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Empirical anharmonic force field and equilibrium structure of hypochlorous acid, HOCl
13 September 1996
ELSEVIER
CHEMICAL PHYSICS LETTERS Chemical PhysicsLetters 259 (1996) 614-618
Empirical anharmonic force field and equilibrium structure of hypochlorous acid, HOC1 R.M. Escribano
a G.
Di Lonardo b, L. Fusina b
a lnstituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain b Dipartimento di Chimica Fisica e lnorganica, Viale Risorgimento 4, 40136 Bologna, Italy
Received 17 June 1996
Abstract The cubic and quartic force fields of HOCI are investigated on the basis of the most recent experimental data on vibration-rotation interaction constants and anharmonicity constants. Some discrepancies with respect to previously reported ab initio results are found and discussed. The geometrical parameters of this molecule are also evaluated from recent data on the equilibrium values of the moments of inertia.
1. Introduction The involvement of hypochlorous acid in atmospheric reactions has prompted a thorough investigation of the spectrum and potential surface of this molecule. The most recent result on the anharmonic force field, as far as we know, is that of Halonen and Ha [1], who performed an ab initio calculation at third-order Moller-Plesset level to determine the equilibrium structure and the anharmonic force field of this molecule. The experimental vibration-rotation data available at the time (up to 1987) were used to calculate some of the quadratic and cubic force constants. Several high-accuracy spectroscopic investigations have been performed since then, covering a large range of vibrational states [2]. The more extensive determination of the vibration-rotation interaction and anharmonicity constants of HO35Cl and HO37C1 is that of Azzolini et al. [3]. This work provides accurate values for the nine a~ parameters (r = 1-3 for the normal modes, b = A, B or C for the rotational constants) and four anharmonicity con-
stants of each molecule, which can be used for the determination of nine cubic and four quartic force constants, respectively, based on empirical data alone. Data on the equilibrium value of the rotational constants A e, Be and C e for HOC1 are more recent and accurate than those of DOCI, which were derived from a lower resolution work [4]. The latter are nonetheless essential if the geometrical parameters of hypochlorous acid are to be estimated on empirical grounds. Since the determination of the anharmonic force constants involves the equilibrium moments of inertia (see equations below), we have evaluated first the geometrical parameters of hypochlorous acid, and then we have used these results in the estimation of the cubic and quartic force fields.
2. Equilibrium geometry Deeley [4] obtained the equilibrium geometrical parameters using a graphical representation of data
0009-2614/96/$12.00 Copyright© 1996Published by Elsevier Science B.V. All rights reserved. PII S0009-2614(96)00774-9
615
R.M. Escribano et al. / Chemical Physics Letters 259 (1996) 614-618
for HOC1 and DOC1, following a procedure first outlined by Deeley and Mills [5] for HOC1 alone. We have explicitly written the equations for the moments of inertia referred to the principal axes, la(~), I~e) and l~ce), in terms of the equilibrium bond distances r(OH), r(OCl) and interatomic angle 0(HOC1). From these equations, we have obtained the algebraic expressions for the derivatives of the moments of inertia with respect to the structural parameters. In this way we can write a system of equations, where the data are the experimental moments of inertia for three molecules (HO35C1, HO37C1 from Ref. [3] and DO35C1 from Ref. [4]) and the unknowns are the equilibrium parameters. The system can be solved using non-linear least-squares techniques, such as the inverse generalized matrix (IGM) method described in some detail below. The experimental moments of inertia of the first two isotopomers were derived from the values of Watson's determinable combinations of the rotational constants of the ground state, to which the vibrational dependence of the rotational constants (parameters a~ and %b where available) were conveniently added. For DO35C1 the moments of inertia Ib~e) are explicitly given in Ref. [4]. Weights have
Table 1 Moments of inertia 1~e) and inertial deflect Ae of HO35CI, HO37C1 and DO35CI (ill /j,/~2) derived from observed data of Refs. [3] and [4], and values calculated in this work through a least-squares fit to the structural parameters of hypochlorous acid Obs. HO35C1 lff ) 34.049008 In(e) 33.226943 la(e) 0.822741 Ae - 0.000676
Weight
Calc.
Obs. - c a l c .
