The absolute intensities of the binary combination bands in the infrared spectrum of SF6

Spectrochimica Acta, Vol. 38A, NO. 8. pp. 841-847, Printed in Great B&am.

1982

0584-8539/82/080841M$03.00/0 @ 1982 Pergamon Press

Ltd

The absolute intensities of the binary combination bands in the infrared spectrum of SF, DOUGLAS S. DUNN, KERIN SCANLON and JOHN OVEREND Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, U.S.A. (Received

16 Nooember 1981)

Abstract-The intensities of all the fundamentals and binary combination bands in the infrared spectrum of SF, have been measured. Estimates of the cubic normal-coordinate force constants previously determined from an anharmonic Urey-Bradley force field have been used to compute the contributions to the combination intensities from the first derivatives of the dipole moment. These, in turn, have been used to obtain experimental values of the second derivatives of the molecular electric dipole moment with respect to the normal coordinates.

INTRODUCTION PERSON and OVEREND [l] have recently considered a model for the quantitative prediction of vibrational strengths of the fundamentals and binary combination transitions in the infrared spectrum of SF,. They were able to compare the results of their predictions with the relative intensities measured in liquid-oxygen solution by BERTSEV, KOLOMIITSEVA and TSYGANENKO [2] but there were, at that time, no experimental measurements of the combination and overtone intensities of SF6 in the gas phase. The latter, of course, give estimates of the electric-dipole transition probabilities of the isolated molecule and hence of the experimental quantities which should properly be compared with the theory. In this paper we report the experimental measurements of the integrated molar absorption coefficients for all the electric-dipole allowed binary combination bands of SF, in the gas phase. These results are then used together with the new estimates of the anharmonic normal-coordinate force constants reported in a previous paper [3] to obtain estimates of the second derivatives of the electric-dipole moment with respect to the normal coordinates. EXPERIMENTALRESULTS The sample of SF, was obtained from Matheson; it had a stated purity of 99.9% and was used without further purification [4]. A glassbodied cell was used for all the measurements. The cell had a path length of 9.2 cm and was fitted with windows of CsI. Sample pressures were measured with a 1 cm bore mercury manometer except in the case of the very intense bands for which the required pressures were 0.2-0.7 torr and for which we used a McLeod gauge. All spectroscopic measurements were made with a PerkinElmer Model 283 double-beam spectrometer which covers the wavenumber range 4000-200 cm-‘. Wavenumber calibration of the spectrometer was carried out at the beginning of the series of measurements using CH,, NHS, HCI, DC], etc.

lines according to the standard calibration scheml for small grating spectrometers [5]. The wavenumber calibration was checked periodically every two or three days throughout the measurements and was found to remain constant. Intensities were measured by recording the spectra directly as absorbance vs wavenumber and integrating the areas under the bands on the spectral record with a polar planimeter. In all cases the base line was determined by superimposing a spectrum of the cell containing just one atmosphere of argon on the spectrum of the sample. Each band was measured at a number of sample pressures and Beer’s law plots were constructed; examples of these are shown in Figs. l-3. The integrated molar absorption coefficients were determined by least-squares analysis of the data used for the Beer’s law plots and are summarized in Tables 1 and 2. We include there the measured intensity of a band at about 300cm-’ which is tentatively assigned to the difference bands (Ye- v2) and (Y*- V& the two bands are expected to be very close together and the measured intensity corresponds to the sum of the intensities for the two bands. No serious problems were encountered in the experimental measurement of the intensities of the binary combination bands. Indeed, all except (Ye+ v~) and (v5 + vg) are well isolated from other bands in the spectrum and their measurement was quite straightforward. The extreme R-branch wing of ( v5+ vg) merges slightly with the P-branch wing of v3. A graphical separation was made and the intensity of (v5 + ug) was determined by integration from 904cm-’ to lower wavenumbers. The (I++ v6) band, which lies to the high wavenumber side of the intense V~fundamental, is slightly overlapped in the wing of the P-branch. Again a graphical separation was made and the intensity of ( v2 + v6) was determined by integration down to 968 cm-‘. We do not believe that the measured intensity of either ( v5 + vg) or ( v2+ vg) is significantly affected by the graphical separation from v3. 841

D. S.

842

DUNN et al.

