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General force field of NSF from microwave and infra-red data
Spectrochimica Acta, 1987,Vol. 23A,pp. 2169to 2186.PeqmmonPressLtd. Printedin NorthernIreland
General force field of NSF from microwave and ix&a-red data A. M. MIEUUand A. GUAEWERI Laboratorio di Spettroscopia a Radiofre&nza
de1 C.N.R. lstituto Chimico
“G. Cia.mician”,UniversitS di Bologna, Bologna, Italy (Received28 October1966) Abstract-The analysis of centrifugal distortion in the rotational spectrum of NSF has been oarried out. The following values for rotational and first order centrifugal distortion constants were obtained (MC/S): A
= 49719.48
B = 8712.34
c = 7393.12
Tbbbb= -0*067700 rLbab= -0.089814
ra*aa= -8.796621 0.437754 Ta.bb =
By combining infrared and microwave data the following force constants were derived:
PII = IO.709 F,,
where F,,,
F,,
=
0.095
F,, F,,
= =
2.871 -0.064
F,, F,,
= 0.978 = 0,031
and E;, are in mdyn@, F,, in mdyn.& E;, and F,, in mdyn.
The internal co-ordinates used are: S, = d(N-S),
S, = 6(S-F),
S, = 6a
small molecules, rotational data on centrifugal distortion constants (C.D. constants) provide very often useful information on the general quadratic force field (G.F.F.). In the case of triatomic molecules of C,, symmetry four independent first order centrifugal distortion constants Ta, ,,,bbb, TMbb,Tab&art3Sticient to determine unambigously the G.F.F. The corresponding calculated vibrational frequencies differ however a few per cent from the experimental values, even in the case of F20 [l], where centrifugal stretching contributions of second and higher order were taken into account. This discrepancy is interpreted [l] as due to anharmonic contributions not only to vibrational frequencies, but also to centrifugal distortion constants. Consequently calculations which are based both on complete consistency between the potential function and experimental vibrational frequencies and on best fit with the rotational spectrum, yield force constants which differ a few per cent from those derived by using only rotational data [l, 21. In the case of molecules of Cd symmetry, microwave rotational data are not sufficient for determining the six force constants of the G.F.F. It is thus not possible to assess the magnitude of the discrepancy between the two methods of calculation. FOR
[I] L.
PIERCE, N. DI CIANNI and R. H. JACKSON, J. Chem. Phya. 38, 730 (1963). [2] G. E. HERBERICH, R. H. JACKSON and D. J. MILLEN, J. Chem. Sot. A, Inwg. Phya. Theoret. 336 (1966). 2159
A. M.
2160
MIRRI
and A.
GUARNIERI
For NOCl and NOBr [3] the disagreement between centrifugal stretching constants calculated from infra-red data only [4] and the experimental values [3] is so large that anharmonicity corrections or the use of higher order approximations in the analysis of rotational transitions represent a refinement which would not account for the main part of the discrepancy. Thus in NOF [5], where higher order centrifugal analysis was carried out, diagonal force constants obtained by i&a-red and microwave data differ considerably from values previously reported [4] as in the case of NOCl and NOBr. Therefore, discrepancy of this order of magnitude must be due to the UreyBradley model or to the very small isotopic shifts on which previous calculations were based [a]. We were therefore attracted to perform a similar analysis on NSF which also belongs to symmetry group G,. Since “b” type rotational transitions of several different sub-branches are observed, [6] this allows the determination of all four first order centrifugal distortion constants. These constants and the three vibrational frequencies of the normal species are expected to lead to a full understanding of the G.F.F. The structure of NSF was completely determined from its microwave spectrum by KIRCHHO~Fand WILSON[6], the sulphur atom is central, so, as far as the molecular force field is concerned, no similarity should be expected with NOF [5]. CENTRIFUGAL STRETCHIX~ANALYSISOF THE ROTATIONAL SPECTRUM The centrifugal analysis was carried out on the lines measured by KIRCHHOFF and WILSON[S] by using the first order treatment of the perturbation Hamiltonian r71:
In the present case, where the transitions measured correspond to J values < 20, higher order effects are expected to be very small [l, 51. First order C.D. constants will thus be practically the same as those obtained from a higher order treatment of centrifugal effects [l]. In any case the lack of information on the anharmonicity of vibrational frequencies will introduce larger uncertainties on the force field which will eventually be obtained. The following expression for the energy of a semi-rigid prolate asymmetric rotator, valid for a planar molecule [8] has been used:
where E, is the rigid rotator energy, Q is a six term column vector which depends on [3] A. M. MIRRI and E. [4] J. P. DEVLIN and I.
