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Raman intensity analysis of cyclic molecules—Cyclohexane
JOURNAL OFMOLECULAR
SPECTROSCOPY
114,336-357 (1985)
Raman Intensity Analysis of Cyclic Molecules-Cyclohexane K. RAVINDRANATH, G. SWARNAKUMARI, AND N. RAJESWARA RAO’ Department
of Physics.Osmania University, Hyderabad 500 007, India
The orientational contribution to Raman or infrared intensity is obtained by the multiplication of a train of matrices J&&u’L-” [Prathibha Naik, V. A. Padma, and N. Rajeshwara Rao.. Pramana 13, Ill- 116( 1979)]. This is very laborious and for a molecule like C,H rz the I& is of the order of 54 X I8 and s’, 18 X 48. It is now shown that, in many cases, J+K+ can be cast into a constant times, the rows of us relating to the angle changes between the bonds. This gives the result of the product directly without having to regularly multiply the matrices. Also, the labor can be confined to one of the groups of the molecule as it is cyclic and the geometry of the group can be used to break up its species. Thus, one derives the formulas dealing with matrices of the order of 3 X 4. Derivation of Raman intensity formulas and intensity analysis is demonstrated
for CbH,,. 0
1985 Academic PEST..hc.
INTRODUCTION
An elegant format for deriving Raman intensity formulas of molecules was developed by Long (I), based on the bond polarizability theory first conceived by Eliashevich and Wolkenstein (2). The polarizability of a molecule was taken to be equal to the sum of the polarizabilities of the constituent bonds. For any tensor component, Long derived,
axy= c [@A- &zxw + &yl, ”
(1)
where ~yf,and CX~ are polarizabilities of the nth bond parallel and perpendicular to it, (assuming cyclindrical symmetry of the bond); 6,, is Kronecker delta function; and nx and ny are direction cosines of the bond. As a molecule is oscillating, the tensor component changes according to the scheme
aQ
aa al ax ._._ ar as = = xy.-.ar
ax dr as ag
Jalc’L
(2)
in matrix form. I is a column matrix of bond lengths and their direction cosines; and s gives Cartesian displacements of atoms, Yinternal coordinates, S symmetry coordinates, and Q normal coordinates. Since I and r have common bond stretches, (2) can be split into aa, -
aQ
_ - J,u’L + J.&Au’L.
’ Present address: 3-6-4 16/2, Himayatnagar, Hyderabad 500 0029, India. 0022-2852/85 $3.00 Copyright 0 1985 by Academic Press. Inc. All rights of reproduction in any form resewed.
336
RAMAN
INTENSITY
ANALYSIS
OF C6HIS
337
Equation (3) shows that the tensor element change is due to two factors, bond stretchings and their orientation given by J, and J*. A. given by x = Ar, is inconvenient to handle and Long (1) uses A = p-‘&g-‘. B is given by r = BX (since A is a rectangular matrix, its inverse cannot be obtained). Substituting in (3) and using LL’ = G, eq. (3) is reduced to
a(u,_aQ
J,u’L + J,K,p-
’ B’u’L-“.
This avoids the g-‘, which is inconvenient to use. The authors (3, 4) from this laboratory have further simplified the formula using the displacements of atoms p in place of the Cartesian triplets (x, y, z), and Wilson’s s vector matrix in place of B. Though thus reduced to l/3 of their sizes, K* and s matrices are still too large when one wants to derive intensity formulas for even moderately large molecules. Taking cyclohexane for example, since it has 18 bonds and 18 atoms, K+ is 54 X 18 and s’ is 18 X 48, as one has to take 48 internal coordinates. Though the molecule has high symmetry these matrices cannot be avoided. We shall now show that (i) it is possible to use the cyclic nature of the molecule and confine the labor only to one group, CH,; (ii) it is possible to avoid a regular multiplication of the matrices J~K,+fu’, and, instead, by suitably casting J,K,, arrive at the result directly; and (iii) the symmetry of the CH2 part can be taken advantage of and split into the Al, and Eg species of cyclohexane, though the point group of this part is not a subgroup of the point group of cyclohexane. (Such a splitting is generally possible only if the point group of a part is a subgroup of the whole molecule.) The derivation of the intensity formulas for this part is shown separately in the Appendix. INTENSITY
FORMULAS
OF THE CH, GROUP
As Fig. 1 shows, the cyclohexane has six CH2 groups arranged in a puckered hexagon. Its point group is Dsd with species distribution 6Al, + 8E, + 5Az, + 8E, + 2A2, + 3A’,; of this AZg and A,, are inactive both in Raman and infrared spectra. We take advantage of the fact that one group can be taken into the other by symmetry operation. Thus, group one goes over to two by Ss to three by C, and to four by inversion. Then, the tensor elements of two, three, and four can be obtained by the symmetry operation, CX’= T’aT 3
(5)
where (Yis the tensor of group one and cy’of 2,3, or 4. T is the corresponding operational matrix, and T’ its transpose. Inversion makes (Y’= (Y.Hence, (Y’of 5 and 6 are identical with cyof 2 and 3, respectively. Now, to obtain the Raman tensor of group one, we shall define internal coordinates as shown in Fig. 1. They are dl, rI, Dl, aI, PI, dl, t%, &, and &. It is advantageous to take 06 also as then the group will have Czv symmetry. Now,
RAVINDRANATH,
338
AND RAJESWARA
SWARNAKUMARL
RAO
8 c,
Group Angles
0 of
Group
0
0 d, I H,C,
@
8,
H,
: H, C, C,
6?:
6, I H2C, C, FIG. I. Cyclohexane-internal
H2C,
0
8,: H,C,C2
C2
coordinates.
+g%RI.+$w, 5
6
AR’ a& 1I&
. . .2da
(6)
AR,, A&, etc., are internal coordinate sets of the various groups. The column matrix R of Eq. (6) can be connected to the symmetry and normal coordinates by R = U’S = u’LQ.
(7)
But, if the L matrix of Eq. (7) is to be properly broken according to the symmetry of the molecule, the various elements of R have to be arranged in the proper manner. For example, taking the bond lengths and angle of H20, the symmetry coordinates are written as SI = 5 (TI + s2
=
s3
=$
r2)
0
(rl
-
r2).
This can be rearranged as rl
SI
r2 =
s2
e
s3
339
RAMAN INTENSITY ANALYSIS OF &HI2
Here S, and S, belong to A, and S, to B, and, correspondingly, the first two columns of U’belong to A, and the third column to B, , The arrangement of internal coordinates on the left cannot be changed if the arrangement of S elements has to be maintained properly. Similarly, the internal coordinates R of Eq. (7) have to be arranged to get the proper arrangement of S and Q elements following the symmetry of the molecule. The elements of column da,,,/~3R also have to be arranged properly. Now, we have tensor elements of only one group, that is, da,,,/ilR, . Every element of this row has to be transformed to get six elements of the other groups by using Eq. (5). Transformed tensors corresponding to & and C, are given in Table I. In these tables every aXYis abbreviated to xy for simplicity. For any internal coordinate of the first group, the transformed tensor element, say XX,changes thus Group 2 Group 5
4
3
XX;
3y_; - 2&xy];
i[(XX +
i[x;u + 3y; + 2&Y_“]
The entries under 2, 3, 4 are identical respectively to 5, 6, 1. To fix the ideas, let us say that these six entries correspond to L&, d6, d, , d2, d,, d4 of the six groups. The peculiar order of the numbering is due to the order in which the symmetry coordinates of Eg (and E,) are taken. The columns of U’ matrix are also arranged following the symmetry of the molecule. Now, there are six symmetry coordinates of A,,, S, = ;
(d, + d2 + d3 + d4 + d5 + d6),
with similar expressions following r, , D, , aI, (0, + &), and (6, + &). Multiplication TABLE I Raman Polarizability Component Changes due to S6 and C, Operations (1)
S6 operat,on: $ xx + 3yy
2/3xy)
.. (r’
xx
$(n
+ (3xx
nyy
+ yy
t
- 2xy)
2/3xy)
“Z ( XL
- 3
nxz
/3yz)
1 yz)
= (Symmetric) ..
(II)
c3
zz
ClperatKrl:
xx
+ 3yy
.. a’
..
+ 2/3xy)
$ (- /3 xx
xx
tJ3yy
+ yy
- 2xy)
2 /3xy)
XL +J3yz)
+ (J3xz
- yz)
= (Symmetric) ..
..
zz
340
RAVINDRANATH,
SWARNAKUMARI, AND RAJESWARA RAO
by these columns is equivalent to adding up the entries under the groups 1 to 6. Thus, the resultant tensor element of A,, species, say xx, is given by
xx = $
(xx + yy).
(8)
Thus, the tensor element XXof the whole molecule is obtained from the tensor elements xx and yy of one group. Similarly, YY=&(xx+yy)
(9)
zz = v%zz
(10)
XY=
(11)
YZ=ZX=O.
Thus, the symmetry of the A Ig species is maintained. Similarly, there are eight symmetry coordinates of Eg type, like S7a
=
-!G
[(-2~‘s + d3 + d,) + (-2dz + ds + dd)]
with similar expressions following rl, D,, aI (d, + 02) (6, + &), (CT+- &), and (6, - I&). Now the other symmetry coordinates are such that they contain either D, + De or D, - De, 0, + f& or 8, - &, and & + 62 or 6r - 62 in place of ar, (Ye,etc., ONSET.The + sign between the brackets refers to El, and the - sign to E, of infrared activity which will be taken up later. Corresponding to this symmetry coordinate we get XX=Js(xx-yy) 2
= -YY
(12)
and
zz = 0
(13)
XY = -&xy
(14)
xz = J3xz
(15)
YZ = &.
