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Chapter 1 Absorption of infrared radiation by molecules
CHAPTER 1
ABSORPTION
OF INFRARED
RADIATION
BY MOLECULES
I.
Theoretical Considerations .................................................................................... 2
II.
Selection Rules For Infrared Absorption .............................................................. 12
m~
A.
Harmonic Oscillator Selection Rules .......................................................... 12
B.
Symmetry Selection Rules .......................................................................... 14
Experimental Determination of Infrared Intensities .............................................. 17
2
GALABOV AND DUDEV
I. T H E O R E T I C A L
CONSIDERATIONS
The probability of absorption of a photon with energy hvn,n- by a molecule per unit of time leading to a transition between a lower energy state n" to higher state n' is given by [1-3]
8~;3 Wn': =-~ (n'[ X ej(u~ rj)
I
n#)2
P(Vn,n,,).
(1.1)
J In expression (1.1) ej and rj are the electric charge and position vector of atom j ill a molecule, rj refers to an arbitrary molecule-fixed Cartesian system, u x is the position vector of the photon with respect to a space-fixed Cartesian system. The quantity p called radiation density is equal to the number of photons with energy hvn,n- per unit volume. It is understood that the Bohr condition En, - F_.n,= hvn,n- must be satisfied. The polarization vector of the photons ux does not affect the molecular wave functions. A quantity called electric transition dipole may, therefore, be defined
P c : =(n'[ E ejrj In">.
(1.2)
J Since the electric dipole moment is given by
p
=
Xejrj,
(1.3)
J Eq. (1.2) becomes
Pn'n" = ( n ' l p In").
(1.4)
Pn'n" has components along the x, y and z axes of the molecule-fixed Cartesian system. The directions of ux and Pn'n" need not coincide since molecules are randomly oriented. There is, therefore, an angle 0 between the vectors ux and Pn'n"- Eq. (1.1) may then be rewritten as
8~;3 n' [ p [ n") 2 COS2 0 P(Vn,n-).
Wn,n- = - - ~ (
(1.5)
cos20 should be taken as an average over all possible orientations of the molecule in space
ABSORPTION OF INFRARED RADIATION
3
2rc COS2 0 = --L1 f cos 2 0 Sin 0 dO dO = _1. 4~ 0 0 3
i
(1.6)
The transition probability associated with absorption of a photon with energy hvn,n,, is then given by
8/I;3
(1.7)
The quantity Bn,n,, = (81t3/3h2)(n'lpln") 2 is the well known Einstein coefficient of absorption. Per unit radiation density it is equal to the transition probability. The Einstein coefficient depends on the molecular structure and may, eventually, be used to characterize molecular properties on the basis of experimentally determined intensities of the respective transitions. In the infrared region each vibrational transition is accompanied by a number of quantum transitions between rotational states of the molecules. In lower resolution spectra or if conditions for considerable broadening of absorption lines are present, the accompanying rotational transitions determine usually a non-symmetric PQR structure of the infrared band. In higher resolution spectra the individual rotational lines are separated. The intensity of an infrared absorption band represents, therefore, a sum over the intensities of all fine structure lines associated with the respective vibrational transition. It is, thus, necessary to describe the factors determining intensities of the component rotational lines and, then, see how these sum up into overall intensity of an infrared band. The probability for a reverse quantum transition from higher (n') to lower energy state (n") is also determined by expression (1.7). Therefore, the probability for absorption of electromagnetic radiation will depend on the number of molecules per unit volume of absorbing medium in the lower (Nn,,) and higher (Nn,) states
8~3 (n' I pin") 2 P(Vn,n-)(Nn--Nn,).
(1.8)
3h 2
For every elementary act of interaction of the electromagnetic radiation with a molecule an energy hvn,n, is absorbed provided some conditions are met. The energy absorbed by a differential element with a cross section equal to unity and thickness dl is equal to 81t3 n"
= vn'n" Sh-- <
Ip I
n") 2
- N
(1.9)
4
GALABOV AND DUDEV
Since the radiation beam intensity is equal to I = c p, we obtain 8~3I
n,,)2
(Nn" - Nn') dl"
(1.10)
or
-dlnI
8~3
=
Vn'n"~
(n' I P In") 2 (Nn" -Nn')dl.
(1.11)
For a given electronic state of the molecule the matrix elements (n' [ p[ n") refer to vibrational-rotational wave functions q~VP,- In the approximation in which vibrational and rotational energies are separable we may write [4]
P [n")2:~
WX~R P WVR dxVR (1.12) t,
=
I,
n
n
~Fv ~FR P~Fv ~FR d'rvR .
The same approximation expressed in terms of quantum numbers yields
Vn'n" = VV,W + VR,R--
(1.13)
It is now of interest to follow how the dipole moment operator acts upon the matrix element components [Eq. (1.12)]. The transition dipole matrix element, as given in expressions (1.1) and (1.5) is defined with respect to a molecule-fixed Cartesian reference system. In this system with axes denoted by x, y and z the dipole moment is expressed as p2 = p2x +p2 +pz2 "
(1.14)
In order to describe the rotational motion of the molecule it is necessary to express p in terms of coordinates referring to a space-fixed Cartesian system. The transformation from molecule-fixed (x, y, z) to space-fixed (X, Y, Z) coordinates is defined as [4] PX = OXx Px + OXy Py + OXz Pz PY = ~Yx Px + ~Yy Py + ~Yz Pz PZ = OZx Px + OZy Py + ~Zz Pz 9
(1.15)
ABSORPTION OF INFRARED RADIATION
5
cl)Fg (F = X, Y, Z; g = x, y, z) are the direction cosines between the respective axes. The transition dipole matrix element may then be written in the form ~ W~r* V~t* (~FxPx + ~ F y p y +~Fzpz)W~W~,dXvR .
(1.16)
F=X,Y,Z
To evaluate the dipole matrix element for a transition between two vibrational states v" -~ v' it is necessary to sum expression (1.16) over all rotational quantum numbers R' and R" associated with the vibrational transition. Considering Eqs. (1.12) and (1.13) and the invariance of molecular dipole moment with respect to orientation in space and, therefore, to rotational coordinates, the following expression for the transition matrix element between vibrational-rotational states n' and n" is obtained [4]
E
(n'lpln")2=
F=X,Y,Z
I E g=x,y,z
(1.17) -
( R' I ~g [R" ) is the rotational wave function matrix element between space - fixed (F) and molecule-fixed (g) Cartesian axes. The matrix elements ( n ' l p l n " ) are primarily associated with vibrational transitions. Since, however, the rotational quanttma numbers R' and R" also change, in a consistent treatment the transition probabilities of all rotational components of a vibrational absorption band must be evaluated. Usually, an approximate approach is adopted [4] and the direction cosine matrix elements replaced by classical averages over the cosines. These are equal to [3-5]
(l)Fg r
= (1/3) ~gg,
(1.18)
where 8gg, is the Kronecker delta symbol (~gg, = 0 if g ~ g' and ~Sgg,= 1 if g = g'). If we now tam back to Eq. (1.8) the population difference (Nn,,- Nn,) must also be approximated analogously to the treatment of the lransitional dipole moment. Using the Boltzmann distribution relations we obtain Nn, = Nn- e
- 0av n, - hvn.) / kT
(1.19) NV,R, = NV-R,, e
- h(Vv, v. + VR,R. ) / kT
6
GALABOV AND DUDEV
In Eq. (1.19) k is the Boltzmann constant and T the absolute temperature. By omitting the rotational terms and taking into account relation (1.18) expression (1.11) becomes 8/i; 3 - dlnI = Vv,v,, 3oh ( V' I p IV" )2 (Nv, ' _ NV,) dl.
(1.20)
Aside from the frequency factor Vv,v,, the remaining part of the expression correctly accounts for the transition probabilities of all rotational components of an infrared absorption band since they have identical vibrational matrix elements. These, however, differ in frequency in view of relation (1.13). In integral form Eq. (1.20) becomes 8/I;31 ln(Io/I) = Vv,v,, 3oh ( V' I p IV" )2 (Nv, _ NV,) "
(1.21)
Io is the intensity of the incident beam. It is a reasonable approximation to accept that the Einstein coefficient is a constant for a quantum jump between two vibrational states [4]. The integration over the entire band will, therefore, yield the following expression A = _1 ~band In (~) dv = v v ' v " 3sn3 - ~ ( v' I p ]w )2 (Nv. _ NV,) 1
(1.22)
If the integration is to be carried out over the individual rotational components of a band the result would be, evidently, slightly different. This inaccuracy may be treated by placing the firequency factor on the left hand side lr = 1 1
and
In
dirty=-3ch
<
l pl
-Nv').
(1.23)
The integrated intensity of the band F becomes, however, frequency dependent. Thus, the relative intensities of different bands in a molecule, or in different molecules, cannot be directly compared. This inconvenience is, very possibly, the reason that in most studies the integrated absorption coefficient A is preferred. It is now necessary to consider in more detail the matrix element and the population factor appearing in the right-hand side of Eqs. (1.22) and (1.23) in order to arrive at an expression that will directly relate the observed integrated infrared absorption band intensities to quantifies characterizing molecular structure. In the usual notation the (V' I p IV") matrix element reads (V'lp
IV") =j'q"-~* p %
dxv.
(1.24)
ABSORPTION OF INFRARED RADIATION
7
Vibrational wave functions may be presented as a product of linear harmonic oscillator wave functions defined with respect to a set of generalized molecule-fixed coordinates called normal vibrational coordinates [4] tFv = q/l(Ql) tF2(Q2) .-- Vk(Qk) -.. tFsN-6(QsN-6)
(1.25)
or
qJV =
3N-6 rI ~i(Oi).