Even though we believe that previous estimates of the intensities of the two i.r. active fundamentals, u3 and v4, are reliable, we redetermined these intensities in this study as a check on the reliability of our measurements. Our new values are compared with the previously determined ones in Table 1. We interpret the excellent agreement as evidence that our new experimental values for the intensities of the combination and difference bands are not encumbered by serious systematic errors. The only band in the spectral region studied other than those given in Tables 1 and 2 is one at cu. 2225 cm-’ which is assigned to (2~~ + v3). The intensity of this band was found to be 0.115 * 0.003 km/mole.

45.(

,

b

4o.c

35.0

30.0

h 25.0 7 E Z P) 20.0 G

Table 1. Experimental absolute integrated molar absorption coefficients, A,, for the i.r. active modes of SF,. y = 948.0 cm-’ and v4 = 615.0 cm-‘. Units are km/mole

15.0

10.0

5.0

1.0

2.0

3.0

bc X lo+

4.0

5.0

8.0

(moleelctn2)

Fig. 2. Beer’s law plots for the following binary combination bands of SF,: (a) (v*+ Q), (b) (y+ v,), (c) (v5+ a& (d) (Y, + VA and (e) (vt + v,). 0.5

bc X 16’ (molee/cm2). Y 3 1.5 2.0 2.5 1.0

3.0 18.C

18.a

14.c

k 2

8.0

9 ;i 8.0

v

4.0

0

5.0

2.c

0

1.0

2.0

1.0 3.0

4.0

5.0

0.0

bc X 16’

bc X 16’ (moles/cm2 ), v,

Fig. 1. Beer’s law plots for the v3 and V, fundamentals SF,.

2.0

of

3.0

4.0

5.0

8.0

(moles/cme)

Fig. 3. Beer’s law plots for the following bands of (b) (vj - MYw,), (c) (IQ+ Q), (d) (2~ + VI).

SF6: (a) (Y,+ 4.

Binary combination bands in the infrared spectrum of SF,

843

Table 2. Experimental absolute integrated molar absorption coefficients, A,, for i.r. active binary combination and difference bands in the vibrational spectrum of gas-phase SF6. Units are km/mole. Statistical dispersions, shown in parentheses, apply to the last digits of the quoted value TKallSltiOn

Wavenumber,

cm-’

a

A,/&

%

Present

work

Bercse", s

1719.2

3.89(7)

5.6 x lO-2

3.8 x 10-z

1388.1

0.46(Z)

6.6 x 10-3

3.0 x 10-3

1587.9

8.38(10)

1.2 x 10-1

9.0 x 10-4

1257.0

1.91(Z)

2.7 x 1O-2

2.0 x 10-z

991.2

13.9703)

2.0 x lo-'

7.7 x 10-z

1456 1140.4 869.9

co.04

(6

x lo-')

a.

b

8.2 x 10-L

0.16(5)

2.3 x 10-3

5.0 x 10-u

4.48(13)

6.4 x 1O-2

4.2 x 1O-2

0.30(l)

4.3 x 10-3

305.9 296

“Ref. [13]. bRef. [2].

