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Relative intensities in the diatomic vibration-rotation Raman spectra: A simple expression for fundamental and higher overtones
J. Quant. Spcctmc.
Radiat. Transfer Vol. 55, No. 2, pp. 225-230, 1996
Pergamon 0022-4073(95)00168-9
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-4073196 $15.00 + 0.00
RELATIVE INTENSITIES IN THE DIATOMIC VIBRATION-ROTATION RAMAN SPECTRA: A SIMPLE EXPRESSION FOR FUNDAMENTAL AND HIGHER OVERTONES M. KOREK“ and H. KOBEISSI*t “Faculty of Science, Beirut Arab University P.O. Box 1l-5020, Beirut, Lebanon and ‘Group of Molecular and Atomic Physics at the National Council for Scientific Research, P.O. Box 11-8281 Beirut, Lebanon (Received
4 May 1995)
Abstract-The study of the relative intensities of the S(J) and O(J) branch lines of the vibration-rotation Raman transitions (uJ ++ v’J’) of a diatomic molecule is considered. These relative intensities are proportional to (z,.)~, where z,,,(J) = (uJly Iu’J + 2)/(vJly )u’J - 2), and y is the polarizability anisotropy. It is shown that the ratio z,,(J) (i) can be expressed as ~UL.1(J)=(GO+GIm,+G2mf)/(G,+G,m,+G2m~),wherem,=-2J+l,m,=25+3andthe coefficients G,, G,, G2 are given by analytical expressions in terms of simple integrals depending on Y(O)(the pure vibration wavefunction) and its successive rotational corrections ‘I’(‘),‘PC’),. . . ; or (ii) can be calculated from the two given experimental values z1 = z,,,(J,) and z2 = z,,“.(J,) by using a simple relation in the form of z,,,(J) = [(A - B - l)z, AZ, + Bz, z2 + l]/[Az, - (A - B - l)z, - z, z2 + B], where the coefficients A and B are given by simple expressions in terms of Ji , J2 and J. The numerical applications to the ground states of I-I2 and CO
molecules show the validity and the accuracy of the present formulation.
INTRODUCTION
In the Raman spectra of diatomic molecules the ratio of the intensities Z’(J) and Z’(J) of the S and 0 branch lines of the vibration-rotation transition (oJ c* u ‘J ‘) originating from the same rotational level J is given by’ ZS(J)/ZO(J) = [(VS)3R(VD)(J+ l)(J + 2)(2J - l)(z,,.,(5))2]/[(VU)3R(v’)(J - l)J(2J + 3)], in which R(v) represents the calibrated spectral response of the spectrometer and detection system at frequency v and z,,,(J) = M::;J+ 2/M;;‘-2=
IU’J + 2)/(vJly
(uJly
IV’J - 2),
(1)
where Mt’ are the matrix elements of the polarizability anisotropy y between the initial and final states.2T3These matrix elements are defined as follows:
M$” = (Y’,,ly I’?,.,.) =
sm
YJr)y(r)Y',,,.dr,
0
where Y, is the rovibrational wavefunction and r is the internuclear distance. The values of the ratio z,,,(J) are needed for various scattering theories and to correct rotational and vibrational-rotational Raman intensities for molecular nonrigidity and vibrational-rotational interaction.4 The aim of this work is to show that z,,,(J) [Eq. (l)] can be obtained in two ways: (i) By using a nonconventional approach of the Rayleigh-Schriidinger perturbation theory, the ratio z,,,(J) can be written as tTo whom all correspondence
should be addressed. 225
226
M. Korek and H. Kobeissi
z,,,(J) = (Go + G, m, + G,mf)/(Go + G, m, + G,mz), where the coefficients G,, G,, Gz are given by analytical expressions in terms of the integrals ( Y,.]u1&) and ( Y,l Y,.), where Y stands for ‘P(O)(the pure vibration wavefunction), or Y(r) (the first rotational perturbative correction to Y(O)),or Yc2)(the second rotation correction). (ii) If two experimental values z,,.,(J,) and z,,,(J,) are available (without any restriction on J, and J,) the calculation of z,,(J) is reduced to a simple relation eliminating the need to evaluate the integrals (vJI y /u/J’), and furthermore, the wavefunctions YL,Jand YL,,J,.Numerical applications to the ground states of H2 and CO molecules are then presented and discussed. THE THEORY 1. Analytic expression of the ratio z,,,(J) In seeking the evaluation of the line intensities in the vibration-rotation Raman spectra of a diatomic molecule, James and Klemperer,3 and later on Buckingham and Szabo,’ derived a formula for the relative intensities of S and 0 branch lines using the first order perturbation theory.6 Tipping and Ogilvie’ preferred to use the Herman-Wallis rotation factor F$” = ((vJly Iv’J’)/(uOly
IdO))*.
