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New analysis of the perturbation of the C2 Ballik-Ramsay system from high-resolution Fourier spectrometry
JOURNAL
OF MOLECULAR
SPECTROSCOPY
109, 334-344 (1985)
New Analysis of the Perturbation of the C2 Ballik-Ramsay System from High-Resolution Fourier Spectrometry F. ROUX,* G. WANNOUS,~
F. MICHAUD,*
AND J. VERGES+
*Laboratoire de Spectro&trie Ionique et Molkdaire. UniversitC Claude Bernard-Lyon I, 43 Boulevard du I I Novembre 1918. 69622 Villeurbanne. France; tFaculti des Sciences, Universite’ Libanaise. Beyrouth-Hadeth. Lebanon: and SLaboratoire Aimh Cotton, CNRS II, Brit. 505, 91405 Orsav. France
The b’Z;-a311, Ballik-Ramsay system of Cz perturbed by the ‘Zgf state was previously partially deperturbed. Therefore, errors were made in the assignment of lines around the perturbation. In this work, the perturbations observed in the b3Z; - u = 0, 1, 2, and 3 levels were completely reduced. The treatment required a theoretical model which includes secondorder terms in the off-diagonal interactions. 0 1985 Academic press, h.
INTRODUCTION
The aim of deperturbing a band system is to obtain parameters having physical significance, which may in turn, be used to predict frequencies of new transitions. Deperturbation techniques are especially important when the interacting state cannot be reached directly. The method of direct fitting of measured wavenumbers in band spectra has become increasingly sophisticated during the past few years. This method first requires the choice of a model Hamiltonian matrix written in terms of molecular constants. The eigenvalues of this matrix, obtained by diagonalization of the model Hamiltonian, are used to compute transition frequencies, which are then compared with experimental values. Then, least-squares adjustments of molecular parameters are made to give eigenvalues which reproduce the observed spectrum. This approach is particularly suitable to the fitting of perturbed spectra. The local perturbing state can be included directly in the Hamiltonian matrix, and this allows the coupling constants describing the interaction, as well as the usual molecular parameters of the perturbed and perturbing state, to be determined simultaneously in a statistically rigorous manner. The present work is concerned with a new Fourier transform study of the b3Z;-a311, Ballik-Ramsay system of C2 discussed in Ref. (I). In this previous analysis, the authors experienced some difficulty in achieving a complete reduction of the perturbations of the 32; state by the ‘2: state. When one type of perturbed level is taken into account, either F, or F3, the reduction is good, but in the 0022-2852185 $3.00 Copyright 0
1985 by Academic Press. Inc.
All rights of reproduction in any form reserved.
334
ANALYSIS
OF THE
BALLIK-RAMSAY
335
SYSTEM
complete fitting the results are less satisfactory and the discrepancies between experimental and calculated wavenumbers lie between 10 X low3 and 80 X 10m3 cm-‘. The purpose of the present work has been to carry out a complete deperturbation of the perturbed Ballik-Ramsay system of Cz. This system has been excited in an electrodeless discharge through CO. The experimental conditions were described in a previous paper (2). The crossing points between the 32; state and the ‘2: state are illustrated in the plots of E, (cm-‘) against J(J t- 1) given in Fig. 1. The magnitudes of perturbations in the rotational levels of the 3Z; state are given in Table I.
I
/’
V=fj
,,$
/’ /
/’ I’
,,I’ /
/’
.’
’
E
,,-c=s
‘CITi’)
5000
10000
5000 1000
2000
3000
4000
J FIG. I. Total perturbations.