(1.26)
i
The coordinates Qi are determined in the process of a semiclassical treatmem of molecular vibrations [3-6]. The principal aim of these calculations is to define the specific coordinates Qi, in the basis of which the Schr6dinger wave functions for the vibrational motion of a molecule are reduced to 3N-6 simple linear harmonic oscillator wave functions. One of the mathematical expressions of this result is Eq. (1.25). 3N-6 is the number of vibrations in an N-atomic molecule (3N-5 in the case of linear molecules). Described in terms of Qi the complex vibrational motion of a molecule is expressed as a superposition of 3N-6 linear harmonic oscillator vibrations, each having specific form as described by Qi and own frequency of oscillation. More comments about normal coordinates will be given in the following section. For a complete description of the theory of normal vibrations the reader is referred to a number of monographs [3-6]. For small vibrations the molecular dipole moment may be expressed as a Taylor series along the displacement coordinates Qi
p=po+E k
1 +'6
0
[
/)3p
,
/
)
0
Q, (1.27)
E (~Qk~--~lOQm Ok Ol Om + --k,l,m 0
For small displacements from the equilibrium configuration, under conditions of mechanical harmonicity (the potential energy is a quadratic function) and electrical harmonicity (the dipole moment is a linear function of vibrational coordinates), the higher terms in Eq. (1.27) are neglected. In the second part of this section we shall discuss in some detail the selection rules that govern vibrational transitions associated with absorption of a photon in the infrared region. We need first to derive, however, an expression relating the measured integrated absorption intensities with quantifies reflecting the electric charge fluctuations taking place in molecules with vibrational distortions. As we shall see, the harmonic oscillator selection rule restricts the allowed transitions only to those involving a change of a single vibrational quantum number by
8
GALABOV AND DUDEV
• For a fundamental transition from ground to one of the first excited vibrational states associated with a normal vibration described by the coordinate Qk the transition dipole matrix element is reduced simply to
J Vk
(Qk) P ~FI~(Qk)dQk =
J Vk
(Qk)Qk ~F~'(Qk)dQk-
(1.28)
0 Substitution with the analytical expression for the wave function qJk(QQ into the integral part of the right-hand side of Eqn (1.28) results in [4] ))1/2 _,v+l h (v k + 1 . J" ~Fk (Qk)Qk V~" (Qk)dQk = 8X2t,Ok
(1.29)
This expression is valid for all excitations involving a change in a vibrational quantum number by 1. Therefore, in the harmonic approximation the integration is carried out also over transitions from higher vibrational states associated with the so-called hot bands. Since these energy levels become populated at higher temperatures, it follows that within the approximations introduced so far, the intensities of absorption bands (Avk = 1) will be temperature independent. In expression (1.29) cok is the harmonic frequency of the vibration. Taking into account relations (1.28) and (1.29) the following expression for the integrated intensity of an infrared absorption band is obtained
A k = ~t'Vk (v k + 1) (Nv,, - N V,) 3cto k
(1.30)
In harmonic approximation, by using the Boltzmann distribution relation, the number of molecules in a given state with vibrational quantum number vk can be expressed as Nvk = Nvo e
(- hVk / kT) Vk
(1.31)
where Nvo is the number of molecules in the ground vibrational state (vibrational quantum number equal to 0). Nv0 is related to the total number of molecules per unit volume N through the relations
ABSORPTION OF INFRARED RADIATION
9
N = Nv0 + Nvl + Nv2 + ... + Nvk + ... = Nvo + Nvo e (-hvk/kT) + Nvo e(-hvk/kT)2 +...+ Nv0 e (-hvk/kT)vk +...
(1.32)
oo
= Nv0
E e(-hvk/kT)vk = Nvo ( 1 - e - h v k / k T ) -1 = Nvo Ok, Vk=0
where Ok = (1 - e
- hv k / kT
)-1
(1.33)
is the vibrational partition function associated with the k-th mode. Thus, from Eqs. (1.31) and (1.32) we have
N
Nvo = - - Ok
(1.34)
Nv k = N e(-hvk/kT)vk.
(1.35)
and
Ok The number of molecules in a higher energy vibrational state with vibrational quantum number vk+l is
N e(_hvk/kT) (Vk+1)
Nv k +1 = Ok
(1.36)
Combining (1.35) and (1.36) we obtain N (1-e-hVk/kT) e(-hvk/kT)vk " Nvk - N v k + l =~-k
(1.37)
Taking into account the result of summation in Eq. (1.32) and that oo
Vk e(-hvk/kT)vk = (1-e-hVk/kT) -2 e-hVk/kT Vk =0 we finally obtain
(1.38)
10
GALABOV AND DUDEV
oo
E (Nvz -Nv k+1)(Vk + 1) v k =0
(1.39)
-__~ Ok
(,-e-~k'kT)
oo
X
e(-hvk/kT)vk (v k + 1)= N.
Vk=0
The integrated absorption coefficient [Eq. (1.30)] is then expressed as
Ak =~1; band in ( ~ ) d v = Nn (V~k_k)(3Q~k)2
(1.40)
The normalized with respect to molar concentration (N = m No) absorption coefficient becomes 1 A~=~I~. '~
Nox v k d~---~ ~ ~
(1.41)
(t3px/tgQk) 2 + (o3p/t3Qk) 2 + (t3pz/CqQk) 2 .
(1.42)
with (t3p/t3Qk) 2 =
In expression (1.41) m is the molar concentration and N Othe Avogadro number. Usually, in practice, wavenumbers are used to determine band positions instead of classical frequencies. Thus, Eq. (1.41)expressed in terms ofwavenumbers (~ = v/c) reads:
A~=~f~ 1.
d~=~ ~
(1.43)
In further discussion vibrational frequencies will be expressed in cm-1. Therefore, for the sake of simplicity hereafter symbols v and o will be used to denote the respective wavenumbers. Since for most polyatomic molecules the harmonic frequencies (Ok) are difficult to determine experimentally, expression (1.43) is approximated by omitting the frequency factor to
(/2
Ak = N0___..~x.,Op 3c 2
it
(1.44)
ABSORPTION OF INFRARED RADIATION
11
The analogous expression for the quantity F k is
(1.45) 3C2Vk For the reasons mentioned above cok is replaced by the frequency of the observed band center Vk. Ak and F k are related by the approximate relation Ak = F k v k .
(1.46)
The approximation comes from the different approach in integrating the observed band areas in evaluating Ak and F k [see expressions (1.22) and (1.23)]. IfEqs. (1.44) and (1.45) are expressed in terms of SI units (with exception of cm-1 for wavenumbers), as suggested first by Steele [7], for A k and F k we obtain:
(1.47) 3c2(4mz0)
(1.48)
Fk = 3c 2 (4~O)V k
where so is the permitfivity of vacuum. Thus, Ak is measured in km mo1-1 and F k in cm 2 tool-1. The contribution of rotational quantization to the integrated absorption coefficient has been treated for symmetric rotor molecules [8]. Summation over rotational quantum numbers for parallel and perpendicular bands introduces a correction factor in the expression for the absorption
Ak = f 1
tI11+exp( 0
<
0,
(149,
In relation (1.49) B is the rotational constant, c the light velocity and VkOthe frequency of the pure vibrational transition. Substitution of typical values for B and Vk~ shows that the error in 8p/SQk dipole moment derivatives due to neglecting rotational quantization may not exceed 5%. For heavier molecules the error will be less than 1 percent. Therefore,
12
GALABOV AND DUDEV
the use of expressions (1.47) and (1.48) in deriving the dipole moment derivatives with respect to normal coordinates is justified in most cases. For molecules with higher symmetry some vibrational transitions may be degenerate. In such cases the following expression relating the observed integrated absorption intensities and the dipole moment derivatives is used
Nor:
f(0Pgx)2
(~pg)2
/~)pg/2 t
(1.5o)
The summation is over all degenerate transitions with the same energy. Vibrations in such symmetric molecules are usually polarized along a single axis, provided that the x, y and z directions are chosen in accordance with the symmetry properties of molecules. For such degenerate vibrations the derivatives 01ag /0Qk, 0p~/0Q k and 0pgz/0Qk are equal. The absolute values of these quantifies may, therefore, be determined without difficulty. It is seen from relations (1.47) and (1.48) that absolute values of dipole moment derivatives can only be evaluated fxom experimental integrated intensifies. Therefore, the directions of charge shifts accompanying particular vibrational distortions remain undetermined. This is a major difficulty in any further reduction of vibrational intensity data. For many years the sign ambiguity problem for the dipole moment derivatives has been a cause for the limited application of vibrational intensities in structural analysis. Another formidable problem arises from difficulties in deriving individual Cartesian components of o~/~:~k derivatives from experimental measurements. For molecules in the gas-phase the components of the t~/c3Qk vector may only be determined for a molecule with sufficient symmetry, such that one of c~p~//~k components only is different from zero. In the general case, it is necessary to establish the direction of polarization of vibrations in order to further rationalize the structural information implicit in the measured absorption intensities. It is, therefore, not surprising that most vibrational intensity studies have been restricted to relatively small and symmetric molecules.
