2.1 General theory

components of light, which had slightly different frequencies compared to the frequency of the incoming light. In a description of the scattering process based on quantum mechanics, the appearance of the shifted frequencies in the Raman scattered light is interpreted in the way that the molecules have shifted quantum state during the process. The shifted frequencies appear symmetrically around the frequency of the exciting light, from which one can conclude that the molecule may either be excited or de-excited during the scattering process. The last requires that the molecules have been excited (e.g., thermally) before the scattering event. This is termed anti-Stokes scattering, while the scattering where the molecules are excited during the process is termed Stokes scattering. The molecules may either shift rotational, vibrational, or electronic state during the scattering, depending on the

The spectroscopic technique based on Raman scattering, where the molecules shift vibrational state, is termed vibrational Raman spectroscopy. A vibrational Raman spectrum contains the unique and highly resolved vibrational signature of the scattering molecule. Normally only the Stokes part of the entire spectrum is measured, since this is more intense than the anti-Stokes part [3]. Vibrational Raman spectroscopy is the Raman technique most widely used in chemical analysis, and it is relevant for the investigation of molecules in solution, biomolecules, and solids (crystals and powders). Since the Raman technique can be performed as a reflection measurement, which requires no or very little sample preparation, it is well suited for the investigation of molecules in their natural environment such as in

Nowadays vibrational Raman spectra are measured by illuminating the sample with polarized laser light with wave numbers either in the near-infrared (NIR), the visible (VIS), or the ultraviolet (UV) and simultaneously monitoring the reflected light. A vibrational Raman spectrum is then obtained by considering the intensity distribution in the Raman scattered light as a function of the so-called Raman shift

Raman scattered light and the laser, respectively. In the case of Stokes scattering,

Ever since the discovery of the Raman effect in 1928, the Achilles heel of Raman spectroscopy has been the low Raman cross section where typically 10<sup>8</sup> incoming photons only generate a single Raman photon. The consequence is that the intensity of the Raman signal becomes very low in general. In the history of Raman spectroscopy, many attempts have been made to overcome this disadvantage. The three most important milestones for the practical application of Raman spectroscopy are

• The development of sensitive charge-coupled devices (CCD-detectors) [6, 7]

Not before the implementation of these three improvements could Raman spectroscopy really compete with the competing spectroscopic techniques IR, NIR, and

A challenge particular in in situ resonance Raman investigations of biomolecules is that the excitation of the resonance Raman process is followed by a simultaneous excitation of fluorescence in either the molecule under investigation or in other molecules present in the sample. Since in general the fluorescence cross section is

, where <sup>e</sup>ν<sup>s</sup> and <sup>e</sup>νlaser are the wave numbers of the

molecule and the specific experimental conditions.

Modern Spectroscopic Techniques and Applications

<sup>Δ</sup>eν<sup>R</sup> defined as <sup>Δ</sup>eν<sup>R</sup> <sup>¼</sup> <sup>e</sup>νlaser � <sup>e</sup>ν<sup>s</sup>

<sup>Δ</sup>eν<sup>R</sup> is positive.

the following:

UV/VIS.

14

the food industry and in medical and environmental applications.

• The invention and development of commercial lasers [4, 5]

• The development of interference filters, i.e., notch and edge filters

A unified treatment of Raman theory can be found in [9, 10] and in [3], where in the last reference a long list of references to the Raman literature is provided. The symmetry aspects, interference phenomena, and polarization properties of resonance Raman scattering have been discussed by Mortensen and Hassing [11] and by Schweitzer-Stenner [12], while the vibronic aspects has been discussed by Siebrand and Zgierski [13].

Raman scattering can be described as a coherent absorption-emission sequence in which a primary photon with wave number <sup>e</sup>ν<sup>p</sup> and polarization vector up is replaced by a scattered photon with wave number <sup>e</sup>ν<sup>s</sup> and polarization vector us. In comparison, fluorescence is an incoherent absorption-emission sequence, i.e., a combination of two independent processes, namely, a real absorption of a primary photon followed by a spontaneous emission of a secondary photon. In fluorescence the initially excited molecule is allowed to decay into other quantum states before the spontaneous emission of light. As well known, the number of vibrations in a molecule is given by the expression 3N 6, where N is the number of atoms in the molecule. Since each vibration can be highly excited, it follows that the total number of vibrational states associated with every electronic state including the

electronic ground state increases strongly with the size of the molecule. This gives many possibilities for the decay and accordingly also for the emission of the light. When the intensity contributions from all the possible radiative transitions are superposed to give the total intensity, the individual spectra overlap with the result that the spectral distribution of the fluorescence becomes broad and without much structure. In contrast, the number of accessible final states in the Raman process is very limited due to the coherent nature of the process. The typical vibrational Raman spectrum consists of narrow Raman lines with the typical FWHM bandwidths of 10–20 cm�<sup>1</sup> .

In the expression for the Raman tensor in Eq. (1), j i a , bj i and j ir is the initial, the final, and the intermediate states of the molecule. In principle, all states are exact eigenstates of the molecular Hamiltonian. ρ, σ is a shorthand notation for the Cartesian components of the electric dipole moment of the molecule divided by the velocity of light and measured relative to a coordinate system in the center of mass of the molecule. <sup>e</sup>νr,a is the wave number difference between the states j i<sup>r</sup> and j i <sup>a</sup> . γ<sup>r</sup> is the damping of the state j ir given as the FWHM width and represents the exponential decay of the state. The matrix elements in the numerators of the two terms in the Raman tensor are given in the Dirac notation [18] and involve mathematically an integration over the coordinates used in the functions describing the molecular states. The terms in the expression for the Raman tensor, where the denominators contain the wave number difference between the state j ir and the laser <sup>e</sup>νp, are called the resonance terms, since their contributions to the Raman intensity increase strongly when the excitation wave number <sup>e</sup>ν<sup>p</sup> becomes equal to their energies <sup>e</sup>νr,a. When this happens the scattering process is called resonance Raman scattering (RRS), while in cases where <sup>e</sup>ν<sup>p</sup> <sup>≪</sup>eνr,a, the process is just called Raman scattering (RS). The difference between RRS and RS and the implications with respect to the kind of molecular information one may obtain in the two

What Is Vibrational Raman Spectroscopy: A Vibrational or an Electronic Spectroscopic…

DOI: http://dx.doi.org/10.5772/intechopen.86838

scattering situations will be discussed in more detail in the following.

Mortensen and Hassing [11] defined the state tensor Sj i<sup>r</sup>

Sj i<sup>r</sup>

j i a and j i b , but we omit these indices for simplicity in writing.

