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A general isotopic sum-rule for frequencies, band intensities, and Coriolis coupling constants
JOURNAL
OF MOLECULAR
A General
SPECTROSCOPY
80,455-458
(1980)
Isotopic Sum-Rule For Frequencies, Band intensities, and Coriolis Coupling Constants
Isotopic sum- and product-rules in a general form involving the vibrational frequencies were first developed by Heicklen (I). Recently, a new isotopic sum-rule related to the frequencies was reported in this Journal (2). This rule was derived in explicit form for symmetrical substitutions, using mass-weighted Cartesian coordinates as a basic set. But, until now, it does not appear to have been realized that a simple isotopic sum-rule valid for symmetrical as well as for unsymmetrical substitutions could be derived in an analogous form for the vibrational frequencies, band intensities (infrared, Raman), and the Coriolis coupling constants. The derivation of such a general sum-rule based on the usual FG matrix formalism (3) is presented in this communication and it will be shown that this rule is a general form of the one published earlier in this Journal (2). lsoropic rules. Tr(GkFR) with respect to the internal or valence coordinates could be expanded to read (3) Tr(A) = Tr(CPFR) = $ 1 (K$)/m,)F,I, 01 I,
(1)
where N is the total number of atoms (I in the isolated molecule, m, is the mass of atom a, Kg’) are the mass-independent coefficients which depend on the equilibrium geometrical parameters, and Fij are the quadratic force constants which are invariant to isotopic substitutions under the Born-Oppenheimer approximation. For the substitution of the ath atom, one obtains from Eq. (1) Tr(AA,) = 1 (Kaj)(-Am,/m,m~))Fi, U In both Eq. (1) and Eq. (2), A is a diagonal matrix related to the frequency (we use the superscript + to denote the isotope). Equation (2) can be modified to read (m,+lAm,)Tr(AA,)
(2) parameters
(3) (3)
= - 1 (K~Vm,)Fi,.
(3)
il
The summation of both sides of Eq. (3) with respect to a and the use of Eq. (1) lead to 5 (&lAm,)Tr(A&)
= -Tr(A),
which provides a relation connecting the isotopic shifts in the vibrational frequencies successive replacement of each atom by its isotope. Since the band intensities (infrared, Raman) obey the sum-rule (3-J)
(4) due to the
where the symmetric matrix B corrected for rotational effects is independent of isotopic substitution (Born-Oppenheimer approximation), Eq. (4) can be modified for intensities to give
i (m,flAm.) 1 AIk,,, = - z:Ik. k k
(6)
cl
In the above relation, Ik may be identified with the integrated infrared intensities or with the squares of the derivatives of mean polarizability or anisotropy with respect to the normal coordinates (i.e., (WlaQ,)*, (ay19Qk)*). It should be noted that the above rule can be directly applied to the experimental data for molecules possessing no permanent dipole moment (infrared) and spherical tops 455
0022-2852/80/040455-04$02.00/O Copyright 0 1980 by Academic Press, Inc. AU rights of reproductioo in any fom reserved.
456
NOTES
(Raman), only. In other cases, the intensity data must be corrected momentum (4, 6). The Coriolis coupling constants 5” satisfy the sum-rule (7)
for vibrational
angular
Tr(C”F) = Tr(I;“A).
(7)
Since C” has elements which satisfy an equation analogous to (1), it is evident that the sum-rule 2 (m,flAm,)Tr(A(<~&))
= -Tr(3”A)
(8)
holds for the Coriolis coupling constants. In many cases, it is advantageous and easier to employ the isotopic substitution of some atoms only, as for example, the isotopic replacement of a set of symmetrically equivalent atoms. Also, in such cases one can obtain a simple isotopic relation. Let us suppose both (Yand p atoms are substituted, simultaneously. Corresponding to this situation, the relation Tr(AA,,p) = 1 Kp'(-Am,lm,m;)Fij+ 1 K~'(-Amplmem~)Fij, (9) iJ ii can be developed from (1). Comparing the RHS of Eq. (9) with those of Eq. (2) and the analogous one related to p, the isotopic relation Tr(AA,,,) can be formulated.
(10)
For the substitution of p atoms, Eq. (10) can be expanded to read TI(AA,,~,
If, (1, P, .
= Tr(AA,) + Tr(AAB)
,,,) = Tr(AA,) + Tr(AA,J + . . * + Tr(AAP).
