JOURNAL OF MOLECULAR SPECTROSCOPY 25, 77--85 (1968)
A Method for Testing and Improving Molecular Constants
of Diatomic Molecules with Special Reference to Br2 ~(~0+)*
ROBERT J .
LERoY* A~D
GEORGE BURNS
Lash Miller Chemical Laboratory, University of Toronto, Toronto, Ontario, Canada
The common procedure of constructing RXR potentials from known spectroscopic data has been reversed in the present work. Instead, spectroscopic
constants, which had often been obtained'by approximate methods, were tested
to determine whether they agree with the expected shape of the RXR potential.
For a special case, the 79.8~Br2 (~s+) molecule, it was found that no spectroscopic constants available in the literature could generate a RXR potential
which is reasonable at all energies, and that some reported constants yield a
highly erroneous potential. A new set of spectroscopic constants has been
calculated. These constants agree with the experimental results and, at the
same time, yield a reasonable I~KR potential They are (in cm-0 : ~ = 324.24,
~x~ = 1.172,
~y~ = 0.342 X 10-2,
~ z , = -0.101 X 10-s,
B~ =
0.081079,
a, = 0.30405 X 10-8, and 7~ - -0.40 X 10-~. The term values
generated from these constants predict that there are eighty vibrational levels
in Br2 0Eg+). The RKR turning points for the lowest thirty-seven vibrational
levels of Br2 0 ~ +) are given.
I. INTRODUCTION
A l t h o u g h it is possible to a p p r o x i m a t e the i n t e r n u c l e a r p o t e n t i a l of' a d i a t o m i c
molecule b y a n a l y t i c functions, such f u n c t i o n s h a v e l i m i t e d use, because t h e y
only reproduce t h e e x p e r i m e n t a l d a t a over a n a r r o w e n e r g y range, t~ees (1),
K l e i n (2), a n d R y d b e r g (3, 4) developed a m e t h o d of c o n s t r u c t i n g a n u m e r i c a l ,
r a t h e r t h a n a n a n a l y t i c , p o t e n t i a l which is free fi'om this d i s a d v a n t a g e . T h e i r
potential, u s u a l l y referred to as the R K R p o t e n t i a l , is o b t a i n e d directly from the
e x p e r i m e n t a l v i b r a t i o n a l a n d r o t a t i o n a l spectroscopic t e r m values, a n d its accuracy d e p e n d s m a i n l y on t h e accuracy of e x p e r i m e n t a l results. T h e R K R p o t e n t i a l
consists of a finite n u m b e r of pairs of i n t e r n u c l e a r distances, the " t u r n i n g
p o i n t s , " a t which the t o t a l energy of t h e d i a t o m i c molecule is equal to its classical p o t e n t i a l energy. A n i n t e r p o l a t i o n b e t w e e n t u r n i n g p o i n t s is necessary to
* This work was supported in part by the Directorate of Chemical Sciences, Air Force
Office of Scientific Research, grant No. AF AFOSR 506-66.
t Present address- Theoretical Chemistry Institute, University of Wisconsin, Madison,
Wisconsin 53706.
77
78
L~ROY AND BURNS
generate the continuous potential. Using the I~KR potential, the eigenfunetions
and eigenvalues of the radial SehrSdinger equation can be obtained ~mmerically
with a high degree of accuracy. On the other hand, the eigenvalues obtained by
solution of the radial SehrSdinger equation using analytic potential functions,
agree with the experimental eigenvMues oMy in the ease of the lower vibrational
levels. For these reasons, Steele, Lippineott, and Vanderslice (5) pointed out that
agreement with the R K R potential is a good quantitative test of any analytic
potential.
The main limitation in the calculation of RKI~ potentials is the lack of sufficient spectroscopic data to define the whole potential. These data are usually expressed in terms of a set of vibrational and rotational constants co~, Be, co~x~,
a~,
• , etc. Their values are chosen in such a way as to generate energies from
the polynomial
•
•
T(v, J) = o~o(v 3. 1/~) _ w~z~(v 3. 1/~)2 3. . . .
(1)
+ J ( J + 1)(Bo - ao(v + }~) + . . . )
which best app~0ximate the observed rotational and vibrational levels. The vi:
brational a n d rotational terms are often separated, using formulas defined
elsewhere (6).