(16)
For the coordinates involving 8, - 0, and 6, - d2, we have vz = 0. Hence we use the value of xz. The other Eg type coordinates are like S76
=
;
[(d,
-
4)
f
(4
-
4K
for which also we obtain XY = J? (xx - yv) 2
(17)
xx=
(18)
-YY=
xz = -J&z
--J&y
(19)
RAMAN
INTENSITY
ANALYSIS
OF &HI2
341
and YZ =
J&z
(20)
zz=o.
(21)
It is satisfying to see that xz and yz of the two degenerate modes appear in the opposite order. INFRARED
INTENSITY
FORMULAS
In deriving infrared intensity formulas we assume that the dipole moment of any molecule is equal to the sum of the dipole moments of the bonds and for any component, (22) P, = C knu, n where u = X, y, z, and I.c,,is the dipole moment of the nth bond. Then, on arguments similar to those advanced for Raman formulas, one can derive formulas similar to Eq. (4); only J, and JQ are different, other matrices being the same. Again, for large molecules, one has to deal with large matrices, but for cyclic molecules we show that the labor can be confined to one group. Since we are dealing with vectors instead of tensors, the dipole moment of the second group can be obtained by p”
Tp:
for S6, p’ = ; (Px + Jsp,) ; <-JjPx
+ PJ
-Pz
for C3, p’= k (-p, ; (-&Px
+ hp,)
- Py)
(23)
PZ
T being the transformation E 5
s6
6
matrix. On transformation, c3
1
px becomes
Z
&i
C3i
2
3
4
Px ; (Px + J5Py) + (-Px + &Py) - Px - ; (P-x + J5p,,
- ; (-Px + &Py).
It is seen at the outset that the sum of these values is zero so that they do not occur in A,,.
342
RAVINDRANATH.
SWARNAKUMARI.
AND
RAJESWARA
RAO
Now, P, for &,, -2d5 + d3 + d,, gives -3p, and (-2d2 + d6 + d4) gives that in Raman the sum of the two combinations cancels while in IR P,=&p,=
4px.
+3p,
so
(24)
Similarly Py = -&py.
(25)
For the degenerate mode (d3 - d,) - (d6 - d4), we get for IR Px = 4py
(26)
Py = J?p,,
(27)
and again a very satisfactory situation showing that PX and Py have equal values of opposite symmetry. It is to be seen that px = 0 for all the coordinates except 0, - o2 and 6, - &, for which py = 0. P, is zero for all the three E,-type modes. For A2,,which have symmetry coordinates of the type, ds + d3 + d, , -d6 - d4 - d2, we easily observe that P, = Py = 0
(28)
P, = tipz.
(29)
and
RAMAN
INTENSITY
ANALYSIS
OF CYCLOHEXANE
We shall now demonstrate how Raman intensity analysis of cyclohexane is done, taking the intensity formulas of CH2 group from the Appendix and using the Raman spectra published in Raman and IR atlas of organic compounds (5). For molecules arranged in an order like in a crystal, one can take a Raman spectrum for each polarizability tensor element. But, for liquids and gases in which the molecules are oriented at random, there can be only two different Raman spectra, one involving 45~~: + 4/?2 and other showing 3P?, where a; = 1/3(c& + & + &)k,
(30)
and ,@ = 1/2[(crL, - a$J2 + (ai,Y - a;;)2 + (& - &J2 + 6(a$ + a$ + (~$1~.
(30a)
The primes denote differentiation with respect to the normal coordinate Qk. The intensity of a Raman line as indicated by the area A (in arbitrary units) under the line is given by (6) 45af + 7p$ = K2Avk(1 - e-h”c/kBT)/(vo - v)4n, (31) where K is a constant, n the degeneracy of the frequency, and hvkC/kB T the Boltzmann factor. The depolarization is (32) From A and p, it is possible to evaluate a? and p’2 k . a? will have a nonzero value only for A,-type oscillations.
343
RAMAN INTENSITY ANALYSIS OF C6HIZ
In the spectra referred to, p is of the order of 0.1. It does not seem to have been evaluated with sufficient accuracy as for CH and C-C stretching lines 45~‘~ % 7p)k. Hence, we do not propose to use the values of p$ in our formulas for A,,. For Ex species, however, the entire intensity is only due to pt. al;i’and pz for A,, and Eg species of cyclohexane thus calculated are given in Table II. Intensity Formulas The intensity formulas in the matrix form, i=Lil, with I = a; (i, j = x, y, and z> are derived and given in the appendix. C& and /?f, as defined earlier, can be calculated using the values of Z*.
TABLE II a; and 02 Values of &HI2 and C6DIl. [These values were published by R. G. Snyder (J. Mol. Spectrosc. 36,204-22 I, (1970)), but he has not given the values for CH lines. The values for the other lines given here compare well with those given by the above author.]
Species
‘k -1
cm
2852
2.63
2105
0.95
802
0.34
724
0.30
1158
0.07
1015
386
0.00
298
4x4 Of*1g
1465 2 x 2 Of Alg
2895 “k
Species
2080 %;t
cm
4x4ofE
II20
-I
4.13
2200
3.03
2924
3.5
2196
2.10
0.52
375
g
427
E
I x I of
E
I x I of
E
8
g
g
-I
2936
785
2xZof
“k cm
635
1029
1.55
796
0.31
1444
1.84
1213
0.17
I267
1.06
937
0.59
I348
1115
344
RAVINDRANATH,
SWARNAKUMARI.
AND RAJESWARA
RAO
Since a:, + & + a>, = 0 for the Je part of the intensity formulas, it is really necessary to evaluate the derivative of cu,, + aYY+ (Y,, as calculated from Eq (1) defining (Y:,= ; (a: + 2&
(33)
Lyj(= JZG,, L,k + &&Lzk.
(34)
Since for A,, oscillations @‘*are not reliable, we are not using even ai._ formulas derived in the appendix. As a:, = -abY = LY&, and a:= = a& for Eg, p;,’ = 3((Y$ + &)/( = 3(&
+ LYE)&
(35)
The formulas for a:,, & and c$ are taken from the appendix and formulas for /3;’ are calculated. (It may be noted that for Vk= 1267 cm-‘, &, = 0 as it belongs to A2 species of the single group.) Now, the purpose of Raman intensity analysis is to evaluate & , ii;, yI , and y2; r’, = dy,/dd and y; = dy2/dD. Suffix one refers to C-H bond and two refers to C-C bond. As explained in earlier papers, there are three shortcomings in the evaluation of accurate values: (i) L matrix elements which are derived from the force constants are not quite reliable, as the latter are generally not determined with sufficient confidence. (ii) The values of tyi, or /3? obtained from the intensities are in error to the extent of 10% for lines of medium intensity. They can be much less reliable for lines of lower intensities. (iii) ai, has sign ambiguity as it is square root of cut, which is derived from the intensity. We have overcome these difficulties by writing the formulas in the form Z’I = A’LL’A = A’GA. (36) This, at once, solves the above difficulties. Since Z’Z= Cl:‘, the sign does not come into the picture, nor does the inaccuracy in obtaining the intensities of weak lines. L Elements are not used, but we have only one equation for each species while electrooptical parameters are larger in number. Hence, assuming that the eops are the same for the isotopic molecules, we obtain cy;f and @p for CsDr2 also to have the necessary number of equations. Another point that we have to face is that the intensities are relative. The constant K2 is also different for the two molecules. Hence these constants also have to be determined. Hence we take G$ and y2 from the literature and determine the rest from the following six equations obtained from the intensity formulas A.20 to A.28 (given in appendix) and using Eq. (36): 2.0952G’r2+ 0.111018G9 - 0.222036&;&
= 3.04K:;
(1)
1.10416;2 + 0.111018&? - 0.222036&;&
= 1.25K;;
(2)
0.08156~2 + 1.9055257, = 0.71K,;
(3)
0.08156~~ + 1.075927~~ = 0.4OKz;
(4)
12.92~;2 + 0.333054~? + 0.6661 ly’ly; + 0.1524y;yi
+ 11.02~: + 2.537: + 2.375~~~~
- (y; + -yi)(lO.lr,
+ 0.505-r~) = (4.5)(11.64)Kf;
(5)
RAMAN
7.2902~‘; + 0.333054r;
INTENSITY
+ 0.66611y;y;
ANALYSIS
OF ChHlz
345
+ 8.93~: + 0.1524~;~~ + 2.53-(:
+ 2.3757,~~ - (y’, + r;)(lO.lri
+ 0.505rJ
= (4.5)(5.63)K:.