16 16 16
34.049364 33.226944 0.822420
- 0.000355 - 0.000001 0.000321
Table 2 Equilibrium structural parameters of hypochlorous acid. Uncertainties are given in units of the last digit This work re(OH) (~)
0.96437(13)
re(OCl) (,~) 0e(HOCI) (°)
Ac
4 4 4
34.646124 33.823677 0,822446
0.000282 0.001123 0.000348
1 1 1
36.683084 35.159344 1.523741
-- 0.001084 0.001056 0.000019
-0.001189
DO35C1 /(cc) 36.682000 /n(e) 35.160400 Ia(e) 1.523760 Ae -0.002160
0.9643(5)
1.68897(2) 102.965(19)
1.6891 (2) 102.96(8)
been assigned to the data proportional to the compliance to the inertial defect rule (at equilibrium, Ia(e) = I~e) + I~C~) should be strictly satisfied). The results of the fit, including the weights, are collected in Table 1, and the best estimates of the structural parameters are listed in Table 2, together with those of Deeley [4]. It can be seen that the graphical method used by this author led to accurate results, which agree with those obtained in the present work within the estimated uncertainty for all three parameters. The importance of the DO35C1 data in the determination of the geometrical parameters is evidenced by the results of a fit based on data of the hydrogen isotopomers only. An unrealistic value for the HOCI angle affected by a large uncertainty (94.5 + 4.5 °) is obtained, Moreover, large correlations ( = 99%) are observed between all parameters.
3. Anharmonic force field
The vibrational dependence of the rotational constants is conveniently expressed as a power expansion in the vibrational coordinates, where the firstorder coefficients are the a~ parameters: b r
HO37C1 1~e) 34.646406 In(t) 33.824800 la~e) 0.822795
Ref. [4]
1
1
r<~s
+...
(1)
In this equation and in the following, the rotational constant B stands for A, B or C. The ar° parameters are made up of a harmonic and an anharmonic contribution [6]:
= ,,
(harm) + ,,r
(anharm).
(2)
The harmonic contribution depends on the equilibrium values of the inertia tensor I~e) and the
R.M. Escribano et aL / Chemical Physics Letters 259 (1996) 614-618
616
Coriolis interaction constants ~'rbs,whereas the anharmonic part contains the cubic force constants ~brst: 2(Bb)2 ~ [~ o/b(harnl) ~
3(a~C)2
t°r
b 2
41~e-'-'----7+ ½E$ ( ~ ' ~ )
x
(3)
, (anharm) -
-
(~)
difficult, because the matrix of the coefficients of the unknowns in the expanded equations is almost singular, and cannot be inverted using standard procedures. Therefore, we have been forced to use some specific algebraic tool, and we have chosen the inverse generalized matrix (IGM) method [8], which consists of the following. For a given matrix A such that ATA (where AT is the transpose of A) contains m zero roots, the inverse generalized matrix is a matrix A ~ which can replace the true inverse Awhen A-I does not exist. For instance, A ~ satisfies: AAIA = A; A I A A I = A I.
1/2
ff --
(5)
(.Or
E
~t)rrsabs b 0 9 : / 2 "
(4)
$
If the harmonic force field and the geometry are known, it is possible to estimate the cubic force constants from the observed values of the rotationvibration interaction constants a~(obs). For each normal mode r, there are three equations like (4), each one relative to a rotational constant B. Expansion of the summation in Eq. (4) also gives three cubic constants qbrrl, ¢~rr2 and ~brr3 for each r, which can therefore be estimated from the data. Thus we can evaluate nine cubic constants, i.e. all cubic constants for an asymmetric bent triatomic molecule [7], but one, ~bl23. In practice, the evaluation of the cubic constants for HOCI turns out to be quite
The singular values (identical to the square root of the eigenvalues if the matrix is not singular) of A l are the reciprocal of the non-null singular values of A, and zero for the m zero roots of A. The interest of this method for our problem lies in the simplicity of its application to the solution of least-squares analyses [8]. The system
Ay = b
(6)
when ATA is singular is indeterminate. The solution of minimum length (smallest modulus 1.91) is .9 = A'b,
(7)
where A l therefore replaces the non-available A -l. The application of the IGM method to the determina-
Table 3 Observed values of the anharmonic contribution to cry of HO35C1 and HO37CI, in 10 -2 crtl- t, and the corresponding calculated values obtained using the IGM method (Set t), and fixing some cubic constants to the results of Ref. [1] (Set 2) HO35C1 A axis ( A e = 20.49760 cm - t )
Ql Q2 Q3
B axis (B e = 0.50735 cm - t )
C axis (C e = 0.49509 c m - J)
obs.
