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Critical analysis of the series expansion of the potential energy function of CO2
JOURNAL OF MOLECULAR
SPECTROSCOPY
(1975)
ST,464479
Analysis of the Series Expansion of the Potential Energy Function of CO,
Critical
I. JOBARD AND A. CH~DIN Laboratoire de Physique Mol~czdaire el d’0ptiqzce AtmosplzCriq~e,~ B(itiment 221, Campus d’Orsay, 91405 Orsay, France
We present in this paper new values of the coefficients involved in the series expansion with respect to internal coordinates of the potential energy function of the COZ molecule. The relative importance of each potential coefficient and the cancellation of some potential coefficients nonsignificantly different from zero, are discussed.
I. INTRODUCTION
In 1971, Cihla and ChCdin (1) proposed an accurate determination of the potential function of the carbon dioxide molecule. This solution has recently been reiterated (2) with a view to fitting additional vibrational levels belonging to more than three isotopic species so as to obtain a more efficient “data-reduction.” By rewriting some of the programs currently used in a slightly different way, one of us (A. C.) has pointed out an error affecting the values of the Aw’s (fourth-order correction to the W’S). Correcting this error has led to smaller values for the Aw’s and a few iterations have significantly improved the fit. This paper is concerned with the presentation of this new solution and
its application
to the calculation
paper numerical
each coefficient involved respect to internal
II.
The potential
of nine isotopic species of COZ. We also present in this
trials which make it possible
to analyze
in the series expansion
the relative
of the potential
importance
energy function
of with
coordinates.
DETERMINATION
energy function
OF THE
POTENTIAL
of linear triatomic
ENERGY
FUNCTION
molecules has been determined
to a
high order of approximation by means of the method carried out by Cihla and ChCdin. Let us recall that this method (f-8) consists in solving algebraically the SchrGdinger equation2 associated
with the motion of the nuclei of the molecule under consideration.
The coefficients occurring in the expansion of the potential energy function are determined through an iterative process which leads to the minimization of the sum of the squares of the deviations between experimental and theoretical energy values of the 1A part of this work has been done Pierre et Marie Curie (Paris VI). 2 Resulting from the Born-Oppenheimer
at the Laboratoire approximation. 464
Copyright
Q 1975 by Academic
All rights of reproduction
Press. Inc.
in any form reserved.
de Spectroscopic
Mol&ulaire,
Universite
POTENTIAL
FUNCTION TABLE
Ii
COEFFICIENTS
OF CO?
I
OF THE POTENTIAL ENERGY FUNCTION WITH
RESPECT TO THE INTERNAL COORDINATES i
a
Identification of Fi
1
b
Fll
2
F22
3
F33
4
F1ll F
5 6
122
F133
7 8
F1lll Fl122
9 10
F1133
F
11
2222
R,&2
AND R3_
Value of Fi (in cm-' 7.10118486
lO+2
3.35544533
10+2
I.23097527
10+3
-4.64853340
IO+'
-2.92561323
IO+'
-2.66177899
IO+~
1 . 68642683
IO+'
8.95554064
lo-'
2.10697178
IO+'
8.31589408
10-l
-1.37333355
1o+o
7.06295807
1o+O
13
-3.14774539
lo+O
14
-4.40575542
1O-2
F2233
12
F3333
6 . 07253274
15 16
-7.71360113
17 18
F1 F
19 20
3333
111111
F111122
21
Fl11133
22 24
F223333
28
F333333
a VaIue of the index
12233
9.63040322
IO-'
?8
lO-3 lo-' Ido 10-I 10-~
7.93249999
lo-2
2.74749816
IO-'
i in the summation of Eq.(l)
‘J/he = FOP example F
10-l 10-l
-2.38249065
F222233
27
7.10695154 4.64850010
1.05571018
F222222
26
10-I IO+'
-1 * 61542514
F113333
25
1.60647895
-5.96705384
'112233
IO+' IO-'
-3.54636163
3.75013123
F112222
23
b
46.5
:
-i Fi Oi(R) i=l
is the coefficient which multiplies
the third order operator RIR$i
vibrational levels.The
theoretical values of the energies are obtained by computing the eigenvalues of the vibrational Hamiltonian transformed by means of two successive contact transformations and expanded up to the fourth order of approximation. This method makes it possible to determine accurately the 28 coefficients involved in the series expansion of the potential energy function of the CO2 molecule. This expansion is
466
JOBARD
AND
TABLE fi
COEFFICIENTS
CHEDIN
II
OF THE POTENTIAL ENERGY FUNCTION
WITH RESPECT TO THE INTERNAL COORDINATES
i
a
Identification
1
2 3 4 5 6
fl73 f122
fl122
13
15 16 17 18 19 20 21 2; 23 24 25 26 27 28
*I113
f1133 f1223 f 2222 f11111 f11113 fll122 fll133 *II223 f12222 f
Value
111111
f111113 *Ill122 f111133 *Ill223 f111333 f112222 '112233 '122223 P 222222
AND 5;
of fi (in cm-'
5.423606
fill
9 IO
14
b
f13 f 22
*Ill1
12
fi
fll
7 8
11
of
6,,&
IO5
8.511384
IO4
1.971324
lo4
-1.485150
lo6
-6.894162
IO4
-3.488122
104
2.338757
IO6
-2.017686
lo4
5.289104
IO3
-6.043205
IO4
7.609672
lo4
2.870283
lo3
-3.497619
106
-3.497362
lo7
1.725030
IO4
-1.348908
10~
-1.037847
105
-5.403060
lo3
5.768660
lo7
4.052455
10'
1.893240
lo6
7.734308
107
1.776057
IO7
7.046218
IO8
-1.136341
IO4
1.480776
lo7
4.405020
lo4
2.140762
IO4
’ Value of the index.i in the summation~of Eq.(2) : V/he = $, fi O;(c) b For example,
'1,223
is thk coefficient which multiplies
the operator
carried out with respect to the three internal coordinates3 RI, Rz, and RB [defined Ref. (I)] which are quasinormal and mass-independent. It is written as
&=2 FiOi(R), i-l
in
(1)
3 Let US call Arl and Arz the variations of the bond lengths, and AtJ the angle of the two bonds. RI, &, and & are linear combinations of the dimensionless coordinates ~~ = Arl/rl", b = ~q t1 = Ar~/rz', where rlc and TP are the equilibrium bond lengths (rIe = rye for a C&like molecule).
POTENTIAL
FUNCTION
TABLE
TO
i
a
Ai
COEFFICIENTS
THE
NORMAL
OF
THE
COORDINATES
b
:dentification of ki
EXPANSION
7
1 2C1 ho
-43.2577 75.2396 -252.a942
111 122
11
1111
13 14
1122 1133 2222
17 18
1 .5321 -11.5804 19.5438 2.6634 -28.0753 6.6889
2233 3335
19 20
11111 11122
22
11133 12222 12233 13333
5.5465 -3.2789
32 34 35 38 39 40
0.61 -0.830”
;;
111111 111122 111133 11222: 112233 113333 222222 222233
46 47
223333 333333 Value
of
D For
12233
example,
ill (q&
are
+ qE2)q;
i
in
snly
the 28
stands
(in
cm-‘)
‘4c’ 60
2
2
676.8553 318.4470 1133.8167
-5771
-39.5923 69.82’0
-43.2577 71 .2124 .239 ..3580
-234.6813 1.3615 -10.4339 17.6088
0.7181
-2.4089 0.6662
-2.4089 0.6466
5.3427 -0.993’ 5.2353 -3.0950
4.9562 -z.g909 5.2216 -3.0869
4.8107 -0.9336 4.9194 -2.9082
0.6996 5.2049 -0.9429 4.9686 -2.9373
0.6154 _‘I.8068 0.8272 0.1146 ci.6109
12.5156 -.‘.7267
0.5156 -0.7054 ‘2.7232 0.1045
0.6154 -0.7860 0.8059 0.1087
rJ. 55?3 -1 -3.0235 0.2572 -0.2683 0.2211
-1.3063 -0.0229 ‘1.2498 -3.2606
18.9875
2.5139 6.3136 -2.7920
.3764
of
:
V/he
k12,,33
,
=
czfficient
L
?>.9920 -2.7920
0.5797
.2517
-3.2934 0.2418
Eq.(3)
18.4977 2.3859 -25.1503
6.1104
-1.3329 -c.0257 0.2813
coefficients).