(6)
The values in these equations are shown to 6 figures for computational consistency. The values of K, , K2, y I , (Y; , y ; , and y 5 are obtained by solving the above equations. To get the values of these parameters we took from literature (7) the values of y2 = 1.32 and 5’2 = 1.67; K, = 1.09, Kz = 1.21; yI = 0.35; Cy’,= 1.35 (1.46); 7; = 2.55 (2.63); 7; = 2.44 (2.4). 7; is comparable with 2.5 of Cardona et al. (7). In these formulas, it may be seen that Z;;, 72, and y$ are multiplied by small quantities. Hence, their contributions to intensity are very small, and are of the same order of magnitude as the error involved in the determination of the intensity. Hence, the value of y$ is not reliable. Values of &, yI , and r’, compare well with those determined earlier in the laboratory (4) using the spectra of CH4 and CD4, which are shown in the brackets. (Though the values are shown to the second decimal place, they are reliable only to two figures.) This is a consequence of the fact that the intensity being proportional to the frequency [see Eq (3 l)]. The contribution of the CH lines, in fact, overshadows the contribution of the C-C lines. If one wants to arrive at reliable values for the eops of C-C bond, one has to take a molecule which does not give very high frequencies compared to the C-C stretching frequency. A similar analysis of infrared intensities is possible, by deriving similar formulas, but we are not including these formulas as intensities are not available. APPENDIX:
DERIVATION
OF INTENSITY FORMULAS A SIMPLIFICATION
OF MOLECULES-
We shall briefly recall Long’s (1) derivation of Raman intensity formulas of molecules, Following Wolkenstein and Eliashewich (2), he starts with the simplifying assumption that the polarizability of a molecule is equal to the sum of the polarizabilities of the bonds. Also, each bond is taken to be of cylindrical symmetry. He derives %KY = c [-Y,nxny +
&L,l.
(A.11
where nx and ny are direction cosines of the nth bond; and yn = CY~ - LYE, difference between the polarizabilities of the nth bond along and perpendicular to it. As the molecule is oscillating, the bonds are stretched and variously oriented, producing an overall change in the value of each of the tensor components. He further derives
aa,_ aQ
ah
d
.
a’. *. 2. dS=
ar ap ar as aQ
Jmu’L
(in
matrix
form),
(A.21
where 1 is a row matrix of bond lengths and their direction cosines, and Yinternal coordinates, S symmetry coordinates, and Q normal coordinates. Since I and r have common bond stretches, (A.2) is split into J&L
+ J,K,Au’L.
(A.31
[It may be recalled that originally Long used Cartesian displacements of the atoms, but Ferigle and Weber (8) suggested using p’s, displacements of the atoms.] Now,
346
RAVINDRANATH,
SWARNAKUMARI,
AND
RAJESWARA
RAO
(A.4) The s vectors, introduced by Wilson (9), are characteristic of his normal coordinate treatment of molecules, s/is’ = g (A-5) is inverse kinetic energy matrix. ugu’ = G (A.6) is symmetric and is block-diagonalized. Since r and p are, in general, not equal in number, s is rectangular. Hence, Crawford (10) suggested that
where ELis a diagonal matrix of reciprocals of masses. The J* part of A.3 now becomes J&&dL-‘I.
64.8)
The multiplication of this train of matrices is often quite laborious. Gussoni et al. (II) modified the & matrix to an extended form, but it does not simplify the matters at all. Now, we shall show that, while it is necessary to form these matrices, it is possible to avoid a regular multiplication. We shall start with Hz0 as an example. Raman Intensity Formulas of Hz0 Let el, e2 be the unit vectors along the bonds rl and r2 and e: and e: unit vectors perpendicular to them (Fig. A.l). Then, s matrix is given by Table A.I. Here pI, p2, p3 are displacements of the atoms Hi, HZ, and 0. It is shown that
ad -=-=
r
elcos 0 - e2 rsin 0
as -=-=
e:
qcos 13- el
aP2
r
~PI
e:
rsin8
FIG. A. 1. H,O-internal
(A.9)
’
coordinates.
(A.lO)
RAMAN
INTENSITY
ANALYSIS
OF C6HlZ
347
TABLE A.1 s Matrix for H20
e: and e: are denoted as unit vectors, as suggested by these formulas. Using the symmetry coordinates, us matrix can be written as in Table A.11. Now, we shall construct the J* and K* matrices (Table A.111).The prime on (Y’Sdenotes differentiation with respect to the normal coordinate (Table A.IV). Similar rows can be guessed for ly and 2-r. Ferigle and Weber (8) suggested a way of evaluating the elements of this matrix. dlx -= dPl
dcos @p1 = -sin a,.-_d% = -sin @ret/r,, dPl aPl
(A.1 1)
where Cpiis angle between r1 and the x axis, and &@,/dp, is a vector shown as e; similar to #@pi = e:/r, . It may be appreciated that e: and e_Tare identical, increasing the angles 0 and 4~) respectively. So (A.12) e? = e?. But, while e: increases the angle 8, it decreases a’2 between rz and the x axis, e: = -e-z. Then, using the relations lx = -2x = -sin e/2 and sin 9, = sin $ = Scos 13/2, TABLE A.11 us Matrix of Hz0
(A.13)
348
RAVINDRANATH,
SWARNAKUMARI,
AND RAJESWARA
RAO
TABLE A.111 J+ Matrix of Hz0
0*x*
Ix
2x
lY
2Y
2$x
2y2x
0
0
CL' YY
0
a' XY
,viY
0
my Ylx
Y2Y
the product J@K+can now be written as
[
27 sin 8/2 cos Of2 $ $
iY2 Y Y2X
-(efr+e:) . 1
The row vector in this expression can be recognised as the second row of us. Therefore, the product of this with PS’U becomes the second row of G matrix and its product with G-’ will be equal to the second row of the unit matrix. Hence, the J+ part of A.3 simply becomes y sin BLz,. (A.14) If we regularly multiply all these matrices together we obtain 7
p. sin* B&j’ + $ sin Qu, + (1 - cos f3)~o]L~1
= y sin B(G2,L;’ + G22Li2’) = 7 sin 8L2I. This is known to the investigators and was pointed out by Long in his earlier papers. But, the point we now make out has been possible because of the recognition of the results given by A. 12 and A. 13, helping convert the K+ matrix into the proper form. It may also be seen that the addition of the Ke matrix rows gives a row which coincides with one of the S’U’matrix columns relating to the angles. This addition is facilitated by multiplying with J+ (In the case of infrared intensities J+ is much simpler).
TABLE A.IV k+ Matrix of H20
1x
2x
0
RAMAN INTENSITY ANALYSIS OF C6H,2
G matrix. The present form of the US matrix helps the determination matrix. The following relations are useful in this respect (Fig. A. 1):
349 of the G
el - e2 = cos 0 e:-e:
= -cos 6
e:-e,
= e!*e2 = -sin 8 = e:.el,
Then, G,, = /JH + /~(l + cos 0); G,2 = - sin f$.&; G22 = ( 1/2d2)[pLH+ PO(1 cos f3)]; and G33 = JQ, + PO(1 - cos 8).
We shall now demonstrate this for a molecule of greater complexity CH2C12. This is of C2, symmetry. As shown in Fig. A.2, the pair of C-Cl bonds is in the plane of the paper and the CH bonds are in the perpendicular plane. There are 4Ai-type oscillations, two relating to the symmetric stretching of the bonds and the other two relating to the angle changes. It will be convenient to arrange the symmetry coordinates such that each pair of bonds is associated with change in the angle between them. Thus
FIG. A.2. CH,C12-intemal
coordinates.
350
RAVINDRANATH,
SWARNAKUMARI,
AND
RAJESWARA
RAO
TABLE A.V s’u’ Matrix of CH2C12
9 C
J2e 7F
s-4 = +
(e, + I92+ 83 + 84 + 02).
Then, the USmatrix is written down for this species. It will be convenient to simplify the s vectors using the following relations among them. All the angles are taken to be tetrahedral for simplicity. Thus, it is seen that el + e2 + e3 + e4 = 0; el + ez = (2/&)e,; e: + e: = -&e, + e2) = (-2&/&)e,; e3 + e4 = (-2/&)e,; e’: + ej e,; and e: + e’: + ef = 0 (Table A.V). = -&(e, + e4) = (2&/J?)
FIG. A.3. CHQ-internal
coordinates.
RAMAN
1NTENSlTY
ANALYSIS
351
OF C6H,,
.I+ for cy,, is given by (2yrlz 2yr2z 2~~3~ 2~~42). yl = (Y~- (Y~of CH and y2 is similar for C-CL With a similar procedure followed for H20, J& becomes -sin p/2 cos p/2 5 e:
IL e: d
[
725
3
et:
2
r22-
(e:
+
e:)
+
g
(e:
+
ej) 11
(
Split them into two rows and cast them into
and \iS(sin $(cos
f)yI[O
0 2
g
gz].
The rows coincide with the columns (3) and (4) of s’u’. Hence on substituting for cos P/2 and sin @/2(cos /$2 = I/&; sin 812 = &I&),
For CL??,, J+ becomes [27, ly 2yr2.v 0 01. Since ly = -sin /3/Z = -2y, J&, -V%yj(sin /3/2)(cos P/2)[j$
zi
becomes,
0 0 $$$I.
The row matrix coincides with the 3rd column of s’u’. Hence (A. 15) Similarly. (A. 16) It is satisfactory to see that alrx + c& + akz = 0.
