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Quasi-linear four-atomic molecules in the vibron model
JOURNAL
OF MOLECULAR
SPECTROSCOPY
156,
190-200 ( 1992)
Quasi-linear Four-Atomic Molecules in the Vibron Model F. IACHELLO, N. MANINI, ’ AND S. Oss2 Centerfor Theoretical Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06511 We study the transition from linear to bent four-atomic molecules within the framework of the vibron model. We consider quasi-linear molecules and analyze the “classical” case of fulminic acid (HCNO). The algebraic force-field constants we obtain can be used to make predictions for further investigations of this molecule. 0 1992Academic PISS, IN. I. INTRODUCTION
It has been recently shown that the vibron model (1-4) provides a description of a linear four-atomic molecule of comparable (or superior) quality to that of Dunham expansions with perturbations. This conclusion is based on the analysis of rigid linear molecules, such as acetylene, HCCH, (5) and monofluoracetylene, HCCF (6). In this article we address the question of whether or not the vibron model provides an equally good description of quasi-linear four-atomic molecules. These molecules stand in between the rigid linear and rigid bent molecules and are therefore difficult to treat by conventional methods ( 7). The “classical” case of this type of molecules is fulminic acid, HCNO. We have therefore done a roto-vibrational analysis of this molecule, the results of which are reported here. 2. VIBRON MODEL
The vibron model for four-atomic molecules has been described in Ref. (5), where it was applied to the study of the linear molecules, HCCH, HCCD, and DCCD. In previous publications (8, 9) we also discussed how the transition from linear to bent triatomic molecules can be described in the vibron model by continuously changing one parameter. We apply the same method for four-atomic molecules. As an example of a quasi-linear molecule we consider HCNO. When applying the vibron model to a four-atomic molecule, one must specify how the bonds are coupled. We number the bonds of HCNO as in Fig. 1, and use the coupling scheme ( 12) 3, i.e., we first couple the HC bond to the CN bond and then the combination to the NO bond. The coupled wavefunctions are still given by the group theoretic pattern, Eq. (2.1) of Ref. (5 ), which we rewrite here in an abbreviated form l[NJ,
[N21,
IN313
(WI,
O),
(wz,O),
(71,
721,
(03,0),
(01,
621,
J, M),
(2.1)
except that now, in view of the labeling of bonds given in Fig. 1, we have the following correspondence between group quantum numbers and local vibrational linear quantum numbers, V,, Vb, V,, I$, Vi: ’ Permanent address: Dipartimento di Fisica, Universita di Trento, 38050 Povo (Trento ) , Italy. 2 Permanent address Dipartimento di Fisica, UniversitA di Trento and INFN, Gruppo Collegato di Trento, 38050 Povo (Trento), Italy. 0022-2852192 $5.00 Copyright
0 1992 by Academic
All rights of reproduction
190 Press, Inc.
in any form reserved.
VIBRON MODEL FOR QUASI-LINEAR
191
MOLECULES
FIG. 1. Bond coordinates of fulminic acid.
w, = 71
=
N, - 2v,,
w3 = Nx - 2v,,
w2 = N2 - 2vb,
72 = Ie,
N, + N2 - (2v, + 2~ + II,),
CT,= N, + N2 + Nj - (2v, +
2Vb + 2V,
3. HAMILTONIAN
+ Vd +
C72 = Id •,- 1,.
Ve),
(2.2)
OPERATOR
The Hamiltonian operator of quasi-linear molecules can be built from the operators discussed already in Ref. (5). The local vibrational Hamiltonian for quasi-linear molecules is H ‘oca’= E,, + A,d, + A&
+ A3t3 + A&,
+ A&,2
+ A,&23
+ /7,23c,23.
(3.1)
This Hamiltonian differs from that of linear molecules, Eq. ( 3.2) of Ref. (5)) by the addition of the two terms b&,2 and A,23C,23. It is diagonal in the basis Eq. (2.1) with eigenvalues E(V,,
Vb, Vc, V& V?) = & - 4A,[(N,
- 4A3[(N3
+
+ l)V,
1)~~ - v:] - A,2[2(N,
- V;] - 4Az[(Nz
+ 1)Vb - vf]
+ Nz + 1)(2v, + 22)b+ v,)
- (2V, + 2Vb + V,)2 - /:] + A,, [ (N, + Nz + 1) - (2V, + 2Vb + V,)] -
A,23[2(N, + N2 + N3 + 1)(2V,
- (2V, + 2Vb +
I’L-1
+ 2Vb + 2V, + Vd + V,)
2v, + v,d+ 2),)2 - (id + &)2]
+ ii,23 [(N, + Nz + N3 + 1) - (2V, + 2Vb + 2V, + 2)d+ V,)] 1id + &I.
(3.2)
It is characterized by seven parameters, A,, AZ, A3, A,2, b,, , A,*., and 2123. In order to separate the contribution of different vibrations, it is convenient to replace the operators C,2 and C,23 by their “reduced’ forms defined as
1 (N +N +lJ+cN3+lj ~~~~=6,23-(N,+N~+N3+1) 1. ~?z=G~-(N,+Nz+~)
6,
c2
(N,+l)+(N2+l)
,
G2
I
C3
2
(3.3)
^
The superscript R characterizes the corresponding Hamiltonian parameters. The addition of the two terms A,2C,2 and A,23C,23 allows one to study departures from the linear geometry toward the bent geometry. When A,2 and A,23 are zero, the energy eigenvalues, Eq. (3.2), are those typical of a linear four-atomic molecule. As A,2 and A,23 vary from zero, one goes toward a bent geometry. The transition is particularly simple if we consider molecules in which only one atom is in a bent equilibrium position. As we change A,2 continuously from zero to A,2 = 2Ay2, we go
192
IACHELLO, MANINI, AND OSS
from a linear configuration to a quasi-linear configuration with one bond bent. The former configuration is characterized by the quantum numbers )I&, vb, vc, vj, vk J, 44) of Eqs. (2.1) and (2.2). The latter configuration can be characterized by the quantum numbers 1v,, vb, vc, vt, vr; K, J, M), where K = ld + 1, is the projection of the angular momentum along the axis of the linear part of the molecule and vf = (v, - lL1)/2. Th’is situation is shown schematically in Fig. 2 and is very similar to that discussed in Ref. (9) for triatomic molecules. 4. HIGHER ORDER TERM
The Hamiltonian ( 3.1) provides the starting point for vibrational analyses of quasilinear molecules. We consider here as quasi-linear molecules those for which the linearity parameter, yo, defined in Ref. ( 7) has an intermediate value between that of the linear limit ,y = - 1.O, and that of the bent limit, y$nf = + 1.O. Fulminic acid is one such molecule, since y. 2 -0.5. Because of this negative value, it is appropriate to use still the notation of linear molecules and label the states by the usual vibrational quantum numbers of linear molecules, Eq. (2.2). The description of quasi-linear molecules, Eq. (3.1)) can be improved in three ways: (i) by adding Majorana-type interactions, as discussed in Ref. (5) for HCCH, H = HI“==’ + X,2ti1z + A&&
+ X2&&.
