Category: Advanced Chemistry

Pure rotational Raman spectra arise from the inelastic scattering of light by molecules undergoing transitions between discrete rotational energy levels, without any accompanying vibrational change. These spectra are an important tool for studying molecular structure and dynamics, providing information about moments of inertia, bond lengths and molecular symmetry, particularly for molecules that may be inactive in microwave spectroscopy.

An ideal pure rotational Raman spectrum is obtained by illuminating a sample with monochromatic visible or near-infrared laser radiation and analysing the frequency shifts of the scattered light. The spectrum consists of a series of lines symmetrically distributed about the incident (Rayleigh) line at , corresponding to rotational transitions between discrete rotational energy levels. For a rigid linear molecule in the absence of fine or hyperfine splitting, the allowed rotational Raman transitions follow the selection rule , reflecting the requirement that the molecular rotation be accompanied by a change in polarisability.

The rotational energy levels are given by eq44 or eq45 and the frequencies of the transitions are directly related to the rotational constant . As such, the spacing between the spectral lines provides information about the moment of inertia and the molecular structure. Using the definition , the wavenumber of the Stokes and anti-Stokes lines are:

and

respectively.

Therefore, the Stokes lines appear at lower wavenumbers than , shifted by for . The anti-Stokes lines appear at higher wavenumbers than , shifted by for , since the lowest anti-Stokes transition is from to (see diagram above).

 

 

The intensity of each transition depends partly on the population of the initial level , which is governed by the Boltzmann distribution, with higher- states being less populated at lower temperatures:

where

is the Boltzmann constant.
is the number of particles in the energy state .
is the total number of particles in the system.

Since represents the fraction of particles in the state ,  the intensity of a spectral line is proportional to , where the value corresponds to a specific energy level. However, there are degenerate states associated with each value of  in a rigid linear molecule. This degeneracy increases the statistical weight of higher levels, redistributing the population towards more degenerate states, which become thermally accessible at typical laboratory temperatures. In other words, at a given temperature, more particles will occupy a higher state with greater degeneracy at equilibrium than they would if the state were non-degenerate. Therefore, when a sample is exposed to a laser radiation, the intensity of each allowed transition in the pure rotational spectrum is proportional to the population of the initial state, which is the product of  and . Because the initial state of an anti-Stokes process is higher than the final state, the intensities of anti-Stokes lines are typically lower than those of Stokes lines.

Other than the population of the initial level, the transition intensity is also dependent on the fourth power of the scattered light frequency . This, together with the Boltzmann distribution, gives:

where is a proportionality constant and is given by eq45.

 

 

The factor in eq32 increases linearly with , while the exponential term decreases exponentially. In pure microwave rotational spectroscopy, the transition intensity is (excluding the factor ) with the maximum intensity occurring at the value of closest to . For Raman spectroscopy, decreases as increases for Stokes lines, while increases as increases for anti-Stokes lines. However, the change in over the visible region of the spectrum is very small (only about 1%) for a typical Raman experiment, because the laser frequency is around 20,000 cm^-1 and is about 1-10 cm^-1. Therefore, the derivative using eq32 still results in the maximum intensity on each side of the Rayleigh line at

For a symmetric rotor, . In a pure rotational Raman spectrum, the selection rule means that the term cancels out when taking differences. As a result, the Raman shifts are identical to those of a linear rotor and form a single series of equally spaced lines with spacing . However, unlike a linear rotor, each allowed transition receives contributions from multiple states. These overlapping contributions increase the intensity of the Raman lines but do not change their positions for an ideal spectrum. Consequently, the overall intensity envelope of the spectrum on each side of the Rayleigh line still resembles a skewed bell curve, with a maximum at the value of corresponding to the highest population.

In practice, the peaks in a real rotational Raman spectrum are not discrete vertical lines but have finite width and shape, appearing as broadened curves. This broadening arises, under typical laboratory conditions, from several effects:

• • Doppler broadening, due to the thermal motion of molecules, causes a spread in observed frequencies as molecules move towards or away from the detector.
  • Instrumental broadening, resulting from the finite resolution of the spectrometer itself.
  • Lifetime broadening, which is based on the energy-time uncertainty relation.