We shall demonstrate this in another molecule, CH3Cl, as a number of new points come into the picture (Fig. A.3). The symmetry coordinates of the A, species are S, = I/&@, + d2 + d,); S2 = D; s3
=
dGP,
matrix can be formed in the The angIes of this molecule are the various vectors are similar to 27, lz 2~~22 2-yr3.z 2~24~. The @I/&, = -sin be:; t34z/ap4 = 0. u’s’
+
02
+
03
-
8,
-
02 -
03).
usual way (Table A.VI). aIso taken to be tetrahedral, and the relations among those discussed for CH2C12, The Ja matrix for (Y,~is K* matrix elements are like 13lz/dp, = -sin /3 X Hence J&, matrix is (since lz = 22 = 32 = -l/3),
352
RAVINDRANATH,
SWARNAKUMARI,
AND RAJESWARA
RAO
TABLE A.VI s’u’ Matrix for CHFI
S1
3
s3
s. ,t
S5a
s4a
Since the row matrix in the brackets coincides with the column under & of s’u’, 4 '
(A.17)
=-Y&3,.
Ly== 3ti
Since this molecule is of CJv symmetry, a:, = c& and (YL + &, + (Y;, = 0. Hence, But, we shall derive the formula for a;,, as it raises some interesting 68 = -l/2&. points. The direction cosines of the bonds are ly = 0; 2y = -\12/3; 3y = l6$; 4y = 0. The J+ matrix is 2x0
(2% 1Y 2~12y 2Y13Y 2~4y) = K
~(0 -1 1 0) a2v
A
apl
~COS~? = ___ aPl
z
-sin
a-
aa
=
-sin
a&.
aPI
Here, we have put 2y = cos +; then sin @ = l/&. Similarly d3y/dp, = -sin +e:. It can be shown that ei* es = -l/2, e; - ei = l/2. J+K+ now becomes
On multiplying this row vector with the third column of s’u’, it is found to be equal to the square of the third column. Then, it can be shown that (A. 18)
RAMAN
INTENSITY ANALYSIS OF C,H,2
353
as expected. It has been possible to deal with a& more directly as the z axis coincides with the symmetry of the molecule and is also along one of the bonds. It will be possible to obtain a similar equation with C& also, but it will be even more indirect than L$ as the direction cosines lx, 2x and 3x do not have a common factor, like in the case of lz, 22, 32, or 1y, 2-v, 3~. The expressions for & and aj3’ show that they are functions of only the third symmetry coordinate, S, . Therefore, one can write
Now
CY,,
=
y,(1z2 + 2z2 + 32*) + 3a2 = y,(cos*P, + cos2p2 + cos*&) + 3cu*. Then
aa,,ap= -.ap
as
-2yl(cos
@,sin PI cos P2sin p2 cos &sin & 0 0 0)~’
(Here u’ is the column matrix of &),
aa,=ap
4
-.-z-.
ap
as
3fi
?‘,
A similar expression can be derived for a&,. Thus in suitable cases, one can avoid even K+ and s’u’ matrices. Now we shall study the s’u’ of the E-type oscillations. It can be shown that es = e: and e; = -e:. Using these values in the K+ matrix, J+K+, after simplification, is obtained as
This, on multiplying with FS’U’,becomes -- 3b4
YdG46
G6
G66)r
which, on multiplying with L-l’, becomes L6r so that 4 Q!;;,a= - 3t/3
ylL6i-
(A.19)
In the case of c& of A,- and E-type oscillations, the row vectors, on multiplication with the last column of s’u’, give a constant times G33 (or G6& indicating that they are geometrically equivalent. Hence, it helps writing down the tensor element directly. This simplification is, however, not possible when the species contains rotation (R, and R, belong to E-type of CH3Ci; for CH2C12, B, contains R, and Bz contains Ry). Their intensity contributions are given by & and LY&.Then, these tensor elements are functions of all the L-” matrix elements (as given in A.8) and to obtain the expressions it is necessary to multiply the train of matrices in a regular way.
354
RAVINDRANATH.
SWARNAKUMARI,
AND RAJESWARA RAO
c6Hl2
Now, we shall pass on to cyclohexane. The s’u’ matrix of the A,, species contains six identical groups, each containing three columns referring to the two hydrogen atoms and one carbon atom of the group. For the Eg species, there are two identical groups [corresponding to a symmetry coordinate like (2d, - C&- d5) and (2d4 - db - dz)]. The S’U’matrices are identical for them. Also, in the group 2dr - d3 - ds , the elements are identical for all the symmetry coordinates except for the coordinates involving 0 and p for which there are small differences. Hence, as an approximation, we have neglected these small differences and have taken the matrix for only one group. There are no such differences in the A Ig group. The matrices are shown in the Table A.VIIa and VIIb The symmetry coordinates of the groups are also given in the Table A.VIII. Now, the G matrix is given by G = us/.~W. It is found that in the A,, species, the G matrix is block-diagonalized into 4 X 4 and 2 X 2. On examining the corresponding symmetry coordinates, we observe that the
TABLE A.VIIa s’u’ of A,, Species of One Group of Cyclohexane
0
TABLE A.VIlb S’u’ of E, Species of One Group of Cyclohexane
n2
0
0
- ef d
h cx -7
f32 d
- e, 7FT
e;:
RAMAN
INTENSITY
ANALYSIS
OF &HI2
355
TABLE A.VIII Symmetry
s1 = /+ S2
s3
S4
Coordinates
(d, + r,)
=
/+
0,
=
/+
‘q+$+6,+62+a)
= &e,
s5
:#G
S6
=
+ D6)
+e* t5] t62 + $6
)
(r, - d,f
+
‘0, +e2 -6, -4’
Symmetry
In S9 and
for Alg Species of one group of C6H,2
5,0
,
Coordinates
for Eg Species of one group of C6H12
B= 0,
6 = 6,
/ 02:
+ 62
group behaves as if it has Cl, symmetry. The 10 coordinates d, , rl , DI , De, 8,, I&, 6,) &. a,, 0, are grouping themselves as 4Ai + A2 + 28, + 2Bz relevant to the CzU symmetry. The six symmetry coordinates of the Ai, species of the molecule (C6Hi2) correspond to 4Ai + 2Bi. In a similar manner, for Eg, the eight symmetry coordinates correspond to 4A1, AZ, 2Bi, and one of the Bl types of C’2”symmetry. The intensity formulas also will be split in the same manner. A,,type oscillations are excited by cu:, and &, = &,. Also, as the J* part follows the rule, CUL+ c& + & = 0, we have derived the formulas only for a:,. &-type oscillations are controlled by ok = -a;,,, (Y& and (Y&.Therefore, though the group shows all the Raman tensor elements, we take only axx, c+, oXz, and (Y,,~, omitting (Y,.and CX,,.(Though the coordinate system is the same for the group as well as the molecule, to differentiate their tensor elements, we use xyz for the group and
356
RAVINDRANATH, SWARNAKUMARI, AND RAJESWARA RAO
XYZ for the molecule.) Both CQ~(or arr) = &/2(& - c&J and LY’Y~ = \j5c& are calculated for 4Ar + 2Br oscillations. A2 is excited only by & = &c&J. The J* part of the intensity formulas are derived by writing down the J*K* matrices for each of the tensor elements. On multiplying with S’U’in the ways explained earlier, the formulas are derived. The J, part is derived following the zero-order approximation, in the way explained in our earlier papers (3, 4). The ultimate formulas (combining J, and J*) are: (i) Polarizability tensor elements for 4 X 4 of A,, species. CC:= Jz(S’lLlj + ijliLzi)
i = 1 to4.
(A.20)
For q there is no J+ part; the J, part is 2 X 4 as there are only two kinds of bonds,
(ii) For 2 X 2 part of A,, species: Here a; = CZk= 0,
(A.22)
i = 5, 6.
(A.23)
i = 1 to 4 and j = 7, 8.
(A.25)
(iii) For 4 X 4 part of the Eg species
(iv) 1 X I part of the Eg species (a%
= [$$+&
+g+y]L;:.
(A.26)
(v) 2 X 2 part of the Eg species 2x! (aLJi=~y;L-ii+~d
&rl
2
-YjPcGY
+
(~+;(;+j&)L$]
(A.27)
(~+;($+;)pc)L&‘]
(A.28)
[
RECEIVED: April 16, 1984
RAMAN INTENSITY ANALYSIS OF CeH,Z
357
REFERENCES D. A. LONG, Proc. Roy. Sot. Lond. 217,203-221 (1953). M. ELIASHEVICHAND WOLKENSTEIN,J. Phys. Moscow 9, 101-326 (1945). G. SWARNAKUMARI AND N. RAJESWARARAO, J. Mol. Specfrosc. 106, 1-l 1 (1984). PRATHIBHANAIK. V. A. PADMA, AND N. RAJESWARARAO, Prumana 13, 111-l 16 (1979). B. S~HRADER AND W. MEIER, “D.M.S. Raman/IR Atlas Organischer Verbindungen of Organic Compounds. Herrausgegeben vom Edited for the Institut fur Spektrochemie und Angerwandte Spektroskopie Dortmund,” Vol. I. Groups (A-E) & Vol. 2 Groups (F-O), Verlag Chemie, Berlin. 6. J. H. BERNSTEINAND G. ALLEN, .J. Opt. Sot. Amer. 45, 237-349 (1955). 7. D. BERMEJO,S. MONTERO, M. CARDONA. AND A. MURAMATSU, Solid State Commun. 42. 153-155
I. 2. 3. 4. 5.
(1982). 8. S. M. FERIGLEAND A. WEBER, Canad. J. Phys. 32,799-807
(1954).
9. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS, “Molecular Vibrations,” McGraw-Hill, New York
1955. 10. B. L. CRAWFORD, J. Chem. Phys. 29, 1042-1045 (1958). 11. M. GUSSONI. S. ABBATE, AND G. ZERBI, J. Raman Spectroscopy, 6, 289-297 (1977).