MAZZARIOL, Spectrochim. Acta 22, 785 (1966). C. HISATSIJNE, Spectrochim. Acta 17, 200 (1961).
[B] R. L. COOK, J. Chews. Phys. 42, 2927 (1965). [6] W. H. I(IRCHHOBF and E. B. WILSON JR., J. Am. Chem.. Sot. 85,1726 [7] D. KIVELSON and E. B. WILSON JR., J. Chem. Phya. 20, 1575(1952). [S] R. A. HILL and T. H. EDWARDS, J. Mol. Spectry 9, 494 (1962).
(1963).
General force field of NSF from microwave and infra-red data
2161
the particular rotational level considered, K is a 6 x 4 matrix, constant for a given molecule, and T is the vector of the four C.D. constants. In order to determine these constants from the rotational spectrum an iterative method for the solution of a linear system of equations by the least square analysis has been programmed for a IBM 7094 digital computer. Each equation is of the type : vi
Vi0 =
-
+
&[Q(J’, 7’) - (QJ, ,)]K .
-
a[E,’ - E,1
AB
+
a[E,’ - E,l
T +
a[E”‘ai EolAA
AC
ac
i3B
where vf is the experimental unsplit frequency of the i-th transition, vi0 the corresponding rigid rotator approximation frequency, J’, r’, and J, T identify the two levels involved in the transition. In the first cycle of calculation values for A, B and C given by Kirchhoff and Wilson were used. Three or four cycles were sufficient for deriving a convergent set of centrifugal and rotational constants. The coefficients of AA, AB and AC are given by the expressions [9]:
aE0 ---C
aB
kk
/
[J(J + 1) + -WI - 2U'zz)l
aEO = J(J + 1) ac
(P z2) -
y aB
where E(k) is the rigid rotator reduced energy expressed as function of Ray’s asymmetry parameter k. The calculation of E(k), (Pz2), (Pz4) was programmed following the usual methods [9]. By using all the transitions listed by Kirchhoff and Wilson an average deviation of 1.94 MC/S between calculated and experimental frequencies was found. Furthermore for two lines the difference between calculated and experimental frequencies was much greater than for all the other lines. Higher order centrifugal distortion effects cannot be responsible for these discrepancies since the two lines do not belong to different sub-branches, moreover the J and k_, values are not higher than those of many other lines used in the calculations. By elimination of the line 14, 12 + 15, 16 and then of the line 164 12 -+ 17, 16, average deviations of 0.73 MC/S and 0.26 MC/S respectively were obtained. The latter deviation cannot be appreciably lowered by eliminating any other line from the input data. In Table 1 are listed experimental and calculated unsplit transition frequencies and centrifugal shifts. In Table 2 are given C.D. and rotational constants obtained from the calculations. The number of significant figures given for the T’S is greater than that required by their precision but it is that needed to reproduce rotational transition frequencies within their experimental errors. [g] M. IV. STRANDBERG, 34
Microwave
Spectroscopy, p. 16. Methuen (1964).
A. M. Mmm and A. Gunmm
2162
Table 1. Rotational
transitions of NSF in Me/s
Observed* unsplit line frequency
Transition
Calculatedunsplit line frequenoy
00o-101 101-110 111-2,t 1,1-203 11o-211 20s-l1 1 21 a-303
16105.42 42324.80 30891.94 32179.25 33530.31 8825.89 8472.50
16105.42 42324.31 30891.86 32179.29 33530.31 8825.74 8472.59
31a-40 1 101-21, 31s-231 614-4, s 61 c-6, 4
26261.79 71897.12 72713.63 34757.97 14897.42
26261.90 71896.89 72713.96 34757.74
7,l3-8, 7 711-624
26438.33 33479.65
81 s-7, 6 91 C8P 6 10110-92,
1111rw
24679.86 17426.92 11890.26 8222.33 9367.67
8
14114-13811
161~15213 17117-162 14 102s-9, , 11, B-10,8 11aI@-10, 7 13,M--12*0 14s1115a14 15,m-l6,15 16,1s-l7a~ 17,14-l% 1, 16,~-1741, 184M-1%17 16sw-15, II 17~14-1%15
