JOURNAL OF APPLIED PHYSICS
VOLUME 84, NUMBER 9
1 NOVEMBER 1998
Statistical–mechanical calculations of thermal properties of diatomic gases
Francisco J. Gordillo-Vázquez
Instituto de Ciencia de Materiales de Madrid (CSIC) Cantoblanco, 28049 Madrid, Spain
Joseph A. Kunca)
Departments of Aerospace Engineering and Physics, University of Southern California, Los Angeles,
California 90089-1191
~Received 6 April 1998; accepted for publication 10 July 1998!
The impact of rotational–vibrational dynamics of molecules on the molecular partition functions,
law of mass action and thermodynamic functions of partially dissociated diatomic gases is
discussed. A group of 11 gases, expected to have their partition functions the most sensitive to the
molecular rotational–vibrational properties, is selected for rigorous and detailed studies, and the
partition functions, dissociation degrees and free energies of the gases are calculated ~using various
models of molecular rotational–vibrational dynamics! and compared in a broad range of
temperature and particle density. © 1998 American Institute of Physics. @S0021-8979~98!00820-2#
I. INTRODUCTION
improved and many of the potentials can now be accurately
calculated in a relatively broad range of the internuclear distance. This allows us to verify the existing models of the
molecular rotational–vibrational dynamics, and subsequently, to perform detailed and rigorous statistical–
mechanical analysis of the thermal properties of many molecular gases. One such analysis was recently done by Liu
et al.1 who investigated in great detail the partition functions
of one molecular gas ~nitrogen! in a broad range of the gas
temperature and density. They found that in most practical
applications where the gas dissociation degree was not large
~that is, when the collisions involving molecules were still
important and the gas temperature was typically below
10 000°), the moderately and highly excited rotational–
vibrational levels of the nitrogen molecule had negligible
impact on the gas partition functions and, consequently, on
the gas dissociation degree and its thermodynamic functions
such as enthalpy, specific heats, etc.
In this paper we study the rotational–vibrational properties of several dozens of partially dissociated diatomic gases
close to local thermal equilibrium, and select a final group of
11 gases which are expected to have strong dependence of
their partition functions on the molecular rotational–
vibrational motion. ~The thermal properties of gases close to
local thermal equilibrium are of importance in many areas of
plasma processing, combustion, astrophysics, and atmospheric and environmental studies.! These 11 gases are studied in great detail, using statistical mechanics, in order to
determine the impact of the molecular rotational–vibrational
dynamics on the gas composition and thermodynamic functions. The impact is demonstrated by calculating and comparing the dissociation degrees and free energies of the gases
in a broad range of temperature and particle density, assuming three different models of the molecular rotational–
vibrational dynamics: ~1! the rotating harmonic oscillator
~the simplest model!, ~2! the rotating Morse oscillator ~the
most popular model!, and ~3! the rotating Tietz–Hua
oscillator2–5 ~the most accurate model!.
The law of mass action and thermodynamic functions of
partially dissociated gases is defined in terms of the translational, electronic, rotational, vibrational and nuclear spin partition functions, which are associated with the translational,
electronic, rotational, vibrational and nuclear spin properties,
respectively, of the gas molecules. Calculations of the atomic
and molecular translational and nuclear spin partition functions are straightforward and accurate. The electronic partition functions of most atoms and atomic ions common in gas
applications can also be calculated with a high degree of
accuracy since reliable atomic electronic levels and their degeneracies are available in the literature. The situation is
more complicated in the case of the molecular electronic
partition functions because energies of higher electronic levels of many molecules are either not known or their accuracy
is poor. However, the temperatures of the molecular gases
which are partially dissociated are usually well below
10 000° so that only a very small number of the electronic
states typically have to be considered in calculations of the
molecular electronic partition functions.
Calculation of the molecular rotational–vibrational partition functions is the most difficult aspect of the applied
statistical mechanics of gases. The functions depend on the
assumed model of the molecular rotational and vibrational
motion, which in turn depend on the molecular internuclear
potentials. In the past, the available ab initio and Rydberg–
Klein–Rees ~RKR! intramolecular potentials were often unreliable ~especially in excited molecules!, and as a result, it
was frequently impossible to determine the accuracy of the
existing models of the molecular rotational–vibrational motion. Therefore, reliable studies of the dependence of the partition functions on the model of the molecular rotational–
vibrational motion were not available in the past. In recent
years, the computational and experimental methods of investigating the molecular intramolecular potentials have greatly
a!
Electronic mail: [email protected]
0021-8979/98/84(9)/4693/11/$15.00
4693
© 1998 American Institute of Physics
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4694
J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
F. J. Gordillo-Vázquez and J. A. Kunc
The present work consists of two parts. In the first part
we study molecular properties of several dozens of diatomic
molecules and select a group of eleven diatomic gases (H2,
Li2, Na2, K2, Cs2, I2, NO, ICl, KI, NaCs and HgI! which
represent a very broad range of molecular spectroscopic constants and which are expected to be the most sensitive to the
model of molecular rotational–vibrational dynamics. In the
second part of the work, we calculate ~using the rigorous
formalism of Refs. 6 and 7! and compare the total partition
functions, dissociation degrees and free energies of the 11
gases selected for final considerations.