Beyond broadening, other phenomena affect the detailed structure and spacing of the spectral lines:

• • Centrifugal distortion causes deviations from the ideal rigid rotor model. As rotational speed increases at higher levels, the molecular bond stretches slightly, increasing the moment of inertia and decreasing the rotational constant . This leads to uneven spacing between lines, especially at higher , and is accounted for using a correction term.

• • Isotopic substitution alters the moment of inertia due to changes in atomic mass, thereby changing the rotational constant . Different isotopologues of the same molecule produce distinct sets of rotational lines, each with slightly different spacings. If multiple isotopes are present in the sample, the resulting spectrum may show clusters or duplications of lines, corresponding to each isotopologue.

Together, these effects lead to a rotational Raman spectrum that is richer and more complex than the idealised, evenly spaced series of lines predicted by the rigid rotor model.

 

Next article: Vibrational raman selection rules

Previous article: Rotational Raman selection rules for symmetric rotors

Content page of Raman spectroscopy

Content page of advanced chemistry

Rotational Raman selection rules for symmetric rotors describe the allowed changes in a symmetric top’s rotational angular momentum states during inelastic light scattering.

A symmetry rotor, such as CH₃Cl, has a permanent electric dipole moment along its symmetry axis. At finite temperatures, the total angular momentum of the rotating molecule is generally not parallel to the symmetry axis and given by:

where is the unit vector along the symmetry axis.

In the absence of external interactions (e.g. an electric field), the vector is conserved. This is because space is isotropic and the physics is unchanged regardless of the orientation of the molecule. It follows that the symmetry axis of the molecule must precess around so that its projection onto stays constant as the molecule rotates (see diagram below).

 

 

In the presence of a static electric field, the interaction of the field with the molecule’s electric dipole moment generates a torque on the molecule given by

where is the angle between and .

This torque changes the molecule’s angular momentum vector according to

Because the torque is perpendicular to the electric field direction (taken as the lab -axis), its component along the field axis is zero. Consequently, the projection of onto that axis cannot change. The only way for to change over time, while keeping a constant projection onto the lab -axis is for it to precess around the direction of the electric field.

As a result, the molecular axis undergoes a compound motion with a fast precession around and a slower precession of around the static electric field line (see above diagram). However, in Raman spectroscopy, the laser produces an electric field that oscillates at about 10¹⁴ to 10¹⁵ Hz, which is much faster than the molecule can physically rotate or precess. The interaction between the field and the molecule’s permanent dipole averages to zero, resulting in no net torque on the permanent dipole and hence no precession of around the electric field line.

While the permanent dipole interaction averages to zero, the induced dipole interaction does not, because it is not confined to the molecular axis. Nevertheless, the interaction between the induced dipole and the electric field is typically very weak, leaving the precession of the molecular axis around as the dominant motion.

In the molecular frame, the polarisability tensor of a symmetric rotor is mathematically identical to that of a linear molecule. This because both linear rotors and symmetric rotors possess cylindrical symmetry (or higher) about their principal axis. It follows that the projection of the molecular induced dipole moment onto the direction of the electric field (laboratory -axis) is given by eq7a, or equivalently,

where .

Substituting the spherical law of cosines (see diagram above), where into eq20 gives:

Expanding the equation and using the identity yields:

where

The selection rules involving the quantum number can be explained semiclassically, with the induced dipole moment given by:

where is the angular frequency of the incident radiation.

Substituting eq21 into eq22 results in:

Using the identity gives:

This equation shows that the induced dipole moment consists of five components: one oscillating at the incident frequency, two at , and two at . Each time-dependent component of the induced dipole moment corresponds to an oscillating dipole, which radiates electromagnetic energy at the frequency at which it oscillates. It follows that the component oscillating at gives rise to Rayleigh-scattered radiation, while those at and produce Raman-scattered radiation.