(11)
3p all form a symmetrically equivalent set, Eq. (11) reduces to
Tr(AAd = nTr(AA,),
(12)
where n might cover, either partially or completely, a set of equivalent relations are general and that they are valid for symmetrical as well stitutions. The relations analogous to (11) for the vibrational band intensities constants read + 1 C Alrc,,p......, = C Al,,,, + 1 Al,,p, + k k !i k and Tr(A(3”A),,p.. .,A = Tr(A(SUNa)+ Tr(AU2AM + . +
atoms. Note that the above as for unsymmetrical suband the Coriolis coupling
AIM,,,
(13)
TtiA(PQJ.
(14)
For atoms belonging to an equivalent set, Eqs. (13) and (14) simplify to C Al,,,,,, = n C AIM,,, k B
(15)
and Tr(A(S”N,,,) = nTr(A(rA),).
(16)
It is naturally evident that in all these relations (for the frequencies, intensities, Coriolis constants), the shifts are defined with respect to a single parent molecule. Again in Eqs. (15) and (16), n covers either some or all the atoms forming a symmetrically equivalent set. As shown below, Eq. (I) of Ref. (2) is a special case of Eqs. (4) and (12). Considering H,O-DIO and H,O-T,O substitutions, let 0 be replaced by O+ in both cases. Then, it follows from Eqs. (4) and (12), that (m&/(m,+ - mO))Tr(AA,,) + (mD/(mD- m,))Tr(AA&
= -Tr(A),
(17a)
(m,$/(m,f - m,))Tr(AA,)
= -Tr(A).
(17b)
and + (mrl(mT- m,))Tr(AA&)
By combining Eqs. (17a) and (17b), it can be easily shown that m”(mr - mD) .X kIO, k
+ mn(mH - m,) 1 kDzO) + Mm,, k
- m,) C hn,, = 0, k
which is identical to Eq. (1) of Ref. (2). In the above relation, At are the elements of A.
(18)
457
NOTES
The utility of the isotopic rules is illustrated in Table I for the Raman intensity data for CH,, CHBD, CH2D2, CHDS, and CD,. The experimental data are talcen from Ref. (8) Eq. (15) for the mean polarizability derivatives shows that Tr(45&,,,
- 45&&,,) = 4 Tr(45&,,,
- 45&J.
(19)
The above result can be rewritten as (20)
45 Tr(&&,,, + 3&$&J = 45 Tr(4&,3DJ. In the same way, Eq. (15) applied to CH,/CH,D,
substitution leads to
Tr(45%;EHzD2~ - 45ti&,,) = 2 Tr(45G,,,
(21)
- 45Gn.J.
which, when employed in conjunction with Eq. (19), reduces to 45 Tr(Gn,,
(22)
+ &&i,J = 45 TT(2G,&.
One obtains from Eq. (15) also the isotopic rules 45 Tr(3&&,, + &‘&,J = 45 Tr(4&&,,,),
(23)
and
The results derived in Table I which pertain to the A, species (common for all the isotopic analogs) indicate that the experimental data (8) satisfy the different isotopic rules (Eqs. (20), (22)-(24)) within the error limits. The principal advantage of the new isotopic rules (Eqs. (4), (6), (B), (1 I), (I3), (14)) is that they can be applied to all molecular types without knowledge of the symmetry or the geometrical parameters of the molecule. These rules are expected to be useful, particularly for radicals and unstable species studied
TABLE I Isotopic Rules for Raman Data for A, Species in CH,, CH,D, CH,D,, CHD,, CD., Experimental
CH,
230.0 + 12
CH,D CH,D, CHD, CD,
194.6 171.0 135.4 115.0
LHS 805 + 345 + 575 2 35.4 k 23.6
data: 45 Tr(ti’2)a,‘r
12 12 12 15 ? 20
RHS 777 5 342 + 542 ” 35.6 + 20.4
15 20 8 20 k 8
+ k + t
15 20 8 8 Equationb,’ (20) (22) (23) (24)
a The derivatives of the mean polarizability were calculated from the intensities and the depolarization ratios listed in Ref. (8). In cases where the experimental values of the depolarization ratios are not listed in (B), the calculated values have been employed. The values are in units of N-’ x 1W2cm-’ g (N = Avogadro number). Ir The error limits are the maximum ones in any one intensity. r See text.