II. ROTATIONAL–VIBRATIONAL EIGENVALUES
The spectroscopic constants of the considered molecules
were taken from Refs. 8 and 9. The RKR data for the intramolecular potentials were taken from Ref. 3 (I2, Li2, Na2,
K2, Cs2 and H2), and Refs. 10 and 11 ~NO!, Refs. 3 and 12
~ICl!, Ref. 13 ~HgI! and Ref. 14 ~NaCs!. The ground electronic state of the KI molecule has ionic bond,15 while some
of the upper electronic states of the molecule have covalent
bonds.16 Due to lack of RKR and ab initio data on the intramolecular potential of the ground electronic state of the KI
molecule ~the potential is known only in the region of its
minimum17,18!, we approximated the Tietz–Hua and Morse
intramolecular potentials of the molecule by a truncated Rittner potential19 ~the latter potential is one of the most accurate
model potentials for the ionic bonds of the alkali halides
molecules20!. In the case of the two upper electronic states of
ICl, we approximated the bottom part of each of the potentials by the potential of the harmonic oscillator; such a procedure allows one to obtain the vibrational constant v e , and
~through the Birge–Sponer relationship! the molecular anharmonicity constant v e x e ~both constants are not available
in the literature!.
The effective internuclear potentials of the considered
molecules in the ith electronic state can be written as
V ~efi ! ~ R ! 5U ~Vi ! ~ R ! 1U ~Ji ! ~ R ! ,
~1!
where R is the internuclear distance, U (i)
V (R) is the vibrational part of the potential and U (i)
(R)
is
the rotational part
J
of the potential. The vibrational part is:
~a! harmonic oscillator:
U hV~ i ! ~ R ! 5 21 k ~ i ! ~ R2R ~ei ! ! 2 ,
~2!
D (i) is the well depth of the potential, R (i)
e is the molecular
bond length, b (i) is the Morse constant and c (i)
h is the Tietz–
Hua parameter. ~Properties of the Tietz–Hua oscillator are
discussed in detail in Refs. 2–5.!
The rotational ~centrifugal! part of the molecular internuclear potential in each considered molecular model is
taken as
U ~Ji ! ~ R ! 5
~i!
2b
~i!
Um
V ~ R ! 5D ~ 12e
~ i ! ~ R2R ~ i ! ! 2
e
~3!
! ,
and
~c! Tietz–Hua oscillator:
~i!
~i!
U th
V ~ R ! 5D
S
~i!
12e 2b h
~i!
~ R2R e !
~ i ! 2b ~hi ! ~ R2R ~ei ! !
E hv ~,Ji ! 5 v ~ei ! ~ v 1 21 ! 1B ~ei ! J ~ J11 ! ,
1
1 2
~i!
~i!
~i!
Em
v ,J 5 v e ~ v 1 2 ! 2 v e x e ~ v 1 2 !
1B ~ei ! J ~ 11J ! 2D ~ei ! J 2 ~ 11J ! 2 ,
12c h e
D
E ~vi,J! th5E ~f i ! l ~viJ! ,
~5!
~9!
where
l ~viJ! 5 21 ~ t ~ i ! ! 2 ~ A ~0i ! 1A ~2i ! !
2 41 ~ r cJ 1 v̄ ! 2 2 41 ~ r cJ 1 v̄ ! 22 ~ t ~ i ! ! 4 ~ A ~2i ! 2A ~0i ! ! 2 ,
A ~0i ! 511
13
B ~ei ! J ~ J11 !
D~i!
~ 12c ~hi ! ! 2
F
~ c ~hi ! ! 2 12c ~hi ! 23
11
~11!
D~i!
DG
S
2c ~hi ! 14
~ c ~hi ! ! 2 21
b ~hi ! R ~ei !
~12!
,
B ~ei ! J ~ J11 !
D~i!
3 ~ c ~hi ! ! 2 22c ~hi ! 21
b ~hi ! R ~ei !
b ~hi ! R ~ei !
,
B ~ei ! J ~ J11 !
~ b ~hi ! ! 2 ~ R ~ei ! ! 2
F
~10!
11
~ 12c ~hi ! ! 2
A ~2i ! 5 ~ P ~ i ! ! 2
1
S
D
~ b ~hi ! ! 2 ~ R ~ei ! ! 2
A ~1i ! 52 P ~ i ! 21
where
b ~hi ! 5 b ~ i ! ~ 12c ~hi ! ! ,
~8!
and an approximate solution of the Schrödinger equation for
the rotating Tietz–Hua oscillator is3
2
~4!
~7!
where v is the molecular vibrational quantum number, and
(i)
v (i)
e and B e are the vibrational constant and the first rotational constant, respectively, of the molecular ith electronic
state.
The molecular rotational–vibrational energies resulting
from an approximate series solution of the Schrödinger equation for the rotating Morse oscillator are:21
16
,
~6!
where m is the reduced mass of the molecule, L
5\ AJ(J11) is the angular momentum of the rotating molecule, and J is the molecular rotational quantum number.
An exact solution of the Schrödinger equation for the
potential, Eq. ~2!, gives the following rotational–vibrational
energies of the rotating harmonic oscillator:
(i)
where k is the molecular force constant;
~b! Morse oscillator:
L2
,
2mR2
S
13
~ c ~hi ! ! 2
~ 12c ~hi ! ! 2
~ b ~hi ! ! 2 ~ R ~ei ! ! 2
DG
,
~13!
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J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
r ~cJi ! 5sign~ c ~hi ! ! r ~Ji ! ,
F. J. Gordillo-Vázquez and J. A. Kunc
i!
r ~ci~!J50 ! 5sign~ c ~hi ! ! r ~J50
,
r ~Ji ! 5 @ 41 1 ~ t ~ i ! ! 2 ~ A ~0i ! 1A ~1i ! 1A ~2i ! !# 1/2,
1
v̄ 5 v 1 ,
2
E ~f i ! 5
~ 2 m D ~ i ! ! 1/2
t ~ i !5
,
\b h
\ 2 ~ b ~hi ! ! 2
2m
B ~ei ! 5
,
P ~ i !5
1
c ~hi !
~a! homonuclear gas:
~14!