SPECTROSCOPY
114,336-357 (1985)
Raman Intensity Analysis of Cyclic Molecules-Cyclohexane K. RAVINDRANATH, G. SWARNAKUMARI, AND N. RAJESWARA RAO’ Department
of Physics.Osmania University, Hyderabad 500 007, India
The orientational contribution to Raman or infrared intensity is obtained by the multiplication of a train of matrices J&&u’L-” [Prathibha Naik, V. A. Padma, and N. Rajeshwara Rao.. Pramana 13, Ill- 116( 1979)]. This is very laborious and for a molecule like C,H rz the I& is of the order of 54 X I8 and s’, 18 X 48. It is now shown that, in many cases, J+K+ can be cast into a constant times, the rows of us relating to the angle changes between the bonds. This gives the result of the product directly without having to regularly multiply the matrices. Also, the labor can be confined to one of the groups of the molecule as it is cyclic and the geometry of the group can be used to break up its species. Thus, one derives the formulas dealing with matrices of the order of 3 X 4. Derivation of Raman intensity formulas and intensity analysis is demonstrated
for CbH,,. 0
1985 Academic PEST..hc.
INTRODUCTION
An elegant format for deriving Raman intensity formulas of molecules was developed by Long (I), based on the bond polarizability theory first conceived by Eliashevich and Wolkenstein (2). The polarizability of a molecule was taken to be equal to the sum of the polarizabilities of the constituent bonds. For any tensor component, Long derived,
axy= c [@A- &zxw + &yl, ”
(1)
where ~yf,and CX~ are polarizabilities of the nth bond parallel and perpendicular to it, (assuming cyclindrical symmetry of the bond); 6,, is Kronecker delta function; and nx and ny are direction cosines of the bond. As a molecule is oscillating, the tensor component changes according to the scheme
aQ
aa al ax ._._ ar as = = xy.-.ar
ax dr as ag
Jalc’L
(2)
in matrix form. I is a column matrix of bond lengths and their direction cosines; and s gives Cartesian displacements of atoms, Yinternal coordinates, S symmetry coordinates, and Q normal coordinates. Since I and r have common bond stretches, (2) can be split into aa, -
aQ
_ - J,u’L + J.&Au’L.
’ Present address: 3-6-4 16/2, Himayatnagar, Hyderabad 500 0029, India. 0022-2852/85 $3.00 Copyright 0 1985 by Academic Press. Inc. All rights of reproduction in any form resewed.
336
RAMAN
INTENSITY
ANALYSIS
OF C6HIS
337
Equation (3) shows that the tensor element change is due to two factors, bond stretchings and their orientation given by J, and J*. A. given by x = Ar, is inconvenient to handle and Long (1) uses A = p-‘&g-‘. B is given by r = BX (since A is a rectangular matrix, its inverse cannot be obtained). Substituting in (3) and using LL’ = G, eq. (3) is reduced to
a(u,_aQ
J,u’L + J,K,p-
’ B’u’L-“.
This avoids the g-‘, which is inconvenient to use. The authors (3, 4) from this laboratory have further simplified the formula using the displacements of atoms p in place of the Cartesian triplets (x, y, z), and Wilson’s s vector matrix in place of B. Though thus reduced to l/3 of their sizes, K* and s matrices are still too large when one wants to derive intensity formulas for even moderately large molecules. Taking cyclohexane for example, since it has 18 bonds and 18 atoms, K+ is 54 X 18 and s’ is 18 X 48, as one has to take 48 internal coordinates. Though the molecule has high symmetry these matrices cannot be avoided. We shall now show that (i) it is possible to use the cyclic nature of the molecule and confine the labor only to one group, CH,; (ii) it is possible to avoid a regular multiplication of the matrices J~K,+fu’, and, instead, by suitably casting J,K,, arrive at the result directly; and (iii) the symmetry of the CH2 part can be taken advantage of and split into the Al, and Eg species of cyclohexane, though the point group of this part is not a subgroup of the point group of cyclohexane. (Such a splitting is generally possible only if the point group of a part is a subgroup of the whole molecule.) The derivation of the intensity formulas for this part is shown separately in the Appendix. INTENSITY
FORMULAS
OF THE CH, GROUP
As Fig. 1 shows, the cyclohexane has six CH2 groups arranged in a puckered hexagon. Its point group is Dsd with species distribution 6Al, + 8E, + 5Az, + 8E, + 2A2, + 3A’,; of this AZg and A,, are inactive both in Raman and infrared spectra. We take advantage of the fact that one group can be taken into the other by symmetry operation. Thus, group one goes over to two by Ss to three by C, and to four by inversion. Then, the tensor elements of two, three, and four can be obtained by the symmetry operation, CX’= T’aT 3
(5)
where (Yis the tensor of group one and cy’of 2,3, or 4. T is the corresponding operational matrix, and T’ its transpose. Inversion makes (Y’= (Y.Hence, (Y’of 5 and 6 are identical with cyof 2 and 3, respectively. Now, to obtain the Raman tensor of group one, we shall define internal coordinates as shown in Fig. 1. They are dl, rI, Dl, aI, PI, dl, t%, &, and &. It is advantageous to take 06 also as then the group will have Czv symmetry. Now,
RAVINDRANATH,
338
AND RAJESWARA
SWARNAKUMARL
RAO
8 c,
Group Angles
0 of
Group
0
0 d, I H,C,
@
8,
H,
: H, C, C,
6?:
6, I H2C, C, FIG. I. Cyclohexane-internal
H2C,
0
8,: H,C,C2
C2
coordinates.
+g%RI.+$w, 5
6
AR’ a& 1I&
. . .2da
(6)
AR,, A&, etc., are internal coordinate sets of the various groups. The column matrix R of Eq. (6) can be connected to the symmetry and normal coordinates by R = U’S = u’LQ.
(7)
But, if the L matrix of Eq. (7) is to be properly broken according to the symmetry of the molecule, the various elements of R have to be arranged in the proper manner. For example, taking the bond lengths and angle of H20, the symmetry coordinates are written as SI = 5 (TI + s2
=
s3
=$
r2)
0
(rl
-
r2).
This can be rearranged as rl
SI
r2 =
s2
e
s3
339
RAMAN INTENSITY ANALYSIS OF &HI2
Here S, and S, belong to A, and S, to B, and, correspondingly, the first two columns of U’belong to A, and the third column to B, , The arrangement of internal coordinates on the left cannot be changed if the arrangement of S elements has to be maintained properly. Similarly, the internal coordinates R of Eq. (7) have to be arranged to get the proper arrangement of S and Q elements following the symmetry of the molecule. The elements of column da,,,/~3R also have to be arranged properly. Now, we have tensor elements of only one group, that is, da,,,/ilR, . Every element of this row has to be transformed to get six elements of the other groups by using Eq. (5). Transformed tensors corresponding to & and C, are given in Table I. In these tables every aXYis abbreviated to xy for simplicity. For any internal coordinate of the first group, the transformed tensor element, say XX,changes thus Group 2 Group 5
4
3
XX;
3y_; - 2&xy];
i[(XX +
i[x;u + 3y; + 2&Y_“]
The entries under 2, 3, 4 are identical respectively to 5, 6, 1. To fix the ideas, let us say that these six entries correspond to L&, d6, d, , d2, d,, d4 of the six groups. The peculiar order of the numbering is due to the order in which the symmetry coordinates of Eg (and E,) are taken. The columns of U’ matrix are also arranged following the symmetry of the molecule. Now, there are six symmetry coordinates of A,,, S, = ;
(d, + d2 + d3 + d4 + d5 + d6),
with similar expressions following r, , D, , aI, (0, + &), and (6, + &). Multiplication TABLE I Raman Polarizability Component Changes due to S6 and C, Operations (1)
S6 operat,on: $ xx + 3yy
2/3xy)
.. (r’
xx
$(n
+ (3xx
nyy
+ yy
t
- 2xy)
2/3xy)
“Z ( XL
- 3
nxz
/3yz)
1 yz)
= (Symmetric) ..
(II)
c3
zz
ClperatKrl:
xx
+ 3yy
.. a’
..
+ 2/3xy)
$ (- /3 xx
xx
tJ3yy
+ yy
- 2xy)
2 /3xy)
XL +J3yz)
+ (J3xz
- yz)
= (Symmetric) ..
..
zz
340
RAVINDRANATH,
SWARNAKUMARI, AND RAJESWARA RAO
by these columns is equivalent to adding up the entries under the groups 1 to 6. Thus, the resultant tensor element of A,, species, say xx, is given by
xx = $
(xx + yy).
(8)
Thus, the tensor element XXof the whole molecule is obtained from the tensor elements xx and yy of one group. Similarly, YY=&(xx+yy)
(9)
zz = v%zz
(10)
XY=
(11)
YZ=ZX=O.
Thus, the symmetry of the A Ig species is maintained. Similarly, there are eight symmetry coordinates of Eg type, like S7a
=
-!G
[(-2~‘s + d3 + d,) + (-2dz + ds + dd)]
with similar expressions following rl, D,, aI (d, + 02) (6, + &), (CT+- &), and (6, - I&). Now the other symmetry coordinates are such that they contain either D, + De or D, - De, 0, + f& or 8, - &, and & + 62 or 6r - 62 in place of ar, (Ye,etc., ONSET.The + sign between the brackets refers to El, and the - sign to E, of infrared activity which will be taken up later. Corresponding to this symmetry coordinate we get XX=Js(xx-yy) 2
= -YY
(12)
and
zz = 0
(13)
XY = -&xy
(14)
xz = J3xz
(15)
YZ = &.
