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Chapter 8 Intensities in Raman spectroscopy
CHAPTER 8
INTENSITIES
IN RAMAN
SPECTROSCOPY
Molecular Polarizability ....................................................................................
190
II.
Intensity of Raman Line ....................................................................................
199
11I.
Raman Intensities and Molecular Symmetry ...................................................... 205
IV.
Resonance Raman Effect ...................................................................................
V.
Experimental Determination of Raman Intensities ............................................. 211
207
A.
Absolute Differential Raman Scattering Cross Section of Nitrogen .......... 212
B.
Differential Raman Scattering Cross Sections of Gaseous Samples .......... 213
189
190
GALABOV AND DUDEV
I. M O L E C U L A R
POLARIZABILITY
When an electric field with strength f is applied to a molecule, an induced dipole ~t is created = ,x f .
(8.0
~t is an important molecular quantity called molecular polarizability. The electrons of the molecule oscillate in accord with the external field applied, thus becoming a source of a secondary radiation which is the scattered radiation. In the general case, the vectors Ix and f do not coincide and Eq. (8.1) can be written in the following form , x = a x x fx + a x Y fv + ~ x z fz
(8.2)
~Y = ~YX fx + ~YV fv + ~ v z fz ~ z = ,xzx fx + a z v fY + a z z fz or
~y
= Otyx
Otyy
Otyz
fy
~z
~zx
~zY
Cxzz
fz
(s.3)
In Eqs. (8.2) and (8.3) IXj and fj (J = X, Y, Z) are the Cartesian components of IX and f, respectively, in a space-fixed coordinate system and tXjK (J, K = X, Y, Z) are the elements of the second rank polarizability tensor or. ct is a symmetrical tensor [CtjK = CtKj (K~J)] and, therefore, has six independent components only. The elements of ct may, in the general case, all be different from zero. The components depend on the space orientation of the molecule but not on the electric field applied. As is known, for each symmetrical tensor a special set of Cartesian axes x', y' and z' exists, such as in respect to which the tensor assumes a unique diagonal form. a acquires the following structure: Otx,x, 0t=/ 0
0 r 0
0 / 0 .
(8.4)
Otz'z'
The induced dipole is parallel to the external field vector. The quantifies ~x'x', Oty,y,and Otz,z, are called principal values of the molecular polarizability, and the axes x', y' and z'
INTENSITIES IN RAMAN SPECTROSCOPY
191
Y
Io
TTT X
/ ~
Direction of propagation
;I•
/••ection
of observation
Fig. 8.1. Principal scheme of scattering experiment
principal polarizability axes. In the general case, ax'x' # Oty,y, ~: Otz,z,. If the molecule possesses isotropic (spherical) polarizability, it is evident that ax,x, = o~y,y,= Otz,z,. For molecules with cylindrical symmetry with z' parallel to the main symmetry axis, Otx,x, = ay,y, ~: Otz,z,. The tensor a is characterized with two invariants with respect to any reorientations of the molecule in space. These are the mean polarizability ~ and the anisotropy 7. and y are defined as = ( a x x + a v y + otzz) / 3
(8.5)
v2 = [ ( a x x - a Y V ) 2 + ( a v Y - a z z ) 2 + ( a z z - a x x ) 2
+ 6 (otXy2 + otXZ2 + a y z 2 ) ] / 2 .
(8.6)
It can easily be seen that ~/vanishes for molecules possessing spherical polarizability. In ordinary scattering experiments a linearly polarized light is usually used to illuminate the sample. The incident beam propagates along one of the laboratory-fixed Cartesian axes (X in our case; Fig. 8.1) and the scattered light is collected in direction perpendicular to that of the incident beam (Z axis). The excitation light may be linearly
192
GALABOV AND DUDEV
polarized along the Y axis. The scattered light may then be depolarized in some extent and contain radiation polarized either parallel (with intensity Ill) or perpendicular (with intensity I_L)with respect to the polarization of the incident beam. Since the intensity of scattered radiation is proportional to the squares of the respective polarizability components [4,5,155,250,251], in accordance with the convention introduced, we have Iii ~ a y y 2 (8.7) Is
~Xy 2 .
Evidently, the intensity of the total scattered light IT = Ill + I L will depend on both a y y and a x y components of the polarizability tensor:
(8.8)
IT -~ (t~Xy2 + t~yy2) .
In gases, the molecules are free and randomly oriented with respect to the space-fixed set of Cartesian axes X, Y and Z. Since all molecules contribute to the intensity of the scattered radiation, it is necessary to average the components of a over all possible orientations of the molecule-fixed axes x, y and z with respect to the inertial (laboratory) Cartesian frame. In analytical form this condition reads [4,5] ajK = ~ ajk tjj tKk. j,k
(8.9)
In Eq. (8.9) tIi (I = X, Y, Z; i = x, y, z) are the direction cosines between the two Cartesian coordinate frames. With respect to the principal axes of the molecule, Eq. (8.9) transforms into (g = x', y' and z') t~jK -- ~ r g
tjg tKg .
(8.10)
Squaring and averaging operations executed upon Eq. (8.10) result in [4]
/2 (OtIK)2 =
tXg tJg tKg (8.11)
= E 0:2 t2g t~g + E egg r g g
tJg tKg tJr tKr.
INTENSITIES IN RAMAN SPECTROSCOPY
193
Since ag remains unchanged under the averaging process, the expressions t2g t ~ and tjg tKg tJr tKr need only be averaged. These are found to be [4] 2g t~gt
{1/5 if J =K = 1/15 if J ~ K
(8.12)
:/15ifJ=K tjg tKg tjr tKr = _ /30 if J r K
(8.13)
Application of Eqs. (8.11), (8.12) and (8.13) to axY and ayy components of the polarizability tensor results in
:(3go+2g
(8.14)
Introducing the invariants of the molecular polarizability ~ and y into Eqs. (8.14) leads to 45 ~2 +4T2 ~ = 45 O~2 = 3,Y2 45 45 ~2 + 7 T2 t~ = 45
(8.15)
An important observation in every scattering experiment is the depolarization ratio p. By definition p is I• P =~Ill
(8.16)
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GALABOV AND DUDEV
Combining Eqs. (8.7), (8.15) and (8.16) yields
P
=
3T 2
(8.17)
45~2 +4.t,2 "
It should be emphasized that the ~2 and ~,2 coefficients in Eq. (8.17) are evaluated for the case of a polyatomic non-linear molecule, linearly polarized incident fight and rectangular experimental setup as shown in Fig. 8.1. Alternative sets of coefficients are to be used for experiments with unpolarized (natural) exciting light and different experimental geometries. These are summarized in Table 8.1. The polarizability terms for 27 most commonly employed scattering geometries [252] are presented. Different coefficients are used in the case of diatomic and linear polyatomic molecules as well [253] 45 ~2 +72 45 iX2 = (3/4)T 2 45
(S.lS)
45 ~2 + (7 / 4)T 2 45 The changes in molecular polarizability during vibrational transitions determine the intensities of Raman lines. The ~ ( polarizability matrix element for a transition from a vibronic state m to a vibronic state n can be presented as follows [4,254-256] (Ct/K)mn = f ~
Ot/K Wm dl;= (n
l ajK I m) (8.19)
h
Ve -Vm - v 0
M(J)em M(K)ne ) . Ve - Vn + v0
In Eq. (8.19) ~m and ~n represent vibronic wave functions of states m and n, respectively, e denotes an intermediate state of the undistorted molecule, v 0 is the frequency of the incident light and h is the Planck's constant. M(J)i 1 and M(K)i I (i, l= m, n, e) are the respective transition moments ( i I ~ I 1 ) and ( i I ~tk I 1 ), with ~ and ~tk the components of the dipole moment operator. It is seen from the above relation that the polarizability of the molecule is frequency dependent through the denominators (% - vm - vo) and (% - v n + v0). It has been shown [255-258] that when the frequency of
INTENSITIES IN RAMAN SPECTROSCOPY
195
TABLE 8.1
Polarizability terms a 2 for the most commonly used scattering geometries a Incident fight Scattered light Polarizability term 2 45 • a i Direction of Direction of Direction of Direction of propagation polarization observation polarization Y X Y 4 5 ~ 2 + 4 y2 X (180 ~)
X
Z
Unpolafized
3~2
Y+Z
45 ~ 2 + 7 72
Y
X
3 ](2
(90 ~
Z
3 y2
X+Z
6 y2
Z
X
3 ](2
(90 ~
Y
45~ 2 + 4 ](2
X+Y
45 ~2 + 7 72
X
Y
3 ](2
(180 ~
Z
4 5 ~ 2 + 4 ](2
Y+Z
4 5 ~ 2 + 7 ~,2
X Z
3 y2 4 5 ~ 2 + 4 y2
X+Z
45 ~ 2 + 7 ](2
Z
X
3 ](2
(90 ~
Y X+Y
3 3,2 6,/2
X
Y
(45 ~2 + 7 ),2)/2
( 180~
Z Y+Z
(45 ~2 + 7 72)/2 45 ~2 + 7 ./2
Y
X
3 ](2
(90 ~
Z
(45~ 2 + 7 ](2)/2
X+Z
(45~ 2 + 13 y2)/2
Z
X
3]( 2
(90 ~
Y
(45~2 + 7 ](2)/2
X+Y
(45~ 2 + 13 ](2)/2
Y (90 ~
X
z
aReprmted from Ref. [252] with permission from Society for Applied Spectroscopy.
196
GALABOV AND DUDEV
exciting light v 0 approaches to, or coincides with, the fxequency of some electronic transition in the molecule, dramatic changes in the magnitude of molecular polarizability, and hence in the intensity of Raman lines, occur. The phenomena are known as preresonance and resonance Raman effects, respectively. The dependence of (aJK)nm on v0 becomes small and can be neglected in the case of ordinary (far-from-resonance) Raman experiment when the following conditions are met: (1) the electronic ground state of the molecule is nondegenerate, and (2) the frequency of exciting radiation v 0 exceeds by far the frequency of the Raman shift Vnm. At the same time v0 must be small as compared with any electronic absorption frequency of the molecule. These conditions are known as Placzek's conditions and in case these are satisfied, the Born-Oppenheimer approximation can be used to justify the separation of the vibronic wave function ~Pi =~F~ o~i vr .
(8.20)
We is the electronic wave function of the ith state and Wvr is the wave function describing the rotovibrational motion for state i. Within the above approximation the electronic wave functions can be excluded from further consideration. Thus, the rotovibrational wave functions of the electronic ground state will only be treated. The left-hand side of Eq. (8.19) assumes the form (O~JK)nm = [ (Wvr) * ~jK Wvr d'l:vr.
Rotational and vibrational distortions in the molecule can also be treated separately which leads to a fitnher splitting of the wave functions: Wvr = ~i v .~Fir '
(8.22)
where superscripts v and r stand for vibrational and rotational wave functions, respectively. Eq. (8.21) transforms into (Ot/K)mn =~ (~Fn)* ( ~ r ) * ( Z j K ~ v ~ F r d f f d z r .