(4-l)
With the introduction of these terms we go from a local description to a normal description. Correspondingly, the local quantum numbers v,, l)b, v,, v$, v: change into vl, v2, v3, vi, v 2. In HCNO, Majorana-type terms do not appear to play a major role, as evidenced by the fact that we obtain fits of equally good quality with and without them. This implies that HCNO behaves as a local molecule, and we can therefore use the quantum numbers v,, ub, vc, v?, v? and VI, 13, ~3, v$, 00 in an interchangeable way, see Table I; (ii) by adding l-splitting operators, as discussed in Refs. (5, 6). Among the various terms of this type, two appear to play an important role in HCNO,
‘Zoo 1
z200----~~----__-_
ZOO’
E(d)BOO -
I’ll--b c
0°3’ 003’=j, ====q*,,\,\\
p\ ’ ‘\ ‘.‘,\ 0°Z2-\\ 1 ‘oO--__-- o o2. --+_~------\
I i near
\
Z20
I:
zoo
co....
o"l
I-o.,....
2,...
1'0 r:o.,.*....
v6dv,Le
FIG. 2. Correlation diagram for the lowest bending states of four-atomic molecules. As the parameter 6,~ changes continuously from AI2 = 0 to 2, z = 2Ayz, the structure of the spectrum changes from that on the left-hand side (linear) to that on the right-hand side (bent).
VIBRON MODEL FOR QUASI-LINEAR
MOLECULES
193
TABLE I Vibrational Quantum Numbers for HCNO Normal
Local
Mode
“1
“a
C-H smtch
3336.1
T
“b
C-N stretch
2195.8
“3
“c
N-O saetch
1253.4
“4 41
Vd&
C-N-O bend
531.2
“5 5
.,p,
H-C-N bend
224.1
Energy (cm-‘)
(4.2) The I-splitting operator CT3 is defined in Ref. (5), while the l-splitting operator d13 is defined in Ref. (6) ; (iii) by introducing higher order terms (powers and products of the c operators). Higher order terms appear to be of crucial importance for a description of HCNO. The second-order terms which appear to play a major role are
As mentioned in Refs. (8, 9), the vibron model is an expansion in terms of Morse roto-vibrators. While the Morse potential provides an excellent description of stretching modes, the potential energy surface in the bending coordinates of a quasi-linear molecule may deviate considerably from that of a single Morse oscillator. This is particularly true in HCNO for the n5 vibration. This implies that if one wants to have a very good description of this mode, one may have to add even higher order terms in 2r5. The behavior in o5 is controlled by the operator d I2. We have thus considered also expansions including cubic terms in c 12, i.e., ( &)3 (sixth-order anharmonicities in v,). 5. VIBRATIONAL ANALYSIS OF HCNO
Fulminic acid, HCNO, and its isotopic substitutions provide a “classical” example of quasi-linear four-atomic molecules. The difference between a linear and a quasiTABLE 11 Comparison between HCCF and HCNO (All Values“ in cm-’ ) HCCF
HCNO
“1
stretch CH
3357
saetch
CH
3336
v2
stretch CC
2239
stretch CN
2195
“3
stretch CF
1061
stretch NO
1254
v4’
bend CCF
367
bend CNO
538
knd
224
+4
2~42 -2~40 bad
v5’ ~52 -2~50
HCC
584
+20
a Rounded to nearest integer.
HCN
-41
194
IACHELLO, MANINI, AND 0%
linear molecule can be seen in Table II, where properties of HCNO and the linear molecule HCCF are compared. While the stretching vibrations of these two molecules are very similar, HCNO is characterized by some unusual properties of the bending vibration, v5. This vibration has a peculiarly low frequency and a negative (and large) value of the A - 2: difference, 2v : - 2~:. As one can see from Fig. 2, these are the peculiar features of a transition from linear to bent. The Hamiltonian operator of the preceding sections allows one to do a vibrational analysis of HCNO. We have done this analysis in two steps. In the first step, we have used the lowest order Hamiltonian, Eq. (3.3), plus the operators (CG)’ and ( c12)*, a total of nine parameters. In the second step we have added eight more higher order terms, as shown in Table III, for a total of 17 parameters. The vibron numbers N, , N2, and N3 characterizing the HC, CN, and CO bonds have been kept fixed to their free values.obtained from the known anharmonicity of the corresponding diatomic molecules through N = (w,/u,x~) - 2. They are .. N1 = 43;
N2 = 156,
N3 = 133.
(5.1)
The data base of vibrational energies of IKNO includes a total of 36 assigned vibrational levels (10-14). Their band origins are listed in Table IV. One can note the abundance of spectral data concerning the v5 vibration. Conversely, spectral data regarding the vl , v2, v3, v4 overtones and combination modes are much fewer. The results of our fits are shown in the same table. Fit I is the nine-parameter fit, while Fit II is the 17-parameter fit. The corresponding rms deviations are A,,,(9 parameters) = 10.8 cm-‘, A,,,( 17 parameters) = 2.0 cm-‘.
TABLE III Values of the Algebraic Parameters (in cm-‘) used in the Anharmonic Analysis of HCNO (the Vibron Numbers Are N, = 43, N2 = 156, and N3 = 133) Q)
0
A;’
0.191819(+02)
-0.193460(+02)
A;’
0.348634(+01)
-0.351006(+01)
A;’
0.235376(+01)
-0.234787(+01)
A::
O.S8578(+00)
-0.55105’6(+00)
I”
0.113747(+00)
-0.139839(+00)
A;;
0.816409(+00)
-0.805192(+Gu)
p
0.150354(-01)
-0.394692(-02)
IS
AZ’
0.146534(-03)
0.182394(-03)
XgJ
0.361415(-04)
-0.146834(-03)
xg;,“’
0.162716(-03)
5 ?
