- Email: [email protected]
Analytical expression for the relative intensities of pure rotational raman spectra
J. Quant. Spectrosc. R&at.
Pergamon SOO22-4073(%)00038-6
Transfer Vol. 56, No. 6, pp. 881-886, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022~4073/96 $15.00 + 0.00
ANALYTICAL EXPRESSION FOR THE RELATIVE INTENSITIES OF PURE ROTATIONAL RAMAN SPECTRA M. KOREKtS
and H. KOBEISSI$
TFaculty of Science, Beirut Arab University, P.O. Box 11-5020, Beirut and §Group of Atomic and Molecular Physics at the National Council for Scientific Research, P.O. Box 11-8281, Beirut, Lebanon (Received 4 December
1995; received for publication 4 March 1996)
Abstract-The relative intensities of the Stokes and anti-Stokes lines of the pure rotational Raman transitions (vJ e, u.J’) of a diatomic molecule are considered; these are proportional to [z,(J)]; where z”(J) = (uJ~y(uJ + Z)/(uJlyluJ - 2) and y is the polarizability anisotropy. By using the Rayleigh-SchrGdinger perturbation theory, it is shown that z”(J) is given by a simple analytical expression for any potential, whether empirical or of the RKR-type and any operator y; this expression implies one coefficient 6, depending on the pure vibrational wavefunction Yy,and its rotational corrections. From this expression another simple formula, totally independent from Yy,,is obtained allowing the calculation of z,(J + j) (j = 1,2,3. . .) when one value z,(J) is provided. This expression is valid for any value of J and j. The numerical application to the ground states of HZ and CO molecules shows the validity and the accuracy of the present formulation. Copyright G 1996 Elsevier Science Ltd
INTRODUCTION According to the Placzek’ polarizability theory, the ratio of S and 0 branch pure rotational Raman spectra of a diatomic molecule is given by?
line intensities
in the
(1) in which R(v) represents the calibrated spectral response of the spectrometer and detection system at frequency v, J is the rotational quantum number and
where u is the vibrational quantum number and A4$ (J’ = J f 2) are the matrix elements of the polarizability anisotropy y . In the infrared (i.r.) vibration-rotation spectra, the rotational effect is commonly characterized by the Herman-Wallis rotational factor ci’ = ((vJlflu’J’>/(~O~flv’O))~ = 1 + Cm + Dm2
(3)
where f is the dipole moment function and m = [J’(J’ + 1) -.J(J + 1)]/2. Recently Korek and Kobeissi4*5 found analytic expressions for F [Eq. (3)] and the Herman-Wallis coefficients C, D, . . . up to the third order, and this for any potential (Dunham or others). More recently, we derived6v7in the vibration-rotation Raman spectra analytic expressions for the rotational factor and the Herman-Wallis coefficients up to the second order. These expressions are valid for any potential (whether empirical or of the RKR-type) and any operator y. In this paper we show that a Herman-Wallis-like formulation can be derived for the pure rotational Raman $To whom all correspondence
should be addressed. 881
882
M. Korek and H. Kobeissi
transitions originating from the same rotational level. A theoretical expression is obtained for z,(J) [Eq. (2)J in terms of the integrals (Y,lr]Y,) and (Y,] Y,), in which Y, and Y2 denote the pure vibrational wavefunction Ye’ or one of its successive perturbative corrections Yu!‘),Yt2’, Yt3),. . . . Moreover by using this expression of z”(J), a “non-integral” simple formula is derived from which thevalueofz,(J+j)(j=1,2,3 ,...) can be obtained only from one experimental value of z,(J) for any value of J or j. THEORY
1. Herman- Wall&like formulation In the Rayleigh-Schrodinger perturbation theory: the eigenvalue E”, and the eignfunction of a rovibrational state (u, J) of a diatomic molecule are given respectively by
YysJ
E,,J= E, i- BJ - DJ2 + HJ3 i- . . .
(4)
YU,= Y$O) + Yy!“1+ Yyc2’#P + Yt3’A3+ . . .
