low-temperature (below a few thousands degrees)
gases are often based on the assumption that the gas particles have finite size. The
molecular 'size' is then characterized by the so-called 'molecular diameter', and the
sum of the molecular diameters of two colliding particles, the 'collision diameter', is
frequently used in studies of transfer of energy and linear momentum between the
particles in the gases.
The model of binary collision as an interaction of two finite-size molecules has been
quite succesful in studies of transport properties (Boltzmann integrals, viscosity, conductivity, etc.) of low-temperature gases in various devices and in Monte-Carlo simulations of low-energy (subsonic) flows [1,2]. Many physical aspects of the model have
* Corresponding author. Fax: +1 213 740 7774; e-mail:[email protected].
l Present address: Department of Aerospace Engineering, University of Southern California, Los Angeles,
CA 90089-1191, USA
0378-4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved
PII S0378-4371 ( 9 7 ) 0 0 3 8 9 - 0
YE. Gorbachev et al. I Physica A 247 (1997) 108-120
109
been studied extensively, and mathematical descriptions of various low-temperature
processes based on the model have been developed. Therefore, the finite-size model of
atom-molecule and molecule-molecule collisions would also be of great advantage in
studies of transport phenomena in high-temperature gases (common in plasma processing, light sources, supersonic and hypersonic flows, etc.) if accurate molecular diameters
in such gases were easily available. However, determination of the molecular diameters
in high-temperature gas is much more difficult than determination of the diameters in
low-temperature gas. In the latter case, the rotational-vibrational energies of most of the
gas molecules are small and the corresponding typical molecular diameters can be given
as simple functions of the molecular bond lengths (the equilibrium internuclear distance) [3,4]. In high-temperature gas, the situation is more complicated because the degree of rotational-vibrational excitation of most of the gas molecules is significant. The
rotational-vibrational energy of the molecules increases with gas temperature and so
are the molecular diameters. Since the molecular scattering cross sections are (roughly)
proportional to square of the collisional diameters, the rotational-vibrational excitation
of molecules has a significant effect on the kinetic properties of high-temperature gases.
In this work, we study molecular diameters of rotationally and vibrationally excited
diatomic molecules assuming that the molecules are either the rotating Morse oscillators
or the rotating Tietz-Hua oscillators (see Ref. [5] which is called hereatter as Paper
I). We derive several analytical expressions (of different accuracies) for the diameters
as functions of molecular vibrational (v) and rotational (J) quantum numbers.
2. Intramolecular potentials
We assume in what follows that diatomic molecules are rotating anharmonic oscillators strecthed by the molecular centrifugal force. The internuclear potential of the
rotationally unexcited molecules is assumed to be either the Morse potential
UM(R) = D [1 - e-#(R-R')] 2 ,
(1)
or the Tietz-Hua potential
1 - e -bh(R-Re) ] 2
UrH(R) = D [ 1 - - c ~ ) J
'
(2)
bh = fl(1 - ch),
(3)
where
D is the well-depth of the potential, Re is the molecular bond length, R is the internuclear distance, fl is the Morse constant and Ch is an optimization parameter obtained
from ab initio and RKR (Rydberg-Klein-Rees) intramolecular potentials. (The rotating Morse and Tietz-Hua oscillators are discussed in detail in Paper I.) The reason for
choosing the Tietz-Hua potential in this work is its remarkably good agreement (within
a broad range of the internuclear distance) with the RKR calculations (Ref. [6] and
110
Y.E. Gorbachev et at,.IPhysica A 247 (1997) 108-120
references therein) and ab initio calculations (Ref. [7] and references therein); see also
discussions in Refs. [5,8,9]. According to these works, the Tietz-Hua potential is more
realistic than the Morse potential, especially in the molecules with moderate and high
values of the rotational and vibrational quantum numbers. (As discussed in Paper I, the
Tietz-Hua potential is one of the very best model potentials available in literature.)
The rotational (centrifugal) part of the molecular internuclear potential is taken as
Uj(R)-
L2
2//R 2 ,
(4)
where # is the reduced mass and L is the angular momentum of the rotating molecule.
Thus, the effective internuclear potential of the diatomic molecules under consideration
is
(5)
Vef(R) = U j ( R ) + U v ( R ) ,
where Uv(R) is given, depending on the assumed oscillator model, by either UM or
UTH.