<sup>α</sup><sup>a</sup>!<sup>b</sup> ρσ <sup>¼</sup> <sup>∑</sup> r

The Raman tensor now becomes

individual contributions is blurred.

The state tensor determines the contribution to the scattering from that partic-

Sj i<sup>r</sup> ρσ <sup>e</sup>νr,a � <sup>e</sup>ν<sup>p</sup> � <sup>i</sup>γ<sup>r</sup>

The introduction of the state tensor is more important in RRS, where only relative few states contribute significantly to the Raman intensity than in RS, where all molecular states contribute with the result that the information of their

One advantage of introducing the state tensor is that it is possible to evaluate the general form of the state tensor in cases, where the symmetry of the molecule is

ρσ depends also on the initial and final states of the Raman process

þ

intermediate state j ir :

ular state. Sj i<sup>r</sup>

17

It follows from Eq. (1) that all the molecular states contribute to the intensity of

ρσ corresponding to the

(3)

ρσ � h i bjρjr h i rjσja (2)

Sj i<sup>r</sup> σρ <sup>e</sup>νr,b <sup>þ</sup> <sup>e</sup>ν<sup>p</sup> <sup>þ</sup> <sup>i</sup>γ<sup>r</sup>

the specific Raman transition j i a ! j i b . It also follows that the first step in the calculation of the intensity is to calculate the Raman tensor by performing the summation over all molecular states and then as the second step calculate the absolute square of the result. The consequence is that the contribution from each state will interfere either constructively or destructively with the contributions from all other states in the Raman intensity. The interference between the individual contributions depends on their magnitude and relative sign. An important issue in the evaluation of the expression for the Raman intensity is to establish the relations between the molecular properties and the Raman process. To achieve this goal, it is appropriate to divide the discussion into two parts: (1) discussion of the contribution to the scattering tensor from a single state and (2) discussion of the interference between the contributions from different states. For that purpose

The basic equation for the theoretical description of Raman scattering is the famous Kramers-Heisenberg (KH) dispersion relation. Kramers and Heisenberg derived the equation by the application of the correspondence principle to the classical dispersion relation. The KH equation expresses the transition probability per second for Raman scattering [14]. The original version did not contain the damping of the scattering system, and thus it did not immediately apply to resonance Raman scattering. Later Weisskopf modified the equation by introducing the damping of the atomic and molecular states with the assumption of an exponential decay of the excited states [15]. Within a modern theoretical framework, the KH relation and the expression for the intensity of the Raman scattered light can be derived by using formal scattering theory [16] or time-dependent second-order perturbation theory. In perturbation theory the interaction energy between the molecule and radiation field, normally considered in the electric dipole approximation, is used as the perturbation, when the time-dependent Schrödinger equation for the total system and molecule plus electromagnetic field is solved (e.g., see [17]).

The basic scattering equations are collected in Eq. (1). The transition probability is expressed through the total differential scattering cross section <sup>d</sup><sup>σ</sup> dΩ � � <sup>¼</sup> IRaman Ilaser . Since the intensity of the laser Ilaser refers to a plane wave and the intensity of the Raman scattered light IRaman is the intensity scattered into the solid angle dΩ, the scattering cross section <sup>d</sup><sup>σ</sup> dΩ � � becomes an area. It follows from Eq. (1) that the intensity of the Raman scattered light corresponding to the Raman transition j i a ! j i b is proportional to the intensity of the laser Ilaser and to the absolute square of the Raman tensor, α<sup>a</sup>!<sup>b</sup> � � � � 2 , where <sup>α</sup><sup>a</sup>!<sup>b</sup> is a tensor of rank 2 represented by a 3 � 3 matrix, which is related to the electric polarizability of the molecule. The expression for the Cartesian components of the Raman tensor <sup>α</sup><sup>a</sup>!<sup>b</sup> ρσ ð Þ ρσ <sup>¼</sup> <sup>x</sup>; <sup>y</sup>; <sup>z</sup> is also given in Eq. (1):

$$I\_{Raman} = \left(\frac{d\sigma}{d\Omega}\right) I\_{laser} d\Omega$$

$$\left(\frac{d\sigma}{d\Omega}\right) = 4\pi \alpha\_{fx}^2 \tilde{\nu}\_s^4 \left| \sum\_{\rho\sigma} u\_{t\rho} \alpha\_{\rho\sigma}^{a \rightarrow b} u\_{p\sigma} \right|^2 \tag{1}$$

$$\alpha\_{\rho\sigma}^{a \rightarrow b} = \sum\_r \frac{\langle b|\rho|r\rangle \langle r|\sigma|a\rangle}{\tilde{\nu}\_{r,a} - \tilde{\nu}\_p - i\gamma\_r} + \frac{\langle b|\sigma|r\rangle \langle r|\rho|a\rangle}{\tilde{\nu}\_{r,b} + \tilde{\nu}\_p + i\gamma\_r}$$

αfsc is the fine structure constant known from atomic physics [17]. Notice that the Raman intensity is proportional to the fourth power of the wave number of the scattered light (i.e., inversely proportional to the fourth power of the wavelength). This means that the intensity of a Raman spectrum measured with a NIR laser with wavelength 1064 nm is decreased with a factor of 16 relative to the intensity of the same spectrum measured with a visible laser with wavelength 532 nm.

What Is Vibrational Raman Spectroscopy: A Vibrational or an Electronic Spectroscopic… DOI: http://dx.doi.org/10.5772/intechopen.86838

In the expression for the Raman tensor in Eq. (1), j i a , bj i and j ir is the initial, the final, and the intermediate states of the molecule. In principle, all states are exact eigenstates of the molecular Hamiltonian. ρ, σ is a shorthand notation for the Cartesian components of the electric dipole moment of the molecule divided by the velocity of light and measured relative to a coordinate system in the center of mass of the molecule. <sup>e</sup>νr,a is the wave number difference between the states j i<sup>r</sup> and j i <sup>a</sup> . γ<sup>r</sup> is the damping of the state j ir given as the FWHM width and represents the exponential decay of the state. The matrix elements in the numerators of the two terms in the Raman tensor are given in the Dirac notation [18] and involve mathematically an integration over the coordinates used in the functions describing the molecular states. The terms in the expression for the Raman tensor, where the denominators contain the wave number difference between the state j ir and the laser <sup>e</sup>νp, are called the resonance terms, since their contributions to the Raman intensity increase strongly when the excitation wave number <sup>e</sup>ν<sup>p</sup> becomes equal to their energies <sup>e</sup>νr,a. When this happens the scattering process is called resonance Raman scattering (RRS), while in cases where <sup>e</sup>ν<sup>p</sup> <sup>≪</sup>eνr,a, the process is just called Raman scattering (RS). The difference between RRS and RS and the implications with respect to the kind of molecular information one may obtain in the two scattering situations will be discussed in more detail in the following.