They obtained an analytical expression for F in terms of the Dunham potential parameters and the coefficients of a general operator y were represented by a power series expansion. Actually, the rotational effect in the rovibrational infrared spectra is commonly characterized by an approximation of the Herman-Wallis rotational factor’ which is given by F$“=((uJ~f~u’J’)/(uO~f~u’0))2=1+Cm+Dm2+~~~,
(3)
where f is the dipole moment function and m = [J’(J’ + 1) - J(J + 1)]/2. The Herman-Wallis coefficients C and D are given in terms of the Dunham anharmonic coefficients a, (up to 7) and for U’ - u < 7.9 Recently Kobeissi and Korek” used a new approach of the Rayleigh-Schrodinger perturbation theory” and found analytical expressions for F$” and for the coefficients C and D for any potential and any operatorf [Eq. (3)]. This work was followed by anotherI one in which Korek and Kobeissi derived analytical expressions for the Herman-Wallis coefficients up to the third order (C, D and E). By using the same approach as the Rayleigh-Schrodinger perturbation theory in the Raman spectra, the wavefunctions for two vibrational-rotational states UJ and u’J’ are given respectively by Y”,=Y’,+1~~+1*~~,+~3~c+...,
(4)
Y,,,, = Y,. + 1’LZ&+ A’%,., + i’3%,, + . . . )
(5)
whereI=J(J+l),,l’=J’(J’+l),Y’,=Y’, co), Y L’I = Y(O)(the pure vibration wavefunctions) and &j = Y’(l) 9 = Y(2) . . . ) a”, = Y’!), g,,, = yw . . aie respectively the Yi’)- and Y$)-rotation c&rect:ons ovfthe iwo considered itates. Thes: iunctfons are well determined with high accuracy” even near dissociation. For a vibration-rotation transition (uJ CI u’J’), in Raman transition, the S and 0 branches are characterized by J’ = J + 2 and J’ = J - 2, respectively. By taking m = (A - ,I’)/2 we obtain for 1 and 1’ 2 =m*/4--m
+3/4,
A’=m2/4+m
+3/4.
(6)
By substituting i and 1’ by their values in Eqs. (4) and (5) one obtainsI Y’, = (Y, + (3/4).@”+ (9/16)9”) - m(a, + (3/2)9,) + m*((1/4)~,+(11/8)~~)+...,
(7)
Relative intensities in the diatomic vibration-rotation
Raman spectra
221
‘I”,,,,.= (‘I’,, + (3/4)W,. + (9/16)9”.) + m(Wu, + (3/2).~S~,,) + mZ((1/4)Lq,, + (1 l/8)9”.) +. . . .
(8)
Then by replacing Yt>,and Y’,,,,.by their expressions [Eqs. (7) and (S)] in the matrix elements I@$’ [Eq. (2)], and by taking into account the normalization factors, we write WY =/((‘y,,I y,,)(y,~,~ly’,.,,~))‘!z and find (see Appendix of Ref. 12)
M$‘/M$’ = G, + G, m + G2m2,
(9)
where
w2 = (‘y,.lY I~,>/t(~“IyD)(~,~I~,~))“2, G,= 1 + (3/4)(<%l~ I‘u,,> + <%lr +(9/16N<~,lY
I Yu, ))/O’v
I Y I Y’,, >
IY I y’, >
- (90 IY I Yd ))/V” G = U/4NWvlyI’J’c,)
I Y I y,, > + <%,,lr
+ (9t.,I Y I y’, >)/Vu, +
1/2((93’,*/9% YV”.
l‘u,>)/W,l~
I’J’,,)
I Y I Y,!, > - l/2(<%
I y’,,, > - @+%IJ%>>/<‘y, I y’, >>I,
+ (11/W(~,ly
IY,,.>
I % )/
(~~,,I~\,,>/<‘y,,lyI’,)11(v9e%lY
This general formulation values
1’3’0,)
I‘u,>
G, = ( - PAY + (312) wns
I’J’,>W’,Ir
I~“,>/<‘y,lY
I ye,>.
(10)
can be used for the S and 0 branches, by giving to m the appropriate m,=25+3;
m,= -2J+
1.
By using this formulation to find an analytic expression for z,,,(J) [Eq. (l)], one can substitute and j,,f”;;‘+2 by their expressions from Eq. (9) and find
Mti-2
z,,,(J) = (G, + G, m, + G,m,2)/(G,,+ G, m, + G,mz).