1.5321 -10.9606
2.4330 -25.6472
0.7451 1.1110 :. i91:
-1 -0.0247 0.2701 -1).2819 0.2322
the
RESPECT
438.058’7 321 1144.9614
-39.5923
-26.5000
summation
SPECIES
13C18,7
?
1 .5321 .2508
non-vanishing for
12~18~
WITH
1.361: -10.7495 ’ 18.1416 2.5825 -27.2225 6.4857
-11
54
FUNC’UO?,
SYMMETRIC
71 a9334 -241.7815
-1 .4582 -0.027@ 0.2946 ‘-0.3073 0.2532 index
VARIOUS
-43.2577 73.0982 -245.6966
0.8514 3.1214 0.647:
there
FOR
638.0587 331 a3063 1179.6018
-1.0526
the
species,
q3
ENERGY
676.8553 326.8799 1163.8418
-2.7920 0.7392 , .4992
23 26 57 28
POTENTIAL
13c’ 60:
2
133
a
AND
33
;;
III THE
q,~21’~~
11 22
1
OF
467
OF COn
c i=l
which
ki
@i(q)
0.2147
. (For
mi1tiplies
the
the
symetri::
#Jperator
.
where the Fi’s are numerical coefficients,4 and the Oi(R)‘s are operators which may be (m, n, and p being integers such that written under the general form RlmRPRP 2 < m + 2n + 2p < 6); they are invariant with respect to the symmetry operations belonging to the point group D-I, of the linear symmetric molecule XYZ. The expansion up to the fourth order of approximation contains 28 operators5 of that kind. III.
IMPROVED
POTENTIAL
ENERGY FUNCTION MOLECULE
OF THE
CARBON
DIOXIDE
Table I gives the set of the 28 “potential coefficients” appearing in the expansion of the potential energy function obtained at the end of the new iterative process. The set of the fi coefficients involved in the expansion
hc
i-l
4The F; coefficients will hereafter be called “potential coefficients.” 6 Let us notice that our programs are actually adapted to the point group C,, and so make it possible to deal with the general linear triatomic molecule, XYZ.
JOBARD
468
AND
CHEDIN
of the potential energy function with respect to the dimensionless internal coordinates 51, [2, and & [also defined in Ref. (I), and recalled in footnote 31, are given in Table II. In Tables III and IV we give the sets of the ki coefficients appearing in the expansion ‘I’ABLE _Iri
COEFFICIENTS
FUNCTION
WITH
zq3 i
a
OF
THE
RESPECT
TO
,FOR
VARIOUS
IV
EXPANSION THE NON
1
11 13 22
3 4
33
2 7 P
SYMMETRIC
SPECIES
11
1111 1113
::
1122
14 15 16
1133 1223
0.7454 2.6284
19 20 21 22
11111 11113 11122
23 24 25 26
11133 11223 11333 12222 12233 13333 22223 22333 33333 111111 111113 117122 111133 111223 111333 112222 112233 113333 12222 3 122333 133333 222222 2222 33 223333 333333
I‘ 1
(in
;.b
See
*
These
footnotes
they fwlrth
order.
16013c170
666.6740
324.2394
325.4845 1158.9482
0.0
-42.3001 2.7924 72.2i48 -242.6949 3.5472 -4.4471
-8.3278
1.4877 -0.2640 -11.3564 19.1611 -0.4490 0.4010
18.2433 -0.R503 0.7768
2.6435 -27.861A 6.6393
2.4793 -26.1207 6.2284
1.4879 -0.2834 -11.0297 18.6089 -0.4515 0.4182 2.4941 -26.2874 6.2644
-2.6867
-2.5880 -0.6490 0.6809
-0.3556 0.69RP
1.4499 -0.5210 -1O.P211
-2.5894 -0.6049 0.7@lj 5.1650 -0.0147 0.7313 -1.0209
0.7195 5.3374 -0.0085 -1.0361
5.0047 -0.0184 0.7618 -0.9626
5.3793 -3.1552 -0.1434 0.2533 -0.1026
5.4594 -3.2203 -0.0762 0.1352 -0.0550
5.0718 -2.9706 -0.1422 0.2543 -0.1038
-0.3312
0.3967
-2.6863
5.1811 -0.0106 0.4137 -0.9774 5.15@3 -3.0368 -0.0756 0.1358 -0.0557
0.5629 *
0.5879 *
-0.7767 C.7981 * *
-0.8023 0.8231 x *
-0.7541 c1.7752 * *
0.1159 0.6150 -1.3AOl * * *
0.1185 0.6312 -1 .4206 * * *
0.5792 -1.2989 * * *
0.5952 -1 .3395 x * *
-0.0267 0.2914
-0.0243 0.2650
-0.3037 0.2497
-0.2753 0.2247
-0.0245 0.2671 -0.2783 0.2287.
-0.0265
0.2892 0.2457
@f Tai,le are
contribute
I
-41.4453 5.1711 71.3741 -2 39.7900 6.6867
74.3522 -249.8844 3.4605 -4.2637
-0.3006
coefficients
do not
cm-‘)
657.2738 0.0
@.587i? Y
0.5627 *
-0.7792 0.7995 * *
O.lC92
0.1113
I
I
a
q,-Lq2,1q22
1154.7075
-42.2988 2.6005
19.7950 -0.8454
-27.6925 6.6020
29 30 31
335.1002 1193.1766
1.4493 -0.4856 -11.1461
2233 3333
27 28
0.0
-7.9893
1333 2222
17 IR
666.6847
6.5247
333
ENERGY
116~13~18~
l16012C170
-41.4409 4.8186 73.5128 -246.9964
133 223
9 IO
POTENTIAL
COORDINATES
657.3123 0.0 333.B910 1189.0427
111 113 122
THE
NORMAL
I~en:~~~pationbl~6012ClEo
2
OF
III. not
considered
t,> the
energy
the
program
expanded
in
~cp to
since the
Eil
Gil
E+O
E-
E-Z
E-?
E-’
-1.‘251506
-1.256150
-i .069564
-4.2920.‘9
‘4.889310
-8.840810
-4.168802
E-l
E-:
1.347
3.9<%
AU,
AW,
E-2
'.955189
Ati;
E-1
1.347743
E-2
E-i
E-2
3.1c19S84
E-l
E-?
2.69859Z
E-.
5.R745"4
6.957749
3.227750
E-2
1.458479
7.136251
Et,
E-?
-3.9244R7
-2.548829
E-i
F-?
-4.7.l4031
E-?
Et0
1.642029
6.35504f
E+1
-1.920453
E-‘
Et-0
Y.413195
E+O
E+?
2.3954725
-;.4,609*
Et2
-..7291162
-~.PRylOb
Ei4
1.353:!0’
-7.390
A3 u
12
A1
wl?2
Y3u
YZLI.
YlLC
y333
y233
y223
y222
y733
y123
y122
y113
Yl12
y111
xLI.
“33
X22 x23
?13
x12
X11
Y
Y
w.
E+O E-3 L-2 E-2 s-1 E-i E-2
-1.052780 -7.9405,4 c.obP*o, 031571 -4.290061 1.375666
* .394?47
EID
E+3
E+*
ii3
E-,
E-:
E+1
El'
E+C
E+!
E-2
E-2
E-3
E-' E-Z E-i
-1.154 1.384 4.010
1.933282
1.078493 E-2
E-1
E-!
E-1
3.156134
2.598968
E-I E-t
I.290773
! .737202 E-2 -1,.010E-2 1.119E-: 3.5’5 E-:
E-1
Eel
-2.458139
Ei'
-2.530556
E--
2.943187
E-2
2.826262
u.892474
2.474485
E-3
,.a,,341 .%.*I7779
c;.o192.35 E-2
2.982558
E-i
.23U4i6
E-i
E-? E-3
1.258390
E-2
E-2
-J.507144
-1.235768
1.282340 E-l r .666,68E-2
-3.646787
-;>.379034
i.202338.E-2
-..V,2642
-'1.8e7457
-:.lt76L*
-'.I59375
'.$20722
-'.'97,79
-',.0503:,5 ElO
-i.721237
2.3094140
...4847874
IQ
E-2
Et,
lzc
1.3145476
ii0
1.364490
E-2
2.35920,6
E+3 El.2
'.<26,268
2 1.27<,I74
‘E.