energy
curves
of the b3Z;
and the ‘Z+g states
(J+l)
of Cz for the location
of the studied
336
ROUX ET AL. TABLE I Perturbations in the Rotational Levels of the ‘2, State Perturbing state b32-
J
X12,’
g
(v)
F,
(v)
Level
50 0=
3
(+
52
0.099)
(-
0.170)
40 lb
4
5
62 (+
42
64
0.329)
(-
0.056)
50
52
(+
0.262)
(-
0.128)
(+
0.099)
(-
1.038)
(+
0.256)
(-
0.133)
(+
0.107)
(-
1.058)
(+
0.095)
(-
0.172)
(+
0.223)
(-
0.087)
(+
0.107)
(-
0.169)
(+
0.229)
(-
0.091)
(+
0.077)
(-
0.124)
(+
0.070)
(-
0.130)
26 zc
Level
F3
28
38
40
I2 gd
CO
6
Numbers
in
parentheses
denote
the
displacement
+ denotes
that
the
level
is
displaced
to
higher
-
that
the
level
is
displaced
to
lower
denotes
a)
Results
from
the
b)
Results
from
the
I-O
and
l-l
bands
c)
Results
from
the
2-O
and
2-1
bands
d)
Results
from
the
3-l
and
3-2
bands
of
the
energies
level
in
I4
cm
-I
:
and
energy
O-O band
THEORETICAL
MODEL
Following previous work (I), the matrix elements of the a311, state have been written taking into account the interactions with the 32:, ‘II,, and ‘2: states. by a ‘22: state, The energy matrix of the upper state, 3Z;, which is perturbed includes off-diagonal terms. To improve the deperturbation, it becomes necessary to include second-order terms in the off-diagonal interaction. The usual form of the first-order spin-orbit interaction is
The spin-spin Freed (3).
interaction
is zero,
according
to
calculations
performed
by
ANALYSIS OF THE BALLIK-RAMSAY
SYSTEM
TABLE II Matrix Elements of the 32; State Perturbed by the ‘2: State of the C2 Molecule
< 3’0
/“13x;
>=9&[2B3
-y Ceff
> = A*
-
4 Dy(x+l) + Hz(6x2+20x+E)]
3zeff
A1 (x+2) + A2x
Definitions of the parameters -----___-----__-___ _-----_-- ~(~ll). poll), s(~JI), o('Z*). o('ll) 20 0,(3lI) = z 311 b?L+)
P,(3w
- 1 3"
2 0 - E(311)]
4(ooBo* + Qo*Bo) [E(3Z+, - E(3WJ
46,2 9/n)
o,(‘n)
= z 3,
[E(3Z+) - E(+I)]
-1
[< -
3z+IHsO11n >]*
1, CE(3r+)-
E(‘il)~
[< 3z+[HSOl~e+ o$r+)
with
=
MS0 =
1 I~+
‘12
- E(‘r+)]
Hamiltonianepin-orbitre
2a, =
[E(3X+)
-<*=
111 AL+ 1 n-
0)
1 11 BL+ II n = 0 )
(convention de Freed)
x = .I (J+l) Upper and lower signs refer to e and f level> req,ertive;y
337
338
ROUX ET AL. TABLE III Laboratory Observations of the C2 Ballik-Ramsay System (cm-‘) v’
vll
0
1
2
5632.1002
(10)
1
7080.2330
(10)
5462.2264
(10)
2
8506.1912
120)
6888.1873
(15)
8292.0522
(16)
0
3 4
3
6697.3655
(14)
8079.2095
(12)*
5
*
7867.7682
unperturbed
(14)*
bands
The two parameters,
A, and AZ, representing
A,
=
c
the second-order
interaction,
are
(‘z”lt AL+132,)(32,1B(r)13Z”,) -6 - J%
U#d
and A2 =
c
(‘z”lmm)(‘&dl~ AL+13Z”,)
U#U
The matrix The matrix in H.
elements of the perturbed element of the perturbing
NUMERICAL
Ev - E, 32; state have been collected in Table II. state, ‘Xl, has been truncated at the term
ANALYSIS AND EXPERIMENTAL
RESULTS
The bands which were studied are listed in Table III. The experimental wavenumbers are deposited with the Editor of the Journal of Molecular Spectroscopy.’ The molecular constants of the perturbed 3Z, state are given in Table IV. A global treatment based on two bands was successfully applied to three vibrational levels (v = 1, 2, and 3). The unperturbed levels u = 4 and 2, = 5 were reached by one
’ Please write to F. Roux if you are interested in obtaining these data.