Calculatedcentrifugal shift -0.04 -2.05 0.16 -0.31 0.10 - 1.80 0.62 - 1.56 - I.95 - 32.07 -22.56
14897.19 26438.81 33479.71
- 14.17 -12.17 - 16.40
24680-02 17426.21
- 12.09 -9.31
11890.49
-9.40
8222.43 9367.88
20475.06 29004.37 36366.67 16079.94 37019.98 10906.73 11268.01 20422.34 28030.09 33902.66 32425.91 12809.66 20649.82 20660.71
-13.96 -72.23
20474.69 29003.73 36366.42 16079.49 37020.67 10907.02 11267.74 20422.30 28030.09 33902.13 32426.30 12809.89 20649.82 20661.16
- 165.53 -232.97 -51.36 - 16.98 -58.03 -23.19 0.53 0.06 11.25 37.91 -88.79 -50.90 - 12.24 - 184.37
* See Ref. [6]. Table 2. Centrifugal distortion, rotational con&ants and average deviation in MC/S
~ssss 7bbbb Tssbb 7ab.b
A B c Average dev. GENERAL
FORCE
-8.796621 -0~067700 0.437754 -0.089814 49719.48 8712.34 7393.12 0.26 FIELD
& f & &
0.032294 0~001178 0.008676 0.002410
DETERMINATION
The G.F.F. constants are related, in the harmonic approximation, C.D. constants through the well known expression [lo] : Tcc,eya=
(1)
-
where T+,~ is in MC/S when force constants [lo]
to first order
corresponding
D. KIVELSON and E. B. WILSON JR. J. Chem. phy8.
to bond stretching
21, 1229 (1953).
and
General force field of NSF from microwave and infra-red data
2163
interaction between bond stretching are in mdyn/A, those corresponding to bending stretching interactions are in mdyn and those corresponding to bending in mdyn. A, masses are in atomic mass units and internuclear distances in A. (F-l)ij is an element Table 3. Calculated C.D. and rotational constants and average deviation in Mc/sec. F,, and F,, in mydn, Best set of force constants: P,,, F,, and E;, in mdynjs, F,, in mdyn. A 49719.89 8712.37 7393.16 0.428
A B c As*
Best set
PI, F,, F,, F,, F,, F,,
10.709 2,871 0.978 0,095 -0.064 0.031
Taaaa Tbbbb Taabb Tabab
-8.929735 -0.057089 0.445180 -0.085918
Richert and Glemser 10.69 2.86 0.99 -
-
* Average deviation.
of the matrix inverse to the force constants matrix, Idao,lBBo,a.s.on are moments-of inertia at the equilibrium positions. The internal co-ordinates are S, = &is, S, = &s, and S, = da, while the terms (H,,/&S,),, a.s.on are partial derivatives at the equilibrium distances of the components of the inertia tensor in respect of the i-th internal co-ordinate. These quantities can be easily calculated from expressions given by %‘vELSON and WILSON[lo], once the molecular structure is known. It would have been desirable to use the method of calculation developed by MILLSand his co-workers [13] since it yields also the probable errors on force constants. In the present case anharmonic contributions to vibrational frequencies are not known, therefore the uncertainty on the weights to be given to the experimental data is larger than in the case of harmonic vibrational frequencies. Furthermore, aa pointed out by MILLS[13], significant results by statistical analysis can be obtained only when the difference between the number of independent observables and that of the quantities to be determined is not too small. For NSF this difference is one. For these reasons a different approach has been used here, based both on complete consistency between experimental vibrational frequencies and force constants and on the best fit with the rotational spectrum [l, 61. By assuming interaction force constants as parameters, a system of three equations with three variables must be solved in order to obtain diagonal force constants from infra-red data. The method of solution here used is described in the Appendix. For each set of force constants which fits the vibrational frequencies, centrifugal [13] J.
ALDOUS
and I. M. Mm~s, Spectrochim.
Acta 18, 1073 (1962).