(11)
Once the coefficients G,, G, , G2 (depending on u and 0’) are calculated, this expression allows one to obtain z,(J) [for any transition (u - u’)] for variable J without repeating the calculation of the integrals of Eq. (2) for each value of J. 2. “Nonintegral” expression for the 2,‘ (J) The evaluation of the coefficients G,, G,,Gz can be avoided by considering three arbitrary values J, , J2, Jj. For these values of J Eq. (11) can be written as G, N, + G2P, = G,,(z, - l),
(12.1)
G, N; + G2P2 = G&2 - l),
(12.2)
G,N,+GzP,=G,,(z,-
(12.3)
l),
in which z,=z,,(Ji)
(i = 1,2,3),
N, = (m, - 4)z, + m,,
(13.1)
M. Korek
228
and H. Kobeissi
N,=(m,+2k--4)z,+m,+2k,
(13.2)
N,=(m,+2n
(13.3)
-4)z,+m,+2n,
+ 8m, - 16)z, + mf,
(14.1)
I’,=(-mf-4k2-4km,+8ms+16k-16)z,+mf+4k2+4km,,
(14.2)
P, = (-rnz
16n - 16)z,+m,2+4n2+4nm,,
I’,=(-mf-4n2-4nm,+8m,+ k = J2 - J, ;
(14.3)
n = J3 - J, .
By eliminating G,, G,,G2 in Eqs. (12) we obtain P, [Nj (~2
-
1)
-
N2h
-
III+
Pz[N,(z,
-
1)
-
N,(z,
-
1)l +Pj[N,(z,
- 1) - N2(z, - l)] = 0.
(15)
If we replace Eqs. (13) and (14) in Eq. (15), we find zj = [(A - B - l)z, - AZ, + Bz, z2 + l]/[Az, - (A - B - l)z, - z, z2 + B],
(16)
where A = (j + k)(j
+ k + n)/[k(k
- n)],
B = (j + n)(j
+ k + n)/[nk(k
- n)],
j=2J,+l.
The coefficients A and B can be given also in terms of J, , J,, and J3 A=(J,+J2+1)(J,+J,+1)lKJ,-J,)(J,-Jdl,
(17.1)
B=(J,+J,+l)(J,+J,+1)lKJ,-J,)(J,-J,)(J,-Jdl
(17.2)
and they are related as B/A =(J,+Jj+l)/[(J,+Jz+l)(Jj-J,)].
(18)
Thus, for two given experimental values z, and z2, i.e. for two lines J, and J,, a third value zj can be calculated from Eq. (16) for any other line J3. The coefficients A and B are given by simple expressions [Eqs. (17)] in terms of J, , J2 and J, with a simple relation between them [Eq. (18)]. RESULTS
AND DISCUSSION
Tipping and Ogilvie’ used theoretical expressions for the Herman-Wallis factors in Raman spectra to calculate the ratio [zO,(J)]* [Eq. (l)] originating from common rotational levels for vibrational-rotational transitions (OJ c-) 1J’); they applied their method to the ground state of the molecule Ht. The results for 1 < J < 10 are displayed in Table 1, column 2. In order to test the validity of Eq. (16) we used the first two values of this data (for J, = 2 and J2 = 3) and we considered Table 1. Values of the ratio [z,,, (J# for the ground state of the molecule H,. The results of the present method are compared to those by Tipping and Ogilvie.’
J
Ref. 7
2 3 4 5 6 7 8 9 10 tRef.
0.639 0.536 0.450 0.380 0.321 0.273 0.233 0.200 0.173 7.
9 5 9 1 6 3 5 7 5
Rq. (16)
Rq. (11)
0.639 0.536 0.450 0.380 0.321 0.273 0.233 0.200 0.173
0.642 0.539 0.454 0.384 0.326 0.279 0.239 0.207 0.180
97 St 86 04 53 23 37 47 32
51 86 91 75 87 17 87 51 86
Relative intensities in the diatomic vibration-rotation
229
Raman spectra
Table 2. Values of [&,(J)12 for the ground state of the molecule CO. The results of the first entry are calculated from Eq. (16) and those of the second entry from Eq. (9). p’
1
0
-
I
3 5 10
I 3 5
tomitted
L’
J=4 0.983 687 67t 1.053 498 12 1.082 537 65 0.978 880 01 1.048 771 16 1.076 843 23
J=5 0.980 090 11 1.065 768 88 1.101 774 39 0.974 248 85 1.059 927 71 1.094 969 45
J=6 0.916 524 27 1.078 179 69 1.121 348 70 0.969 639 93 1.071 200 02 1.112 836 63
J=7 0.972 963 07 1.090 735 01 1.141 261 05 0.965 052 25 1.082 592 19 1.131 275 13
J=8 0.969 415 16 1.103 435 27 I.161 520 188 0.960 486 90 1.094 103 35 1.150 009 28
figures are identical to those of the first entry.