II, S E L E C T I O N R U L E S F O R I N F R A R E D A B S O R P T I O N
A. Harmonic Oscillator Selection Rules In deriving the relations between infrared absorption intensities and dipole moment derivatives we have restricted the treatment to transitions involving a change of a single
ABSORPTION OF INFRARED RADIATION
13
vibrational quantum number by +1. The respective selection rule arises from the properties of the linear harmonic oscillator wave function. In general, the selection rules indicate: (a) allowed transitions with absorption bands observed in the spectrum; (b) transitions that are forbidden with absorption bands of zero or small intensity. As mentioned, some selection rules are determined by properties of the harmonic oscillator wave functions. A second set of selection rules are associated with the symmetry properties of vibrations. The harmonic oscillator selection rules for vibrational transition can be evaluated using the expression of the dipole moment as a power series with respect to normal coordinates. On the basis of expressions (1.25) - (1.27) the dipole moment matrix element for a transition between vibrational states V' and V" may be written as
(v'l pIv")=IVv" p % d~v = P0 I"I I ~Fk ~Fk dQk k
.+
Z r
f C Q~ ~ dQ~FIf v;',ei' dQ1
k
1,k
l Z 2p /c)Qk2 )
+2 1
o
(1.51)
I ~Ij'k*Q2 ~k dQk H I ~PI*~I'11' dQl l~k
[~2P/(c~QkaQl)~I~FkQk ~Fk dQk f~FI*QI~FI'dQI I-[ I ~ W " m dQm m#l#k
+ higher terms. The first term in expression (1.51) is equal to zero except for the case with Vk'= Vk". This is determined by the orthogonality of vibrational eigenfunctions. In fact, it is zero for any transition between different vibrational states. The linear term is not zero for transitions involving a change of a single quantum number by + 1 only. Because of orthogonality of the function ~Pl the integral PPl'* ~Pl" dQl will be zero if any other vibrational quantum number except Vk is changed (Vl, l~k). The harmonic approximation restricts the dipole moment expansion to the constant and linear terms. Thus, the selection rule associated with the approximation of electrical harmonicity, states that transitions involving a change by + 1 of just one of the vibrational quantum numbers of the linear harmonic oscillator functions defining the vibrational states of molecules are only allowed. Aside from this, for a transition to take place at least one of the Cartesian components of the (0p/0Qk) 0 derivatives should differ from
14
GALABOV AND DUDEV
zero. We shall later see how the symmetry of vibrations determines the selection rules associated with these derivatives. The third term in expression (1.51) considers transitions associated with a change in a vibrational quantum number by 2. It governs the intensities of the usually weak overtone bands. The possibility of observing such transitions is determined by the fact that vibrations in real molecules are not strictly harmonic, both with respect to potential energy and dipole moment functions. When one quantum number is changed by 2, the linear term has also a f ~ t e , usually small, contribution to the matrix element. The fourth term is associated with the intensities of the weak combination bands (vibrational sum or difference bands). These bands are due to transitions involving changes by • 1 of two vibrational quantum numbers. Absorption bands associated with higher terms in the dipole moment function expansion are also observed. As expected, their intensities are orders of magnitude lower than the intensities of fundamental transitions. Nevertheless, studies on higher overtone transitions flourished during the past fifteen years [9,10]. The main reason for these developments is, very possibly, the unexpected "local mode" rather than "normal mode" properties for some of these transitions. Interesting opportunities for studying properties of individual chemical bonds in complex molecules have been revealed. Applications have mostly concentrated on higher overtones of C-H stretching modes [9,10].
B. Symmetry Selection Rules As stated before, a transition between different vibrational states may take place ff the respective coefficients in the dipole moment expansion as a power series to normal coordinates are not equal to zero. For a fundamental transition at least one of the components/~px//R~, i ~ p y / ~ and/)pz//~Qk must not be zero. For overtone transitions a non-zero value is required for at least one of the derivatives ~px/C3~ 2, ~ p y / ~ 2 and oa2pz/~2. A group theoretical analysis may determine the infrared active transitions by considering the symmetry properties of the vibrational wave functions of the interacting states and of molecular dipole moment. In most general terms, a transition between vibrational states V' and V" will take place if at least one of the component matrix elements differs from zero ( V ' [ Px IV")
(V' [ py IV")
( V ' [ Pz I V " ) .
(1.52)
If all three dipole matrix elements are zero the transition will be symmetry forbidden or infrared inactive.
ABSORPTION OF INFRARED RADIATION
15
The symmetry selection rules are derived by studying the effect of symmetry operations on the matrix elements. The rules stem from the property of dipole moment matrix elements to be invariant with respect to a symmetry operation (R) R ( ( V ' I p g [ V " > ) = (V' I pg IV")
(g = x , y , z ) .
(1.53)
Since the transition dipole is a physical observable, it is evident that its value should be independent under symmetry operations. In other words, the intramolecular charge distribution and fluctuations are invariant with respect to symmetry operations. The representation of transition dipole moment element is given by the direct product of the representations of the respective vibrational wave functions and dipole moment component rk = rv, • rpg • r w
(g = x, y, z)
(1.54)
where k is an index of the k-th normal mode. Since the dipole moment is a vector with components directed along the axes of the reference Cartesian system, it is clear that under a symmetry operation its components will transform as the respective x, y and z Cartesian axes. The same arguments hold for the igpx/0Qk,/gpy/~ and C3pz/~ dipole derivatives. Therefore, the symmetry representations of the dipole moment components, as present in expression (1.52) coincide with the representations of the respective Cartesian axes. The symmetry properties of vibrational wave functions are treated in detail elsewhere [3] and will not be discussed here. The representation of the ground vibrational state wave function belongs to the point group of the molecule at equilibrium configuration. The direct product F k coincides with one of the irreducible representations of the molecular point group. The component matrix element (V'[pg[V") will be different from zero only if the resulting F k coincides with the totally symmetric representation of the point group of the molecule F 0. In the case of degenerate vibrations F k is a reducible representation. The infrared active (allowed) transitions must have the totally symmetric irreducible representation in the structure of F k associated with the respective degenerate mode. The synunetry selection rule may also be expressed in alternative ways. An inflated transition is not forbidden only in the case where the direct product of the presentations of the two interacting states Fv, XFv,, coincides with the representation of at least one of the dipole moment Cartesian components. For a fundamental transition (v' k = 1, v" k = 0) the above requirement concerns the irreducible representation of the excited level (v'). The selection rule for such transitions is simply r v, • rpg=
to.
(1.55)
16
GALABOV AND DUDEV
Similar symmetry restrictions also apply for overtone and combination bands. As already discussed, these transitions are not aUowed under the harmonic oscillator selection rules. R should be pointed out that even ff a given transition is not forbidden under both symmetry and harmonic oscillator selection rules, it may have a very low intensity. This will be determined by the particular form of the vibration and the electronic structure of the molecule. The assignment of a given band to infrared active or forbidden transition is, therefore, not always a straightforward task. The derivation of expressions relating observed integrated infrared absorption coefficients with dipole moment matrix elements for the respective transitions shows that the experimental quantifies contain important structural information. It is related with the distribution and dynamics of electric charges in molecules. We should bear in mind, however, that there are a number of restrictions associated with the possibility to determine dipole moment derivatives with respect to normal coordinates. In many regions of the observed infrared spectrum for any molecule of a medium size, a strong overlap of closely situated bands is usuaUy present. Thus, individual intensifies for all fundamental transitions may not be accurately determined. Lately, with the development of advanced software for band deconvolution and curve fitting this difficulty has been, to some extent, overcome. As already mentioned, another formidable problem arises from the necessity to know the exact direction of polarization for each vibrational mode, so that individual Cartesian components of the Op/0Qk derivatives are evaluated. So far, no general approach to solve this problem experimentally for molecules in the gas-phase has been developed. Thus, in most cases, an elaborate molecular analysis of observed vibrational absorption intensities is only possible for higher symmetry molecules. Existing perturbations in the spectra associated with Fermi resonances, Coriolis interactions and strong anhannonicity effects may often hamper the interpretation of experimental intensity data. The sign ambiguity problem for dipole moment derivatives, as already discussed, is also present. Finally, it should be emphasized that the quantifies 0p/0Qk contain in a rather obscure form the structural information sought. This is due to the very complex nature of normal coordinates. It is, therefore, essential to further reduce the experimental 0p//~)k derivatives into quantifies characterizing electrical properties of molecular sub-units -atomic groupings, chemical bonds or individual atoms. Various theoretical formulations for analysis of vibrational intensities have been put forward. The approaches developed a r e quite analogous to the analysis of vibrational frequencies in terms of force constants. As known, force constants may be associated with properties of molecular sub-units. If such a rationalization of intensity data is successfully performed, another important aim of spectroscopy studies may become possible: quantitative prediction of vibrational intensities by transferring intensity parameters between molecules containing the same
ABSORPTION OF INFRARED 1LM)IATION
17
structural units in a similar environment. The analogy with transferable force constants should again be underlined.
III.
EXPERIMENTAL DETERMINATION OF INFRARED INTENSITIES
The theoretical models for interpretation of infrared intensities presented in the subsequent chapters have been largely applied in analyzing gas-phase experimental data. Gas-phase intensities provide an unique opportunity to study in a uniform approach the mterrelatiom between molecular structure and intensity parameters. This is due to the fact that, in contrast to vibrational frequencies, the absorption coefficients depend strongly on the phase state and on solvent effects. Intensities of different modes of the same molecule are not influenced in a systematic way by the solvent. The variations of absorption coefficients may reach tens and hundreds percent. Accurately determined gasphase intensities are, therefore, of fundamental importance as a source of experimental information on intramolecularproperties. Past difficulties in experimental measurements of integrated infrared intensities have been associated mostly with the low resolving power of spectrometers, poor accuracy on the ordinate and absence of computer facilities for band integration, deconvolution and curve fitting in overlap parts of the spectra. It is clear that presently we have far better experimental means for accurate determination of the integrated intensities of individual absorption bands. Still, however, careful considerations of a number of possible sources of errors are needed in order to obtain sufficiently accurate intensity data. Some of these problems will be discussed later on. It is interesting that the methods developed for experimental determination of vibrational intensities in the gas-phase were aimed at resolving problems arising mostly from the low resolution power of the available spectrometers at the time. It may appear that nowdays, when the researchers have access to instruments with resolution of the order of a few hundredths or even few thousandths of a wavenumber, these techniques may be of lesser importance. Although this is partly true, the current experimental approaches for experimental determination of vibrational intensities fully rely on the original developments. This is determined by the fact that these methods not only compensate for the effect of low resolution on intensities but also provide criteria for the accuracy of measurements and the influence of such phenomena as adsorption of sample gas or slow diffusion process. Thus, the extrapolation method of Wilson and Wells [ 11], further developed by Penner and Weber [12], is the standard approach for experimental intensity studies.