(16)
For the coordinates involving 8, - 0, and 6, - d2, we have vz = 0. Hence we use the value of xz. The other Eg type coordinates are like S76
=
;
[(d,
-
4)
f
(4
-
4K
for which also we obtain XY = J? (xx - yv) 2
(17)
xx=
(18)
-YY=
xz = -J&z
--J&y
(19)
RAMAN
INTENSITY
ANALYSIS
OF &HI2
341
and YZ =
J&z
(20)
zz=o.
(21)
It is satisfying to see that xz and yz of the two degenerate modes appear in the opposite order. INFRARED
INTENSITY
FORMULAS
In deriving infrared intensity formulas we assume that the dipole moment of any molecule is equal to the sum of the dipole moments of the bonds and for any component, (22) P, = C knu, n where u = X, y, z, and I.c,,is the dipole moment of the nth bond. Then, on arguments similar to those advanced for Raman formulas, one can derive formulas similar to Eq. (4); only J, and JQ are different, other matrices being the same. Again, for large molecules, one has to deal with large matrices, but for cyclic molecules we show that the labor can be confined to one group. Since we are dealing with vectors instead of tensors, the dipole moment of the second group can be obtained by p”
Tp:
for S6, p’ = ; (Px + Jsp,) ; <-JjPx
+ PJ
-Pz
for C3, p’= k (-p, ; (-&Px
+ hp,)
- Py)
(23)
PZ
T being the transformation E 5
s6
6
matrix. On transformation, c3
1
px becomes
Z
&i
C3i
2
3
4
Px ; (Px + J5Py) + (-Px + &Py) - Px - ; (P-x + J5p,,
- ; (-Px + &Py).
It is seen at the outset that the sum of these values is zero so that they do not occur in A,,.
342
RAVINDRANATH.
SWARNAKUMARI.
AND
RAJESWARA
RAO
Now, P, for &,, -2d5 + d3 + d,, gives -3p, and (-2d2 + d6 + d4) gives that in Raman the sum of the two combinations cancels while in IR P,=&p,=
4px.
+3p,
so
(24)
Similarly Py = -&py.
(25)
For the degenerate mode (d3 - d,) - (d6 - d4), we get for IR Px = 4py
(26)
Py = J?p,,
(27)
and again a very satisfactory situation showing that PX and Py have equal values of opposite symmetry. It is to be seen that px = 0 for all the coordinates except 0, - o2 and 6, - &, for which py = 0. P, is zero for all the three E,-type modes. For A2,,which have symmetry coordinates of the type, ds + d3 + d, , -d6 - d4 - d2, we easily observe that P, = Py = 0
(28)
P, = tipz.
(29)
and
RAMAN
INTENSITY
ANALYSIS
OF CYCLOHEXANE
We shall now demonstrate how Raman intensity analysis of cyclohexane is done, taking the intensity formulas of CH2 group from the Appendix and using the Raman spectra published in Raman and IR atlas of organic compounds (5). For molecules arranged in an order like in a crystal, one can take a Raman spectrum for each polarizability tensor element. But, for liquids and gases in which the molecules are oriented at random, there can be only two different Raman spectra, one involving 45~~: + 4/?2 and other showing 3P?, where a; = 1/3(c& + & + &)k,
(30)
and ,@ = 1/2[(crL, - a$J2 + (ai,Y - a;;)2 + (& - &J2 + 6(a$ + a$ + (~$1~.
(30a)
The primes denote differentiation with respect to the normal coordinate Qk. The intensity of a Raman line as indicated by the area A (in arbitrary units) under the line is given by (6) 45af + 7p$ = K2Avk(1 - e-h”c/kBT)/(vo - v)4n, (31) where K is a constant, n the degeneracy of the frequency, and hvkC/kB T the Boltzmann factor. The depolarization is (32) From A and p, it is possible to evaluate a? and p’2 k . a? will have a nonzero value only for A,-type oscillations.
343
RAMAN INTENSITY ANALYSIS OF C6HIZ
In the spectra referred to, p is of the order of 0.1. It does not seem to have been evaluated with sufficient accuracy as for CH and C-C stretching lines 45~‘~ % 7p)k. Hence, we do not propose to use the values of p$ in our formulas for A,,. For Ex species, however, the entire intensity is only due to pt. al;i’and pz for A,, and Eg species of cyclohexane thus calculated are given in Table II. Intensity Formulas The intensity formulas in the matrix form, i=Lil, with I = a; (i, j = x, y, and z> are derived and given in the appendix. C& and /?f, as defined earlier, can be calculated using the values of Z*.
TABLE II a; and 02 Values of &HI2 and C6DIl. [These values were published by R. G. Snyder (J. Mol. Spectrosc. 36,204-22 I, (1970)), but he has not given the values for CH lines. The values for the other lines given here compare well with those given by the above author.]
Species
‘k -1
cm
2852
2.63
2105
0.95
802
0.34
724
0.30
1158
0.07
1015
386
0.00
298
4x4 Of*1g
1465 2 x 2 Of Alg
2895 “k
Species
2080 %;t
cm
4x4ofE
II20
-I
4.13
2200
3.03
2924
3.5
2196
2.10
0.52
375
g
427
E
I x I of
E
I x I of
E
8
g
g
-I
2936
785
2xZof
“k cm
635
1029
1.55
796
0.31
1444
1.84
1213
0.17
I267
1.06
937
0.59
I348
1115
344
RAVINDRANATH,
SWARNAKUMARI.
AND RAJESWARA
RAO
Since a:, + & + a>, = 0 for the Je part of the intensity formulas, it is really necessary to evaluate the derivative of cu,, + aYY+ (Y,, as calculated from Eq (1) defining (Y:,= ; (a: + 2&
(33)
Lyj(= JZG,, L,k + &&Lzk.
(34)
Since for A,, oscillations @‘*are not reliable, we are not using even ai._ formulas derived in the appendix. As a:, = -abY = LY&, and a:= = a& for Eg, p;,’ = 3((Y$ + &)/( = 3(&
+ LYE)&
(35)
The formulas for a:,, & and c$ are taken from the appendix and formulas for /3;’ are calculated. (It may be noted that for Vk= 1267 cm-‘, &, = 0 as it belongs to A2 species of the single group.) Now, the purpose of Raman intensity analysis is to evaluate & , ii;, yI , and y2; r’, = dy,/dd and y; = dy2/dD. Suffix one refers to C-H bond and two refers to C-C bond. As explained in earlier papers, there are three shortcomings in the evaluation of accurate values: (i) L matrix elements which are derived from the force constants are not quite reliable, as the latter are generally not determined with sufficient confidence. (ii) The values of tyi, or /3? obtained from the intensities are in error to the extent of 10% for lines of medium intensity. They can be much less reliable for lines of lower intensities. (iii) ai, has sign ambiguity as it is square root of cut, which is derived from the intensity. We have overcome these difficulties by writing the formulas in the form Z’I = A’LL’A = A’GA. (36) This, at once, solves the above difficulties. Since Z’Z= Cl:‘, the sign does not come into the picture, nor does the inaccuracy in obtaining the intensities of weak lines. L Elements are not used, but we have only one equation for each species while electrooptical parameters are larger in number. Hence, assuming that the eops are the same for the isotopic molecules, we obtain cy;f and @p for CsDr2 also to have the necessary number of equations. Another point that we have to face is that the intensities are relative. The constant K2 is also different for the two molecules. Hence these constants also have to be determined. Hence we take G$ and y2 from the literature and determine the rest from the following six equations obtained from the intensity formulas A.20 to A.28 (given in appendix) and using Eq. (36): 2.0952G’r2+ 0.111018G9 - 0.222036&;&
= 3.04K:;
(1)
1.10416;2 + 0.111018&? - 0.222036&;&
= 1.25K;;
(2)
0.08156~2 + 1.9055257, = 0.71K,;
(3)
0.08156~~ + 1.075927~~ = 0.4OKz;
(4)
12.92~;2 + 0.333054~? + 0.6661 ly’ly; + 0.1524y;yi
+ 11.02~: + 2.537: + 2.375~~~~
- (y; + -yi)(lO.lr,
+ 0.505-r~) = (4.5)(11.64)Kf;
(5)
RAMAN
7.2902~‘; + 0.333054r;
INTENSITY
+ 0.66611y;y;
ANALYSIS
OF ChHlz
345
+ 8.93~: + 0.1524~;~~ + 2.53-(:
+ 2.3757,~~ - (y’, + r;)(lO.lri
+ 0.505rJ
= (4.5)(5.63)K:.