4.41&63
-3.690712
~-3
Et,
-!.244124
-4.728260
E+1
-1.238089
Is+'
-1.845113 =+o
E+O
7.6,6867
E+O
Et3
2.3780854
-5.226719
E+7
-2.718761
Et3
1.3146245 1.6778200
lZc
ljc;
ir,
E12
E+j
1.:33
7.117
-4.000
I,747882
E-’
E-2
E-2
1.4217,6
2.976968
,.521415
-6.4R6529
2.922749
c .240222
',.260209
'.731840
'.316589
-3.626153
-3.241860
..429991
1.297013
~-2
E-i
E-l
E-1
E+?
E-L
E-3
E-2
E-3
E-2
6-2
E-3
E-?
E-l
E-2
E-2
-1.'778103 -j.600551
E-2
E-J
E-l
E+i
E+1
6+i)
E+1
E+C
tro
l’io
1.137119
-4.871597
-9.97723!;
-,.17269?
-'.i05WG
i.'33554
-'.P336Pj
-',.150423
-2.799597
2.317R9t‘l
i.5096894
1.33334'0
lto
1.2761174
E-2 E-3
.'02
E-2 '.244
-8.420
1.721855
1.068291
2.877846
2.382910
-2.405647
2.722552
G.116854
5.3*5058
7.166399
I.265405
-3.366298
-4.~84027
-,.8x789
1.223004
-3.770944
-:'.604161
1.473154
-1.595523
-9.787065
-i.r50544
-'.14700h
i.501660
-1.727238
-n.886403
-1.564109
‘.2899228
: .4315428
E-2
E-l
E-l
E-l
II+,
E-e
E-3
E-2
U-j
E-2
E-2
E-3
E-2
E-l
E-2
~-2
E-2
E-3
E-I
E,l
E+i
EtL?
E+,
Et0
E+O
E+j
Et2
E+,
1.3537107
3.178
Q.980
2.390
E-l
E-3
E-2
1.594051
'.727305
2.832418
2.465502
->.452454
2.875891
5.738091
4.685648
i.050121
’ .268297
-3.578870
-,.86115?
,,.168650
1.243815
-,.066174
-7.941239
j-364055
-I.P92079
-9.516932
-1.113e30
-1.105444
".464516
-1.831650
-c.098828
-'.885406
2.2676334
- .3689398
E-L
E-7
E-'
E-l
El,
E-2
E-,
E-i
E-2
E-Z
E-L
E-3
E-2
E-l
E-l
E-2
E-2
E-l
E-l
6+1
E+l
E+O
E+i
E+O
E+O
Et,
E+L
E+3
JOBARD AND CHEDIN
470
of the potential energy function with respect to the dimensionless normal ccordinates (ql, 421,q22,43) of each one of the nine isotopic species we have studied. This expansion is written as
(3) with e(q)
We have computed,
=
qlm(q212
+
with this new solution,
q222)"q3".
the values of the spectroscopic
constants
TABLE VI COMPARISON BETWEEN EXPERIMENTAL AND CALCULATED VALUES OF SOME ENERGY LEVELS (in cm-')
W60, vibrational level
(1) -E
E exp
(1000)1 (1000),1
(11)
’
CFilC~
E
exP
-E
oc7
0.002
0.006
-0.001
-0.
(I)
i ribrational talc 1
level
OllU
E
exP
-E
(II) talc )
E
-E exP
-0. DC11
-0.007
I:ll'o)I
ca1c
0.020
-c.o05
-0.020
0.007
0220
0.040
0.002
0002
0.030
O.cill
0221
O.O,?O
(21’1 II
0.040
0.003
oo"l
0.001
-0.001
0.010
-0.004
-0.020
0.003
oo"3
0.06C
0.020
(21’1),, (21'1) III
I:11'0),,
’ %’ 602
o.002
16012c180 -¶-
vibrational level
0001
(11)
(1)
IEexp -Ecalc rE
i exP -Ecalc
-0.060
V'ibrational
level
-E ’ exp talc'
-0.040
-0.005
ooO2
-0.c50
O.Cl4
( IO01 1I
-0.050
-0.011
-0.070
-0.033
-0.020
0.004
-0.120
-0.011
-0.019
(10°1)11
-0.02C
-0.013
-0.06c
0.019
( loo1
(2221 111
-0.29o
0.017
01'2
(2221 JIII
-0.430
0.050
01'
(I) : value of 1971 (Ref.(l))
column (II): new value.
(II 1 E
oo"l
-0.1 YO
column
(I) -E ) exp talc
0.026
(lOOl)I
0112
E
II
3
POTENTIAL. FUNCTION OF CO,
471
TABLE VII
VALUES OF THE NORM GIVEN BY Eq.(4) FOR 9 SPECIES OF CO2
Isotopic species
Norm t0N201 (in cm-*)
'%'60*
2.3
10-4
'%'60*
I.7
10-3
16012c~80
1.2
10-3
16012c170
2.7
10~~
~2~18~ 2
8.0
IO-~
16013c180
2.6
1O-3 1
I
--I 10-l
occurring in the expression6 of the vibrational energy, referred to the minimum of the potential ; they are given in Table V for the nine species. In order to illustrate, by a few significant examples, the improvement brought by these new iterations, we compare in Table VI the new values of the discrepancies between experimental and calculated values of some representative vibrational levels for 12C1602, WsOz and 16012C1s0 molecules with the values of the discrepancies obtained in 1971 (1). The calculation, for each isotopic species respectively, of the “norm” N2, given by the expression X2 = 5 AE,2Pi/ 2 Pi i=l
i=l
(4)
(that is to say, the weighted sum of the squares of the deviations AE; between experimental and calculated energy values of a set of n observed levels), leads to the nine numerical values of Table VII. These values correspond to a mean deviation of a few hundredths of cm-‘, which is in very good agreement with the present accuracy of the experimental data. Particularly, the best levels of the principal isotopic molecule 12C1602 are reproduced within a few thousandths of cm-‘. 6 See Ref. (2). Our X’s are defined as being one-half of the X’s of Ref. (1).
472
JOBARD IV. CRITICAL
A. Relative Importance
AND
ANALYSIS
of the “Potential
CHEDIN OF THE
SOLUTION
Coejicients”
The numerical tests which were carried out with the new set of the 28 “potential coefficients” Fi (given in Table I) aimed at examining the influence of a modification of the value of one coefficient on the mean value of the deviations (or better, on the “norm”) between theory and experiment. We have represented some of the results in Figs. l-4. For each of the 28 potential coefficients, the percentage of modification of the coeffkient value is plotted along the X-axis, and the change of the “norm,” characterized by the factor multiplying the reference norm obtained with the nonmodified value of the coefficient, is plotted along the Y-axis. For example, Fig. 1 shows that a 1% increase of the value of the F33 coefficient leads to a “norm” equal to 3.10 f6 times the reference norm. We must notice that the scales on each figure are different, but most of the 28 curves reveal an important increase of the norm for a rather small change of the coefficient value, except the curves of Fig. 4, relative to the variations of the coefficients F 1~22 and FwB~~. Nevertheless, all these curves have the same shape : they are almost exactly parabolas, having as their minimum a 0% variation of the coefficient. This means that the reference solutionthat is to say, the new potential energy function-constitutes a minimum stable with respect to the modification of the value of one of the coefficients. IJ~(x% ),‘N* (0%)
-2% FIG. 1. The relative
potential
coefficient
variation value.
-1%
of the norm
0%
t1%
+2%
[see Eq. (4)] as a function
OF/F
of relative
variations
of one
POTENTIAL
FUNCTION
473
OF COz
t
-2% FIG.
potential
2. The relative variation coefficient value.
-1%
of the norm
c!%
+2%
+1%
[see Eq.