ANALYSIS OF THE BALLIK-RAMSAY
339
SYSTEM
TABLE IV Molecular Constants of the ‘2; State of C2 (cm-‘) v-0
3
2
1
O.LO32
(16)
0.1044
(10)
3 (14)
0.1075
0.1090
f? (15)
0.1078
5a (20)
0.1092
(20)
Lff B3
(21
1.44154
(2)
1.49049
(2)
1.47419
(2)
1.45784
0.6210
(8)
0.6223
(7)
0.6226
0.17
(6)
- 0.13
(4)
- 0.13
(5)
- 0.20
(6)
0.23
(9)
0.26
(9)
0.20
(14)
0.50
(20)
1.42514
(3)
1.40878
(3)
bff DL x 105
‘(3
x
IO2
-
(10)
0.6257
(14)
0.6238
- 0.392
(10)
0)
0.6261
- 0.410
Lff 11 HxX
10
B.rndS
2-o 2-I
1-o 1-1
o-o
0
0
3-1 3-2
4-2
5-3
b RMS
N
=
3.4
3.2
3.4
3.4
2.8
2.7
293
507
415
402
131
95
TABLE V Parameters of the C, b3Z; State (cm-‘)
*a&
5632.1039
we
1470.374
WeXe
(10) (7) (3)
11.143
WeYe
0.0128
(4)
BCe
1.49864
(5)
%
0.01629
(3)
YB
-
Dde
x
BeJO
0.6200
(18)
0.0015
(8)
0.1019
(12)
0.0021
(5)
-
Ee e %
-
(5)
0.000009
105
2 aT b
"
=
The
Te
+ we
origin
of
("4)
T
is
-w
x ee
v=
0
("+$)
level
3 + WeY,
of
a3nu
("++
state
2 c
Bv
=
Be
-
aB
(v+$
+
eB
(v++)
3 +
yB
+
...
2 dD
e
" E"
= De
=
Ee
+
-
O-D (v+$
aF
'v+$
1
+
8,
iv+
+
....
4 + WeLe
("+$
+
....
(Y+$
+
....
(10)
(5)
340
ROUX ET AL. TABLE VI Parameters (cm-‘) between the b’Z; and XlF&f States of C2
InteWtiOn Perturbing state x1x+
This
work
(12)
AMIOT
3
0
2.36
1
2.73
(61
x IO4
Al
CY?
4
b
a
ff
b3C-
t”;
x 104
*2
2.72
(30)
1.39
(4)
1.19
(4)
2.63
(40)
1.25
(3)
1.07
(2)
(4)
0.49
(3)
5
2
2.05
(5)
2.23
(10)
0.57
6
3
0.82
(2)
0.35
(10)
0
0
a Value
obtained
when
A2
is
set
equal
to
zero
Value
obtained
when
A,
is
set
equal
to
zero
zero
when
b
A,
or
A2 are
significant
set
equal
to
it
is
not
possible
to
obtain
value.
TABLE VII Rotational Perturbation* in the F, u = 2 Level (2-1 Band) Observed Minus Calculated Wavenumbers (cm-‘) P
J
R
a
-*
b
21 22 + 0.026
23
-
+ 0.041
25
-
-
0.000
+ 0.046
-
o.oM)
0.004 + 0.107
26 27
+ 0.108
+ 0.001
-
0.176
-
-
0.037
+ 0.005
-
0.024
-
-
28 29
-
31
0.169
0.042
+ 0.021
32 33
-
+ 0.029
+ 0.003
+ 0.040
-
+ 0.105
+ 0.001
-
0.174
+ 0.000
-
0.044
+ 0.000
-
0.030
-
x
10
+ 0.002
-
-
0.001
0.002
0.001
-L l
a
F1e F2. F3
before
simultaneously
reduction
b after
reduction
fitted
0.001
0.002
0.001
30
-
+ 0.031 0.004
24
b
a
(RMS
:
3,4
-3
cm
-1)
0.007
ANALYSIS OF THE BALLIK-RAMSAY
341
SYSTEM
TABLE VIII Rotational Perturbation* in the F3 u = 2 Level (2-1 Band) Observed Minus Calculated Wavenumbers (cm-‘) P
J
R b
a
33 34 35 36 37
+ 0.036
+0.001
+ 0.060
+ 0.002
+ 0.001
+ 0.036
+0.058
+0.001
+0.226
-0.001
-a.091
+0.003
+0.229
38 39
43
-0.035
-0.018
l
a b
Fl,F before after
2 ’
F3
simultaneously
+0.054
*O.OOl
+0.221
-0.003
-0.091
+ 0.005
- 0.038
+o.ooi
-0.024
-0.001
+ 0.001
+0.002
44
+ 0.002
+0.001
-0.034
42
+0.033
+o.ooo
-0.091
40 41
b
a
+0.001
fitted
(RMS
:
3,4
x
IO
-3
cm
-1)
reduction reduction
