A polarization analyzer between the sample and the detector can be used to distinguish the polarization of scattered light. If the polarization in the scattered light is the same as the incident we refer to this as I_||, I_ZZ or I_pol. If the polarization in the scattered light is perpendicular to the incident we refer to this as I_^, I_XZ or I_dep. The depolarization ratio r is r = I_dep/I_pol.

Although we discuss the light scattered along Y it must be understood that light is in fact scattered into all directions. The solid angle of a sphere is 4p stearadians and the solid angle collected is dW where dW < 4p. For example, if the f-number of a collection lens is 1 then the geometry for collection is

In this case since D = F we have q = arctan(1/2) = 26.56^o = 0.147p radians. The solid angle here can be calculated by integrating the differential volume element dW = dfsinqdq over the limits 0 to 2p and 0 to 0.147p.

When we evaluate the integral over the limits shown we find that the second term is approximately 0.1. So the solid angle defined by f/1 collection optics is 0.4p stearadians. This arrangement leads to collection of about 10% of the total light scattered from the sample.

The part of the cross-section ds that contributes to the detected scattered intensity is the ratio of the power at the detector dP to the incident intensity I₀. From dP = I₀ds we obtain

The differential power dP is proportional to the solid angle dW subtended by the detector. The scattered irradiance is the power per stearadian at the detector such that dP = I_sdW. Note the difference between irradiance and intensity.

Incident light dP = I₀ds where I₀ is in units of W/cm².

Scattered light dP = I_sdW where I_s is in units of W/sr.

Based on this definition we can determine the total Raman scattering cross section sr by integrating over all angles in 4p stearadians.

The transition polarizability a_rs is expressed as a function of the incident radiation frequency w₀. The transition polarizability is a Cartesian tensor where r and s are direction x,y,z in the molecular frame. The first term is an anti-resonant term and the second term is a resonant term. Both terms are important for off-resonance or non-resonant Raman scattering. Only the second term is important for resonant Raman scattering. In the resonant term the energy denominator would approach infinity were it not for the damping term iG_n. G_n arises due to the finite lifetime of the intermediate state. The shorter the lifetime in the intermediate state, the smaller the Raman cross section. This is particularly important for resonant Raman scattering. The states involved can be defined in terms of all 3N - 6 vibrational modes by writing:

v à v''

{v₁,v₂,…v_3N-6} à {v''₁,v₂,…v_3N-6}

In general, polarizability arises due to state mixing. The KHD expression shows mixing of the ground state with one or more higher electronic states. In other words the states |iñ and |fñ refer to two different vibrational states in the ground electronic state. The states |nñ refer to a set of vibrational states in one or more electronic state. When we consider resonant Raman scattering we will consider resonance with a single electronic state. For non-resonant Raman scattering the intermediate must be a superposition. In practice, we cannot calculate the magnitude of the non-resonant Raman cross section due to the complex nature of a superposition state. We can, however, determine the polarization, symmetry properties, and selection rules for non-resonant Raman scattering.

If the frequency of exciting radiation is far removed from the resonant frequency, i.e. w₀ << w_ge then vibrational energy terms in the energy denominator can be ignored compared to w₀ - w_ge and w₀ + w_ge. The transition polarizability becomes

The quantities á0|vñ and áv|v''ñ are vibrational overlaps. The square of a vibrational overlap is a Franck-Condon factor so the Raman excitation profile bears a defined relationship to the absorption spectrum. Here the v quantum numbers refer to the intermediate state vibrational energy levels. Since excitation is off-resonance there are in pricnciple many vibrational and electronic states that can contribute. The above approach is a sum-over-states approach. We can use the closure relation S_v|vñáv| = 1 to simplify the expressions.

The bottom equation describes Rayleigh scattering. The initial and final vibrational states are the same in Rayleigh scattering. There are no selection rules. All molecules are active Rayleigh scatterers. The Kroenecker delta in the top equation is d_0v'' = á0|v''ñ where d_0v'' = 0 if 0 ¹ v''. Thus the first term and the second term are the same here. This means that non-resonant Raman scattering will not occur within the Condon approximation. In order to explain non-resonant Raman scattering we must consider the coordinate dependence to the transition dipole moment, e.g. expand the transition dipole moment in a power series

and keep only the first term we find that the coordinate dependence of the transition moment can play a role. The reason for this is that even though á0|v''ñ = 0 for 0 ¹ v'', in general á0|Q|v''ñ does not need to be zero. Thus, for Raman scattering to be allowed we use the linear term above and make the substitution

The selection rules arise from the requirement that á0|Q|v''ñ does not vanish. We can define a polarizability derivative such that the transition polarizability is

where a'_rs is the polarizability derivative also called the derived polarizability. The terms a'_rs and (¶a_rs/¶Q_i) are equivalent. For a harmonic oscillator áv|Q|v''ñ vanishes except when v'' = v± 1. Thus, the selection rule of Dv = ± 1 applies to non-resonant Raman scattering as well as infrared spectroscopy (within the harmonic approximation).

If the incident frequency w₀ is in resonance with an electronic transition of the molecule the anti-resonant term (with w₀ + w_eg in the denominator) can be neglected and only the resonant term contributes to Raman scattering. If we keep terms up to linear in Q, we may express the transition polarizability as a sum of two terms

vanish except the m⁰_ge term that does not depend on Q_i.

In this expression the energy of an incident photon is equal to that of the energy difference between a ground state vibrational energy level gv' and an excited state level ev. The term iG_ev is a phenomenological damping term. This term arises from dephasing and lifetime broadening in the excited state levels. One can envision the contribution of G as an energy width to each of the excited state energy levels.