The vibrational constants of the Morse oscillator are
=
,
(6)
and
h/~ 2
O,)eXe -- 87Z2C]2 ,
(7)
where h is Planck's constant, c is the speed of light, fDe and fOeXe are in cm -1 and the
rest of the quantities are in units of the c.g.s, system. In the present work, we calculate
fl from the relationship given by Eq. (6) (see discussion in Paper I) and the constants
(De, OgeXe and D (see Table 1) are the 'best' available in literature.
3. The mean internuclear distance in the excited molecules
Porter et al. derived [10], using the Hamilton-Jacoby theory and the Bohr-Sommerfeld
quantization rule, the time dependence of the radial internuclear distance of a rotating
oscillator [7]. We apply below their approach, in order to determine the molecular
diameters, to the rotating Tietz-Hua oscillator and to the rotating Morse oscillator.
3.1. The rotating Tietz-Hua oscillator
The Hamiltonian of the rotating Tietz-Hua oscillator with angular momentum L is
n-
(phR)2
L2
-- 2--7- + ~ 2
[ l_e_bh(R_Re ) ]2
+ D l1 : - - c ~ , ) J
'
whore phR = #(dR/dt) is the linear momentum of the oscillator radial motion.
(8)
Y.E. Gorbachevet al. IPhysica A 247 (1997) 108-120
111
The equilibrium internuclear distance R, of a diatomic molecule in the Jth rotational
state can be obtained from the obvious relationship
dVef(R) _ O,
dR
(9)
where
L2
Vef(R) = 21d~---i + UrH(R).
(10)
After introducing a new variable
C:-
b2L2
(ll)
21xD'
the derivative
(Eq. (9)) becomes
dVef(R)
[1 -e-y-eye ]
yeCj
dR
= (1--c--he_Y.ey---~)3 (1--Ch)--eY*e - -if=0,y.
(12)
y. = bhR.
(13)
where
and
Ye = bhRe ,
and where values of the parameter bh for a number of diatomic molecules can be found
in Paper I.
Eq. (12) can be solved through asymptotic analysis in terms of the small parameter
Ch (see Paper I). The resulting solution is
Y, = Ye + BhCj + Dh C2 ,
(14)
where
Bh -- (1 - Ch)2
y3e
3 2 3 B2 + 3BhCh(1 -- Ch)
and Dh= ~B h - Ye
y3
(15)
In order to apply the semiclassical quantization, we introduce a new variable, ~h(R) =
e x p [ - b h ( R - R.)], and use the following series expansion of 1/(bhR) 2 about Ch. = 1:
1
1
2
(bhR)----~ = (bhR,)-----~ + ~ ( ~ ,
h
1(
- 1)
(bhR,) 3
3)
1 - b~,
(~h, _ 1)2 + . . . .
(16)
Substituting this expansion into Eq. (8) and solving the equation for p~ at constant
value of the rotational-vibrational energy Ev.rj
~ = H gives
p~ = ±(2~t) 1/2 [Ev.Tff--L2(ArH + BrH~h, -- CrH(~h,)2)
-D
(\ 1
h h
2 U2
h]
~--ch~h.¢~------,) ]
'
(17)
112
Y.E. Gorbachev et al. I Physica A 247 (1997) 108-120
where
~h, = e-bh(R, -R~) ,
(18)
L2=h2[J(J+I)-A2],
(19)
At14 =
E
3
]
(20)
Cr/-/,
(21)
1 + bhR,(bhR, - 3) bhR, Crl4 ,
BrI-I = 2
bhR. )
1 + bh-~, Z 3
C r ~ I - 2#bhR3
1-
,
(22)
and A is the quantum number for the axial component of the molecular electronic
angular momentum.
The Bohr-Sommerfeld quantization rule is
I f p Rh d R =
N v = ~-~n
( v~+ )
h,
(23)
where, as before, v is the vibrational quantum number. Taking the positive root of
Eq. (17) and changing variable R to ~h, gives
N~ --
(2//) 1/2 f
h
h
K~,j(¢.) d~. ,
7zbh J
(24)
where
I'&j( ~h)
{[Ev,~ - LZ(Am.~_BTH~h
h h 2 - D ( 1 - . , :h~he*,~2"[j 1/2
- CTH(~h)2)](I --Ch~.¢e,)
~h,(1 - ch~h~he,)
(25)
and Ch,< and ~h,> are roots of the equation ph = 0. After rearranging the terms in
Eq. (17), the equation p~ = 0 can be written as the following fourth-order polynomial:
f l ( ~ , ) = ath(~h) 4 + bth(~h) 3 + Cth(~h,) 2 + dth~ h + eth = 0,
(26)
with
ath =
2 h
Ch(
~e, )2CTHL2 ,
bth = BrI-IL 2Ch(~e,)
2 h 2+
(27)
2Ch~he,CTHL2
(28)
Y.E. Gorbachev et al. I Physica A 247 (1997) 108-120
Cth :
111 C 2h ( ~ eh, ) 2 (E~,j
113
ATHL 2) "k- 2Bll.lL2ch~he, 4- CTHL 2 _ (~h
,. e* J~2D
(29)
~-- 2BrHL2Ch~he, + C r n L2 -- (~e,)h 2D ,
dth = 2~eh,D- - BrHL 2 __ 2 C h ~ eh, ( E v , j
-- ATH L2)
~
2~eh,D -
BTH L2 ,
(30)
and
(31)
ArHL 2 - D ,
eth = E TM -
where the simplification of the coefficients Cth and dth resulted from the fact that eh is,
in general, much smaller than one.