band only. The equilibrium values of the molecular constants for the ‘2; state are presented in Table V. The interaction parameters between the 3E; and ‘2; states are listed in Table VI. This table also gives a comparison between the values of the first-order spin-orbit terms obtained in this work and those found by Amiot et al. (I). The values are of the same magnitude, and they fulfill the theoretical predictions made by Clementi (4) and Veseth (5). It should be noted that only the F3 level of u = 3 is observed (as shown in Table I), which explains why the quoted uncertainty of this level is relatively large. A simultaneous determination of the second-order parameters, A, and AZ, is not possible. Various tests show that they are of the same order of magnitude, and that they are of equal importance as regards the improvements they make to the theoretical model. Tables VII and VIII show the contribution of these parameters for the F, and F3 levels, respectively, of the vibrational level o = 2 of the 3x; state close to the perturbation. To conclude this work, we have calculated the coefficients for the development of the total wavefunction of the 3Z; state which is perturbed by the pure F,,F3, and ‘2; wavefunctions. From this we have deduced the mixing coefficients of F,,
342
ROUX ET AL.
FIG. 2. Variations against J of the percentage of the character F, (- - -), F3 (p), the perturbed F, (u = 2) level.
and ‘Zg+(-a-) in
Fj, and ‘Zi, respectively, with all the other states. Figures 2, 3 and 4 illustrate the variation of these coefficients. CONCLUSION
We have improved the study of the perturbation of the ‘2; state by the ‘Zl state. The introduction of a second-order parameter allows the model to be somewhat refined. The noticeable effect is that the perturbations are reduced to the limit imposed by the accuracy of the experiments (2 or 3 X 10m3cm-‘). The results given by Amiot et al. (I) arise from a fit taken from one level alone (F,or F3), and the root mean square of the residuals was given as 7.3 X 10m3cm-‘. Similarly, a fit including the whole system gave a standard deviation of 80 X lop3 cm-]. A slightly more detailed investigation of the experimental results for the different transitions has shown that these anormalies stem both from an incomplete model for the deperturbation of the band (it included only the first-order terms in the off-diagonal interaction) and from errors in the assignment of lines around the
FIG. 3. Variations against J of the percentage the perturbed Fj (u = 2) level.
of the character
F, (- - -), F, (-),
and I&Z (-.
FIG. 4. Variations against J of the percentage the perturbing ‘2: (u = 5) level.
of the character
F, (- - -). F2 (-),
and ‘Zb
(-
-) in
.
-) in
344
ROUX
ET AL.
perturbations. Furthermore, the results confirm observations in the Phillips system b’II,-X12,+ (6). A new deperturbation of this system is currently in progress. RECEIVED:
June
22,
1984 REFERENCES
I. C. AMIOT, J. CHAUVILLE, AND J. P. MAILLARD, J. Mol. Spectrosc. 75, 19-40 (1979). 2. C. EFFANTIN, F. MICHAUD, F. Roux, J. d’INCAN, AND J. VERGES, J. Mol. Spectrosc. 92, 349-362 ( 1982). 3. K. F. FREED, J. Chem. Phys. 45,42 14-4241 (1966). 4. E. CLEMENTI AND K. S. PITZER, J. Chem. Phys. 32,656-662 (1960). 5. L. VESETH, Canad. J. Phys 53,299-302 (1975). 6. J. CHAUVILLE, J. P. MAILLARD, AND A. W. MANTZ, J. Mol. Spectrosc. 68, 399-411 (1977).