2164
A. M. MIRRI and A. GUARNIEYRI
contributions Av, to rotational lines were obtained and a system of linear equations of the following type was solved by the least square method:
where vi are experimental frequencies and vi0 rigid frequencies calculated with the rotational constants A, B and C listed in Table 2. Average deviations between experimental and calculated transition frequencies, i.e. vie = via’ + Avi (where vi” is the i-th rigid rotator frequency calculated with the new rotational constants A + AA, B + AB and C + AC) were then obtained for the many different set of force constants. By this method a minimum average deviation of O-27MC/S resulted. This is nearly the same as that obtained by least square determination of C.D. constants from the rotational spectrum. However the discrepancy between experimental and calculated J = 0 --+ 1 and J = 1 + 2 transition frequencies was too large considering that centrifugal shifts for the levels involved is very small and higher order contributions are negligible. In particular for the 1o1-+ l,, transition the deviation is O-6MC/S. In the calculations the condition was then included that, for the above transitions, deviations should not exceed O-1MC/S. This value is perhaps two or three times the experimental error involved in frequencies measurements. However a fit of the low J transitions within the experimental error does not seem justified, since centrifugal shifts are calculated starting from force constants consistent with anharmonic vibrational frequencies. It may be added that a better fit for transitions with low J values yields larger average deviations, but the molecular force field corresponding to the best fit is not appreciably changed. Increments for F,,, F,, and F,, were fixed on the basis of preliminary results [ 111. Starting with input data for the parameters, corresponding to 1 > F,, > - 1, 05 > F,, > -0.5 and 0.6 > F,, > -0.6 it was possible, by inspection of the results, to narrow progressively the ranges of variation for the interaction force constants and to use smaller increments AF,,, AF,, and AF,,. The minimum average deviation compatible with the magnitude of the increments AF,, corresponds to AE = 0.428 MC/S. In the final calculations we used AF,, = 0.01 mdyn/A, AF,, = 0*0002 mdyn and AF,, = 0.001 mdyn. To each of these increments corresponds a change of about 0.005 MC/Sin AE. Therefore those set of force constants which yield AE values differing by less than 0.005 MC/S from the minimum average deviation ought to be considered equally valid in the present calculations. These sets are included in the following ranges for each force constant: -0~005/0~235 mdyn/A -0*096/-0.017 mdyn F,, 0~039/0~021mdyn F,, 10*66/10*77mdyn/A F11 2*881/2.858 mdyn/A F,, 0.979/0*978 mdyn . A F,, The indeterminacy expressed by the above ranges is due to the practical necessity of using finite intervals for the parameters. Fl2
[ll]
A. M. MIRRI and A. GUARNIERI,Rend. hmzd. iVaz.
Lincei40, 641 (1966).
General force field of NSF from microwave and infrared data
2105
The rotational spectrum is not very sensitive to the value of the interaction force constants, which are however very small. That affecting most the rotational spectrum is P, as in the case of the nitrosyl halides [4,2]. The best set corresponding to the minimum average deviation between the experimental and the calculated rotational spectrum is listed in Table 2 together with calculated rotational and C.D. constants. Diagonal force constants are very similar to those obtained by RIUHERT and GLEMSER [12] under the assumption F,, = F,,= F,,= 0,which is now justified by the present results. APPENDIX By assuming arbitrary values for the three interaction force constants the following system of equations in the variables Fll, F,, and F,, is to be solved:
A,+zy
+ A,pz
G,,x + G,y + Gsez - D = 0 x_~z- xF,,= - yFls2 - zF$ -C = 0 + -,%,~a - 2A,,Fr;8y - 2A,,E;$ - 2A,,F,$
(1) (2)
-
B = 0
(3)
where: x = FIX, y = F,,,
z = F,,.
D = L, + AZ + 1, C = vz4l IGI
B = &Is + I,&
+ I,&
+ &F~z’
-
+ A,sE;s2
2Gl,,Fl,
-
2Gl,Fl,
-
2G,,F,,
2Fl,F,,F,, + A,,F,s2
-
2A,,FuF,,
-
2AI,Fl,F,, -
2A23F12F13
and A,, is the cofaotor of the Gi4element of ]G]. Choice between all the possible solutions of this system can be msde since the order of magnitude of the solutionswhich have a physical me8ning is known. From (1) and (3) equation (4) is obtained: az2 + b(y)2 + 4~) = 0
(4)
fQA12- 4ac(Y) > 0
(5)
The condition expressed by (5) : is satisfied by y < ye where y, is a solution of equation (6):
[WI2 - 4MY) = 0
(5)
Only one is found to be positive for the values of the parameterswhich have physical meaning. Starting from y = y. a solution x0, zo is found from (1) and (3) (the other correspondsto c x0). 9 This set is not, of course, generally consistent with equation (2). By iterations with y values decrementedby Ay, after ten cycles it is usually possible to f?nd a set of values for x, y and z which fit equation (2). The solution for y is obtained by linear interpolation between the two values of y which correspond to quantities of opposite sign in the left member of (2). By preliminary desk computing it was found that by assuming Ay = -0~005 mdyn/A it is possible to fit equation (2) within less than 0.01 per cent in &,1,1,. [12] G. RICFCERT and 0. GLEIVEIEZL, 2. Anorg. Allgem. Chem. 807,328 (1981).