(5)
where 1= J(J + l), Y$‘), Yi2), Yi3),. . . are the Y-corrections and B,, D,, H,, . . . are the E-corrections which are obtained by successive use of the equation’*”
(6)
4: + Ed)”= S” where
with K, = l/r2 - (+,]1/r2]&) and cV= E, - U (r is the internuclear distance and U is the rotationless potential). For the pure rotational Raman transitions (vJ c-) vJ’) with J’ = J + 2 for the branch S and J’ = J - 2 for the branch 0, the expressions for 1= J(J + 1) and I’ = J’(J’ + 1) in terms of m = (A’ - 1)/2 are given by A= m(m - 1); il’ = m(m + 1) In order to obtain the wavefunctions and 1’ in (5), and find
YDJand YVsin terms of one parameter m, we substitute 1
Yd
=
a - pm + qm2 + . . .
(7a)
YY, = a + flrn + qm2 + . . .
WI
a = y(o) + 5 yyc3 " ” + z4 y(1) "
@a)
where
$ y$*)
(8b)
q = yy!u+ ‘2’ye’
PC)
p
=
Iy!" +
By replacing Y,, and YU [Eqs. (711 in the matrix elements of the polarizability NT = we obtain
(~“~lvl~“~)l(
anisotropy (9)
Relative intensities of pure rotational Raman spectra
883
where
and 8b + 44~ - 10d 16~ + 24b + 18~ + 9d
6=
(lib)
with a=
(Yul”‘lyIYu$‘)/(Yt”‘lYuF’),
(124
b=
( Y~“lyIY~“‘)/(Y~o’~Y~o’),
Wb)
C=
(
d=
(Y~"lyIY~'))/(Y~"'~Yf')
Y~2)lyIY~o))/(Y~)lY~o)) -f a( Y~‘qY~‘))/(Y:lY:),
(W
and
(124
Thus, for the pure rotational transitions in Raman spectra, analytic expressions are obtained for the Herman-Wallis-like coefficients p and 6 without any restriction on the potential function U(r) or on the operator y. 2. Analytic expression for the ratio z,(J) For the branches 0 (J’ - J = -2) given respectively by [Eq. (lo)]
and S (J’ - J = +2) the corresponding
matrix elements are
M;;-2 = /A(1+ 682 + . . .)
(134
Ki.2
W)
and =
p(1 + da2+ . . .),
where 13=mO=
-2J+l
and
a=ms=2J+3
By replacing Eqs. (13) in Eq. (2) we obtain 1 + f&r2+ . * . zv(J)= 1 + (332 + ...
(14)
Therefore, for a given value of J, the relative intensities of S and 0 branch lines in the pure rotational Raman spectra is given by an analytical expression in terms of one coefficient 6 which is given in terms of a, b, c, d, i.e., of integrals of the form (Y,lyj Y2) and ( YIIY,), where Y, and Yz stand for the pure-vibrational wavefunction Yu$’or for one of its successive perturbative corrections Yt”, YL”, . . . . 3. “Non-integral”
expression for the ratio z,,(J)
For two arbitrary values J, and J2, and by using Eq. (14) we obtain the following relation between the two corresponding values z1 = zV(J1)and z2 = zy(J2) 6=
1
ZI -
a: -
@z, =
z2 - 1 a: - eiz2
with al = 25, + 3; &=
-2J,+
a2 = 2J2 + 3 1; &= -2J2+
1
M. Korek and H. Kobeissi Table 1. Values of the coefficients u, b, c, d, and 6 for the ground state of the molecule HI. a = 2.13628601 b=2.1117519x 10-l c = -1.252931 8 x lo-’ d = 2.457 891 5 x lO-5 6 = 4.702 407 0 x lo-’
Eq. Eq. Eq. Eq. Eq.