18
GALABOV AND DUDEV
Early attempts for experimental measurements of infrared intensities [13,14] resulted in greatly divergent values for the same molecule. It was soon realized that most of the difficulties were associated with the low resolving power of the spectrometers used [15]. The problems arise from the fact that the incident infrared beam emerging from the monochromator is not strictly monochromatic, but contains a band of frequencies around the frequency v' determined by the slit function g(v,v'). There is, thus, a perfectly good chance for the intensity of the transmitted radiation I by a cell I = Io exp (- ~ p 1)
(1.56)
to differ from the value corresponding to a monochromatic beam with frequency v'. This is particularly possible if the measurements are carried over rotational f'me structure of an infrared band where sharp variations of the absorption coefficient ct are expected. In expression (1.56) p is the pressure of the sample gas, 1is the optical path length of the cell and I0 is the intensity of the incident radiation. The quantity of principal interest in intensity measurements is the integrated absorption coefficient A as defined by Eqs. (1.43), (1.44) and (1.47). For gas samples the respective expression is (1.57)
Due to the finite width of the slit function g(v,v') the measured intensity at a setting v' will not be equal to the true absorption intensity I. An apparent intensity T(v') will be in fact determined. It is given by the integral oo
r ( v ' ) = I I(v)g(v,v')dv.
(1.58)
g(v,v') is the portion of the light of frequency v that reaches the detector when the spectrometer is set at frequency v'. The integration can be carried out from --oo to +oo since g(v,v') is different from zero only in the immediate vicinity of v'. The width of the frequency band is of the order of the spectrometer resolution. Analogous expression can be written for T o oo
T0(v') = I Io (v)g(v,v')dv.
The apparent absorption coefficient J3(v') can, therefore, be defined as
(1.59)
ABSORPTION OF INFRARED RADIATION
]3(v') = (1/pl) In [To(v')/T(v')].
19
(1.60)
The integration over the entire interval of an vibration-rotation band will produce the apparent integrated absorption coefficient B
B : Ibaad~ dr'
:lSb=d pl
(1.61)
The Wilson and Wells [ 11] extrapolation theorem proves that Lira B = A. pl-~
(1.62)
Since both p and 1 can be varied in experimental conditions, the theorem (1.62) provides a convenient way of determining the true absorption coefficient A. The entire derivation of Wilson and Wells [ 11] will be presented since important considerations associated with the accuracy of intensity measurements emerge at different stages of proving the theorem. The difference (A-B) is examined
1 Sbandhl(TIO/dv'= 1 ~bandin( f(v') ~ f(v)g(v,v')_:_v/ It T0I ) p-I ~ g(v, v')dv) d r ' .
A - B = p--~
(1.63)
In expression (1.63) f(v) is f(v) = e x p ( - t x p l )
- I/I o
(1.64)
and I0 is assumed to be constant over the frequency range of the slit width. In presence of foreign gases, such as H20 and CO 2, this assumption may not hold. It is of interest to consider here the function f. Let us suppose that f is constant over the frequency range of the slit width. It becomes a constant multiplier of the inner integral and the entire expression (1.63) vanishes. B will then be equal to A. If, however, the width of individual rotational lines is of the order of the slit width, the exponential function f can vary considerably and B may be very different from A. The limit of A-B with p l y 0 needs now be considered. Expression (1.63) is differentiated with respect to pl to yield [ctfgdv l ~ g d__v1j dr'. Lira(A-B)= Lim I {c~-'.~Tg?~ J dv'= I ~/ct - SIgdv
(1.65)
20
GALABOV AND DUDEV
The function f = exp (- a p 1) approaches unity at the limit. Expression (1.65) will vanish and the theorem proven under certain conditions. First, it will vanish if the absorption coefficient {x is constant over the integration range which is the slit width. Note that expression (1.63) vanishes under the condition that f as an exponent of a, is a constant. In Eq. (1.65) the analogous requirement is for c~. Small variations in ~ can result in much larger changes in f. On the second place, expression (1.65) may vanish if the resolving power of the monochromator is constant over the width of the band regardless of {z. In mathematical terms this condition is expressed as g(v,v') = g(v - v')
(1.66)
g ( v - v') = g(v'- v).
(1.67)
and
If conditions (1.66) and (1.67) are satisfied, we have g(v - v') dv = ~ g(v - v') d(v-v') = G.
(1.68)
G is independent of v'. Another simplification follows f~{xgdvdv' = ~c~fgdvdv' : Gfr
(1.69)
Substituting (1.68) and (1.69) into (1.65) we obtain ~ct dv' - ~ctdv = 0 .
(1.70)
Consequently, Lim B = A pl-,0
(1.71)
if a number of conditions are met. The first is that I 0 is a constant in a resolved range. This can be achieved by removing external absorption fxom atmospheric H20 and CO 2. It is also clear that working under conditions of higher resolution betters the constancy of I0 in the narrower resolved range. In addition, the theorem (1.71) will hold if {z does not vary over the range of the resolution of the spectrometer, or if the resolution is constant over the entire vibrational-rotational band. If appropriate care is taken so that the above
ABSORPTION OF INFRARED RADIATION
21
o:0r V
J
F Bcl
I
50~1
0
I
I
I
1
1
I
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 c/,equivolent poth length in cm at S.T.P.
Fig. 1.1. The dependence Bpl/pl for the infrared active modes of methane (Reproduced from Ref. [ 16] with permission).
mentioned conditions are satisfied, Eq. (1.71) provides a convenient approach of determining the ~ue absorption coefficient A by extrapolating B to zero value of pl. It should be noted that the requirement for constant resolution over the integration range of a band is a specific property of the insmxment. It may not always be fully satisfied. For small molecules in the gas-phase at low pressure with clearly expressed fine rotational structure, the variations of r and f = exp(-~pl) at each rotational line is so extreme that the conditions for accurate extrapolation measurement may not be present. Following an early approach of Bartholome [14], Wilson and Wefts [11] recommended that for such molecules a foreign non-absorbing gas under higher pressure is introduced in the sample cell. It will induce a collision broadening of rotational lines and at sufficiently high pressure will lead to a collapse of the fine structure. Obviously, under such conditions the extrapolation method may be used with confidence. In the original method of Wilson and Wells the true integrated absorption coefficient A is determined as the slope of tangent at the origin of the dependence Bpl versus pl. Typical plots of this dependence are shown in Fig. 1.1. There is, however, certain arbiuminess in choosing the exact direction of the tangent at the origin. The method also requires that the measurements be done at very low partial pressures of the absorbing gas where the accuracy is low. Considerable errors can, therefore, result. The problem is resolved by working at sufficiently high pressures of the inert transparent
22
GALABOV AND DUDEV
160 iO.lcm CH4 ~i 120
,
.
....
U
T uE 8 0
4.95 cm CH4
O I
o ,e-
d~ 4o
0
.
.
.
.
.
2".50 cm
CH 4
100 200 300 400 Pressure of nitrogen inotmospheres
Fig. 1.2. Dependence between the apparent integrated absorption of the v 3 band of CH4 and the pressure of added nitrogen (Reproduced from Ref. [ 16] with permission). 1.2
O.6
In(1~
------,0.6 ---~1,0.4 I
! ---4o.2
~lOAtrn/
i
~OAt~~ 42Atm~
OAtm 2800
L
___---- ~ ~ ' N ~ 3000
1 3200
1~ ~C m -I
Fig. 1.3. v 3 band of methane under different pressures of nitrogen used as an external gas. The partial pressure of methane is unchanged (Reproduced from Ref. [ 16]
with permission).
ABSORPTION OF INFRARED RADIATION
23
foreign gas (Ar, N2, etc.) so that the Beer's low plot becomes a straight line passing through the origin. This procedure has been suggested by Penner and Weber [ 12]. Depending on the size of the molecules, the type of rotational fme structure or band shape, the adequate pressure may vary significantly. For larger molecules a pressure of one atmosphere may be sufficient. For small molecules a pressure of up to I00 atm may be needed to reach the linear region of the dependence Bpl/pl. An example is shown in Fig. 1.2. The high pressures used may cause, however, some additional complications. For larger molecules under high pressure a certain amount of the gas sample may deposit on the cell walls, though in absence of foreign gas when the partial pressure is measured, the sample is in a gas-phase. This is a particular property of the molecule under study and each case needs careful consideration so that appropriate conditions for measurements are chosen. Overend [16] has pointed out that pressure-induced absorption can affect the apparent absorption coefficient value. The effect is attributed to intermolecular interaction. It is manifested in the slow rise of the apparent absorption coefficient B as the pressure is increased. The phenomenon is clearly shown in Fig. 1.3. If such effects are present, the pressure-induced absorption has to be eliminated. This is achieved by extrapolating the linear part of the curve to zero pressure of the external gas for each B value determined.