Expanding fl(~h.) in power series about ~h = 1 and retaining only the terms which
are linear or quadratic in ~h leads to
fl(~h.) --~ f2(~h.)
=
(01(~h) 2 q - (02~h q- (03,
(32)
where
(01 = 6ath + 3bth + cth,
(02 = --8ath -- 3bth + dth,
(03 = 3ath + bth + eth . (33)
After some algebra, we obtain the following vibrational-rotational energies of diatomic
molecules represented by the rotating Tietz-Hua oscillators:
E~r~J = D + L Z B j B ~ - (FoXl - F[' - F~"L 2 - F ~ t t L 4 ) 2 ,
(34)
where the functions B j , B'J, Fo, F[', F~" and F~" are given in Paper I.
The expression for the radial orbit of the Tietz-Hua oscillator is (see Eq. (57) and
Refs. [5,6,10]):
Rth((0t h ) =
R , + b ; 1 [6th + In(1 - eth COS(0th)],
(35)
where the angle variable (0th changes from 0 to re,
6th=--ln(--2(03~
\
,
(36)
(02 f
and
eth=
1
4(03(01 "~ 1/2
-~
j
(37)
The mean internuclear distance of the oscillator can be obtained from the orbit
(Eq. (35)) by averaging the internuclear distance Rth over the orbit period T,
1/
T
(Rv,J)th = -f
1/
2n
Rth(t)d t = ~
0
=R.+b~-I
Rth((0th) d(0th
0
6th+ln
2
'
Y.E. Gorbachev et al. IPhysica A 247 (1997) 108-120
114
where t is time. Thus, the relative increase of the mean internuclear distance caused
by molecular rotational-vibrational excitation can be given as
(Rv,l)th - Re
Pth --
~th + ln[(1 + V/1 - e2h)/2] + y , -- Ye
(39)
-Re
Ye
(Rv,J)th must be a real and positive number. This requires that 2q~3/tP2~<0. When
2q~3/q~2 = 0, (Rv,J)th ---+oo since then 6th ~ oo, and the molecule becomes an unstable
diatomic system. According to Eq. (33),
q)3 = 3ath + bth + eth :
E~ - D
- L2BjB~
(40)
= 0.
When 2q~3/q~2 = 0, Eqs. (34) and (40) lead to the following second-order equation:
(41)
~Xlx 2 + ~X2X1 + ~3 = 0 ,
where Xl = v + 0.5, and
~i = - v ~ o l ,
. 2 = F0 - v ~ o 0 ,
,it 4
o~3 = -(v/-D + F~"L 2 + F 3 L ).
(42)
Solution of Eq. (41) leads to the v - J contour for the vibrational and rotational
quantum numbers at which, according to the Tietz-Hua model, diatomic molecule
dissociates. The corresponding value of v at a given J is the integer closest to the
smaller value of the following expression:
l)th :
- ~ 2 + x/~22 - 4~1~3
1
2al
2
(43)
3.2. The rotatin9 Morse oscillator
In this case the effective internuclear potential is
Vef(R)-
-
L2
-+ D [1 -e-/~(R-R')] 2 .
2p.R 2
(44)
In order to apply the semiclassical quantization to the rotating Morse oscillator, we
use the expansion given by Eq. (16) (with ch = 0) about ~.m = 1 (~m = exp[-fl(R R.)]). After some algebra one obtains
Nv=~(v-bl)
= ~ f1p R d Rm
:
7V~(2~__Cm__X/Z--~m )
(45)
where
are = E M
v,J
- D - AML 2 ,
bm ----2D¢~'. - BML 2 ,
Cm
=
-- D ¢( em* )
2 -~- CM L2
(46)
(47)
(48)
Y.E. Gorbachev et al. IPhysica A 247 (1997) 108-120
(49)
~m = e - # ( R . - R , )
,[ 3(
e~
AM-2~ ~ 1-~-2.