INTERPRETATION

Before embarking on the interpretation of the intensities of the combination bands we need values of the first derivatives of the molecular electric dipole moment with respect to q3 and q4. From the values of A4 given in Table 1 we determined a simple-averaged value, A4 = 69 km/mole. The value of A3 reported by KIM, MCDOWELL and KING [6] seems to be out of line with the other values. The real difficulty in measuring A, arises because it is a very intense band and one must use either very low partial pressures of SF, or very short pathlengths. In the present study we used low pressures measured with a calibrated McLeod gauge. SCHACHTSCHNEIDER[7] used low-gas densities determined by expanding a weighed quantity of SFs into a calibrated volume and BRODBECKet al. [8] used pressures in the range 30-100 torr with a very short cell, the pathlength of which was determined interferometrically. SCHATZ and HORNIG [9] and KIM et al. [61 used dilution techniques to measure the low-partial pressure of SF, in their sample. Our conclusion after discussion the problem with Professor King is that a systematic error in the dilution is responsible for the high value of A3 measured by KIM et al. and for this reason we have neglected their result in arriving at our average value of A3 = 1073 km/mole. The signs of ap/aq3 and +ddq4 were taken from the transferred polar tensors and are the same as those previously adopted by PERSON and OVEREND [l, 101. The measured intensities were then used to compute experimental values of the dipole transition matrix elements. Since the mechanical vibrational anharmonicities were expressed in a Cartesian representation of the normal coordinates

[l l] it is convenient at this point to express the quantum states for the dipole matrix elements in a Cartesian notation rather than in the polar notation which is more familiar and in which the states were described in the preceding and in Table 2. The correspondence between the two notations is summarized in Table 3. The important transition moments available from the observed intensities have been summarized in Table 4. As is well known, there are contributions to the electric-dipole vibrational transition moments for two-quantum transitions (overtones, binary combination bands, and binary difference bands) from mechanical anharmonicity and from electrical anharmonicity, i.e. the second derivatives of the dipole moment with respect to the normal coordinates. In their earlier treatment, PERSON and OVEREND [l] focused attention on the first contributions and did not consider the second. Our new treatment of the mechanical anharmonicity in SF, with the anharmonic Urey-Bradley force field [3] gives significantly better estimates of the anharmonic contributions to the vibrational energies than were given by the anharmonic simplevalence force field on which PERSON and OVEREND [l] based their interpretation. We therefore believe that we are now in a position to make a reasonable estimate of the contributions to the dipole-moment matrix elements from mechanical anharmonicity and hence to use the experimental results to learn something about the electrical anharmonicity in SFs. The calculated contributions from mechanical anharmonicity are summarized in Table 4; under column (i) we show the results calculated from the normal-coordinate force constants given in ref. [3]. Under column (ii) we show the results calculated

844

D. S. DUNN et al. Table 3. Notation for the normal modes of SF6; correspondence between the polar and Cartesian representations of the normal coordinates Polar

symetry

Polar

Cartesian

Table 4. Observed electric dipole matrix elements for the binary combination bands in the i.r. spectrum of SF, compared with the calculated contributions from mechanical anharmonicity. A Cartesian representation of the normal coordinates is used, cf. Table 3 s

x104,

i

51 Experlme"tala

Calculated (i)

contribution

from mechanical

anharmonic~tyb

(Ii)

(Id)

(IV)

1

4

t35.5

f21.6

+23.0

+20.9

1

5

t13.3

-15.3

-15.7

-18.7

+21.6 -15.3

2

4

t3a.o

+29.8

+31.7

+32.9

+29.8

2

5

i20.0

-9.5

-10.1

-16.1

-9.5

2

13

256.4

-120.1

-120.1

-50.1

-120.1

6

12

i2.7=

f6.8

+6.8

C4.5

+10.3

7

12

t6.0

-11.3

-11.3

-R.4

+2.8

11

15

t33.5

-48.6

-48.6

-175.3

-22.2

“The sign is indeterminate since A, - l(Op(v#. b(i) using the normal-coordinate force constants from ref. [3]. (ii) Using normal-coordinate force constants from the internal-coordinate force constants of ref. [3] but with 0 = 1.8 A-‘. (iii) Using an anharmonic Urey-Bradley field based on the quadratic field given in ref. [12]. (iv) Same as (i) but with an assumed value of H,,, = -0.5 md A/rad’. ‘These experimental values are upper bounds, cf. Table 2.