G(v) = c%(v 3- }~) -- co~x~(v 3- },~)'~ 3- cooy~(v 3- }~)3 3- . . . ,
AGo+t/~ ~ G(v 3. 1) - G(v),
(2)
(3)
and
B~ = B~ -- ao(v + }~) + y~(v + 1/~)2 . . . .
(4)
The calculation of the R K R turning points is involved, and hence these poten, tials have not been extensively used in the past. However, with the advent of
large digital computers, this problem has become relatively simple. There are
now programs available both for calculating the RKI~ turning points of diatomie
molecules, and for solving the radial SehrSdinger equations for these systems (7).
The relative ease with which R K R potentials can be generated now allows us
t o turn the problem around. We are now able to test whether spectroscopic constants are compatible with the expected shape of an R K R potential. This ap,preach tests the approximations which sometimes are used to obtain spectroscopic constants in the absence of sufficient experimental data. Moreover, it m a y
yield approximate values of new higher order spectroscopic constants.
I t is the purpose of the present paper to demonstrate the usefulness of such
an approach.
II. METHOD
In the calculation of an R K R potential, the vibrational constants determine
the distance between the two turning points for a given vibrational level. On the
METHOD FOR TESTING AND IMPROVING MOLECULAR CONSTANTS
79
other hand, the rotational constants determiue the internuclear distance con:esponding to the average of the turning points for a given vibrational level. Therefore, an error in the vibrational constants affects the shape of the calculated R K H
potentiM in a different way than does an error in the rotational constants. More
specifically, if the zXG~+i/2 value for a given energy is too small, the two turaing
points will be too far apart. Analogously, if the rotational constants yield Coo
large a By for a given vibrational level, the values of both turning points will be
too large, although their difference may be correct.
The method described here tests whether the spectroscopic constants yield an
I~KR potential which satisfies the following requirements:
(1) The outer turning points should asymptotically approach tile kaown dissoeiation limit of the given electronic state. This is equivalent to the requirement
that in equations (2) and (3) zXG~+~/2 = 0 and G(v) = D, for the same value of
v, where D is the known dissociation energy.
(2) The slope of the inner portion of the t l K R potential must be negative.
(3) Over the range of the inner turning points, the IRKR potential must be
come steeper with decreasing internuclear distance. In other words, the second
derivative of the potential with respect to internuclear distance, d2U(r)/dr 2, must
be positive in this region.
In order to satisfy these requirements, the values of the vibrational and r o t a
tional constants reported in literature may need to be changed. However, the
adjusted constants must also be consistent with the existing spectroscopic data.
The lower order rotational and vibrational constants are usually fairly accurately determined b y the readily available vibrational energies and B,,'s for the
lowest few vibrational levels. On the other hand, accurate values of the higher
order constants can only be experimentally determined if the energies and B~
values are known for a large number of vibrational levels. In the absence of this
data, the present method may not only improve the existing values of the
constants, but may also yield values of new higher order constants.
The methods ,of deriving these constants are similar for both rotational and
vibrational eases. First, a trial value of the highest order constant is selected. The
term value equation [(2) or (4)] which includes the new constant must yield the
best possible least squares fit to the experimental data. This requirement defines
new values of all lower order constants. The new set of constants is tested to determine whether it abides b y the requirements (1)-(3). If a number of such trial
values are used, the best combination of constants can be found. It should be
emphasized that for an arbitrary choice of the new higher order constant, all the
others are uniquely determined b y the exp.erimental data.
When testing the existing constants, the vibrational constants should be considered first. This is because their compliance with requirement (1) is relatively
independent of the choice of rotational constants, while the latter's fulfillment of
requirements (2) and (3) depends on the values of the vibrational constants.
80
LEROY AND BURNS
..'"//
12,00C
--
/"
_~"
/
~
/
.:;~
./
It_qr--
.'"
~
/
J
./
. , , ~ /.
...~/
_
.Zf"
"'
i
2"0
A .....
re
2-5
s.o
3-5
r(A)
FIe. 1. RKR potentials calculated for ground-state 79,S~Br.2using different sets of spectroscopic constants. The curves were generated from the constants in Table I. Curve A
was obtained from the R&V constants and Curve D from our derived best constants.