(12a) (12b) (12~) (12d) (lib)
or 22 =
A- -pzi
-
4 - tz,
where
with j=J2-J,;
$=J,+J2+
1
1=25,+j+
Thus, knowing one experimental value zI = zy(JI), the calculation of the wavefunctions !Pvr, YVJ and the integrals ( Y,]r] Y2) and
APPLICATION
In the present formulation, the determination of the relative intensities Is/P of the pure rotational Raman transitions is reduced to that of calculating one coefficient 6 [Eq. (11 b)]. The computation of the coefficient requires those of a, b, c, d [Eqs. (12)], i.e., those of the constants9 B,, D, and the wavefunctions Y,(O) , !P)y , YC2) V . These functions are determined successively from one unique differential equation. lo The integration of this differential equation is done by using the algorithm described by Korek and Kobeissi.9 Once Y,(O) , Y(l) Yi2) for a given vibrational level u are obtained, the calculation of the integrals ( Y&l Y2) and ;‘Y,] Y2) [Eqs. (12)J becomes easy and the determination of the coefficient 6 [Eq. (1 lb)] can be done without any restriction on the operator y, the potential function U(r) and the vibrational level v. In order to make useful comparisons, we apply the present formulation [Eq. (14)] to the Dunham potential” of the ground state of HZ with a polynomial operator in (r - re)/rc up to the fourth order.12 For the transitions (OJ * OJ’) we give in Table 1 the coefficients a, b, c, d, and 6. By using this value of the coefficient 6, we give in Table 2 (column 2) the values of the ratio z”(J) from Eq. (14) for 2 G J G 10. We compare our results with those calculated by a numerical method2 Table 2. Values of the ratio z,(J) (u = 0) for the ground state of the molecule HZ. The results from Eq. (14) and those from Eq. (15) are compared to those calculated by a numerical method.2 J
Eq. (14)
2 3 4 5 6 7 8 9 10
1.018 7304 1.026 027 6 1.033 094 8 1.039 862 9 1.046 272 3 1.052 274 7 1.057 833 8 1.062 925 2 1.067 535 9
Eq. (15) 1.018 730 4 1.026 027 5 1.033 094 8 1.039 862 8 1.046 272 2 1.052 274 6 1.057 833 7 1.062 925 1 1.067 535 8
Ref. 2 1.018 9 1.026 3 1.033 5 1.040 5 1.0472 1.053 6 1.059 6 1.065 2 1.070 5
Relative intensities of pure rotational Raman spectra
885
Table 3. Values of the ratio z,(J) for the ground state of the molecule CO.$ The results from Eq. (14) (first entry) are compared to those calculated by a numerical method” (second entry). l.’
J=2
J=4
cl
1.OOO125 703
2
1.000 122 233 9 1.000 119091 7 1.000 116 223 0 1.000 113 550 6 1.OOO110 946 51
8t 4 6 8 10
J=6
1.000 226 237 69 1.000 219 992 20 024 1.000 214 388 70 I.000 209 176 207 1.000 204 367 98 1.000 199 680 709
1.000 326 712 806 I.000 317 697 791 I.000 309 533 636 I.000 302 080 166 1.000 295 137 97 1.OOO288 369 456
J=8 1.000 427 100 308 1.000415 318 526 1.000 404 649 857 I.000 394 909 987 1.000 385 834 6 036 1.OOO376 982 7 173
tOmitted figures are identical to those of the first entry.