General force field of NSF from microwave and ix&a-red data A. M. MIEUUand A. GUAEWERI Laboratorio di Spettroscopia a Radiofre&nza
de1 C.N.R. lstituto Chimico
“G. Cia.mician”,UniversitS di Bologna, Bologna, Italy (Received28 October1966) Abstract-The analysis of centrifugal distortion in the rotational spectrum of NSF has been oarried out. The following values for rotational and first order centrifugal distortion constants were obtained (MC/S): A
= 49719.48
B = 8712.34
c = 7393.12
Tbbbb= -0*067700 rLbab= -0.089814
ra*aa= -8.796621 0.437754 Ta.bb =
By combining infrared and microwave data the following force constants were derived:
PII = IO.709 F,,
where F,,,
F,,
=
0.095
F,, F,,
= =
2.871 -0.064
F,, F,,
= 0.978 = 0,031
and E;, are in mdyn@, F,, in mdyn.& E;, and F,, in mdyn.
The internal co-ordinates used are: S, = d(N-S),
S, = 6(S-F),
S, = 6a
small molecules, rotational data on centrifugal distortion constants (C.D. constants) provide very often useful information on the general quadratic force field (G.F.F.). In the case of triatomic molecules of C,, symmetry four independent first order centrifugal distortion constants Ta, ,,,bbb, TMbb,Tab&art3Sticient to determine unambigously the G.F.F. The corresponding calculated vibrational frequencies differ however a few per cent from the experimental values, even in the case of F20 [l], where centrifugal stretching contributions of second and higher order were taken into account. This discrepancy is interpreted [l] as due to anharmonic contributions not only to vibrational frequencies, but also to centrifugal distortion constants. Consequently calculations which are based both on complete consistency between the potential function and experimental vibrational frequencies and on best fit with the rotational spectrum, yield force constants which differ a few per cent from those derived by using only rotational data [l, 21. In the case of molecules of Cd symmetry, microwave rotational data are not sufficient for determining the six force constants of the G.F.F. It is thus not possible to assess the magnitude of the discrepancy between the two methods of calculation. FOR
[I] L.
PIERCE, N. DI CIANNI and R. H. JACKSON, J. Chem. Phya. 38, 730 (1963). [2] G. E. HERBERICH, R. H. JACKSON and D. J. MILLEN, J. Chem. Sot. A, Inwg. Phya. Theoret. 336 (1966). 2159
A. M.
2160
MIRRI
and A.
GUARNIERI
For NOCl and NOBr [3] the disagreement between centrifugal stretching constants calculated from infra-red data only [4] and the experimental values [3] is so large that anharmonicity corrections or the use of higher order approximations in the analysis of rotational transitions represent a refinement which would not account for the main part of the discrepancy. Thus in NOF [5], where higher order centrifugal analysis was carried out, diagonal force constants obtained by i&a-red and microwave data differ considerably from values previously reported [4] as in the case of NOCl and NOBr. Therefore, discrepancy of this order of magnitude must be due to the UreyBradley model or to the very small isotopic shifts on which previous calculations were based [a]. We were therefore attracted to perform a similar analysis on NSF which also belongs to symmetry group G,. Since “b” type rotational transitions of several different sub-branches are observed, [6] this allows the determination of all four first order centrifugal distortion constants. These constants and the three vibrational frequencies of the normal species are expected to lead to a full understanding of the G.F.F. The structure of NSF was completely determined from its microwave spectrum by KIRCHHO~Fand WILSON[6], the sulphur atom is central, so, as far as the molecular force field is concerned, no similarity should be expected with NOF [5]. CENTRIFUGAL STRETCHIX~ANALYSISOF THE ROTATIONAL SPECTRUM The centrifugal analysis was carried out on the lines measured by KIRCHHOFF and WILSON[S] by using the first order treatment of the perturbation Hamiltonian r71:
In the present case, where the transitions measured correspond to J values < 20, higher order effects are expected to be very small [l, 51. First order C.D. constants will thus be practically the same as those obtained from a higher order treatment of centrifugal effects [l]. In any case the lack of information on the anharmonicity of vibrational frequencies will introduce larger uncertainties on the force field which will eventually be obtained. The following expression for the energy of a semi-rigid prolate asymmetric rotator, valid for a planar molecule [8] has been used:
where E, is the rigid rotator energy, Q is a six term column vector which depends on [3] A. M. MIRRI and E. [4] J. P. DEVLIN and I.