 

 

Raman selection rules are often discussed for thermally populated rotational levels, where is typically large (the classical limit). For large , the magnitude of the quantum angular momentum of the molecule is . Substituting this approximation into gives:

is the angular frequency of a Stokes-scattered photon, which is associated with the resultant quantum rotational transitional frequency of . This corresponds to the selection rule , because at large ,

Similarly, the angular frequency (anti-Stokes scattering) leads to the selection rule , while corresponds to . For the selection rules involving the remaining quantum numbers and , we refer to the time-dependent perturbation theory, which states that the transition probability can be found by evaluating:

where and are the Wigner D-functions.

The integrals and are non-zero only if and . This confirms that we require and for a probable transition to occur ().

Therefore, the combined selection rules for rotational Raman transition of a symmetric rotor are:

 

Next article: Pure rotational raman spectra

Previous article: Rotational Raman selection rules for linear rotors

Content page of Raman spectroscopy

Content page of advanced chemistry

Rotational Raman selection rules describe the allowed changes in a molecule’s rotational angular momentum states during inelastic light scattering.

The explicit rotational Raman selection rules for a linear rotor can be derived by considering the molecule with a molecular frame interacting with an external electric field applied along the laboratory -axis ().

The angle is the polar angle between the molecular -axis and the laboratory -axis, while is the azimuthal angle in the molecular -plane, measured from the -axis towards the -axis (see diagram above).

The first step of the derivation involves expressing the induced dipole moment in terms of the molecular -axis before projecting it onto the direction of the electric field (laboratory -axis).

Simple geometry shows that the direction cosines (the projections of the unit vector of the laboratory -axis onto the molecular axes) are:

Similarly, the electric field applied along the laboratory -axis has the following components in the molecular frame:

In the molecular frame, the induced dipole moment is a vector with three components :

where are the unit vectors along the molecular axes.

Likewise, the laboratory unit vector can be written in terms of the molecular frame as . Since the projection of any vector onto a unit vector is given by the dot product between the two vectors (see diagram below), the induced dipole moment along the laboratory -axis can be expressed by projecting onto as follows:

Substituting eq4 into eq6 yields:

For a linear molecule, the molecular axes coincide with the principal axes of the polarisability tensor, so that off-diagonal components vanish. Combining ,  and with eq5 and eq7 results in:

Replacing and with , and with (since the molecular -axis is the parallel axis of rotation, while the molecular – and -axes are the perpendicular axes of rotation), and using , gives:

As explained in the previous article, the transition probability associated with scattered radiation is evaluated using , which in this case, is:

where and are the un-normalised spherical harmonics.

The integral is equal to zero and when and  respectively. Thus, we require for a probable transition to occur (), resulting in:

where

is non-zero when either or is non-zero, or when both are non-zero. For the first integral, let and , giving:

which is non-zero only if , or equivalently,

This corresponds to Rayleigh scattering.

The second integral, with and , is:

Substituting the recurrence relation into eq10 yields:

where

Letting in the same recurrence relation and substituting it into eq11 results in:

where

for a linear rotor because . and are non-zero only if , which gives the same Rayleigh scattering selection rule as eq9. For to be non-zero, we require or equivalently, . The selection rule corresponding to is . Therefore, the selection rules for rotational Raman transition of a linear rotor are:

 

 

The selection rules can also be explained semiclassically, with the induced dipole moment given by:

where is the angular frequency of the incident radiation.

If the linear molecule is rotating an angular frequency , the value of its polarisability repeats twice per revolution cycle (see diagram above):

where is the constant baseline (average) polarisability of the molecule.

Substituting eq13 into eq12 gives:

Using the identity yields:

Eq14 shows that the induced dipole moment consists of three components: one oscillating at the incident frequency, and two at . Each time-dependent component of the induced dipole moment corresponds to an oscillating dipole, which radiates electromagnetic energy at the frequency at which it oscillates. It follows that the components oscillating at and give rise to Rayleigh-scattered radiation and Raman-scattered radiation respectively.

 

 

Raman selection rules are often discussed for thermally populated rotational levels, where is typically large (the classical limit). For large , the magnitude of the quantum angular momentum of the molecule is . Substituting this approximation into gives:

is the angular frequency of a Stokes-scattered photon, which is associated with the resultant quantum rotational transitional frequency of . This corresponds to the selection rule , because at large ,

Similarly, the angular frequency (anti-Stokes scattering) leads to the selection rule .