ŝ5c X ŝ X 1c X 2 ŝ X 2 ,
2 m ~ R ~ei ! ! 2
ê5c X ê X 1c X 2 ê X 2 ,
~19!
f̂ 5c X f̂ X 1c X 2 f̂ X 2 ,
,
~b! heteronuclear gas:
~15!
\2
4695
ŝ5c X8 ŝ X 1c 8Y ŝ Y 1c XY ŝ XY ,
,
(i)
and where v e x (i)
e and D e are the anharmonic constant and
the second rotational constant, respectively, of the molecular
ith electronic state.
Equation ~8! is a truncated series expansion solution of
the Schrödinger equation for the rotating Morse oscillator.21
Even though more accurate ~higher-order! expansions of the
solution have been discussed in the literature, the expression,
Eq. ~8!, remains the most common in applications because of
its simplicity, high accuracy for low rotational–vibrational
levels, and because of a lack of reliable spectroscopic constants associated with the higher terms in the higher-order
expansions. In molecules with moderate and large values of
the rotational and vibrational quantum numbers, the energy
equation ~8! becomes inaccurate. Then, much more reliable
values of the higher rotational–vibrational levels can be obtained from Eq. ~9! ~see discussions in Refs. 2 and 3!.
ê5c X8 ê X 1c 8Y ê Y 1c XY ê XY ,
~20!
f̂ 5c X8 f̂ X 1c 8Y f̂ Y 1c XY f̂ XY ,
where c X , c X8 , c X 2 , c 8Y and c XY are the corresponding mass
fractions of the gas,
c X5
NX aNA
5
5a,
NA
NA
c X25
2N X 2
NA
~21!
2N A ~ 12 a ! /2
5
512 a ,
NA
and
c X8 5c 8Y 5
c XY 5
N X8
N A8
2N XY
N A8
5
5
N 8Y
N A8
5
a N A8 /2 a
5 ,
2
N A8
2N A8 ~ 12 a ! /2
N A8
~22!
512 a ,
where
III. THERMODYNAMIC FUNCTIONS
Thermodynamic functions ~enthalpy, specific heats, etc.!
of each gas component of a gas mixture containing atoms X,
Y and molecules XY ~or atoms X and molecules X2 in the
case of a homonuclear diatomic gas! in a partially dissociated gas can be obtained from the component’s four primary
thermodynamic parameters ~temperature T, pressure p, volume V and entropy S) and its two primary thermodynamic
potentials ~internal energy E and free energy F). The entropy, internal energy, and free energy of the sth gas component of the mixture can be given, respectively, as
S D
S
D
S
D
S s ~ T ! 5N s kT ln
Q s~ T !
Ns
d ln Q s ~ T !
1N s kT
d T
E s ~ T ! 5N s kT 2
and
d ln Q s ~ T !
d T
and
N A8 52N XY 1N X8 1N 8Y ,
~23!
with a and N A ~or N A8 ) being the dissociation degree and the
~constant! total number of atoms ~both free and bound in
molecules! in the considered gas, respectively.
IV. PARTITION FUNCTIONS
A. Atomic partition functions
1N s kT,
~16!
,
~17!
V
V
FS D G
F s ~ T ! 52N s kT ln
N A 52N X 2 1N X
Q s~ T !
11 ,
Ns
~18!
where the derivatives are taken at constant volume V, N s is
the molecular ~atomic! density of the sth gas component, and
Q s is the total partition function of the molecules ~atoms! of
the sth component.
The specific entropy ŝ, specific internal energy ê, and
specific free energy f̂ of a partially dissociated diatomic gas
can be given as:
The atomic partition functions are calculated in the way
discussed in Refs. 6 and 7. The total atomic partition function Z is the product of the atomic translational partition
function Z tr , the atomic electronic partition function Z el , and
the atomic nuclear-spin partition function Z (l)
n ~for the lth
atomic isotope!
Z5Z trZ elZ ~nl !
S
~ 2 p mkT ! 3/2V
5
h3
DF (
i5i m
i51
S DG
g i exp 2
ei
kT
~ 2I ~nl ! 11 ! ,
~24!
where V is the volume of the gas, h is Planck’s constant, k is
the Boltzmann constant, m is the atomic mass, e i and g i are
the energy and electronic degeneracy, respectively, of the ith
atomic level, i m is the number of the atomic levels, and I (l)
n is
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4696
J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
F. J. Gordillo-Vázquez and J. A. Kunc
TABLE I. The spectroscopic constants and other parameters of the electronic states of the ICl molecule used in
the present work.
Molecule
a
v TH
max
M a
v max
a
J TH
max
M a
J max
c hb
b hb
Dc
m d/10223
y me
e
v ef
v ex eg
B eh
D eh
R ei
bj
DU THk
DU M k
E max /Dl
T em
a thn
b thn
c thn
In ( 127I) o
In ( 35Cl) o
ICl(X 1 S 1
g )
ICl(A 3 P 1 )
ICl(A 8 3 P 2 )
ICl(B 3 P 0 1 )
ICl(B 8 O 1 )
86
60
545
331
20.086 212
2.008 578
17 557
4.552 37
4.291 713
384.27
1.49
0.114 157
4.029 9031028
2.3209
1.849 159
1.97
5.68
0.997
0
86.034
21.070331021
21.008831024
5/2
3/2
31
23
320
179
20.167 208
2.542 557
3814.7
4.552 37
5.848 799
211.0
2.12
0.085 29
5.574 2831028
2.6850
2.178 324
0.76
8.42
0.993
13 742
31.238
28.575831022
24.189531025
5/2
3/2
38
28
350
198
20.157 361
2.373 450
4875
4.552 37
5.465 235
224.57
1.882
0.086 53
5.138 7331028
2.6650
2.050 745
0.98
7.73
0.990
12 682
38.268
29.104831022
25.627531025
5/2
3/2
5
5
215
79
20.522 091
7.123 749
907
4.552 37
12.449 436
221.1
9.62
0.0872
5.425431028
2.660
4.680 239
8.59
13.97
0.990
17 381
4.8224
22.907431022
2.544131025
5/2
3/2
7
9
95
55
20.279 25
2.287 209
176
4.552 37
6.684 355
37.2663
1.9656
0.044 919 8
2.610631027
3.7386
1.787 93
3.39
6.74
0.990
18 152
7.0816
27.677631022
4.350131025
5/2
3/2
a TH
v max ,
M
M
and J TH
v max
max , J max are the maximum vibrational and rotational quantum numbers for the rotating
Tietz–Hua and Morse oscillators, respectively.