from

an anharmonic

force

field identical

to that

used in ref. [3] except that the Morse parameter, a, was increased from 1.7 to 1.8 As-‘. The results shown under column (iii) were obtained using a completely different anharmonic Urey-Bradley force field which was constructed by taking the quadratic part from the Urey-Bradley force field of LABONVILLE et al. [12], assuming a value of the van der Waals F-F distance to be 2.708, and assuming the Morse parameter, a, to be 1.7 k’. In all the anharmonic force fields used to calculate the results shown in columns (i), (ii), and (iii) we assumed the potential energy in the valence-anglebending internal coordinate to be purely harmonic. This assumption was made because we have no

convenient model for the anharmonic part of the bending potential. In order to test the sensitivity of our results to this assumption we carried out another calculation of the contributions of mechanical anharmonicity to the dipole transition moments using the force constants of ref. [3] plus an assumed bending force constant, H,,, = -0.5 md &ad’. The results of this calculation are shown in Table 4 under column (iv). These several calculations shown in columns (ii)served to give a rough estimate of the reliability of the values given in column (i) which really represent the best estimates we are able to make at this time. Note, in particular, that the results in columns (ii) and (iv) are complementary

Binary combination

in that the contributions from mechanical anharmonicity to (u, + vJ, (v, + Y& (v2 + vq), and (~2 + vs) are sensitive to anharmonicity in the bondstretching coordinates but not to anharmonicity in the valence-angle bending coordinate. In contrast, the contributions to ( v6 + vlZ), (v, + z+), and (vI1 + Y,~) are sensitive to anharmonicity in the bending coordinate but not to anharmonicity in the bondstretching coordinate. The contributions to ( v2 + v,J are sensitive neither to changes in the cubic stretching force constant nor to the introduction of a cubic bending force constant. It is apparent from comparison of the calculated values in columns (i) and (iii) that the contributions

of mechanical anharmonicity to most of the matrix elements are not strongly dependent on the details of the force field. The most glaring exceptions are (vz+ yt3) and (v,, + Y,~), both of which lie very close to the intense SF stretching fundamental and are strongly coupled with it. The mixing of these two combination states with the fundamental depends critically on the cubic force constants and the calculated energies and hence the dipole strengths borrowed from the fundamental by the two combination bands in the calculation of the effects of mechanical anharmonicity is critically dependent on the details of the force field. For this reason we shall forego further discussion of the intensities of these two bands. At this point we turn to the band at -3OOcm-’ which we showed in Table 2 with a measured intensity of 0.30 + 0.01 km/mole. This is at a lower wavenumber than any of the fundamentals of SF6 [8] and we have assigned it as a difference band. From its observed wavenumber it must be either (u) - vz), which, using the wavenumbers of the fundamentals given by c

J

#

+ 360

340

320

Wavenumber,

300

cm-

280

845

bands in the infrared spectrum of SF6

260

1

Fig. 4. The difference band at ca. 3OOcm-’ in the i.r. spectrum of SF,. Resolution: 0.25 cm-‘, sample pressure: 767.7 torr, path length: 9 cm.

Table 5. Values of (#p”/aq,aq,.) determined

from the experimental intensities and the set of contributions from mechanical anharmonicity shown under column (i) in Table 4

1

4

C27.8

or

-114.1

1

5

+4.0

or

+57.2

2

4

+16.3

or

-135.5

2

5

-21.0

or

6

12 a

-8.3

or

-19.0

7

12

+10.6

or

+34.5

+5l3.e

“These values are calculated on the assumption that the experimental value of j(Ol~“lu,,= I, vu= I)]= 2.7; if this value is reduced, the value of (dzpx/3q68q12) will be changed correspondingly.