(4)] as a function
.&F/F
of relative
variations
of one
If we define the “importance” of a coefficient as being the magnitude of the increase of the norm yielded by a 1% variation of the accepted coefficient values, it is then possible to classify the 28 coefficients, for instance, in decreasing order of importance. This classification is given in Table VIII: -The column.
index giving
the order of importance
of each coefficient
is given in the first
-The coefficients appear in separate columns according to whether they belong to the terms of order 0, 1,2,3, or 4 in the series expansion of the potential energy function. --In the last column are given the values of the relative increases of the “norm.” B. Discussiot~ In this theoretical frame, let us recall that, by the expansion of the potential energy function order n - 2 to the energy.’ Thus, as Tlo, VI, Vz, ators of degrees 2, 3, 4, 5, and 6, they contribute, 7Actually, it is sometimes difficult energy of a nondiagonal operator.
to evaluate
assumption, an operator of degree n in is supposed to give a contribution of ‘v3, and V4 contain, respectively, operrespectively, to the energies of orders
the order
of magnitude
of the contribution
to the
474
JOBARD
AND CHEDIN
0, 1, 2, 3, and 4. Furthermore, when the expansion of the potential is carried out with respect to internal quasinormal coordinates and if we consider only low values of the vibrational quantum numbers, the convergence of the potential expansion is ensured by the values of the coefficients Fi. So, it is important to check that the classification of the “potential coefficients F;,” in decreasing order of their absolute values, is in agreement with these assumptions. This classification, given in Table IX, shows that it is generally verified. If we compare Table IX with Table VIII, which gives the classification of the potential we can see that they are, as a whole, coefhcients in decreasing order of “importance,” slightly different. The essential differences between the two classifications are set forth on Table IX by arrows pointing out the rows occupied by the same coefficients in Table VIII. So, we must notice the partrcular importance of the coefficients F22, F2222, F222222,and Fa3, F 3333, Fr3333, F333333.That importance can essentially be explained by experimental reasons regarding the characteristics of the set of levels whose agreement between theory and experiment is studied : (a) Among the 100 levels of the principal isotopic species which compose the set of levels we use, the vibrational levels involving the quantum number 7~2are associated with statistical weights, the sum of which is three times greater than the sum of the statistical weights associated with the levels involving or, and four times greater than
~*(x%)/‘N~(o%) ,L
4
0%
-10%
0%
+10%
+20%
AF/F
FIG. 3. The relative variation of the norm [see Eq. (411 as a function of relative variations Jotential coefficient value.
of one
POTENTIAL FUNCTION OF COP
L
I
-20%
I
I
I
I
-10%
0%
+10%
+20%
475
I
.
AF/F
FIG. 4. The relative variation of the norm [see Eq. (4)] as a function of relative variations of one potential coefficient value.
the sum of the weights associated with the levels involving vs. So, the choice of the statistical weights, which are to some extent related to the accuracy of the experimental data, favors the peculiar importance of the coefficients Fz, FXW, and FKWXW (b) The other characteristic is that the set of levels used contains highly excited levels : -First, there are levels involving high values of ~3; seven levels are such that v3 is larger or equal to 5, while vr is not involved in any level with a value greater than 4; those levels are, particularly, levels belonging to the ooOv3series* (the ~3 value going up to 9). This explains the peculiar importance of the F 3333,FLM, and F333333coefficients : effectively, the successive contact transformations to which the original Hamiltonian is submitted are such that these coefficients, for instance, strongly contribute to the spectroscopic constant ~333; for example, a variation of ~333in the energy expression of the level CNY9is multiplied by 729 (va3 = 9”) and contributes to the deviation (observed value minus calculated value) which occurs itself in the norm by its square. The fact 8 See Ref. (9).
JOBARD AND CHEDIN
476
TABLE VIII CLASSIFICATION
OF THE Fi COEFFICIENTS WITH RESPECT TO
THEIR "IMPORTANCE".
(DECREASING ORDER).
coefficients
27 28 a
_-.U_*.~~_-~__._.______ 11122 71 I_-_~~___-__^___~______I_ 112222 1.1
For exemple , 12233 stands for the coefficient F multiplies
the operator
R R2R2 123
12233 which in the potential expansion.
that there are levels involving high ~3 values explains in this way the importance of the coefficients mentioned before, and greatly contributes to their accurate determination. -Second, the set of levels also contains many levels which are strongly coupled by the Fermi resonance, and consequently involve rather large values of the “quantum
POTENTIAL
FUNCTION
01: CO2
477
number ~2” compared to the mean value of zlrfor the whole set of levels. This contributes as well to emphasizing the importance of the F22, F2222, Fzzzzzz coefficients. In conclusion, we see that the order of importance of the ‘Lpotential coefficients,” defined by reference to the criterion of their respective influences on the norm of the discrepancies between theoretical and experimental values, takes into account not only the respective values of the coefficients, but also the accuracy of experimental data, and the nature of the vibrational levels constituting the set of levels we use. TABLE IX CLASSIFICATION
OF THE
Fi COEFFICIENTS WITH RESPECT To
THEIR ASSOLUTE VALUE. (DECREASING OXDER)
Fi coefficients 0th
lSt
xder
order
1
33b
2
11
3
22'
IFiI
a
rd 3 oMer
2nd order
th 4 order
I
(in cm-I)
230.975 710.118 335.545
4
133
5 6
111 122
266.178
-+--
46.485
\
29.256
t _
-
21.070
1133
7 8
'7.063
3333
9
11133
10
13333
II
11111
12 _____ 13
_
3.148 -~-_ 1.686 113333
1.615
117733 .---_.___._
1.373 ---.... 0.963
14 -.~
6.073 __-__. _~_3.546
0.896
------_-.___ :lI;ii::
0.832 ____ 0.711 0.597 ___.. 0.465
I
0.275
I
0.161
0.106 0.077
0.044
a See footnote of Table VIII
222233
0.024
112222
0.004
JOBARD
478
AND CHEDIN TABLE X
ITERATIONS CARRIED OUT WITH LESS THAN 28 NON-ZERO
Fi COEFFICIENTS
26 COEFFICIENTS
27 COEFFICIEN
I No
of the iteration 167. 2.2 1.6
F222233
I
1 2 3
2.1 1.5
25 COEFFICIENTS
+I F
223333
C. Cancellation
11. 3.8 2.3 1.5
of the Potential
i 3 4
Coqqicients
24 COEFFICIENTS
Nonsignificantly
Dijerent
from Zero
Some of the less important coefficients (in the lower part of Table VIII) are not necessary for the accurate determination of the vibrational energies of the present set of levels. It is indeed possible to obtain a similar agreement between theory and experiment when one of these coefficients is constrained to zero during several iterative cycles. Likewise, a satisfactory norm is reached when different combinations of two, three, four, or even five unimportant coefficients are simultaneously set equal to zero. As examples, the results of some of these trials are given in Table X.g The coefficients whose values are kept equal to zero during iterations are set down in the first column; the values of the ratios (N*/N*n,r) of the norms N*, obtained at the end of each iterative cycle, to the reference norm N2nor, obtained by working with the full set of the 28 accepted potential coefficient values, are given in the second column. It appears that several solutions with 27,26,2.5, or 24 coefficients are satisfactory and, more precisely, lead to an agreement between theory and experiment, close to the agreement yielded by using the full set of 28 coefficients. This means that the values of five or six coefficients (Fww, F11122,F222233,FwB~, F12222,F223333)are not significantly different from zero. *We could not intend, for obvious reasons regarding computation time, to try all the possible combinations of coefficients to be cancelled out simultaneously; nevertheless the tests, restricted in numbers, that we carried out are conclusive.
POTENTIAL
FUNCTION
Ok’ COr
479
It seems reasonable to state that the values of these coefficients will be better defined when it is possible to work with a set of levels at the same time more numerous and more accurate, that is to say, composed of levels highly excited in wl, vz, and zrS,and systematically measured with better accuracy. ACKNOWLEDGMENT
We wish to express our gratitude to Professor G. Amat and to Dr. Z. Cihla for helpful discussions throughout the course of this work. RECEIVED :
March 31, 1975 REFERENCES
1. Z. CIHLAANDA. CH~DIN,J. Mol. Spectrosc. 40, 337 (1971). 2. I. JOBARD,Thesis, University of Paris, 1974.
3. A. C&DIN, Thesis, University of Paris, 1971. J. A. CHBDINANDZ. CIHLA,J. Mol. Spectrosc. 45, 475 (1973). 5. Z. CIHLAANDA. CH~DIN, J. Mol. Spectrosc. 47, 531 (1973). 6. A. C&DIN ANDZ. CIHLA:J. Mol. Spedrosc. 47, 542 (1973). 7. A. CH~DINANDZ. Crrry J. Mol. Spectrosc. 49, 289 (1974). 8. A. CHBDINANDZ. C~FILA,J. Mol. Spectrosc. 47, 554 (1973). 9. T. K. MCCURBIN,JR., J. PL~VA,R. PULFREY,W. TELFAIR, ANDT. TODD,J. Mol. Spectrosc. 49, 136 (1974).