b
c h and b h ~in Å 21 ) are the parameters in the potential, Eq. ~4!.
c
D is the well depth of the intramolecular potential ~in cm21).
d
m is the reduced mass of the molecule ~in g!.
e m
y e 5 b (i) R (i)
e .
f
v e is the molecular vibrational constant ~in cm21).
g
v e x e is the anharmonicity constant ~in cm21).
h
B e and D e are the rotational constants ~in cm21).
i
R e is the molecular bond length ~in Å!.
j
b is the Morse constant ~in Å 21 ).
k
DU i 5 u U i 2U u /D ~in %! is the average deviation of the intramolecular potential U i from the corresponding
‘exact’ potential U, where U i is the Tietz–Hua potential ~denoted by the subscript TH! or the Morse potential
~denoted by the subscript M ), and U is the potential obtained from either ab initio or RKR data ~see Refs. 25,
26, 27 and 12!.
l
E max is the highest molecular energy assumed in the calculations of the parameter c h .
m
T e is the molecular electronic energy ~in cm21).
n
a th , b th , c th are the coefficients in Eq. ~39!.
o
In ( 127I) and In ( 35Cl) are the spin quantum numbers of the molecular nuclei.
the spin quantum number of the atomic nucleus of the isotope ~values of I (l)
n used in this paper are listed in Tables
I–IV!.
B. Molecular partition functions
The translational partition function Q tr is given by an
expression similar to that for the partition function Z tr @see
relationship ~24!# with m now replaced by the molecular
mass (2m or m X 1m Y ).
The molecular internal partition function is given by
1. Rotating Morse and Tietz – Hua oscillators
i!
The molecular partition functions are calculated using
the same general formalism as that discussed in Refs. 6 and
7.
The total molecular partition function Q tot can be written
as
Q tot5Q trQ in ,
~25!
where Q tr and Q in are the molecular translational and internal
partition functions, respectively.
1
Q int5
s
i,J !
~
~
i m J max v max
(( (
i51 J50 v 50
S
3exp 2
g i ~ 2I ~nl ! 11 !~ 2I ~nk ! 11 !~ 2J11 !
D
hc ~ i, v ,J ! ~ i51,v 50,J50 !
2e
! ,
~e
kT
~26!
where e (i, v ,J) is the energy of the (i, v ,J)th molecular level
~in cm21), c is the speed of light, s is a symmetry factor
~equal to two for homonuclear and equal to one for hetero-
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J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
F. J. Gordillo-Vázquez and J. A. Kunc
4697
TABLE II. The spectroscopic constants and other parameters of the electronic states of the NO molecule used in the present work. The internuclear potential
curves were taken from ab initio calculations reported in Refs. 10 and 11. The meaning of the symbols is the same as in Table I.
Molecule
v TH
max
M
v max
TH
J max
M
J max
ch
bh
D
m /10223
y me
ve
v ex e
Be
De
Re
b
DU TH
DU M
E max /D
Te
a th
b th
c th
In ( 14N)
In ( 17O)
NO(X 2 P r )
NO(a 4 P i )
NO(B 2 P r )
NO(L 8 2 f )
NO(b 4 S 2 )
NO(1 2 S 1 )
58
40
228
133
0.013 727
2.715 59
53 341
1.249
3.169 163
1904.2
14.07
1.672
5.15631026
1.151
2.7534
6.19
7.22
0.9999
0
58.121
21.592131021
24.412031024
1
5/2
35
25
158
91
0.008 200 3
2.408 413
16 361
1.249
3.523 501
930
11
1.071
5.681431026
1.451
2.428 326
3.87
3.95
0.990
35 661
35.470
21.211931021
26.538331024
1
5/2
28
27
198
107
20.482 743
3.426 50
22 722
1.249
3.299 998
1043
7.70
1.106
4.974531026
1.428
2.310 923
8.90
11.16
0.990
43 414
28.166
21.204531021
21.210931024
1
5/2
29
31
148
85
20.073 021
2.737 96
14 501
1.249
3.702 436
920
14.59
1.116
6.568631026
1.451
2.551 645
8.30
11.38
0.990
52 281
28.977
21.192631021
25.167631024
1
5/2
33
26
158
95
20.085 078
3.015 38
21 183
1.249
3.662 665
1211
15.00
1.298
5.964731026
1.318
2.778 957
4.36
6.67
0.990
45 357
33.050
21.334931021
24.085931024
1
5/2
10
25
188
81
20.959 787
8.156 35
24 510
1.249
4.426 097
1950.89
38.8194
1.9817
8.179 1531026
1.063 49
4.161 861
4.96
11.93
0.990
45 137
10.441
26.726531022
5.642531025
1
5/2
nuclear molecules!, i m is the number of the molecular elec(i,J)
tronic levels, J (i)
max and v max are the maximum values of the
accessible rotational and vibrational quantum numbers, re(k)
spectively, and I (l)
n and I n are the spin quantum numbers of
the molecular nuclei. ~The quantum number J (i)
max for a given
TABLE III. The spectroscopic constants and other parameters of the
H2(X 1 S 1
g ) molecule used in the present work. The RKR potential for the
molecule was taken from Ref. 3. The meaning of the symbols is the same as
in Table I.
