# 13: Molecular Spectroscopy

- Page ID
- 51078

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**Sorry, but all current instructors at UCD use the commercial textbook for this course.**

Molecular spectroscopy is the study of absorption of light by molecules. In the gas phase at low pressures, molecules exhibit absorption in narrow lines which are very characteristic of the molecule as well as the temperature and pressure of its environment.

- 13.1: The Electromagnetic Spectrum
- Electromagnetic radiation—light—is a form of energy whose behavior is described by the properties of both waves and particles. Some properties of electromagnetic radiation, such as its refraction when it passes from one medium to another are explained best by describing light as a wave. Other properties, such as absorption and emission, are better described by treating light as a particle.

- 13.2: Rotational Transitions Accompany Vibrational Transitions
- Each of the modes of vibration of diatomic molecules in the gas phase also contains closely-spaced (1-10 cm-1 difference) energy states attributable to rotational transitions that accompany the vibrational transitions. A molecule’s rotation can be affected by its vibrational transition because there is a change in bond length, so these rotational transitions are expected to occur. Since vibrational energy states are on the order of 1000 cm-1, the rotational energy states can be superimposed upon

- 13.3: The Vibration-Rotation Spectrum
- Vibration-rotation interaction describes the inversely proportional relationship of the rotational constant and the vibrational state. The rotational constants decrease as the vibrational states increase, and their interaction influences the frequencies at where the lines of R and P branches occurred. This effect can be observed well on a rotational-vibrational spectrum, and the vibration-rotation relationship also has its own mathematical form.

- 13.4: The Pure Rotational Spectrum has Unequal Spacings
- Vibrational energy which is a consequence of the oscillations/ vibrations of the nuclei along inter nuclear axis, is possible only when the distance between the nuclei is not fixed/ rigid; that means the separation between the two nuclei is flexible/ elastic (non-rigid rotor). Consequently, centrifugal force, when the molecule is rotating, tends to fly the reduced mass μ away from the axis of rotation.

- 13.5: Vibrational Overtones
- Although the harmonic oscillator proves useful at lower energy levels, like n=1, it fails at higher numbers of n, failing not only to properly model atomic bonds and dissociations, but also unable to match spectra showing additional lines than is accounted for in the harmonic oscillator model.

- 13.6: Electronic Spectra Contain Electronic, Vibrational, and Rotational Information
- Molecules can also undergo changes in electronic transitions during microwave and infrared absorptions. The energy level differences are usually high enough that it falls into the visible to UV range; in fact, most emissions in this range can be attributed to electronic transitions.

- 13.7: The Franck-Condon Principle
- The Franck-Condon Principle describes the intensities of vibronic transitions, or the absorption or emission of a photon. It states that when a molecule is undergoing an electronic transition, such as ionization, the nuclear configuration of the molecule experiences no significant change. This is due in fact that nuclei are much more massive than electrons and the electronic transition takes place faster than the nuclei can respond. When the nucleus realigns itself with with the new electronic c

- 13.8: Rotational Spectra of Polyatomic Molecules
- To consider the rotational energy of molecules, it is useful to divided molecules into five categories: Diatomic, linear, symmetric tops, spherical tops, and asymmetric tops. The principle moments of inertial of polyatomic molecules: Rotation of the molecule can take places about any axis passing through the center of mass. There are two unique axes that are at 90º of each other, and about which the moment of inertia is a minimum or a maximum.

- 13.9: Normal Modes in Polyatomic Molecules
- Normal modes are used to describe the different vibrational motions in molecules. Each mode can be characterized by a different type of motion and each mode has a certain symmetry associated with it. Group theory is a useful tool in order to determine what symmetries the normal modes contain and predict if these modes are IR and/or Raman active. Consequently, IR and Raman spectroscopy is often used for vibrational spectra.

- 13.10: Irreducible Representation of Point Groups
- Each of these coordinates belongs to an irreducible representation of the point the molecule under investigation. Vibrational wavefunctions associated with vibrational energy levels share this property as well. The normal coordinates and the vibration wavefunction can be categorized further according to the point group they belong to. From the character table predictions can be made for which symmetries can exist.

- 13.11: Time-Dependent Perturbation Theory
- Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian. Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. Thus, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory.

- 13.12: The Selection Rule for the Rigid Rotator
- A selection rule describes how the probability of transitioning from one level to another cannot be zero. Selection rules only permit transitions between consecutive rotational levels: ΔJ=J±1, and require the molecule to contain a permanent dipole moment. Due to the dipole requirement, molecules such as HF and HCl have pure rotational spectra and molecules such as hydrogen and nitrogen are rotationally inactive.

- 13.13: The Harmonic Oscillator Selection Rule
- Transitions with Δv= ±1, ±2, ... are all allowed for anharmonic potential, but the intensity of the peaks become weaker as Δv increases. v=0 to v=1 transition is normally called the fundamental vibration, while those with larger Δv are called overtones. Δv=0 transition is allowed between the lower and upper electronic states with energy E1 and E2 are involved, i.e. (E1, v''=n) →→ (E2, v'=n), where the double prime and prime indicate the lower and upper quantum state.

- 13.14: Group Theory Determines Infrared Activity
- Group theory makes it easy to predict which normal modes will be IR and/or Raman active. If the symmetry label of a normal mode corresponds to x, y, or z, then the fundamental transition for this normal mode will be IR active. If the symmetry label of a normal mode corresponds to products of x, y, or z (such as \(x^2\) or yz) then the fundamental transition for this normal mode will be Raman active.

- 13.E: Molecular Spectroscopy (Exercises)
- These are exercises for Chapter 13 of the McQuarrie and Simon Textmap for Physical Chemistry.