It follows from Eq. (1) that all the molecular states contribute to the intensity of the specific Raman transition j i a ! j i b . It also follows that the first step in the calculation of the intensity is to calculate the Raman tensor by performing the summation over all molecular states and then as the second step calculate the absolute square of the result. The consequence is that the contribution from each state will interfere either constructively or destructively with the contributions from all other states in the Raman intensity. The interference between the individual contributions depends on their magnitude and relative sign. An important issue in the evaluation of the expression for the Raman intensity is to establish the relations between the molecular properties and the Raman process. To achieve this goal, it is appropriate to divide the discussion into two parts: (1) discussion of the contribution to the scattering tensor from a single state and (2) discussion of the interference between the contributions from different states. For that purpose Mortensen and Hassing [11] defined the state tensor Sj i<sup>r</sup> ρσ corresponding to the intermediate state j ir :

$$\mathcal{S}^{|r\rangle}\_{\rho\sigma} \equiv \langle b|\rho|r\rangle \langle r|\sigma|a\rangle \tag{2}$$

The state tensor determines the contribution to the scattering from that particular state. Sj i<sup>r</sup> ρσ depends also on the initial and final states of the Raman process j i a and j i b , but we omit these indices for simplicity in writing.

The Raman tensor now becomes

$$\alpha^{a \to b}\_{\rho \sigma} = \sum\_{r} \frac{\mathbf{S}^{|r\rangle}\_{\rho \sigma}}{\widetilde{\nu}\_{r,a} - \widetilde{\nu}\_p - i\gamma\_r} + \frac{\mathbf{S}^{|r\rangle}\_{\sigma \rho}}{\widetilde{\nu}\_{r,b} + \widetilde{\nu}\_p + i\gamma\_r} \tag{3}$$

The introduction of the state tensor is more important in RRS, where only relative few states contribute significantly to the Raman intensity than in RS, where all molecular states contribute with the result that the information of their individual contributions is blurred.

One advantage of introducing the state tensor is that it is possible to evaluate the general form of the state tensor in cases, where the symmetry of the molecule is

electronic ground state increases strongly with the size of the molecule. This gives many possibilities for the decay and accordingly also for the emission of the light. When the intensity contributions from all the possible radiative transitions are superposed to give the total intensity, the individual spectra overlap with the result that the spectral distribution of the fluorescence becomes broad and without much structure. In contrast, the number of accessible final states in the Raman process is very limited due to the coherent nature of the process. The typical vibrational Raman spectrum consists of narrow Raman lines with the typical FWHM band-

The basic equation for the theoretical description of Raman scattering is the famous Kramers-Heisenberg (KH) dispersion relation. Kramers and Heisenberg derived the equation by the application of the correspondence principle to the classical dispersion relation. The KH equation expresses the transition probability per second for Raman scattering [14]. The original version did not contain the damping of the scattering system, and thus it did not immediately apply to resonance Raman scattering. Later Weisskopf modified the equation by introducing the damping of the atomic and molecular states with the assumption of an exponential decay of the excited states [15]. Within a modern theoretical framework, the KH relation and the expression for the intensity of the Raman scattered light can be derived by using formal scattering theory [16] or time-dependent second-order perturbation theory. In perturbation theory the interaction energy between the molecule and radiation field, normally considered in the electric dipole approximation, is used as the perturbation, when the time-dependent Schrödinger equation for the total system and molecule plus electromagnetic field is solved (e.g.,

The basic scattering equations are collected in Eq. (1). The transition probability

� � becomes an area. It follows from Eq. (1) that the intensity of the

, where <sup>α</sup><sup>a</sup>!<sup>b</sup> is a tensor of rank 2 represented by a 3 � 3 matrix,

IlaserdΩ

usρα<sup>a</sup>!<sup>b</sup> ρσ up<sup>σ</sup>

<sup>þ</sup> h i <sup>b</sup>jσj<sup>r</sup> h i <sup>r</sup>jρj<sup>a</sup> <sup>e</sup>νr,b <sup>þ</sup> <sup>e</sup>ν<sup>p</sup> <sup>þ</sup> <sup>i</sup>γ<sup>r</sup>

� � � � �

2

the intensity of the laser Ilaser refers to a plane wave and the intensity of the Raman scattered light IRaman is the intensity scattered into the solid angle dΩ, the scattering

Raman scattered light corresponding to the Raman transition j i a ! j i b is proportional to the intensity of the laser Ilaser and to the absolute square of the Raman

which is related to the electric polarizability of the molecule. The expression for the Cartesian components of the Raman tensor <sup>α</sup><sup>a</sup>!<sup>b</sup> ρσ ð Þ ρσ <sup>¼</sup> <sup>x</sup>; <sup>y</sup>; <sup>z</sup> is also given in

> dΩ � �

> > � � � � �

αfsc is the fine structure constant known from atomic physics [17]. Notice that the Raman intensity is proportional to the fourth power of the wave number of the scattered light (i.e., inversely proportional to the fourth power of the wavelength). This means that the intensity of a Raman spectrum measured with a NIR laser with wavelength 1064 nm is decreased with a factor of 16 relative to the intensity of the

IRaman <sup>¼</sup> <sup>d</sup><sup>σ</sup>

<sup>¼</sup> <sup>4</sup>πα<sup>2</sup> fsceν 4 <sup>s</sup> ∑ ρσ

same spectrum measured with a visible laser with wavelength 532 nm.

h i bjρjr h i rjσja <sup>e</sup>νr,a � <sup>e</sup>ν<sup>p</sup> � <sup>i</sup>γ<sup>r</sup>

dσ dΩ � �

<sup>α</sup><sup>a</sup>!<sup>b</sup> ρσ <sup>¼</sup> <sup>∑</sup> r

dΩ � � <sup>¼</sup> IRaman

Ilaser . Since

(1)

is expressed through the total differential scattering cross section <sup>d</sup><sup>σ</sup>

widths of 10–20 cm�<sup>1</sup>

see [17]).

cross section <sup>d</sup><sup>σ</sup>

tensor, α<sup>a</sup>!<sup>b</sup> � � � � 2

Eq. (1):

16

dΩ

.