Set 1
Set 2
obs.
Set 1
Set 2
obs.
Set 1
Set 2
141.077 - 53.781 4.652
141.041 - 53.778 4.650
140.832 - 68.724 2.649
0.078 0.190 0.672
0.078 0.194 0.670
0.078 0.234 0.675
0.156 0.159 0.639
0.156 0.154 0.641
0.157 0.183 0.644
HO37C1 A axis (A e = 20.49694 c m - J)
Q~ Q2 Q3
B axis (B e = 0.49840 c m - l)
C axis (C c = 0.48657 c m - l)
obs.
Set 1
Set 2
obs.
Set 1
Set 2
obs.
Set 1
Set 2
140.907 -53.801 4.607
140.907 -53.796 4.605
140.706 -68.687 2.680
0.075 0.186 0.654
0.074 0.190 0.652
0.075 0.229 0.657
0.150 0.156 0.622
0.150 0.151 0.624
0.150 0.179 0.627
617
R.M. Escribano et al,/ Chemical Physics Letters 259 (1996) 614-618
tion of force constants was developed by Gellai and Jancso [9]. It has been recently used for the evaluation of the force field of the SiEHsCI molecule [10]. In the HOC1 problem under study, a test of the method is provided by a comparison of the observed and calculated values of the a b. These values should match within the experimental error if the matrix of the coefficients can be inverted. We have calculated the harmonic contribution to the a ~ , as well as the derivatives of the moments of inertia with respect to the normal modes, arbe, using the ASYM40 Program by Hedberg and Mills [6]. The structural parameters of HOC1 derived in the present work, r(OH) = 0.96437 ~,, r(OCl) = 1.68897 /~ and 0 ( H O C 1 ) = 102.965 °, and the harmonic force field of Deeley and Mills [5], have been adopted in the calculations. The observed values of arb were taken form Ref. [3]. Table 3 compares the anharmonic contribution to otb, given as a b ( o b s ) a f ( h a r m ) , and the corresponding calculated values obtained using the IGM method outlined above, listed as Set 1. The agreement is good, which provides support to the validity of this method. As a check of the program employed in this calculation, we have verified that the constants determined by
Halonen and Ha [1] reproduce the results listed in Table 3 of Ref. [ 1]. The upper part of Table 4 lists the cubic force constants of HO35C1 and HO37C1 estimated with this method (Set 1), together with the results of Halonen and Ha [1]. It can be noticed that the values of some constants are quite different, although both sets reproduce well the observations. The diagonal force constants ~bll I and ~b333 a r e negative in either case, as is usually found for bond stretching anharmonicities [11]. The large discrepancies probably arise from the fact that four out of ten cubic constants were fixed to their ab initio values in Ref. [1], while in this work all nine parameters in Set 1 were obtained from the experimental data. These data may not provide sufficient independent information to determine all parameters uniquely. In an attempt to shed more light on this problem, we carried out additional calculations, in which three force constants at a time, one for each normal mode, were fixed to the values of Ref. [1]. The best results, labeled Set 2, are also shown in Tables 3 and 4. Whereas the cubic constants in this Set are now closer to the values of Halonen and Ha, the reproduction of the observations is less satisfactory than with Set 1. We believe
Table 4 Cubic and quartic force constants(in cm- i ) of HO35CIand HO37C1. Set 1: calculated.usingthe IGM method; Set 2: calculatedfixing some force constants(in brackets) to the values of Ref. [1] HO35CI HO37C1 Set 1
Set 2
Ref. [1]
Set 1
Set 2
- 613.2 1061.0 136.5 244.7
- 2473.5 [-2.2] - 26.7 1002.9 [ - 156.9] - 78.1 97.0 [ - 21.0] - 264.1
- 2605.8 -2.2 - 25.4 837.5 - 156.9 - 63.7 54.3 - 21.0 - 267.2