(column 4). The comparison between these values shows an overall agreement which is considered as confirmation of the validity and the accuracy of the present formula [Eq. (14)]. We notice that, this accuracy decreases with the increase of the value of J. This may be explained by the perturbation aspect (second order correction) of Eq. (10). Because of the large value of the ratio Be/o, of the molecule H2, higher order corrections are probably needed for large value of J. Knowing one experimental value z”(J), we derived a “non-integral” formula [Eq. (15)] where the calculation of z,(J + j) (j = * 1, + 2, . . .) becomes much easier and faster. By using the value z,(J = 2) of the molecule’ H? in Eq. (15), we display in Table 2, column 3 the values of z,(2 -t-j) forj=l, 2,..., 8. The comparison of these values to those obtained from Eq. (14) shows an excellent accuracy up to seven significant figures for all the considered values of J. By taking z,(J) from column 2 for any other J, the values of column 3 (z&I + j)) remain the same. Due to its inferior sensitivity and limited resolution, the overtone transitions are rarely observed in Raman spectroscopy. To show the generality of the present formulation, we present in Table 3, for the ground state of the molecule CO,’ the calculated values of z,(J) (first entry) by using Eq. (14) for the transitions 0 < u < 10 and J < 9. The comparison of these values to those calculated by a numerical methodI (second entry) shows the good accuracy of the present formula for the different considered values of v and J. For a higher precision in the case of a large value of J, additional terms cm’, (m4 in Eq. (10) are needed. The calculation of the coefficients c, 5, . . . is undertaken. By replacing the values of z,(2) for 0 < u < 10 calculated from Eq. (14) (first entry of Table 3) in Eq. (15) we obtain z,(J) for 0 < u Q 10 and J = 4, 6, 8 (not shown). The comparison of these values to those calculated from Eq. (14) shows an excellent accuracy up to 12 significant figures for all considered values of u and J. CONCLUSION
The study of the relative intensity of the branches S and 0 in the pure rotational Raman transitions (uJ c-) ~7) of a diatomic molecule is reduced to that of the ratio z,(J) [Eq. (2)]. Using a numerical method for calculating this ratio for a pure rotational transition for a variable value of J, one has to calculate the integrals of Eq. (2) for each value of J. While by using Eq. (14), once the only coefficient 6 is determined, for a given value of u, the calculation of z,(J) for variable J is easier and faster, (6 depends, for a given v, on Y, and its rotational corrections). Moreover if one value of z,(J) is available, by using Eq. (15) the coefficient 6 is eliminated and the calculation of z,(J) for variable J becomes much more easier and faster because the calculation of the wavefunctions Y,,,, Y,,J and the integrals (Y,]r] Y2) and ( YIJY2) [Eq. (12)] is avoided; the calculation of z,(J) is thus reduced to an elementary calculation of the coefficients p, q, s. t [Eqs. (16)] without any restriction on the values of J or V. The numerical application to the ground states of H2 and CO molecules with Dunham potential shows the accuracy and the validity of the formulae (14) and (15) for the fundamental and higher overtones. Applications to other types of potentials like that of Morse and RKR-type and other types of operators like y(x) = exp(ax) showed similar results (not shown).
886
M. Korek and H. Kobeissi REFERENCES
1. G. Placzek, Handbuch der Radiologie, Vol. VI, Part 2, p. 205, E. Marx, ed., Akademic (1934). 2. L. M. Cheung, D. M. Bishop, D. L. Drapcho, and G. M. Rosenblatt, Chem. Phys. Left. SO, 445 (1981). 3. R. Herman and R. F. Wallis, J. Chem. Phys. 23, 637 (1955). 4. H. Kobeissi and M. Korek, J. Phys. B: At. Mol. Opt. Phys. 27, 3653 (1994). 5. M. Korek and H. Kobeissi, JQSRT 52, 631 (1994). 6. M. Korek and H. Kobeissi, Can. J. Phys. 73, 559 (1995). 7. M. Korek and H. Kobeissi, JQSRT 55, 225 (1996). 8. A. Messiah, M&unique Quantique, Tome II, Dunod, Paris (1972). 9. M. Korek and H. Kobeissi, J. Comput. Chem. 13, 1103 (1992). 10. H. Kobeissi and M. Korek, J. Phys. B: At. Mol. Opt. Phys. 26, L-35 (1993). 11. C. L. Bekel and F. Wu, paper presented at the 27th Symp. on Molecular Structure and Spectroscopy, paper AA9, Columbus, OH (1972). 12. R. H. Tipping and J. F. Ogilvie, J. Ram. Spectrosc. 15, 38 (1984). 13. H. Kobeissi, M. Dagher, A. El-Haj, and M. Kobeissi, J. Cornput. Chem. 10, 358 (1989). 14. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, Toronto (1950).