-0.125577(-03)
AE
-0.378280(-03)
AZ
-0.526957(-04) 0.447574(-04)
A9 A”,’ WT
-0.858478(-04)
A”,’ um
0.209769(-05)
AZ
0.242097(-07)
Am
(36kvds)
10.8
2.0
(5.2)
VIBRON MODEL FOR QUASI-LINEAR
195
MOLECULES
TABLE IV Vibron Model Fits to HCNO (All Energies in cm-‘)
Z+ x+ cx+ x+ c+ FL+ IC+ c z+ YL+ Z+ IZ+ n II n 11 n1 n n
n n n n n A A A A A A a 0 a
0002w 000 1111 000 1’1’ 0000020 OlOOw 1ooOw 0010%0 0001’31 0001131 0000040 0100?20 1 ooOc2o 002Ow 000 1100 OOOO%l 0001’22 000 1’20 0101’00 1001w 0000031
010001’ 100001’ 001001l 002@11 000 1’1’ 000OQ~ 000 1131 OOOO”42 OlOOJ?22 1000022 000 1’22 0000033 0000053
l-
000
l-
0000044
1133
H
0000055
exp
tit I
exp-I
fit II
1069.946 e 758.510 e 760.151 e 541.764 e 2195.761 b 3336.110 a 1254.227 b 1405.143 e 1407.109 e 1247.745 c 27 14.457 b 3836.130 a 2498.292 b
1084.2 764.6 764.6 539.6 2178.3
14.2
1071.7 758.2 759.3 546.6 2193.0
537.567 e 224.485 c 1033.210 c 1077.826 d 2722.940 b 3870.570 a 872.506 C 2401.419b 3532.489 b 1482.851 b 2733.226 d 762.790 = 500.598 c 1409.669 e 1227.018 c 2662.462 b 3785.050 d 1039.208 e 817.504 = 1605.740 c 1356.200 C 1168.8OoC 1550.200 C
6.1 4.5 -2.1 -17.5
3309.4 1257.0 1392.7 1392.7 1261.1 2716.2 3850.4
-26.7 2.8 -12.5 -14.4 13.4
2505.0 547.1 227.5 1035.0 1083.4 2722.0 3853.2
6.7 9.5 3.1 1.8 5.6 -0.9 -17.4
858.8 2405.6
-13.7 4.2
3538.3 1481.3 2725.9
5.8 -1.6 -7.3 8.5 -2.7 -10.4
771.3 497.9 1399.3 1219.0 2674.9 3809.1 1038.4 809.8 1618.9 1347.0 1161.9 1553.0
1.7 14.2
-8.0 12.5 24.1 -0.8 -7.7 13.1 -9.2 -6.9 2.8
3336.6 1253.8 1404.9 1407.1 1247.8 2711.9 3833.2 2498.5 535.4 222.6 1032.8 1082.2 2725.2 3868.7 869.7 2402.5 3535.3 1482.6 2733.3 760.0 498.6 1407.2 1226.5 2664.5 3786.6 1037.8 817.8 1607.9 1358.6 1170.5 1547.8
exp-II
1.8 -0.3 -0.9 4.8 -2.8 0.5 -0.4 -0.2 0.0 0.0 -2.6 -2.9 0.2 -2.2 -1.9 -0.4 4.4 2.2 -1.9 -2.8 1.1 2.8 -0.2 0.1 -2.8 -2.0 -2.5 -0.5 2.0 1.5 -1.4 0.3 2.1 2.4 1.7 -2.4
aRef. (10); bRef. (II); CRef. (12); dRef. (13); eRef. (14).
A further test of the quality of these fits is provided by the available measurements of two Q-branch hot transitions ( 12). These were not included in the fits, since the energy of the lower state is unknown. Comparing the experimental data with the calculations we find (all values in cm-’ ): (i) (00021):i - (OO020)2. (ii) (00021)3 - (00020)*.
Exp.: 22 1.6; Exp.: 223.5;
Calc. I: 218.7; Calc. I: 219.0;
Calc. II: 222.5. Calc. II: 224.5.
One can see again that the deviation of fit II is less than 2.0 cm-‘. In view of the nonstandard character of the u5 vibration it is instructive to plot the quantity
196
IACHELLO, MANINI, AND OSS
AG(v) =
G(u + l)n - G(v),,
v = even,
G(u + 1)~ - G(n)n,
v = odd,
(5.3)
where G(V) are the band origins and 2)= u5. This quantity has a typical behavior for harmonic and anharmonic vibrations of a linear molecule and for harmonic and anharmonic vibrations of a bent molecule, shown in the top portion of Fig. 3. In HCNO the quantity AG( u) has a rather peculiar behavior, shown in the bottom part of Fig. 3, where it is compared with the results of the calculation. This behavior is neither that of linear nor of a bent molecule, but it is nonetheless well reproduced by the calculation. 6. HAMILTONIAN
OPERATOR INCLUDING
ROTATIONS
The vibron model allows one to do also rotational analyses of spectra. Since this aspect has not been discussed in Ref. (5)) we address it specifically here. In the conventional approach, the rotational coefficient B, is expanded as (6.1)
where n is the total number of vibrations (n = 5 in linear and quasi-linear four-atomic molecules). A similar expansion can be written in the vibron model, by considering the rotational Hamiltonian EP* = [B() + 2 AjJ”d&P,
(6.2)
k
AG 6)
Lineor
Harmonic
Linear
Anhornonic
Bent
Anhornonic
FIG, 3. (a) Behavior of the quantity AG( u) of Eq. (5.3) for harmonic and anharmonic (Morse) vibrations of a linear and a bent molecule. (b) The quantity AG( u) as measured in HCNO (squares) and as calculated in fit II (continuous line).
VIBRON
MODEL
FOR QUASI-LINEAR
MOLECULES
197
’ where ck denotes generically one of the operators of Eq. ( 3.1) and the superscript RV is introduced in order to distinguish the coefficients in (6.2) from those in ( 3.1). We have used this Hamiltonian to analyze the measured A& values AB, = B,-
Bo.
(6.3)
The method of analysis is as follows. We first do a purely vibrational analysis, as discussed in the previous sections, using the Hamiltonian H = c A,&,
or its higher order counterpart. the Hamiltonian
(6.4)
This gives the parameters, Ak. We next diagnonalize H’ = H + Hrot,
(6.5)
for a given value of J. Since the values of A& are usually rather small ( 10d3- 10e4 cm-’ ), in order to amplify the effect of the rotational terms, thus avoiding a “noise” problem, we use a large value of J. Since the eigenvalues of the j2 operator are J( J + l), one has 1).