MAZZARIOL, Spectrochim. Acta 22, 785 (1966). C. HISATSIJNE, Spectrochim. Acta 17, 200 (1961).
[B] R. L. COOK, J. Chews. Phys. 42, 2927 (1965). [6] W. H. I(IRCHHOBF and E. B. WILSON JR., J. Am. Chem.. Sot. 85,1726 [7] D. KIVELSON and E. B. WILSON JR., J. Chem. Phya. 20, 1575(1952). [S] R. A. HILL and T. H. EDWARDS, J. Mol. Spectry 9, 494 (1962).
(1963).
General force field of NSF from microwave and infra-red data
2161
the particular rotational level considered, K is a 6 x 4 matrix, constant for a given molecule, and T is the vector of the four C.D. constants. In order to determine these constants from the rotational spectrum an iterative method for the solution of a linear system of equations by the least square analysis has been programmed for a IBM 7094 digital computer. Each equation is of the type : vi
Vi0 =
-
+
&[Q(J’, 7’) - (QJ, ,)]K .
-
a[E,’ - E,1
AB
+
a[E,’ - E,l
T +
a[E”‘ai EolAA
AC
ac
i3B
where vf is the experimental unsplit frequency of the i-th transition, vi0 the corresponding rigid rotator approximation frequency, J’, r’, and J, T identify the two levels involved in the transition. In the first cycle of calculation values for A, B and C given by Kirchhoff and Wilson were used. Three or four cycles were sufficient for deriving a convergent set of centrifugal and rotational constants. The coefficients of AA, AB and AC are given by the expressions [9]:
aE0 ---C
aB
kk
/
[J(J + 1) + -WI - 2U'zz)l
aEO = J(J + 1) ac
(P z2) -
y aB
where E(k) is the rigid rotator reduced energy expressed as function of Ray’s asymmetry parameter k. The calculation of E(k), (Pz2), (Pz4) was programmed following the usual methods [9]. By using all the transitions listed by Kirchhoff and Wilson an average deviation of 1.94 MC/S between calculated and experimental frequencies was found. Furthermore for two lines the difference between calculated and experimental frequencies was much greater than for all the other lines. Higher order centrifugal distortion effects cannot be responsible for these discrepancies since the two lines do not belong to different sub-branches, moreover the J and k_, values are not higher than those of many other lines used in the calculations. By elimination of the line 14, 12 + 15, 16 and then of the line 164 12 -+ 17, 16, average deviations of 0.73 MC/S and 0.26 MC/S respectively were obtained. The latter deviation cannot be appreciably lowered by eliminating any other line from the input data. In Table 1 are listed experimental and calculated unsplit transition frequencies and centrifugal shifts. In Table 2 are given C.D. and rotational constants obtained from the calculations. The number of significant figures given for the T’S is greater than that required by their precision but it is that needed to reproduce rotational transition frequencies within their experimental errors. [g] M. IV. STRANDBERG, 34
Microwave
Spectroscopy, p. 16. Methuen (1964).
A. M. Mmm and A. Gunmm
2162
Table 1. Rotational
transitions of NSF in Me/s
Observed* unsplit line frequency
Transition
Calculatedunsplit line frequenoy
00o-101 101-110 111-2,t 1,1-203 11o-211 20s-l1 1 21 a-303
16105.42 42324.80 30891.94 32179.25 33530.31 8825.89 8472.50
16105.42 42324.31 30891.86 32179.29 33530.31 8825.74 8472.59
31a-40 1 101-21, 31s-231 614-4, s 61 c-6, 4
26261.79 71897.12 72713.63 34757.97 14897.42
26261.90 71896.89 72713.96 34757.74
7,l3-8, 7 711-624
26438.33 33479.65
81 s-7, 6 91 C8P 6 10110-92,
1111rw
24679.86 17426.92 11890.26 8222.33 9367.67
8
14114-13811
161~15213 17117-162 14 102s-9, , 11, B-10,8 11aI@-10, 7 13,M--12*0 14s1115a14 15,m-l6,15 16,1s-l7a~ 17,14-l% 1, 16,~-1741, 184M-1%17 16sw-15, II 17~14-1%15
Calculatedcentrifugal shift -0.04 -2.05 0.16 -0.31 0.10 - 1.80 0.62 - 1.56 - I.95 - 32.07 -22.56
14897.19 26438.81 33479.71
- 14.17 -12.17 - 16.40
24680-02 17426.21
- 12.09 -9.31
11890.49
-9.40
8222.43 9367.88
20475.06 29004.37 36366.67 16079.94 37019.98 10906.73 11268.01 20422.34 28030.09 33902.66 32425.91 12809.66 20649.82 20660.71
-13.96 -72.23
20474.69 29003.73 36366.42 16079.49 37020.67 10907.02 11267.74 20422.30 28030.09 33902.13 32426.30 12809.89 20649.82 20661.16
- 165.53 -232.97 -51.36 - 16.98 -58.03 -23.19 0.53 0.06 11.25 37.91 -88.79 -50.90 - 12.24 - 184.37
* See Ref. [6]. Table 2. Centrifugal distortion, rotational con&ants and average deviation in MC/S
~ssss 7bbbb Tssbb 7ab.b
A B c Average dev. GENERAL
FORCE
-8.796621 -0~067700 0.437754 -0.089814 49719.48 8712.34 7393.12 0.26 FIELD
& f & &
0.032294 0~001178 0.008676 0.002410
DETERMINATION
The G.F.F. constants are related, in the harmonic approximation, C.D. constants through the well known expression [lo] : Tcc,eya=
(1)
-
where T+,~ is in MC/S when force constants [lo]