Pergamon SOO22-4073(%)00038-6
Transfer Vol. 56, No. 6, pp. 881-886, 19% Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022~4073/96 $15.00 + 0.00
ANALYTICAL EXPRESSION FOR THE RELATIVE INTENSITIES OF PURE ROTATIONAL RAMAN SPECTRA M. KOREKtS
and H. KOBEISSI$
TFaculty of Science, Beirut Arab University, P.O. Box 11-5020, Beirut and §Group of Atomic and Molecular Physics at the National Council for Scientific Research, P.O. Box 11-8281, Beirut, Lebanon (Received 4 December
1995; received for publication 4 March 1996)
Abstract-The relative intensities of the Stokes and anti-Stokes lines of the pure rotational Raman transitions (vJ e, u.J’) of a diatomic molecule are considered; these are proportional to [z,(J)]; where z”(J) = (uJ~y(uJ + Z)/(uJlyluJ - 2) and y is the polarizability anisotropy. By using the Rayleigh-SchrGdinger perturbation theory, it is shown that z”(J) is given by a simple analytical expression for any potential, whether empirical or of the RKR-type and any operator y; this expression implies one coefficient 6, depending on the pure vibrational wavefunction Yy,and its rotational corrections. From this expression another simple formula, totally independent from Yy,,is obtained allowing the calculation of z,(J + j) (j = 1,2,3. . .) when one value z,(J) is provided. This expression is valid for any value of J and j. The numerical application to the ground states of HZ and CO molecules shows the validity and the accuracy of the present formulation. Copyright G 1996 Elsevier Science Ltd
INTRODUCTION According to the Placzek’ polarizability theory, the ratio of S and 0 branch pure rotational Raman spectra of a diatomic molecule is given by?
line intensities
in the
(1) in which R(v) represents the calibrated spectral response of the spectrometer and detection system at frequency v, J is the rotational quantum number and
where u is the vibrational quantum number and A4$ (J’ = J f 2) are the matrix elements of the polarizability anisotropy y . In the infrared (i.r.) vibration-rotation spectra, the rotational effect is commonly characterized by the Herman-Wallis rotational factor ci’ = ((vJlflu’J’>/(~O~flv’O))~ = 1 + Cm + Dm2
(3)
where f is the dipole moment function and m = [J’(J’ + 1) -.J(J + 1)]/2. Recently Korek and Kobeissi4*5 found analytic expressions for F [Eq. (3)] and the Herman-Wallis coefficients C, D, . . . up to the third order, and this for any potential (Dunham or others). More recently, we derived6v7in the vibration-rotation Raman spectra analytic expressions for the rotational factor and the Herman-Wallis coefficients up to the second order. These expressions are valid for any potential (whether empirical or of the RKR-type) and any operator y. In this paper we show that a Herman-Wallis-like formulation can be derived for the pure rotational Raman $To whom all correspondence
should be addressed. 881
882
M. Korek and H. Kobeissi
transitions originating from the same rotational level. A theoretical expression is obtained for z,(J) [Eq. (2)J in terms of the integrals (Y,lr]Y,) and (Y,] Y,), in which Y, and Y2 denote the pure vibrational wavefunction Ye’ or one of its successive perturbative corrections Yu!‘),Yt2’, Yt3),. . . . Moreover by using this expression of z”(J), a “non-integral” simple formula is derived from which thevalueofz,(J+j)(j=1,2,3 ,...) can be obtained only from one experimental value of z,(J) for any value of J or j. THEORY
1. Herman- Wall&like formulation In the Rayleigh-Schrodinger perturbation theory: the eigenvalue E”, and the eignfunction of a rovibrational state (u, J) of a diatomic molecule are given respectively by
YysJ
E,,J= E, i- BJ - DJ2 + HJ3 i- . . .
(4)
YU,= Y$O) + Yy!“1+ Yyc2’#P + Yt3’A3+ . . .