458
NOTES
by matrix isolation techniques, where the exact structure is not known. It is evident that for symmetrical substitutions, Eqs. (12), (15), and (16) implicitly require a knowledge of the symmetry of the molecule. The relations presented here are the compact forms of all first order sum-rules (9). Needless to say, for symmetrical substitutions, the isotopic rules are valid for each irreducible representation, separately. In contrast to Crawford’s rules (4), Eqs. (6), (13), (15) can be employed to check the intensity data without knowledge of the inverse frequency parameters. The isotopic rule derived using the first-order perturbation theory for heavy atom substitutions (9,ZO) shows that within this approximation, Eq. (4) holds for the shifts associated with each fundamental vibration, separately. With little effort, the results presented in Ref. (5) could be employed to show that this is the case with the intensity data, as well. ACKNOWLEDGMENTS We thank the referee for his valuable comments. The financial support provided by the Fonds der Chemischen Industrie, the Deutsche Forschungsgemeinschaft, and the Minister fur Wissenschaft und Forschung (Landesamt fur Forschung-NRW) is gratefully acknowledged. REFERENCES 1. J. HEICKLEN, J. Chem. Phys. 36,721-726 (1962). 2. P. BABURAO, K. V. SIVA SARMA,ANDK. SREERAMAMURTY,J.Mol. Spectrosc. 73,503-X)4(1978). 3. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS,“Molecular Vibrations,” pp. 61,64, 187, 192, McGraw-Hill, New York, 1955. 4. B. L. CRAWFORD,JR., J. Chem. Phys. 20, 977-981 (1952). 5. N. MOHAN, A. J. P. ALIX, AND A. MOLLER, Mol. Phys. 33, 319-329 (1977); A. MULLER AND N. MOHAN, J. Chem. Phys. 67, 1918-1925 (1977). 6. J. H. G. BODE, W. M. A. SMIT, AND A. J. VAN STRATEN, J. Mol. Spectrosc. 75,478-484 (1979); W. M. A. SMIT, J. H. G. BODE, AND A. J. VAN STRATEN,J. Mol. Spectrosc. 75,485-494 (1979). 7. S. J. CYVIN, “Molecular Vibrations and Mean Square Amplitudes,” p. 84, Elsevier, Amsterdam, 1968. 8. D. BERMWO, R. ESCRIBANO,AND J. M. ORZA, J. Mol. Spectrosc. 65, 345-353 (1977). Trends” (A. J. Barnes and W. J. Orville9. A. MOLLER, in “Vibrational Spectroscopy-Modern Thomas, Eds.), pp. 139- 166, Elsevier, Amsterdam, 1977. 10. N. MOHAN AND A. MDLLER, J. Chem. Phys. 67, 295-302 (1977). N. MOHAN AND A. MOLLER Faculty of Chemistry University of Bielefeld 4800 Bielefeld I West Germany Received July 6, 1979
OF MOLECULAR
A General
SPECTROSCOPY
80,455-458
(1980)
Isotopic Sum-Rule For Frequencies, Band intensities, and Coriolis Coupling Constants
Isotopic sum- and product-rules in a general form involving the vibrational frequencies were first developed by Heicklen (I). Recently, a new isotopic sum-rule related to the frequencies was reported in this Journal (2). This rule was derived in explicit form for symmetrical substitutions, using mass-weighted Cartesian coordinates as a basic set. But, until now, it does not appear to have been realized that a simple isotopic sum-rule valid for symmetrical as well as for unsymmetrical substitutions could be derived in an analogous form for the vibrational frequencies, band intensities (infrared, Raman), and the Coriolis coupling constants. The derivation of such a general sum-rule based on the usual FG matrix formalism (3) is presented in this communication and it will be shown that this rule is a general form of the one published earlier in this Journal (2). lsoropic rules. Tr(GkFR) with respect to the internal or valence coordinates could be expanded to read (3) Tr(A) = Tr(CPFR) = $ 1 (K$)/m,)F,I, 01 I,
(1)
where N is the total number of atoms (I in the isolated molecule, m, is the mass of atom a, Kg’) are the mass-independent coefficients which depend on the equilibrium geometrical parameters, and Fij are the quadratic force constants which are invariant to isotopic substitutions under the Born-Oppenheimer approximation. For the substitution of the ath atom, one obtains from Eq. (1) Tr(AA,) = 1 (Kaj)(-Am,/m,m~))Fi, U In both Eq. (1) and Eq. (2), A is a diagonal matrix related to the frequency (we use the superscript + to denote the isotope). Equation (2) can be modified to read (m,+lAm,)Tr(AA,)
(2) parameters
(3) (3)
= - 1 (K~Vm,)Fi,.