(8.23)
In order to assess the contributions of the two wave functions ~i v and ~Fir to the intensity of the Raman line, a molecule-fLxed Cartesian coordinate system (x, y, z) which is able to rotate together with the molecule is necessary to introduce. The CqK can then be expressed in terms of polarizability tensor components in the molecule-fixed coordinate
INTENSITIES IN RAMAN SPECTROSCOPY
197
system and the respective direction cosines between the two Cartesian coordinate systems. Combination of Eqs. (8.9) and (8.23) leads to (OqK)mn = ~ ~ (~FV)* (~Fr) * Otjk tjj tKk Wv ~Fr dxV dxr j,k (8.24) =Z ~ j,k
(~r),
tjj tKk ~ r dxr I
(,,.i.,#),O~jk,.i.,vdxV
Thus, it is clear that the polarizability of a molecule, and hence the intensity of a Raman line, depend on the properties of the two integrals appearing in Eq. (8.24). These must not be zero if the transition m ~ n is to be Raman active. Examination of the first integral leads to the rotational selection rules which govern the appearance of rotational lines in the Raman spectra. These are discussed in detail elsewhere [4]. The properties of the vibrational integral
(".~")m,, =S (,IV)* '~k"I": <,.,:v
(8.25)
will be considered. For small molecular vibrations, and in case the Placzek's conditions (1) and (2) are satisfied, the polarizability of the molecule can be expanded in a Taylor series along the normal coordinates Qi
0~=0~ O+
~r
~176 0 Qr
~r
Z r,s
0
Qr Qs (8.26)
+ -- ~ ' ' Qr esQt + . . . . 6 r,s,t OQr~Qs~Qt 0 Here ot0 is the polarizability tensor in an equilibrium non-perturbed state and the subscript o in the polarizability derivative terms means that these are taken at the equilibrium geometry of the molecule. Combining Eqs. (8.25) and (8.26) and neglecting higher terms in the expansion (8.26), the following expression is obtained for the ith normal mode
(",~)~-(",~)o s (~v), ~v ~v (8.27)
+(c)(Zjk/c)Qi)0 S (~Fv)* Qi ~v dl:V.
198
GALABOV AND DUDEV
The vibrational wave functions ~Pmv and ~pv can be presented as a product of the wave functions of the individual normal coordinates tpv =WVl (QI) ~Pv2 (Q2) --. ~PV(3N_6)(Q3N-6)
(8.28)
~pv =~PnVl(Q1) ~PnV2(Q2) "--~Fn(3N_6)(Q3N-6) 9
(8.29)
Eq. (8.27) transforms to (O~Jk)mn = (O~jk)o {f [~nVl(Q1)]* tpv, (Q1) dQ1--X f [~pv (Q1)]* ~Pv i (Qi)dQi'" "}
(8.30) + (~O~jk/OQi )o {~ [Wnvl(QI)]* w v 1 (Q1)dQ1 "'" •
[~Pm (Qi)]* QiWVi(Qi)dQi-- } 9
The vibrational quantum numbers associated with vibrational levels mi and ni (i = 1 to 3N6, where'N is the number of atoms in the molecule) may be denoted as v mi and v ai, respectively. The first term in the fight-hand side of Eq. (8.30) is zero unless vml = v nl, v m2 = vn2, . . ., vmi = v n i , . . . , etc. because of the orthogonality of vibrational wave functions. In other words, this term does not vanish ordy for elastic light scattering when the molecule does not change its vibrational state. The first term in Eq. (8.30) is, therefore, responsible for the Rayleigh scattering. The second term that gives rise to an inelastic scattering will not vanish if: (a) the polarizability derivative (a~k//gQi)o is not zero (this condition requires ~k to change when the molecule undergoes a particular vibrational transition), and (b) the integral is not zero. In the framework of the harmonic approximation, the last condition is satisfied if the vibrational quantum numbers of the ith transition v mi and v ni only differ by a factor of +1 Av = v m i - v ni = +1 .
(8.31)
All other vibrational quantum numbers vml, vial, vm2, v n 2 , . . . , vm(i-l), vn(i-l), vm(i+l), vn(i+l), . . . , etc. must not change. Under far-from-resonance conditions there are six independent elements of the polarizability tensor: axx, axy, axz, ayy, O~yz and CZzz. Therefore, six equations analogous to Eq. (8.30) exist for each normal mode i. At least one of the six (CZjl0mn(j, k = x, y, z) components must be non-zero if the ith vibration is to
INTENSITIES IN RAMAN SPECTROSCOPY
199
be Raman active. According to Eq. (8.31) fundamental transitions between vibrational energy levels are only allowed. Due to anhannonicity effects, overtone and combination bands appear in the Raman spectra, though typically with lower intensity. In analyzing the anharmonicity effects, higher terms in the Taylor series expansion [Eq. (8.26)] need to be considered. The selection rule (8.31) was discussed in detail in Chapter 1 and we will not elaborate further here. The symmetry selection role will be discussed in Section 8.3. If the vibrational wave functions qJmiV(Qi) and q~niV(Qi) are substituted with the respective analytical expressions, the relation given by Eq. (1.29) (Chapter 1) is obtained. The first integral in Eq. (8.30) does not contribute to the intensity of a Raman band. The change in Ctjk due to vibrational transition i can, then, be expressed as (Ctjl0i = (0Ctjk/c3Qi)0 [b 2 (vi + 1)189,
(8.32)
bE=
(8.33)
where
h 8g2toi
is the zero-point amplitude.
II. I N T E N S I T Y O F R A M A N L I N E The intensity of ith Stokes-shifted Raman line is given by the relation [250,255, 259] 27x5 )4 2 Ii = c------32 ~- I0 Nvi (vo - v i X (~Jk)i j,k
(8.34)
where I0 and v 0 are the intensity and frequency of the incident light, respectively, v i is the vibrational frequency, c is the velocity of light, and Nvi is the number of molecules in vibrational state with quantum number vi. By combining Eqs (8.32) and (8.34) the following expression is obtained Ii = 27/1:5 32c4 I o ( v o - v i )4 Nvi (vi+ 1) b 29 X (c)O~jk / t)Qi)20 9 j,k
(8.35)
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GALABOV AND DUDEV
The selection rule for Raman intensities (b) discussed in the previous section requires that the vibrational quantum number changes by + 1 for the Stokes lines in the Raman spectra. This is, however, true for the fundamental lines, corresponding to transition 0--->1, and for the "hot bands" related to transitions of the type 1-->2, 2---~3, etc. These appear at the same frequency in the spectra as the fundamental bands. Thus, the frequencies associated with fundamental and "hot band" transitions are hardly discernible. All such transitions contribute to the intensity of a Raman band. A summation over all vibrational quantum numbers v i from zero to infinity is needed to account for all contributions to the intensity Ii. Following the considerations given in Section 1.1 [Eqs. (1.31) and (1.39)], the expression is obtained
Z N vi(Vi + 1 ) = ~ -i vi=0 vi=O
e(-hvi/kT)vi (vi + 1) (8.36)
N 1_ e-hvi/kT ' where N is the total number of scattering molecules per unit volume. Therefore, 27/I; 5 Ii=
)4
32C4 IoN(v0-v i
9b 2 1-
2
e_-~i/k T ~ (~0tjk/~Qi)0 . j,k
(8.37)
This equation can also be written in the following form Ii = I0 N o i ,
(8.38)
where 27~ 5 )4 b2 ffi = ~c--"--'~-32(v0 - vi 1 e-hVi/kT X (~~ j,k
2 (8.39)
-
is the total Raman scattering cross section for full solid angle 4~. If wave numbers are used instead of frequencies, Eq. (8.39) transforms into
ffi :
27/1:5 )4 32 (V0 -Vi
b2 2 - hc/kT X (OO~jk/oQi)0, 1 - e -vi j,k
(8.40)
where v0 and~i are the wave numbers of the incident light and of the vibrational transition, respectively. Since the number of scattering molecules N and the intensity of
INTENSITIES IN RAMAN SPECTROSCOPY
201
irradiation Io are quantifies that are difficult to measure experimentally, o i appears more appropriately observable than the Raman intensity Ii. The polarizability derivatives c3Ctjk/0Qiappearing in Eqs. (8.35), (8.37), (8.39) and (8.40) form a three dimensional tensor of size 3•215 For the sake of simplicity i t can be presented in rectangular form. Thus, the supertensor CtQ can be expressed as:
O~Q =
(
~)(Zxx/OQI OCtxy/()QI OCtxz/OQ1 ... ()~yy/t)Ql bCtyz/c)Q1 ... \symmetrical bazz / bQ1 ... (8.41) OCtxx / ~)Q3N-6 ~)Ctxy/ ~)Q3N-6 ~)(Zxz/ ~Q3N-6"~ ~O~yy/OQ3N_6 ~Ctyz/~Q3N_6] 9 symmetrical 3azz / ~)Q3N-6
The last term in Eq. (8.40) containing elements of the supertensor aQ Can be expressed in terms of the invariants of the molecular polarizability ~ and ), with respect to normal coordinates: --' (ti
=
~
bet
bQi
=
-
l(bax x Oayy 0azz )
3 [ ~)Qi + ~)Qi
+
'
(8.42)
~)Qi
1 I(3axx t~Qi
~,~i
OQi
~.~-"Qi
+6 /)Qi
~)Qi )
OQi
21}
(8.43)
"
Unlike the equilibrium mean polarizabilities ~0 which are non-zero quantifies, their derivatives with respect to normal coordinates may equal zero. These quantifies vanish for non-totally symmetric vibrations. In such cases the anisotropic part of the tensor tZQ contributes only to the intensity of the Raman line. Introducing ~ and "/i' as well as the degree of degeneracy gi in Eq. (8.40) leads to [2601 27~ 5 )4 b2 [ 2 ,)2] t~i = 32 (V0-~i 1- e_~ihc/kT gi 3(~[) 2 +-~0'i 9
(8.44)
202
GALABOV AND DUDEV
Thus, the total Raman scattering cross section of the Stokes-shifted line into a full solid angle 4n is expressed in terms of derivatives of the molecular polarizability invariants with respect to normal vibrational coordinates. Only a portion of the scattered by the sample light is, however, collected. Usually the scattered radiation is detected into a fixed direction with a narrow collection angle. Another quantity, the differential Raman scattering cross section into a given direction (dt~/d~)i, is employed in Raman intensity measurements [254,260]. If the rectangular experimental setup, as shown in Fig. 8.1, is employed, the differential Raman scattering cross section is given by [260]
"~ i
24454 ( v 0 - v i ,4 l_e_~ihc/k T gi E45(~[ ,2 +7 (121 Ti -
(8.45)
The quantity (da/d~)i is called absolute differential Raman scattering cross section. It is often termed absolute Raman intensity as well. From Eq. (8.45) it is clear that (do/d.O)i depends on several factors such as ~0 and ~i and the absolute temperature T. In order to operate with comparable quantifies that are independent from the experimental conditions, the so-called "standard intensity" or "scattering coefficient" Si is commonly used [73,253,260-263] Si = gi [ 4 5 ( ~ ) 2+ 7(Ti')21 -
(8.46)
Eq. (8.45) assumes the following form
do) = "~ i
b2 24 g4 )4 Si " 4-'--~ (T0 - v i 1_ e_Vi-hc/kT
(8.47)
Another experimental observable in the Rarnan intensity experiment is the depolarization ratio Pi of the vibrational line 3(~) 2 Pi
4 5 ( ~ ) 2+4(T~) 2 -
(8.48/
Pi provides a unique possibility for examining the symmetry of vibrational transitions. Since ~ vanishes for distortions belonging to a non-fully symmetric species, it easily can be realized that Pmax = 3/4. Depolarization ratio assumes its minimum value for the totally-symmetric vibrations of spherical top molecules. In these cases the Raman line is fully polarized and Ti' and Pi are zero.