Molecule
v TH
max
M
v max
J TH
max
M
J max
ch
bh
D
m /10223
y me
ve
v ex e
Be
De
Re
b
DU TH
DU M
E max /D
Te
a th
b th
c th
In ( 1 H)
H2(X
1
S1
g )
21
15
39
22
0.170 066
1.613 939
38 318
0.083 69
1.441 848
4403.21
121.33
60.853
0.0465
0.74144
1.944 660
2.65
5.76
0.9922
0
20.785
23.266331021
25.832931023
1/2
electronic state is limited by the value of the molecular dissociation energy for the state.! The energies e (i, v ,J) are given
in Eqs. ~7! ~the rotating harmonic oscillator!, ~8! ~the rotating
Morse oscillator! and ~9! ~the rotating Tietz–Hua oscillator!.
TABLE IV. The spectroscopic constants and other parameters of the
KI(X 1 S 1
g ) molecule used in the present work. The intramolecular potential
of the ground electronic state of the molecule is approximated by a truncated
Rittner potential ~see Ref. 19!. The meaning of the symbols is the same as in
Table I.
Molecule
KI(X 1 S 1
g )
v TH
max
J TH
max
ch
bh
D
m /10223
y me
ve
v ex e
Be
De
Re
364
905
0.246 642
0.570 852 5
26 790
4.950 18
2.309 454
186.53
0.574
0.060 874 7
2.593431028
3.0478
0.757 744 76
7.90
20.57
0.9900
0
36.463
21.9439
5.503831023
26.974631026
3.066531029
3/2
5/2
b
DU TH
DU M
E max /D
Te
a th
b th
c th
d th
e th
In ( 39K)
In ( 127I)
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4698
J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
F. J. Gordillo-Vázquez and J. A. Kunc
From Eqs. ~8!, ~27! and ~28!, one has
In order to obtain values of the quantum numbers v (i,J)
max
and J (i)
max for the rotating Morse oscillator, we use the obvious
relationship,
~i!
~i!
Em
v ,J 5D J ,
~27!
a ~1i ! x 21 1 a ~2i ! x 1 1 a ~3i ! 50,
where
a ~1i ! 52 v e x ~ei ! ,
where
*
~28!
is the internuclear distance of the
and where R (i)
J
rotationally–vibrationally excited molecule, at which the
molecule dissociates, and R (i) is the equilibrium internuclear
*
distance of the rotationally excited molecule. Subsequently,
(i)
the distance R J is found from the numerical solution of the
following equation:
~i!
dV m
eff
dR
R5R ~Ji !
52D ~ i ! b ~ i ! $ exp@ 2 b ~ i ! ~ R ~Ji ! 2R ~ei ! !#
2exp@ 22 b ~ i ! ~ R ~Ji ! 2R ~ei ! !# %
L2
2
m ~ R ~Ji ! ! 3
50.
~29!
An approximate analytical solution of Eq. ~29! can be
obtained using the assumption ~valid in many diatomic mol(i)
ecules! that b (i) (R (i)
J 2R e ).3. Then, the second exponential function in Eq. ~29! can be neglected and the equation
reduces to:
~ h ~ i ! ! 3 exp~ 2 h ~ i ! ! 5C ~Ji ! ,
where h 5 b
(i)
(i)
R (i)
J
~i! 2
~30!
and
~b ! L
exp~ 2 b ~ i ! R ~ei ! ! .
2mD~i!
C ~Ji ! 5
2
~31!
Relationship ~30! can be rewritten as
~ h ~ i ! ! s ~ h ~ i ! ! p exp~ 2 h ~ i ! ! 5C ~Ji ! ,
~32!
where s532 p. In the range of the constants b (i) and R (i)
e
considered in this paper, the values of the expression
( h (i) ) p exp(2h(i)) are close to P d '1700. Subsequently, one
can write
R ~Ji ! 5
S D
Pd
1
~i!
b
C ~Ji !
2/9
'
5.22
b ~ C ~Ji ! ! 2/9
~i!
~33!
.
The equilibrium position of the rotationally excited molecule is given by2
R ~ i !5
*
1
@ y m ~ i ! 1B ~mi ! C ~mi ! 1D ~mi ! ~ C ~mi ! ! 2 # ,
b~i! e
D ~mi ! 5
3 ~ B ~mi ! ! 2
2
2
B ~mi ! 5
i,J !
5
v ~max
~i! 3
!
~ym
e
,
C ~mi ! 5
b ~ i !L 2
.
2mD~i!
2 a ~2i ! 1 Aa ~2i ! 2 24 a ~1i ! a ~3i !
2 a ~1i !
~35!
1
2 ,
2
~38!
which allows one to derive the values for v (i,J)
max ~as a function
(i)
(i,J
)
max
of J) and J (i)
max ~for v max 50), respectively. @Equation ~38!
has poor accuracy in the case of the KI molecule because the
truncated Rittner19 ionic bond potential cannot be approximated very accurately ~see Table IV! by the Morse function#.
(i)th
The values of v (i,J)th
max and J max of the rotating Tietz–Hua
oscillator were obtained from the numerical solution of the
molecular Schrödinger equation22 ~see Tables I–IV!. For
practical reasons, these values can be approximated by the
following polynomials:
i,J ! th
5a th1b thJ1c thJ 2 ,
v ~max
~NO, ICl, H2! molecules,
~39!
and
i,J ! th
5a th1b thJ1c thJ 2 1d thJ 3 1e thJ 4 ,
v ~max
~KI! molecule,
~40!
where the coefficients a th , b th , c th and d th , e th are given in
Tables I–IV.