(6)
The values in these equations are shown to 6 figures for computational consistency. The values of K, , K2, y I , (Y; , y ; , and y 5 are obtained by solving the above equations. To get the values of these parameters we took from literature (7) the values of y2 = 1.32 and 5’2 = 1.67; K, = 1.09, Kz = 1.21; yI = 0.35; Cy’,= 1.35 (1.46); 7; = 2.55 (2.63); 7; = 2.44 (2.4). 7; is comparable with 2.5 of Cardona et al. (7). In these formulas, it may be seen that Z;;, 72, and y$ are multiplied by small quantities. Hence, their contributions to intensity are very small, and are of the same order of magnitude as the error involved in the determination of the intensity. Hence, the value of y$ is not reliable. Values of &, yI , and r’, compare well with those determined earlier in the laboratory (4) using the spectra of CH4 and CD4, which are shown in the brackets. (Though the values are shown to the second decimal place, they are reliable only to two figures.) This is a consequence of the fact that the intensity being proportional to the frequency [see Eq (3 l)]. The contribution of the CH lines, in fact, overshadows the contribution of the C-C lines. If one wants to arrive at reliable values for the eops of C-C bond, one has to take a molecule which does not give very high frequencies compared to the C-C stretching frequency. A similar analysis of infrared intensities is possible, by deriving similar formulas, but we are not including these formulas as intensities are not available. APPENDIX:
DERIVATION
OF INTENSITY FORMULAS A SIMPLIFICATION
OF MOLECULES-
We shall briefly recall Long’s (1) derivation of Raman intensity formulas of molecules, Following Wolkenstein and Eliashewich (2), he starts with the simplifying assumption that the polarizability of a molecule is equal to the sum of the polarizabilities of the bonds. Also, each bond is taken to be of cylindrical symmetry. He derives %KY = c [-Y,nxny +
&L,l.
(A.11
where nx and ny are direction cosines of the nth bond; and yn = CY~ - LYE, difference between the polarizabilities of the nth bond along and perpendicular to it. As the molecule is oscillating, the bonds are stretched and variously oriented, producing an overall change in the value of each of the tensor components. He further derives
aa,_ aQ
ah
d
.
a’. *. 2. dS=
ar ap ar as aQ
Jmu’L
(in
matrix
form),
(A.21
where 1 is a row matrix of bond lengths and their direction cosines, and Yinternal coordinates, S symmetry coordinates, and Q normal coordinates. Since I and r have common bond stretches, (A.2) is split into J&L
+ J,K,Au’L.
(A.31
[It may be recalled that originally Long used Cartesian displacements of the atoms, but Ferigle and Weber (8) suggested using p’s, displacements of the atoms.] Now,
346
RAVINDRANATH,
SWARNAKUMARI,
AND
RAJESWARA
RAO
(A.4) The s vectors, introduced by Wilson (9), are characteristic of his normal coordinate treatment of molecules, s/is’ = g (A-5) is inverse kinetic energy matrix. ugu’ = G (A.6) is symmetric and is block-diagonalized. Since r and p are, in general, not equal in number, s is rectangular. Hence, Crawford (10) suggested that
where ELis a diagonal matrix of reciprocals of masses. The J* part of A.3 now becomes J&&dL-‘I.
64.8)
The multiplication of this train of matrices is often quite laborious. Gussoni et al. (II) modified the & matrix to an extended form, but it does not simplify the matters at all. Now, we shall show that, while it is necessary to form these matrices, it is possible to avoid a regular multiplication. We shall start with Hz0 as an example. Raman Intensity Formulas of Hz0 Let el, e2 be the unit vectors along the bonds rl and r2 and e: and e: unit vectors perpendicular to them (Fig. A.l). Then, s matrix is given by Table A.I. Here pI, p2, p3 are displacements of the atoms Hi, HZ, and 0. It is shown that
ad -=-=
r
elcos 0 - e2 rsin 0
as -=-=
e:
qcos 13- el
aP2
r
~PI
e:
rsin8
FIG. A. 1. H,O-internal
(A.9)
’
coordinates.
(A.lO)
RAMAN
INTENSITY
ANALYSIS
OF C6HlZ
347
TABLE A.1 s Matrix for H20
e: and e: are denoted as unit vectors, as suggested by these formulas. Using the symmetry coordinates, us matrix can be written as in Table A.11. Now, we shall construct the J* and K* matrices (Table A.111).The prime on (Y’Sdenotes differentiation with respect to the normal coordinate (Table A.IV). Similar rows can be guessed for ly and 2-r. Ferigle and Weber (8) suggested a way of evaluating the elements of this matrix. dlx -= dPl
dcos @p1 = -sin a,.-_d% = -sin @ret/r,, dPl aPl
(A.1 1)
where Cpiis angle between r1 and the x axis, and &@,/dp, is a vector shown as e; similar to #@pi = e:/r, . It may be appreciated that e: and e_Tare identical, increasing the angles 0 and 4~) respectively. So (A.12) e? = e?. But, while e: increases the angle 8, it decreases a’2 between rz and the x axis, e: = -e-z. Then, using the relations lx = -2x = -sin e/2 and sin 9, = sin $ = Scos 13/2, TABLE A.11 us Matrix of Hz0
(A.13)
348
RAVINDRANATH,
SWARNAKUMARI,
AND RAJESWARA
RAO
TABLE A.111 J+ Matrix of Hz0
0*x*
Ix
2x
lY
2Y
2$x
2y2x
0
0
CL' YY
0
a' XY
,viY
0
my Ylx
Y2Y
the product J@K+can now be written as
[
27 sin 8/2 cos Of2 $ $
iY2 Y Y2X
-(efr+e:) . 1
The row vector in this expression can be recognised as the second row of us. Therefore, the product of this with PS’U becomes the second row of G matrix and its product with G-’ will be equal to the second row of the unit matrix. Hence, the J+ part of A.3 simply becomes y sin BLz,. (A.14) If we regularly multiply all these matrices together we obtain 7
p. sin* B&j’ + $ sin Qu, + (1 - cos f3)~o]L~1
= y sin B(G2,L;’ + G22Li2’) = 7 sin 8L2I. This is known to the investigators and was pointed out by Long in his earlier papers. But, the point we now make out has been possible because of the recognition of the results given by A. 12 and A. 13, helping convert the K+ matrix into the proper form. It may also be seen that the addition of the Ke matrix rows gives a row which coincides with one of the S’U’matrix columns relating to the angles. This addition is facilitated by multiplying with J+ (In the case of infrared intensities J+ is much simpler).
TABLE A.IV k+ Matrix of H20
1x
2x
0
RAMAN INTENSITY ANALYSIS OF C6H,2
G matrix. The present form of the US matrix helps the determination matrix. The following relations are useful in this respect (Fig. A. 1):
349 of the G
el - e2 = cos 0 e:-e:
= -cos 6
e:-e,
= e!*e2 = -sin 8 = e:.el,
Then, G,, = /JH + /~(l + cos 0); G,2 = - sin f$.&; G22 = ( 1/2d2)[pLH+ PO(1 cos f3)]; and G33 = JQ, + PO(1 - cos 8).
We shall now demonstrate this for a molecule of greater complexity CH2C12. This is of C2, symmetry. As shown in Fig. A.2, the pair of C-Cl bonds is in the plane of the paper and the CH bonds are in the perpendicular plane. There are 4Ai-type oscillations, two relating to the symmetric stretching of the bonds and the other two relating to the angle changes. It will be convenient to arrange the symmetry coordinates such that each pair of bonds is associated with change in the angle between them. Thus
FIG. A.2. CH,C12-intemal
coordinates.
350
RAVINDRANATH,
SWARNAKUMARI,
AND
RAJESWARA
RAO
TABLE A.V s’u’ Matrix of CH2C12
9 C
J2e 7F
s-4 = +
(e, + I92+ 83 + 84 + 02).
Then, the USmatrix is written down for this species. It will be convenient to simplify the s vectors using the following relations among them. All the angles are taken to be tetrahedral for simplicity. Thus, it is seen that el + e2 + e3 + e4 = 0; el + ez = (2/&)e,; e: + e: = -&e, + e2) = (-2&/&)e,; e3 + e4 = (-2/&)e,; e’: + ej e,; and e: + e’: + ef = 0 (Table A.V). = -&(e, + e4) = (2&/J?)
FIG. A.3. CHQ-internal
coordinates.
RAMAN
1NTENSlTY
ANALYSIS
351
OF C6H,,
.I+ for cy,, is given by (2yrlz 2yr2z 2~~3~ 2~~42). yl = (Y~- (Y~of CH and y2 is similar for C-CL With a similar procedure followed for H20, J& becomes -sin p/2 cos p/2 5 e:
IL e: d
[
725
3
et:
2
r22-
(e:
+
e:)
+
g
(e:
+
ej) 11
(
Split them into two rows and cast them into
and \iS(sin $(cos
f)yI[O
0 2
g
gz].
The rows coincide with the columns (3) and (4) of s’u’. Hence on substituting for cos P/2 and sin @/2(cos /$2 = I/&; sin 812 = &I&),
For CL??,, J+ becomes [27, ly 2yr2.v 0 01. Since ly = -sin /3/Z = -2y, J&, -V%yj(sin /3/2)(cos P/2)[j$
zi
becomes,
0 0 $$$I.
The row matrix coincides with the 3rd column of s’u’. Hence (A. 15) Similarly. (A. 16) It is satisfactory to see that alrx + c& + akz = 0.
We shall demonstrate this in another molecule, CH3Cl, as a number of new points come into the picture (Fig. A.3). The symmetry coordinates of the A, species are S, = I/&@, + d2 + d,); S2 = D; s3
=
dGP,
matrix can be formed in the The angIes of this molecule are the various vectors are similar to 27, lz 2~~22 2-yr3.z 2~24~. The @I/&, = -sin be:; t34z/ap4 = 0. u’s’
+
02
+
03
-
8,
-
02 -
03).
usual way (Table A.VI). aIso taken to be tetrahedral, and the relations among those discussed for CH2C12, The Ja matrix for (Y,~is K* matrix elements are like 13lz/dp, = -sin /3 X Hence J&, matrix is (since lz = 22 = 32 = -l/3),
352
RAVINDRANATH,
SWARNAKUMARI,
AND RAJESWARA
RAO
TABLE A.VI s’u’ Matrix for CHFI
S1
3
s3
s. ,t
S5a
s4a
Since the row matrix in the brackets coincides with the column under & of s’u’, 4 '
(A.17)
=-Y&3,.