INTENSITIES IN RAMAN SPECTROSCOPY
203
By combining Eqs. (8.46) and (8.48) expressions for the absolute values of ~ and ~'i' are obtained: 1
I :1=
Si0-40i)] 1/2
3g (iu
SiPi 11/2 ['f:[=
3gi(l+pi)J
(8.49)
(8.50)
As in the case of dipole moment derivatives with respect to normal coordinates, absolute values of ~ and )'i' only can be evaluated from the experiment. Extended basis set ab mitio MO calculations of polarizability derivatives are usually employed in solving the sign ambiguity problem. Eq. (8.45) expresses the dependence of the absolute Raman differential cross section on ( ~ ) 2 and (~'i')2. As already mentioned, the scattered light consists of two components: III -~ 4 5 ( ~ ) 2 + 40,i')2
(8.51)
I_1_-~ 3('/i')2 .
(8.52)
and
These quantifies can be measured separately. Thus, the so-called trace and anisotropy spectra can be obtained from experiment [264]. The trace spectra are, as a rule, simpler and are due to fully symmetric vibrational transitions only. The lines are narrow, well separated and easy to analyze. Anisotropy spectra originate from non-totally symmetric vibrational distortions in the molecule and usually represent a complicated superposition of strongly overlapped bands. The interpretation of spectra in the gas-phase is also hampered by the presence of a complex rotational structure of the vibrational bands. Trace (It) and anisotropy (Ia) spectra can be obtained after careful polarization experiments using the following equations [264]: It = Ill - (4/3) I_L
(8.53)
Ia = (7/3) l_t- .
(8.54)
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GALABOV AND DUDEV
Thus, the respective absolute trace and anisotropy Raman differential cross sections are functions of a single polarizability invariant derivative only (-~)
=2484 ( V o - v i ) 4 i
b2 (~)2 1 - e -'~i'ihe/kT
(8.55)
drya ) = 7 2484 (Vo-Vi) 4 b? )2 d~Ji 4-5 1_ e-~i hc/kT gi()'~ "
(8.56)
Equations (8.45), (8.55) and (8.56) are derived for the CGS system of units" (d6/d.Q)i is expressed in cm2/sr, the molecular polarizability ct in cm 3, 0~OQi in cm2/g 89 and the scattering coefficients Si in cm4/g. Martin and Montero [263] first suggested using SI units in Raman intensity measurements. In adapting Eq. (8.45) to the new system of units the factor (47re0)2 has to be introduced in the denominator of the equation where to = 8.8542• C V -1 m -1 is the permitivity of vacuum (C - Colomb, V - Volt). Following the considerations given in Ref. [263] Eq. (8.45) can be further simplified if a dimensionless normal coordinate qi instead of the mass-weighted Qi is used [263] (8.57)
qi = (47t 2 c ~ i / h ) 8 9
where ~i is the harmonic vibrational wavenumber. Eq. (8.45) transforms into [263]
(do/
82
(~0 -~i) 4
[45(~)2 + 7(~/~)2
" ~ i =9~"~'2 1-e_~i-ffc~ T gi
].
(8.58)
In Eq. (8.58) (da/d.Q)i is expressed in m2/sr, ct and t3ntx/aqin in C V -1 m 2 and Si in C 2 V -2 m 4. The conversion factors between different unit systems for these quantifies are as follows 1 cm2/sr = 10--4 m2/sr 1 ~3 = 10-24 cm 3 = 1.112644• 10--40 C V -1 m 2 .
(8.59)
The invariants ~ and ~i' represent combinations between the six independent components a~k/0Qi (j, k = x, y, z) of the polarizability tensor t:tQ [Eqs. (8.42) and (8.43)]. These quantifies can be determined from experiment for very few small and highly symmetric molecules only. Typically, the number of intensity parameters to be evaluated exceeds by far the number of experimental observations. The XY 2 bent
INTENSITIES IN RAMAN SPECTROSCOPY
205
molecule is a good example in this respect. It has three normal vibrations, both infrared and Raman active, grouped as follows: FV = 2A 1 + B 2.
(8.60)
Suppose the molecule is oriented along its principal axes with x axis coinciding with the C 2 symmetry axis and z axis lying in the plane of the molecule. The supertensor CtQ has then the following structure:
Q1 o tZQ=
0
Q2 o
o
o
o
1 {Xzz
0
d.r
0
Q3 o
o
o
o
o
o
o.
2 Otzz
ct 3
0
0
(8.61)
~ik (i = 1, 2, 3; j, k = x, y, z) denotes the derivatives of the polarizability components ff-jk with respect to the ith normal mode. Since for such a molecule r • ctlyy ~ Ctlzz and O~Zxx ~ r , ~2zz, the number of independent polarizability derivatives is 7. The number of experimental quantifies, however, is just 5 ( ~ , ~'1', ~ , ~'2', ~'3'). It is evident that even for such simple and symmetric molecules the analysis meets serious challenges. In comparison, infi'ared intensities for C2v molecules can, in principle, be transformed into dipole moment derivatives since there is only one non-zero component of the dipole moment derivative vector for each normal mode. The great complexity in acquiring experimental polarizability derivatives as compared with dipole moment derivatives is probably the reason for the relatively limited use of Raman intensities in analyzing molecular properties.
III. R A M A N I N T E N S I T I E S A N D M O L E C U L A R
SYMMETRY
According to the harmonic oscillator selection rule, as briefly discussed in Section 8.1, fundamental transitions are only allowed in Raman spectra. In practice, it means that the fundamental lines will appear with higher intensity than the overtone and combination lines. In this section the relationship between molecular symmetry and the intensity of the Raman line will be considered. Eq. (8.25) shows that the intensity of each fundamental transition in the Raman spectrum is governed by integrals of the following type
206
GALABOV AND DUDEV
( n I ajk I m )
(j, k =x, y, z) .
(8.62)
Since, in the ordinary Raman experiment, the polarizability tensor has six independent components, six integrals with such a structure exist ( n l CXxxI m )
(nl~lm)
( n [ cxzzl m )
( n l ~ y I m>
(nl~lm>
( n I Cry, I m>
.
(8.63)
A vibrational transition is allowed if at least one of these integrals differs from zero. The symmetry selection rule states that the integral ( n I ~k I m ) is not zero if the direct product of representations for the ith vibrational mode F i = F n x F~k • F m
(j, k =x, y, z)
(8.64)
belongs to the fully symmetric species of the molecular point group. Since the ground state wave function is always fully symmetric, the behavior of the product F i will depend on the symmetry properties of the polarizability tensor components and of the vibrational wave function of the first excited state. Hence, if that wave function belongs to the same symmetry species as the respective polarizability component, the direct product F i will be fully symmetric. In this case, the ith transition will be Raman active. An interesting case are molecules possessing a center of symmetry. According to the symmetry selection rule, a given vibrational transition is infrared active ff the direct product between the two vibrational wave function representations and that of the dipole moment vector component is totally syaunetric (Section 1.2.2). The ground state vibrational wave function, as mentioned, is always totally symmetric. It belongs to even or gerade representation since it is not affected by inversion operation with respect to the center of symmetry. The dipole moment vector, however, changes sign when the inversion operation is executed. It, consequently, belongs to odd or ungerade representation. Thus, it is easy to deduce that the representation of the first vibrational excited state wave function must be odd (ungerade)(belonging to A u, B u, etc. synunetry species) if the fundamental transition is to be infrared active. The molecular polarizability tensor ct is defined in relation with two vectors: the induced dipole ~t and the electric field vector f [Eq. (8.1)]. Hence, the representation of ct can be expressed as a direct product of the representations of the respective vectors [265] F a = F~t • I f .
(8.65)
INTENSITIES IN RAMAN SPECTROSCOPY
207
Since the inversion operation alters the sign of the two vectors, according to Eq. (8.65), ct remains unaffected. It, therefore, belongs to an even symmetry species. It is evident that the excited state wave function must possess the same symmetry properties as ct if the transition is to be Raman active. Thus, for molecules with center of synanetry fundamental transitions to excited states belonging to even syrmnetry species (Ag, Bg, etc.) are only active. The above discussion outlines the so-called rule of mutual exclusion in vibrational spectroscopy which reads: for molecules with a center of synanetry vibrations that are Raman active are infrared inactive and v i c e v e r s a .
IV. R E S O N A N C E
RAMAN EFFECT
The relations and conclusions drawn in the previous sections were deduced for the non-resonance Raman experiment performed with frequency of the exciting light laying far from the frequency of any electronic transition in the molecule. In such a case the differential Raman scattering cross section (do/dD)i depends on the incident light frequency Vo through the term (v0 - v i ) 4 [Eqs. (8.45) and (8.58)]. If the Placzek's conditions (section 8.1) are satisfied, the dependence of molecular polarizability on Vo is negligible. In the treatment presented in the preceding sections, vibrational wave functions of the ground electronic state were only considered. If, however, the excitation radiation frequency approaches to Ore-resonance conditions) or coincides with (resonance conditions) the frequency of some electronic transition in the molecule, the application of the ground-state approach is not justifiable any more. The dependence of molecular polarizability on the incident light frequency has to be taken into account. As was already mentioned, under pre-resonance and resonance conditions some Raman lines are enhanced hundreds and thousands of times. In this section a brief outline of the basic theory of resonance Raman effect is presented. Before proceeding further, it is necessary to introduce the following notation: (a) the ground electronic state is denoted by g; vibrational levels belonging to this state are designated as i (initial vibrational state) and f (final vibrational state). The respective vibronic wave functions will be labeled as gi and gf; (b) the higher-energy electronic levels assume symbols r s and t; transitions g--,e and g--,s are considered allowed and g--,t - forbidden; and (c) vibrational levels belonging to e, s and t are denoted by v, I and u, respectively. In the subsequent derivations all electronic and vibrational states are considered as non-degenerate.