 

Next article: rotational raman selection rules for symmetric rotors

Previous article: General selection rules for scattered radiation

Content page of Raman spectroscopy

Content page of advanced chemistry

Selection rules for scattered radiation describe the allowed changes in a molecule’s state during light scattering.

According to the time-dependent perturbation theory, the transition probability between the orthogonal states and is proportional to the square of the corresponding transition matrix element . For example, in conventional rotational spectroscopy, this involves the electric dipole moment operator , and transitions occur only for molecules with a permanent dipole moment.

Transitions associated with scattering radiation, however, arise from an induced dipole moment rather than a permanent one. It follows, with reference to eq1, that the transition probability is given by:

Thus, a necessary condition for scattered-radiation transitions is that the matrix element be non-zero. To examine this condition further, we consider how the molecular polarisability depends on the molecular coordinate , which may represent a rotational angle or a vibrational displacement. This dependence can be expressed through a Taylor series about the equilibrium configuration:

Substituting this series (and neglecting higher-order terms) into gives:

Because the wavefunctions are orthogonal, the first term vanishes unless , which corresponds to Rayleigh scattering. Consequently, a Raman transition can occur only if , i.e. the polarisability of the molecule must change as it undergoes rotational or vibrational motion.

When , we have , which is the transition probability for Rayleigh scattering. When , we have the transition probability for Raman scattering:

Since represents a small perturbation of the electron cloud of the molecule, . Furthermore, the magnitude of the transition matrix element is never large: for vibrational Raman scattering it is proportional to the small zero-point vibrational amplitude, while for rotational Raman scattering it is a dimensionless angular matrix element bounded by unity. Consequently,

 

Next article: rotational raman selection rules for linear rotors

Previous article: Raman spectroscopy (instrumentation)

Content page of Raman spectroscopy

Content page of advanced chemistry

Raman spectroscopy relies on a compact yet sophisticated set of instruments designed to probe molecular rovibrational states through photon-matter interactions.

The laser provides monochromatic, coherent and intense light required to induce Raman scattering. Common laser types include diode lasers, Nd:YAG lasers and argon-ion lasers, operating at wavelengths ranging from the ultraviolet to the near-infrared region. The sample compartment is designed to accommodate solids, liquids or gases with minimal preparation. Raman instruments may use backscattering (180°), right-angle (90°) or forward-scattering geometries, depending on the application. In the above diagram, scattered radiation perpendicular to the laser beam is focused onto a pinhole.

The pinhole acts as a spatial filter that controls the amount of light arriving at the detector and helps define the resolution of the final spectrum. After passing through the slit, the scattered radiation still contains a mixture of Rayleigh, Stokes and anti-Stokes frequencies and is therefore dispersed into its individual wavelength components using a reflective diffraction grating. Each groove on the grating acts as a secondary source of light, generating spherical wavefronts in accordance with Huygen’s principle. Neighbouring reflected wavefronts interfere with one another to produce constructive diffraction patterns as defined by:

where

is the order of the diffraction (with the detector typically positioned to record ).
is the wavelength of the light.
is the spacing between the grooves.
is the angle between the incident ray and grating’s normal vector.
is the angle between the diffracted ray and grating’s normal vector.

 

 

Since the groove spacing is fixed, each wavelength is diffracted at a unique angle, causing light reflected from different grooves to travel slightly different path lengths. Constructive interference occurs when these wavefronts arrive in phase, producing intensity maxima at specific angles, while destructive interference reduces or cancels the signal at other angles. This process results in the spatial separation of wavelengths at the focal plane of the instrument.

A charge-coupled device (CCD) is positioned at this focal plane to detect the dispersed light. Its location is precisely calculated to receive the first-order () diffracted rays. The CCD consists of a two-dimensional array of light-sensitive pixels fabricated on a silicon substrate and is widely used due to its high sensitivity, low-noise quality, and ability to simultaneously record a broad spectral range. When photons strike a pixel, they are absorbed by the silicon, generating electron–hole pairs via the photoelectric effect. The number of electrons produced is proportional to the intensity of the incident light. Because each wavelength is focused onto a distinct position on the detector, individual pixels record the intensity corresponding to a single Raman frequency, yielding the final spectrum of intensity versus frequency (or versus Raman shift ).