- 613.1 1060.8 - 132.6 244.9 -432.5 - 34.0 12.1 - 46.9 - 255.5
- 2471.7 [-2.1] - 25.8 1002.7 [ - 156.2] - 77.5 94.3 [ - 20.2] - 260.2
1439.8 - 1164.0 -- 75.6 1010.6
1411.7 - 887.4 -- 40.4 613.1
616.2 - 379.5 -- 10.0 194.4
1434.7 - 1162.0 -- 73.6 1009.9
4h Il ~bn2 I3
--
~221 q~222 q~223
--432.5 -- 33.3
I
~b33t tk332 ~333
12.3 - 48.3 -- 259.1
~llll
619.5
t~1122
-- 379.9
qblt33
-- 9.6 193.3
~2222
Ref. [1] -
2605.9 -2.1 -- 25.3 839.2 - 156.2 - 63.5 53.2 - 20.2 - 263.8 1411.7 - 889.1 -- 39.4 615.6
618
R.M. Escribano et al. / Chemical Physics Letters 259 (1996) 614-618
that, until more evidence is available, it may be meaningless to assess which method, either ab initio calculations or a purely empirical fit, gives a more correct representation of the cubic force field of this molecule. The anharmonicity constants Xrs are the coefficients of the second-order term of the vibrational term values: 1
c(v)= E°,r(Vr+½) + r
I
r~s
cm- I ) or zero. The results of Sets 1 and 2 are shown in the lower part of Table 4, compared to the values of Ref. [1]. Again we find large discrepancies between these calculations, our values of Set 1 being systematically smaller as a consequence of the smaller values of some of the cubic constants determined in this work, notably ~b~l~ and ~b22~. The signs of all parameters coincide in all calculations, and, as expected, the results are very close for the 35C1 and 37C1 isotopic species.
(8)
+...
The Xrs contain contributions from the cubic and the quartic force constants. The formulas relating these parameters are [11]: (8o,,2
-
+ ~-~s~)r2rs~---~0~
~b.,r = 1 6 x .
t
+2E
3o4)
~--'~s ) '
(9)
Acknowledgements This work was carried out within the framework of a Cooperation Program between CNR of Italy and CSIC of Spain. RE also acknowledges support from the Spanish DGICYT, project PB93-0138.
tot
o,,(o,?
l
-
References
-
r
- 4[ A( ~rst +
,
(lO)
[1] L. Halonen and T.-K. Ha, J. Chem. Phys. 88 (1988) 3775. [2] B. Abel, H.H. Hamann, A.A. Kachanov and J. Troe, J. Chem. Phys. 104 (1996) 3189, and references therein. [3] C. Azzolini, F. Cavazza, G. Crovetti, G. Di Lonardo, R. Frulla, R. Escribano and L. Fusina, J. Mol. Spectrosc. 168
(1994) 494.
where a , , = (,Or + O,, + , o , ) ( , o ~ - - ~o, -- o,,) x ( - ,,,, + o.,, -
o.,,)( - ,,,, -
o,, + ,,,,).
With the nine cubic force constants determined above and the available experimental values of Xll, xl2, xl3 and x22 [3,12], we have derived the quartic constants ~bzlll, (~2222' (])1122 and (~1133" The value of the cubic constant ~bl23, necessary to evaluate the last two parameters, was taken from the ab initio work of Halonen and Ha [1]. In practice, the effect of (~123 is only noticeable over the (])1133 parameter, which changes between = - 1 0 and = - 1 cm -l when ~b123 is given the value of Ref. [1] ( - 198.4
[4] C.M. Deeley, J. Mol. Spectrosc. 122 (1987) 481. [5] C.M. Deeley and I.M. Mills, J. Mol. Spectrosc. 114 (1985) 368. [6] L. Hedberg and I.M. Mills, J. Mol. Spectrose. 160 (1993) 117. [7] I.M. Mills, in: Theoretical chemistry, specialist periodical report, Vol. I. Quantum chemistry (London, 1974). [8] G.H. Golub and C. Reinsch, Numer. Math. 14 (1970) 403. [9] B. Gellai and G. Jancso, J. Mol. Struct. 12 (1972) 478. [10] A. Ben Altabef and R. Escribano, Spectrochim. Acta 47A (1991) 455. [I1] I.M. Mills, in: Molecular spectroscopy: modem research, Voi. I. eds. K. Narahari Rao and C.W. Mathews (Academic Press, New York, 1972). [12] F. Cavazza, G. Di Lonardo and L. Fusina, Gaz. Chim. Ital., in press.