Modern Spectroscopic Techniques and Applications

known. First, we notice that when the molecule has no symmetry, the state tensors are determined exclusively by the physical properties of the specific molecule considered, and all state tensor components may a priori be nonvanishing. The same holds for the Raman tensor, since it is a superposition of state tensors scaled with the associated energy factors. However, when the molecule exhibits some symmetry, the state tensors will be determined both by the physics and by the constraints imposed by the symmetry. The consequence of the symmetry is that some tensor components vanish. When the symmetry gets higher, the number of vanishing state tensor components increases. It is also important to notice that the structure of each state tensor contributing to a specific Raman transition can be different and different from the structure of the Raman tensor. This is because the state tensors are determined by the symmetry and the physical properties of the individual states, which contribute to the scattering, while the Raman tenor is a superposition of the state tensors. As we shall see later, this is particularly important in resonance Raman scattering of molecules containing a chromophore with high symmetry such as the molecules containing the heme group.

to rotation. In the Raman case, these are called the rotational invariants of the

What Is Vibrational Raman Spectroscopy: A Vibrational or an Electronic Spectroscopic…

properties of resonance Raman have been discussed in [20]. It is demonstrated that the amount of molecular information, which may be extracted from a resonance Raman experiment, can be increased considerably by measuring the polarizationresolved Raman signals in addition to the unpolarized signals. Specifically the results of two case studies are discussed. As already mentioned the first focuses on the aggregation of hemoglobin inside RBCs of human blood, and the second is an in vitro study of the stability of dye-sensitized solar cells. Here we shall only define the polarization properties relevant for the discussion that follows below, and we

In polarization-resolved Raman experiments, two spectra are measured from

The polarization properties in Raman scattering are expressed through the

DPR �

The polarization-resolved Raman measurements are illustrated in Figure 1, which shows the 180° scattering geometry (reflection measurement) being the geometry mostly used. The molecule (M) is placed in the center of a space-fixed coordinate system, and the laser light, which is linearly polarized along the Z � axis, is propagating along the X � axis in the negative direction. The Raman signals back reflected from the molecule are measured with a detector placed on the X � axis and with the polarization analyzers directed along the Z � axis for measurement of

Backscattering geometry illustrating polarized resolved Raman measurements of the parallel (along Z) and

! and k !

dΩ 

dσ dΩ ⟘ dσ dΩ ∥

<sup>∥</sup> and perpendicular <sup>d</sup><sup>σ</sup>

, and Σ2. Recently the polarization

<sup>⟘</sup> scattering cross sections can be

<sup>2</sup> (marked red in Figure 1) and along the <sup>Y</sup> � axis for

are the polarization and k-vector of the laser. M is the

(4)

Raman tensor, and they are denoted as Σ0, Σ<sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86838

refer to [11, 20] for further details.

dΩ 

depolarization ratio (DPR) defined as

the parallel component α<sup>a</sup>!<sup>b</sup>

Figure 1.

19

perpendicular (along Y) polarized data. E<sup>0</sup>

randomly oriented molecule.

ZZ 

determined by using the relation in Eq. (1).

which the parallel <sup>d</sup><sup>σ</sup>

Traditionally the symmetry of a molecule with well-defined configuration has been described by using point groups and group representation theory, and from the early days of quantum physics, this has been used to derive selection rules, i.e., to give the conditions under which a particular matrix element must vanish. In spectroscopy the symmetry-based selection rules determine when a transition does not appear in the spectrum, i.e., it is a symmetry-forbidden transition.

In cases where the molecular point group has a threefold rotation axis or axes of higher order, one must, however, as demonstrated in [11] and by Mortensen in [19], apply the so-called non-commuting generator (NCG) approach to molecular symmetry in order to be able to evaluate the general form of the state tensors. In [19] the NCG approach is explained in detail, while in [11] it is shown how the method can be applied to calculate the structure of the state tensors both for molecules with integer and half integer spin. Besides all the possible state tensors for molecules with integer spin in the most important point groups have been evaluated and collected as an appendix. The appendix containing the state tensors is also reproduced in [3].

Recently [20] the NCG method has been extended and applied to develop the state and Raman tensors for molecular aggregates. Specifically the tensors for the Htype dimer of two coupled monomers with D4<sup>h</sup> symmetry are evaluated, and the result has been applied to interpret the experimental results of polarized resolved RRS spectra of a diluted solution of red blood cells (RBCs) of human blood. The main conclusion from this study is that aggregation between heme-protein molecules inside the RBC can be studied in vivo by polarized RRS, which opens the possibility of monitoring the effects on the aggregation of drugs added to the blood.

#### 2.2 Polarization properties of Raman scattering

The polarization is a unique property of Raman scattering, which distinguishes the Raman technique from the UV/VIS and IR spectroscopy. In general, the polarization of the Raman scattered light is different from the polarization of the incoming laser light. This property is valid for oriented molecules (crystals), but perhaps more surprisingly, it is also valid for randomly oriented molecules like molecules in solutions and in powders. The reason for this is that the Raman process is controlled by a tensor (the Raman tensor) and not by a vector (the electric dipole vector) like in UV/VIS and IR absorption. While a vector only has one quantity, namely, its length, which is not changed (invariant) under rotation of the molecules, a tensor of rank 2 has three combinations of the nine tensor components, which are invariant

What Is Vibrational Raman Spectroscopy: A Vibrational or an Electronic Spectroscopic… DOI: http://dx.doi.org/10.5772/intechopen.86838

to rotation. In the Raman case, these are called the rotational invariants of the Raman tensor, and they are denoted as Σ0, Σ<sup>1</sup> , and Σ2. Recently the polarization properties of resonance Raman have been discussed in [20]. It is demonstrated that the amount of molecular information, which may be extracted from a resonance Raman experiment, can be increased considerably by measuring the polarizationresolved Raman signals in addition to the unpolarized signals. Specifically the results of two case studies are discussed. As already mentioned the first focuses on the aggregation of hemoglobin inside RBCs of human blood, and the second is an in vitro study of the stability of dye-sensitized solar cells. Here we shall only define the polarization properties relevant for the discussion that follows below, and we refer to [11, 20] for further details.

In polarization-resolved Raman experiments, two spectra are measured from which the parallel <sup>d</sup><sup>σ</sup> dΩ <sup>∥</sup> and perpendicular <sup>d</sup><sup>σ</sup> dΩ <sup>⟘</sup> scattering cross sections can be determined by using the relation in Eq. (1).