z, = z(2) and z2 = z(3). To obtain zj = z(4) (for J3 = 4) we used this equation, with k = 1 and n = 2. The same calculation is repeated to obtain z3 = z(5) by taking z, = z(3) and a2 = z(4) (already calculated), and so on for other values z(J) till J = 10. These results are displayed in Table 1, column 3. The comparison of these values of zO,(J) calculated from the present formulation zP to those calculated by Tipping and Ogilvie,’ zTo, shows an overall agreement. We consider this agreement to be a confirmation of the validity and the accuracy of the present formula [Eq. (16)]. By calculating 6 = [(zP)’ - (z~O)~]/(Zr”)‘, we notice that the relative discrepancy 6 increases with the increase of the value of J (except for J = 10). This may be explained by the order of correction (second order) used in Eq. (9); since for large value of J higher order corrections are probably needed for the H, molecule (having a large value of the ratio B&X,). Relation (16) is presented in a general form, in order that Z~can be calculated for different k and n and different “initial values” z, and z?. By using the data of column 2 of Table 1, we calculated, for the same transition (OJ ++ 1J ‘), the values of [z(J)]’ for the H, molecule from Eq. (16) by changing the different parameters z, , z2, k, and n of this equation in three different ways: (i) The procedure is described above. (ii) z, = z(2), z? = z(3), k = 1 and n = 2, 3,4,. . . (iii) ;, = z(2), z2 = z(J) (J = 3,4, 5,. . .), k = 1,2,3,. . . and n = 2, 3,4,. . . (the values of z> are those calculated successively from Eq. (16). We noticed that the values of [z(J)]’ calculated by the three procedures are the same up to six significant figures. We present in the same table the values of [z(J)12 calculated by using Eq. (11) (implying the coefficients G,, G, , G2) for the same transition (OJ c* 1J ‘) and the same molecule Hz (column 4). The comparison between the values of column 4 to those of column 2 showed an overall agreement with an increasing relative discrepancy with the increase of the value of J. This may be explained as in the previous application (values of column 3). In the Raman spectroscopy, the overtone transitions are rarely observed due to its inferior sensitivity and limited resolution. Seeking to show the generality of Eq. (16), we present in Table 2 (for CO-ground state) the value of [z,,.(J)]’ (first entry) calculated by using Eqs. (11) for the transitions L” - L’= 1, 3,5 with c = 0, 10 and J < 9.” As in the previous application (Table 1) and by using Eq. (16) with the first two values _] - = z(2) and z2 = z(3) for different transitions u - t” we calculated [z,,~.(J)]’ for J < 9 (second entry of Table 2). The comparison between the first and the second entry of this table is another proof of the validity and the accuracy of Eq. (16) for the considered transitions (rJ ++ v’J’). Similar results are obtained for the transitions U’- z: = 2,4, 6,7 with v = 0, 10, 20. If we compare the values of Table 1 to those of Table 2 for Au = 1 we find that a better accuracy is obtained in the case of the CO molecule (Table 2). To explain this, we may notice that, in perturbation theory, the accuracy of the calculation of M$“/M:z [Eq. (9)] increases when Be/o, decreases. The ratio Be/o, for the molecule H, isI 0.013 833; while for the CO molecule is 0.000889 9.
230
M. Korek and H. Kobeissi CONCLUSION
In the Raman vibrational-rotational transition (uJ c-) u/J’) of diatomic molecules, the study of the relative intensity, of the branches S and 0, is reduced to that of the ratio z,,,,(J) for any value of J. Z,,..(J) can be calculated either (i) from Eq. (I 1) where the calculation of the coefficients G,, G, and G, leads to an easy and fast calculation of z(J) for variable .I or (ii) from Eq. (16) when two experimental values z,,,(J,) and z,.(J,) are available. In this case the calculation of z?,.,(J)(for variable J) is easier and faster because the calculation of the wavefunctions Y,, and Y,.,,, and the integrals (Y, 1y 1Y,, ) and (Y, 1‘I”,,.) is avoided and an elementary calculation becomes sufficient to determine the line intensities. The numerical applications to the ground states of the molecules H, and CO show the accuracy and the validity of the present formulation for the fundamental and the higher overtones. REFERENCES 1. L. M. Cheung, D. M. Bishop, D. L. Drapcho, and G. M. Rosenblatt, Chem. Phys. Lett. 80, 445 (1981). 2. G. Placzek, Hundbuch der Radilogie Vol. VI part 2 (Ed. E. Marx), p. 205, Akademic, Berlin (1934). 3. T. C. James and W. Klemperer, .I. them. Whys. 31, 130 (1959). 4. C. Asawaroengchai and G. M. Rosenblatt, J. them. Phys. 72, 2664 (1980). 5. A. D. Buckingham and A. Szabo, J. Ram. Spectrosc. 7, 46 (1978). 6. A. Messiah, Mecanique &antique, Tome II, Dumond, Paris (1972). 7. R. H. Tipping and J. F. Ogilvie, J. Ram. Spectrosc. 15, 38 (1984). 8. R. Herman and R. Wallis, J. them. Phys. 23, 637 (1955). 9. R. Tipping and R. M. Herman, J. Molec. Spectrosc. 36, 404 (1970). 10. H. Kobeissi and M. Korek, J. Phys. B 27, 3653 (1994). 11. M. Korek and H. Kobeissi, Can. J. Chem. 71, 331 (1993). 12. M. Korek and H. Kobeissi, JQSRT 52, 631 (1994). 13. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, Toronto (1950).