2. Rotating harmonic oscillator
The total molecular partition function of the rotating harmonic oscillator can be written as23
i !h ~ i !h h
Q htot5Q htrQ helQ ~vib
Q rot Q n ,
(Q htr)
~41!
(Q hel)
and electronic
partition
where the translational
functions are calculated in the usual way. The nuclear spin
partition function Q hn is
Q hn 5 ~ 2I ~nl ! 11 !~ 2I ~nk ! 11 ! ,
~42!
and the rotational partition function of the oscillator is
Q ~roti ! h 5
.
1
3 ~ B ~mi ! ! 2
~i! ,
ym
e
The solution of Eq. ~36! gives the ( v ,J) contour for the
rotating Morse oscillator,
`
~34!
with
~i!
ym
5 b ~ i ! R ~ei ! ,
e
~37!
x 1 5 ~ v 1 12 ! .
with
S D
a ~2i ! 5 v ~ei ! ,
a ~3i ! 5B ~ei ! J ~ J11 ! 2D ~ei ! @ J ~ J11 !# 2 2D ~Ji ! ,
m~ i !
~i!
~i!
~i!
D ~Ji ! 5V m
eff ~ R5R J ! 2V eff ~ R5R !
i!
D ~J50
5D ~ i ! ,
~36!
( ~ 2J11 ! exp
J50
1 kT
,
s hcB ~ei !
S
2J ~ J11 !
hcB ~ei !
kT
D
~43!
where the last approximation should not be applied to hydrogen gas at temperatures below 900 K because the rotational
temperature of the gas is 88 K.
The vibrational partition function of the oscillator is
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J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
F. J. Gordillo-Vázquez and J. A. Kunc
4699
TABLE V. Constant values of n d ~in cm23) for several homonuclear gases.
Gas
H2
7.498 94310
nd
`
i !h
Q ~vib
5
(
v 50
Cs2
~i!
e v hc v e
/kT
23
10
1
5
~i!
12e 2hc v e
/kT
K2
22
Na2
1.412 53310
22
2.371 37310
3.311 31310
S D
a2
m 3/2 kT
5
12 a 2n A 2 p \ 2
3/2
3
3
~ Z ~nl ! 2 ! X ~ Z 2el! X
~ Q int! X 2
e
2D ~ i51 ! /kT
a5
NX
,
NA
~46!
~ Z ~nl ! ! X ~ Z ~nl ! ! Y ~ Z el! X ~ Z el! Y
~ Q int! XY
e 2D
~ i51 ! /kT
,
where n A8 5N A8 /V,
,
where n A 5N A /V is the particle density of the X atoms, N A
is the total number of X atoms ~both free and bound in molecules! in the gas, m is the reduced mass of the molecule X2,
Q int is the molecular internal partition function, D (i51) is the
dissociation energy of the X2 molecule in the ground
electronic–rotational–vibrational state and the partition
functions (Z) X and (Q tot)X2 are given by Eqs. ~24! and ~25!,
respectively. The gas dissociation degree is defined by
3/2
~50!
N X8 5 a N TX 5 a
~45!
3.715 3531023
S D
V. DISSOCIATION DEGREE
In the case of diatomic gases, the law of mass action
reduces to:
~a! homonuclear gas:
I2
22
a2
m 3/2 kT
5
4 ~ 12 a ! 2n A8 2 p \ 2
~44!
.
Li2
22
N A8
2
and
N 8Y 5 a N TY 5 a
N A8
2
~51!
,
N A8 5N TX 1N TY 5N X8 1N 8Y 12N XY 5 a N A8 12N XY ,
N XY 5
N A8 ~ 12 a !
2
~52!
~53!
,
and N TX and N TY are total numbers of the X and Y atoms ~free
and bound in molecules!, respectively.
An approximate expression, similar to Eq. ~48!, for the
dissociation degree of an heteronuclear gas is
n 8d
a2
~ i51 ! /kT
5 e 2D
,
4 ~ 12 a ! n A8
N A 52N X 2 1N X 5const
and
~54!
with
N X25
N A ~ 12 a !
,
2
~47!
where N X and N X 2 are numbers of atoms X and molecules
X2 in the gas, respectively,
A simple approximate expression for a can be obtained
from
a2
nd
~ i51 ! /kT
5 e 2D
,
12 a n A
~48!
where
n d5
S D
m 3/2 kT
2 2p\2
3/2
3
~ Z ~nl ! 2 ! X ~ Z 2el! X
~ Q int! X 2
~49!
is a weak function of the gas temperature, so often it can be
assumed to be constant in practical applications. The constant values of n d for several gases are given in Table V.
~b! heteronuclear gas:
The mass action law for mixture of X and Y atoms and
XY molecules leads to:
n 8d 5
S D
m 3/2 kT
2 2p\2
3/2
3
~ Z ~nl ! ! X ~ Z ~nl ! ! Y ~ Z el! X ~ Z el! Y
~ Q int! XY
,
~55!
which is also a weak function of temperature. Approximate
~constant! values of n 8d for several gases are shown in Table
VI.
VI. FREE ENERGY
According to expressions ~18!, ~19! and ~20!, the specific
free energy ~that is, the value of the free energy per unit
volume! of a diatomic gas is:
~a! homonuclear gas:
TABLE VI. Constant values of n 8d ~in cm23) for several heteronuclear
gases.