The thickness of the blue excited state levels is dependent upon excited state lifetimes and dephasing processes. The various levels have different energy widths to illustrate the fact that the dephasing rate can depend on vibrational state. Without the dephasing rate G_ev the resonance term would approach when w_ev,gv' = w₀. The larger the dephasing terms iG_ev the smaller the overall resonant Raman cross section.

The terms áv'|vñ and áv|v''ñ are Franck-Condon factors. In fact, these are the same Franck-Condon factors found in absorption spectroscopy. Just as in absorption spectroscopy there must be displacement along a normal mode coordinate upon electronic excitation in order for it to be Franck-Condon active.

The Albrecht B-term describes resonance Raman scattering of a vibronically active vibrational mode. In electronic absorption a vibronic mode is one which causes a transition to be allowed by distortion of the molecule to lower the symmetry. In resonant Raman scattering a vibronic mode has a resonance enhancement pattern that is different from a Franck-Condon active mode for the same reason. The Franck-Condon active modes are the totally symmetric modes of the molecule and the vibronic modes are the non-totally symmetry modes of the molecule. The B term is more complicated than the A term. In the A term the transition moment for the ground to excited state electronic transition m_ge is contributes and in B-term scattering it is the terms in (¶m_ge/¶Q_i)₀Q_i that contribute. Recall that the Herzberg-Teller expansion is:

For a rank 2 tensor (e.g. transition polarizability) we can write down three rotational invariants, S^J, that are linear combinations of the a^J_M that are independent of reference frame where J = 0, 1, 2… and M = 0 , ± 1 ,…, ± J. The rotational invariants are:

Each S^J is called an invariant because it is independent of orientation. The length of a vector is independent of its orientation. That is the same thing as saying that for the vector m, the combination m_x² + m_y² + m_z² is a rotational invariant. A second rank tensor has three invariants,

S⁰ isotropic part

S¹ anti-symmetric anisotropy

S² symmetric anisotropy

The isotropic part of the polarizability is proportional to the square of the trace of the polarizability tensor S⁰ = (Tra)²/3. The trace of tensor a (written as Tra) is the sum of diagonal elements of the tensor.

Therefore, S² represents the deviation of the polarizability from spherical symmetry.

The form of a in the molecular frame depends on the symmetry of the vibration. For non-resonant Raman the polarizability tensor is symmetric and therefore the anti-symmetric anisotropy S¹ is zero. Inspection of the anti-symmetric anisotropy shows that it is zero when a_rs = a_sr. S² depends on non-zero off diagonal terms and on differences in the diagonal terms. It is not necessarily zero in non-resonant Raman scattering.

Symmetric top molecules have two equal components, so a = b ¹ c.

Asymmetric top molecules have a ¹ b ¹ c.

Thus, S¹ = S² = 0 and r = 0.

and the tensor b is the anisotropy of the polarizability:

This equation assumes that the Raman tensor is symmetric (and is only valid for non-resonant Raman scattering). Non-symmetric molecules b can be non-zero even for totally symmetric modes.

Non-totally symmetric modes are the modes that responsible for vibronic coupling or Herzberg-Teller coupling in absorption spectroscopy. For these modes Tra vanishes and only b contributes to the non-resonant Raman scattering cross section. Since Tra vanishes for non-totally symmetric modes we also have S⁰ = 0. Taken together with the fact that S¹ = 0 in non-resonant Raman we have r = 3/4 for non-totally symmetric modes. In non-resonant Raman scattering r is never larger than 3/4. . For a totally symmetric mode S⁰ ¹ 0 and so r < 3/4. For molecules symmetric enough for the x, y, and z directions to be equivalent r = 0 since S¹ = S² = 0.

The polarizability depends on the symmetry of the electronic transition. For example, in a z-polarized transition a totally symmetric A-term mode has only one non-zero tensor component, a_zz. From this consideration we can readily calculate that r = 1/3. A doubly degenerate resonant electronic state (i.e. a state that x,y polarized such as in porphyrins) results in two equal diagonal tensor components, e.g. a_xx = a_yy, which leads to r = 1/8. If the electronic transition is triply degenerate (i.e. if the molecule is spherically symmetric such as SF₆) then a_xx = a_yy = a_zz. In this case r = 0.

In resonant Raman scattering the Raman tensor is not necessarily symmetric. B term enhancement can lead to anomalous polarization in which r > 3/4. Vibronic coupling of two states can lead to Raman tensors in which a_sr ¹ a_rs. A nonzero value of S¹ can lead to a depolarization ratio of greater than 3/4.

Anomalous polarization can be explained using the Herzberg-Teller approach. Suppose that the electronic transition g à e is x polarized and the transition g à r is y polarized leading to a Raman activity of the fundamental transition of a non-totally symmetric vibration of symmetry G_v = G_xG_y. The transition is resonant with |e0ñ and |e1ñ intermediate states. Herzberg-Teller coupling requires that vibronic coupling to state mix |e0ñ with |r1ñ and |e1ñ with |r0ñ. The vibronic intermediate states are:

The a_xy component dominates when the incident frequency is resonant with the g0 à e1 transition, while the a_yx component is resonant with the g0 à e0 transition. If the energy levels of the states e and r are well separated then the energy denominators are nearly equal:

If w₀ is far from resonance then a_xy » a_yx. This is a non-resonant condition where the Raman tensor is symmetric, S¹ = 0, and r = 3/4. On the other hand, close to resonance with the 0-1 transition, we have a_xy >> a_yx and for w₀ close to the frequency of the 0-0 transition, a_yx >> a_xy. When the exciting radiation is midway between the 0-0 and 0-1 resonances, the relationship a_xy » -a_yx results. This leads to S¹ ¹ 0, while S⁰ = S² = 0. At this frequency the depolarization ratio approaches infinity. Anomalous polarization was first observed in the vibronic bands of hemes.