H’“’ = [B. + c Af”&]J(J+
(6.6)
k
Subtracting the constant term B. J( J + 1)) one can rewrite H’ in the form H’ = c A;&
(6.7)
The new fit to the experimental values of & + ( A&,) J( J + 1) gives A j,. The rotationvibration parameters are obviously given by RV
Ak
_
-
AL-Art
(6.8)
J(J+
1) ’
Eblc -
-%c
and the calculated A& values by AB,=
The same procedure nal terms.
J(J+
1)
’
can be used if one includes higher order and/or 7. ROTATIONAL
ANALYSIS
(6.9)
nondiago-
OF HCNO
Several rotational constants of HCNO have been measured. We have performed an analysis of the 30 available data with nine parameters, as in Fit I. The rotational parameters are shown in Table V and the corresponding fit in Table VI. The rms deviation is A,,,(9 parameters)
= 0.80 X 10e4 cm-‘.
(7.1)
The 17-parameter fit was not attempted here in view of the fact that the data base is not very extensive. 8. PREDICTIONS
The values of the algebraic parameters, Ak and A;“, can be used to make predictions for further studies of HCNO. As an example of this predictive power, we show in Fig.
198
IACHELLO, MANINI, AND OS TABLE V Values of the Roto-Vibrational Parameters (in 10e4 cm-‘) of HCNO in the Vibron Model (the Vibron Numbers are N, = 43, N2 = 156, and N, = 133) 0.609926(-01) 0.397793(-01) 0.313236(-01) -0.229058(-01) 0.621068(-02) -0.768865(-02) 0.972172(-03) -0.510487(-05) -0.371330(-05)
Am
0.80
(30 levels)
4 all Z states expected in the region 6500-6600 cm-’ with up to o1 + v2 + v3 + ( l/ 2)( u4 + v5) = 7 quanta, surrounding the C-H stretching overtone ( 20000)“, predicted at 6533.5 cm-‘. For these states, as well as for II, A, . . . , states falling in this region, we have also calculated A& values, which can be obtained from us upon request.
TABLE VI Vibron Model Fit to the Roto-vibrational Constants of HCNO (All Values in LO-”cm-‘)
A&p
Abit
exp-flt
15.23 e
14.51
0.72
14.78 c
14.51
0.27
15.56 e -24.40 d
15.02 -24.78
0.54
-10.33 d
-10.15
-0.18
-17.08 *
-16.62
-0.46
25.87 =
26.23
-0.36
25.61 =
26.23
-0.62
25.43 *
23.56
1.87
-11.84 *
-9.84
-2.00
0.38
3.73 *
4.68
-0.95
-32.90 *
-32.89
-0.01
4.80 C
5.43
-0.63
lo.05 *
9.73
0.32
20.54 C
20.41
0.13
-18.81 b
-19.38
0.57
20.81*
21.48
-0.67
-14.75 *
-15.12
0.37
0.32 *
-0.54
0.86
-6.61 *
-6.92
0.31
-23.15 *
-23.23
0.08
13.72 =
15.13
-1.41 -0.90
16.60 *
17.50
28.20 e
26.84
1.36
26.00 *
26.05
-0.05
-6.70 *
-7.38 7.15
0.68
27.67 *
23.37 28.79
-0.09 -1.12
28.00 C
27.40
0.60
7.42 * 23.28 *
“Ref. (II); CRef. (IZ): *Ref. (13); cRct(l4).
0.27
VIBRON MODEL FOR QUASI-LINEAR
6600
-
Old ;--_0035Y~3.031
MOLECULES
199
p--0023Y~0.561
1102~
E(d)
loorP~-[0.731
-
0106T 02029 r-_
6550
-
000 do 2000~
0011Pc0.251
f-
0113Yr0.031
1021Y~1.031
y--ook2i2ro.031 __010&1.431 t=0202i2~0.271
OOlldlp 1006V
;~OOOlO~
030080
6500
’
OOlS%D
L3.161 lOO# ;-ro.00:
FIG. 4. Energy level diagram showing Z bands with up to seven quanta that we calculate in the region 6500-6600 cm-’ of HCNO. The splitting (in cm-‘) between 2+ and Z- state is shown in square brackets.
9. CONCLUSIONS
We have discussed here how to modify the Hamiltonian of linear four-atomic molecules in order to describe quasi-linear four-atomic molecules within the framework of the vibron model. As long as one remains with quasi-linear molecules, the modification is straightforward, and we have used it to analyze vibrations and rotations of fulminic acid, HCNO. This molecule is a “classical” example of quasi-linear molecules. The vibron model appears to describe the corresponding data well. It provides a description of the full dynamics of HCNO within one model Hamiltonian. This Hamiltonian is purely local, in the sense that no operators of the Majorana type have been used. This is in part due to the lack of experimental information on states that are strongly affected by Majorana operators (stretching overtones). The values of the algebraic force field constants, Ak parameters, that we have extracted are based on 36 assigned bands. We note that the data base is much smaller than that of acetylene, C2H2, and monofluoracetylene, HCCF, two molecules that we have analyzed previously. Therefore, our force-field constants may not be so reliable as those of the other two molecules. From this point of view, it would be interesting to extend the measurements to other vibrational states. We note in particular the absence of experimental data on the overtones of the v4 vibration, for example, ( 00020)“. These overtones are important to determine whether or not the v4 bender is strictly linear or quasi-linear. In conclusion, results of the analysis presented here appear to indicate that the vibron model can be safely extended to quasi-linear four-atomic molecules, thus providing a tool to describe these systems. ACKNOWLEDGMENTS This work was supported in part by D.O.E. Grant DE-FGO2-9 1ER40608. We acknowledge useful correspondence with B. P. Winnewisser. RECEIVED:
April 20, 1992
200
IACHELLO, MANINI, AND OSS REFERENCES
1. F. IACHELLO,Chem. Phys. Lett. 78,581-585 (1981). 2. F. IACHELLOAND R. D. LEVINE, J. Chem. Phys. 77,3046-3055 ( 1982). S. 0. S. VAN ROOSMALEN, A. E. L. DIEPERINK,AND F. IACHELLO,Chem. Phys. Left. 85, 32-36 ( 1985). 4. 0. S. VAN ROOSMALEN, F. IACHELLO,R. D. LEVINE, AND A. E. L. DIEPERINK, J. Chem. Phys. 79, 2515-2536 (1983). 5. F. IACHELLO,S. Oss, AND R. LEMUS, J. Mol. Spectrosc. 149, 132-151 ( 1991). 6. F. IACHELLO,S. Oss, AND L. VIOLA, Mol. Phys., in press. 7. B. P. WINNEWISSER,in “Molecular Spectroscopy: Modem Research” (K. Narahari Rao, Ed.), Vol. III, pp. 321-419, Academic Press, New York, 1985. 8. F. IACHELLOAND S. Oss, J. Mol. Spectrosc. 142, 85-107 ( 1990). 9. F. IACHELLO,S. Oss, AND R. LEMUS, J. Mol. Spectrosc. 146, 56-78 ( 1991). 10. B. P. WINNEWISSER,M. WINNEWISSER,AND F. WINTHER, .I. Mol. Spectrosc. 51, 65-96 (1974). 1 I. E. L. FERRETTIAND K. NARAHARI RAO, J. Mol. Spectrosc. 51,97-106 ( 1974). 12. F. WINTHER, J. Mol. Spectrosc. 62, 232-296 ( 1976). 13. P. R. BUNKER, B. M. LANDSBERG,AND B. P. WINNEWISSER,J. Mol. Spectrosc. 79,9-25 ( 1979). 14. B. P. WINNEWISSERAND P. JENSEN,J. Mol. Spectrosc. 101,408-421 (1983).