ELSEVIER
CHEMICAL PHYSICS LETTERS Chemical PhysicsLetters 259 (1996) 614-618
Empirical anharmonic force field and equilibrium structure of hypochlorous acid, HOC1 R.M. Escribano
a G.
Di Lonardo b, L. Fusina b
a lnstituto de Estructura de la Materia, CSIC, Serrano 123, 28006 Madrid, Spain b Dipartimento di Chimica Fisica e lnorganica, Viale Risorgimento 4, 40136 Bologna, Italy
Received 17 June 1996
Abstract The cubic and quartic force fields of HOCI are investigated on the basis of the most recent experimental data on vibration-rotation interaction constants and anharmonicity constants. Some discrepancies with respect to previously reported ab initio results are found and discussed. The geometrical parameters of this molecule are also evaluated from recent data on the equilibrium values of the moments of inertia.
1. Introduction The involvement of hypochlorous acid in atmospheric reactions has prompted a thorough investigation of the spectrum and potential surface of this molecule. The most recent result on the anharmonic force field, as far as we know, is that of Halonen and Ha [1], who performed an ab initio calculation at third-order Moller-Plesset level to determine the equilibrium structure and the anharmonic force field of this molecule. The experimental vibration-rotation data available at the time (up to 1987) were used to calculate some of the quadratic and cubic force constants. Several high-accuracy spectroscopic investigations have been performed since then, covering a large range of vibrational states [2]. The more extensive determination of the vibration-rotation interaction and anharmonicity constants of HO35Cl and HO37C1 is that of Azzolini et al. [3]. This work provides accurate values for the nine a~ parameters (r = 1-3 for the normal modes, b = A, B or C for the rotational constants) and four anharmonicity con-
stants of each molecule, which can be used for the determination of nine cubic and four quartic force constants, respectively, based on empirical data alone. Data on the equilibrium value of the rotational constants A e, Be and C e for HOC1 are more recent and accurate than those of DOCI, which were derived from a lower resolution work [4]. The latter are nonetheless essential if the geometrical parameters of hypochlorous acid are to be estimated on empirical grounds. Since the determination of the anharmonic force constants involves the equilibrium moments of inertia (see equations below), we have evaluated first the geometrical parameters of hypochlorous acid, and then we have used these results in the estimation of the cubic and quartic force fields.
2. Equilibrium geometry Deeley [4] obtained the equilibrium geometrical parameters using a graphical representation of data
0009-2614/96/$12.00 Copyright© 1996Published by Elsevier Science B.V. All rights reserved. PII S0009-2614(96)00774-9
615
R.M. Escribano et al. / Chemical Physics Letters 259 (1996) 614-618
for HOC1 and DOC1, following a procedure first outlined by Deeley and Mills [5] for HOC1 alone. We have explicitly written the equations for the moments of inertia referred to the principal axes, la(~), I~e) and l~ce), in terms of the equilibrium bond distances r(OH), r(OCl) and interatomic angle 0(HOC1). From these equations, we have obtained the algebraic expressions for the derivatives of the moments of inertia with respect to the structural parameters. In this way we can write a system of equations, where the data are the experimental moments of inertia for three molecules (HO35C1, HO37C1 from Ref. [3] and DO35C1 from Ref. [4]) and the unknowns are the equilibrium parameters. The system can be solved using non-linear least-squares techniques, such as the inverse generalized matrix (IGM) method described in some detail below. The experimental moments of inertia of the first two isotopomers were derived from the values of Watson's determinable combinations of the rotational constants of the ground state, to which the vibrational dependence of the rotational constants (parameters a~ and %b where available) were conveniently added. For DO35C1 the moments of inertia Ib~e) are explicitly given in Ref. [4]. Weights have
Table 1 Moments of inertia 1~e) and inertial deflect Ae of HO35CI, HO37C1 and DO35CI (ill /j,/~2) derived from observed data of Refs. [3] and [4], and values calculated in this work through a least-squares fit to the structural parameters of hypochlorous acid Obs. HO35C1 lff ) 34.049008 In(e) 33.226943 la(e) 0.822741 Ae - 0.000676
Weight
Calc.
Obs. - c a l c .