Radiat. Transfer Vol. 55, No. 2, pp. 225-230, 1996
Pergamon 0022-4073(95)00168-9
Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-4073196 $15.00 + 0.00
RELATIVE INTENSITIES IN THE DIATOMIC VIBRATION-ROTATION RAMAN SPECTRA: A SIMPLE EXPRESSION FOR FUNDAMENTAL AND HIGHER OVERTONES M. KOREK“ and H. KOBEISSI*t “Faculty of Science, Beirut Arab University P.O. Box 1l-5020, Beirut, Lebanon and ‘Group of Molecular and Atomic Physics at the National Council for Scientific Research, P.O. Box 11-8281 Beirut, Lebanon (Received
4 May 1995)
Abstract-The study of the relative intensities of the S(J) and O(J) branch lines of the vibration-rotation Raman transitions (uJ ++ v’J’) of a diatomic molecule is considered. These relative intensities are proportional to (z,.)~, where z,,,(J) = (uJly Iu’J + 2)/(vJly )u’J - 2), and y is the polarizability anisotropy. It is shown that the ratio z,,(J) (i) can be expressed as ~UL.1(J)=(GO+GIm,+G2mf)/(G,+G,m,+G2m~),wherem,=-2J+l,m,=25+3andthe coefficients G,, G,, G2 are given by analytical expressions in terms of simple integrals depending on Y(O)(the pure vibration wavefunction) and its successive rotational corrections ‘I’(‘),‘PC’),. . . ; or (ii) can be calculated from the two given experimental values z1 = z,,,(J,) and z2 = z,,“.(J,) by using a simple relation in the form of z,,,(J) = [(A - B - l)z, AZ, + Bz, z2 + l]/[Az, - (A - B - l)z, - z, z2 + B], where the coefficients A and B are given by simple expressions in terms of Ji , J2 and J. The numerical applications to the ground states of I-I2 and CO
molecules show the validity and the accuracy of the present formulation.
INTRODUCTION
In the Raman spectra of diatomic molecules the ratio of the intensities Z’(J) and Z’(J) of the S and 0 branch lines of the vibration-rotation transition (oJ c* u ‘J ‘) originating from the same rotational level J is given by’ ZS(J)/ZO(J) = [(VS)3R(VD)(J+ l)(J + 2)(2J - l)(z,,.,(5))2]/[(VU)3R(v’)(J - l)J(2J + 3)], in which R(v) represents the calibrated spectral response of the spectrometer and detection system at frequency v and z,,,(J) = M::;J+ 2/M;;‘-2=
IU’J + 2)/(vJly
(uJly
IV’J - 2),
(1)
where Mt’ are the matrix elements of the polarizability anisotropy y between the initial and final states.2T3These matrix elements are defined as follows:
M$” = (Y’,,ly I’?,.,.) =
sm
YJr)y(r)Y',,,.dr,
0
where Y, is the rovibrational wavefunction and r is the internuclear distance. The values of the ratio z,,,(J) are needed for various scattering theories and to correct rotational and vibrational-rotational Raman intensities for molecular nonrigidity and vibrational-rotational interaction.4 The aim of this work is to show that z,,,(J) [Eq. (l)] can be obtained in two ways: (i) By using a nonconventional approach of the Rayleigh-Schriidinger perturbation theory, the ratio z,,,(J) can be written as tTo whom all correspondence
should be addressed. 225
226
M. Korek and H. Kobeissi
z,,,(J) = (Go + G, m, + G,mf)/(Go + G, m, + G,mz), where the coefficients G,, G,, Gz are given by analytical expressions in terms of the integrals ( Y,.]u1&) and ( Y,l Y,.), where Y stands for ‘P(O)(the pure vibration wavefunction), or Y(r) (the first rotational perturbative correction to Y(O)),or Yc2)(the second rotation correction). (ii) If two experimental values z,,.,(J,) and z,,,(J,) are available (without any restriction on J, and J,) the calculation of z,,(J) is reduced to a simple relation eliminating the need to evaluate the integrals (vJI y /u/J’), and furthermore, the wavefunctions YL,Jand YL,,J,.Numerical applications to the ground states of H2 and CO molecules are then presented and discussed. THE THEORY 1. Analytic expression of the ratio z,,,(J) In seeking the evaluation of the line intensities in the vibration-rotation Raman spectra of a diatomic molecule, James and Klemperer,3 and later on Buckingham and Szabo,’ derived a formula for the relative intensities of S and 0 branch lines using the first order perturbation theory.6 Tipping and Ogilvie’ preferred to use the Herman-Wallis rotation factor F$” = ((vJly Iv’J’)/(uOly
IdO))*.