ABSORPTION
OF INFRARED
RADIATION
BY MOLECULES
I.
Theoretical Considerations .................................................................................... 2
II.
Selection Rules For Infrared Absorption .............................................................. 12
m~
A.
Harmonic Oscillator Selection Rules .......................................................... 12
B.
Symmetry Selection Rules .......................................................................... 14
Experimental Determination of Infrared Intensities .............................................. 17
2
GALABOV AND DUDEV
I. T H E O R E T I C A L
CONSIDERATIONS
The probability of absorption of a photon with energy hvn,n- by a molecule per unit of time leading to a transition between a lower energy state n" to higher state n' is given by [1-3]
8~;3 Wn': =-~ (n'[ X ej(u~ rj)
I
n#)2
P(Vn,n,,).
(1.1)
J In expression (1.1) ej and rj are the electric charge and position vector of atom j ill a molecule, rj refers to an arbitrary molecule-fixed Cartesian system, u x is the position vector of the photon with respect to a space-fixed Cartesian system. The quantity p called radiation density is equal to the number of photons with energy hvn,n- per unit volume. It is understood that the Bohr condition En, - F_.n,= hvn,n- must be satisfied. The polarization vector of the photons ux does not affect the molecular wave functions. A quantity called electric transition dipole may, therefore, be defined
P c : =(n'[ E ejrj In">.
(1.2)
J Since the electric dipole moment is given by
p
=
Xejrj,
(1.3)
J Eq. (1.2) becomes
Pn'n" = ( n ' l p In").
(1.4)
Pn'n" has components along the x, y and z axes of the molecule-fixed Cartesian system. The directions of ux and Pn'n" need not coincide since molecules are randomly oriented. There is, therefore, an angle 0 between the vectors ux and Pn'n"- Eq. (1.1) may then be rewritten as
8~;3 n' [ p [ n") 2 COS2 0 P(Vn,n-).
Wn,n- = - - ~ (
(1.5)
cos20 should be taken as an average over all possible orientations of the molecule in space
ABSORPTION OF INFRARED RADIATION
3
2rc COS2 0 = --L1 f cos 2 0 Sin 0 dO dO = _1. 4~ 0 0 3
i
(1.6)
The transition probability associated with absorption of a photon with energy hvn,n,, is then given by
8/I;3
(1.7)
The quantity Bn,n,, = (81t3/3h2)(n'lpln") 2 is the well known Einstein coefficient of absorption. Per unit radiation density it is equal to the transition probability. The Einstein coefficient depends on the molecular structure and may, eventually, be used to characterize molecular properties on the basis of experimentally determined intensities of the respective transitions. In the infrared region each vibrational transition is accompanied by a number of quantum transitions between rotational states of the molecules. In lower resolution spectra or if conditions for considerable broadening of absorption lines are present, the accompanying rotational transitions determine usually a non-symmetric PQR structure of the infrared band. In higher resolution spectra the individual rotational lines are separated. The intensity of an infrared absorption band represents, therefore, a sum over the intensities of all fine structure lines associated with the respective vibrational transition. It is, thus, necessary to describe the factors determining intensities of the component rotational lines and, then, see how these sum up into overall intensity of an infrared band. The probability for a reverse quantum transition from higher (n') to lower energy state (n") is also determined by expression (1.7). Therefore, the probability for absorption of electromagnetic radiation will depend on the number of molecules per unit volume of absorbing medium in the lower (Nn,,) and higher (Nn,) states
8~3 (n' I pin") 2 P(Vn,n-)(Nn--Nn,).
(1.8)
3h 2
For every elementary act of interaction of the electromagnetic radiation with a molecule an energy hvn,n, is absorbed provided some conditions are met. The energy absorbed by a differential element with a cross section equal to unity and thickness dl is equal to 81t3 n"
= vn'n" Sh-- <
Ip I
n") 2
- N
(1.9)
4
GALABOV AND DUDEV
Since the radiation beam intensity is equal to I = c p, we obtain 8~3I
n,,)2
(Nn" - Nn') dl"
(1.10)
or
-dlnI
8~3
=
Vn'n"~
(n' I P In") 2 (Nn" -Nn')dl.
(1.11)
For a given electronic state of the molecule the matrix elements (n' [ p[ n") refer to vibrational-rotational wave functions q~VP,- In the approximation in which vibrational and rotational energies are separable we may write [4]
P [n")2:~
WX~R P WVR dxVR (1.12) t,
=
I,
n
n
~Fv ~FR P~Fv ~FR d'rvR .
The same approximation expressed in terms of quantum numbers yields
Vn'n" = VV,W + VR,R--
(1.13)
It is now of interest to follow how the dipole moment operator acts upon the matrix element components [Eq. (1.12)]. The transition dipole matrix element, as given in expressions (1.1) and (1.5) is defined with respect to a molecule-fixed Cartesian reference system. In this system with axes denoted by x, y and z the dipole moment is expressed as p2 = p2x +p2 +pz2 "
(1.14)
In order to describe the rotational motion of the molecule it is necessary to express p in terms of coordinates referring to a space-fixed Cartesian system. The transformation from molecule-fixed (x, y, z) to space-fixed (X, Y, Z) coordinates is defined as [4] PX = OXx Px + OXy Py + OXz Pz PY = ~Yx Px + ~Yy Py + ~Yz Pz PZ = OZx Px + OZy Py + ~Zz Pz 9
(1.15)
ABSORPTION OF INFRARED RADIATION
5
cl)Fg (F = X, Y, Z; g = x, y, z) are the direction cosines between the respective axes. The transition dipole matrix element may then be written in the form ~ W~r* V~t* (~FxPx + ~ F y p y +~Fzpz)W~W~,dXvR .
(1.16)
F=X,Y,Z
To evaluate the dipole matrix element for a transition between two vibrational states v" -~ v' it is necessary to sum expression (1.16) over all rotational quantum numbers R' and R" associated with the vibrational transition. Considering Eqs. (1.12) and (1.13) and the invariance of molecular dipole moment with respect to orientation in space and, therefore, to rotational coordinates, the following expression for the transition matrix element between vibrational-rotational states n' and n" is obtained [4]
E
(n'lpln")2=
F=X,Y,Z
I E g=x,y,z
(1.17) -
( R' I ~g [R" ) is the rotational wave function matrix element between space - fixed (F) and molecule-fixed (g) Cartesian axes. The matrix elements ( n ' l p l n " ) are primarily associated with vibrational transitions. Since, however, the rotational quanttma numbers R' and R" also change, in a consistent treatment the transition probabilities of all rotational components of a vibrational absorption band must be evaluated. Usually, an approximate approach is adopted [4] and the direction cosine matrix elements replaced by classical averages over the cosines. These are equal to [3-5]
(l)Fg r
= (1/3) ~gg,
(1.18)
where 8gg, is the Kronecker delta symbol (~gg, = 0 if g ~ g' and ~Sgg,= 1 if g = g'). If we now tam back to Eq. (1.8) the population difference (Nn,,- Nn,) must also be approximated analogously to the treatment of the lransitional dipole moment. Using the Boltzmann distribution relations we obtain Nn, = Nn- e
- 0av n, - hvn.) / kT
(1.19) NV,R, = NV-R,, e
- h(Vv, v. + VR,R. ) / kT
6
GALABOV AND DUDEV
In Eq. (1.19) k is the Boltzmann constant and T the absolute temperature. By omitting the rotational terms and taking into account relation (1.18) expression (1.11) becomes 8/i; 3 - dlnI = Vv,v,, 3oh ( V' I p IV" )2 (Nv, ' _ NV,) dl.
(1.20)
Aside from the frequency factor Vv,v,, the remaining part of the expression correctly accounts for the transition probabilities of all rotational components of an infrared absorption band since they have identical vibrational matrix elements. These, however, differ in frequency in view of relation (1.13). In integral form Eq. (1.20) becomes 8/I;31 ln(Io/I) = Vv,v,, 3oh ( V' I p IV" )2 (Nv, _ NV,) "
(1.21)
Io is the intensity of the incident beam. It is a reasonable approximation to accept that the Einstein coefficient is a constant for a quantum jump between two vibrational states [4]. The integration over the entire band will, therefore, yield the following expression A = _1 ~band In (~) dv = v v ' v " 3sn3 - ~ ( v' I p ]w )2 (Nv. _ NV,) 1
(1.22)
If the integration is to be carried out over the individual rotational components of a band the result would be, evidently, slightly different. This inaccuracy may be treated by placing the firequency factor on the left hand side lr = 1 1
and
In
dirty=-3ch
<
l pl
-Nv').
(1.23)
The integrated intensity of the band F becomes, however, frequency dependent. Thus, the relative intensities of different bands in a molecule, or in different molecules, cannot be directly compared. This inconvenience is, very possibly, the reason that in most studies the integrated absorption coefficient A is preferred. It is now necessary to consider in more detail the matrix element and the population factor appearing in the right-hand side of Eqs. (1.22) and (1.23) in order to arrive at an expression that will directly relate the observed integrated infrared absorption band intensities to quantifies characterizing molecular structure. In the usual notation the (V' I p IV") matrix element reads (V'lp
IV") =j'q"-~* p %
dxv.
(1.24)
ABSORPTION OF INFRARED RADIATION
7
Vibrational wave functions may be presented as a product of linear harmonic oscillator wave functions defined with respect to a set of generalized molecule-fixed coordinates called normal vibrational coordinates [4] tFv = q/l(Ql) tF2(Q2) .-- Vk(Qk) -.. tFsN-6(QsN-6)
(1.25)
or
qJV =
3N-6 rI ~i(Oi).