8M--.~R3
1
115
(50)
1-~2- *
2~R,
'
(51)
,
(52)
and
cM - 2U/~R----~, 1 -
where E ~ is the vibrational-rotational energy of the rotating Morse oscillator. The
energy E ~ is obtained from solution of Eq. (45) with respect to a,.. The solution is
am= - (--~N, v
2
b~m~--~m)2
'
(53)
which together with Eq. (46) gives
EvM,j =
D + tZ2AMJ(J +
[~h(v+ l/2)
1)
2DeU'-u*-hZBMJ(J + l) ] 2
2[De2(U,-U.)_ - - ~ + ~]U2 '
-~1-~
(54)
where
Ue=flRe
u.=flR.=ue+BthCd+OthC2,
and
(55)
with
,
1
Bh-~-U--~ and
3
V~= 2(B~) 2 -
3 (B~)2
Ue .
(56)
The expression for the radial orbit of the Morse oscillator is
Rm(~P,n)= R, +/~-1 [6m + In( 1 -
~mcos~pm)],
(57)
where
6m=-ln(-2am~
(58)
\bin)'
and
~m =
1
4amCm ~ 1/2
b2,, /
.
(59)
Subsequently, the mean internuclear distance of the rotating Morse oscillator can be
obtained in the way similar to that leading to the relationship given by Eq. (38),
+,
,60,
116
Y.E. Gorbachev et al. IPhysica A 247 (1997) 108-120
80
~
"
".
c~,(x'~, )
. 2o
\\
lO
%
I
\
I
100
200
300
4()0
Rotational quantum number
s0o
Fig. 1. The v--J contours showing the values of the vibrational quantum number v and the rotational quantum
number J at which molecules C12(XIE+), N2(XIE +) and NO(X2IIr) dissociate: solid lines (Eq. (43) (the rotating Tietz-Hua oscillator)), dashed lines (Eq. (63) - (the rotating Morse oscillator)).
or
Pm=
(RvJ)M -- Re
'
Re
=
Om + ln[(1 + V/1 - e2)/2] + u. - ue
Ue
(61)
Within the framework of the Morse oscillator model, the condition for molecular
dissociation is am = 0, or, according to Eq. (46),
(62)
E M j -- D - A M L 2 = 0 .
The relationships Eqs. (54) and (62) give the v - J contour for the vibrational and
rotational quantum numbers at which diatomic molecule dissociates. The corresponding
value of v at a given J is the greatest integer number smaller than
21r . ,. "0/2
Vm = - ~ (t~t'm)
1
(63)
~ .
Examples of contours (Eqs. (43) and (63)) for the C12(XIE+), N2(X1E +) and
NO(X2IXr) molecules (which represent a broad range of molecular spectroscopic
constants) are shown in Fig. 1.
The maximum values of the rotational quantum numbers Jmax of diatomic molecules
can be found from the v - J contours similar to those shown in Fig. 1, or from solution
of the following set of equations:
V~f(R) = d V ed fR( R ) = 0
and
,
Vef(R
) -- d 2 V e f ( R ) __ O,
dR 2
(64)
Y.E. Gorbachev et al. I Physica A 247 (1997) 108-120
117
which leads to
U'(R) =
L2
~
3L2
tr'(R)=
-3L 2
U " ( R ) - ]./R 4
and
(65)
where U'(R) and U"(R) are the first and the second derivative, respectively, of the
vibrational parts of the corresponding (Tietz-Hua or Morse) potentials with respect to
R. Thus, in order to obtain Jmax one has to solve the equation
3L2
3
~r"(R) + - ~ = U"(R) + -~U'(R) = O.
(66)
K
Numerical solution of Eq. (66) gives Rmax, the internuclear distance corresponding to
J = J,no~. For the rotating Morse oscillator, this distance can also be found from the
following semi-empirical expression:
flRmax ~- fiR e -'[-2 In 2 -- 1 +
(67)
fie e q- 2 In 2 -- 1/2 "
(The accuracy of the approximation (Eq, (67)) is better, for all diatomic molecules,
than 4%.)
Taking the above into account, Jmax can be obtained from Eq. (65) as the greatest
integer number smaller than
1/
Jmax : ~
-1"}-
i
/
I"F-
(68)
The quantum numbers Jmax obtained from Eq. (68) (the exact values) and those
from the contours (Eqs. (43,63)) (the approximate values) are slightly different due
to use of the expansion given by Eq. (16) in derivation of the relationship given
by Eq. (43). The values of ,/max shown in Table 1 are those obtained from
Eq. (68).