208
GALABOV AND DUDEV
The theory of intensities of resonantly-enhanced Raman lines is based on the Kramers-Heisenberg-Dirac dispersion equation [266,267]. The problem is analyzed in terms of vibronic interactions in the molecule. The JKth matrix component of the molecular polarizability for the gi~gf transition in the Raman spectrum can be expressed as follows [256,257]
(0:JK)gi,gf= E / evg:gi
Eev-Egi-E0 (8.66)
(evlMj ] E[g- i)eEgf (gflvM - E0 Klev)
"
Mj and M K are the respective dipole moment operators and E 0 is the excitation light energy. In resonance conditions the first term in Eq. (8.66) becomes dominant since the denominator (Eev - Egi - E0) rapidly decreases. In this case, a phenomenological damping constant F is introduced in the denominator expression. Thus, retaining the resonant term in Eq. (8.66) only, we obtain
( .(ev[M-K!gi)(-gflMJ lev)) . )gi,gf = evg:gi E ~, Eev - Egi - E0 + iF
(8.67)
This equation can be further developed if the electronic wave function is expanded in a Taylor series along the normal coordinates Qa of the molecule. This is known as the Herzberg-Teller expansion [268] and its application to Eq. (8.67) yields [255,256]
(ctji0gi,gf = A + B + C ,
(8.68)
A = ~ ~ (Mj)ge ( MK)eg (gf[ ev) (ev Igi) e~g v Eev - Egi - E0 + iF
(8.69)
where
INTENSITIES IN RA_MAN SPECTROSCOPY
X X
E
209
(gflQ~lev) (ev Igi)
~., ( (MJ)gshae(MK)eq
e~g v s~e a
E e - Es
Eev - Egi - E0 + iF (8.70)
(Mj)ge heas(MK)sg
E e ~ Es
C
_
Eev - Egi - E0 + iF
(M j )ge (M K )et h~g
(g~Qalev) (ev[gi)
Eg-Et
Eev - Egi - E0 + iF
_
~ ~v ~a e~g tag
(gfl ev)(eqQalgi) /
J
(8.71)
h~ (Mj)te (M K )eg
(gfl ev)(eqQ~lgi) / i
Eg - E t
Eev - Egi - E0 + iF
J
In these equations (Mp)s~ = [~0] Mpl e0] (p = J, K; ~, ~ = g, e, s, t) are the electric transition dipole moments at the equilibrium molecular geometry and lk~a= [~0[ (aH/0Qa)ol so]. (aH/SQa)0 is the vibronic coupling operator for the normal mode a. H is the electronic Hamiltonian of the molecule. The contribution of each term A, B and C to the intensity of a Raman line is considered as follows. A term. One excited electronic state e is included in this term. With v 0 approaching v e the Raman line associated with a totally symmetric mode derives its intensity from this term through the Franck-Condon mechanism. This term does not depend on vibronic mixing of state e with other electronic states of the molecule. If the two electronic states g and e lie far below the next electronic levels and, hence, the probability of vibronic coupling between state e and the other excited electronic states is very low, the A term dominates in magnitude over the other two terms in Eq. (8.68). B term. This term comprises allowed transitions between the ground electronic state and two electronic states e and s (E e < Es). It determines to a great extent the intensity of a Raman line associated with a non-fully symmetric transition. In analyzing the properties of the B term the role of vibronic coupling in generating Raman intensifies becomes clear. Analysis shows that those normal vibrations which can couple two excited electronic states (e and s) undergo a striking intensity enhancement if the frequency of incident radiation is tuned on the frequency of the lowest electronic transition g---~. The nearer the state s to state e, the more pronounced is the effect of interaction. C term. It considers transitions to two electronic states e and t. Transition g~t is
not necessarily allowed. Since, in the general case, E e - E s < E g - E t, the magnitude of
210
GALABOV AND DUDEV
TABLE 8.2 Intensities of some Raman lines of pyrazine relative to the intensity of 940 cm-1 Raman line of benzene-d6 obtained at different excitation fight wavelengths (Reprinted from Ref. [274] with permission) Relative intensity a
Excitation wavelength
(in.m)
703cm-1 v 4 (b2g)
754cm- 1 v5 (b2g)
925cm -1 Vl0a (big)
514.5
0.38
0.041
0.046
457.9
0.47
0.082
0.12
363.8
0.57
0.20
1.14
aIn liquid phase.
this term will be small compared with the B term [Eqs. (8.70) and (8.71)]. Therefore, the contribution of C term to the overall intensity of the Raman line is not expected to be significant. This term is important for molecules possessing low-lying forbidden electronic states. Eqs. (8.68) through (8.71) were first derived by Albrecht [255,256]. His predictions have proved correct in many cases [269-277]. An example will demonstrate the role of vibronic coupling in generating resonance Raman intensities as derived in Albrecht's theory 03 term). It has been shown in a series of publications of Ito et al. [270, 271,274] that the Vl0a non-totally symmetric vibration of pyrazine (C-H out-of-plane bending) demonstrates a remarkable enhancement in Raman spectrum when the exciting radiation frequency approaches the frequency of the lowest-lying electronic transition at 323 nm. The Vl0a vibrational transition belongs to blg symmetry species and appears at the 925 cm-1 in the Raman spectnma. Results presented in Table 8.2 illustrate quite clearly the resonance effect on the intensity of this line. Analysis of the data accumulated reveals that the coupling between the lowest-lying allowed electronic state IB3u (n--+n*) and the next excited electronic state IB2u (rc---~*) through non-fully syuunetric big vibrational mode is responsible for the enhancement of 925 cm-1 Raman line. Low energy separation between the two excited electronic states, as seen from Fig. 8.2, favors this process. Vibronic coupling between IB3u and the two 1Blu electronic states will be less effective due to the higher energy separation. Therefore, vibrational modes (v 4 and v5) that are able to couple these electronic states are not expected to be significantly enhanced. The data collected in Table 8.2 illustrate this conclusion. Recently Okamoto [257] evaluated additional second-order terms to Eq. (8.68), thus expanding the applicability of Albrecht's theory. The emphasis is laid on the role of forbidden electronic transitions in generating resonance Raman intensities. It has been shown that: (1) the Raman line can derive intensity from a forbidden electronic transition
INTENSITIES IN RAMAN SPECTROSCOPY
I
211
60700
IBlu (n-~r*), IB2u (~-->x*)
50900
1Blu (n--~r*)
37800
1B2u (a:-~*)
30900
1B3u (n-~x*)
z:
0
Ag
Fig. 8.2. Energy diagram of the low-lying electronic states of pyrazine.
even if the excitation radiation is tuned in resonance with an allowed electronic transition, and (2) enhancement may be detected if the excitation is in resonance with a vibronically allowed, though electronically forbidden, transition.
V. E X P E R I M E N T A L D E T E R M I N A T I O N OF RAMAN INTENSITIES Eq. (8.45) shows that for an ordinary Raman experiment the absolute differential Raman scattering cross sections can be expressed in terms of derivatives of the molecular polarizability invariants ~ and V with respect to normal coordinates. These derivatives contain valuable information about the variation of molecular polarizability with vibrational motion. Gas-phase Raman scattering cross sections are most suited for intensity analysis since at low partial pressure of the sampling gas these quantities are not influenced by effects of intermolecular interactions, thus reflecting properties of individual molecules.
212
G.M.,ABOV AND DUDEV
Direct determination of absolute Raman differential cross sections is quite difficult and tedious work often leading to incorrect results. It is easier to measure cross sections relative to some standard. The absolute differential Raman scattering cross sections of the sample can then be straightforwardly obtained.
A. Absolute Differential Raman Scattering Cross Section of Nitrogen The absolute scattering cross section of the Q-branch of the Raman band of nitrogen at 2331 cm-1 has been chosen as a standard in Raman intensity experiments. The special role of this molecule in gas-phase Raman spectroscopy is based on several reasons: (1) nitrogen is a relatively inert gas and does not react with the gas sample; (2) it quickly forms mixtures with the sampling gas; (3) the region of the vibrational specmma where the nitrogen Raman line appears is very low populated and, for many gases, this line does not overlap with sample gas lines; and (4) since the absorption band of nitrogen lies in the far ultraviolet, laser excitation beams with wavelengths corresponding to near ultraviolet or visible light do not cause resonance enhancement of the nitrogen Raman line. Its intensity depends on the excitation frequency through the term (V 0 - v i ) 4 only. Thus, the differential scattering cross section of nitrogen can be used as a standard for a wide range of excitation laser lines. The absolute differential scattering cross section of the standard needs to be determined as precisely as possible. A number of measurements has been performed over the past forty years [260,278-287]. The introduction of lasers in Raman spectroscopy and of computer processing of spectral data has improved highly the accuracy of Raman intensity measurements. As a result, the absolute differential Raman scattering cross section of nitrogen reported from different laboratories deviates within a few percent only [260,284,285,287]. One of the possible approaches in determining (dtr/d.Q)Q,N2 is to use the absolute differential cross section of the strongest purely rotational Raman line J = 1--->3 (J the rotational quantum number) of hydrogen at 587 em-1 as a standard. It is given by [260] __~] rot,H2
= 24 g4 )4 3(J + 1)(J + 2) 7,y20 4---~ (~0 -Vrot 2(2J + 3)(2J + 1) "
(8.72)
70 is the anisotropy at the equilibrium geometry. The rectangular experimental setup, as shown in Fig. 8.1, is considered. Both experiment and theory have provided a reliable value for the hydrogen anisotropy. Moreover, its dependence on the excitation wavelength has been thoroughly established. As a result, the absolute differential cross section of the hydrogen rotational line (do/d~)rot, H2 has been accurately determined. The estimated value has been employed in determining the absolute differential cross
INTENSITIES IN RAMAN SPECTROSCOPY
213
section of the Q-branch of the vibrational Raman line of nitrogen. The following value has been obtained [260] (do/d.Q)Q,N2 = (5.05 4- 0.1)x 10-48 (~0 - 233 lcm-l) 4 cm6 sr-1.
(8.73)
It can be applied for a wide range of laser excitation wavelengths coveting the visible as well as the near ultraviolet up to 330 nm [260].
B. Differential Raman Scattering Cross Sections of Gaseous Samples The differential Raman scattering cross section of the ith line of a gas sample relative to that of the 2331 cm -1 line of nitrogen is given by [260] (do
/ d.Q)i
(v0 - vi )4
2331 cm -1
(~0 - 2331 cm-1)4
Vi (1- e-~ihe/kT)
~_
(do / m)Q,N 2
(8.74) gi [45(~)2 + 7gi ('Y[)2 ]
I45 (~)22
+7ZN2 (Ylq2
)2 I
"
In Eq. (8.74) li is the portion of the anisotropic scattering localized in the Q-branch of the respective Raman line. Its value has been determined for a number of molecules. In the case of linear molecules with small rotational constants gi is 0.25 [280,281]. Since the Bolzmann factor for the nitrogen molecule is very small at room temperature, it has been neglected in deriving Eq. (8.74). It is seen fxom the above expression that the relative differential scattering cross section depends on the excitation light wavenumber and on the absolute temperature. To make the measured quantifies comparable, a relative normalized differential Raman scattering cross section Ei has been defmed [260] (d~ / d'Q)i
.
(~0 -Vi) -4
Ei = (do / dn)Q,N2 (% _ 2331 r
(1-e-Vihr (8.75)
2331 cm-1 Vi
gi [45(~) 2 + 7gi (,y~)2] 7%N2('~N2
]
214
GALABOV AND DUDEV
The integrated intensity of the ith Raman line of a gaseous sample for a rectangular experimental setup can be expressed as I i = I0 p (do/d.O)i ,
(8.76)
where p is the partial pressure of the sampling gas. Combining Eq. (8.76) with the respective expression for the integrated intensity of the 2331 cm -1 Raman line of nitrogen
IN2 = IO
(8.77)
(da/d ) z
the following relation is obtained Ii
~2
=
(do/d.O)i
P
(do/d.O)N2
PN2
.
(8.78)
The relative differential Raman scattering cross section of the ith line of the sample can, therefore, be determined experimentally by employing the expression: (do / d.Q)i (do / d.O)N2
Ii
IN2
PN2 P
(8.79)
The absolute differential Raman scattering cross section of the ith line of the sample can be obtained l~om the relative value by using the absolute scattering cross section of nitrogen as given in Eq. (8.73).