(1.26)
i
The coordinates Qi are determined in the process of a semiclassical treatmem of molecular vibrations [3-6]. The principal aim of these calculations is to define the specific coordinates Qi, in the basis of which the Schr6dinger wave functions for the vibrational motion of a molecule are reduced to 3N-6 simple linear harmonic oscillator wave functions. One of the mathematical expressions of this result is Eq. (1.25). 3N-6 is the number of vibrations in an N-atomic molecule (3N-5 in the case of linear molecules). Described in terms of Qi the complex vibrational motion of a molecule is expressed as a superposition of 3N-6 linear harmonic oscillator vibrations, each having specific form as described by Qi and own frequency of oscillation. More comments about normal coordinates will be given in the following section. For a complete description of the theory of normal vibrations the reader is referred to a number of monographs [3-6]. For small vibrations the molecular dipole moment may be expressed as a Taylor series along the displacement coordinates Qi
p=po+E k
1 +'6
0
[
/)3p
,
/
)
0
Q, (1.27)
E (~Qk~--~lOQm Ok Ol Om + --k,l,m 0
For small displacements from the equilibrium configuration, under conditions of mechanical harmonicity (the potential energy is a quadratic function) and electrical harmonicity (the dipole moment is a linear function of vibrational coordinates), the higher terms in Eq. (1.27) are neglected. In the second part of this section we shall discuss in some detail the selection rules that govern vibrational transitions associated with absorption of a photon in the infrared region. We need first to derive, however, an expression relating the measured integrated absorption intensities with quantifies reflecting the electric charge fluctuations taking place in molecules with vibrational distortions. As we shall see, the harmonic oscillator selection rule restricts the allowed transitions only to those involving a change of a single vibrational quantum number by
8
GALABOV AND DUDEV
• For a fundamental transition from ground to one of the first excited vibrational states associated with a normal vibration described by the coordinate Qk the transition dipole matrix element is reduced simply to
J Vk
(Qk) P ~FI~(Qk)dQk =
J Vk
(Qk)Qk ~F~'(Qk)dQk-
(1.28)
0 Substitution with the analytical expression for the wave function qJk(QQ into the integral part of the right-hand side of Eqn (1.28) results in [4] ))1/2 _,v+l h (v k + 1 . J" ~Fk (Qk)Qk V~" (Qk)dQk = 8X2t,Ok
(1.29)
This expression is valid for all excitations involving a change in a vibrational quantum number by 1. Therefore, in the harmonic approximation the integration is carried out also over transitions from higher vibrational states associated with the so-called hot bands. Since these energy levels become populated at higher temperatures, it follows that within the approximations introduced so far, the intensities of absorption bands (Avk = 1) will be temperature independent. In expression (1.29) cok is the harmonic frequency of the vibration. Taking into account relations (1.28) and (1.29) the following expression for the integrated intensity of an infrared absorption band is obtained
A k = ~t'Vk (v k + 1) (Nv,, - N V,) 3cto k
(1.30)
In harmonic approximation, by using the Boltzmann distribution relation, the number of molecules in a given state with vibrational quantum number vk can be expressed as Nvk = Nvo e
(- hVk / kT) Vk
(1.31)
where Nvo is the number of molecules in the ground vibrational state (vibrational quantum number equal to 0). Nv0 is related to the total number of molecules per unit volume N through the relations
ABSORPTION OF INFRARED RADIATION
9
N = Nv0 + Nvl + Nv2 + ... + Nvk + ... = Nvo + Nvo e (-hvk/kT) + Nvo e(-hvk/kT)2 +...+ Nv0 e (-hvk/kT)vk +...
(1.32)
oo
= Nv0
E e(-hvk/kT)vk = Nvo ( 1 - e - h v k / k T ) -1 = Nvo Ok, Vk=0
where Ok = (1 - e
- hv k / kT
)-1
(1.33)
is the vibrational partition function associated with the k-th mode. Thus, from Eqs. (1.31) and (1.32) we have
N
Nvo = - - Ok
(1.34)
Nv k = N e(-hvk/kT)vk.
(1.35)
and
Ok The number of molecules in a higher energy vibrational state with vibrational quantum number vk+l is
N e(_hvk/kT) (Vk+1)
Nv k +1 = Ok
(1.36)
Combining (1.35) and (1.36) we obtain N (1-e-hVk/kT) e(-hvk/kT)vk " Nvk - N v k + l =~-k
(1.37)
Taking into account the result of summation in Eq. (1.32) and that oo
Vk e(-hvk/kT)vk = (1-e-hVk/kT) -2 e-hVk/kT Vk =0 we finally obtain
(1.38)
10
GALABOV AND DUDEV
oo
E (Nvz -Nv k+1)(Vk + 1) v k =0
(1.39)
-__~ Ok
(,-e-~k'kT)
oo
X
e(-hvk/kT)vk (v k + 1)= N.
Vk=0
The integrated absorption coefficient [Eq. (1.30)] is then expressed as
Ak =~1; band in ( ~ ) d v = Nn (V~k_k)(3Q~k)2
(1.40)
The normalized with respect to molar concentration (N = m No) absorption coefficient becomes 1 A~=~I~. '~
Nox v k d~---~ ~ ~
(1.41)
(t3px/tgQk) 2 + (o3p/t3Qk) 2 + (t3pz/CqQk) 2 .
(1.42)
with (t3p/t3Qk) 2 =
In expression (1.41) m is the molar concentration and N Othe Avogadro number. Usually, in practice, wavenumbers are used to determine band positions instead of classical frequencies. Thus, Eq. (1.41)expressed in terms ofwavenumbers (~ = v/c) reads:
A~=~f~ 1.
d~=~ ~
(1.43)
In further discussion vibrational frequencies will be expressed in cm-1. Therefore, for the sake of simplicity hereafter symbols v and o will be used to denote the respective wavenumbers. Since for most polyatomic molecules the harmonic frequencies (Ok) are difficult to determine experimentally, expression (1.43) is approximated by omitting the frequency factor to
(/2
Ak = N0___..~x.,Op 3c 2
it
(1.44)
ABSORPTION OF INFRARED RADIATION
11
The analogous expression for the quantity F k is
(1.45) 3C2Vk For the reasons mentioned above cok is replaced by the frequency of the observed band center Vk. Ak and F k are related by the approximate relation Ak = F k v k .
(1.46)
The approximation comes from the different approach in integrating the observed band areas in evaluating Ak and F k [see expressions (1.22) and (1.23)]. IfEqs. (1.44) and (1.45) are expressed in terms of SI units (with exception of cm-1 for wavenumbers), as suggested first by Steele [7], for A k and F k we obtain:
(1.47) 3c2(4mz0)
(1.48)
Fk = 3c 2 (4~O)V k
where so is the permitfivity of vacuum. Thus, Ak is measured in km mo1-1 and F k in cm 2 tool-1. The contribution of rotational quantization to the integrated absorption coefficient has been treated for symmetric rotor molecules [8]. Summation over rotational quantum numbers for parallel and perpendicular bands introduces a correction factor in the expression for the absorption
Ak = f 1
tI11+exp( 0
<
0,
(149,
In relation (1.49) B is the rotational constant, c the light velocity and VkOthe frequency of the pure vibrational transition. Substitution of typical values for B and Vk~ shows that the error in 8p/SQk dipole moment derivatives due to neglecting rotational quantization may not exceed 5%. For heavier molecules the error will be less than 1 percent. Therefore,
12
GALABOV AND DUDEV
the use of expressions (1.47) and (1.48) in deriving the dipole moment derivatives with respect to normal coordinates is justified in most cases. For molecules with higher symmetry some vibrational transitions may be degenerate. In such cases the following expression relating the observed integrated absorption intensities and the dipole moment derivatives is used
Nor:
f(0Pgx)2
(~pg)2
/~)pg/2 t
(1.5o)
The summation is over all degenerate transitions with the same energy. Vibrations in such symmetric molecules are usually polarized along a single axis, provided that the x, y and z directions are chosen in accordance with the symmetry properties of molecules. For such degenerate vibrations the derivatives 01ag /0Qk, 0p~/0Q k and 0pgz/0Qk are equal. The absolute values of these quantifies may, therefore, be determined without difficulty. It is seen from relations (1.47) and (1.48) that absolute values of dipole moment derivatives can only be evaluated fxom experimental integrated intensifies. Therefore, the directions of charge shifts accompanying particular vibrational distortions remain undetermined. This is a major difficulty in any further reduction of vibrational intensity data. For many years the sign ambiguity problem for the dipole moment derivatives has been a cause for the limited application of vibrational intensities in structural analysis. Another formidable problem arises from difficulties in deriving individual Cartesian components of o~/~:~k derivatives from experimental measurements. For molecules in the gas-phase the components of the t~/c3Qk vector may only be determined for a molecule with sufficient symmetry, such that one of c~p~//~k components only is different from zero. In the general case, it is necessary to establish the direction of polarization of vibrations in order to further rationalize the structural information implicit in the measured absorption intensities. It is, therefore, not surprising that most vibrational intensity studies have been restricted to relatively small and symmetric molecules.