16 16 16
34.049364 33.226944 0.822420
- 0.000355 - 0.000001 0.000321
Table 2 Equilibrium structural parameters of hypochlorous acid. Uncertainties are given in units of the last digit This work re(OH) (~)
0.96437(13)
re(OCl) (,~) 0e(HOCI) (°)
Ac
4 4 4
34.646124 33.823677 0,822446
0.000282 0.001123 0.000348
1 1 1
36.683084 35.159344 1.523741
-- 0.001084 0.001056 0.000019
-0.001189
DO35C1 /(cc) 36.682000 /n(e) 35.160400 Ia(e) 1.523760 Ae -0.002160
0.9643(5)
1.68897(2) 102.965(19)
1.6891 (2) 102.96(8)
been assigned to the data proportional to the compliance to the inertial defect rule (at equilibrium, Ia(e) = I~e) + I~C~) should be strictly satisfied). The results of the fit, including the weights, are collected in Table 1, and the best estimates of the structural parameters are listed in Table 2, together with those of Deeley [4]. It can be seen that the graphical method used by this author led to accurate results, which agree with those obtained in the present work within the estimated uncertainty for all three parameters. The importance of the DO35C1 data in the determination of the geometrical parameters is evidenced by the results of a fit based on data of the hydrogen isotopomers only. An unrealistic value for the HOCI angle affected by a large uncertainty (94.5 + 4.5 °) is obtained, Moreover, large correlations ( = 99%) are observed between all parameters.
3. Anharmonic force field
The vibrational dependence of the rotational constants is conveniently expressed as a power expansion in the vibrational coordinates, where the firstorder coefficients are the a~ parameters: b r
HO37C1 1~e) 34.646406 In(t) 33.824800 la~e) 0.822795
Ref. [4]
1
1
r<~s
+...
(1)
In this equation and in the following, the rotational constant B stands for A, B or C. The ar° parameters are made up of a harmonic and an anharmonic contribution [6]:
= ,,
(harm) + ,,r
(anharm).
(2)
The harmonic contribution depends on the equilibrium values of the inertia tensor I~e) and the
R.M. Escribano et aL / Chemical Physics Letters 259 (1996) 614-618
616
Coriolis interaction constants ~'rbs,whereas the anharmonic part contains the cubic force constants ~brst: 2(Bb)2 ~ [~ o/b(harnl) ~
3(a~C)2
t°r
b 2
41~e-'-'----7+ ½E$ ( ~ ' ~ )
x
(3)
, (anharm) -
-
(~)
difficult, because the matrix of the coefficients of the unknowns in the expanded equations is almost singular, and cannot be inverted using standard procedures. Therefore, we have been forced to use some specific algebraic tool, and we have chosen the inverse generalized matrix (IGM) method [8], which consists of the following. For a given matrix A such that ATA (where AT is the transpose of A) contains m zero roots, the inverse generalized matrix is a matrix A ~ which can replace the true inverse Awhen A-I does not exist. For instance, A ~ satisfies: AAIA = A; A I A A I = A I.
1/2
ff --
(5)
(.Or
E
~t)rrsabs b 0 9 : / 2 "
(4)
$
If the harmonic force field and the geometry are known, it is possible to estimate the cubic force constants from the observed values of the rotationvibration interaction constants a~(obs). For each normal mode r, there are three equations like (4), each one relative to a rotational constant B. Expansion of the summation in Eq. (4) also gives three cubic constants qbrrl, ¢~rr2 and ~brr3 for each r, which can therefore be estimated from the data. Thus we can evaluate nine cubic constants, i.e. all cubic constants for an asymmetric bent triatomic molecule [7], but one, ~bl23. In practice, the evaluation of the cubic constants for HOCI turns out to be quite
The singular values (identical to the square root of the eigenvalues if the matrix is not singular) of A l are the reciprocal of the non-null singular values of A, and zero for the m zero roots of A. The interest of this method for our problem lies in the simplicity of its application to the solution of least-squares analyses [8]. The system
Ay = b
(6)
when ATA is singular is indeterminate. The solution of minimum length (smallest modulus 1.91) is .9 = A'b,
(7)
where A l therefore replaces the non-available A -l. The application of the IGM method to the determina-
Table 3 Observed values of the anharmonic contribution to cry of HO35C1 and HO37CI, in 10 -2 crtl- t, and the corresponding calculated values obtained using the IGM method (Set t), and fixing some cubic constants to the results of Ref. [1] (Set 2) HO35C1 A axis ( A e = 20.49760 cm - t )
Ql Q2 Q3
B axis (B e = 0.50735 cm - t )
C axis (C e = 0.49509 c m - J)
obs.