SPECTROSCOPY
(1975)
ST,464479
Analysis of the Series Expansion of the Potential Energy Function of CO,
Critical
I. JOBARD AND A. CH~DIN Laboratoire de Physique Mol~czdaire el d’0ptiqzce AtmosplzCriq~e,~ B(itiment 221, Campus d’Orsay, 91405 Orsay, France
We present in this paper new values of the coefficients involved in the series expansion with respect to internal coordinates of the potential energy function of the COZ molecule. The relative importance of each potential coefficient and the cancellation of some potential coefficients nonsignificantly different from zero, are discussed.
I. INTRODUCTION
In 1971, Cihla and ChCdin (1) proposed an accurate determination of the potential function of the carbon dioxide molecule. This solution has recently been reiterated (2) with a view to fitting additional vibrational levels belonging to more than three isotopic species so as to obtain a more efficient “data-reduction.” By rewriting some of the programs currently used in a slightly different way, one of us (A. C.) has pointed out an error affecting the values of the Aw’s (fourth-order correction to the W’S). Correcting this error has led to smaller values for the Aw’s and a few iterations have significantly improved the fit. This paper is concerned with the presentation of this new solution and
its application
to the calculation
paper numerical
each coefficient involved respect to internal
II.
The potential
of nine isotopic species of COZ. We also present in this
trials which make it possible
to analyze
in the series expansion
the relative
of the potential
importance
energy function
of with
coordinates.
DETERMINATION
energy function
OF THE
POTENTIAL
of linear triatomic
ENERGY
FUNCTION
molecules has been determined
to a
high order of approximation by means of the method carried out by Cihla and ChCdin. Let us recall that this method (f-8) consists in solving algebraically the SchrGdinger equation2 associated
with the motion of the nuclei of the molecule under consideration.
The coefficients occurring in the expansion of the potential energy function are determined through an iterative process which leads to the minimization of the sum of the squares of the deviations between experimental and theoretical energy values of the 1A part of this work has been done Pierre et Marie Curie (Paris VI). 2 Resulting from the Born-Oppenheimer
at the Laboratoire approximation. 464
Copyright
Q 1975 by Academic
All rights of reproduction
Press. Inc.
in any form reserved.
de Spectroscopic
Mol&ulaire,
Universite
POTENTIAL
FUNCTION TABLE
Ii
COEFFICIENTS
OF CO?
I
OF THE POTENTIAL ENERGY FUNCTION WITH
RESPECT TO THE INTERNAL COORDINATES i
a
Identification of Fi
1
b
Fll
2
F22
3
F33
4
F1ll F
5 6
122
F133
7 8
F1lll Fl122
9 10
F1133
F
11
2222
R,&2
AND R3_
Value of Fi (in cm-' 7.10118486
lO+2
3.35544533
10+2
I.23097527
10+3
-4.64853340
IO+'
-2.92561323
IO+'
-2.66177899
IO+~
1 . 68642683
IO+'
8.95554064
lo-'
2.10697178
IO+'
8.31589408
10-l
-1.37333355
1o+o
7.06295807
1o+O
13
-3.14774539
lo+O
14
-4.40575542
1O-2
F2233
12
F3333
6 . 07253274
15 16
-7.71360113
17 18
F1 F
19 20
3333
111111
F111122
21
Fl11133
22 24
F223333
28
F333333
a VaIue of the index
12233
9.63040322
IO-'
?8
lO-3 lo-' Ido 10-I 10-~
7.93249999
lo-2
2.74749816
IO-'
i in the summation of Eq.(l)
‘J/he = FOP example F
10-l 10-l
-2.38249065
F222233
27
7.10695154 4.64850010
1.05571018
F222222
26
10-I IO+'
-1 * 61542514
F113333
25
1.60647895
-5.96705384
'112233
IO+' IO-'
-3.54636163
3.75013123
F112222
23
b
46.5
:
-i Fi Oi(R) i=l
is the coefficient which multiplies
the third order operator RIR$i
vibrational levels.The
theoretical values of the energies are obtained by computing the eigenvalues of the vibrational Hamiltonian transformed by means of two successive contact transformations and expanded up to the fourth order of approximation. This method makes it possible to determine accurately the 28 coefficients involved in the series expansion of the potential energy function of the CO2 molecule. This expansion is
466
JOBARD
AND
TABLE fi
COEFFICIENTS
CHEDIN
II
OF THE POTENTIAL ENERGY FUNCTION
WITH RESPECT TO THE INTERNAL COORDINATES
i
a
Identification
1
2 3 4 5 6
fl73 f122
fl122
13
15 16 17 18 19 20 21 2; 23 24 25 26 27 28
*I113
f1133 f1223 f 2222 f11111 f11113 fll122 fll133 *II223 f12222 f
Value
111111
f111113 *Ill122 f111133 *Ill223 f111333 f112222 '112233 '122223 P 222222
AND 5;
of fi (in cm-'
5.423606
fill
9 IO
14
b
f13 f 22
*Ill1
12
fi
fll
7 8
11
of
6,,&
IO5
8.511384
IO4
1.971324
lo4
-1.485150
lo6
-6.894162
IO4
-3.488122
104
2.338757
IO6
-2.017686
lo4
5.289104
IO3
-6.043205
IO4
7.609672
lo4
2.870283
lo3
-3.497619
106
-3.497362
lo7
1.725030
IO4
-1.348908
10~
-1.037847
105
-5.403060
lo3
5.768660
lo7
4.052455
10'
1.893240
lo6
7.734308
107
1.776057
IO7
7.046218
IO8
-1.136341
IO4
1.480776
lo7
4.405020
lo4
2.140762
IO4
’ Value of the index.i in the summation~of Eq.(2) : V/he = $, fi O;(c) b For example,
'1,223
is thk coefficient which multiplies
the operator
carried out with respect to the three internal coordinates3 RI, Rz, and RB [defined Ref. (I)] which are quasinormal and mass-independent. It is written as
&=2 FiOi(R), i-l
in
(1)
3 Let US call Arl and Arz the variations of the bond lengths, and AtJ the angle of the two bonds. RI, &, and & are linear combinations of the dimensionless coordinates ~~ = Arl/rl", b = ~q t1 = Ar~/rz', where rlc and TP are the equilibrium bond lengths (rIe = rye for a C&like molecule).
POTENTIAL
FUNCTION
TABLE
TO
i
a
Ai
COEFFICIENTS
THE
NORMAL
OF
THE
COORDINATES
b
:dentification of ki
EXPANSION
7
1 2C1 ho
-43.2577 75.2396 -252.a942
111 122
11
1111
13 14
1122 1133 2222
17 18
1 .5321 -11.5804 19.5438 2.6634 -28.0753 6.6889
2233 3335
19 20
11111 11122
22
11133 12222 12233 13333
5.5465 -3.2789
32 34 35 38 39 40
0.61 -0.830”
;;
111111 111122 111133 11222: 112233 113333 222222 222233
46 47
223333 333333 Value
of
D For
12233
example,
ill (q&
are
+ qE2)q;
i
in
snly
the 28
stands
(in
cm-‘)
‘4c’ 60
2
2
676.8553 318.4470 1133.8167
-5771
-39.5923 69.82’0
-43.2577 71 .2124 .239 ..3580
-234.6813 1.3615 -10.4339 17.6088
0.7181
-2.4089 0.6662
-2.4089 0.6466
5.3427 -0.993’ 5.2353 -3.0950
4.9562 -z.g909 5.2216 -3.0869
4.8107 -0.9336 4.9194 -2.9082
0.6996 5.2049 -0.9429 4.9686 -2.9373
0.6154 _‘I.8068 0.8272 0.1146 ci.6109
12.5156 -.‘.7267
0.5156 -0.7054 ‘2.7232 0.1045
0.6154 -0.7860 0.8059 0.1087
rJ. 55?3 -1 -3.0235 0.2572 -0.2683 0.2211
-1.3063 -0.0229 ‘1.2498 -3.2606
18.9875
2.5139 6.3136 -2.7920
.3764
of
:
V/he
k12,,33
,
=
czfficient
L
?>.9920 -2.7920
0.5797
.2517
-3.2934 0.2418
Eq.(3)
18.4977 2.3859 -25.1503
6.1104
-1.3329 -c.0257 0.2813
coefficients).