II, S E L E C T I O N R U L E S F O R I N F R A R E D A B S O R P T I O N
A. Harmonic Oscillator Selection Rules In deriving the relations between infrared absorption intensities and dipole moment derivatives we have restricted the treatment to transitions involving a change of a single
ABSORPTION OF INFRARED RADIATION
13
vibrational quantum number by +1. The respective selection rule arises from the properties of the linear harmonic oscillator wave function. In general, the selection rules indicate: (a) allowed transitions with absorption bands observed in the spectrum; (b) transitions that are forbidden with absorption bands of zero or small intensity. As mentioned, some selection rules are determined by properties of the harmonic oscillator wave functions. A second set of selection rules are associated with the symmetry properties of vibrations. The harmonic oscillator selection rules for vibrational transition can be evaluated using the expression of the dipole moment as a power series with respect to normal coordinates. On the basis of expressions (1.25) - (1.27) the dipole moment matrix element for a transition between vibrational states V' and V" may be written as
(v'l pIv")=IVv" p % d~v = P0 I"I I ~Fk ~Fk dQk k
.+
Z r
f C Q~ ~ dQ~FIf v;',ei' dQ1
k
1,k
l Z 2p /c)Qk2 )
+2 1
o
(1.51)
I ~Ij'k*Q2 ~k dQk H I ~PI*~I'11' dQl l~k
[~2P/(c~QkaQl)~I~FkQk ~Fk dQk f~FI*QI~FI'dQI I-[ I ~ W " m dQm m#l#k
+ higher terms. The first term in expression (1.51) is equal to zero except for the case with Vk'= Vk". This is determined by the orthogonality of vibrational eigenfunctions. In fact, it is zero for any transition between different vibrational states. The linear term is not zero for transitions involving a change of a single quantum number by + 1 only. Because of orthogonality of the function ~Pl the integral PPl'* ~Pl" dQl will be zero if any other vibrational quantum number except Vk is changed (Vl, l~k). The harmonic approximation restricts the dipole moment expansion to the constant and linear terms. Thus, the selection rule associated with the approximation of electrical harmonicity, states that transitions involving a change by + 1 of just one of the vibrational quantum numbers of the linear harmonic oscillator functions defining the vibrational states of molecules are only allowed. Aside from this, for a transition to take place at least one of the Cartesian components of the (0p/0Qk) 0 derivatives should differ from
14
GALABOV AND DUDEV
zero. We shall later see how the symmetry of vibrations determines the selection rules associated with these derivatives. The third term in expression (1.51) considers transitions associated with a change in a vibrational quantum number by 2. It governs the intensities of the usually weak overtone bands. The possibility of observing such transitions is determined by the fact that vibrations in real molecules are not strictly harmonic, both with respect to potential energy and dipole moment functions. When one quantum number is changed by 2, the linear term has also a f ~ t e , usually small, contribution to the matrix element. The fourth term is associated with the intensities of the weak combination bands (vibrational sum or difference bands). These bands are due to transitions involving changes by • 1 of two vibrational quantum numbers. Absorption bands associated with higher terms in the dipole moment function expansion are also observed. As expected, their intensities are orders of magnitude lower than the intensities of fundamental transitions. Nevertheless, studies on higher overtone transitions flourished during the past fifteen years [9,10]. The main reason for these developments is, very possibly, the unexpected "local mode" rather than "normal mode" properties for some of these transitions. Interesting opportunities for studying properties of individual chemical bonds in complex molecules have been revealed. Applications have mostly concentrated on higher overtones of C-H stretching modes [9,10].
B. Symmetry Selection Rules As stated before, a transition between different vibrational states may take place ff the respective coefficients in the dipole moment expansion as a power series to normal coordinates are not equal to zero. For a fundamental transition at least one of the components/~px//R~, i ~ p y / ~ and/)pz//~Qk must not be zero. For overtone transitions a non-zero value is required for at least one of the derivatives ~px/C3~ 2, ~ p y / ~ 2 and oa2pz/~2. A group theoretical analysis may determine the infrared active transitions by considering the symmetry properties of the vibrational wave functions of the interacting states and of molecular dipole moment. In most general terms, a transition between vibrational states V' and V" will take place if at least one of the component matrix elements differs from zero ( V ' [ Px IV")
(V' [ py IV")
( V ' [ Pz I V " ) .
(1.52)
If all three dipole matrix elements are zero the transition will be symmetry forbidden or infrared inactive.
ABSORPTION OF INFRARED RADIATION
15
The symmetry selection rules are derived by studying the effect of symmetry operations on the matrix elements. The rules stem from the property of dipole moment matrix elements to be invariant with respect to a symmetry operation (R) R ( ( V ' I p g [ V " > ) = (V' I pg IV")
(g = x , y , z ) .
(1.53)
Since the transition dipole is a physical observable, it is evident that its value should be independent under symmetry operations. In other words, the intramolecular charge distribution and fluctuations are invariant with respect to symmetry operations. The representation of transition dipole moment element is given by the direct product of the representations of the respective vibrational wave functions and dipole moment component rk = rv, • rpg • r w
(g = x, y, z)
(1.54)
where k is an index of the k-th normal mode. Since the dipole moment is a vector with components directed along the axes of the reference Cartesian system, it is clear that under a symmetry operation its components will transform as the respective x, y and z Cartesian axes. The same arguments hold for the igpx/0Qk,/gpy/~ and C3pz/~ dipole derivatives. Therefore, the symmetry representations of the dipole moment components, as present in expression (1.52) coincide with the representations of the respective Cartesian axes. The symmetry properties of vibrational wave functions are treated in detail elsewhere [3] and will not be discussed here. The representation of the ground vibrational state wave function belongs to the point group of the molecule at equilibrium configuration. The direct product F k coincides with one of the irreducible representations of the molecular point group. The component matrix element (V'[pg[V") will be different from zero only if the resulting F k coincides with the totally symmetric representation of the point group of the molecule F 0. In the case of degenerate vibrations F k is a reducible representation. The infrared active (allowed) transitions must have the totally symmetric irreducible representation in the structure of F k associated with the respective degenerate mode. The synunetry selection rule may also be expressed in alternative ways. An inflated transition is not forbidden only in the case where the direct product of the presentations of the two interacting states Fv, XFv,, coincides with the representation of at least one of the dipole moment Cartesian components. For a fundamental transition (v' k = 1, v" k = 0) the above requirement concerns the irreducible representation of the excited level (v'). The selection rule for such transitions is simply r v, • rpg=
to.
(1.55)
16
GALABOV AND DUDEV
Similar symmetry restrictions also apply for overtone and combination bands. As already discussed, these transitions are not aUowed under the harmonic oscillator selection rules. R should be pointed out that even ff a given transition is not forbidden under both symmetry and harmonic oscillator selection rules, it may have a very low intensity. This will be determined by the particular form of the vibration and the electronic structure of the molecule. The assignment of a given band to infrared active or forbidden transition is, therefore, not always a straightforward task. The derivation of expressions relating observed integrated infrared absorption coefficients with dipole moment matrix elements for the respective transitions shows that the experimental quantifies contain important structural information. It is related with the distribution and dynamics of electric charges in molecules. We should bear in mind, however, that there are a number of restrictions associated with the possibility to determine dipole moment derivatives with respect to normal coordinates. In many regions of the observed infrared spectrum for any molecule of a medium size, a strong overlap of closely situated bands is usuaUy present. Thus, individual intensifies for all fundamental transitions may not be accurately determined. Lately, with the development of advanced software for band deconvolution and curve fitting this difficulty has been, to some extent, overcome. As already mentioned, another formidable problem arises from the necessity to know the exact direction of polarization for each vibrational mode, so that individual Cartesian components of the Op/0Qk derivatives are evaluated. So far, no general approach to solve this problem experimentally for molecules in the gas-phase has been developed. Thus, in most cases, an elaborate molecular analysis of observed vibrational absorption intensities is only possible for higher symmetry molecules. Existing perturbations in the spectra associated with Fermi resonances, Coriolis interactions and strong anhannonicity effects may often hamper the interpretation of experimental intensity data. The sign ambiguity problem for dipole moment derivatives, as already discussed, is also present. Finally, it should be emphasized that the quantifies 0p/0Qk contain in a rather obscure form the structural information sought. This is due to the very complex nature of normal coordinates. It is, therefore, essential to further reduce the experimental 0p//~)k derivatives into quantifies characterizing electrical properties of molecular sub-units -atomic groupings, chemical bonds or individual atoms. Various theoretical formulations for analysis of vibrational intensities have been put forward. The approaches developed a r e quite analogous to the analysis of vibrational frequencies in terms of force constants. As known, force constants may be associated with properties of molecular sub-units. If such a rationalization of intensity data is successfully performed, another important aim of spectroscopy studies may become possible: quantitative prediction of vibrational intensities by transferring intensity parameters between molecules containing the same
ABSORPTION OF INFRARED 1LM)IATION
17
structural units in a similar environment. The analogy with transferable force constants should again be underlined.
III.
EXPERIMENTAL DETERMINATION OF INFRARED INTENSITIES
The theoretical models for interpretation of infrared intensities presented in the subsequent chapters have been largely applied in analyzing gas-phase experimental data. Gas-phase intensities provide an unique opportunity to study in a uniform approach the mterrelatiom between molecular structure and intensity parameters. This is due to the fact that, in contrast to vibrational frequencies, the absorption coefficients depend strongly on the phase state and on solvent effects. Intensities of different modes of the same molecule are not influenced in a systematic way by the solvent. The variations of absorption coefficients may reach tens and hundreds percent. Accurately determined gasphase intensities are, therefore, of fundamental importance as a source of experimental information on intramolecularproperties. Past difficulties in experimental measurements of integrated infrared intensities have been associated mostly with the low resolving power of spectrometers, poor accuracy on the ordinate and absence of computer facilities for band integration, deconvolution and curve fitting in overlap parts of the spectra. It is clear that presently we have far better experimental means for accurate determination of the integrated intensities of individual absorption bands. Still, however, careful considerations of a number of possible sources of errors are needed in order to obtain sufficiently accurate intensity data. Some of these problems will be discussed later on. It is interesting that the methods developed for experimental determination of vibrational intensities in the gas-phase were aimed at resolving problems arising mostly from the low resolution power of the available spectrometers at the time. It may appear that nowdays, when the researchers have access to instruments with resolution of the order of a few hundredths or even few thousandths of a wavenumber, these techniques may be of lesser importance. Although this is partly true, the current experimental approaches for experimental determination of vibrational intensities fully rely on the original developments. This is determined by the fact that these methods not only compensate for the effect of low resolution on intensities but also provide criteria for the accuracy of measurements and the influence of such phenomena as adsorption of sample gas or slow diffusion process. Thus, the extrapolation method of Wilson and Wells [ 11], further developed by Penner and Weber [12], is the standard approach for experimental intensity studies.
