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On the nature of isotope shifts in YXY′ molecules
JOURNAL
OF MOLECULAR
SPECTROSCOPY
On the Nature
47, 353-354
(1973)
of Isotope Shifts in YXY’
Molecules*
The widespread use of matrix-isolation spectroscopy and its inherent simplification of the vibrational spectra of polyatomic molecules have greatly increased the amount of data on isotope shifts in triaotmic and other molecules. Although the Wilson F and G matrix formalism (1) predicts the frequency relationships of YXY, Y’XY’, and YXY’ to high accuracy when harmonic frequencies are used, one finds that approximations developed by the early workers E. 0. Salant and J. E. Rosenthal (2) and quoted in standard reference works such as that of Herzberg (3) are still being used to make qualitative interpretations of observed spectra. In cases where the change in isotopic mass Amr is small compared with nzr, the expectation that the observed shift in the asymmetric stretching frequency between YXY and YXY’ will be half that between YXY and Y’XY’ has led to unnecessary confusion and in some cases misassignment of observed frequencies (4). The formula developed by Salant and Rosenthal for the molecule Y(m+Am)-X(M)Y(m+Am), viz. Aw~“/wI = -
MAm/2(m
+ Am)[2m
sin20 + M]
(1)
(where 0 is the half the bond angle) is actually in complete agreement with the predictions of the Wilson formalism to the extent that (ur”/03)~ can be expanded as 1 + ~Aw~“/cQ, neglecting the (Aw~“/u$ term. It is otherwise accurate regardless of the value of Am/m. For the molecule Ycm)XcM)Ycm+““). Salant and Rosenthal give Aoi/wg
= - MAm/2(2m
+ Am)[2m
sin% f M]
(2)
(5) to i.e., nearly half the shift given by Eq. (1). M ore accurate formulas were developed by Rosenthal treat the case in which (Am/m)” is not <
MAm(M
= _~~ 2(m
W%
+
Am)[4m?
•k m)
cos28 sin% + 2mM + M2]
(3
This is accurate to the extent that (wr’w31/~1wr)r may be expanded as 1 + ZAmi’/tir + 2Aw3)/w3, neglecting higher order terms. The key point is that Eq. (3) may be further simplified if, and only if, the low-frequency approximation may be applied to WI. That being the case, one calculates for YXY’ exactly half the shift predicted by Eq. (1). Although this low-frequency approximation may be satisfactory in some cases where M i: m, such as in the CO* molecule or even for the first transition row dihalides, it is practically never the case for molecules in which M >> m. In the former case the normal stretching modes of the YXY’ molecule are close in form to the symmetric and antisymmetric stretching modes of the unsubstituted YXY molecule. * Work performed
under
the auspices
of the U. S. Atomic 353
Copyright All rights
@
1973 by Academic
of reproduction
Press,
in any form
Inc. reserved.
Energy
Commission
In the latter case, the normal modes of the 1~~Yl~’molecule are more nearI>. characterized as S--I and S-Y’ stretches, with the X-Y stretch, and the symmetric and antisymmetric stretching frequencies of the I’XY molecule, being quite C!OSZin frequency, and the X-Y stretch and the stretches of the IY’.YIv’ molecule forming a second group of frequencies. This is precisely the case for H,O, D20, and HOD (6’) in which Am is not <
I, UE-ol= WI
(,4J
fiZ(l + cos2B) + PU’ 4 fiZ(l + cos28) + llv I
’
(5)
c = w1’cA13’ _ (P. + PLJ(k + PI!‘) - P2 cos22L9 + (/& + cc,)2 - fic1.2cos228 WWQ [ 1 ’ the p’s are the reciprocal masses, and w, w’, and 0” refer to the frequencies of I’XE’, YXY’, and Y’XE”, respectively. We have successfully used Eqs. (4-6) to predict the frequencies of isotopically substituted species reported in the literature and in our unpublished results for CeOn, Pro*, and TbOs, even when observed frequencies rather than harmonic frequencies are used (in cases of expected small anharmonicities and interaction force constants such as for the heavy-metal oxides and halides). RECEIVED:
May 9, 1973
I. E. B. WILSON, JK., J. Chem. Phys. 7, 1047 (1939); 15, 736 (1947 J. 2. E. 0. SALANT AND J. E. ROSENTHAL, Phys. Rev. 42,812 (1932). 3. G. HERZBERG, “Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules,” p. 231, Van Nostrand, Princeton, NJ, 1945. 4:. R. L. DEKOCK AND W. WELTNER, JR., J. Pkys. C/rem. 75, 514 (1971). 5. J. E. ROSENTHAL, Phys. Rev. 45, 426 (1934). 6. E. B. WILSON, JR., J. C. DECIUS,AND P. C. CROSS, “Molecular Vibrations,” p. 183, McGraw-Hill, NY, 19.55 and references cited’therein. 7. W. S. BENEDICT,N. GAILAR,AND E. K. PLYLER,J. L’kem.Phys. 24, 1139 (1956). 8. Ref. (6), p. 74 and ‘references cited therein. 9. W. F. LIBBY, J. Chem. Phys. 11, 101 (1943). 10. S. D. GABELNICK,G. ‘I’. REEDY, AND M. G. CIIASANOV,C/rem.Phys. Letl. 19, 90 (1973) ; J. Chew Phys. 58, 4468 (1973). 11. J. W. HASTIE, R. H. HAUGE, AND J. L. MARGRAVE,IIigh Temp. Sci. 3, 56 (1971) 12. C. W. DEKOCK, R. D. WESLEY, AND D. D. RADTKE,High Temp. Sci. 4,41 (1972). S. D. GABEWICK Chemical Engineering Division Argonne National Laboratory 9700 South Cass Avenue Argonne, Illinois 60436