1.5321 -10.9606
2.4330 -25.6472
0.7451 1.1110 :. i91:
-1 -0.0247 0.2701 -1).2819 0.2322
the
RESPECT
438.058’7 321 1144.9614
-39.5923
-26.5000
summation
SPECIES
13C18,7
?
1 .5321 .2508
non-vanishing for
12~18~
WITH
1.361: -10.7495 ’ 18.1416 2.5825 -27.2225 6.4857
-11
54
FUNC’UO?,
SYMMETRIC
71 a9334 -241.7815
-1 .4582 -0.027@ 0.2946 ‘-0.3073 0.2532 index
VARIOUS
-43.2577 73.0982 -245.6966
0.8514 3.1214 0.647:
there
FOR
638.0587 331 a3063 1179.6018
-1.0526
the
species,
q3
ENERGY
676.8553 326.8799 1163.8418
-2.7920 0.7392 , .4992
23 26 57 28
POTENTIAL
13c’ 60:
2
133
a
AND
33
;;
III THE
q,~21’~~
11 22
1
OF
467
OF COn
c i=l
which
ki
@i(q)
0.2147
. (For
mi1tiplies
the
the
symetri::
#Jperator
.
where the Fi’s are numerical coefficients,4 and the Oi(R)‘s are operators which may be (m, n, and p being integers such that written under the general form RlmRPRP 2 < m + 2n + 2p < 6); they are invariant with respect to the symmetry operations belonging to the point group D-I, of the linear symmetric molecule XYZ. The expansion up to the fourth order of approximation contains 28 operators5 of that kind. III.
IMPROVED
POTENTIAL
ENERGY FUNCTION MOLECULE
OF THE
CARBON
DIOXIDE
Table I gives the set of the 28 “potential coefficients” appearing in the expansion of the potential energy function obtained at the end of the new iterative process. The set of the fi coefficients involved in the expansion
hc
i-l
4The F; coefficients will hereafter be called “potential coefficients.” 6 Let us notice that our programs are actually adapted to the point group C,, and so make it possible to deal with the general linear triatomic molecule, XYZ.
JOBARD
468
AND
CHEDIN
of the potential energy function with respect to the dimensionless internal coordinates 51, [2, and & [also defined in Ref. (I), and recalled in footnote 31, are given in Table II. In Tables III and IV we give the sets of the ki coefficients appearing in the expansion ‘I’ABLE _Iri
COEFFICIENTS
FUNCTION
WITH
zq3 i
a
OF
THE
RESPECT
TO
,FOR
VARIOUS
IV
EXPANSION THE NON
1
11 13 22
3 4
33
2 7 P
SYMMETRIC
SPECIES
11
1111 1113
::
1122
14 15 16
1133 1223
0.7454 2.6284
19 20 21 22
11111 11113 11122
23 24 25 26
11133 11223 11333 12222 12233 13333 22223 22333 33333 111111 111113 117122 111133 111223 111333 112222 112233 113333 12222 3 122333 133333 222222 2222 33 223333 333333
I‘ 1
(in
;.b
See
*
These
footnotes
they fwlrth
order.
16013c170
666.6740
324.2394
325.4845 1158.9482
0.0
-42.3001 2.7924 72.2i48 -242.6949 3.5472 -4.4471
-8.3278
1.4877 -0.2640 -11.3564 19.1611 -0.4490 0.4010
18.2433 -0.R503 0.7768
2.6435 -27.861A 6.6393
2.4793 -26.1207 6.2284
1.4879 -0.2834 -11.0297 18.6089 -0.4515 0.4182 2.4941 -26.2874 6.2644
-2.6867
-2.5880 -0.6490 0.6809
-0.3556 0.69RP
1.4499 -0.5210 -1O.P211
-2.5894 -0.6049 0.7@lj 5.1650 -0.0147 0.7313 -1.0209
0.7195 5.3374 -0.0085 -1.0361
5.0047 -0.0184 0.7618 -0.9626
5.3793 -3.1552 -0.1434 0.2533 -0.1026
5.4594 -3.2203 -0.0762 0.1352 -0.0550
5.0718 -2.9706 -0.1422 0.2543 -0.1038
-0.3312
0.3967
-2.6863
5.1811 -0.0106 0.4137 -0.9774 5.15@3 -3.0368 -0.0756 0.1358 -0.0557
0.5629 *
0.5879 *
-0.7767 C.7981 * *
-0.8023 0.8231 x *
-0.7541 c1.7752 * *
0.1159 0.6150 -1.3AOl * * *
0.1185 0.6312 -1 .4206 * * *
0.5792 -1.2989 * * *
0.5952 -1 .3395 x * *
-0.0267 0.2914
-0.0243 0.2650
-0.3037 0.2497
-0.2753 0.2247
-0.0245 0.2671 -0.2783 0.2287.
-0.0265
0.2892 0.2457
@f Tai,le are
contribute
I
-41.4453 5.1711 71.3741 -2 39.7900 6.6867
74.3522 -249.8844 3.4605 -4.2637
-0.3006
coefficients
do not
cm-‘)
657.2738 0.0
@.587i? Y
0.5627 *
-0.7792 0.7995 * *
O.lC92
0.1113
I
I
a
q,-Lq2,1q22
1154.7075
-42.2988 2.6005
19.7950 -0.8454
-27.6925 6.6020
29 30 31
335.1002 1193.1766
1.4493 -0.4856 -11.1461
2233 3333
27 28
0.0
-7.9893
1333 2222
17 IR
666.6847
6.5247
333
ENERGY
116~13~18~
l16012C170
-41.4409 4.8186 73.5128 -246.9964
133 223
9 IO
POTENTIAL
COORDINATES
657.3123 0.0 333.B910 1189.0427
111 113 122
THE
NORMAL
I~en:~~~pationbl~6012ClEo
2
OF
III. not
considered
t,> the
energy
the
program
expanded
in
~cp to
since the
Eil
Gil
E+O
E-
E-Z
E-?
E-’
-1.‘251506
-1.256150
-i .069564
-4.2920.‘9
‘4.889310
-8.840810
-4.168802
E-l
E-:
1.347
3.9<%
AU,
AW,
E-2
'.955189
Ati;
E-1
1.347743
E-2
E-i
E-2
3.1c19S84
E-l
E-?
2.69859Z
E-.
5.R745"4
6.957749
3.227750
E-2
1.458479
7.136251
Et,
E-?
-3.9244R7
-2.548829
E-i
F-?
-4.7.l4031
E-?
Et0
1.642029
6.35504f
E+1
-1.920453
E-‘
Et-0
Y.413195
E+O
E+?
2.3954725
-;.4,609*
Et2
-..7291162
-~.PRylOb
Ei4
1.353:!0’
-7.390
A3 u
12
A1
wl?2
Y3u
YZLI.
YlLC
y333
y233
y223
y222
y733
y123
y122
y113
Yl12
y111
xLI.
“33
X22 x23
?13
x12
X11
Y
Y
w.
E+O E-3 L-2 E-2 s-1 E-i E-2
-1.052780 -7.9405,4 c.obP*o, 031571 -4.290061 1.375666
* .394?47
EID
E+3
E+*
ii3
E-,
E-:
E+1
El'
E+C
E+!
E-2
E-2
E-3
E-' E-Z E-i
-1.154 1.384 4.010
1.933282
1.078493 E-2
E-1
E-!
E-1
3.156134
2.598968
E-I E-t
I.290773
! .737202 E-2 -1,.010E-2 1.119E-: 3.5’5 E-:
E-1
Eel
-2.458139
Ei'
-2.530556
E--
2.943187
E-2
2.826262
u.892474
2.474485
E-3
,.a,,341 .%.*I7779
c;.o192.35 E-2
2.982558
E-i
.23U4i6
E-i
E-? E-3
1.258390
E-2
E-2
-J.507144
-1.235768
1.282340 E-l r .666,68E-2
-3.646787
-;>.379034
i.202338.E-2
-..V,2642
-'1.8e7457
-:.lt76L*
-'.I59375
'.$20722
-'.'97,79
-',.0503:,5 ElO
-i.721237
2.3094140
...4847874
IQ
E-2
Et,
lzc
1.3145476
ii0
1.364490
E-2
2.35920,6
E+3 El.2
'.<26,268
2 1.27<,I74
‘E.