(3)
il
The summation of both sides of Eq. (3) with respect to a and the use of Eq. (1) lead to 5 (&lAm,)Tr(A&)
= -Tr(A),
which provides a relation connecting the isotopic shifts in the vibrational frequencies successive replacement of each atom by its isotope. Since the band intensities (infrared, Raman) obey the sum-rule (3-J)
(4) due to the
where the symmetric matrix B corrected for rotational effects is independent of isotopic substitution (Born-Oppenheimer approximation), Eq. (4) can be modified for intensities to give
i (m,flAm.) 1 AIk,,, = - z:Ik. k k
(6)
cl
In the above relation, Ik may be identified with the integrated infrared intensities or with the squares of the derivatives of mean polarizability or anisotropy with respect to the normal coordinates (i.e., (WlaQ,)*, (ay19Qk)*). It should be noted that the above rule can be directly applied to the experimental data for molecules possessing no permanent dipole moment (infrared) and spherical tops 455
0022-2852/80/040455-04$02.00/O Copyright 0 1980 by Academic Press, Inc. AU rights of reproductioo in any fom reserved.
456
NOTES
(Raman), only. In other cases, the intensity data must be corrected momentum (4, 6). The Coriolis coupling constants 5” satisfy the sum-rule (7)
for vibrational
angular
Tr(C”F) = Tr(I;“A).
(7)
Since C” has elements which satisfy an equation analogous to (1), it is evident that the sum-rule 2 (m,flAm,)Tr(A(<~&))
= -Tr(3”A)
(8)
holds for the Coriolis coupling constants. In many cases, it is advantageous and easier to employ the isotopic substitution of some atoms only, as for example, the isotopic replacement of a set of symmetrically equivalent atoms. Also, in such cases one can obtain a simple isotopic relation. Let us suppose both (Yand p atoms are substituted, simultaneously. Corresponding to this situation, the relation Tr(AA,,p) = 1 Kp'(-Am,lm,m;)Fij+ 1 K~'(-Amplmem~)Fij, (9) iJ ii can be developed from (1). Comparing the RHS of Eq. (9) with those of Eq. (2) and the analogous one related to p, the isotopic relation Tr(AA,,,) can be formulated.
(10)
For the substitution of p atoms, Eq. (10) can be expanded to read TI(AA,,~,
If, (1, P, .
= Tr(AA,) + Tr(AAB)
,,,) = Tr(AA,) + Tr(AA,J + . . * + Tr(AAP).
(11)
3p all form a symmetrically equivalent set, Eq. (11) reduces to
Tr(AAd = nTr(AA,),
(12)
where n might cover, either partially or completely, a set of equivalent relations are general and that they are valid for symmetrical as well stitutions. The relations analogous to (11) for the vibrational band intensities constants read + 1 C Alrc,,p......, = C Al,,,, + 1 Al,,p, + k k !i k and Tr(A(3”A),,p.. .,A = Tr(A(SUNa)+ Tr(AU2AM + . +
atoms. Note that the above as for unsymmetrical suband the Coriolis coupling
AIM,,,
(13)
TtiA(PQJ.
(14)
For atoms belonging to an equivalent set, Eqs. (13) and (14) simplify to C Al,,,,,, = n C AIM,,, k B
(15)
and Tr(A(S”N,,,) = nTr(A(rA),).
(16)
It is naturally evident that in all these relations (for the frequencies, intensities, Coriolis constants), the shifts are defined with respect to a single parent molecule. Again in Eqs. (15) and (16), n covers either some or all the atoms forming a symmetrically equivalent set. As shown below, Eq. (I) of Ref. (2) is a special case of Eqs. (4) and (12). Considering H,O-DIO and H,O-T,O substitutions, let 0 be replaced by O+ in both cases. Then, it follows from Eqs. (4) and (12), that (m&/(m,+ - mO))Tr(AA,,) + (mD/(mD- m,))Tr(AA&
= -Tr(A),
(17a)
(m,$/(m,f - m,))Tr(AA,)
= -Tr(A).
(17b)
and + (mrl(mT- m,))Tr(AA&)
By combining Eqs. (17a) and (17b), it can be easily shown that m”(mr - mD) .X kIO, k
+ mn(mH - m,) 1 kDzO) + Mm,, k
- m,) C hn,, = 0, k
which is identical to Eq. (1) of Ref. (2). In the above relation, At are the elements of A.
(18)
457
NOTES
The utility of the isotopic rules is illustrated in Table I for the Raman intensity data for CH,, CHBD, CH2D2, CHDS, and CD,. The experimental data are talcen from Ref. (8) Eq. (15) for the mean polarizability derivatives shows that Tr(45&,,,
- 45&&,,) = 4 Tr(45&,,,
- 45&J.
(19)
The above result can be rewritten as (20)
45 Tr(&&,,, + 3&$&J = 45 Tr(4&,3DJ. In the same way, Eq. (15) applied to CH,/CH,D,
substitution leads to
Tr(45%;EHzD2~ - 45ti&,,) = 2 Tr(45G,,,
(21)
- 45Gn.J.
which, when employed in conjunction with Eq. (19), reduces to 45 Tr(Gn,,
(22)
+ &&i,J = 45 TT(2G,&.