OF MOLECULAR
SPECTROSCOPY
109, 334-344 (1985)
New Analysis of the Perturbation of the C2 Ballik-Ramsay System from High-Resolution Fourier Spectrometry F. ROUX,* G. WANNOUS,~
F. MICHAUD,*
AND J. VERGES+
*Laboratoire de Spectro&trie Ionique et Molkdaire. UniversitC Claude Bernard-Lyon I, 43 Boulevard du I I Novembre 1918. 69622 Villeurbanne. France; tFaculti des Sciences, Universite’ Libanaise. Beyrouth-Hadeth. Lebanon: and SLaboratoire Aimh Cotton, CNRS II, Brit. 505, 91405 Orsav. France
The b’Z;-a311, Ballik-Ramsay system of Cz perturbed by the ‘Zgf state was previously partially deperturbed. Therefore, errors were made in the assignment of lines around the perturbation. In this work, the perturbations observed in the b3Z; - u = 0, 1, 2, and 3 levels were completely reduced. The treatment required a theoretical model which includes secondorder terms in the off-diagonal interactions. 0 1985 Academic press, h.
INTRODUCTION
The aim of deperturbing a band system is to obtain parameters having physical significance, which may in turn, be used to predict frequencies of new transitions. Deperturbation techniques are especially important when the interacting state cannot be reached directly. The method of direct fitting of measured wavenumbers in band spectra has become increasingly sophisticated during the past few years. This method first requires the choice of a model Hamiltonian matrix written in terms of molecular constants. The eigenvalues of this matrix, obtained by diagonalization of the model Hamiltonian, are used to compute transition frequencies, which are then compared with experimental values. Then, least-squares adjustments of molecular parameters are made to give eigenvalues which reproduce the observed spectrum. This approach is particularly suitable to the fitting of perturbed spectra. The local perturbing state can be included directly in the Hamiltonian matrix, and this allows the coupling constants describing the interaction, as well as the usual molecular parameters of the perturbed and perturbing state, to be determined simultaneously in a statistically rigorous manner. The present work is concerned with a new Fourier transform study of the b3Z;-a311, Ballik-Ramsay system of C2 discussed in Ref. (I). In this previous analysis, the authors experienced some difficulty in achieving a complete reduction of the perturbations of the 32; state by the ‘2: state. When one type of perturbed level is taken into account, either F, or F3, the reduction is good, but in the 0022-2852185 $3.00 Copyright 0
1985 by Academic Press. Inc.
All rights of reproduction in any form reserved.
334
ANALYSIS
OF THE
BALLIK-RAMSAY
335
SYSTEM
complete fitting the results are less satisfactory and the discrepancies between experimental and calculated wavenumbers lie between 10 X low3 and 80 X 10m3 cm-‘. The purpose of the present work has been to carry out a complete deperturbation of the perturbed Ballik-Ramsay system of Cz. This system has been excited in an electrodeless discharge through CO. The experimental conditions were described in a previous paper (2). The crossing points between the 32; state and the ‘2: state are illustrated in the plots of E, (cm-‘) against J(J t- 1) given in Fig. 1. The magnitudes of perturbations in the rotational levels of the 3Z; state are given in Table I.
I
/’
V=fj
,,$
/’ /
/’ I’
,,I’ /
/’
.’
’
E
,,-c=s
‘CITi’)
5000
10000
5000 1000
2000
3000
4000
J FIG. I. Total perturbations.