(5)
where 1= J(J + l), Y$‘), Yi2), Yi3),. . . are the Y-corrections and B,, D,, H,, . . . are the E-corrections which are obtained by successive use of the equation’*”
(6)
4: + Ed)”= S” where
with K, = l/r2 - (+,]1/r2]&) and cV= E, - U (r is the internuclear distance and U is the rotationless potential). For the pure rotational Raman transitions (vJ c-) vJ’) with J’ = J + 2 for the branch S and J’ = J - 2 for the branch 0, the expressions for 1= J(J + 1) and I’ = J’(J’ + 1) in terms of m = (A’ - 1)/2 are given by A= m(m - 1); il’ = m(m + 1) In order to obtain the wavefunctions and 1’ in (5), and find
YDJand YVsin terms of one parameter m, we substitute 1
Yd
=
a - pm + qm2 + . . .
(7a)
YY, = a + flrn + qm2 + . . .
WI
a = y(o) + 5 yyc3 " ” + z4 y(1) "
@a)
where
$ y$*)
(8b)
q = yy!u+ ‘2’ye’
PC)
p
=
Iy!" +
By replacing Y,, and YU [Eqs. (711 in the matrix elements of the polarizability NT = we obtain
(~“~lvl~“~)l(
anisotropy (9)
Relative intensities of pure rotational Raman spectra
883
where
and 8b + 44~ - 10d 16~ + 24b + 18~ + 9d
6=
(lib)
with a=
(Yul”‘lyIYu$‘)/(Yt”‘lYuF’),
(124
b=
( Y~“lyIY~“‘)/(Y~o’~Y~o’),
Wb)
C=
(
d=
(Y~"lyIY~'))/(Y~"'~Yf')
Y~2)lyIY~o))/(Y~)lY~o)) -f a( Y~‘qY~‘))/(Y:lY:),
(W
and
(124
Thus, for the pure rotational transitions in Raman spectra, analytic expressions are obtained for the Herman-Wallis-like coefficients p and 6 without any restriction on the potential function U(r) or on the operator y. 2. Analytic expression for the ratio z,(J) For the branches 0 (J’ - J = -2) given respectively by [Eq. (lo)]
and S (J’ - J = +2) the corresponding
matrix elements are
M;;-2 = /A(1+ 682 + . . .)
(134
Ki.2
W)
and =
p(1 + da2+ . . .),
where 13=mO=
-2J+l
and
a=ms=2J+3
By replacing Eqs. (13) in Eq. (2) we obtain 1 + f&r2+ . * . zv(J)= 1 + (332 + ...
(14)
Therefore, for a given value of J, the relative intensities of S and 0 branch lines in the pure rotational Raman spectra is given by an analytical expression in terms of one coefficient 6 which is given in terms of a, b, c, d, i.e., of integrals of the form (Y,lyj Y2) and ( YIIY,), where Y, and Yz stand for the pure-vibrational wavefunction Yu$’or for one of its successive perturbative corrections Yt”, YL”, . . . . 3. “Non-integral”
expression for the ratio z,,(J)
For two arbitrary values J, and J2, and by using Eq. (14) we obtain the following relation between the two corresponding values z1 = zV(J1)and z2 = zy(J2) 6=
1
ZI -
a: -
@z, =
z2 - 1 a: - eiz2
with al = 25, + 3; &=
-2J,+
a2 = 2J2 + 3 1; &= -2J2+
1
M. Korek and H. Kobeissi Table 1. Values of the coefficients u, b, c, d, and 6 for the ground state of the molecule HI. a = 2.13628601 b=2.1117519x 10-l c = -1.252931 8 x lo-’ d = 2.457 891 5 x lO-5 6 = 4.702 407 0 x lo-’
Eq. Eq. Eq. Eq. Eq.