Set 1
Set 2
obs.
Set 1
Set 2
obs.
Set 1
Set 2
141.077 - 53.781 4.652
141.041 - 53.778 4.650
140.832 - 68.724 2.649
0.078 0.190 0.672
0.078 0.194 0.670
0.078 0.234 0.675
0.156 0.159 0.639
0.156 0.154 0.641
0.157 0.183 0.644
HO37C1 A axis (A e = 20.49694 c m - J)
Q~ Q2 Q3
B axis (B e = 0.49840 c m - l)
C axis (C c = 0.48657 c m - l)
obs.
Set 1
Set 2
obs.
Set 1
Set 2
obs.
Set 1
Set 2
140.907 -53.801 4.607
140.907 -53.796 4.605
140.706 -68.687 2.680
0.075 0.186 0.654
0.074 0.190 0.652
0.075 0.229 0.657
0.150 0.156 0.622
0.150 0.151 0.624
0.150 0.179 0.627
617
R.M. Escribano et al,/ Chemical Physics Letters 259 (1996) 614-618
tion of force constants was developed by Gellai and Jancso [9]. It has been recently used for the evaluation of the force field of the SiEHsCI molecule [10]. In the HOC1 problem under study, a test of the method is provided by a comparison of the observed and calculated values of the a b. These values should match within the experimental error if the matrix of the coefficients can be inverted. We have calculated the harmonic contribution to the a ~ , as well as the derivatives of the moments of inertia with respect to the normal modes, arbe, using the ASYM40 Program by Hedberg and Mills [6]. The structural parameters of HOC1 derived in the present work, r(OH) = 0.96437 ~,, r(OCl) = 1.68897 /~ and 0 ( H O C 1 ) = 102.965 °, and the harmonic force field of Deeley and Mills [5], have been adopted in the calculations. The observed values of arb were taken form Ref. [3]. Table 3 compares the anharmonic contribution to otb, given as a b ( o b s ) a f ( h a r m ) , and the corresponding calculated values obtained using the IGM method outlined above, listed as Set 1. The agreement is good, which provides support to the validity of this method. As a check of the program employed in this calculation, we have verified that the constants determined by
Halonen and Ha [1] reproduce the results listed in Table 3 of Ref. [ 1]. The upper part of Table 4 lists the cubic force constants of HO35C1 and HO37C1 estimated with this method (Set 1), together with the results of Halonen and Ha [1]. It can be noticed that the values of some constants are quite different, although both sets reproduce well the observations. The diagonal force constants ~bll I and ~b333 a r e negative in either case, as is usually found for bond stretching anharmonicities [11]. The large discrepancies probably arise from the fact that four out of ten cubic constants were fixed to their ab initio values in Ref. [1], while in this work all nine parameters in Set 1 were obtained from the experimental data. These data may not provide sufficient independent information to determine all parameters uniquely. In an attempt to shed more light on this problem, we carried out additional calculations, in which three force constants at a time, one for each normal mode, were fixed to the values of Ref. [1]. The best results, labeled Set 2, are also shown in Tables 3 and 4. Whereas the cubic constants in this Set are now closer to the values of Halonen and Ha, the reproduction of the observations is less satisfactory than with Set 1. We believe
Table 4 Cubic and quartic force constants(in cm- i ) of HO35CIand HO37C1. Set 1: calculated.usingthe IGM method; Set 2: calculatedfixing some force constants(in brackets) to the values of Ref. [1] HO35CI HO37C1 Set 1
Set 2
Ref. [1]
Set 1
Set 2
- 613.2 1061.0 136.5 244.7
- 2473.5 [-2.2] - 26.7 1002.9 [ - 156.9] - 78.1 97.0 [ - 21.0] - 264.1
- 2605.8 -2.2 - 25.4 837.5 - 156.9 - 63.7 54.3 - 21.0 - 267.2