energy
curves
of the b3Z;
and the ‘Z+g states
(J+l)
of Cz for the location
of the studied
336
ROUX ET AL. TABLE I Perturbations in the Rotational Levels of the ‘2, State Perturbing state b32-
J
X12,’
g
(v)
F,
(v)
Level
50 0=
3
(+
52
0.099)
(-
0.170)
40 lb
4
5
62 (+
42
64
0.329)
(-
0.056)
50
52
(+
0.262)
(-
0.128)
(+
0.099)
(-
1.038)
(+
0.256)
(-
0.133)
(+
0.107)
(-
1.058)
(+
0.095)
(-
0.172)
(+
0.223)
(-
0.087)
(+
0.107)
(-
0.169)
(+
0.229)
(-
0.091)
(+
0.077)
(-
0.124)
(+
0.070)
(-
0.130)
26 zc
Level
F3
28
38
40
I2 gd
CO
6
Numbers
in
parentheses
denote
the
displacement
+ denotes
that
the
level
is
displaced
to
higher
-
that
the
level
is
displaced
to
lower
denotes
a)
Results
from
the
b)
Results
from
the
I-O
and
l-l
bands
c)
Results
from
the
2-O
and
2-1
bands
d)
Results
from
the
3-l
and
3-2
bands
of
the
energies
level
in
I4
cm
-I
:
and
energy
O-O band
THEORETICAL
MODEL
Following previous work (I), the matrix elements of the a311, state have been written taking into account the interactions with the 32:, ‘II,, and ‘2: states. by a ‘22: state, The energy matrix of the upper state, 3Z;, which is perturbed includes off-diagonal terms. To improve the deperturbation, it becomes necessary to include second-order terms in the off-diagonal interaction. The usual form of the first-order spin-orbit interaction is
The spin-spin Freed (3).
interaction
is zero,
according
to
calculations
performed
by
ANALYSIS OF THE BALLIK-RAMSAY
SYSTEM
TABLE II Matrix Elements of the 32; State Perturbed by the ‘2: State of the C2 Molecule
< 3’0
/“13x;
>=9&[2B3
-y Ceff
> = A*
-
4 Dy(x+l) + Hz(6x2+20x+E)]
3zeff
A1 (x+2) + A2x
Definitions of the parameters -----___-----__-___ _-----_-- ~(~ll). poll), s(~JI), o('Z*). o('ll) 20 0,(3lI) = z 311 b?L+)
P,(3w
- 1 3"
2 0 - E(311)]
4(ooBo* + Qo*Bo) [E(3Z+, - E(3WJ
46,2 9/n)
o,(‘n)
= z 3,
[E(3Z+) - E(+I)]
-1
[< -
3z+IHsO11n >]*
1, CE(3r+)-
E(‘il)~
[< 3z+[HSOl~e+ o$r+)
with
=
MS0 =
1 I~+
‘12
- E(‘r+)]
Hamiltonianepin-orbitre
2a, =
[E(3X+)
-<*=
111 AL+ 1 n-
0)
1 11 BL+ II n = 0 )
(convention de Freed)
x = .I (J+l) Upper and lower signs refer to e and f level> req,ertive;y
337
338
ROUX ET AL. TABLE III Laboratory Observations of the C2 Ballik-Ramsay System (cm-‘) v’
vll
0
1
2
5632.1002
(10)
1
7080.2330
(10)
5462.2264
(10)
2
8506.1912
120)
6888.1873
(15)
8292.0522
(16)
0
3 4
3
6697.3655
(14)
8079.2095
(12)*
5
*
7867.7682
unperturbed
(14)*
bands
The two parameters,
A, and AZ, representing
A,
=
c
the second-order
interaction,
are
(‘z”lt AL+132,)(32,1B(r)13Z”,) -6 - J%
U#d
and A2 =
c
(‘z”lmm)(‘&dl~ AL+13Z”,)
U#U
The matrix The matrix in H.
elements of the perturbed element of the perturbing
NUMERICAL
Ev - E, 32; state have been collected in Table II. state, ‘Xl, has been truncated at the term
ANALYSIS AND EXPERIMENTAL
RESULTS
The bands which were studied are listed in Table III. The experimental wavenumbers are deposited with the Editor of the Journal of Molecular Spectroscopy.’ The molecular constants of the perturbed 3Z, state are given in Table IV. A global treatment based on two bands was successfully applied to three vibrational levels (v = 1, 2, and 3). The unperturbed levels u = 4 and 2, = 5 were reached by one
’ Please write to F. Roux if you are interested in obtaining these data.