MCDOWELL, ALDRIDGE and HOLLAND [13] has an expected wavenumber of 305.9 cm-’ or (u* - vg) which has an expected wavenumber of 296.1 cm-‘. The spectrum of this band, recorded with a Digilab FTS-20 spectrometer, is illustrated in Fig. 4. The most obvious interpretation is that it is a single band centered at -305 cm-’ and since the calculated contributions from mechanical anharmonicity to (u2 = 1, u4 = 01px1v2 = 0,04= 1) is in all cases much larger than that to (u. = 1, u2 = Olpx1u,3 = 0,u2= l), cf. Table 6, we assigned the observed intensity to (vg - YJ and concluded that (vz- ug) is vanishingly weak. In Table 6 we show the experimental electric dipole matrix elements derived from the observed intensity of the difference band at 305 cm-’ and the calculated contributions from mechanical anharmonicity. We show two sets of experimental numbers; the first based on the assumption that all the intensity of the 305 cm-’ band is ascribable to ( vj - vz), the second based on the assumption that all the intensity is ascribable to ( v1 - r+J. The large differences in the experimental values of the transition moment occur because of the large differences in the populations of the lower state corresponding to each assignment. If we accept the assignment of the 305 wavenumber band to ( V~- u2), we conclude that (dzpx/dq2dq4) is equal to -65.1 x lo-“ or +291.2 x 10m4eA, a result at variance with the other estimates of the quantity given in Table 5. DISCUSSION In an attempt to resolve the apparent inconsistency in that we obtained different values of (d2px/dq2dq4) from (I+ + Y*)and (v3 - v2) we resorted to a remeasurement of the intensities of all the bands using a Fourier-transform spectrometer as described by SCANLON, LAUX and OVEREND 1141. The remeasured integrated intensities in km/mole were as follows: A(vj) = 859 f 36, A(v.,) = 80 & 3, A(v,+ v3) = 3.45 f 0.08, A(v1+ v.,) = 0.42 ? 0.01,

D. S. DUNN et al.

846

Table 6. Observed electric dipole matrix elements for the binary difference bands in the i.r. spectrum of SF, compared with the calculated contributions from mechanical anharmonicity. A Cartesian representation of the normal coordinates is used

(1)

(11)

-56.5

-60.1

+3.4

f3.4

(lil)

(iv)

/ 4

2

2

13

‘89.1

t3a.2

-59.8

-56.5

+1.?3

+1.4

“The value of k89.1 is calculated on the assumption that all the intensity is ascribable to (Q - v~), the value of t38.2 on the basis that all the intensity is ascribable to ( v2- vg). bCf. footnote (b), Table 4.

A(u2+ ~,)=7.79?0.21, A(v2+ v6) 13.02kO.14, A( vj + v6) = 3.73 +- 0.15,

A(v2+ v.J= 1.72kO.05, A(v.,+ ~,)=0.14~0.01, A( v3 - v2) = 0.27. and Since the new values of A(v,) and A(v4) do not match particularly well the other measurements summarized in Table 1 and since we have been experiencing some problems with our Fouriertransform spectrometer, we prefer to use the experimental results given in Tables 1 and 2 but, the fact that the remeasured combination intensities are in reasonable agreement with those in Table 2 does reassure us that our experimental intensities are not seriously in error. Our conclusion is that the apparent inconsistency between the two values of (d2px/dq2dq4) is not the result of gross error in measurement. Deferring consideration of this particular difficulty, we turn to the values of the second derivatives of the dipole moment given in Table 5. Most of these are reasonably large and make significant contributions to the matrix element (u[~Ju’), cf. Table 4. We are fortunate in having available ab initio values of some of these derivatives [15] calculated with a reasonably large basis set and in which we have some confidence. The essential results of these ab initio calculations are summarized in Table 7. Although, as we expected, the experimental values of the first derivatives of the dipole moment and the ab initio calculated ones do not agree exactly, the magnitudes are reasonably close-certainly within a factor of two. Similarly, the ab initio value of the second derivative, (d2px/dq1dq4), matches in a satisfactory way the one experimental value (#px/aq,dq4) = 28 x 10m4eA but not the other, (#px/dq1dq4) = -114 x 10m4eA and leads us to accept the former as the correct experimental value. In the case of (d2px/dq28q4), the value obtained in the ab initio calculation agrees with none of the possible experimental values although it appears that the one experimental value (a*p’/aq,dq,) = +16x lo-“eA is the closest to the ab initio result.