(12a) (12b) (12~) (12d) (lib)
or 22 =
A- -pzi
-
4 - tz,
where
with j=J2-J,;
$=J,+J2+
1
1=25,+j+
Thus, knowing one experimental value zI = zy(JI), the calculation of the wavefunctions !Pvr, YVJ and the integrals ( Y,]r] Y2) and
APPLICATION
In the present formulation, the determination of the relative intensities Is/P of the pure rotational Raman transitions is reduced to that of calculating one coefficient 6 [Eq. (11 b)]. The computation of the coefficient requires those of a, b, c, d [Eqs. (12)], i.e., those of the constants9 B,, D, and the wavefunctions Y,(O) , !P)y , YC2) V . These functions are determined successively from one unique differential equation. lo The integration of this differential equation is done by using the algorithm described by Korek and Kobeissi.9 Once Y,(O) , Y(l) Yi2) for a given vibrational level u are obtained, the calculation of the integrals ( Y&l Y2) and ;‘Y,] Y2) [Eqs. (12)J becomes easy and the determination of the coefficient 6 [Eq. (1 lb)] can be done without any restriction on the operator y, the potential function U(r) and the vibrational level v. In order to make useful comparisons, we apply the present formulation [Eq. (14)] to the Dunham potential” of the ground state of HZ with a polynomial operator in (r - re)/rc up to the fourth order.12 For the transitions (OJ * OJ’) we give in Table 1 the coefficients a, b, c, d, and 6. By using this value of the coefficient 6, we give in Table 2 (column 2) the values of the ratio z”(J) from Eq. (14) for 2 G J G 10. We compare our results with those calculated by a numerical method2 Table 2. Values of the ratio z,(J) (u = 0) for the ground state of the molecule HZ. The results from Eq. (14) and those from Eq. (15) are compared to those calculated by a numerical method.2 J
Eq. (14)
2 3 4 5 6 7 8 9 10
1.018 7304 1.026 027 6 1.033 094 8 1.039 862 9 1.046 272 3 1.052 274 7 1.057 833 8 1.062 925 2 1.067 535 9
Eq. (15) 1.018 730 4 1.026 027 5 1.033 094 8 1.039 862 8 1.046 272 2 1.052 274 6 1.057 833 7 1.062 925 1 1.067 535 8
Ref. 2 1.018 9 1.026 3 1.033 5 1.040 5 1.0472 1.053 6 1.059 6 1.065 2 1.070 5
Relative intensities of pure rotational Raman spectra
885
Table 3. Values of the ratio z,(J) for the ground state of the molecule CO.$ The results from Eq. (14) (first entry) are compared to those calculated by a numerical method” (second entry). l.’
J=2
J=4
cl
1.OOO125 703
2
1.000 122 233 9 1.000 119091 7 1.000 116 223 0 1.000 113 550 6 1.OOO110 946 51
8t 4 6 8 10
J=6
1.000 226 237 69 1.000 219 992 20 024 1.000 214 388 70 I.000 209 176 207 1.000 204 367 98 1.000 199 680 709
1.000 326 712 806 I.000 317 697 791 I.000 309 533 636 I.000 302 080 166 1.000 295 137 97 1.OOO288 369 456
J=8 1.000 427 100 308 1.000415 318 526 1.000 404 649 857 I.000 394 909 987 1.000 385 834 6 036 1.OOO376 982 7 173
tOmitted figures are identical to those of the first entry.
