Chapter 8. Molecular Motion and Spectroscopy Translation Rotation Vibration Review from Chap. 2 Energy levels Degrees of Freedom microwave infrared UV/vis Rotational Spectroscopy Vibrational Spectroscopy Electronic Transitions

Ch 8. Molecular Motion and Spectroscopy • Study of Interaction of Matter and Light (Photon) • Molecular Spectroscopy  Information about molecules such as geometry and energy levels are obtained by the interaction of molecules and photons • Molecular motions: Translation, Rotation, Vibration  determines the energy levels for the absorption or emission of photons

Spectroscopy In Ch 2, spectroscopy of atoms: transition between energy levels

8.1 Degrees of Freedom: Translation, Rotation, and Vibration 4 8.1 Degrees of Freedom: Translation, Rotation, and Vibration Consider a single Ar atom moving in 3-D space: - Moving motion is referred to as Translation To analyze the translation of an Ar, we need to know position (x, y, z) and momentum (px, py, pz) Where it is Where it is headed Each coordinate-momentum pair [for example, (x,px)] is referred to as a Degree of Freedom (DF) An Ar atom moving through 3-D space has three DFs N argon atoms possesses 3N DFs: All translational DFs

5 Internal Motions - A collection of N atoms possesses 3N DFs - If N atoms happen to be bonded together into an N-atom molecule, the number of DFs is still 3N. - But, atoms in a molecule cannot translate independently of each other No 3N translational DFs Contribution of DFs to Internal Motions - Two types: Rotation and Vibration

Center of Mass (Balanced Point) 6 Center of Mass (Balanced Point) - A point mass that can represent the molecule - Motion of the center of mass requires 3 DFs to describe it - In general, regardless of its size or complexity, a molecule has 3 translational DFs - Thus, (3N – 3) DFs for the internal motions of rotation and vibration

Figure 8.1

Figure 8.2

Rotational and vibrational DFs 9 Rotational and vibrational DFs N atomic Linear Molecule Non-Linear Molecule Rotation 2 DFs 3 DFs Vibration 3N – 5 3N - 6

Table 8.1

Etotal = Enuclear + Eelectron = Etrans + Evib + Erot + Eelectron = Etrans + Einternal

Electronic, Vibrational, and Rotational Energy Levels of a Diatomic Molecule Exercise: Indicate the molecular state in which it is electronically in the ground state, vibrationally in the first excited state, and rotationally in the ground state.

8.2 Rotational Motion: Defining a Molecular Day re = rA + rB Rotational Kinetic Energy (Erot) where ω is angular velocity, dθ/dt

Rotational Kinetic Energy (Erot) reduced mass moment of inertia (compare this with ) Using angular moment (compare this with ) the rotational kinetic energy becomes (compare this with )

Rotational period For Krot = kBT For a typical molecule at room temperature (one picosecond)

Quantization of Rotational Energy V = 0 cyclic boundary condition: Ψ(2π + θ) = Ψ(θ) By solving Schrodinger equation for rotational motion, the rotational energy levels are Rotational energy levels in wavenumber (cm-1)

Spacing between adjacent rotational levels j and j-1,

Rotational Spectroscopy (1) Bohr postulate (2) Selection Rule

The molecule should have a permanent electric dipole moment Requirements for a molecule to show a pure rotational spectrum (absorption or emission): The molecule should have a permanent electric dipole moment Review from Ch 6 • The electric dipole moment, μ = q r distance between charges magnitude of the charge separation

far-infrared to the microwave spectral regions (microwave spectroscopy)

8.3 Vibrational Motion: Molecular Calisthenics Harmonic oscillator A molecule vibrates ~50 times during a molecular day (one rotation)

Quantization of Vibrational Energy By solving Schrodinger equation for vibrational motion, Vibrational energy level where is a vibrational quantum number Zero point energy Spacing between adjacent vibrational sates

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Infrared spectral region

Vibrational Spectroscopy Vibrational selection rule

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8.4 Electronic Transitions in Molecules Molecular Orbital (MO) Theory for C2H4 molecule, UV or Visible spectral region

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We will skip Photoelectron Spectroscopy !!