One obtains from Eq. (15) also the isotopic rules 45 Tr(3&&,, + &‘&,J = 45 Tr(4&&,,,),
(23)
and
The results derived in Table I which pertain to the A, species (common for all the isotopic analogs) indicate that the experimental data (8) satisfy the different isotopic rules (Eqs. (20), (22)-(24)) within the error limits. The principal advantage of the new isotopic rules (Eqs. (4), (6), (B), (1 I), (I3), (14)) is that they can be applied to all molecular types without knowledge of the symmetry or the geometrical parameters of the molecule. These rules are expected to be useful, particularly for radicals and unstable species studied
TABLE I Isotopic Rules for Raman Data for A, Species in CH,, CH,D, CH,D,, CHD,, CD., Experimental
CH,
230.0 + 12
CH,D CH,D, CHD, CD,
194.6 171.0 135.4 115.0
LHS 805 + 345 + 575 2 35.4 k 23.6
data: 45 Tr(ti’2)a,‘r
12 12 12 15 ? 20
RHS 777 5 342 + 542 ” 35.6 + 20.4
15 20 8 20 k 8
+ k + t
15 20 8 8 Equationb,’ (20) (22) (23) (24)
a The derivatives of the mean polarizability were calculated from the intensities and the depolarization ratios listed in Ref. (8). In cases where the experimental values of the depolarization ratios are not listed in (B), the calculated values have been employed. The values are in units of N-’ x 1W2cm-’ g (N = Avogadro number). Ir The error limits are the maximum ones in any one intensity. r See text.
458
NOTES
by matrix isolation techniques, where the exact structure is not known. It is evident that for symmetrical substitutions, Eqs. (12), (15), and (16) implicitly require a knowledge of the symmetry of the molecule. The relations presented here are the compact forms of all first order sum-rules (9). Needless to say, for symmetrical substitutions, the isotopic rules are valid for each irreducible representation, separately. In contrast to Crawford’s rules (4), Eqs. (6), (13), (15) can be employed to check the intensity data without knowledge of the inverse frequency parameters. The isotopic rule derived using the first-order perturbation theory for heavy atom substitutions (9,ZO) shows that within this approximation, Eq. (4) holds for the shifts associated with each fundamental vibration, separately. With little effort, the results presented in Ref. (5) could be employed to show that this is the case with the intensity data, as well. ACKNOWLEDGMENTS We thank the referee for his valuable comments. The financial support provided by the Fonds der Chemischen Industrie, the Deutsche Forschungsgemeinschaft, and the Minister fur Wissenschaft und Forschung (Landesamt fur Forschung-NRW) is gratefully acknowledged. REFERENCES 1. J. HEICKLEN, J. Chem. Phys. 36,721-726 (1962). 2. P. BABURAO, K. V. SIVA SARMA,ANDK. SREERAMAMURTY,J.Mol. Spectrosc. 73,503-X)4(1978). 3. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS,“Molecular Vibrations,” pp. 61,64, 187, 192, McGraw-Hill, New York, 1955. 4. B. L. CRAWFORD,JR., J. Chem. Phys. 20, 977-981 (1952). 5. N. MOHAN, A. J. P. ALIX, AND A. MOLLER, Mol. Phys. 33, 319-329 (1977); A. MULLER AND N. MOHAN, J. Chem. Phys. 67, 1918-1925 (1977). 6. J. H. G. BODE, W. M. A. SMIT, AND A. J. VAN STRATEN, J. Mol. Spectrosc. 75,478-484 (1979); W. M. A. SMIT, J. H. G. BODE, AND A. J. VAN STRATEN,J. Mol. Spectrosc. 75,485-494 (1979). 7. S. J. CYVIN, “Molecular Vibrations and Mean Square Amplitudes,” p. 84, Elsevier, Amsterdam, 1968. 8. D. BERMWO, R. ESCRIBANO,AND J. M. ORZA, J. Mol. Spectrosc. 65, 345-353 (1977). Trends” (A. J. Barnes and W. J. Orville9. A. MOLLER, in “Vibrational Spectroscopy-Modern Thomas, Eds.), pp. 139- 166, Elsevier, Amsterdam, 1977. 10. N. MOHAN AND A. MDLLER, J. Chem. Phys. 67, 295-302 (1977). N. MOHAN AND A. MOLLER Faculty of Chemistry University of Bielefeld 4800 Bielefeld I West Germany Received July 6, 1979