Curves B and C illustrate the effect of variation in the rotational constants on the potential.
A n y given set of vibrational constants is tested b y calculating AG~+I/2for increasing v, until it becomes zero or negative. If this occurs when G(v) is less than
2 cm -1 smaller than the dissociation energy, we consider the constants to be
satisfactory. I n most cases it is necessary to change the existing constants before
this requirement can be satisfied. I n general, this requires the introduction of an
additional higher order vibrational constant. For each trial value of the highest
order constant, the corresponding lower order constants were defined by a fit to
the experimental data, as described above. A judicious choice of trial values will
m a k e this iteration converge very quickly.
The first step in the testing of any given set of rotational constants is the construction of an R K R potential b y combining the rotational constants with the
best set of vibrational constants. If the first and second derivatives of the inner
portion of this potential satisfy requirements (2) and (3) at all energies up to the
dissociation limit, the original constants are satisfactory. If the existing rotational constants are found to be inadequate, a new higher order constant usually
needs to be introduced. R K R potentials are constructed and tested for several
trial values of the highest order constant until its best value is found. At low
energies, there will be a relatively wide range of acceptable values of the highest
order rotational constant, but this range becomes progressively more restricted
at higher energies.
M E T H O D FOR T E S T I N G AND I M P R O V I N G M O L E C U L A R C O N S T A N T S
S1
TABLE I
VIBRATIONAL AND ROTATIONAL CONSTANTS USED TO GENERATE R K R POTENTIALS 1N
FIG. 1. ALL CONSTANTS ARE EXPRESSED IN WAVE NUMBERS. COLUMN A ARE TfIE
R & V CONSTANTS AND COLUMN D ARE OUR DERIVED CONSTANTS
Curve
Constant
A
w~
w~xe
~ye
o~zo
B~
a~
78
323.21
1.0282
--0.97 X 10-3
--0.417 X 10 4
0.08106
0.265 X 10-3
--1.394 X 10-~
324.24
1.172
0.342 X 10-3
-0.101 X 10-3
0.081074
0.29605 X 10 3
--0.60 X 10-~
C
D
324.24
1.172
0.342 X 10-2
--0.101 X 10-3
0.081085
0.31205 X 10-3
--0.20 X 10-3
324.24
1.172
0.342 X 10-~
--0.101 X 10-a
0.081079
0.30405 X 10 3
- 0 . 4 0 X 10
0-082
~x.
I
K
v
(.9
>
0-080
"'--(8L
I
I
t
I
I
5
(v+
81)
I
5
1
FIG. 2. B~ values for the three isotopic forms of ground-state Br2 (12~+). ® Results of
Brown (10). X Results of Horsley and Barrow (11).
III. S P E C T R O S C O P I C C O N S T A N T S F O R G R O U N D S T A T E OF 7Y,SlBr2(l~g+)
I n o u r w o r k , all R K R t u r n i n g p o i n t s w e r e c a l c u l a t e d u s i n g t h e c o m p u t e r p r o g r a m r e c e n t l y d e v e l o p e d b y Z a r e (7, 8). T h e F o r t r a n I V v e r s i o n of his p r o g r a m
was m o d i f i e d s l i g h t l y f o r use in t h e p r e s e n t c a l c u l a t i o n s . T h i s p r o g r a m c a n y i e l d
t u r n i n g p o i n t s for all v a l u e s of t h e v i b r a t i o n a l q u a n t u m n u m b e r v. T h e c a l c u l a t i o n of t h e t u r n i n g p o i n t s for o n e h u n d r e d l e v e l s r e q u i r e d less t h a n 0.3 m i n u t e
e x e c u t i o n of a n I B M 7094 I I . T h e first a n d s e c o n d d e r i v a t i v e s of t h e i n n e r port i o n of t h e p o t e n t i a l a t t h e e n e r g y of a g i v e n v i b r a t i o n a l l e v e l v w e r e o b t a i n e d
82
LBROY AND BURNS
De
12,000
'IE
(0
v
c
Ill 6 , 0 0 0
o
I
+2
I
I
I
I
I
-4
I0 ° x ~e
I
I
I
-I0
(cm-')
FIG. 3. Effect of w on the derivatives of the inner portion of RKR potentials calculated
for ground-state 79.S~Br2.The ordinate is energy above the potential minimum. In region A,
the first derivat~ive is positive, contradicting condition (2). In region B, the second derivative is negative, contradicting condition (3). In region C, both conditions (2) and (3) are
satisfied. The vibrational constants from column D in Table I were used in obtaining the
data shown here.