They obtained an analytical expression for F in terms of the Dunham potential parameters and the coefficients of a general operator y were represented by a power series expansion. Actually, the rotational effect in the rovibrational infrared spectra is commonly characterized by an approximation of the Herman-Wallis rotational factor’ which is given by F$“=((uJ~f~u’J’)/(uO~f~u’0))2=1+Cm+Dm2+~~~,
(3)
where f is the dipole moment function and m = [J’(J’ + 1) - J(J + 1)]/2. The Herman-Wallis coefficients C and D are given in terms of the Dunham anharmonic coefficients a, (up to 7) and for U’ - u < 7.9 Recently Kobeissi and Korek” used a new approach of the Rayleigh-Schrodinger perturbation theory” and found analytical expressions for F$” and for the coefficients C and D for any potential and any operatorf [Eq. (3)]. This work was followed by anotherI one in which Korek and Kobeissi derived analytical expressions for the Herman-Wallis coefficients up to the third order (C, D and E). By using the same approach as the Rayleigh-Schrodinger perturbation theory in the Raman spectra, the wavefunctions for two vibrational-rotational states UJ and u’J’ are given respectively by Y”,=Y’,+1~~+1*~~,+~3~c+...,
(4)
Y,,,, = Y,. + 1’LZ&+ A’%,., + i’3%,, + . . . )
(5)
whereI=J(J+l),,l’=J’(J’+l),Y’,=Y’, co), Y L’I = Y(O)(the pure vibration wavefunctions) and &j = Y’(l) 9 = Y(2) . . . ) a”, = Y’!), g,,, = yw . . aie respectively the Yi’)- and Y$)-rotation c&rect:ons ovfthe iwo considered itates. Thes: iunctfons are well determined with high accuracy” even near dissociation. For a vibration-rotation transition (uJ CI u’J’), in Raman transition, the S and 0 branches are characterized by J’ = J + 2 and J’ = J - 2, respectively. By taking m = (A - ,I’)/2 we obtain for 1 and 1’ 2 =m*/4--m
+3/4,
A’=m2/4+m
+3/4.
(6)
By substituting i and 1’ by their values in Eqs. (4) and (5) one obtainsI Y’, = (Y, + (3/4).@”+ (9/16)9”) - m(a, + (3/2)9,) + m*((1/4)~,+(11/8)~~)+...,
(7)
Relative intensities in the diatomic vibration-rotation
Raman spectra
221
‘I”,,,,.= (‘I’,, + (3/4)W,. + (9/16)9”.) + m(Wu, + (3/2).~S~,,) + mZ((1/4)Lq,, + (1 l/8)9”.) +. . . .
(8)
Then by replacing Yt>,and Y’,,,,.by their expressions [Eqs. (7) and (S)] in the matrix elements I@$’ [Eq. (2)], and by taking into account the normalization factors, we write WY =
M$‘/M$’ = G, + G, m + G2m2,
(9)
where
w2 = (‘y,.lY I~,>/t(~“IyD)(~,~I~,~))“2, G,= 1 + (3/4)(<%l~ I‘u,,> + <%lr +(9/16N<~,lY
I Yu, ))/O’v
I Y I Y’,, >
IY I y’, >
- (90 IY I Yd ))/V” G = U/4NWvlyI’J’c,)
I Y I y,, > + <%,,lr
+ (9t.,I Y I y’, >)/Vu, +
1/2((93’,*/9% YV”.
l‘u,>)/W,l~
I’J’,,)
I Y I Y,!, > - l/2(<%
I y’,,, > - @+%IJ%>>/<‘y, I y’, >>I,
+ (11/W(~,ly
IY,,.>
I % )/
(~~,,I~\,,>/<‘y,,lyI’,)11(v9e%lY
This general formulation values
1’3’0,)
I‘u,>
G, = (
I’J’,>W’,Ir
I~“,>/<‘y,lY
I ye,>.
(10)
can be used for the S and 0 branches, by giving to m the appropriate m,=25+3;
m,= -2J+
1.
By using this formulation to find an analytic expression for z,,,(J) [Eq. (l)], one can substitute and j,,f”;;‘+2 by their expressions from Eq. (9) and find
Mti-2
z,,,(J) = (G, + G, m, + G,m,2)/(G,,+ G, m, + G,mz).
(11)
Once the coefficients G,, G, , G2 (depending on u and 0’) are calculated, this expression allows one to obtain z,(J) [for any transition (u - u’)] for variable J without repeating the calculation of the integrals of Eq. (2) for each value of J. 2. “Nonintegral” expression for the 2,‘ (J) The evaluation of the coefficients G,, G,,Gz can be avoided by considering three arbitrary values J, , J2, Jj. For these values of J Eq. (11) can be written as G, N, + G2P, = G,,(z, - l),
(12.1)
G, N; + G2P2 = G&2 - l),
(12.2)
G,N,+GzP,=G,,(z,-
(12.3)
l),
in which z,=z,,(Ji)
(i = 1,2,3),
N, = (m, - 4)z, + m,,
(13.1)
M. Korek
228
and H. Kobeissi
N,=(m,+2k--4)z,+m,+2k,
(13.2)
N,=(m,+2n
(13.3)
-4)z,+m,+2n,
+ 8m, - 16)z, + mf,
(14.1)
I’,=(-mf-4k2-4km,+8ms+16k-16)z,+mf+4k2+4km,,
(14.2)
P, = (-rnz
16n - 16)z,+m,2+4n2+4nm,,
I’,=(-mf-4n2-4nm,+8m,+ k = J2 - J, ;
(14.3)
n = J3 - J, .