Gas
NaCs
HgI
ICl
NO
n d8
8.709 6331021
4.216 9631022
3.388 4431023
4.508 1631024
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4700
J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
F. J. Gordillo-Vázquez and J. A. Kunc
FIG. 1. The internal partition function, Q int , of the NO molecule represented by the rotating Tietz–Hua oscillator ~solid line!, the rotating Morse
oscillator ~dashed line! and the rotating harmonic oscillator ~dashed-dot
line!. The dotted line gives the values of the total density n A8 ~in cm23) ~of
atoms which are free and bound in molecules! at which the gas is almost
fully dissociated ( a 50.95) at a given temperature T.
H F S D GJ
H FS
where
Zi
N 8i
ZX
f̂ 5 a 2R X T ln
11
NX
1 ~ 12 a ! 2R X 2 T ln
FIG. 2. The internal partition function, Q int , of the ICl molecule. The meaning of the symbols is the same as in Fig. 1.
~ Q tot! X 2
N X2
D GJ
11
and
,
~56!
5
~ Z elZ n Z tr! i
N 8i
S D
2 ~ Z n Z el! i m i kT
5
anA
2p\2
F
~ Q tot! XY 2 ~ Q int! XY ~ m X 1m Y ! kT
5
N XY
2p\2
~ 12 a ! n A8
G
3/2
~ i[X,Y ! ,
~62!
3/2
.
~63!
with
VII. RESULTS AND DISCUSSION
k
R X5
,
mX
and
R X25
k
,
m X2
~57!
where R X and R X 2 are the gas constants for the atomic and
molecular components, respectively,
S D
Z X ~ Z elZ n Z tr! X ~ Z n Z el! X m X kT
5
5
NX
NX
anA
2p\2
and
~ Q tot! X 2
N X2
5
2 ~ Q int! X 2
~ 12 a ! n A
S D
m X 2 kT
2p\2
3/2
,
~58!
3/2
The main conclusion of this work is that the internal
partition functions of partially dissociated diatomic gases are
weakly dependent on the molecular model of the rotational–
vibrational motion, and that the model of the rotating harmonic oscillator ~with the highest rotational–vibrational
level being close to the molecular dissociation energy! is
quite appropriate to use in statistical–mechanical studies of
the gases close to local thermal equilibrium. This can be seen
in Figs. 1–4 where the internal partition functions of several
discussed molecules ~representing a broad range of
rotational–vibrational properties! are shown together with
the values of the particle density and temperature at which
the gases are almost completely dissociated. ~One should re-
~59!
;
~b! heteronuclear gas:
f̂ 5
H F S D GJ H F S D GJ
H F S D GJ
a
ZX
2R X T ln
11
2
N X8
1 ~ 12 a ! 2R XY T ln
1
a
ZY
2R Y T ln
11
2
N 8Y
~ Q tot! XY
11
N XY
,
~60!
with
R X5
k
,
mX
RY5
k
,
mY
and
R XY 5
k
,
m XY
~61!
FIG. 3. The internal partition function, Q int , of the H2 molecule. The meaning of the symbols is the same as in Fig. 1, except for n A8 which is now
replaced by n A .
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J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
FIG. 4. The internal partition function, Q int , of the KI molecule. The meaning of the symbols is the same as in Fig. 1.
member when viewing Figs. 1–4 that some of the discussed
substances may not be in the gaseous phase at certain temperatures and densities.!
It seems that the above conclusion also includes gases of
diatomic molecules with ionic bonds even though the calculations of the internal partition functions of such molecules
have an accuracy which is expected to be worse than the
accuracy of calculations similar to the diatomic molecules
with covalent bonds. This is because the approximation of
the Rittner potential by the rotating Morse oscillator is more
of a crude necessity than an accurate physical picture of the
intramolecular dynamics of the molecules with ionic bonds.
The substantial difference between DU TH and DU M ~see
Table IV and discussions in Refs. 2–3! clearly indicates that
the Tietz–Hua potential is a much better approximation to
the truncated Rittner potential than the Morse potential.
However, the accuracy of the replacement of the Rittner potential of the KI molecule by the Tietz–Hua potential cannot
be precisely established in the entire range of the interaction
distance because neither ab initio nor RKR data for the intramolecular potentials of the ionic-bond molecules exist in
the literature. ~This is the reason why the partition function
of the KI molecule treated as the rotating Morse oscillator is
not shown in Fig. 4.! The calculated partition functions of
the ionic molecules discussed24 ~KI, KBr, CsCl, CsI, etc.!
represented by the rotating Tietz–Hua potential are close to
FIG. 5. The dissociation degree a of the NO gas where molecules are
represented by the rotating Tietz–Hua oscillator ~solid line!, the rotating
Morse oscillator ~dashed line! and the rotating harmonic oscillator ~dasheddot line!. n 8A and T are particle density and temperature, respectively.
F. J. Gordillo-Vázquez and J. A. Kunc
4701
FIG. 6. The dissociation degree a of the ICl gas. The meaning of the
symbols is the same as in Fig. 5.
the molecule partition functions obtained here from the
model of the rotating harmonic oscillator. This is in support
of our inclusion of the molecules with ionic bonds in our
statement that the impact of the model of the molecular
rotational–vibrational dynamics on the partition functions of
partially dissociated diatomic gases is small ~less than 10%!
and that, for practical purposes, the partition functions in
such gases can be calculated using the rotating harmonic
oscillator as the model of molecular rotational–vibrational
motion.
The usefulness of the rotating harmonic oscillator for
statistical–mechanical studies of thermodynamic properties
of partially dissociated diatomic gases results from the relationship between the molecular anharmonicity and the molecular vibrational eigenvalues. At low vibrational quantum
numbers, the impact of the anharmonicity can be neglected
in all common models of molecular vibrational dynamics.