18
GALABOV AND DUDEV
Early attempts for experimental measurements of infrared intensities [13,14] resulted in greatly divergent values for the same molecule. It was soon realized that most of the difficulties were associated with the low resolving power of the spectrometers used [15]. The problems arise from the fact that the incident infrared beam emerging from the monochromator is not strictly monochromatic, but contains a band of frequencies around the frequency v' determined by the slit function g(v,v'). There is, thus, a perfectly good chance for the intensity of the transmitted radiation I by a cell I = Io exp (- ~ p 1)
(1.56)
to differ from the value corresponding to a monochromatic beam with frequency v'. This is particularly possible if the measurements are carried over rotational f'me structure of an infrared band where sharp variations of the absorption coefficient ct are expected. In expression (1.56) p is the pressure of the sample gas, 1is the optical path length of the cell and I0 is the intensity of the incident radiation. The quantity of principal interest in intensity measurements is the integrated absorption coefficient A as defined by Eqs. (1.43), (1.44) and (1.47). For gas samples the respective expression is (1.57)
Due to the finite width of the slit function g(v,v') the measured intensity at a setting v' will not be equal to the true absorption intensity I. An apparent intensity T(v') will be in fact determined. It is given by the integral oo
r ( v ' ) = I I(v)g(v,v')dv.
(1.58)
g(v,v') is the portion of the light of frequency v that reaches the detector when the spectrometer is set at frequency v'. The integration can be carried out from --oo to +oo since g(v,v') is different from zero only in the immediate vicinity of v'. The width of the frequency band is of the order of the spectrometer resolution. Analogous expression can be written for T o oo
T0(v') = I Io (v)g(v,v')dv.
The apparent absorption coefficient J3(v') can, therefore, be defined as
(1.59)
ABSORPTION OF INFRARED RADIATION
]3(v') = (1/pl) In [To(v')/T(v')].
19
(1.60)
The integration over the entire interval of an vibration-rotation band will produce the apparent integrated absorption coefficient B
B : Ibaad~ dr'
:lSb=d pl
(1.61)
The Wilson and Wells [ 11] extrapolation theorem proves that Lira B = A. pl-~
(1.62)
Since both p and 1 can be varied in experimental conditions, the theorem (1.62) provides a convenient way of determining the true absorption coefficient A. The entire derivation of Wilson and Wells [ 11] will be presented since important considerations associated with the accuracy of intensity measurements emerge at different stages of proving the theorem. The difference (A-B) is examined
1 Sbandhl(TIO/dv'= 1 ~bandin( f(v') ~ f(v)g(v,v')_:_v/ It T0I ) p-I ~ g(v, v')dv) d r ' .
A - B = p--~
(1.63)
In expression (1.63) f(v) is f(v) = e x p ( - t x p l )
- I/I o
(1.64)
and I0 is assumed to be constant over the frequency range of the slit width. In presence of foreign gases, such as H20 and CO 2, this assumption may not hold. It is of interest to consider here the function f. Let us suppose that f is constant over the frequency range of the slit width. It becomes a constant multiplier of the inner integral and the entire expression (1.63) vanishes. B will then be equal to A. If, however, the width of individual rotational lines is of the order of the slit width, the exponential function f can vary considerably and B may be very different from A. The limit of A-B with p l y 0 needs now be considered. Expression (1.63) is differentiated with respect to pl to yield [ctfgdv l ~ g d__v1j dr'. Lira(A-B)= Lim I {c~-'.~Tg?~ J dv'= I ~/ct - SIgdv
(1.65)
20
GALABOV AND DUDEV
The function f = exp (- a p 1) approaches unity at the limit. Expression (1.65) will vanish and the theorem proven under certain conditions. First, it will vanish if the absorption coefficient {x is constant over the integration range which is the slit width. Note that expression (1.63) vanishes under the condition that f as an exponent of a, is a constant. In Eq. (1.65) the analogous requirement is for c~. Small variations in ~ can result in much larger changes in f. On the second place, expression (1.65) may vanish if the resolving power of the monochromator is constant over the width of the band regardless of {z. In mathematical terms this condition is expressed as g(v,v') = g(v - v')
(1.66)
g ( v - v') = g(v'- v).
(1.67)
and
If conditions (1.66) and (1.67) are satisfied, we have g(v - v') dv = ~ g(v - v') d(v-v') = G.
(1.68)
G is independent of v'. Another simplification follows f~{xgdvdv' = ~c~fgdvdv' : Gfr
(1.69)
Substituting (1.68) and (1.69) into (1.65) we obtain ~ct dv' - ~ctdv = 0 .
(1.70)
Consequently, Lim B = A pl-,0
(1.71)
if a number of conditions are met. The first is that I 0 is a constant in a resolved range. This can be achieved by removing external absorption fxom atmospheric H20 and CO 2. It is also clear that working under conditions of higher resolution betters the constancy of I0 in the narrower resolved range. In addition, the theorem (1.71) will hold if {z does not vary over the range of the resolution of the spectrometer, or if the resolution is constant over the entire vibrational-rotational band. If appropriate care is taken so that the above
ABSORPTION OF INFRARED RADIATION
21
o:0r V
J
F Bcl
I
50~1
0
I
I
I
1
1
I
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 c/,equivolent poth length in cm at S.T.P.
Fig. 1.1. The dependence Bpl/pl for the infrared active modes of methane (Reproduced from Ref. [ 16] with permission).
mentioned conditions are satisfied, Eq. (1.71) provides a convenient approach of determining the ~ue absorption coefficient A by extrapolating B to zero value of pl. It should be noted that the requirement for constant resolution over the integration range of a band is a specific property of the insmxment. It may not always be fully satisfied. For small molecules in the gas-phase at low pressure with clearly expressed fine rotational structure, the variations of r and f = exp(-~pl) at each rotational line is so extreme that the conditions for accurate extrapolation measurement may not be present. Following an early approach of Bartholome [14], Wilson and Wefts [11] recommended that for such molecules a foreign non-absorbing gas under higher pressure is introduced in the sample cell. It will induce a collision broadening of rotational lines and at sufficiently high pressure will lead to a collapse of the fine structure. Obviously, under such conditions the extrapolation method may be used with confidence. In the original method of Wilson and Wells the true integrated absorption coefficient A is determined as the slope of tangent at the origin of the dependence Bpl versus pl. Typical plots of this dependence are shown in Fig. 1.1. There is, however, certain arbiuminess in choosing the exact direction of the tangent at the origin. The method also requires that the measurements be done at very low partial pressures of the absorbing gas where the accuracy is low. Considerable errors can, therefore, result. The problem is resolved by working at sufficiently high pressures of the inert transparent
22
GALABOV AND DUDEV
160 iO.lcm CH4 ~i 120
,
.
....
U
T uE 8 0
4.95 cm CH4
O I
o ,e-
d~ 4o
0
.
.
.
.
.
2".50 cm
CH 4
100 200 300 400 Pressure of nitrogen inotmospheres
Fig. 1.2. Dependence between the apparent integrated absorption of the v 3 band of CH4 and the pressure of added nitrogen (Reproduced from Ref. [ 16] with permission). 1.2
O.6
In(1~
------,0.6 ---~1,0.4 I
! ---4o.2
~lOAtrn/
i
~OAt~~ 42Atm~
OAtm 2800
L
___---- ~ ~ ' N ~ 3000
1 3200
1~ ~C m -I
Fig. 1.3. v 3 band of methane under different pressures of nitrogen used as an external gas. The partial pressure of methane is unchanged (Reproduced from Ref. [ 16]
with permission).
ABSORPTION OF INFRARED RADIATION
23
foreign gas (Ar, N2, etc.) so that the Beer's low plot becomes a straight line passing through the origin. This procedure has been suggested by Penner and Weber [ 12]. Depending on the size of the molecules, the type of rotational fme structure or band shape, the adequate pressure may vary significantly. For larger molecules a pressure of one atmosphere may be sufficient. For small molecules a pressure of up to I00 atm may be needed to reach the linear region of the dependence Bpl/pl. An example is shown in Fig. 1.2. The high pressures used may cause, however, some additional complications. For larger molecules under high pressure a certain amount of the gas sample may deposit on the cell walls, though in absence of foreign gas when the partial pressure is measured, the sample is in a gas-phase. This is a particular property of the molecule under study and each case needs careful consideration so that appropriate conditions for measurements are chosen. Overend [16] has pointed out that pressure-induced absorption can affect the apparent absorption coefficient value. The effect is attributed to intermolecular interaction. It is manifested in the slow rise of the apparent absorption coefficient B as the pressure is increased. The phenomenon is clearly shown in Fig. 1.3. If such effects are present, the pressure-induced absorption has to be eliminated. This is achieved by extrapolating the linear part of the curve to zero pressure of the external gas for each B value determined.