By eliminating G,, G,,G2 in Eqs. (12) we obtain P, [Nj (~2
-
1)
-
N2h
-
III+
Pz[N,(z,
-
1)
-
N,(z,
-
1)l +Pj[N,(z,
- 1) - N2(z, - l)] = 0.
(15)
If we replace Eqs. (13) and (14) in Eq. (15), we find zj = [(A - B - l)z, - AZ, + Bz, z2 + l]/[Az, - (A - B - l)z, - z, z2 + B],
(16)
where A = (j + k)(j
+ k + n)/[k(k
- n)],
B = (j + n)(j
+ k + n)/[nk(k
- n)],
j=2J,+l.
The coefficients A and B can be given also in terms of J, , J,, and J3 A=(J,+J2+1)(J,+J,+1)lKJ,-J,)(J,-Jdl,
(17.1)
B=(J,+J,+l)(J,+J,+1)lKJ,-J,)(J,-J,)(J,-Jdl
(17.2)
and they are related as B/A =(J,+Jj+l)/[(J,+Jz+l)(Jj-J,)].
(18)
Thus, for two given experimental values z, and z2, i.e. for two lines J, and J,, a third value zj can be calculated from Eq. (16) for any other line J3. The coefficients A and B are given by simple expressions [Eqs. (17)] in terms of J, , J2 and J, with a simple relation between them [Eq. (18)]. RESULTS
AND DISCUSSION
Tipping and Ogilvie’ used theoretical expressions for the Herman-Wallis factors in Raman spectra to calculate the ratio [zO,(J)]* [Eq. (l)] originating from common rotational levels for vibrational-rotational transitions (OJ c-) 1J’); they applied their method to the ground state of the molecule Ht. The results for 1 < J < 10 are displayed in Table 1, column 2. In order to test the validity of Eq. (16) we used the first two values of this data (for J, = 2 and J2 = 3) and we considered Table 1. Values of the ratio [z,,, (J# for the ground state of the molecule H,. The results of the present method are compared to those by Tipping and Ogilvie.’
J
Ref. 7
2 3 4 5 6 7 8 9 10 tRef.
0.639 0.536 0.450 0.380 0.321 0.273 0.233 0.200 0.173 7.
9 5 9 1 6 3 5 7 5
Rq. (16)
Rq. (11)
0.639 0.536 0.450 0.380 0.321 0.273 0.233 0.200 0.173
0.642 0.539 0.454 0.384 0.326 0.279 0.239 0.207 0.180
97 St 86 04 53 23 37 47 32
51 86 91 75 87 17 87 51 86
Relative intensities in the diatomic vibration-rotation
229
Raman spectra
Table 2. Values of [&,(J)12 for the ground state of the molecule CO. The results of the first entry are calculated from Eq. (16) and those of the second entry from Eq. (9). p’
1
0
-
I
3 5 10
I 3 5
tomitted
L’
J=4 0.983 687 67t 1.053 498 12 1.082 537 65 0.978 880 01 1.048 771 16 1.076 843 23
J=5 0.980 090 11 1.065 768 88 1.101 774 39 0.974 248 85 1.059 927 71 1.094 969 45
J=6 0.916 524 27 1.078 179 69 1.121 348 70 0.969 639 93 1.071 200 02 1.112 836 63
J=7 0.972 963 07 1.090 735 01 1.141 261 05 0.965 052 25 1.082 592 19 1.131 275 13
J=8 0.969 415 16 1.103 435 27 I.161 520 188 0.960 486 90 1.094 103 35 1.150 009 28
figures are identical to those of the first entry.