The polarization properties in Raman scattering are expressed through the depolarization ratio (DPR) defined as

$$DPR \equiv \frac{\left(\frac{d\sigma}{d\Omega}\right)\_\perp}{\left(\frac{d\sigma}{d\Omega}\right)\_\parallel} \tag{4}$$

The polarization-resolved Raman measurements are illustrated in Figure 1, which shows the 180° scattering geometry (reflection measurement) being the geometry mostly used. The molecule (M) is placed in the center of a space-fixed coordinate system, and the laser light, which is linearly polarized along the Z � axis, is propagating along the X � axis in the negative direction. The Raman signals back reflected from the molecule are measured with a detector placed on the X � axis and with the polarization analyzers directed along the Z � axis for measurement of the parallel component α<sup>a</sup>!<sup>b</sup> ZZ <sup>2</sup> (marked red in Figure 1) and along the <sup>Y</sup> � axis for

#### Figure 1.

known. First, we notice that when the molecule has no symmetry, the state tensors are determined exclusively by the physical properties of the specific molecule considered, and all state tensor components may a priori be nonvanishing. The same holds for the Raman tensor, since it is a superposition of state tensors scaled with the associated energy factors. However, when the molecule exhibits some symmetry, the state tensors will be determined both by the physics and by the constraints imposed by the symmetry. The consequence of the symmetry is that some tensor components vanish. When the symmetry gets higher, the number of vanishing state tensor components increases. It is also important to notice that the structure of each state tensor contributing to a specific Raman transition can be different and different from the structure of the Raman tensor. This is because the state tensors are determined by the symmetry and the physical properties of the individual states, which contribute to the scattering, while the Raman tenor is a superposition of the state tensors. As we shall see later, this is particularly important in resonance Raman scattering of molecules containing a chromophore with high

Traditionally the symmetry of a molecule with well-defined configuration has been described by using point groups and group representation theory, and from the early days of quantum physics, this has been used to derive selection rules, i.e., to give the conditions under which a particular matrix element must vanish. In spectroscopy the symmetry-based selection rules determine when a transition does

In cases where the molecular point group has a threefold rotation axis or axes of higher order, one must, however, as demonstrated in [11] and by Mortensen in [19], apply the so-called non-commuting generator (NCG) approach to molecular symmetry in order to be able to evaluate the general form of the state tensors. In [19] the NCG approach is explained in detail, while in [11] it is shown how the method can be applied to calculate the structure of the state tensors both for molecules with integer and half integer spin. Besides all the possible state tensors for molecules with integer spin in the most important point groups have been evaluated and collected as an appendix. The appendix containing the state tensors is also

Recently [20] the NCG method has been extended and applied to develop the state and Raman tensors for molecular aggregates. Specifically the tensors for the Htype dimer of two coupled monomers with D4<sup>h</sup> symmetry are evaluated, and the result has been applied to interpret the experimental results of polarized resolved RRS spectra of a diluted solution of red blood cells (RBCs) of human blood. The main conclusion from this study is that aggregation between heme-protein molecules inside the RBC can be studied in vivo by polarized RRS, which opens the possibility of monitoring the effects on the aggregation of drugs added to the blood.

The polarization is a unique property of Raman scattering, which distinguishes the Raman technique from the UV/VIS and IR spectroscopy. In general, the polarization of the Raman scattered light is different from the polarization of the incoming laser light. This property is valid for oriented molecules (crystals), but perhaps more surprisingly, it is also valid for randomly oriented molecules like molecules in solutions and in powders. The reason for this is that the Raman process is controlled by a tensor (the Raman tensor) and not by a vector (the electric dipole vector) like in UV/VIS and IR absorption. While a vector only has one quantity, namely, its length, which is not changed (invariant) under rotation of the molecules, a tensor of rank 2 has three combinations of the nine tensor components, which are invariant

symmetry such as the molecules containing the heme group.

Modern Spectroscopic Techniques and Applications

2.2 Polarization properties of Raman scattering

reproduced in [3].

18

not appear in the spectrum, i.e., it is a symmetry-forbidden transition.

Backscattering geometry illustrating polarized resolved Raman measurements of the parallel (along Z) and perpendicular (along Y) polarized data. E<sup>0</sup> ! and k ! are the polarization and k-vector of the laser. M is the randomly oriented molecule.

measurement of the perpendicular polarized component αa!<sup>b</sup> YZ � � � � <sup>2</sup> (marked green in Figure 1). For randomly oriented molecules, i.e., solutions and powders, we have to perform an average calculation of these scattering quantities with respect to orientation of the molecule. The average quantities αa!<sup>b</sup> ZZ � � � � <sup>2</sup> D E and <sup>α</sup>a!<sup>b</sup> YZ � � � � <sup>2</sup> D E have been calculated in [11] by the application of angular momentum theory, and we refer to [11] for details. The result for the DPR is as follows:

$$DPR = \frac{\left\langle \left| \sigma\_{YZ}^{a \rightarrow b} \right|^2 \right\rangle}{\left\langle \left| \sigma\_{ZZ}^{a \rightarrow b} \right|^2 \right\rangle} = \frac{5\Sigma^1 + 3\Sigma^2}{10\Sigma^0 + 4\Sigma^2} \tag{5}$$

vibrational Raman spectroscopy of larger molecules, i.e., either crystals or solutions and powders. In these molecules, the rotational spectra are not resolved, which means that it is not necessary to consider the rotational motion explicitly. In the adiabatic Born-Oppenheimer (ABO) approximation (e.g., see [17]), each molecular eigenstate is therefore, as a first step, approximated by a product of an electronic state and a vibrational state, where the ladder is normally considered within the harmonic approximation. Within the ABO approximation, the states in the state and Raman tensors are replaced by the product states: j i a ¼ j ig j i va , bj i ¼ j ig j i vb and j ir ¼ j ie ve j i where g and e are the electronic ground state and an excited electronic state, respectively. ve j i ¼ j i v1, v2, v3, v4,……v3N�<sup>6</sup> is the vibrational sub-state associated with the electronic state ei, where in the harmonic approximation the vibrational state is also factorized in the product of 3N � 6, and one-dimensional harmonic oscillator states, each of which describing a normal vibration. N is the number of atoms in the molecule. However there are situations where the ABO approximation breaks down the product functions that serve in general as good approximations to the molecular wavefunctions. The most important situation for resonance Raman scattering, where it is necessary to go beyond the ABO approximation, is the one where an excited electronic state is coupled to the high-lying vibrational states belonging to the electronic ground state. The nonadiabatic coupling is mainly provided by the <sup>T</sup>^ <sup>N</sup> � operator, which is the operator describing the kinetic energy of the nuclei. This gives rise to radiationless transitions, which leads to an increase of the damping constant of the excited electronic state and explains essentially why the magnitude of the bandwidths observed in experimental UV/VIS absorption spectra is larger than the radiation damping. For the discussion given here, it is sufficient to associate the bandwidths estimated in UV/VIS spectra with the constant γ<sup>r</sup>¼ev used in the Raman tensor. See [25, 26] for further discussions of

What Is Vibrational Raman Spectroscopy: A Vibrational or an Electronic Spectroscopic…

DOI: http://dx.doi.org/10.5772/intechopen.86838

the origin of the bandwidths found in Raman and UV/VIS experiments.

function for the vibrational motion.