INTENSITIES
IN RAMAN
SPECTROSCOPY
Molecular Polarizability ....................................................................................
190
II.
Intensity of Raman Line ....................................................................................
199
11I.
Raman Intensities and Molecular Symmetry ...................................................... 205
IV.
Resonance Raman Effect ...................................................................................
V.
Experimental Determination of Raman Intensities ............................................. 211
207
A.
Absolute Differential Raman Scattering Cross Section of Nitrogen .......... 212
B.
Differential Raman Scattering Cross Sections of Gaseous Samples .......... 213
189
190
GALABOV AND DUDEV
I. M O L E C U L A R
POLARIZABILITY
When an electric field with strength f is applied to a molecule, an induced dipole ~t is created = ,x f .
(8.0
~t is an important molecular quantity called molecular polarizability. The electrons of the molecule oscillate in accord with the external field applied, thus becoming a source of a secondary radiation which is the scattered radiation. In the general case, the vectors Ix and f do not coincide and Eq. (8.1) can be written in the following form , x = a x x fx + a x Y fv + ~ x z fz
(8.2)
~Y = ~YX fx + ~YV fv + ~ v z fz ~ z = ,xzx fx + a z v fY + a z z fz or
~y
= Otyx
Otyy
Otyz
fy
~z
~zx
~zY
Cxzz
fz
(s.3)
In Eqs. (8.2) and (8.3) IXj and fj (J = X, Y, Z) are the Cartesian components of IX and f, respectively, in a space-fixed coordinate system and tXjK (J, K = X, Y, Z) are the elements of the second rank polarizability tensor or. ct is a symmetrical tensor [CtjK = CtKj (K~J)] and, therefore, has six independent components only. The elements of ct may, in the general case, all be different from zero. The components depend on the space orientation of the molecule but not on the electric field applied. As is known, for each symmetrical tensor a special set of Cartesian axes x', y' and z' exists, such as in respect to which the tensor assumes a unique diagonal form. a acquires the following structure: Otx,x, 0t=/ 0
0 r 0
0 / 0 .
(8.4)
Otz'z'
The induced dipole is parallel to the external field vector. The quantifies ~x'x', Oty,y,and Otz,z, are called principal values of the molecular polarizability, and the axes x', y' and z'
INTENSITIES IN RAMAN SPECTROSCOPY
191
Y
Io
TTT X
/ ~
Direction of propagation
;I•
/••ection
of observation
Fig. 8.1. Principal scheme of scattering experiment
principal polarizability axes. In the general case, ax'x' # Oty,y, ~: Otz,z,. If the molecule possesses isotropic (spherical) polarizability, it is evident that ax,x, = o~y,y,= Otz,z,. For molecules with cylindrical symmetry with z' parallel to the main symmetry axis, Otx,x, = ay,y, ~: Otz,z,. The tensor a is characterized with two invariants with respect to any reorientations of the molecule in space. These are the mean polarizability ~ and the anisotropy 7. and y are defined as = ( a x x + a v y + otzz) / 3
(8.5)
v2 = [ ( a x x - a Y V ) 2 + ( a v Y - a z z ) 2 + ( a z z - a x x ) 2
+ 6 (otXy2 + otXZ2 + a y z 2 ) ] / 2 .
(8.6)
It can easily be seen that ~/vanishes for molecules possessing spherical polarizability. In ordinary scattering experiments a linearly polarized light is usually used to illuminate the sample. The incident beam propagates along one of the laboratory-fixed Cartesian axes (X in our case; Fig. 8.1) and the scattered light is collected in direction perpendicular to that of the incident beam (Z axis). The excitation light may be linearly
192
GALABOV AND DUDEV
polarized along the Y axis. The scattered light may then be depolarized in some extent and contain radiation polarized either parallel (with intensity Ill) or perpendicular (with intensity I_L)with respect to the polarization of the incident beam. Since the intensity of scattered radiation is proportional to the squares of the respective polarizability components [4,5,155,250,251], in accordance with the convention introduced, we have Iii ~ a y y 2 (8.7) Is
~Xy 2 .
Evidently, the intensity of the total scattered light IT = Ill + I L will depend on both a y y and a x y components of the polarizability tensor:
(8.8)
IT -~ (t~Xy2 + t~yy2) .
In gases, the molecules are free and randomly oriented with respect to the space-fixed set of Cartesian axes X, Y and Z. Since all molecules contribute to the intensity of the scattered radiation, it is necessary to average the components of a over all possible orientations of the molecule-fixed axes x, y and z with respect to the inertial (laboratory) Cartesian frame. In analytical form this condition reads [4,5] ajK = ~ ajk tjj tKk. j,k
(8.9)
In Eq. (8.9) tIi (I = X, Y, Z; i = x, y, z) are the direction cosines between the two Cartesian coordinate frames. With respect to the principal axes of the molecule, Eq. (8.9) transforms into (g = x', y' and z') t~jK -- ~ r g
tjg tKg .
(8.10)
Squaring and averaging operations executed upon Eq. (8.10) result in [4]
/2 (OtIK)2 =
tXg tJg tKg (8.11)
= E 0:2 t2g t~g + E egg r g g
tJg tKg tJr tKr.
INTENSITIES IN RAMAN SPECTROSCOPY
193
Since ag remains unchanged under the averaging process, the expressions t2g t ~ and tjg tKg tJr tKr need only be averaged. These are found to be [4] 2g t~gt
{1/5 if J =K = 1/15 if J ~ K
(8.12)
:/15ifJ=K tjg tKg tjr tKr = _ /30 if J r K
(8.13)
Application of Eqs. (8.11), (8.12) and (8.13) to axY and ayy components of the polarizability tensor results in
:(3go+2g
(8.14)
Introducing the invariants of the molecular polarizability ~ and y into Eqs. (8.14) leads to 45 ~2 +4T2 ~ = 45 O~2 = 3,Y2 45 45 ~2 + 7 T2 t~ = 45
(8.15)
An important observation in every scattering experiment is the depolarization ratio p. By definition p is I• P =~Ill
(8.16)
194
GALABOV AND DUDEV
Combining Eqs. (8.7), (8.15) and (8.16) yields
P
=
3T 2
(8.17)
45~2 +4.t,2 "
It should be emphasized that the ~2 and ~,2 coefficients in Eq. (8.17) are evaluated for the case of a polyatomic non-linear molecule, linearly polarized incident fight and rectangular experimental setup as shown in Fig. 8.1. Alternative sets of coefficients are to be used for experiments with unpolarized (natural) exciting light and different experimental geometries. These are summarized in Table 8.1. The polarizability terms for 27 most commonly employed scattering geometries [252] are presented. Different coefficients are used in the case of diatomic and linear polyatomic molecules as well [253] 45 ~2 +72 45 iX2 = (3/4)T 2 45
(S.lS)
45 ~2 + (7 / 4)T 2 45 The changes in molecular polarizability during vibrational transitions determine the intensities of Raman lines. The ~ ( polarizability matrix element for a transition from a vibronic state m to a vibronic state n can be presented as follows [4,254-256] (Ct/K)mn = f ~
Ot/K Wm dl;= (n
l ajK I m) (8.19)
h
Ve -Vm - v 0
M(J)em M(K)ne ) . Ve - Vn + v0
In Eq. (8.19) ~m and ~n represent vibronic wave functions of states m and n, respectively, e denotes an intermediate state of the undistorted molecule, v 0 is the frequency of the incident light and h is the Planck's constant. M(J)i 1 and M(K)i I (i, l= m, n, e) are the respective transition moments ( i I ~ I 1 ) and ( i I ~tk I 1 ), with ~ and ~tk the components of the dipole moment operator. It is seen from the above relation that the polarizability of the molecule is frequency dependent through the denominators (% - vm - vo) and (% - v n + v0). It has been shown [255-258] that when the frequency of
INTENSITIES IN RAMAN SPECTROSCOPY
195
TABLE 8.1
Polarizability terms a 2 for the most commonly used scattering geometries a Incident fight Scattered light Polarizability term 2 45 • a i Direction of Direction of Direction of Direction of propagation polarization observation polarization Y X Y 4 5 ~ 2 + 4 y2 X (180 ~)
X
Z
Unpolafized
3~2
Y+Z
45 ~ 2 + 7 72
Y
X
3 ](2
(90 ~
Z
3 y2
X+Z
6 y2
Z
X
3 ](2
(90 ~
Y
45~ 2 + 4 ](2
X+Y
45 ~2 + 7 72
X
Y
3 ](2
(180 ~
Z
4 5 ~ 2 + 4 ](2
Y+Z
4 5 ~ 2 + 7 ~,2
X Z
3 y2 4 5 ~ 2 + 4 y2
X+Z
45 ~ 2 + 7 ](2
Z
X
3 ](2
(90 ~
Y X+Y
3 3,2 6,/2
X
Y
(45 ~2 + 7 ),2)/2
( 180~
Z Y+Z
(45 ~2 + 7 72)/2 45 ~2 + 7 ./2
Y
X
3 ](2
(90 ~
Z
(45~ 2 + 7 ](2)/2
X+Z
(45~ 2 + 13 y2)/2
Z
X
3]( 2
(90 ~
Y
(45~2 + 7 ](2)/2
X+Y
(45~ 2 + 13 ](2)/2
Y (90 ~
X
z
aReprmted from Ref. [252] with permission from Society for Applied Spectroscopy.
196
GALABOV AND DUDEV
exciting light v 0 approaches to, or coincides with, the fxequency of some electronic transition in the molecule, dramatic changes in the magnitude of molecular polarizability, and hence in the intensity of Raman lines, occur. The phenomena are known as preresonance and resonance Raman effects, respectively. The dependence of (aJK)nm on v0 becomes small and can be neglected in the case of ordinary (far-from-resonance) Raman experiment when the following conditions are met: (1) the electronic ground state of the molecule is nondegenerate, and (2) the frequency of exciting radiation v 0 exceeds by far the frequency of the Raman shift Vnm. At the same time v0 must be small as compared with any electronic absorption frequency of the molecule. These conditions are known as Placzek's conditions and in case these are satisfied, the Born-Oppenheimer approximation can be used to justify the separation of the vibronic wave function ~Pi =~F~ o~i vr .
(8.20)
We is the electronic wave function of the ith state and Wvr is the wave function describing the rotovibrational motion for state i. Within the above approximation the electronic wave functions can be excluded from further consideration. Thus, the rotovibrational wave functions of the electronic ground state will only be treated. The left-hand side of Eq. (8.19) assumes the form (O~JK)nm = [ (Wvr) * ~jK Wvr d'l:vr.
Rotational and vibrational distortions in the molecule can also be treated separately which leads to a fitnher splitting of the wave functions: Wvr = ~i v .~Fir '
(8.22)
where superscripts v and r stand for vibrational and rotational wave functions, respectively. Eq. (8.21) transforms into (Ot/K)mn =~ (~Fn)* ( ~ r ) * ( Z j K ~ v ~ F r d f f d z r .