- 613.1 1060.8 - 132.6 244.9 -432.5 - 34.0 12.1 - 46.9 - 255.5
- 2471.7 [-2.1] - 25.8 1002.7 [ - 156.2] - 77.5 94.3 [ - 20.2] - 260.2
1439.8 - 1164.0 -- 75.6 1010.6
1411.7 - 887.4 -- 40.4 613.1
616.2 - 379.5 -- 10.0 194.4
1434.7 - 1162.0 -- 73.6 1009.9
4h Il ~bn2 I3
--
~221 q~222 q~223
--432.5 -- 33.3
I
~b33t tk332 ~333
12.3 - 48.3 -- 259.1
~llll
619.5
t~1122
-- 379.9
qblt33
-- 9.6 193.3
~2222
Ref. [1] -
2605.9 -2.1 -- 25.3 839.2 - 156.2 - 63.5 53.2 - 20.2 - 263.8 1411.7 - 889.1 -- 39.4 615.6
618
R.M. Escribano et al. / Chemical Physics Letters 259 (1996) 614-618
that, until more evidence is available, it may be meaningless to assess which method, either ab initio calculations or a purely empirical fit, gives a more correct representation of the cubic force field of this molecule. The anharmonicity constants Xrs are the coefficients of the second-order term of the vibrational term values: 1
c(v)= E°,r(Vr+½) + r
I
r~s
cm- I ) or zero. The results of Sets 1 and 2 are shown in the lower part of Table 4, compared to the values of Ref. [1]. Again we find large discrepancies between these calculations, our values of Set 1 being systematically smaller as a consequence of the smaller values of some of the cubic constants determined in this work, notably ~b~l~ and ~b22~. The signs of all parameters coincide in all calculations, and, as expected, the results are very close for the 35C1 and 37C1 isotopic species.
(8)
+...
The Xrs contain contributions from the cubic and the quartic force constants. The formulas relating these parameters are [11]: (8o,,2
-
+ ~-~s~)r2rs~---~0~
~b.,r = 1 6 x .
t
+2E
3o4)
~--'~s ) '
(9)
Acknowledgements This work was carried out within the framework of a Cooperation Program between CNR of Italy and CSIC of Spain. RE also acknowledges support from the Spanish DGICYT, project PB93-0138.
tot
o,,(o,?
l
-
References
-
r
- 4[ A( ~rst +
,
(lO)
[1] L. Halonen and T.-K. Ha, J. Chem. Phys. 88 (1988) 3775. [2] B. Abel, H.H. Hamann, A.A. Kachanov and J. Troe, J. Chem. Phys. 104 (1996) 3189, and references therein. [3] C. Azzolini, F. Cavazza, G. Crovetti, G. Di Lonardo, R. Frulla, R. Escribano and L. Fusina, J. Mol. Spectrosc. 168
(1994) 494.
where a , , = (,Or + O,, + , o , ) ( , o ~ - - ~o, -- o,,) x ( - ,,,, + o.,, -
o.,,)( - ,,,, -
o,, + ,,,,).
With the nine cubic force constants determined above and the available experimental values of Xll, xl2, xl3 and x22 [3,12], we have derived the quartic constants ~bzlll, (~2222' (])1122 and (~1133" The value of the cubic constant ~bl23, necessary to evaluate the last two parameters, was taken from the ab initio work of Halonen and Ha [1]. In practice, the effect of (~123 is only noticeable over the (])1133 parameter, which changes between = - 1 0 and = - 1 cm -l when ~b123 is given the value of Ref. [1] ( - 198.4
[4] C.M. Deeley, J. Mol. Spectrosc. 122 (1987) 481. [5] C.M. Deeley and I.M. Mills, J. Mol. Spectrosc. 114 (1985) 368. [6] L. Hedberg and I.M. Mills, J. Mol. Spectrose. 160 (1993) 117. [7] I.M. Mills, in: Theoretical chemistry, specialist periodical report, Vol. I. Quantum chemistry (London, 1974). [8] G.H. Golub and C. Reinsch, Numer. Math. 14 (1970) 403. [9] B. Gellai and G. Jancso, J. Mol. Struct. 12 (1972) 478. [10] A. Ben Altabef and R. Escribano, Spectrochim. Acta 47A (1991) 455. [I1] I.M. Mills, in: Molecular spectroscopy: modem research, Voi. I. eds. K. Narahari Rao and C.W. Mathews (Academic Press, New York, 1972). [12] F. Cavazza, G. Di Lonardo and L. Fusina, Gaz. Chim. Ital., in press.