ANALYSIS OF THE BALLIK-RAMSAY
339
SYSTEM
TABLE IV Molecular Constants of the ‘2; State of C2 (cm-‘) v-0
3
2
1
O.LO32
(16)
0.1044
(10)
3 (14)
0.1075
0.1090
f? (15)
0.1078
5a (20)
0.1092
(20)
Lff B3
(21
1.44154
(2)
1.49049
(2)
1.47419
(2)
1.45784
0.6210
(8)
0.6223
(7)
0.6226
0.17
(6)
- 0.13
(4)
- 0.13
(5)
- 0.20
(6)
0.23
(9)
0.26
(9)
0.20
(14)
0.50
(20)
1.42514
(3)
1.40878
(3)
bff DL x 105
‘(3
x
IO2
-
(10)
0.6257
(14)
0.6238
- 0.392
(10)
0)
0.6261
- 0.410
Lff 11 HxX
10
B.rndS
2-o 2-I
1-o 1-1
o-o
0
0
3-1 3-2
4-2
5-3
b RMS
N
=
3.4
3.2
3.4
3.4
2.8
2.7
293
507
415
402
131
95
TABLE V Parameters of the C, b3Z; State (cm-‘)
*a&
5632.1039
we
1470.374
WeXe
(10) (7) (3)
11.143
WeYe
0.0128
(4)
BCe
1.49864
(5)
%
0.01629
(3)
YB
-
Dde
x
BeJO
0.6200
(18)
0.0015
(8)
0.1019
(12)
0.0021
(5)
-
Ee e %
-
(5)
0.000009
105
2 aT b
"
=
The
Te
+ we
origin
of
("4)
T
is
-w
x ee
v=
0
("+$)
level
3 + WeY,
of
a3nu
("++
state
2 c
Bv
=
Be
-
aB
(v+$
+
eB
(v++)
3 +
yB
+
...
2 dD
e
" E"
= De
=
Ee
+
-
O-D (v+$
aF
'v+$
1
+
8,
iv+
+
....
4 + WeLe
("+$
+
....
(Y+$
+
....
(10)
(5)
340
ROUX ET AL. TABLE VI Parameters (cm-‘) between the b’Z; and XlF&f States of C2
InteWtiOn Perturbing state x1x+
This
work
(12)
AMIOT
3
0
2.36
1
2.73
(61
x IO4
Al
CY?
4
b
a
ff
b3C-
t”;
x 104
*2
2.72
(30)
1.39
(4)
1.19
(4)
2.63
(40)
1.25
(3)
1.07
(2)
(4)
0.49
(3)
5
2
2.05
(5)
2.23
(10)
0.57
6
3
0.82
(2)
0.35
(10)
0
0
a Value
obtained
when
A2
is
set
equal
to
zero
Value
obtained
when
A,
is
set
equal
to
zero
zero
when
b
A,
or
A2 are
significant
set
equal
to
it
is
not
possible
to
obtain
value.
TABLE VII Rotational Perturbation* in the F, u = 2 Level (2-1 Band) Observed Minus Calculated Wavenumbers (cm-‘) P
J
R
a
-*
b
21 22 + 0.026
23
-
+ 0.041
25
-
-
0.000
+ 0.046
-
o.oM)
0.004 + 0.107
26 27
+ 0.108
+ 0.001
-
0.176
-
-
0.037
+ 0.005
-
0.024
-
-
28 29
-
31
0.169
0.042
+ 0.021
32 33
-
+ 0.029
+ 0.003
+ 0.040
-
+ 0.105
+ 0.001
-
0.174
+ 0.000
-
0.044
+ 0.000
-
0.030
-
x
10
+ 0.002
-
-
0.001
0.002
0.001
-L l
a
F1e F2. F3
before
simultaneously
reduction
b after
reduction
fitted
0.001
0.002
0.001
30
-
+ 0.031 0.004
24
b
a
(RMS
:
3,4
-3
cm
-1)
0.007
ANALYSIS OF THE BALLIK-RAMSAY
341
SYSTEM
TABLE VIII Rotational Perturbation* in the F3 u = 2 Level (2-1 Band) Observed Minus Calculated Wavenumbers (cm-‘) P
J
R b
a
33 34 35 36 37
+ 0.036
+0.001
+ 0.060
+ 0.002
+ 0.001
+ 0.036
+0.058
+0.001
+0.226
-0.001
-a.091
+0.003
+0.229
38 39
43
-0.035
-0.018
l
a b
Fl,F before after
2 ’
F3
simultaneously
+0.054
*O.OOl
+0.221
-0.003
-0.091
+ 0.005
- 0.038
+o.ooi
-0.024
-0.001
+ 0.001
+0.002
44
+ 0.002
+0.001
-0.034
42
+0.033
+o.ooo
-0.091
40 41
b
a
+0.001
fitted
(RMS
:
3,4
x
IO
-3
cm
-1)
reduction reduction