OF MOLECULAR
SPECTROSCOPY
On the Nature
47, 353-354
(1973)
of Isotope Shifts in YXY’
Molecules*
The widespread use of matrix-isolation spectroscopy and its inherent simplification of the vibrational spectra of polyatomic molecules have greatly increased the amount of data on isotope shifts in triaotmic and other molecules. Although the Wilson F and G matrix formalism (1) predicts the frequency relationships of YXY, Y’XY’, and YXY’ to high accuracy when harmonic frequencies are used, one finds that approximations developed by the early workers E. 0. Salant and J. E. Rosenthal (2) and quoted in standard reference works such as that of Herzberg (3) are still being used to make qualitative interpretations of observed spectra. In cases where the change in isotopic mass Amr is small compared with nzr, the expectation that the observed shift in the asymmetric stretching frequency between YXY and YXY’ will be half that between YXY and Y’XY’ has led to unnecessary confusion and in some cases misassignment of observed frequencies (4). The formula developed by Salant and Rosenthal for the molecule Y(m+Am)-X(M)Y(m+Am), viz. Aw~“/wI = -
MAm/2(m
+ Am)[2m
sin20 + M]
(1)
(where 0 is the half the bond angle) is actually in complete agreement with the predictions of the Wilson formalism to the extent that (ur”/03)~ can be expanded as 1 + ~Aw~“/cQ, neglecting the (Aw~“/u$ term. It is otherwise accurate regardless of the value of Am/m. For the molecule Ycm)XcM)Ycm+““). Salant and Rosenthal give Aoi/wg
= - MAm/2(2m
+ Am)[2m
sin% f M]
(2)
(5) to i.e., nearly half the shift given by Eq. (1). M ore accurate formulas were developed by Rosenthal treat the case in which (Am/m)” is not <
MAm(M
= _~~ 2(m
W%
+
Am)[4m?
•k m)
cos28 sin% + 2mM + M2]
(3
This is accurate to the extent that (wr’w31/~1wr)r may be expanded as 1 + ZAmi’/tir + 2Aw3)/w3, neglecting higher order terms. The key point is that Eq. (3) may be further simplified if, and only if, the low-frequency approximation may be applied to WI. That being the case, one calculates for YXY’ exactly half the shift predicted by Eq. (1). Although this low-frequency approximation may be satisfactory in some cases where M i: m, such as in the CO* molecule or even for the first transition row dihalides, it is practically never the case for molecules in which M >> m. In the former case the normal stretching modes of the YXY’ molecule are close in form to the symmetric and antisymmetric stretching modes of the unsubstituted YXY molecule. * Work performed
under
the auspices
of the U. S. Atomic 353
Copyright All rights
@
1973 by Academic
of reproduction
Press,
in any form
Inc. reserved.
Energy
Commission
In the latter case, the normal modes of the 1~~Yl~’molecule are more nearI>. characterized as S--I and S-Y’ stretches, with the X-Y stretch, and the symmetric and antisymmetric stretching frequencies of the I’XY molecule, being quite C!OSZin frequency, and the X-Y stretch and the stretches of the IY’.YIv’ molecule forming a second group of frequencies. This is precisely the case for H,O, D20, and HOD (6’) in which Am is not <
I, UE-ol= WI
(,4J
fiZ(l + cos2B) + PU’ 4 fiZ(l + cos28) + llv I
’
(5)
c = w1’cA13’ _ (P. + PLJ(k + PI!‘) - P2 cos22L9 + (/& + cc,)2 - fic1.2cos228 WWQ [ 1 ’ the p’s are the reciprocal masses, and w, w’, and 0” refer to the frequencies of I’XE’, YXY’, and Y’XE”, respectively. We have successfully used Eqs. (4-6) to predict the frequencies of isotopically substituted species reported in the literature and in our unpublished results for CeOn, Pro*, and TbOs, even when observed frequencies rather than harmonic frequencies are used (in cases of expected small anharmonicities and interaction force constants such as for the heavy-metal oxides and halides). RECEIVED:
May 9, 1973
I. E. B. WILSON, JK., J. Chem. Phys. 7, 1047 (1939); 15, 736 (1947 J. 2. E. 0. SALANT AND J. E. ROSENTHAL, Phys. Rev. 42,812 (1932). 3. G. HERZBERG, “Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules,” p. 231, Van Nostrand, Princeton, NJ, 1945. 4:. R. L. DEKOCK AND W. WELTNER, JR., J. Pkys. C/rem. 75, 514 (1971). 5. J. E. ROSENTHAL, Phys. Rev. 45, 426 (1934). 6. E. B. WILSON, JR., J. C. DECIUS,AND P. C. CROSS, “Molecular Vibrations,” p. 183, McGraw-Hill, NY, 19.55 and references cited’therein. 7. W. S. BENEDICT,N. GAILAR,AND E. K. PLYLER,J. L’kem.Phys. 24, 1139 (1956). 8. Ref. (6), p. 74 and ‘references cited therein. 9. W. F. LIBBY, J. Chem. Phys. 11, 101 (1943). 10. S. D. GABELNICK,G. ‘I’. REEDY, AND M. G. CIIASANOV,C/rem.Phys. Letl. 19, 90 (1973) ; J. Chew Phys. 58, 4468 (1973). 11. J. W. HASTIE, R. H. HAUGE, AND J. L. MARGRAVE,IIigh Temp. Sci. 3, 56 (1971) 12. C. W. DEKOCK, R. D. WESLEY, AND D. D. RADTKE,High Temp. Sci. 4,41 (1972). S. D. GABEWICK Chemical Engineering Division Argonne National Laboratory 9700 South Cass Avenue Argonne, Illinois 60436