 

Next article: General selection rules for scattered radiation

Previous article: Raman scattering

Content page of Raman spectroscopy

Content page of advanced chemistry

Raman scattering is the inelastic scattering of light by molecules in which the exchanged energy reveals information about the molecules’ vibrational or rotational states.

This effect arises from the interaction between incident light and the internal energy structure of molecules, and is best understood by contrasting it with ordinary absorption. When a photon collides with a molecule and its frequency corresponds exactly to the energy difference between two stationary molecular states, the molecule can absorb the photon and undergo a real electronic, vibrational or rotational transition. In this situation, the photon’s energy matches the gap between two eigenstates of the unperturbed molecular Hamiltonian, making true absorption possible.

However, most photons do not have energies that coincide with any allowed transition. In such cases, absorption cannot occur, yet the photon may still interact with the molecule. This is the domain of Raman scattering, which is an inelastic scattering process rather than a resonant absorption phenomenon. The mechanism begins when the electric field associated with the incident photon interacts with the molecule’s charge distribution. Although the photon is detected as a particle, it propagates as an electromagnetic wave, and its oscillating electric field transiently polarises the molecule. In other words, an applied electric field can distort the molecule and induce an electric dipole moment , with a stronger field resulting in a larger dipole moment for a given molecule:

where is the polarisability of the molecule, which is is a measure of the degree to which the electrons in the molecule can be displaced relative to the nuclei.

Since and , eq1 can also be written in matrix form as:

Here, is the -component of , where denotes the direction of the applied electric field and denotes the direction of the induced dipole moment component.

 

 

The redistributed electrons transform the molecule from its ground state into a short-lived virtual energy state. Unlike a real excited state, this state is an eigenstate of the perturbed Hamiltonian. Because it exists only during the presence of the perturbing field, the virtual state lasts on the order of femtoseconds, with its existence permitted by the time–energy uncertainty principle. Since the virtual state is not stable, the molecule quickly relaxes. If it relaxes back to the same stationary state it occupied before the interaction, the photon re-emitted has the same energy as the incident photon. This elastic process is known as Rayleigh scattering, and it accounts for the vast majority of scattered photons (see diagram below).

But if the molecule relaxes to a different stationary vibrational or rotational state than the one it started in, energy must be conserved by adjusting the energy of the scattered photon accordingly. If the molecule ends up in a state of higher internal energy, the scattered photon necessarily loses that amount of energy and emerges with a lower frequency (Stokes scattering). Conversely, if the molecule relaxes to a lower one, the scattered photon gains the excess internal energy and is emitted with a higher frequency (anti-Stokes scattering). These two types of scattering are collectively known as the Raman effect or Raman scattering.

Thus, Raman scattering provides a window into the vibrational and rotational structure of molecules: the energy shifts — Stokes and anti-Stokes — encode the quantised energy differences between molecular states. By measuring these shifts, Raman spectroscopy reveals the “fingerprint” of molecular vibrations, enabling powerful chemical identification and structural analysis without requiring the incident light to resonate with any real molecular transition.

However, due to the instantaneous orientation of the molecule, different photons colliding with it may lead to different extents of polarisation according to the polarisation tensor. The transition probability for Raman scattering is even lower, at about one in ten million. Therefore, Raman spectroscopy requires a high-power laser source, with the complete instrumentation described in the next article.

 

Next article: Raman spectroscopy (instrumentation)

Content page of Raman spectroscopy

Content page of advanced chemistry

Character tables provide a systematic group-theoretical framework for determining whether molecular vibrational modes are infrared– or Raman-active by examining how those modes transform relative to the electric dipole moment and polarisability operators.

The first step is to work out the symmetries of the normal modes of vibration of a molecule. For example, H₂O lying in the -plane belongs to the point group. Two of its three normal modes (symmetric stretch and bend), as explained in an earlier article, transform according to the irreducible representation, while the remaining one (asymmetric stretch) transforms according to .