(8.23)
In order to assess the contributions of the two wave functions ~i v and ~Fir to the intensity of the Raman line, a molecule-fLxed Cartesian coordinate system (x, y, z) which is able to rotate together with the molecule is necessary to introduce. The CqK can then be expressed in terms of polarizability tensor components in the molecule-fixed coordinate
INTENSITIES IN RAMAN SPECTROSCOPY
197
system and the respective direction cosines between the two Cartesian coordinate systems. Combination of Eqs. (8.9) and (8.23) leads to (OqK)mn = ~ ~ (~FV)* (~Fr) * Otjk tjj tKk Wv ~Fr dxV dxr j,k (8.24) =Z ~ j,k
(~r),
tjj tKk ~ r dxr I
(,,.i.,#),O~jk,.i.,vdxV
Thus, it is clear that the polarizability of a molecule, and hence the intensity of a Raman line, depend on the properties of the two integrals appearing in Eq. (8.24). These must not be zero if the transition m ~ n is to be Raman active. Examination of the first integral leads to the rotational selection rules which govern the appearance of rotational lines in the Raman spectra. These are discussed in detail elsewhere [4]. The properties of the vibrational integral
(".~")m,, =S (,IV)* '~k"I": <,.,:v
(8.25)
will be considered. For small molecular vibrations, and in case the Placzek's conditions (1) and (2) are satisfied, the polarizability of the molecule can be expanded in a Taylor series along the normal coordinates Qi
0~=0~ O+
~r
~176 0 Qr
~r
Z r,s
0
Qr Qs (8.26)
+ -- ~ ' ' Qr esQt + . . . . 6 r,s,t OQr~Qs~Qt 0 Here ot0 is the polarizability tensor in an equilibrium non-perturbed state and the subscript o in the polarizability derivative terms means that these are taken at the equilibrium geometry of the molecule. Combining Eqs. (8.25) and (8.26) and neglecting higher terms in the expansion (8.26), the following expression is obtained for the ith normal mode
(",~)~-(",~)o s (~v), ~v ~v (8.27)
+(c)(Zjk/c)Qi)0 S (~Fv)* Qi ~v dl:V.
198
GALABOV AND DUDEV
The vibrational wave functions ~Pmv and ~pv can be presented as a product of the wave functions of the individual normal coordinates tpv =WVl (QI) ~Pv2 (Q2) --. ~PV(3N_6)(Q3N-6)
(8.28)
~pv =~PnVl(Q1) ~PnV2(Q2) "--~Fn(3N_6)(Q3N-6) 9
(8.29)
Eq. (8.27) transforms to (O~Jk)mn = (O~jk)o {f [~nVl(Q1)]* tpv, (Q1) dQ1--X f [~pv (Q1)]* ~Pv i (Qi)dQi'" "}
(8.30) + (~O~jk/OQi )o {~ [Wnvl(QI)]* w v 1 (Q1)dQ1 "'" •
[~Pm (Qi)]* QiWVi(Qi)dQi-- } 9
The vibrational quantum numbers associated with vibrational levels mi and ni (i = 1 to 3N6, where'N is the number of atoms in the molecule) may be denoted as v mi and v ai, respectively. The first term in the fight-hand side of Eq. (8.30) is zero unless vml = v nl, v m2 = vn2, . . ., vmi = v n i , . . . , etc. because of the orthogonality of vibrational wave functions. In other words, this term does not vanish ordy for elastic light scattering when the molecule does not change its vibrational state. The first term in Eq. (8.30) is, therefore, responsible for the Rayleigh scattering. The second term that gives rise to an inelastic scattering will not vanish if: (a) the polarizability derivative (a~k//gQi)o is not zero (this condition requires ~k to change when the molecule undergoes a particular vibrational transition), and (b) the integral is not zero. In the framework of the harmonic approximation, the last condition is satisfied if the vibrational quantum numbers of the ith transition v mi and v ni only differ by a factor of +1 Av = v m i - v ni = +1 .
(8.31)
All other vibrational quantum numbers vml, vial, vm2, v n 2 , . . . , vm(i-l), vn(i-l), vm(i+l), vn(i+l), . . . , etc. must not change. Under far-from-resonance conditions there are six independent elements of the polarizability tensor: axx, axy, axz, ayy, O~yz and CZzz. Therefore, six equations analogous to Eq. (8.30) exist for each normal mode i. At least one of the six (CZjl0mn(j, k = x, y, z) components must be non-zero if the ith vibration is to
INTENSITIES IN RAMAN SPECTROSCOPY
199
be Raman active. According to Eq. (8.31) fundamental transitions between vibrational energy levels are only allowed. Due to anhannonicity effects, overtone and combination bands appear in the Raman spectra, though typically with lower intensity. In analyzing the anharmonicity effects, higher terms in the Taylor series expansion [Eq. (8.26)] need to be considered. The selection rule (8.31) was discussed in detail in Chapter 1 and we will not elaborate further here. The symmetry selection role will be discussed in Section 8.3. If the vibrational wave functions qJmiV(Qi) and q~niV(Qi) are substituted with the respective analytical expressions, the relation given by Eq. (1.29) (Chapter 1) is obtained. The first integral in Eq. (8.30) does not contribute to the intensity of a Raman band. The change in Ctjk due to vibrational transition i can, then, be expressed as (Ctjl0i = (0Ctjk/c3Qi)0 [b 2 (vi + 1)189,
(8.32)
bE=
(8.33)
where
h 8g2toi
is the zero-point amplitude.
II. I N T E N S I T Y O F R A M A N L I N E The intensity of ith Stokes-shifted Raman line is given by the relation [250,255, 259] 27x5 )4 2 Ii = c------32 ~- I0 Nvi (vo - v i X (~Jk)i j,k
(8.34)
where I0 and v 0 are the intensity and frequency of the incident light, respectively, v i is the vibrational frequency, c is the velocity of light, and Nvi is the number of molecules in vibrational state with quantum number vi. By combining Eqs (8.32) and (8.34) the following expression is obtained Ii = 27/1:5 32c4 I o ( v o - v i )4 Nvi (vi+ 1) b 29 X (c)O~jk / t)Qi)20 9 j,k
(8.35)
200
GALABOV AND DUDEV
The selection rule for Raman intensities (b) discussed in the previous section requires that the vibrational quantum number changes by + 1 for the Stokes lines in the Raman spectra. This is, however, true for the fundamental lines, corresponding to transition 0--->1, and for the "hot bands" related to transitions of the type 1-->2, 2---~3, etc. These appear at the same frequency in the spectra as the fundamental bands. Thus, the frequencies associated with fundamental and "hot band" transitions are hardly discernible. All such transitions contribute to the intensity of a Raman band. A summation over all vibrational quantum numbers v i from zero to infinity is needed to account for all contributions to the intensity Ii. Following the considerations given in Section 1.1 [Eqs. (1.31) and (1.39)], the expression is obtained
Z N vi(Vi + 1 ) = ~ -i vi=0 vi=O
e(-hvi/kT)vi (vi + 1) (8.36)
N 1_ e-hvi/kT ' where N is the total number of scattering molecules per unit volume. Therefore, 27/I; 5 Ii=
)4
32C4 IoN(v0-v i
9b 2 1-
2
e_-~i/k T ~ (~0tjk/~Qi)0 . j,k
(8.37)
This equation can also be written in the following form Ii = I0 N o i ,
(8.38)
where 27~ 5 )4 b2 ffi = ~c--"--'~-32(v0 - vi 1 e-hVi/kT X (~~ j,k
2 (8.39)
-
is the total Raman scattering cross section for full solid angle 4~. If wave numbers are used instead of frequencies, Eq. (8.39) transforms into
ffi :
27/1:5 )4 32 (V0 -Vi
b2 2 - hc/kT X (OO~jk/oQi)0, 1 - e -vi j,k
(8.40)
where v0 and~i are the wave numbers of the incident light and of the vibrational transition, respectively. Since the number of scattering molecules N and the intensity of
INTENSITIES IN RAMAN SPECTROSCOPY
201
irradiation Io are quantifies that are difficult to measure experimentally, o i appears more appropriately observable than the Raman intensity Ii. The polarizability derivatives c3Ctjk/0Qiappearing in Eqs. (8.35), (8.37), (8.39) and (8.40) form a three dimensional tensor of size 3•215 For the sake of simplicity i t can be presented in rectangular form. Thus, the supertensor CtQ can be expressed as:
O~Q =
(
~)(Zxx/OQI OCtxy/()QI OCtxz/OQ1 ... ()~yy/t)Ql bCtyz/c)Q1 ... \symmetrical bazz / bQ1 ... (8.41) OCtxx / ~)Q3N-6 ~)Ctxy/ ~)Q3N-6 ~)(Zxz/ ~Q3N-6"~ ~O~yy/OQ3N_6 ~Ctyz/~Q3N_6] 9 symmetrical 3azz / ~)Q3N-6
The last term in Eq. (8.40) containing elements of the supertensor aQ Can be expressed in terms of the invariants of the molecular polarizability ~ and ), with respect to normal coordinates: --' (ti
=
~
bet
bQi
=
-
l(bax x Oayy 0azz )
3 [ ~)Qi + ~)Qi
+
'
(8.42)
~)Qi
1 I(3axx t~Qi
~,~i
OQi
~.~-"Qi
+6 /)Qi
~)Qi )
OQi
21}
(8.43)
"
Unlike the equilibrium mean polarizabilities ~0 which are non-zero quantifies, their derivatives with respect to normal coordinates may equal zero. These quantifies vanish for non-totally symmetric vibrations. In such cases the anisotropic part of the tensor tZQ contributes only to the intensity of the Raman line. Introducing ~ and "/i' as well as the degree of degeneracy gi in Eq. (8.40) leads to [2601 27~ 5 )4 b2 [ 2 ,)2] t~i = 32 (V0-~i 1- e_~ihc/kT gi 3(~[) 2 +-~0'i 9
(8.44)
202
GALABOV AND DUDEV
Thus, the total Raman scattering cross section of the Stokes-shifted line into a full solid angle 4n is expressed in terms of derivatives of the molecular polarizability invariants with respect to normal vibrational coordinates. Only a portion of the scattered by the sample light is, however, collected. Usually the scattered radiation is detected into a fixed direction with a narrow collection angle. Another quantity, the differential Raman scattering cross section into a given direction (dt~/d~)i, is employed in Raman intensity measurements [254,260]. If the rectangular experimental setup, as shown in Fig. 8.1, is employed, the differential Raman scattering cross section is given by [260]
"~ i
24454 ( v 0 - v i ,4 l_e_~ihc/k T gi E45(~[ ,2 +7 (121 Ti -
(8.45)
The quantity (da/d~)i is called absolute differential Raman scattering cross section. It is often termed absolute Raman intensity as well. From Eq. (8.45) it is clear that (do/d.O)i depends on several factors such as ~0 and ~i and the absolute temperature T. In order to operate with comparable quantifies that are independent from the experimental conditions, the so-called "standard intensity" or "scattering coefficient" Si is commonly used [73,253,260-263] Si = gi [ 4 5 ( ~ ) 2+ 7(Ti')21 -
(8.46)
Eq. (8.45) assumes the following form
do) = "~ i
b2 24 g4 )4 Si " 4-'--~ (T0 - v i 1_ e_Vi-hc/kT
(8.47)
Another experimental observable in the Rarnan intensity experiment is the depolarization ratio Pi of the vibrational line 3(~) 2 Pi
4 5 ( ~ ) 2+4(T~) 2 -
(8.48/
Pi provides a unique possibility for examining the symmetry of vibrational transitions. Since ~ vanishes for distortions belonging to a non-fully symmetric species, it easily can be realized that Pmax = 3/4. Depolarization ratio assumes its minimum value for the totally-symmetric vibrations of spherical top molecules. In these cases the Raman line is fully polarized and Ti' and Pi are zero.