band only. The equilibrium values of the molecular constants for the ‘2; state are presented in Table V. The interaction parameters between the 3E; and ‘2; states are listed in Table VI. This table also gives a comparison between the values of the first-order spin-orbit terms obtained in this work and those found by Amiot et al. (I). The values are of the same magnitude, and they fulfill the theoretical predictions made by Clementi (4) and Veseth (5). It should be noted that only the F3 level of u = 3 is observed (as shown in Table I), which explains why the quoted uncertainty of this level is relatively large. A simultaneous determination of the second-order parameters, A, and AZ, is not possible. Various tests show that they are of the same order of magnitude, and that they are of equal importance as regards the improvements they make to the theoretical model. Tables VII and VIII show the contribution of these parameters for the F, and F3 levels, respectively, of the vibrational level o = 2 of the 3x; state close to the perturbation. To conclude this work, we have calculated the coefficients for the development of the total wavefunction of the 3Z; state which is perturbed by the pure F,,F3, and ‘2; wavefunctions. From this we have deduced the mixing coefficients of F,,
342
ROUX ET AL.
FIG. 2. Variations against J of the percentage of the character F, (- - -), F3 (p), the perturbed F, (u = 2) level.
and ‘Zg+(-a-) in
Fj, and ‘Zi, respectively, with all the other states. Figures 2, 3 and 4 illustrate the variation of these coefficients. CONCLUSION
We have improved the study of the perturbation of the ‘2; state by the ‘Zl state. The introduction of a second-order parameter allows the model to be somewhat refined. The noticeable effect is that the perturbations are reduced to the limit imposed by the accuracy of the experiments (2 or 3 X 10m3cm-‘). The results given by Amiot et al. (I) arise from a fit taken from one level alone (F,or F3), and the root mean square of the residuals was given as 7.3 X 10m3cm-‘. Similarly, a fit including the whole system gave a standard deviation of 80 X lop3 cm-]. A slightly more detailed investigation of the experimental results for the different transitions has shown that these anormalies stem both from an incomplete model for the deperturbation of the band (it included only the first-order terms in the off-diagonal interaction) and from errors in the assignment of lines around the
FIG. 3. Variations against J of the percentage the perturbed Fj (u = 2) level.
of the character
F, (- - -), F, (-),
and I&Z (-.
FIG. 4. Variations against J of the percentage the perturbing ‘2: (u = 5) level.
of the character
F, (- - -). F2 (-),
and ‘Zb
(-
-) in
.
-) in
344
ROUX
ET AL.
perturbations. Furthermore, the results confirm observations in the Phillips system b’II,-X12,+ (6). A new deperturbation of this system is currently in progress. RECEIVED:
June
22,
1984 REFERENCES
I. C. AMIOT, J. CHAUVILLE, AND J. P. MAILLARD, J. Mol. Spectrosc. 75, 19-40 (1979). 2. C. EFFANTIN, F. MICHAUD, F. Roux, J. d’INCAN, AND J. VERGES, J. Mol. Spectrosc. 92, 349-362 ( 1982). 3. K. F. FREED, J. Chem. Phys. 45,42 14-4241 (1966). 4. E. CLEMENTI AND K. S. PITZER, J. Chem. Phys. 32,656-662 (1960). 5. L. VESETH, Canad. J. Phys 53,299-302 (1975). 6. J. CHAUVILLE, J. P. MAILLARD, AND A. W. MANTZ, J. Mol. Spectrosc. 68, 399-411 (1977).