The character table (see above table) shows that the linear functions (or Cartesian coordinates) and also transform according to and respectively. This correspondence means that the collective atomic displacements associated with each of the three H₂O normal modes transform in the same way under the symmetry operations of the group as the corresponding Cartesian vector component.

The second step is to recognise that an IR-active vibrational transition requires a change in the molecular dipole moment during the vibration. The dipole moment is a vector quantity whose non-zero components transform as the linear functions , and . If a normal mode transforms according to the same irreducible representation as one of these components, then the corresponding component of the dipole moment changes as the atoms are displaced from their equilibrium positions.

Therefore, the modes that transform according to and are IR-active.

For vibrations to be Raman-active, we refer to the quadratic functions because the components of the polarisability tensor transform in the same way as these functions. In this case, both the and  modes are Raman-active.

 

 

 

Previous article: Fourier-transform infrared spectroscopy

Content page of vibrational spectroscopy

Content page of advanced chemistry

The descent of symmetry method offers an efficient approach to determining the normal modes of CO₂.

The conventional procedure for determining the normal modes of a molecule involves the following steps:

1. 1. Forming a basis set of unit displacement vectors representing the instantaneous motions of the atoms.
   2. Determining geometrically the transformed vectors for each symmetry operation associated with the point group of the molecule.
   3. Constructing the symmetry operation matrix such that .
   4. Evaluating the trace of each matrix to obtain the character of the reducible representation, and decomposing each representation to determine the irreducible representations of the normal modes of the molecule.

However, CO₂ is a linear molecule belonging to the ​ point group, whose infinitely many symmetry elements render the above procedure impractical. The descent of symmetry method circumvents this difficulty by replacing ​ with a suitable finite subgroup, while preserving the essential symmetry properties of all degrees of freedom. This is justified because

• • the symmetry operations of both the parent group and its subgroup return the molecule to physically indistinguishable configurations; and
  • both the parent group and its subgroup act on the same set of basis functions (e.g. the Cartesian displacement vectors), from which reducible representations of either group are generated and subsequently decomposed to obtain the irreducible representations of the respective group.

As a result, all the normal modes of a molecule that transform according to certain irreducible representations of a parent group must also transform according to irreducible representations of the subgroup. The final step is to match the irreducible representations of both groups through a process known as correlation.

To proceed, we select an appropriate subgroup of for constructing the transformation matrices. While not strictly required, it is generally advantageous for the chosen subgroup to contain symmetry elements belonging to the same classes as those of the parent group, in order to minimise ambiguity during the correlation step. A common and convenient choice is the point group, which includes the key symmetry operations of , , and  preserved from the parent group.

Next, we follow steps 1 to 3 by imposing the symmetry operations on the set of nine unit displacement vectors . For example, the transformation matrix for inversion is:

where .

The eight transformation matrices form a reducible representation of the point group with the following traces:

Using eq27a, is decomposed to . Subtracting the translational modes () and the rotational modes () gives the vibrational representation:

The correlation step involves comparing the characters of symmetry operations common to both and , using their respective character tables. In particular:

• • Since rotation about the -axis (intermolecular axis) is not a physical degree of freedom for a linear molecule, characters associated with in are not used in the correlation. Instead, the characters of in are compared with those of and in .
  • Only the reflection planes and in  correspond to in ( does not).

This leads to the following correlation between the irreducible representations of the two groups:

It is evident that the vibrational mode transforming according to the totally symmetric irreducible representation in , and correspondingly according to (also totally symmetric) in , represents the symmetric stretching motion of CO₂.

Two of the four vibrational modes are degenerate and transform according to when the full symmetry of CO₂ is considered. These two modes describe bending motions in mutually perpendicular planes and share the same vibrational frequency. Upon descending from to , this degeneracy is lifted, and the doubly degenerate bending mode splits into two symmetry-distinct components transforming as and .

The remaining vibrational mode transforms according to and is antisymmetric with respect to inversion, identifying it as the antisymmetric stretching mode.