numerically from the turning points at (v - 1), v, and (v -4- 1). Any other choice
of the two nearby points should yield equivalent results. However, due to the
limited precision of our computer program, the use of smaller increments, such
as 0.2, sometimes yields erroneous derivatives.
I n this work, R K R potentials for the ground electronic state of Brs were calculated using a number of different sets of rotational and vibrational constants.
Some of these potentials are shown in Fig. 1, and the corresponding constants are
listed in Table I. The values of all necessary constants not listed in the table
were taken from Herzberg (6).
The most recent literature values of the spectroscopic constants for groundstate 7°,SlBr2 are those obtained b y Horsley and Barrow (9) by interpolation of
their absorption data for isotopically pure 7°,79Br2 and sl,SlBr2 . They measured
By values for the first four vibrational levels of the two isotopically pure species.
When their Bv's for each level of the two species were mass-corrected and aver-
METttOI) FOR TESTING AND 1MPI~OVING MOLECULAR CONSTANTS
,'~3
TABLE II
HKR TURNING POINTS FOR GROUND-STATE79,S~Br2 ( ~ + ) . E Is (]IVEN IN tilt -1, f IN ~\
v
E
r ~n)
r (out)
v
0
l
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
161.8
483.7
803.3
1120.6
1435.6
1748.3
2058.8
2367.0
2672.9
2976.6
3278.0
3577.2
3874.1
4168.7
4461.0
4751.1
5038.8
5324.1
5607.1
2.2324
2.1988
2.1768
2.1595
2.1451
2.1326
2.1216
2.1117
2.1026
2.0943
2.0867
2.0796
2.0731
2.0670
2.0612
2.0559
2.0508
2.0461
2.0417
2.3345
2.3763
2.4066
2.4323
2.4554
2.4767
2.4968
2.5159
2.5343
2.5521
2.5694
2.5864
2.6031
2.6195
2.6357
2.6518
2.6677
2.6835
2.6993
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
E
5887.8
6166.0
6441.8
6715.1
6986.0
7254.3
7520.0
7783.2
8043.6
8301.4
8556.5
8808.7
9058.1
9304.6
9548.1
9788.7
10026.1
10260.4
r (in)
r (out)
2.0375
2.0336
2.0299
2.0265
2.0232
2.0202
2.0173
2.0147
2.0122
2.0098
2.0076
2.0056
2.0037
2.0020
2.0004
1.9989
1.9975
1.9963
2.7150
2.7306
2.7463
2.7619
2.7776
2.7933
2.8091
2.8249
2.8409
2.8569
2.8731
2.8893
2.9057
2.9223
2.9391
2.9560
2.9732
2.9905
aged, we o b t a i n e d four empirical values which are c o n s i d e r a b l y more a c c u r a t e
t h a n the earlier results (10) (see Fig. 2). Previously, R a o a n d V e n k a t e s w a r l u (11)
(hereafter referred to as R & V ) h a d m e a s u r e d a long series of b a n d progressions
in the ultraviolet, a n d d e t e r m i n e d a p p r o x i m a t e values of t h e first four v i b r a t i o u a l
c o n s t a n t s (ae, ~0eXe, oaye, a n d C0eZe). However, B~ values could n o t be o b t a i n e d
directly from t h e i r e x p e r i m e n t a l data. Therefore, t h e y derived r o t a t i o n a l cons t a n t s from their v i b r a t i o n a l c o n s t a n t s using a n a p p r o x i m a t e Pekeris e q u a t i o n
which was derived for a Morse p o t e n t i a l t h r o u g h t h e use of p e r t u r b a t i o n methods.