4.41&63
-3.690712
~-3
Et,
-!.244124
-4.728260
E+1
-1.238089
Is+'
-1.845113 =+o
E+O
7.6,6867
E+O
Et3
2.3780854
-5.226719
E+7
-2.718761
Et3
1.3146245 1.6778200
lZc
ljc;
ir,
E12
E+j
1.:33
7.117
-4.000
I,747882
E-’
E-2
E-2
1.4217,6
2.976968
,.521415
-6.4R6529
2.922749
c .240222
',.260209
'.731840
'.316589
-3.626153
-3.241860
..429991
1.297013
~-2
E-i
E-l
E-1
E+?
E-L
E-3
E-2
E-3
E-2
6-2
E-3
E-?
E-l
E-2
E-2
-1.'778103 -j.600551
E-2
E-J
E-l
E+i
E+1
6+i)
E+1
E+C
tro
l’io
1.137119
-4.871597
-9.97723!;
-,.17269?
-'.i05WG
i.'33554
-'.P336Pj
-',.150423
-2.799597
2.317R9t‘l
i.5096894
1.33334'0
lto
1.2761174
E-2 E-3
.'02
E-2 '.244
-8.420
1.721855
1.068291
2.877846
2.382910
-2.405647
2.722552
G.116854
5.3*5058
7.166399
I.265405
-3.366298
-4.~84027
-,.8x789
1.223004
-3.770944
-:'.604161
1.473154
-1.595523
-9.787065
-i.r50544
-'.14700h
i.501660
-1.727238
-n.886403
-1.564109
‘.2899228
: .4315428
E-2
E-l
E-l
E-l
II+,
E-e
E-3
E-2
U-j
E-2
E-2
E-3
E-2
E-l
E-2
~-2
E-2
E-3
E-I
E,l
E+i
EtL?
E+,
Et0
E+O
E+j
Et2
E+,
1.3537107
3.178
Q.980
2.390
E-l
E-3
E-2
1.594051
'.727305
2.832418
2.465502
->.452454
2.875891
5.738091
4.685648
i.050121
’ .268297
-3.578870
-,.86115?
,,.168650
1.243815
-,.066174
-7.941239
j-364055
-I.P92079
-9.516932
-1.113e30
-1.105444
".464516
-1.831650
-c.098828
-'.885406
2.2676334
- .3689398
E-L
E-7
E-'
E-l
El,
E-2
E-,
E-i
E-2
E-Z
E-L
E-3
E-2
E-l
E-l
E-2
E-2
E-l
E-l
6+1
E+l
E+O
E+i
E+O
E+O
Et,
E+L
E+3
JOBARD AND CHEDIN
470
of the potential energy function with respect to the dimensionless normal ccordinates (ql, 421,q22,43) of each one of the nine isotopic species we have studied. This expansion is written as
(3) with e(q)
We have computed,
=
qlm(q212
+
with this new solution,
q222)"q3".
the values of the spectroscopic
constants
TABLE VI COMPARISON BETWEEN EXPERIMENTAL AND CALCULATED VALUES OF SOME ENERGY LEVELS (in cm-')
W60, vibrational level
(1) -E
E exp
(1000)1 (1000),1
(11)
’
CFilC~
E
exP
-E
oc7
0.002
0.006
-0.001
-0.
(I)
i ribrational talc 1
level
OllU
E
exP
-E
(II) talc )
E
-E exP
-0. DC11
-0.007
I:ll'o)I
ca1c
0.020
-c.o05
-0.020
0.007
0220
0.040
0.002
0002
0.030
O.cill
0221
O.O,?O
(21’1 II
0.040
0.003
oo"l
0.001
-0.001
0.010
-0.004
-0.020
0.003
oo"3
0.06C
0.020
(21’1),, (21'1) III
I:11'0),,
’ %’ 602
o.002
16012c180 -¶-
vibrational level
0001
(11)
(1)
IEexp -Ecalc rE
i exP -Ecalc
-0.060
V'ibrational
level
-E ’ exp talc'
-0.040
-0.005
ooO2
-0.c50
O.Cl4
( IO01 1I
-0.050
-0.011
-0.070
-0.033
-0.020
0.004
-0.120
-0.011
-0.019
(10°1)11
-0.02C
-0.013
-0.06c
0.019
( loo1
(2221 111
-0.29o
0.017
01'2
(2221 JIII
-0.430
0.050
01'
(I) : value of 1971 (Ref.(l))
column (II): new value.
(II 1 E
oo"l
-0.1 YO
column
(I) -E ) exp talc
0.026
(lOOl)I
0112
E
II
3
POTENTIAL. FUNCTION OF CO,
471
TABLE VII
VALUES OF THE NORM GIVEN BY Eq.(4) FOR 9 SPECIES OF CO2
Isotopic species
Norm t0N201 (in cm-*)
'%'60*
2.3
10-4
'%'60*
I.7
10-3
16012c~80
1.2
10-3
16012c170
2.7
10~~
~2~18~ 2
8.0
IO-~
16013c180
2.6
1O-3 1
I
--I 10-l
occurring in the expression6 of the vibrational energy, referred to the minimum of the potential ; they are given in Table V for the nine species. In order to illustrate, by a few significant examples, the improvement brought by these new iterations, we compare in Table VI the new values of the discrepancies between experimental and calculated values of some representative vibrational levels for 12C1602, WsOz and 16012C1s0 molecules with the values of the discrepancies obtained in 1971 (1). The calculation, for each isotopic species respectively, of the “norm” N2, given by the expression X2 = 5 AE,2Pi/ 2 Pi i=l
i=l
(4)
(that is to say, the weighted sum of the squares of the deviations AE; between experimental and calculated energy values of a set of n observed levels), leads to the nine numerical values of Table VII. These values correspond to a mean deviation of a few hundredths of cm-‘, which is in very good agreement with the present accuracy of the experimental data. Particularly, the best levels of the principal isotopic molecule 12C1602 are reproduced within a few thousandths of cm-‘. 6 See Ref. (2). Our X’s are defined as being one-half of the X’s of Ref. (1).
472
JOBARD IV. CRITICAL
A. Relative Importance
AND
ANALYSIS
of the “Potential
CHEDIN OF THE
SOLUTION
Coejicients”
The numerical tests which were carried out with the new set of the 28 “potential coefficients” Fi (given in Table I) aimed at examining the influence of a modification of the value of one coefficient on the mean value of the deviations (or better, on the “norm”) between theory and experiment. We have represented some of the results in Figs. l-4. For each of the 28 potential coefficients, the percentage of modification of the coeffkient value is plotted along the X-axis, and the change of the “norm,” characterized by the factor multiplying the reference norm obtained with the nonmodified value of the coefficient, is plotted along the Y-axis. For example, Fig. 1 shows that a 1% increase of the value of the F33 coefficient leads to a “norm” equal to 3.10 f6 times the reference norm. We must notice that the scales on each figure are different, but most of the 28 curves reveal an important increase of the norm for a rather small change of the coefficient value, except the curves of Fig. 4, relative to the variations of the coefficients F 1~22 and FwB~~. Nevertheless, all these curves have the same shape : they are almost exactly parabolas, having as their minimum a 0% variation of the coefficient. This means that the reference solutionthat is to say, the new potential energy function-constitutes a minimum stable with respect to the modification of the value of one of the coefficients. IJ~(x% ),‘N* (0%)
-2% FIG. 1. The relative
potential
coefficient
variation value.
-1%
of the norm
0%
t1%
+2%
[see Eq. (4)] as a function
OF/F
of relative
variations
of one
POTENTIAL
FUNCTION
473
OF COz
t
-2% FIG.
potential
2. The relative variation coefficient value.
-1%
of the norm
c!%
+2%
+1%
[see Eq.