21

Since the ABO approximation only allows a partial separation of the electronic and nuclear motions, the functions describing the electronic states will depend on both the electronic coordinates and on the nuclear coordinates, while the vibrational functions only depend on the nuclear coordinates. However, the vibrational motion depends indirectly on the electronic state, since the electronic eigenvalue is a function of the nuclear coordinates and is found to play the role of the potential energy

The vibronic version of the state tensor in Eq. (2) may now be written as

cules are not known. In fact, one important task in optical spectroscopy is to provide knowledge of these functions. Since also their dependence on the nuclear coordinates is not known, the formal integrations over the nuclear coordinates in the state tensor given in Eq. (6) are performed by introducing an appropriate Taylor expansion in the nuclear coordinates. The expansion of the state tensors may be performed in two different ways: (1) the electronic functions are expanded using perturbation theory and then the integration over the nuclear coordinates are performed (e.g., see [27]), and (2) the electronic transition moments

ρð Þ¼ Q h i g qð Þj ; Q ρð Þj q e qð Þ ; Q and σeg ð Þ¼ Q h i e qð Þj ; Q σð Þj q g qð Þ ; Q defined through

<sup>S</sup>j i<sup>e</sup> ve j i ρσ <sup>¼</sup> h i g qð Þ ; <sup>Q</sup> vbð Þj <sup>Q</sup> <sup>ρ</sup>ð Þj <sup>q</sup> e qð Þ ; <sup>Q</sup> veð Þ <sup>Q</sup> h i e qð Þ ; <sup>Q</sup> veð Þj <sup>Q</sup> <sup>σ</sup>ð Þj <sup>q</sup> g qð Þ ; <sup>Q</sup> vað Þ <sup>Q</sup> (6)

where q and Q symbolize the set of electronic and nuclear position coordinates, respectively. Thus each of the two matrix elements in the state tensor is evaluated by integrating formally over both the electronic and the nuclear coordinates. Besides, according to the adiabatic idea, the integration over the electronic coordinates should be performed before the integration over the nuclear coordinates. In general, the functions describing the excited electronic states in larger mole-

where Σ0, Σ<sup>1</sup> , and Σ<sup>2</sup> are the rotational invariants mentioned above and each of them is a combination of the absolute squares of the Raman tensor components given in molecule-fixed coordinates [11]. Σ<sup>0</sup> is the absolute square of the trace of the tensor, Σ<sup>2</sup> is the symmetric anisotropy, and Σ<sup>1</sup> refers to the antisymmetric part of the Raman tensor, i.e., it contains terms of the form <sup>α</sup><sup>a</sup>!<sup>b</sup> ρσ � <sup>α</sup><sup>a</sup>!<sup>b</sup> σρ � � � � � � 2 .

#### 2.3 Vibronic expansion of the state and Raman tensors

From the beginning of the history of Raman spectroscopy, it has been an important task to transform the basic and general scattering expressions into a form suitable for the interpretation of experimental Raman data. It follows from Eq. (1) that an important problem is that the calculation of the Raman intensity involves a summation over all molecular states, which are not known. Another problem is that in a molecule, the motions of the electrons and nuclei are not independent, which means that the functions describing the molecular states depend upon coordinates defining the positions of both the electrons and nuclei. It follows that any evaluation of the expression for the Raman intensity requires the introduction of approximations. It seems that there exist two different kinds of approach: the first approach introduces a number of radical approximations in one step, which leads to simple results, but with limited applicability. This approach was first followed by Placzek in 1934 with the development of his polarizability theory, which is valid for nonresonance Raman scattering [9]. The polarizability theory has been reconsidered by Long in 2002 [3] and is also briefly discussed below. The second approach introduces a number of approximations in a stepwise way directed toward Raman scattering in specific molecular systems. This kind of approach became relevant after the invention of the laser and development of commercial lasers, in particular tunable lasers in the 1960s and 1970s by which it became possible to perform resonance Raman experiments where the wave number of the laser was scanned through the visible absorption band of a large molecule. In particular, resonance Raman studies of biological molecules containing a chromophore came into focus [21–24]. Accordingly, it became necessary to transform the basic equations into a form suitable for the interpretation and extraction of the molecular information obtained under resonance conditions.

Almost all theoretical treatments of optical spectroscopic processes in molecules, including UV/VIS and Raman spectroscopy, are formulated within the adiabatic formalism. This formalism is based upon the assumption that in a molecule the motions of the light electrons are only weakly correlated to the motions of the heavy nuclei, so that it is possible to approximately separate these motions, when the Schrödinger equation for the molecule is solved. In this chapter we focus on

What Is Vibrational Raman Spectroscopy: A Vibrational or an Electronic Spectroscopic… DOI: http://dx.doi.org/10.5772/intechopen.86838

vibrational Raman spectroscopy of larger molecules, i.e., either crystals or solutions and powders. In these molecules, the rotational spectra are not resolved, which means that it is not necessary to consider the rotational motion explicitly. In the adiabatic Born-Oppenheimer (ABO) approximation (e.g., see [17]), each molecular eigenstate is therefore, as a first step, approximated by a product of an electronic state and a vibrational state, where the ladder is normally considered within the harmonic approximation. Within the ABO approximation, the states in the state and Raman tensors are replaced by the product states: j i a ¼ j ig j i va , bj i ¼ j ig j i vb and j ir ¼ j ie ve j i where g and e are the electronic ground state and an excited electronic state, respectively. ve j i ¼ j i v1, v2, v3, v4,……v3N�<sup>6</sup> is the vibrational sub-state associated with the electronic state ei, where in the harmonic approximation the vibrational state is also factorized in the product of 3N � 6, and one-dimensional harmonic oscillator states, each of which describing a normal vibration. N is the number of atoms in the molecule. However there are situations where the ABO approximation breaks down the product functions that serve in general as good approximations to the molecular wavefunctions. The most important situation for resonance Raman scattering, where it is necessary to go beyond the ABO approximation, is the one where an excited electronic state is coupled to the high-lying vibrational states belonging to the electronic ground state. The nonadiabatic coupling is mainly provided by the <sup>T</sup>^ <sup>N</sup> � operator, which is the operator describing the kinetic energy of the nuclei. This gives rise to radiationless transitions, which leads to an increase of the damping constant of the excited electronic state and explains essentially why the magnitude of the bandwidths observed in experimental UV/VIS absorption spectra is larger than the radiation damping. For the discussion given here, it is sufficient to associate the bandwidths estimated in UV/VIS spectra with the constant γ<sup>r</sup>¼ev used in the Raman tensor. See [25, 26] for further discussions of the origin of the bandwidths found in Raman and UV/VIS experiments.