Table 7. Summary of results of ab initio calculations of the electric dipolemoment derivatives of SF, with respect to dimensionless normal coordinates.’ Units are eA X 10m4b ab initio -~

aCtas,

-624

a2uxias,as, a2ux/as,as a2,‘/aq2aq,

-1133

-1388

a~x/aq,

‘t

c d

CXp.XilW"t

-358

+21

+28 or -114

-5

+16 or -136

-5

-65 or +291

“Determined with a double-zeta basis set plus polarization functions on the sulfur atom, cf. ref. [15]. bathe conversion to derivatives with respect to normal coordinates, Q, which have dimensions amu”‘A is achieved by multiplying ap/dql by (27~w,/Fi)“* = 0.1722192 q”* and a*k/ulaq,dq,.by (2~cw,/~)“2(2Pcw~,/~)“*. ‘Determined experimentally from (v2 + I+). dDetermined experimentally from (IQ- v*).

We believe that the apparently anomalous value of (8*px/8q2dq4) derived from the difference band results from an underestimation of the anharmonic contributions to the intensity of (vg - Y*).We note that (v2 + v6) lies close to the v3 fundamental; indeed it appears to be reasonantly coupled with it as evidenced by the intensity of ( v2 + V& cf. Table 2, and the strong dependence of the contribution to (OICLIJVZ = 1, ~5~= 1) on the details of the anharmonic force field, cf. Table 4. Since (t+ + v6) is the lower state of the principal hot band of (v~- vZ), we may expect the lower states of most of the other hot bands of (v, - v2) to be similarly coupled to the corresponding states in the v3 manifold (e.g. 102= 3, us = 1, u13= 2) will be coupled with (us = 2, u4 = 1, u5 = 1, u13= 1) which will also be connected by the same operator with ju2 = 1, tr4 = 2, uJ = 1, IJ,~= 0)) [16]. This resonant coupling may result in a breakdown of perturbation

Binary combination

bands in the infrared spectrum of SF,

theory and, as a consequence, the experimental estimate of (u, = 0, v2 = 1(/1”Ju, = 1, u2 = 0) may be in error since it was determined from the experimental intensities by using a perturbation treatment [17,18] to correct for hot band contributions to the band intensity. Furthermore, since the vibrational partition function of SF, at room temperature is about three, only about one third of the molecules contributing to the intensity of ( V~- v2) at room temperature do so through the principal u2 = 1 + Us= 1 transition and the derived value of (04=0, v2= 1]~x]04= 1, u2=O) may be in serious error if the perturbation treatment of the hot bands breaks down. For this reason we discount the value of (a2~X/&@q.+) determined from the experimental intensity of ( vj - v2). The most important result of the present study is the establishment of the magnitudes and signs of the second derivatives of the dipole moment, (#~*/aq1aq4) and (#1*.*/aq28q4). Since the anharmanic contributions to the dipole strengths of the (v, + vq) and (Ye+ vq) transitions are sensitive to principal anharmonic force constants in the bondstretching coordinates but not to principal anharmanic force constants in the bending coordinates [19] and since the anharmonic force field was designed specifically to model the anharmonicity in the stretching coordinates, the experimental values of (a2~x/8q18q4) and (82px/8q28q4) should be reasonably good. We have not been able to obtain a systematic estimate of the likely errors but we believe these to be not more than *Xl%. Also, as we already pointed out, the calculated contributions from mechanical anharmonicity to (v, + v5), ( v2+ Q), and (v2 + I+) are not changed when we arbitrarily introduce a principal cubic bending force constant and these numbers should be reasonably well determined. By analogy with the ab inifio calculations of (8’px/aq18q4) and (a2pxlaq2aq4) we choose the values of (a2px/aq1aq5) and (azcL’/aq2aq5) with the smallest magnitudes as being most likely to be the correct (a*px/aq,aqs) = +4.0x 10m4eA and ones, i.e. (a2px/aqzaq5)= -2l.O~lO-~eA. We have somewhat less confidence in the moment, second derivatives of the dipole (a*px/aq,aq,,) and (a2px/aq,aq,2), largely because we have less confidence in the anharmonic bending potential and therefore in the correct contributions to (O]pX(v, = 1, u,, = 1) from mechanical anharmonicity, cf. Table 4, columns (iii) and (iv). We expect the results described in this paper to be a useful contribution to the ongoing programatic study aimed at developing an understanding of the higher derivatives of the dipole moments of simple polyatomic molecules.