However, at moderate and high vibrational quantum numbers the vibrational eigenvalues obtained from the ~different!
models differ from one another. In addition, all reasonable
models of the anharmonic oscillators ~such as the Morse and
Tietz–Hua oscillators! have more vibrational levels between
the molecular vibrational ground state and its dissociation
continuum than the corresponding harmonic oscillators.
~This is caused by the fact that the molecular anharmonicity
always decreases vibrational energy of the molecule!. Thus,
the density of the moderately and highly excited vibrational
states is higher in an anharmonic oscillator than in the cor-
FIG. 7. The dissociation degree a of the H2 gas. The meaning of the symbols is the same as in Fig. 5, except for n 8A which is now replaced by n A .
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4702
J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
FIG. 8. The temperature dependence of the particle densities n d ~in cm23) in
H2, I2, Li2, Na2, K2 and Cs2 gases where molecules are represented by the
rotating Tietz–Hua oscillator.
responding harmonic oscillator. Therefore, molecular anharmonicity increases the reliability of the assumption that the
molecular states with moderate and high quantum numbers v
have continuum distribution of their vibrational energies.
@Since statistical weights of harmonic and anharmonic levels
are the same ~one! they do not impact the reliability#. This is
true even in the H2 molecule in the ground electronic state
where the anharmonicity is large and the energy gaps between the upper vibrational levels are also ~relatively! large.
However, the excitation energies of these upper levels are
too high for the levels to have a meaningful effect on the
rotational–vibrational partition functions of partially dissociated hydrogen gas. This can be seen in Figs. 3, 7 and 12
where the partition functions, dissociation degree and free
energy of partially dissociated hydrogen gas are shown as
functions of the gas temperature and particle density. Although thermal properties of the gas depend somewhat on
the model of the rotational–vibrational dynamics of the H2
molecules, the dependence is rather weak as long as the gas
is partially dissociated. Therefore, one can say that the rotating harmonic oscillator can also be used in practical studies
of thermal properties of partially dissociated gases with large
values of spectroscopic constants v e and v e x e . ~However,
one should be aware of the remark following Eq. ~43! when
the temperatures of such gases are low.!
FIG. 9. The temperature dependence of the particle densities n 8d ~in cm23)
in NO, ICl, HgI and NaCs gases where molecules are represented by the
rotating Tietz–Hua oscillator.
F. J. Gordillo-Vázquez and J. A. Kunc
FIG. 10. The specific free energy f̂ of NO gas where molecules are represented by the rotating Tietz–Hua oscillator ~solid line!, the rotating Morse
oscillator ~dashed line! and the rotating harmonic oscillator ~dashed-dot
line!; the solid, dashed and dashed-dot lines overlap.
Dissociation degrees of several discussed gases, calculated assuming different models of molecular dynamics, are
shown in Figs. 5–7. The same trend is seen in all the cases
considered: as the gas density increases, the gas dissociation
degrees predicted by the rotating harmonic oscillator, the rotating Morse oscillator and the rotating Tietz–Hua oscillator
models start to differ slightly but the differences are always
less than 10% if the gas remains partially dissociated.
Figures 8 and 9 show the temperature dependence of the
functions n d and n 8d @Eqs. ~49! and ~55!, respectively# obtained from the model of the rotating Tietz–Hua oscillator.
In general, the variation of n d and n 8d with temperature is not
strong, being almost constant at moderate and high temperatures. Approximate ~constant! values of n d and n 8d for several
diatomic gases discussed are shown in Tables V and VI,
respectively. The range of gas temperature where the constancy of these values can be assumed is: T>2000 K (H2,
Li2, Na2, I2, NO, ICl! and T>1000 K (K2, Cs2, NaCs, HgI!.
As can be seen in Figs. 10–12, the agreement of the
calculated free energies of several selected gases for the
three models of the rotating oscillators ~harmonic, Morse’s
and Tietz–Hua’s! is within a few percent in the entire range
of temperature and density considered as long as the gas
remains partially dissociated. ~Similar agreement has been
obtained for the other thermodynamic functions of the gases
shown in Figs. 10–12 and for the Li2, Na2, K2, Cs2, I2, KI,
FIG. 11. The specific free energy f̂ of ICl gas. The meaning of the symbols
is the same as in Fig. 10; the solid, dashed and dashed-dot lines overlap.
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J. Appl. Phys., Vol. 84, No. 9, 1 November 1998
F. J. Gordillo-Vázquez and J. A. Kunc
4703
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1
2
FIG. 12. The specific free energy f̂ of H2 gas. The meaning of the symbols
is the same as in Fig. 10; the solid, dashed and dashed-dot lines overlap.
NaCs, and HgI gases which are not shown in the figures.!
Thus, one can say that the rotating harmonic oscillator is an
acceptable model of molecular rotational–vibrational dynamics in statistical–mechanical calculations of degrees of
dissociation and thermodynamic functions of partially dissociated gases close to local thermal equilibrium. Values of
some thermodynamic functions for some gases discussed
here were tabulated, in limited ranges of particle density and
~mainly low and moderate! temperature, in Refs. 28–31.
They agree well ~typically within 10%! with the corresponding values of the functions obtained in the present work: the
small differences result from the variations in the spectroscopic constants, values and numbers of the molecular levels
and dissociation energies considered that were used by different authors.
ACKNOWLEDGMENTS
The authors would like to thank Francis Gilmore, Robert
Gordon and Leo Kaledin for a number of helpful discussions. This work was supported by the Air Force Office for
Scientific Research, Grant No. F-49620-1-0373 and Contract
No. 23019B, and by the Phillips Laboratory, Air Force Material Command, through the use of the MHPCC under Grant
No. F-29601-2-0001. One of us, F.J.G.V., thanks the Spanish
CICYT for financial support while a postdoctoral researcher
at USC, and support given by Grant No. MAT96-0529C02-01 at ICMM.
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