to first order
corresponding
D. KIVELSON and E. B. WILSON JR. J. Chem. phy8.
to bond stretching
21, 1229 (1953).
and
General force field of NSF from microwave and infra-red data
2163
interaction between bond stretching are in mdyn/A, those corresponding to bending stretching interactions are in mdyn and those corresponding to bending in mdyn. A, masses are in atomic mass units and internuclear distances in A. (F-l)ij is an element Table 3. Calculated C.D. and rotational constants and average deviation in Mc/sec. F,, and F,, in mydn, Best set of force constants: P,,, F,, and E;, in mdynjs, F,, in mdyn. A 49719.89 8712.37 7393.16 0.428
A B c As*
Best set
PI, F,, F,, F,, F,, F,,
10.709 2,871 0.978 0,095 -0.064 0.031
Taaaa Tbbbb Taabb Tabab
-8.929735 -0.057089 0.445180 -0.085918
Richert and Glemser 10.69 2.86 0.99 -
-
* Average deviation.
of the matrix inverse to the force constants matrix, Idao,lBBo,a.s.on are moments-of inertia at the equilibrium positions. The internal co-ordinates are S, = &is, S, = &s, and S, = da, while the terms (H,,/&S,),, a.s.on are partial derivatives at the equilibrium distances of the components of the inertia tensor in respect of the i-th internal co-ordinate. These quantities can be easily calculated from expressions given by %‘vELSON and WILSON[lo], once the molecular structure is known. It would have been desirable to use the method of calculation developed by MILLSand his co-workers [13] since it yields also the probable errors on force constants. In the present case anharmonic contributions to vibrational frequencies are not known, therefore the uncertainty on the weights to be given to the experimental data is larger than in the case of harmonic vibrational frequencies. Furthermore, aa pointed out by MILLS[13], significant results by statistical analysis can be obtained only when the difference between the number of independent observables and that of the quantities to be determined is not too small. For NSF this difference is one. For these reasons a different approach has been used here, based both on complete consistency between experimental vibrational frequencies and force constants and on the best fit with the rotational spectrum [l, 61. By assuming interaction force constants as parameters, a system of three equations with three variables must be solved in order to obtain diagonal force constants from infra-red data. The method of solution here used is described in the Appendix. For each set of force constants which fits the vibrational frequencies, centrifugal [13] J.
ALDOUS
and I. M. Mm~s, Spectrochim.
Acta 18, 1073 (1962).