(4)] as a function
.&F/F
of relative
variations
of one
If we define the “importance” of a coefficient as being the magnitude of the increase of the norm yielded by a 1% variation of the accepted coefficient values, it is then possible to classify the 28 coefficients, for instance, in decreasing order of importance. This classification is given in Table VIII: -The column.
index giving
the order of importance
of each coefficient
is given in the first
-The coefficients appear in separate columns according to whether they belong to the terms of order 0, 1,2,3, or 4 in the series expansion of the potential energy function. --In the last column are given the values of the relative increases of the “norm.” B. Discussiot~ In this theoretical frame, let us recall that, by the expansion of the potential energy function order n - 2 to the energy.’ Thus, as Tlo, VI, Vz, ators of degrees 2, 3, 4, 5, and 6, they contribute, 7Actually, it is sometimes difficult energy of a nondiagonal operator.
to evaluate
assumption, an operator of degree n in is supposed to give a contribution of ‘v3, and V4 contain, respectively, operrespectively, to the energies of orders
the order
of magnitude
of the contribution
to the
474
JOBARD
AND CHEDIN
0, 1, 2, 3, and 4. Furthermore, when the expansion of the potential is carried out with respect to internal quasinormal coordinates and if we consider only low values of the vibrational quantum numbers, the convergence of the potential expansion is ensured by the values of the coefficients Fi. So, it is important to check that the classification of the “potential coefficients F;,” in decreasing order of their absolute values, is in agreement with these assumptions. This classification, given in Table IX, shows that it is generally verified. If we compare Table IX with Table VIII, which gives the classification of the potential we can see that they are, as a whole, coefhcients in decreasing order of “importance,” slightly different. The essential differences between the two classifications are set forth on Table IX by arrows pointing out the rows occupied by the same coefficients in Table VIII. So, we must notice the partrcular importance of the coefficients F22, F2222, F222222,and Fa3, F 3333, Fr3333, F333333.That importance can essentially be explained by experimental reasons regarding the characteristics of the set of levels whose agreement between theory and experiment is studied : (a) Among the 100 levels of the principal isotopic species which compose the set of levels we use, the vibrational levels involving the quantum number 7~2are associated with statistical weights, the sum of which is three times greater than the sum of the statistical weights associated with the levels involving or, and four times greater than
~*(x%)/‘N~(o%) ,L
4
0%
-10%
0%
+10%
+20%
AF/F
FIG. 3. The relative variation of the norm [see Eq. (411 as a function of relative variations Jotential coefficient value.
of one
POTENTIAL FUNCTION OF COP
L
I
-20%
I
I
I
I
-10%
0%
+10%
+20%
475
I
.
AF/F
FIG. 4. The relative variation of the norm [see Eq. (4)] as a function of relative variations of one potential coefficient value.
the sum of the weights associated with the levels involving vs. So, the choice of the statistical weights, which are to some extent related to the accuracy of the experimental data, favors the peculiar importance of the coefficients Fz, FXW, and FKWXW (b) The other characteristic is that the set of levels used contains highly excited levels : -First, there are levels involving high values of ~3; seven levels are such that v3 is larger or equal to 5, while vr is not involved in any level with a value greater than 4; those levels are, particularly, levels belonging to the ooOv3series* (the ~3 value going up to 9). This explains the peculiar importance of the F 3333,FLM, and F333333coefficients : effectively, the successive contact transformations to which the original Hamiltonian is submitted are such that these coefficients, for instance, strongly contribute to the spectroscopic constant ~333; for example, a variation of ~333in the energy expression of the level CNY9is multiplied by 729 (va3 = 9”) and contributes to the deviation (observed value minus calculated value) which occurs itself in the norm by its square. The fact 8 See Ref. (9).
JOBARD AND CHEDIN
476
TABLE VIII CLASSIFICATION
OF THE Fi COEFFICIENTS WITH RESPECT TO
THEIR "IMPORTANCE".
(DECREASING ORDER).
coefficients
27 28 a
_-.U_*.~~_-~__._.______ 11122 71 I_-_~~___-__^___~______I_ 112222 1.1
For exemple , 12233 stands for the coefficient F multiplies
the operator
R R2R2 123
12233 which in the potential expansion.
that there are levels involving high ~3 values explains in this way the importance of the coefficients mentioned before, and greatly contributes to their accurate determination. -Second, the set of levels also contains many levels which are strongly coupled by the Fermi resonance, and consequently involve rather large values of the “quantum
POTENTIAL
FUNCTION
01: CO2
477
number ~2” compared to the mean value of zlrfor the whole set of levels. This contributes as well to emphasizing the importance of the F22, F2222, Fzzzzzz coefficients. In conclusion, we see that the order of importance of the ‘Lpotential coefficients,” defined by reference to the criterion of their respective influences on the norm of the discrepancies between theoretical and experimental values, takes into account not only the respective values of the coefficients, but also the accuracy of experimental data, and the nature of the vibrational levels constituting the set of levels we use. TABLE IX CLASSIFICATION
OF THE
Fi COEFFICIENTS WITH RESPECT To
THEIR ASSOLUTE VALUE. (DECREASING OXDER)
Fi coefficients 0th
lSt
xder
order
1
33b
2
11
3
22'
IFiI
a
rd 3 oMer
2nd order
th 4 order
I
(in cm-I)
230.975 710.118 335.545
4
133
5 6
111 122
266.178
-+--
46.485
\
29.256
t _
-
21.070
1133
7 8
'7.063
3333
9
11133
10
13333
II
11111
12 _____ 13
_
3.148 -~-_ 1.686 113333
1.615
117733 .---_.___._
1.373 ---.... 0.963
14 -.~
6.073 __-__. _~_3.546
0.896
------_-.___ :lI;ii::
0.832 ____ 0.711 0.597 ___.. 0.465
I
0.275
I
0.161
0.106 0.077
0.044
a See footnote of Table VIII
222233
0.024
112222
0.004
JOBARD
478
AND CHEDIN TABLE X
ITERATIONS CARRIED OUT WITH LESS THAN 28 NON-ZERO
Fi COEFFICIENTS
26 COEFFICIENTS
27 COEFFICIEN
I No
of the iteration 167. 2.2 1.6
F222233
I
1 2 3
2.1 1.5
25 COEFFICIENTS
+I F
223333
C. Cancellation
11. 3.8 2.3 1.5
of the Potential
i 3 4
Coqqicients
24 COEFFICIENTS
Nonsignificantly
Dijerent
from Zero
Some of the less important coefficients (in the lower part of Table VIII) are not necessary for the accurate determination of the vibrational energies of the present set of levels. It is indeed possible to obtain a similar agreement between theory and experiment when one of these coefficients is constrained to zero during several iterative cycles. Likewise, a satisfactory norm is reached when different combinations of two, three, four, or even five unimportant coefficients are simultaneously set equal to zero. As examples, the results of some of these trials are given in Table X.g The coefficients whose values are kept equal to zero during iterations are set down in the first column; the values of the ratios (N*/N*n,r) of the norms N*, obtained at the end of each iterative cycle, to the reference norm N2nor, obtained by working with the full set of the 28 accepted potential coefficient values, are given in the second column. It appears that several solutions with 27,26,2.5, or 24 coefficients are satisfactory and, more precisely, lead to an agreement between theory and experiment, close to the agreement yielded by using the full set of 28 coefficients. This means that the values of five or six coefficients (Fww, F11122,F222233,FwB~, F12222,F223333)are not significantly different from zero. *We could not intend, for obvious reasons regarding computation time, to try all the possible combinations of coefficients to be cancelled out simultaneously; nevertheless the tests, restricted in numbers, that we carried out are conclusive.
POTENTIAL
FUNCTION
Ok’ COr
479
It seems reasonable to state that the values of these coefficients will be better defined when it is possible to work with a set of levels at the same time more numerous and more accurate, that is to say, composed of levels highly excited in wl, vz, and zrS,and systematically measured with better accuracy. ACKNOWLEDGMENT
We wish to express our gratitude to Professor G. Amat and to Dr. Z. Cihla for helpful discussions throughout the course of this work. RECEIVED :
March 31, 1975 REFERENCES
1. Z. CIHLAANDA. CH~DIN,J. Mol. Spectrosc. 40, 337 (1971). 2. I. JOBARD,Thesis, University of Paris, 1974.
3. A. C&DIN, Thesis, University of Paris, 1971. J. A. CHBDINANDZ. CIHLA,J. Mol. Spectrosc. 45, 475 (1973). 5. Z. CIHLAANDA. CH~DIN, J. Mol. Spectrosc. 47, 531 (1973). 6. A. C&DIN ANDZ. CIHLA:J. Mol. Spedrosc. 47, 542 (1973). 7. A. CH~DINANDZ. Crrry J. Mol. Spectrosc. 49, 289 (1974). 8. A. CHBDINANDZ. C~FILA,J. Mol. Spectrosc. 47, 554 (1973). 9. T. K. MCCURBIN,JR., J. PL~VA,R. PULFREY,W. TELFAIR, ANDT. TODD,J. Mol. Spectrosc. 49, 136 (1974).