Ly== 3ti
Since this molecule is of CJv symmetry, a:, = c& and (YL + &, + (Y;, = 0. Hence, But, we shall derive the formula for a;,, as it raises some interesting 68 = -l/2&. points. The direction cosines of the bonds are ly = 0; 2y = -\12/3; 3y = l6$; 4y = 0. The J+ matrix is 2x0
(2% 1Y 2~12y 2Y13Y 2~4y) = K
~(0 -1 1 0) a2v
A
apl
~COS~? = ___ aPl
z
-sin
a-
aa
=
-sin
a&.
aPI
Here, we have put 2y = cos +; then sin @ = l/&. Similarly d3y/dp, = -sin +e:. It can be shown that ei* es = -l/2, e; - ei = l/2. J+K+ now becomes
On multiplying this row vector with the third column of s’u’, it is found to be equal to the square of the third column. Then, it can be shown that (A. 18)
RAMAN
INTENSITY ANALYSIS OF C,H,2
353
as expected. It has been possible to deal with a& more directly as the z axis coincides with the symmetry of the molecule and is also along one of the bonds. It will be possible to obtain a similar equation with C& also, but it will be even more indirect than L$ as the direction cosines lx, 2x and 3x do not have a common factor, like in the case of lz, 22, 32, or 1y, 2-v, 3~. The expressions for & and aj3’ show that they are functions of only the third symmetry coordinate, S, . Therefore, one can write
Now
CY,,
=
y,(1z2 + 2z2 + 32*) + 3a2 = y,(cos*P, + cos2p2 + cos*&) + 3cu*. Then
aa,,ap= -.ap
as
-2yl(cos
@,sin PI cos P2sin p2 cos &sin & 0 0 0)~’
(Here u’ is the column matrix of &),
aa,=ap
4
-.-z-.
ap
as
3fi
?‘,
A similar expression can be derived for a&,. Thus in suitable cases, one can avoid even K+ and s’u’ matrices. Now we shall study the s’u’ of the E-type oscillations. It can be shown that es = e: and e; = -e:. Using these values in the K+ matrix, J+K+, after simplification, is obtained as
This, on multiplying with FS’U’,becomes -- 3b4
YdG46
G6
G66)r
which, on multiplying with L-l’, becomes L6r so that 4 Q!;;,a= - 3t/3
ylL6i-
(A.19)
In the case of c& of A,- and E-type oscillations, the row vectors, on multiplication with the last column of s’u’, give a constant times G33 (or G6& indicating that they are geometrically equivalent. Hence, it helps writing down the tensor element directly. This simplification is, however, not possible when the species contains rotation (R, and R, belong to E-type of CH3Ci; for CH2C12, B, contains R, and Bz contains Ry). Their intensity contributions are given by & and LY&.Then, these tensor elements are functions of all the L-” matrix elements (as given in A.8) and to obtain the expressions it is necessary to multiply the train of matrices in a regular way.
354
RAVINDRANATH.
SWARNAKUMARI,
AND RAJESWARA RAO
c6Hl2
Now, we shall pass on to cyclohexane. The s’u’ matrix of the A,, species contains six identical groups, each containing three columns referring to the two hydrogen atoms and one carbon atom of the group. For the Eg species, there are two identical groups [corresponding to a symmetry coordinate like (2d, - C&- d5) and (2d4 - db - dz)]. The S’U’matrices are identical for them. Also, in the group 2dr - d3 - ds , the elements are identical for all the symmetry coordinates except for the coordinates involving 0 and p for which there are small differences. Hence, as an approximation, we have neglected these small differences and have taken the matrix for only one group. There are no such differences in the A Ig group. The matrices are shown in the Table A.VIIa and VIIb The symmetry coordinates of the groups are also given in the Table A.VIII. Now, the G matrix is given by G = us/.~W. It is found that in the A,, species, the G matrix is block-diagonalized into 4 X 4 and 2 X 2. On examining the corresponding symmetry coordinates, we observe that the
TABLE A.VIIa s’u’ of A,, Species of One Group of Cyclohexane
0
TABLE A.VIlb S’u’ of E, Species of One Group of Cyclohexane
n2
0
0
- ef d
h cx -7
f32 d
- e, 7FT
e;:
RAMAN
INTENSITY
ANALYSIS
OF &HI2
355
TABLE A.VIII Symmetry
s1 = /+ S2
s3
S4
Coordinates
(d, + r,)
=
/+
0,
=
/+
‘q+$+6,+62+a)
= &e,
s5
:#G
S6
=
+ D6)
+e* t5] t62 + $6
)
(r, - d,f
+
‘0, +e2 -6, -4’
Symmetry
In S9 and
for Alg Species of one group of C6H,2
5,0
,
Coordinates
for Eg Species of one group of C6H12
B= 0,
6 = 6,
/ 02:
+ 62
group behaves as if it has Cl, symmetry. The 10 coordinates d, , rl , DI , De, 8,, I&, 6,) &. a,, 0, are grouping themselves as 4Ai + A2 + 28, + 2Bz relevant to the CzU symmetry. The six symmetry coordinates of the Ai, species of the molecule (C6Hi2) correspond to 4Ai + 2Bi. In a similar manner, for Eg, the eight symmetry coordinates correspond to 4A1, AZ, 2Bi, and one of the Bl types of C’2”symmetry. The intensity formulas also will be split in the same manner. A,,type oscillations are excited by cu:, and &, = &,. Also, as the J* part follows the rule, CUL+ c& + & = 0, we have derived the formulas only for a:,. &-type oscillations are controlled by ok = -a;,,, (Y& and (Y&.Therefore, though the group shows all the Raman tensor elements, we take only axx, c+, oXz, and (Y,,~, omitting (Y,.and CX,,.(Though the coordinate system is the same for the group as well as the molecule, to differentiate their tensor elements, we use xyz for the group and
356
RAVINDRANATH, SWARNAKUMARI, AND RAJESWARA RAO
XYZ for the molecule.) Both CQ~(or arr) = &/2(& - c&J and LY’Y~ = \j5c& are calculated for 4Ar + 2Br oscillations. A2 is excited only by & = &c&J. The J* part of the intensity formulas are derived by writing down the J*K* matrices for each of the tensor elements. On multiplying with S’U’in the ways explained earlier, the formulas are derived. The J, part is derived following the zero-order approximation, in the way explained in our earlier papers (3, 4). The ultimate formulas (combining J, and J*) are: (i) Polarizability tensor elements for 4 X 4 of A,, species. CC:= Jz(S’lLlj + ijliLzi)
i = 1 to4.
(A.20)
For q there is no J+ part; the J, part is 2 X 4 as there are only two kinds of bonds,
(ii) For 2 X 2 part of A,, species: Here a; = CZk= 0,
(A.22)
i = 5, 6.
(A.23)
i = 1 to 4 and j = 7, 8.
(A.25)
(iii) For 4 X 4 part of the Eg species
(iv) 1 X I part of the Eg species (a%
= [$$+&
+g+y]L;:.
(A.26)
(v) 2 X 2 part of the Eg species 2x! (aLJi=~y;L-ii+~d
&rl
2
-YjPcGY
+
(~+;(;+j&)L$]
(A.27)
(~+;($+;)pc)L&‘]
(A.28)
[
RECEIVED: April 16, 1984
RAMAN INTENSITY ANALYSIS OF CeH,Z
357
REFERENCES D. A. LONG, Proc. Roy. Sot. Lond. 217,203-221 (1953). M. ELIASHEVICHAND WOLKENSTEIN,J. Phys. Moscow 9, 101-326 (1945). G. SWARNAKUMARI AND N. RAJESWARARAO, J. Mol. Specfrosc. 106, 1-l 1 (1984). PRATHIBHANAIK. V. A. PADMA, AND N. RAJESWARARAO, Prumana 13, 111-l 16 (1979). B. S~HRADER AND W. MEIER, “D.M.S. Raman/IR Atlas Organischer Verbindungen of Organic Compounds. Herrausgegeben vom Edited for the Institut fur Spektrochemie und Angerwandte Spektroskopie Dortmund,” Vol. I. Groups (A-E) & Vol. 2 Groups (F-O), Verlag Chemie, Berlin. 6. J. H. BERNSTEINAND G. ALLEN, .J. Opt. Sot. Amer. 45, 237-349 (1955). 7. D. BERMEJO,S. MONTERO, M. CARDONA. AND A. MURAMATSU, Solid State Commun. 42. 153-155
I. 2. 3. 4. 5.
(1982). 8. S. M. FERIGLEAND A. WEBER, Canad. J. Phys. 32,799-807
(1954).
9. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS, “Molecular Vibrations,” McGraw-Hill, New York
1955. 10. B. L. CRAWFORD, J. Chem. Phys. 29, 1042-1045 (1958). 11. M. GUSSONI. S. ABBATE, AND G. ZERBI, J. Raman Spectroscopy, 6, 289-297 (1977).