Since the ABO approximation only allows a partial separation of the electronic and nuclear motions, the functions describing the electronic states will depend on both the electronic coordinates and on the nuclear coordinates, while the vibrational functions only depend on the nuclear coordinates. However, the vibrational motion depends indirectly on the electronic state, since the electronic eigenvalue is a function of the nuclear coordinates and is found to play the role of the potential energy function for the vibrational motion.

The vibronic version of the state tensor in Eq. (2) may now be written as

$$S^{|\varepsilon|\langle v\_{\varepsilon}\rangle}\_{\rho\sigma} = \langle \mathfrak{g}(q,\mathbb{Q})v\_{b}(\mathbb{Q})|\rho(q)|e(q,\mathbb{Q})v\_{\varepsilon}(\mathbb{Q})\rangle\langle e(q,\mathbb{Q})v\_{\varepsilon}(\mathbb{Q})|\sigma(q)|\mathfrak{g}(q,\mathbb{Q})v\_{a}(\mathbb{Q})\rangle \tag{6}$$

where q and Q symbolize the set of electronic and nuclear position coordinates, respectively. Thus each of the two matrix elements in the state tensor is evaluated by integrating formally over both the electronic and the nuclear coordinates. Besides, according to the adiabatic idea, the integration over the electronic coordinates should be performed before the integration over the nuclear coordinates.

In general, the functions describing the excited electronic states in larger molecules are not known. In fact, one important task in optical spectroscopy is to provide knowledge of these functions. Since also their dependence on the nuclear coordinates is not known, the formal integrations over the nuclear coordinates in the state tensor given in Eq. (6) are performed by introducing an appropriate Taylor expansion in the nuclear coordinates. The expansion of the state tensors may be performed in two different ways: (1) the electronic functions are expanded using perturbation theory and then the integration over the nuclear coordinates are performed (e.g., see [27]), and (2) the electronic transition moments ρð Þ¼ Q h i g qð Þj ; Q ρð Þj q e qð Þ ; Q and σeg ð Þ¼ Q h i e qð Þj ; Q σð Þj q g qð Þ ; Q defined through

measurement of the perpendicular polarized component αa!<sup>b</sup>

tation of the molecule. The average quantities αa!<sup>b</sup>

Modern Spectroscopic Techniques and Applications

[11] for details. The result for the DPR is as follows:

where Σ0, Σ<sup>1</sup>

DPR ¼

2.3 Vibronic expansion of the state and Raman tensors

obtained under resonance conditions.

20

Figure 1). For randomly oriented molecules, i.e., solutions and powders, we have to perform an average calculation of these scattering quantities with respect to orien-

calculated in [11] by the application of angular momentum theory, and we refer to

<sup>2</sup> D E <sup>¼</sup> <sup>5</sup>Σ<sup>1</sup> <sup>þ</sup> <sup>3</sup>Σ<sup>2</sup>

αa!<sup>b</sup> YZ � � � � 2 D E

> αa!<sup>b</sup> ZZ � � � �

of the Raman tensor, i.e., it contains terms of the form <sup>α</sup><sup>a</sup>!<sup>b</sup> ρσ � <sup>α</sup><sup>a</sup>!<sup>b</sup> σρ

them is a combination of the absolute squares of the Raman tensor components given in molecule-fixed coordinates [11]. Σ<sup>0</sup> is the absolute square of the trace of the tensor, Σ<sup>2</sup> is the symmetric anisotropy, and Σ<sup>1</sup> refers to the antisymmetric part

From the beginning of the history of Raman spectroscopy, it has been an impor-

Almost all theoretical treatments of optical spectroscopic processes in molecules, including UV/VIS and Raman spectroscopy, are formulated within the adiabatic formalism. This formalism is based upon the assumption that in a molecule the motions of the light electrons are only weakly correlated to the motions of the heavy nuclei, so that it is possible to approximately separate these motions, when the Schrödinger equation for the molecule is solved. In this chapter we focus on

tant task to transform the basic and general scattering expressions into a form suitable for the interpretation of experimental Raman data. It follows from Eq. (1) that an important problem is that the calculation of the Raman intensity involves a summation over all molecular states, which are not known. Another problem is that in a molecule, the motions of the electrons and nuclei are not independent, which means that the functions describing the molecular states depend upon coordinates defining the positions of both the electrons and nuclei. It follows that any evaluation of the expression for the Raman intensity requires the introduction of approximations. It seems that there exist two different kinds of approach: the first approach introduces a number of radical approximations in one step, which leads to simple results, but with limited applicability. This approach was first followed by Placzek in 1934 with the development of his polarizability theory, which is valid for nonresonance Raman scattering [9]. The polarizability theory has been reconsidered by Long in 2002 [3] and is also briefly discussed below. The second approach introduces a number of approximations in a stepwise way directed toward Raman scattering in specific molecular systems. This kind of approach became relevant after the invention of the laser and development of commercial lasers, in particular tunable lasers in the 1960s and 1970s by which it became possible to perform resonance Raman experiments where the wave number of the laser was scanned through the visible absorption band of a large molecule. In particular, resonance Raman studies of biological molecules containing a chromophore came into focus [21–24]. Accordingly, it became necessary to transform the basic equations into a form suitable for the interpretation and extraction of the molecular information

ZZ � � � � 2 D E

, and Σ<sup>2</sup> are the rotational invariants mentioned above and each of

� � �

YZ � � � �

and αa!<sup>b</sup> YZ � � � � 2 D E

<sup>10</sup>Σ<sup>0</sup> <sup>þ</sup> <sup>4</sup>Σ<sup>2</sup> (5)

� � � 2 .

