Chemistry 330

Vibrational and Rotational Spectroscopy

The Electromagnetic Spectrum

The electromagnetic spectrum and the classification of the spectral regions.

The band at the bottom of the illustration indicates the types of transitions that absorb or emit in the various regions.

The Electromagnetic Spectrum (Cont’d)

Transition Intensity

The intensity of a transition is the area under a plot of the molar absorption coefficient against the wavenumber of the incident radiation.

Absorption and Emission

Absorption and emission of radiation and the attainment of thermal equilibrium. The excited state can return to the lower state– spontaneously – stimulated by radiation

already present at the transition frequency.

Forbidden Transitions

When a 1s electron becomes a 2s electron, there is a spherical migration of charge

There is no dipole moment associated with this migration of charge

This transition is electric-dipole forbidden

Forbidden Transitions

When a 1s electron becomes a 2p electron, there is a dipole associated with the charge migration

This transition is allowed.

Doppler Broadening

The shape of a Doppler-broadened spectral line The distribution reflects the Maxwell distribution of speeds in the sample Lines broaden as T increases

The Definition of Moment of Inertia

In this molecule – three identical atoms

attached to the B atom– three different but

mutually identical atoms attached to the C atom.

Centre of mass lies on the C3 axisPerpendicular distances are measured from the axis passing through the B and C atoms.

Asymmetric Rotor

An asymmetric rotor has three different moments of inertia; all three rotation axes coincide at the centre of mass of the molecule.

Types of Rigid Rotors

A schematic illustration of the classification of rigid rotors.

Spherical Rotors

The rotational energy levels of a linear or spherical rotor. Note that the energy separation between neighbouring levels increases as J increases.

The significance of the quantum number K.

When |K| is close to its maximum value, J, most of the molecular rotation is around the principal axis. When K = 0 the molecule has no angular momentum about its principal axis: it is undergoing end-over-end rotation.

The significance of the quantum number MJ.

When MJ is close to its maximum value, J, most of the molecular rotation is around the laboratory z-axis. An intermediate value of MJ.

When MJ = 0 the molecule has no angular momentum about the z-axis.

Linear Rotor

The effect of an electric field on the energy levels of a polar linear rotor. All levels are doubly degenerate except that with MJ = 0.

Absorption and Emission

The processes that account for absorption and emission of radiation and the attainment of thermal equilibrium.

The excited state can return to the lower state– spontaneously – by stimulated emission

Centrifugal Distortion

The effect of rotation on a molecule. The centrifugal force arising from rotation distorts the molecule, opening out bond angles and stretching bonds slightly. The effect is to increase the moment of inertia of the molecule and hence to decrease its rotational constant.

The Gross Selection Rule for Rotations

A rotating polar molecule looks like an oscillating dipole which can stir the electromagnetic field into oscillation. Classical origin of the gross selection rule for rotational transitions.

Photon Absorption

When a photon is absorbed, the angular momentum of the combined system is conserved.

If the molecule is rotating in the same sense as the spin of the incoming photon, then J increases by 1.

The Linear Rotor

The transitions allowed by the selection rule J = 1

The intensities reflect the populations of the initial level in each case and the strengths of the transition dipole moments.

Polarizability

An electric field applied to a molecule results in its distortion, and the distorted molecule acquires a contribution to its dipole moment

Polarizability (cont’d)

The polarizability may be different when the field is applied – parallel – perpendicular to the molecular axis

The molecule has an anisotropic polarizability.

The Raman Selection Rules

The distortion induced in a molecule by an applied electric field returns to its initial value after a rotation of only 180 Origin of the J = 2 selection rule in rotational Raman spectroscopy.

A Rotational Raman Spectrum

The rotational energy levels of a linear rotor and the transitions allowed by the J = 2 Raman selection rules. The form of a typical rotational Raman spectrum

A molecular potential energy curve

The energy may be approximated by a parabola near the bottom of the well.

The parabolic potential leads to harmonic oscillations.

The Definition of the Force Constant

The force constant measures of the curvature of the potential energy close to the equilibrium extension of the bond.

Nonpolar Species

The oscillation of a molecule, even if it is nonpolar, may result in an oscillating dipole that can interact with the electromagnetic field.

The electric dipole moment of a heteronuclear diatomic molecule varies as shown by the green curve. For small displacements the change in dipole moment is proportional to the displacement.

Morse Potentials

The Morse potential energy curve reproduces the general shape of a molecular potential energy curve.The number of bound levels is finite. – Note the relation

between the dissociation energy, D0, and the minimum energy, De, of the curve.

The Dissociation Energy

The dissociation energy is the sum of the separations of the vibrational energy levels up to the dissociation limit just as the length of a ladder is the sum of the separations of its rungs.

Birge-Sponer Plot

The area under a plot of transition wavenumber against vibrational quantum number is equal to the dissociation energy of the molecule. The Birge-Sponer extrapolation.

The HCl Spectrum

A high-resolution vibration-rotation absorption spectrum of HCl.

The lines appear in pairs because H35Cl and H37Cl both contribute!

Note - no Q branch,

P, Q, R Branches

The formation of P, Q, and R branches in a vibration-rotation spectrum. The intensities reflect the populations of the initial rotational levels.

O, Q, S Branches

The formation of O, Q, and S branches in a vibration-rotation Raman spectrum of a linear rotor. Note the frequency scale runs in the opposite direction to those of the P, Q, R branches.

The Vibrations of CO2.

The stretching modes are not independent, and if one CO group is excited the other begins to vibrate. The symmetric and antisymmetric stretches are independent, and one can be excited without affecting the other: they are normal modes.The two perpendicular bending motions are also normal modes.

The Normal Modes of Water

The three normal modes of H2O. The

mode v2 is

predominantly bending, and occurs at lower wavenumber than the other two.

Symmetry and Normal Modes

The atomic displacements of CH4 and the

symmetry elements used to calculate the characters.