OF MOLECULAR
SPECTROSCOPY
156,
190-200 ( 1992)
Quasi-linear Four-Atomic Molecules in the Vibron Model F. IACHELLO, N. MANINI, ’ AND S. Oss2 Centerfor Theoretical Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06511 We study the transition from linear to bent four-atomic molecules within the framework of the vibron model. We consider quasi-linear molecules and analyze the “classical” case of fulminic acid (HCNO). The algebraic force-field constants we obtain can be used to make predictions for further investigations of this molecule. 0 1992Academic PISS, IN. I. INTRODUCTION
It has been recently shown that the vibron model (1-4) provides a description of a linear four-atomic molecule of comparable (or superior) quality to that of Dunham expansions with perturbations. This conclusion is based on the analysis of rigid linear molecules, such as acetylene, HCCH, (5) and monofluoracetylene, HCCF (6). In this article we address the question of whether or not the vibron model provides an equally good description of quasi-linear four-atomic molecules. These molecules stand in between the rigid linear and rigid bent molecules and are therefore difficult to treat by conventional methods ( 7). The “classical” case of this type of molecules is fulminic acid, HCNO. We have therefore done a roto-vibrational analysis of this molecule, the results of which are reported here. 2. VIBRON MODEL
The vibron model for four-atomic molecules has been described in Ref. (5), where it was applied to the study of the linear molecules, HCCH, HCCD, and DCCD. In previous publications (8, 9) we also discussed how the transition from linear to bent triatomic molecules can be described in the vibron model by continuously changing one parameter. We apply the same method for four-atomic molecules. As an example of a quasi-linear molecule we consider HCNO. When applying the vibron model to a four-atomic molecule, one must specify how the bonds are coupled. We number the bonds of HCNO as in Fig. 1, and use the coupling scheme ( 12) 3, i.e., we first couple the HC bond to the CN bond and then the combination to the NO bond. The coupled wavefunctions are still given by the group theoretic pattern, Eq. (2.1) of Ref. (5 ), which we rewrite here in an abbreviated form l[NJ,
[N21,
IN313
(WI,
O),
(wz,O),
(71,
721,
(03,0),
(01,
621,
J, M),
(2.1)
except that now, in view of the labeling of bonds given in Fig. 1, we have the following correspondence between group quantum numbers and local vibrational linear quantum numbers, V,, Vb, V,, I$, Vi: ’ Permanent address: Dipartimento di Fisica, Universita di Trento, 38050 Povo (Trento ) , Italy. 2 Permanent address Dipartimento di Fisica, UniversitA di Trento and INFN, Gruppo Collegato di Trento, 38050 Povo (Trento), Italy. 0022-2852192 $5.00 Copyright
0 1992 by Academic
All rights of reproduction
190 Press, Inc.
in any form reserved.
VIBRON MODEL FOR QUASI-LINEAR
191
MOLECULES
FIG. 1. Bond coordinates of fulminic acid.
w, = 71
=
N, - 2v,,
w3 = Nx - 2v,,
w2 = N2 - 2vb,
72 = Ie,
N, + N2 - (2v, + 2~ + II,),
CT,= N, + N2 + Nj - (2v, +
2Vb + 2V,
3. HAMILTONIAN
+ Vd +
C72 = Id •,- 1,.
Ve),
(2.2)
OPERATOR
The Hamiltonian operator of quasi-linear molecules can be built from the operators discussed already in Ref. (5). The local vibrational Hamiltonian for quasi-linear molecules is H ‘oca’= E,, + A,d, + A&
+ A3t3 + A&,
+ A&,2
+ A,&23
+ /7,23c,23.
(3.1)
This Hamiltonian differs from that of linear molecules, Eq. ( 3.2) of Ref. (5)) by the addition of the two terms b&,2 and A,23C,23. It is diagonal in the basis Eq. (2.1) with eigenvalues E(V,,
Vb, Vc, V& V?) = & - 4A,[(N,
- 4A3[(N3
+
+ l)V,
1)~~ - v:] - A,2[2(N,
- V;] - 4Az[(Nz
+ 1)Vb - vf]
+ Nz + 1)(2v, + 22)b+ v,)
- (2V, + 2Vb + V,)2 - /:] + A,, [ (N, + Nz + 1) - (2V, + 2Vb + V,)] -
A,23[2(N, + N2 + N3 + 1)(2V,
- (2V, + 2Vb +
I’L-1
+ 2Vb + 2V, + Vd + V,)
2v, + v,d+ 2),)2 - (id + &)2]
+ ii,23 [(N, + Nz + N3 + 1) - (2V, + 2Vb + 2V, + 2)d+ V,)] 1id + &I.