<sup>2</sup> (marked green in

have been

the formal integration over the electronic coordinates are Taylor expanded in the nuclear coordinates (e.g., see [11]). After the expansion of the state tensors in Eq. (6), using either scheme 1 or 2, the result is inserted in Eq. (1) in order to obtain the Raman intensity. Applying (2) to the first order (being in most cases sufficient), the state tensor in Eq. (6) becomes

$$\begin{split} \mathbf{S}\_{\rho\sigma}^{\left| \epsilon \right> \left| v\_{\epsilon} \right>} &= \langle \nu\_{b} \left| \nu\_{\epsilon} \right> \langle v\_{\epsilon} \left| \nu\_{a} \right> \rho\_{\rm g\epsilon}^{0} \sigma\_{\rm g\rm g}^{0} + \sum\_{k=1}^{3N-6} \langle \nu\_{b} \left| \mathbf{Q}\_{k} \left| \nu\_{\epsilon} \right> \langle v\_{\epsilon} \left| \nu\_{a} \rangle \left( \frac{\partial \rho\_{\rm g\epsilon}}{\partial \mathbf{Q}\_{k}} \right) \sigma\_{\rm g\epsilon}^{0} \right. \tag{7} \\ &+ \langle \nu\_{b} \left| \nu\_{\epsilon} \right> \langle v\_{\epsilon} \left| \mathbf{Q}\_{k} \left| \nu\_{a} \right> \rho\_{\rm g\epsilon}^{0} \left( \frac{\partial \sigma\_{\rm g\epsilon}}{\partial \mathbf{Q}\_{k}} \right)\_{0} \end{split} \tag{7}$$

<sup>α</sup>^ρσ <sup>Q</sup>;eνvb, va ;eν<sup>p</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86838

To obtain Eq. (9) "the closure rule," i.e., 1

<sup>α</sup>^ρσ <sup>Q</sup>;eνvb, va ;eν<sup>p</sup>

� � ffi <sup>α</sup>^<sup>0</sup>

therefore more convenient in this case:

numbers of the Raman bands <sup>e</sup>νvb, va

23

� � <sup>¼</sup> <sup>∑</sup>

e6¼g

What Is Vibrational Raman Spectroscopy: A Vibrational or an Electronic Spectroscopic…

expansion of the molecular transition polarizability tensor <sup>α</sup>^ρσ <sup>Q</sup>;eνvb, va

of states j ii , has been applied to the vibrational sub-states of every electronic state, and the electronic transition moments defined above (expansion scheme 2) have been introduced. Because Eq. (9) involves a summation over all excited electronic states, a Taylor expansion of the electronic transition moments ρgeð Þ Q and σeg ð Þ Q would result in too many parameters, which cannot be determined. A direct Taylor

> ρσ þ ∑ 3N�6 k¼1

where Qk is the kth normal coordinate. To calculate the Raman tensor, the expansion in Eq. (10) is inserted in Eq. (8), and the result is inserted in Eq. (1) in order to obtain the intensity of the Raman signal of the Raman transition j i a ! j i b . Conclusion: It follows that the Raman signal only depends on variables related to

the electronic ground state, i.e., the set of normal coordinates Q and the wave

However, the spectral distribution in the Raman spectra is independent of the laser wave number, which only has influence on the absolute intensity of the Raman signal. As seen from Eq. (9), all information about the individual electronically excited states is "washed out" in the calculation (and in the experimental signals), which means that we cannot obtain any information on individual excited electronic states by measuring the Raman signals under these conditions. Nonresonance Raman scattering (or RS for short) becomes therefore a vibrational spectroscopic technique just like IR and NIR. There are however two essential differences between RS and IR and NIR: first, the Raman signals are obtained by the scattering of laser light with wave numbers typically in the visible region and therefore much higher than the wave numbers of the IR and NIR photons, which are directly absorbed in IR and NIR spectroscopy. Second, since the Raman signal is controlled by a tensor instead of by a vector, the spectral selection rules of RS become different from those of IR and NIR. The selection rules for RS can easily be derived by considering Eq. (10) inserted in Eq. (8). Assuming room temperature and considering only the Stokes spectrum, it follows that Raman spectra only contain the fundamental transitions, i.e., j i va ¼ 0 ! j i vb ¼ 1 , where the intensity is provided by the first derivative of the polarizability tensor. Besides for smaller molecules (e.g., benzene) having a high symmetry, the polarization enables one to distinguish between totally symmetric and asymmetric vibrations in the Raman spectrum. The DPR for symmetric vibrations becomes 0 ≤ DPR < 0:75, which follows directly from Eq. (5) and the fact that the Raman tensor is symmetric in nonresonance (i.e., <sup>Σ</sup><sup>1</sup> <sup>¼</sup> 0). For asymmetric vibrations the DPR <sup>¼</sup> <sup>0</sup>:75, since the tensor, because of the symmetry, has no trace (i.e., <sup>Σ</sup><sup>0</sup> <sup>¼</sup> 0) for these modes. For molecules without symmetry, all vibrations are of course in principle totally symmetric. Due to the generally narrow bandwidths of the Raman bands, Raman spectra represent in most cases a very well-defined vibrational signature of the molecule. For this reason and since the Raman signal can be measured in a reflection geometry and because no sample preparation is really needed, Raman signals are

∂α^ρσ ∂Qk � �

0

, besides the wave number of the laser <sup>e</sup>νp.

ρgeð Þ Q σeg ð Þ Q <sup>e</sup>νe, <sup>g</sup> � <sup>e</sup>ν<sup>p</sup>

^ <sup>¼</sup> <sup>∑</sup><sup>i</sup>

<sup>þ</sup> <sup>σ</sup>geð Þ <sup>Q</sup> <sup>ρ</sup>eg ð Þ <sup>Q</sup> <sup>e</sup>νe, <sup>g</sup> <sup>þ</sup> <sup>e</sup>ν<sup>p</sup>

j ii hij valid for any complete set

;eνp � � is

Qk þ … (10)

(9)

where all vibrational states are functions of the set of normal coordinate Q. The first and second lines are the state tensor in the Franck-Condon and the Herzberg-Teller approximations, respectively. Comparing Eq. (7) with the result in [27], it follows that the second expansion scheme leads to the simplest parametrization of the expression for the Raman intensity; although taken to infinite order, which is of course impossible in practice, the two expansion schemes are equivalent.

The vibronic models developed for RRS, which are found in the Raman literature, differ essentially from each other in two ways: (1) whether expansion scheme 1 or 2 has been applied and (2) the degree of approximation that has been introduced.