Table 1
The molecular spectroscopic constants used in the present calculations: ¢Oe (the vibrational constant, in
cm-1); rOeXe (the anharmonicity constant, in era-l); Re (the molecular bond length, in .~.); Vmax (the
maximum value of the molecular vibrational quantum number): vth,~ax(Eq. (43)) and ~
(Eq. (63)); Jmax
(the maximum value of the molecular rotational quantum number): both Jthax and Jr,~ax are obtained from
Eq. (68)
Molecule
goe
O)eXe
Re
vtmhax
~max
gthax
g~mx
CI2(X 1~ + )
N2(X l ~+ )
NO(X 2IIr)
559.7
2358.6
1904.2
2.67
14.32
14.07
1.987
1.097
1.151
62
66
56
72
68
55
400
266
223
392
265
224
118
Y.E. Gorbachev et aLI Physica A 247 (1997) 108-120
4. Diameters of the excited molecules
The mean kinetic diameter (din) of a diatomic molecule in the electronic ground
state and excited to vth vibrational and Jth rotational level can be given as [3,4]
(dm) "~ (Rv,j) + 2.25,
(69)
where (dm) and ( R u ) are in .~. (In the case of electronically excited molecules, the
relationship given by Eq. (69) should be treated as a crude estimate.)
5. Results and discussion
Typical v- and J-dependence of functions Pth and Pm for several diatomic molecules
are shown in Figs. 2-4. Clearly, the rotational-vibrational excitation of diatomic molecules has a significant impact on the molecular diameters. The diameters increase with
increase of the molecular quantum numbers v and J, and the dependence of the diameters on the vibrational quantum number is stronger than on the rotational quantum
number. Also, the molecular diameters predicted by the Tietz-Hua model are, in
general, larger than those obtained from the Morse model.
As discussed in Paper I, the Tietz-Hua potential is much more realistic model of
intramolecular potential in diatomic molecules than the Morse potential when molecular
rotational and vibrational quantum numbers are not small. Therefore, the diameters of
rotationally and vibrationally excited diatomic molecules predicted by the model of the
3.5
i
2.51-/
i
/
/
,
,
60
lJ
5o
,
v=lO
i~ //
o
,
1O0
,
I
,
200
300
Rotational quantum number
4
500
Fig. 2. Dependenceof the ratios pth(solid lines - Eq. (39)) and Pm (dashedlines - Eq. (61)) on vibrational
and rotationalquantum numbers of the CI2(XI~+) molecule.
Y.E. Gorbachev et al. IPhysica A 247 (1997) 108-120
119
3.5
I
j
80
/
Y
2
1.s
o"
0
/
• -- --
/
'
"
,'
50
,
/
r
X"L.,'
.-"
/I
30
/,'
1,'
l//
s50
-7
1 O0
/'
11
//v=,ot
-'7-150
,
200
Rotational quantum number
,
250
/
300
Fig. 3. Dependence of the ratios pth(solid lines - Eq. (39)) and Pm (dashed lines -Eq. (61)) on vibrational
and rotational quantum numbers of the NO(X2IIr) molecule.
2.5
50 il
;,
k-p,/ ,0:1
:/
X'~I ~° :1
;'/ ~ 1
,'/
,'/
a.1.5
,
.../
,,'/
~,1
0.5
I
50
I
I
i
1O0
150
200
Rotational quantum number
250
300
Fig. 4. Dependence of the ratios Pth(solid lines - Eq. (39)) and Pm (dashed lines - Eq. (61)) on vibrational
and rotational quantum numbers of the N2(X ] E+ ) molecule.
120
Y.E. Gorbachev et al. IPhysica A 247 (1997) 108-120
rotating Tietz-Hua oscillator are more realistic than the diameters obtained from the
model of the rotating Morse oscillator.
Acknowledgements
We thank Alex Dalgarno and Evgueni E. Nikitin for valuable comments. This
work was supported by a Soros Foundation Grant, by a grant from the Intemational Institute for lnterphase Interactions, by the Air Force Office for Scientific Research Grant F-49620-93-1-0373 and Contract 23019B, and by the Phillips Laboratory,
Air Force Material Command, USAF, through the use of the MHPCC under Grant
F-29601-93-2-0001. One of us, FJGV, thanks the Spanish CICYT for financial support
while a postdoctoral researcher at USC.
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