(3.2)
It is characterized by seven parameters, A,, AZ, A3, A,2, b,, , A,*., and 2123. In order to separate the contribution of different vibrations, it is convenient to replace the operators C,2 and C,23 by their “reduced’ forms defined as
1 (N +N +lJ+cN3+lj ~~~~=6,23-(N,+N~+N3+1) 1. ~?z=G~-(N,+Nz+~)
6,
c2
(N,+l)+(N2+l)
,
G2
I
C3
2
(3.3)
^
The superscript R characterizes the corresponding Hamiltonian parameters. The addition of the two terms A,2C,2 and A,23C,23 allows one to study departures from the linear geometry toward the bent geometry. When A,2 and A,23 are zero, the energy eigenvalues, Eq. (3.2), are those typical of a linear four-atomic molecule. As A,2 and A,23 vary from zero, one goes toward a bent geometry. The transition is particularly simple if we consider molecules in which only one atom is in a bent equilibrium position. As we change A,2 continuously from zero to A,2 = 2Ay2, we go
192
IACHELLO, MANINI, AND OSS
from a linear configuration to a quasi-linear configuration with one bond bent. The former configuration is characterized by the quantum numbers )I&, vb, vc, vj, vk J, 44) of Eqs. (2.1) and (2.2). The latter configuration can be characterized by the quantum numbers 1v,, vb, vc, vt, vr; K, J, M), where K = ld + 1, is the projection of the angular momentum along the axis of the linear part of the molecule and vf = (v, - lL1)/2. Th’is situation is shown schematically in Fig. 2 and is very similar to that discussed in Ref. (9) for triatomic molecules. 4. HIGHER ORDER TERM
The Hamiltonian ( 3.1) provides the starting point for vibrational analyses of quasilinear molecules. We consider here as quasi-linear molecules those for which the linearity parameter, yo, defined in Ref. ( 7) has an intermediate value between that of the linear limit ,y = - 1.O, and that of the bent limit, y$nf = + 1.O. Fulminic acid is one such molecule, since y. 2 -0.5. Because of this negative value, it is appropriate to use still the notation of linear molecules and label the states by the usual vibrational quantum numbers of linear molecules, Eq. (2.2). The description of quasi-linear molecules, Eq. (3.1)) can be improved in three ways: (i) by adding Majorana-type interactions, as discussed in Ref. (5) for HCCH, H = HI“==’ + X,2ti1z + A&&
+ X2&&.
(4-l)
With the introduction of these terms we go from a local description to a normal description. Correspondingly, the local quantum numbers v,, l)b, v,, v$, v: change into vl, v2, v3, vi, v 2. In HCNO, Majorana-type terms do not appear to play a major role, as evidenced by the fact that we obtain fits of equally good quality with and without them. This implies that HCNO behaves as a local molecule, and we can therefore use the quantum numbers v,, ub, vc, v?, v? and VI, 13, ~3, v$, 00 in an interchangeable way, see Table I; (ii) by adding l-splitting operators, as discussed in Refs. (5, 6). Among the various terms of this type, two appear to play an important role in HCNO,
‘Zoo 1
z200----~~----__-_
ZOO’
E(d)BOO -
I’ll--b c
0°3’ 003’=j, ====q*,,\,\\
p\ ’ ‘\ ‘.‘,\ 0°Z2-\\ 1 ‘oO--__-- o o2. --+_~------\
I i near
\
Z20
I:
zoo
co....
o"l
I-o.,....
2,...
1'0 r:o.,.*....
v6dv,Le
FIG. 2. Correlation diagram for the lowest bending states of four-atomic molecules. As the parameter 6,~ changes continuously from AI2 = 0 to 2, z = 2Ayz, the structure of the spectrum changes from that on the left-hand side (linear) to that on the right-hand side (bent).
VIBRON MODEL FOR QUASI-LINEAR
MOLECULES
193
TABLE I Vibrational Quantum Numbers for HCNO Normal
Local
Mode
“1
“a
C-H smtch
3336.1
T
“b
C-N stretch
2195.8
“3
“c
N-O saetch
1253.4
“4 41
Vd&
C-N-O bend
531.2
“5 5
.,p,
H-C-N bend
224.1
Energy (cm-‘)
(4.2) The I-splitting operator CT3 is defined in Ref. (5), while the l-splitting operator d13 is defined in Ref. (6) ; (iii) by introducing higher order terms (powers and products of the c operators). Higher order terms appear to be of crucial importance for a description of HCNO. The second-order terms which appear to play a major role are
As mentioned in Refs. (8, 9), the vibron model is an expansion in terms of Morse roto-vibrators. While the Morse potential provides an excellent description of stretching modes, the potential energy surface in the bending coordinates of a quasi-linear molecule may deviate considerably from that of a single Morse oscillator. This is particularly true in HCNO for the n5 vibration. This implies that if one wants to have a very good description of this mode, one may have to add even higher order terms in 2r5. The behavior in o5 is controlled by the operator d I2. We have thus considered also expansions including cubic terms in c 12, i.e., ( &)3 (sixth-order anharmonicities in v,). 5. VIBRATIONAL ANALYSIS OF HCNO
Fulminic acid, HCNO, and its isotopic substitutions provide a “classical” example of quasi-linear four-atomic molecules. The difference between a linear and a quasiTABLE 11 Comparison between HCCF and HCNO (All Values“ in cm-’ ) HCCF
HCNO
“1
stretch CH
3357
saetch
CH
3336
v2
stretch CC
2239
stretch CN
2195
“3
stretch CF
1061
stretch NO
1254
v4’
bend CCF
367
bend CNO
538
knd
224
+4
2~42 -2~40 bad
v5’ ~52 -2~50
HCC
584
+20
a Rounded to nearest integer.
HCN
-41
194
IACHELLO, MANINI, AND 0%
linear molecule can be seen in Table II, where properties of HCNO and the linear molecule HCCF are compared. While the stretching vibrations of these two molecules are very similar, HCNO is characterized by some unusual properties of the bending vibration, v5. This vibration has a peculiarly low frequency and a negative (and large) value of the A - 2: difference, 2v : - 2~:. As one can see from Fig. 2, these are the peculiar features of a transition from linear to bent. The Hamiltonian operator of the preceding sections allows one to do a vibrational analysis of HCNO. We have done this analysis in two steps. In the first step, we have used the lowest order Hamiltonian, Eq. (3.3), plus the operators (CG)’ and ( c12)*, a total of nine parameters. In the second step we have added eight more higher order terms, as shown in Table III, for a total of 17 parameters. The vibron numbers N, , N2, and N3 characterizing the HC, CN, and CO bonds have been kept fixed to their free values.obtained from the known anharmonicity of the corresponding diatomic molecules through N = (w,/u,x~) - 2. They are .. N1 = 43;
N2 = 156,
N3 = 133.
(5.1)
The data base of vibrational energies of IKNO includes a total of 36 assigned vibrational levels (10-14). Their band origins are listed in Table IV. One can note the abundance of spectral data concerning the v5 vibration. Conversely, spectral data regarding the vl , v2, v3, v4 overtones and combination modes are much fewer. The results of our fits are shown in the same table. Fit I is the nine-parameter fit, while Fit II is the 17-parameter fit. The corresponding rms deviations are A,,,(9 parameters) = 10.8 cm-‘, A,,,( 17 parameters) = 2.0 cm-‘.