INTENSITIES IN RAMAN SPECTROSCOPY
203
By combining Eqs. (8.46) and (8.48) expressions for the absolute values of ~ and ~'i' are obtained: 1
I :1=
Si0-40i)] 1/2
3g (iu
SiPi 11/2 ['f:[=
3gi(l+pi)J
(8.49)
(8.50)
As in the case of dipole moment derivatives with respect to normal coordinates, absolute values of ~ and )'i' only can be evaluated from the experiment. Extended basis set ab mitio MO calculations of polarizability derivatives are usually employed in solving the sign ambiguity problem. Eq. (8.45) expresses the dependence of the absolute Raman differential cross section on ( ~ ) 2 and (~'i')2. As already mentioned, the scattered light consists of two components: III -~ 4 5 ( ~ ) 2 + 40,i')2
(8.51)
I_1_-~ 3('/i')2 .
(8.52)
and
These quantifies can be measured separately. Thus, the so-called trace and anisotropy spectra can be obtained from experiment [264]. The trace spectra are, as a rule, simpler and are due to fully symmetric vibrational transitions only. The lines are narrow, well separated and easy to analyze. Anisotropy spectra originate from non-totally symmetric vibrational distortions in the molecule and usually represent a complicated superposition of strongly overlapped bands. The interpretation of spectra in the gas-phase is also hampered by the presence of a complex rotational structure of the vibrational bands. Trace (It) and anisotropy (Ia) spectra can be obtained after careful polarization experiments using the following equations [264]: It = Ill - (4/3) I_L
(8.53)
Ia = (7/3) l_t- .
(8.54)
204
GALABOV AND DUDEV
Thus, the respective absolute trace and anisotropy Raman differential cross sections are functions of a single polarizability invariant derivative only (-~)
=2484 ( V o - v i ) 4 i
b2 (~)2 1 - e -'~i'ihe/kT
(8.55)
drya ) = 7 2484 (Vo-Vi) 4 b? )2 d~Ji 4-5 1_ e-~i hc/kT gi()'~ "
(8.56)
Equations (8.45), (8.55) and (8.56) are derived for the CGS system of units" (d6/d.Q)i is expressed in cm2/sr, the molecular polarizability ct in cm 3, 0~OQi in cm2/g 89 and the scattering coefficients Si in cm4/g. Martin and Montero [263] first suggested using SI units in Raman intensity measurements. In adapting Eq. (8.45) to the new system of units the factor (47re0)2 has to be introduced in the denominator of the equation where to = 8.8542• C V -1 m -1 is the permitivity of vacuum (C - Colomb, V - Volt). Following the considerations given in Ref. [263] Eq. (8.45) can be further simplified if a dimensionless normal coordinate qi instead of the mass-weighted Qi is used [263] (8.57)
qi = (47t 2 c ~ i / h ) 8 9
where ~i is the harmonic vibrational wavenumber. Eq. (8.45) transforms into [263]
(do/
82
(~0 -~i) 4
[45(~)2 + 7(~/~)2
" ~ i =9~"~'2 1-e_~i-ffc~ T gi
].
(8.58)
In Eq. (8.58) (da/d.Q)i is expressed in m2/sr, ct and t3ntx/aqin in C V -1 m 2 and Si in C 2 V -2 m 4. The conversion factors between different unit systems for these quantifies are as follows 1 cm2/sr = 10--4 m2/sr 1 ~3 = 10-24 cm 3 = 1.112644• 10--40 C V -1 m 2 .
(8.59)
The invariants ~ and ~i' represent combinations between the six independent components a~k/0Qi (j, k = x, y, z) of the polarizability tensor t:tQ [Eqs. (8.42) and (8.43)]. These quantifies can be determined from experiment for very few small and highly symmetric molecules only. Typically, the number of intensity parameters to be evaluated exceeds by far the number of experimental observations. The XY 2 bent
INTENSITIES IN RAMAN SPECTROSCOPY
205
molecule is a good example in this respect. It has three normal vibrations, both infrared and Raman active, grouped as follows: FV = 2A 1 + B 2.
(8.60)
Suppose the molecule is oriented along its principal axes with x axis coinciding with the C 2 symmetry axis and z axis lying in the plane of the molecule. The supertensor CtQ has then the following structure:
Q1 o tZQ=
0
Q2 o
o
o
o
1 {Xzz
0
d.r
0
Q3 o
o
o
o
o
o
o.
2 Otzz
ct 3
0
0
(8.61)
~ik (i = 1, 2, 3; j, k = x, y, z) denotes the derivatives of the polarizability components ff-jk with respect to the ith normal mode. Since for such a molecule r • ctlyy ~ Ctlzz and O~Zxx ~ r , ~2zz, the number of independent polarizability derivatives is 7. The number of experimental quantifies, however, is just 5 ( ~ , ~'1', ~ , ~'2', ~'3'). It is evident that even for such simple and symmetric molecules the analysis meets serious challenges. In comparison, infi'ared intensities for C2v molecules can, in principle, be transformed into dipole moment derivatives since there is only one non-zero component of the dipole moment derivative vector for each normal mode. The great complexity in acquiring experimental polarizability derivatives as compared with dipole moment derivatives is probably the reason for the relatively limited use of Raman intensities in analyzing molecular properties.
III. R A M A N I N T E N S I T I E S A N D M O L E C U L A R
SYMMETRY
According to the harmonic oscillator selection rule, as briefly discussed in Section 8.1, fundamental transitions are only allowed in Raman spectra. In practice, it means that the fundamental lines will appear with higher intensity than the overtone and combination lines. In this section the relationship between molecular symmetry and the intensity of the Raman line will be considered. Eq. (8.25) shows that the intensity of each fundamental transition in the Raman spectrum is governed by integrals of the following type
206
GALABOV AND DUDEV
( n I ajk I m )
(j, k =x, y, z) .
(8.62)
Since, in the ordinary Raman experiment, the polarizability tensor has six independent components, six integrals with such a structure exist ( n l CXxxI m )
(nl~lm)
( n [ cxzzl m )
( n l ~ y I m>
(nl~lm>
( n I Cry, I m>
.
(8.63)
A vibrational transition is allowed if at least one of these integrals differs from zero. The symmetry selection rule states that the integral ( n I ~k I m ) is not zero if the direct product of representations for the ith vibrational mode F i = F n x F~k • F m
(j, k =x, y, z)
(8.64)
belongs to the fully symmetric species of the molecular point group. Since the ground state wave function is always fully symmetric, the behavior of the product F i will depend on the symmetry properties of the polarizability tensor components and of the vibrational wave function of the first excited state. Hence, if that wave function belongs to the same symmetry species as the respective polarizability component, the direct product F i will be fully symmetric. In this case, the ith transition will be Raman active. An interesting case are molecules possessing a center of symmetry. According to the symmetry selection rule, a given vibrational transition is infrared active ff the direct product between the two vibrational wave function representations and that of the dipole moment vector component is totally syaunetric (Section 1.2.2). The ground state vibrational wave function, as mentioned, is always totally symmetric. It belongs to even or gerade representation since it is not affected by inversion operation with respect to the center of symmetry. The dipole moment vector, however, changes sign when the inversion operation is executed. It, consequently, belongs to odd or ungerade representation. Thus, it is easy to deduce that the representation of the first vibrational excited state wave function must be odd (ungerade)(belonging to A u, B u, etc. synunetry species) if the fundamental transition is to be infrared active. The molecular polarizability tensor ct is defined in relation with two vectors: the induced dipole ~t and the electric field vector f [Eq. (8.1)]. Hence, the representation of ct can be expressed as a direct product of the representations of the respective vectors [265] F a = F~t • I f .
(8.65)
INTENSITIES IN RAMAN SPECTROSCOPY
207
Since the inversion operation alters the sign of the two vectors, according to Eq. (8.65), ct remains unaffected. It, therefore, belongs to an even symmetry species. It is evident that the excited state wave function must possess the same symmetry properties as ct if the transition is to be Raman active. Thus, for molecules with center of synanetry fundamental transitions to excited states belonging to even syrmnetry species (Ag, Bg, etc.) are only active. The above discussion outlines the so-called rule of mutual exclusion in vibrational spectroscopy which reads: for molecules with a center of synanetry vibrations that are Raman active are infrared inactive and v i c e v e r s a .
IV. R E S O N A N C E
RAMAN EFFECT
The relations and conclusions drawn in the previous sections were deduced for the non-resonance Raman experiment performed with frequency of the exciting light laying far from the frequency of any electronic transition in the molecule. In such a case the differential Raman scattering cross section (do/dD)i depends on the incident light frequency Vo through the term (v0 - v i ) 4 [Eqs. (8.45) and (8.58)]. If the Placzek's conditions (section 8.1) are satisfied, the dependence of molecular polarizability on Vo is negligible. In the treatment presented in the preceding sections, vibrational wave functions of the ground electronic state were only considered. If, however, the excitation radiation frequency approaches to Ore-resonance conditions) or coincides with (resonance conditions) the frequency of some electronic transition in the molecule, the application of the ground-state approach is not justifiable any more. The dependence of molecular polarizability on the incident light frequency has to be taken into account. As was already mentioned, under pre-resonance and resonance conditions some Raman lines are enhanced hundreds and thousands of times. In this section a brief outline of the basic theory of resonance Raman effect is presented. Before proceeding further, it is necessary to introduce the following notation: (a) the ground electronic state is denoted by g; vibrational levels belonging to this state are designated as i (initial vibrational state) and f (final vibrational state). The respective vibronic wave functions will be labeled as gi and gf; (b) the higher-energy electronic levels assume symbols r s and t; transitions g--,e and g--,s are considered allowed and g--,t - forbidden; and (c) vibrational levels belonging to e, s and t are denoted by v, I and u, respectively. In the subsequent derivations all electronic and vibrational states are considered as non-degenerate.