The r o t a t i o n a l c o n s t a n t s o b t a i n e d in this w a y were t h e n corrected to m a k e t h e m
consistent with the empirical B~ values reported b y B r o w n (10) for a few of the
lowest v i b r a t i o n a l levels. T h e details of this procedure are discussed b y R a o a n d
V e n k a t e s w a r l u (12). T h e B K R p o t e n t i a l we c o n s t r u c t e d from these workers'
c o n s t a n t s is curve A i n Fig. 1. I t shows t h a t the tZ&V r o t a t i o n a l c o n s t a n t s are
i n a d e q u a t e , since the slope of the i n n e r t u r n i n g points is positive above 6000
em -I, c o n t r a d i c t i n g our r e q u i r e m e n t (2). I n addition, the values of AG~+~/2generated from the lZ&V c o n s t a n t s did n o t go t 9 zero u n t i l G(v) was a p p r o x i m a t e l y
1000 em -~ above t h e k n o w n (9) dissociation limit, a n d this c o n t r a d i c t s our first
r e q u i r e m e n t for a r e a s o n a b l e set of c o n s t a n t s .
I n general, c o n d i t i o n (1) m a y require the i n t r o d u c t i o n of one more v i b r a t i o n a l
c o n s t a n t t h a n is w a r r a n t e d b y t h e e x p e r i m e n t a l G(v) data. However, in the
special ease considered here this h a p p e n e d to be u n n e c e s s a r y . F o r the r o t a t i o n a l
84
LEROY AND BURNS
constant, o n the other hand, out" method requires the evaluation of one more
constant than was reported b y Horsley and Barrow (9).
Our improved spectroscopic constants for the ground state of 79,StBr~ w e r e obtained in the following way. First tIorsley and Barrow's (9) two rotational constants were applied to the raw R & V d a t a to yield G(v) values. F r o m these we
then obtained a set of vibrational constants consistent with requirement (1).
Next, R K R potentiMs were generated from these vibrational constants and a
number of trial values of 7~. For each potential, the first and second derivatives
of the inner turning points were calculated over a wide energy range. Figure 3
shows how we found the set of rotational constants which satisfy requirements
(2) and (3) over the widest possible energy range. These new constants were then
applied to the R & V data to yield an improved set of G(v) values, and hence
better vibrational constants. When this cycle was repeated two more times, the
following self-consistent molecular constants were obtained, all in era-l:
cos = 324.24,
co~x~ = 1.172,
~0eye ---- 0.342 X 10 -2,
C0~Z~ = --0.101 X 10-3,
B~ = 0.081079,
o~ = 0.30405 X 10 -3 ,
~e = --0.40 X 10 -~,
and
r~ = 2.2814 A.
These constants predict the existence of eighty vibrational levels for ground-state
79.SlBr2 (~Zg+). The potential generated from these constants is curve D in Fig. 1,
and the turning points for the first 37 levels are given in Table II.
The standard deviation between the emission frequencies predicted b y our
constants and those observed b y R & V is 4-1.0 em -~. This is quite acceptable,
since R &V estimated an accuracy of 4-0.7 em -~ for only strong sharp lines.
CONCLUSIONS
For diatomie molecules in a given electronic state, the spectroscopic constants
must produce an R K R potential which satisfies at least the following three requirements:
(1) T h e outer turning points should approach the known dissociation limit
asymptotically.
(2) T h e slope of the inner portion of the R K R potential must be negative.
(3) T h e second derivative of the inner portion of the potential must always be
positive.
The conditions m a y be used in conjunction with experimental data to improve
the existing vibrational and rotational constants.
The above requirements were applied to the ground state of 79,S~Br2 (t~g+) and
an improved set of four vibrational and three rotational constants were generated. Turning points are reported for 37 vibrational levels within the range of
the existing experimental data (9, 11).
M E T H O D F O g T E S T I N G AND I M P R O V I N G M O L E C U L A R C O N S T A N T S
,45
ACKNOWLEDGMENTS
One of us (R.J.L.) is i n d e b t e d to the Province of Ontario for a G r a d u a t e F e l l o w s h i p
We are grateful to t h e I n s t i t u t e of C o m p u t e r Science of the University of Toronto for use
of their facilities.
RECEIVED: M a r c h 4, 1967
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2.
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4'
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