TABLE III Values of the Algebraic Parameters (in cm-‘) used in the Anharmonic Analysis of HCNO (the Vibron Numbers Are N, = 43, N2 = 156, and N3 = 133) Q)
0
A;’
0.191819(+02)
-0.193460(+02)
A;’
0.348634(+01)
-0.351006(+01)
A;’
0.235376(+01)
-0.234787(+01)
A::
O.S8578(+00)
-0.55105’6(+00)
I”
0.113747(+00)
-0.139839(+00)
A;;
0.816409(+00)
-0.805192(+Gu)
p
0.150354(-01)
-0.394692(-02)
IS
AZ’
0.146534(-03)
0.182394(-03)
XgJ
0.361415(-04)
-0.146834(-03)
xg;,“’
0.162716(-03)
5 ?
-0.125577(-03)
AE
-0.378280(-03)
AZ
-0.526957(-04) 0.447574(-04)
A9 A”,’ WT
-0.858478(-04)
A”,’ um
0.209769(-05)
AZ
0.242097(-07)
Am
(36kvds)
10.8
2.0
(5.2)
VIBRON MODEL FOR QUASI-LINEAR
195
MOLECULES
TABLE IV Vibron Model Fits to HCNO (All Energies in cm-‘)
Z+ x+ cx+ x+ c+ FL+ IC+ c z+ YL+ Z+ IZ+ n II n 11 n1 n n
n n n n n A A A A A A a 0 a
0002w 000 1111 000 1’1’ 0000020 OlOOw 1ooOw 0010%0 0001’31 0001131 0000040 0100?20 1 ooOc2o 002Ow 000 1100 OOOO%l 0001’22 000 1’20 0101’00 1001w 0000031
010001’ 100001’ 001001l 002@11 000 1’1’ 000OQ~ 000 1131 OOOO”42 OlOOJ?22 1000022 000 1’22 0000033 0000053
l-
000
l-
0000044
1133
H
0000055
exp
tit I
exp-I
fit II
1069.946 e 758.510 e 760.151 e 541.764 e 2195.761 b 3336.110 a 1254.227 b 1405.143 e 1407.109 e 1247.745 c 27 14.457 b 3836.130 a 2498.292 b
1084.2 764.6 764.6 539.6 2178.3
14.2
1071.7 758.2 759.3 546.6 2193.0
537.567 e 224.485 c 1033.210 c 1077.826 d 2722.940 b 3870.570 a 872.506 C 2401.419b 3532.489 b 1482.851 b 2733.226 d 762.790 = 500.598 c 1409.669 e 1227.018 c 2662.462 b 3785.050 d 1039.208 e 817.504 = 1605.740 c 1356.200 C 1168.8OoC 1550.200 C
6.1 4.5 -2.1 -17.5
3309.4 1257.0 1392.7 1392.7 1261.1 2716.2 3850.4
-26.7 2.8 -12.5 -14.4 13.4
2505.0 547.1 227.5 1035.0 1083.4 2722.0 3853.2
6.7 9.5 3.1 1.8 5.6 -0.9 -17.4
858.8 2405.6
-13.7 4.2
3538.3 1481.3 2725.9
5.8 -1.6 -7.3 8.5 -2.7 -10.4
771.3 497.9 1399.3 1219.0 2674.9 3809.1 1038.4 809.8 1618.9 1347.0 1161.9 1553.0
1.7 14.2
-8.0 12.5 24.1 -0.8 -7.7 13.1 -9.2 -6.9 2.8
3336.6 1253.8 1404.9 1407.1 1247.8 2711.9 3833.2 2498.5 535.4 222.6 1032.8 1082.2 2725.2 3868.7 869.7 2402.5 3535.3 1482.6 2733.3 760.0 498.6 1407.2 1226.5 2664.5 3786.6 1037.8 817.8 1607.9 1358.6 1170.5 1547.8
exp-II
1.8 -0.3 -0.9 4.8 -2.8 0.5 -0.4 -0.2 0.0 0.0 -2.6 -2.9 0.2 -2.2 -1.9 -0.4 4.4 2.2 -1.9 -2.8 1.1 2.8 -0.2 0.1 -2.8 -2.0 -2.5 -0.5 2.0 1.5 -1.4 0.3 2.1 2.4 1.7 -2.4
aRef. (10); bRef. (II); CRef. (12); dRef. (13); eRef. (14).
A further test of the quality of these fits is provided by the available measurements of two Q-branch hot transitions ( 12). These were not included in the fits, since the energy of the lower state is unknown. Comparing the experimental data with the calculations we find (all values in cm-’ ): (i) (00021):i - (OO020)2. (ii) (00021)3 - (00020)*.
Exp.: 22 1.6; Exp.: 223.5;
Calc. I: 218.7; Calc. I: 219.0;
Calc. II: 222.5. Calc. II: 224.5.
One can see again that the deviation of fit II is less than 2.0 cm-‘. In view of the nonstandard character of the u5 vibration it is instructive to plot the quantity
196
IACHELLO, MANINI, AND OSS
AG(v) =
G(u + l)n - G(v),,
v = even,
G(u + 1)~ - G(n)n,
v = odd,
(5.3)
where G(V) are the band origins and 2)= u5. This quantity has a typical behavior for harmonic and anharmonic vibrations of a linear molecule and for harmonic and anharmonic vibrations of a bent molecule, shown in the top portion of Fig. 3. In HCNO the quantity AG( u) has a rather peculiar behavior, shown in the bottom part of Fig. 3, where it is compared with the results of the calculation. This behavior is neither that of linear nor of a bent molecule, but it is nonetheless well reproduced by the calculation. 6. HAMILTONIAN
OPERATOR INCLUDING
ROTATIONS
The vibron model allows one to do also rotational analyses of spectra. Since this aspect has not been discussed in Ref. (5)) we address it specifically here. In the conventional approach, the rotational coefficient B, is expanded as (6.1)
where n is the total number of vibrations (n = 5 in linear and quasi-linear four-atomic molecules). A similar expansion can be written in the vibron model, by considering the rotational Hamiltonian EP* = [B() + 2 AjJ”d&P,
(6.2)
k
AG 6)
Lineor
Harmonic
Linear
Anhornonic
Bent
Anhornonic
FIG, 3. (a) Behavior of the quantity AG( u) of Eq. (5.3) for harmonic and anharmonic (Morse) vibrations of a linear and a bent molecule. (b) The quantity AG( u) as measured in HCNO (squares) and as calculated in fit II (continuous line).