2164
A. M. MIRRI and A. GUARNIEYRI
contributions Av, to rotational lines were obtained and a system of linear equations of the following type was solved by the least square method:
where vi are experimental frequencies and vi0 rigid frequencies calculated with the rotational constants A, B and C listed in Table 2. Average deviations between experimental and calculated transition frequencies, i.e. vie = via’ + Avi (where vi” is the i-th rigid rotator frequency calculated with the new rotational constants A + AA, B + AB and C + AC) were then obtained for the many different set of force constants. By this method a minimum average deviation of O-27MC/S resulted. This is nearly the same as that obtained by least square determination of C.D. constants from the rotational spectrum. However the discrepancy between experimental and calculated J = 0 --+ 1 and J = 1 + 2 transition frequencies was too large considering that centrifugal shifts for the levels involved is very small and higher order contributions are negligible. In particular for the 1o1-+ l,, transition the deviation is O-6MC/S. In the calculations the condition was then included that, for the above transitions, deviations should not exceed O-1MC/S. This value is perhaps two or three times the experimental error involved in frequencies measurements. However a fit of the low J transitions within the experimental error does not seem justified, since centrifugal shifts are calculated starting from force constants consistent with anharmonic vibrational frequencies. It may be added that a better fit for transitions with low J values yields larger average deviations, but the molecular force field corresponding to the best fit is not appreciably changed. Increments for F,,, F,, and F,, were fixed on the basis of preliminary results [ 111. Starting with input data for the parameters, corresponding to 1 > F,, > - 1, 05 > F,, > -0.5 and 0.6 > F,, > -0.6 it was possible, by inspection of the results, to narrow progressively the ranges of variation for the interaction force constants and to use smaller increments AF,,, AF,, and AF,,. The minimum average deviation compatible with the magnitude of the increments AF,, corresponds to AE = 0.428 MC/S. In the final calculations we used AF,, = 0.01 mdyn/A, AF,, = 0*0002 mdyn and AF,, = 0.001 mdyn. To each of these increments corresponds a change of about 0.005 MC/Sin AE. Therefore those set of force constants which yield AE values differing by less than 0.005 MC/S from the minimum average deviation ought to be considered equally valid in the present calculations. These sets are included in the following ranges for each force constant: -0~005/0~235 mdyn/A -0*096/-0.017 mdyn F,, 0~039/0~021mdyn F,, 10*66/10*77mdyn/A F11 2*881/2.858 mdyn/A F,, 0.979/0*978 mdyn . A F,, The indeterminacy expressed by the above ranges is due to the practical necessity of using finite intervals for the parameters. Fl2
[ll]
A. M. MIRRI and A. GUARNIERI,Rend. hmzd. iVaz.
Lincei40, 641 (1966).
General force field of NSF from microwave and infrared data
2105
The rotational spectrum is not very sensitive to the value of the interaction force constants, which are however very small. That affecting most the rotational spectrum is P, as in the case of the nitrosyl halides [4,2]. The best set corresponding to the minimum average deviation between the experimental and the calculated rotational spectrum is listed in Table 2 together with calculated rotational and C.D. constants. Diagonal force constants are very similar to those obtained by RIUHERT and GLEMSER [12] under the assumption F,, = F,,= F,,= 0,which is now justified by the present results. APPENDIX By assuming arbitrary values for the three interaction force constants the following system of equations in the variables Fll, F,, and F,, is to be solved:
A,+zy
+ A,pz
G,,x + G,y + Gsez - D = 0 x_~z- xF,,= - yFls2 - zF$ -C = 0 + -,%,~a - 2A,,Fr;8y - 2A,,E;$ - 2A,,F,$
(1) (2)
-
B = 0
(3)
where: x = FIX, y = F,,,
z = F,,.
D = L, + AZ + 1, C = vz4l IGI
B = &Is + I,&
+ I,&
+ &F~z’
-
+ A,sE;s2
2Gl,,Fl,
-
2Gl,Fl,
-
2G,,F,,
2Fl,F,,F,, + A,,F,s2
-
2A,,FuF,,
-
2AI,Fl,F,, -
2A23F12F13
and A,, is the cofaotor of the Gi4element of ]G]. Choice between all the possible solutions of this system can be msde since the order of magnitude of the solutionswhich have a physical me8ning is known. From (1) and (3) equation (4) is obtained: az2 + b(y)2 + 4~) = 0
(4)
fQA12- 4ac(Y) > 0
(5)
The condition expressed by (5) : is satisfied by y < ye where y, is a solution of equation (6):
[WI2 - 4MY) = 0
(5)
Only one is found to be positive for the values of the parameterswhich have physical meaning. Starting from y = y. a solution x0, zo is found from (1) and (3) (the other correspondsto c x0). 9 This set is not, of course, generally consistent with equation (2). By iterations with y values decrementedby Ay, after ten cycles it is usually possible to f?nd a set of values for x, y and z which fit equation (2). The solution for y is obtained by linear interpolation between the two values of y which correspond to quantities of opposite sign in the left member of (2). By preliminary desk computing it was found that by assuming Ay = -0~005 mdyn/A it is possible to fit equation (2) within less than 0.01 per cent in &,1,1,. [12] G. RICFCERT and 0. GLEIVEIEZL, 2. Anorg. Allgem. Chem. 807,328 (1981).