841

Acknowledgement-Acknowledgement is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of the research. REFERENCES

[l] W. B. PERSON and J. OVEREND,J. C/tern. Phys. 66, 1442 (1977). [2] V. V: BE~TSEV, T. D. KOLOMIITSEVAand N. M. TSYGANENKO,Opt. Spectrosc. 31, 263 (1974). [3] I. SUZUKI and J. OVEREND,Spectrochim. Acta 37.4, 1093 (1981). [4] A relatively weak band, with an apparent intensity A - 2 km/mole. at 1283 cm-’ in the i.r. soectrum of SF, has been ascribed to CF, impurity at about 0.2% concentration. This band did not interfere with the spectroscopic measurements and no corrections have been made to the experimental data for this impurity. [5] Tables of wavenumbers for the calibration of infrared spectrometers, ZUPAC Commission on Molecular Structure and Spectroscopy. Butterworth, Washington (l%l). [6] K. KIM, R. S. MCDOWELLand W. T. KING, J. C/tern. Phys. 73, 36 (1980). [7] J. H. SCHACHTSCHNEIDER, dissertation, University of Minnesota (l%O). [8] C. BRODBECK,I. ROSSI, H. STRAPELIASand J. P. BOUANICH,Chem. Phys. 54, 1 (1980). [9] P. N. SCHATZand D. F. HORNIG,J. Chem. Phys. 21, 1516 (1953). [lo] The absolute sign of +/U.q is arbitrary and depends on the phase of the normal coordinate. However, if the sign is defined by a polar tensor, all signs are maintained consistent with the phase of the normal coordinates adopted in the present work. [ttl J. OVEREND,Spkctrochim. kta 32A, 1581 (1976). f121 P. LABONVILLE,J. R. FERRARO,M. C. WALL and L. J. BASILE, Coord. Chem. Rev. 7, 257 (1972). 1131R. S. MCDOWELL,J. P. ALDRIDGEand R. F. HOLLAND,J. Phys. Chem. 80, 1203 (1976). 1141K. SCANLON, L. LAUX and J. OVEREND, Appl. Spectrosc. 33, 346 (1979). it51 K. SCANLON, R. A. EADES and D. A. DIXON, Spectrochimica Acta 38A. 849 (1982). [t61 Since each hot band contributes only a fraction of the intensity of the principal transition, the very large contribution of hot bands to the intensity implies that there are significant contributions from a very large number of hot bands. rt71 S. J. YAO. dissertation. Universitv of Minnesota (1%6). ft81 S. J. YAO and J. OVEREND,Spectrochim. Acta 32A, 1059 (1976). 1191This effect is traceable to the dependence of the normal-coordinate force constants on principal cubic internal-coordinate force constants. By symmetry, vi and v2 are purely stretching vibrations and V~and vs are purely bending vibrations; Y, and v, are mixtures of bending and stretching vibrations. The principal internal-coordinate cubic stretching force constants can contribute only to those cubic normalcoordinate force constants for which all the normal coordinates contain stretching motion and hence only to normal-coordinate force constants of the etc. The same type klu, kllS, etc. but not to k4,6.12. argument applies, mutatis mutandis, to the anharmanic bending force constants.