VIBRON
MODEL
FOR QUASI-LINEAR
MOLECULES
197
’ where ck denotes generically one of the operators of Eq. ( 3.1) and the superscript RV is introduced in order to distinguish the coefficients in (6.2) from those in ( 3.1). We have used this Hamiltonian to analyze the measured A& values AB, = B,-
Bo.
(6.3)
The method of analysis is as follows. We first do a purely vibrational analysis, as discussed in the previous sections, using the Hamiltonian H = c A,&,
or its higher order counterpart. the Hamiltonian
(6.4)
This gives the parameters, Ak. We next diagnonalize H’ = H + Hrot,
(6.5)
for a given value of J. Since the values of A& are usually rather small ( 10d3- 10e4 cm-’ ), in order to amplify the effect of the rotational terms, thus avoiding a “noise” problem, we use a large value of J. Since the eigenvalues of the j2 operator are J( J + l), one has 1).
H’“’ = [B. + c Af”&]J(J+
(6.6)
k
Subtracting the constant term B. J( J + 1)) one can rewrite H’ in the form H’ = c A;&
(6.7)
The new fit to the experimental values of & + ( A&,) J( J + 1) gives A j,. The rotationvibration parameters are obviously given by RV
Ak
_
-
AL-Art
(6.8)
J(J+
1) ’
Eblc -
-%c
and the calculated A& values by AB,=
The same procedure nal terms.
J(J+
1)
’
can be used if one includes higher order and/or 7. ROTATIONAL
ANALYSIS
(6.9)
nondiago-
OF HCNO
Several rotational constants of HCNO have been measured. We have performed an analysis of the 30 available data with nine parameters, as in Fit I. The rotational parameters are shown in Table V and the corresponding fit in Table VI. The rms deviation is A,,,(9 parameters)
= 0.80 X 10e4 cm-‘.
(7.1)
The 17-parameter fit was not attempted here in view of the fact that the data base is not very extensive. 8. PREDICTIONS
The values of the algebraic parameters, Ak and A;“, can be used to make predictions for further studies of HCNO. As an example of this predictive power, we show in Fig.
198
IACHELLO, MANINI, AND OS TABLE V Values of the Roto-Vibrational Parameters (in 10e4 cm-‘) of HCNO in the Vibron Model (the Vibron Numbers are N, = 43, N2 = 156, and N, = 133) 0.609926(-01) 0.397793(-01) 0.313236(-01) -0.229058(-01) 0.621068(-02) -0.768865(-02) 0.972172(-03) -0.510487(-05) -0.371330(-05)
Am
0.80
(30 levels)
4 all Z states expected in the region 6500-6600 cm-’ with up to o1 + v2 + v3 + ( l/ 2)( u4 + v5) = 7 quanta, surrounding the C-H stretching overtone ( 20000)“, predicted at 6533.5 cm-‘. For these states, as well as for II, A, . . . , states falling in this region, we have also calculated A& values, which can be obtained from us upon request.
TABLE VI Vibron Model Fit to the Roto-vibrational Constants of HCNO (All Values in LO-”cm-‘)
A&p
Abit
exp-flt
15.23 e
14.51
0.72
14.78 c
14.51
0.27
15.56 e -24.40 d
15.02 -24.78
0.54
-10.33 d
-10.15
-0.18
-17.08 *
-16.62
-0.46
25.87 =
26.23
-0.36
25.61 =
26.23
-0.62
25.43 *
23.56
1.87
-11.84 *
-9.84
-2.00
0.38
3.73 *
4.68
-0.95
-32.90 *
-32.89
-0.01
4.80 C
5.43
-0.63
lo.05 *
9.73
0.32
20.54 C
20.41
0.13
-18.81 b
-19.38
0.57
20.81*
21.48
-0.67
-14.75 *
-15.12
0.37
0.32 *
-0.54
0.86
-6.61 *
-6.92
0.31
-23.15 *
-23.23
0.08
13.72 =
15.13
-1.41 -0.90
16.60 *
17.50
28.20 e
26.84
1.36
26.00 *
26.05
-0.05
-6.70 *
-7.38 7.15
0.68
27.67 *
23.37 28.79
-0.09 -1.12
28.00 C
27.40
0.60
7.42 * 23.28 *
“Ref. (II); CRef. (IZ): *Ref. (13); cRct(l4).
0.27
VIBRON MODEL FOR QUASI-LINEAR
6600
-
Old ;--_0035Y~3.031
MOLECULES
199
p--0023Y~0.561
1102~
E(d)
loorP~-[0.731
-
0106T 02029 r-_
6550
-
000 do 2000~
0011Pc0.251
f-
0113Yr0.031
1021Y~1.031
y--ook2i2ro.031 __010&1.431 t=0202i2~0.271
OOlldlp 1006V
;~OOOlO~
030080
6500
’
OOlS%D
L3.161 lOO# ;-ro.00:
FIG. 4. Energy level diagram showing Z bands with up to seven quanta that we calculate in the region 6500-6600 cm-’ of HCNO. The splitting (in cm-‘) between 2+ and Z- state is shown in square brackets.
9. CONCLUSIONS
We have discussed here how to modify the Hamiltonian of linear four-atomic molecules in order to describe quasi-linear four-atomic molecules within the framework of the vibron model. As long as one remains with quasi-linear molecules, the modification is straightforward, and we have used it to analyze vibrations and rotations of fulminic acid, HCNO. This molecule is a “classical” example of quasi-linear molecules. The vibron model appears to describe the corresponding data well. It provides a description of the full dynamics of HCNO within one model Hamiltonian. This Hamiltonian is purely local, in the sense that no operators of the Majorana type have been used. This is in part due to the lack of experimental information on states that are strongly affected by Majorana operators (stretching overtones). The values of the algebraic force field constants, Ak parameters, that we have extracted are based on 36 assigned bands. We note that the data base is much smaller than that of acetylene, C2H2, and monofluoracetylene, HCCF, two molecules that we have analyzed previously. Therefore, our force-field constants may not be so reliable as those of the other two molecules. From this point of view, it would be interesting to extend the measurements to other vibrational states. We note in particular the absence of experimental data on the overtones of the v4 vibration, for example, ( 00020)“. These overtones are important to determine whether or not the v4 bender is strictly linear or quasi-linear. In conclusion, results of the analysis presented here appear to indicate that the vibron model can be safely extended to quasi-linear four-atomic molecules, thus providing a tool to describe these systems. ACKNOWLEDGMENTS This work was supported in part by D.O.E. Grant DE-FGO2-9 1ER40608. We acknowledge useful correspondence with B. P. Winnewisser. RECEIVED:
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