z, = z(2) and z2 = z(3). To obtain zj = z(4) (for J3 = 4) we used this equation, with k = 1 and n = 2. The same calculation is repeated to obtain z3 = z(5) by taking z, = z(3) and a2 = z(4) (already calculated), and so on for other values z(J) till J = 10. These results are displayed in Table 1, column 3. The comparison of these values of zO,(J) calculated from the present formulation zP to those calculated by Tipping and Ogilvie,’ zTo, shows an overall agreement. We consider this agreement to be a confirmation of the validity and the accuracy of the present formula [Eq. (16)]. By calculating 6 = [(zP)’ - (z~O)~]/(Zr”)‘, we notice that the relative discrepancy 6 increases with the increase of the value of J (except for J = 10). This may be explained by the order of correction (second order) used in Eq. (9); since for large value of J higher order corrections are probably needed for the H, molecule (having a large value of the ratio B&X,). Relation (16) is presented in a general form, in order that Z~can be calculated for different k and n and different “initial values” z, and z?. By using the data of column 2 of Table 1, we calculated, for the same transition (OJ ++ 1J ‘), the values of [z(J)]’ for the H, molecule from Eq. (16) by changing the different parameters z, , z2, k, and n of this equation in three different ways: (i) The procedure is described above. (ii) z, = z(2), z? = z(3), k = 1 and n = 2, 3,4,. . . (iii) ;, = z(2), z2 = z(J) (J = 3,4, 5,. . .), k = 1,2,3,. . . and n = 2, 3,4,. . . (the values of z> are those calculated successively from Eq. (16). We noticed that the values of [z(J)]’ calculated by the three procedures are the same up to six significant figures. We present in the same table the values of [z(J)12 calculated by using Eq. (11) (implying the coefficients G,, G, , G2) for the same transition (OJ c* 1J ‘) and the same molecule Hz (column 4). The comparison between the values of column 4 to those of column 2 showed an overall agreement with an increasing relative discrepancy with the increase of the value of J. This may be explained as in the previous application (values of column 3). In the Raman spectroscopy, the overtone transitions are rarely observed due to its inferior sensitivity and limited resolution. Seeking to show the generality of Eq. (16), we present in Table 2 (for CO-ground state) the value of [z,,.(J)]’ (first entry) calculated by using Eqs. (11) for the transitions L” - L’= 1, 3,5 with c = 0, 10 and J < 9.” As in the previous application (Table 1) and by using Eq. (16) with the first two values _] - = z(2) and z2 = z(3) for different transitions u - t” we calculated [z,,~.(J)]’ for J < 9 (second entry of Table 2). The comparison between the first and the second entry of this table is another proof of the validity and the accuracy of Eq. (16) for the considered transitions (rJ ++ v’J’). Similar results are obtained for the transitions U’- z: = 2,4, 6,7 with v = 0, 10, 20. If we compare the values of Table 1 to those of Table 2 for Au = 1 we find that a better accuracy is obtained in the case of the CO molecule (Table 2). To explain this, we may notice that, in perturbation theory, the accuracy of the calculation of M$“/M:z [Eq. (9)] increases when Be/o, decreases. The ratio Be/o, for the molecule H, isI 0.013 833; while for the CO molecule is 0.000889 9.
230
M. Korek and H. Kobeissi CONCLUSION
In the Raman vibrational-rotational transition (uJ c-) u/J’) of diatomic molecules, the study of the relative intensity, of the branches S and 0, is reduced to that of the ratio z,,,,(J) for any value of J. Z,,..(J) can be calculated either (i) from Eq. (I 1) where the calculation of the coefficients G,, G, and G, leads to an easy and fast calculation of z(J) for variable .I or (ii) from Eq. (16) when two experimental values z,,,(J,) and z,.(J,) are available. In this case the calculation of z?,.,(J)(for variable J) is easier and faster because the calculation of the wavefunctions Y,, and Y,.,,, and the integrals (Y, 1y 1Y,, ) and (Y, 1‘I”,,.) is avoided and an elementary calculation becomes sufficient to determine the line intensities. The numerical applications to the ground states of the molecules H, and CO show the accuracy and the validity of the present formulation for the fundamental and the higher overtones. REFERENCES 1. L. M. Cheung, D. M. Bishop, D. L. Drapcho, and G. M. Rosenblatt, Chem. Phys. Lett. 80, 445 (1981). 2. G. Placzek, Hundbuch der Radilogie Vol. VI part 2 (Ed. E. Marx), p. 205, Akademic, Berlin (1934). 3. T. C. James and W. Klemperer, .I. them. Whys. 31, 130 (1959). 4. C. Asawaroengchai and G. M. Rosenblatt, J. them. Phys. 72, 2664 (1980). 5. A. D. Buckingham and A. Szabo, J. Ram. Spectrosc. 7, 46 (1978). 6. A. Messiah, Mecanique &antique, Tome II, Dumond, Paris (1972). 7. R. H. Tipping and J. F. Ogilvie, J. Ram. Spectrosc. 15, 38 (1984). 8. R. Herman and R. Wallis, J. them. Phys. 23, 637 (1955). 9. R. Tipping and R. M. Herman, J. Molec. Spectrosc. 36, 404 (1970). 10. H. Kobeissi and M. Korek, J. Phys. B 27, 3653 (1994). 11. M. Korek and H. Kobeissi, Can. J. Chem. 71, 331 (1993). 12. M. Korek and H. Kobeissi, JQSRT 52, 631 (1994). 13. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, Toronto (1950).