 

 

Next article: Correlation diagram

Previous article: Subgroup (descent of symmetry)

Content page of group theory

Content page of advanced chemistry

Main content page

The exclusion rule in vibrational spectroscopy states that, for molecules with a centre of symmetry, a vibrational mode that is infrared-active is Raman-inactive, and a mode that is Raman-active is infrared-inactive.

From a group‐theoretical perspective, the exclusion rule follows from the symmetry properties of the operators that govern infrared and Raman transitions. A vibrational mode is IR-active if its irreducible representation transforms like one of the linear basis functions (Cartesian coordinates) , , or , because the electric dipole moment operator has the same symmetry. It is Raman-active if its irreducible representation appears in the symmetry of the quadratic basis functions , , , , , or , which correspond to components of the polarisability tensor.

A point group to which a centrosymmetric molecule belongs always includes the inversion symmetry operator. Each of the basis functions , , or  changes sign (e.g. ) when it is inverted through the origin. On the other hand, quadratic basis functions, which are products of two linear functions, do not change sign upon inversion because both linear functions constituting the quadratic function change sign (e.g. and ).

This implies that the character corresponding to the inversion operation of an irreducible representation associated with a linear basis function is always different from that associated with a quadratic basis function. It follows that irreducible representations that are ungerade (u) can be IR-active but cannot be Raman-active, while irreducible representations that are gerade (g) can be Raman-active but cannot be IR-active. In other words, no vibrational mode of centrosymmetric molecules can be both IR- and Raman-active, which is the essence of the mutual exclusion rule.

This principle is clearly illustrated by the vibrational spectrum of CO₂​, which has four vibrational modes belonging to the point group. The symmetric stretching vibration transforms as the ​ irreducible representation and is therefore Raman-active but IR-inactive. In contrast, the asymmetric stretching vibration transforms as ​, which only has the same symmetry as the function, making it IR-active but Raman-inactive. The two bending vibrational modes are doubly degenerate and transform as the ​ irreducible representation, which corresponds to the and functions. Consequently, the bending modes are also IR-active and Raman-inactive. Thus, no mode is simultaneously IR- and Raman-active.

Accordingly, the IR spectrum of CO₂​ contains absorption bands corresponding to the asymmetric stretch and the bending vibrations, whereas the Raman spectrum shows a strong band associated with the symmetric stretch (see diagram below).

 

 

Previous article: Why components of the polarisability tensor transform like quadratic functions

Content page of Raman spectroscopy

Content page of advanced chemistry

The tight-binding model incorporates the Hückel approximation to describe the electronic structure of solids by treating electrons as localised around atoms and hopping between neighbouring sites.

Consider a linear chain of identical atoms located at position , with representing the s-orbital of atom . According to the Hückel approximation, the Hamiltonian is defined by two parameters:

1) is the energy of an electron localised on a single atom.

2) is the energy associated with an electron hopping between adjacent atoms.

3) All other matrix elements are zero.

To find the eigenvalues and eigenstates satisfying the Schrödinger equation , we adopt the linear combination of atomic orbitals (LCAO) approach, in which

Multiplying the Schrödinger equation on the left by the bra gives:

Substituting eq1 into eq2, and noting that the states  are orthonormal, yields:

Expanding the first summation in eq3 and applying the Hückel approximation results in:

Substituting the trial solution into eq4, and imposing the boundary conditions of , gives:

This rearranges to:

Using the trigonometric identity , with  and yields:

which simplifies to:

Since , we have . So, , which when substituted into eq5 gives:

where .

Although is an integer due to the condition , its specific range in eq6 follows the fact that each eigenvalue is associated with one of linearly independent eigenstates described by eq1.

 

 

An electron localised on an atom resides in a bound state with energy measured relative to the vacuum level (). Therefore, the conventional value of is negative. For s-orbitals, is also negative because corresponds to an integral of the form (positive wavefunction) (negative potential) (positive wavefunction). It follows that the lower-energy states occurs when is small in eq6 (so that the cosine term is close to +1 for large ). In other words, and represent the lowest and highest energy states of the system respectively (see diagram below).

One important application of the tight-binding model is its role in explaining band theory, which will be explored in the next article.

 

Next article: Band theory

Content page of solid-state chemistry

Content page of advanced chemistry

Main content page