(column 4). The comparison between these values shows an overall agreement which is considered as confirmation of the validity and the accuracy of the present formula [Eq. (14)]. We notice that, this accuracy decreases with the increase of the value of J. This may be explained by the perturbation aspect (second order correction) of Eq. (10). Because of the large value of the ratio Be/o, of the molecule H2, higher order corrections are probably needed for large value of J. Knowing one experimental value z”(J), we derived a “non-integral” formula [Eq. (15)] where the calculation of z,(J + j) (j = * 1, + 2, . . .) becomes much easier and faster. By using the value z,(J = 2) of the molecule’ H? in Eq. (15), we display in Table 2, column 3 the values of z,(2 -t-j) forj=l, 2,..., 8. The comparison of these values to those obtained from Eq. (14) shows an excellent accuracy up to seven significant figures for all the considered values of J. By taking z,(J) from column 2 for any other J, the values of column 3 (z&I + j)) remain the same. Due to its inferior sensitivity and limited resolution, the overtone transitions are rarely observed in Raman spectroscopy. To show the generality of the present formulation, we present in Table 3, for the ground state of the molecule CO,’ the calculated values of z,(J) (first entry) by using Eq. (14) for the transitions 0 < u < 10 and J < 9. The comparison of these values to those calculated by a numerical methodI (second entry) shows the good accuracy of the present formula for the different considered values of v and J. For a higher precision in the case of a large value of J, additional terms cm’, (m4 in Eq. (10) are needed. The calculation of the coefficients c, 5, . . . is undertaken. By replacing the values of z,(2) for 0 < u < 10 calculated from Eq. (14) (first entry of Table 3) in Eq. (15) we obtain z,(J) for 0 < u Q 10 and J = 4, 6, 8 (not shown). The comparison of these values to those calculated from Eq. (14) shows an excellent accuracy up to 12 significant figures for all considered values of u and J. CONCLUSION
The study of the relative intensity of the branches S and 0 in the pure rotational Raman transitions (uJ c-) ~7) of a diatomic molecule is reduced to that of the ratio z,(J) [Eq. (2)]. Using a numerical method for calculating this ratio for a pure rotational transition for a variable value of J, one has to calculate the integrals of Eq. (2) for each value of J. While by using Eq. (14), once the only coefficient 6 is determined, for a given value of u, the calculation of z,(J) for variable J is easier and faster, (6 depends, for a given v, on Y, and its rotational corrections). Moreover if one value of z,(J) is available, by using Eq. (15) the coefficient 6 is eliminated and the calculation of z,(J) for variable J becomes much more easier and faster because the calculation of the wavefunctions Y,,,, Y,,J and the integrals (Y,]r] Y2) and ( YIJY2) [Eq. (12)] is avoided; the calculation of z,(J) is thus reduced to an elementary calculation of the coefficients p, q, s. t [Eqs. (16)] without any restriction on the values of J or V. The numerical application to the ground states of H2 and CO molecules with Dunham potential shows the accuracy and the validity of the formulae (14) and (15) for the fundamental and higher overtones. Applications to other types of potentials like that of Morse and RKR-type and other types of operators like y(x) = exp(ax) showed similar results (not shown).
886
M. Korek and H. Kobeissi REFERENCES
1. G. Placzek, Handbuch der Radiologie, Vol. VI, Part 2, p. 205, E. Marx, ed., Akademic (1934). 2. L. M. Cheung, D. M. Bishop, D. L. Drapcho, and G. M. Rosenblatt, Chem. Phys. Left. SO, 445 (1981). 3. R. Herman and R. F. Wallis, J. Chem. Phys. 23, 637 (1955). 4. H. Kobeissi and M. Korek, J. Phys. B: At. Mol. Opt. Phys. 27, 3653 (1994). 5. M. Korek and H. Kobeissi, JQSRT 52, 631 (1994). 6. M. Korek and H. Kobeissi, Can. J. Phys. 73, 559 (1995). 7. M. Korek and H. Kobeissi, JQSRT 55, 225 (1996). 8. A. Messiah, M&unique Quantique, Tome II, Dunod, Paris (1972). 9. M. Korek and H. Kobeissi, J. Comput. Chem. 13, 1103 (1992). 10. H. Kobeissi and M. Korek, J. Phys. B: At. Mol. Opt. Phys. 26, L-35 (1993). 11. C. L. Bekel and F. Wu, paper presented at the 27th Symp. on Molecular Structure and Spectroscopy, paper AA9, Columbus, OH (1972). 12. R. H. Tipping and J. F. Ogilvie, J. Ram. Spectrosc. 15, 38 (1984). 13. H. Kobeissi, M. Dagher, A. El-Haj, and M. Kobeissi, J. Cornput. Chem. 10, 358 (1989). 14. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, Toronto (1950).