208
GALABOV AND DUDEV
The theory of intensities of resonantly-enhanced Raman lines is based on the Kramers-Heisenberg-Dirac dispersion equation [266,267]. The problem is analyzed in terms of vibronic interactions in the molecule. The JKth matrix component of the molecular polarizability for the gi~gf transition in the Raman spectrum can be expressed as follows [256,257]
(0:JK)gi,gf= E / evg:gi
Eev-Egi-E0 (8.66)
(evlMj ] E[g- i)eEgf (gflvM - E0 Klev)
"
Mj and M K are the respective dipole moment operators and E 0 is the excitation light energy. In resonance conditions the first term in Eq. (8.66) becomes dominant since the denominator (Eev - Egi - E0) rapidly decreases. In this case, a phenomenological damping constant F is introduced in the denominator expression. Thus, retaining the resonant term in Eq. (8.66) only, we obtain
( .(ev[M-K!gi)(-gflMJ lev)) . )gi,gf = evg:gi E ~, Eev - Egi - E0 + iF
(8.67)
This equation can be further developed if the electronic wave function is expanded in a Taylor series along the normal coordinates Qa of the molecule. This is known as the Herzberg-Teller expansion [268] and its application to Eq. (8.67) yields [255,256]
(ctji0gi,gf = A + B + C ,
(8.68)
A = ~ ~ (Mj)ge ( MK)eg (gf[ ev) (ev Igi) e~g v Eev - Egi - E0 + iF
(8.69)
where
INTENSITIES IN RA_MAN SPECTROSCOPY
X X
E
209
(gflQ~lev) (ev Igi)
~., ( (MJ)gshae(MK)eq
e~g v s~e a
E e - Es
Eev - Egi - E0 + iF (8.70)
(Mj)ge heas(MK)sg
E e ~ Es
C
_
Eev - Egi - E0 + iF
(M j )ge (M K )et h~g
(g~Qalev) (ev[gi)
Eg-Et
Eev - Egi - E0 + iF
_
~ ~v ~a e~g tag
(gfl ev)(eqQalgi) /
J
(8.71)
h~ (Mj)te (M K )eg
(gfl ev)(eqQ~lgi) / i
Eg - E t
Eev - Egi - E0 + iF
J
In these equations (Mp)s~ = [~0] Mpl e0] (p = J, K; ~, ~ = g, e, s, t) are the electric transition dipole moments at the equilibrium molecular geometry and lk~a= [~0[ (aH/0Qa)ol so]. (aH/SQa)0 is the vibronic coupling operator for the normal mode a. H is the electronic Hamiltonian of the molecule. The contribution of each term A, B and C to the intensity of a Raman line is considered as follows. A term. One excited electronic state e is included in this term. With v 0 approaching v e the Raman line associated with a totally symmetric mode derives its intensity from this term through the Franck-Condon mechanism. This term does not depend on vibronic mixing of state e with other electronic states of the molecule. If the two electronic states g and e lie far below the next electronic levels and, hence, the probability of vibronic coupling between state e and the other excited electronic states is very low, the A term dominates in magnitude over the other two terms in Eq. (8.68). B term. This term comprises allowed transitions between the ground electronic state and two electronic states e and s (E e < Es). It determines to a great extent the intensity of a Raman line associated with a non-fully symmetric transition. In analyzing the properties of the B term the role of vibronic coupling in generating Raman intensifies becomes clear. Analysis shows that those normal vibrations which can couple two excited electronic states (e and s) undergo a striking intensity enhancement if the frequency of incident radiation is tuned on the frequency of the lowest electronic transition g---~. The nearer the state s to state e, the more pronounced is the effect of interaction. C term. It considers transitions to two electronic states e and t. Transition g~t is
not necessarily allowed. Since, in the general case, E e - E s < E g - E t, the magnitude of
210
GALABOV AND DUDEV
TABLE 8.2 Intensities of some Raman lines of pyrazine relative to the intensity of 940 cm-1 Raman line of benzene-d6 obtained at different excitation fight wavelengths (Reprinted from Ref. [274] with permission) Relative intensity a
Excitation wavelength
(in.m)
703cm-1 v 4 (b2g)
754cm- 1 v5 (b2g)
925cm -1 Vl0a (big)
514.5
0.38
0.041
0.046
457.9
0.47
0.082
0.12
363.8
0.57
0.20
1.14
aIn liquid phase.
this term will be small compared with the B term [Eqs. (8.70) and (8.71)]. Therefore, the contribution of C term to the overall intensity of the Raman line is not expected to be significant. This term is important for molecules possessing low-lying forbidden electronic states. Eqs. (8.68) through (8.71) were first derived by Albrecht [255,256]. His predictions have proved correct in many cases [269-277]. An example will demonstrate the role of vibronic coupling in generating resonance Raman intensities as derived in Albrecht's theory 03 term). It has been shown in a series of publications of Ito et al. [270, 271,274] that the Vl0a non-totally symmetric vibration of pyrazine (C-H out-of-plane bending) demonstrates a remarkable enhancement in Raman spectrum when the exciting radiation frequency approaches the frequency of the lowest-lying electronic transition at 323 nm. The Vl0a vibrational transition belongs to blg symmetry species and appears at the 925 cm-1 in the Raman spectnma. Results presented in Table 8.2 illustrate quite clearly the resonance effect on the intensity of this line. Analysis of the data accumulated reveals that the coupling between the lowest-lying allowed electronic state IB3u (n--+n*) and the next excited electronic state IB2u (rc---~*) through non-fully syuunetric big vibrational mode is responsible for the enhancement of 925 cm-1 Raman line. Low energy separation between the two excited electronic states, as seen from Fig. 8.2, favors this process. Vibronic coupling between IB3u and the two 1Blu electronic states will be less effective due to the higher energy separation. Therefore, vibrational modes (v 4 and v5) that are able to couple these electronic states are not expected to be significantly enhanced. The data collected in Table 8.2 illustrate this conclusion. Recently Okamoto [257] evaluated additional second-order terms to Eq. (8.68), thus expanding the applicability of Albrecht's theory. The emphasis is laid on the role of forbidden electronic transitions in generating resonance Raman intensities. It has been shown that: (1) the Raman line can derive intensity from a forbidden electronic transition
INTENSITIES IN RAMAN SPECTROSCOPY
I
211
60700
IBlu (n-~r*), IB2u (~-->x*)
50900
1Blu (n--~r*)
37800
1B2u (a:-~*)
30900
1B3u (n-~x*)
z:
0
Ag
Fig. 8.2. Energy diagram of the low-lying electronic states of pyrazine.
even if the excitation radiation is tuned in resonance with an allowed electronic transition, and (2) enhancement may be detected if the excitation is in resonance with a vibronically allowed, though electronically forbidden, transition.
V. E X P E R I M E N T A L D E T E R M I N A T I O N OF RAMAN INTENSITIES Eq. (8.45) shows that for an ordinary Raman experiment the absolute differential Raman scattering cross sections can be expressed in terms of derivatives of the molecular polarizability invariants ~ and V with respect to normal coordinates. These derivatives contain valuable information about the variation of molecular polarizability with vibrational motion. Gas-phase Raman scattering cross sections are most suited for intensity analysis since at low partial pressure of the sampling gas these quantities are not influenced by effects of intermolecular interactions, thus reflecting properties of individual molecules.
212
G.M.,ABOV AND DUDEV
Direct determination of absolute Raman differential cross sections is quite difficult and tedious work often leading to incorrect results. It is easier to measure cross sections relative to some standard. The absolute differential Raman scattering cross sections of the sample can then be straightforwardly obtained.
A. Absolute Differential Raman Scattering Cross Section of Nitrogen The absolute scattering cross section of the Q-branch of the Raman band of nitrogen at 2331 cm-1 has been chosen as a standard in Raman intensity experiments. The special role of this molecule in gas-phase Raman spectroscopy is based on several reasons: (1) nitrogen is a relatively inert gas and does not react with the gas sample; (2) it quickly forms mixtures with the sampling gas; (3) the region of the vibrational specmma where the nitrogen Raman line appears is very low populated and, for many gases, this line does not overlap with sample gas lines; and (4) since the absorption band of nitrogen lies in the far ultraviolet, laser excitation beams with wavelengths corresponding to near ultraviolet or visible light do not cause resonance enhancement of the nitrogen Raman line. Its intensity depends on the excitation frequency through the term (V 0 - v i ) 4 only. Thus, the differential scattering cross section of nitrogen can be used as a standard for a wide range of excitation laser lines. The absolute differential scattering cross section of the standard needs to be determined as precisely as possible. A number of measurements has been performed over the past forty years [260,278-287]. The introduction of lasers in Raman spectroscopy and of computer processing of spectral data has improved highly the accuracy of Raman intensity measurements. As a result, the absolute differential Raman scattering cross section of nitrogen reported from different laboratories deviates within a few percent only [260,284,285,287]. One of the possible approaches in determining (dtr/d.Q)Q,N2 is to use the absolute differential cross section of the strongest purely rotational Raman line J = 1--->3 (J the rotational quantum number) of hydrogen at 587 em-1 as a standard. It is given by [260] __~] rot,H2
= 24 g4 )4 3(J + 1)(J + 2) 7,y20 4---~ (~0 -Vrot 2(2J + 3)(2J + 1) "
(8.72)
70 is the anisotropy at the equilibrium geometry. The rectangular experimental setup, as shown in Fig. 8.1, is considered. Both experiment and theory have provided a reliable value for the hydrogen anisotropy. Moreover, its dependence on the excitation wavelength has been thoroughly established. As a result, the absolute differential cross section of the hydrogen rotational line (do/d~)rot, H2 has been accurately determined. The estimated value has been employed in determining the absolute differential cross
INTENSITIES IN RAMAN SPECTROSCOPY
213
section of the Q-branch of the vibrational Raman line of nitrogen. The following value has been obtained [260] (do/d.Q)Q,N2 = (5.05 4- 0.1)x 10-48 (~0 - 233 lcm-l) 4 cm6 sr-1.
(8.73)
It can be applied for a wide range of laser excitation wavelengths coveting the visible as well as the near ultraviolet up to 330 nm [260].
B. Differential Raman Scattering Cross Sections of Gaseous Samples The differential Raman scattering cross section of the ith line of a gas sample relative to that of the 2331 cm -1 line of nitrogen is given by [260] (do
/ d.Q)i
(v0 - vi )4
2331 cm -1
(~0 - 2331 cm-1)4
Vi (1- e-~ihe/kT)
~_
(do / m)Q,N 2
(8.74) gi [45(~)2 + 7gi ('Y[)2 ]
I45 (~)22
+7ZN2 (Ylq2
)2 I
"
In Eq. (8.74) li is the portion of the anisotropic scattering localized in the Q-branch of the respective Raman line. Its value has been determined for a number of molecules. In the case of linear molecules with small rotational constants gi is 0.25 [280,281]. Since the Bolzmann factor for the nitrogen molecule is very small at room temperature, it has been neglected in deriving Eq. (8.74). It is seen fxom the above expression that the relative differential scattering cross section depends on the excitation light wavenumber and on the absolute temperature. To make the measured quantifies comparable, a relative normalized differential Raman scattering cross section Ei has been defmed [260] (d~ / d'Q)i
.
(~0 -Vi) -4
Ei = (do / dn)Q,N2 (% _ 2331 r
(1-e-Vihr (8.75)
2331 cm-1 Vi
gi [45(~) 2 + 7gi (,y~)2] 7%N2('~N2
]
214
GALABOV AND DUDEV
The integrated intensity of the ith Raman line of a gaseous sample for a rectangular experimental setup can be expressed as I i = I0 p (do/d.O)i ,
(8.76)
where p is the partial pressure of the sampling gas. Combining Eq. (8.76) with the respective expression for the integrated intensity of the 2331 cm -1 Raman line of nitrogen
IN2 = IO
(8.77)
(da/d ) z
the following relation is obtained Ii
~2
=
(do/d.O)i
P
(do/d.O)N2
PN2
.
(8.78)
The relative differential Raman scattering cross section of the ith line of the sample can, therefore, be determined experimentally by employing the expression: (do / d.Q)i (do / d.O)N2
Ii
IN2
PN2 P
(8.79)
The absolute differential Raman scattering cross section of the ith line of the sample can be obtained l~om the relative value by using the absolute scattering cross section of nitrogen as given in Eq. (8.73).