Rotational Spectra Simplest Case: Diatomic or Linear Polyatomic molecule Rigid Rotor Model: Two nuclei joined by a weightless rod J = Rotational quantum number (J = 0, 1, 2, …) I = Moment of inertia =  r 2  = reduced mass = m 1 m 2 / (m 1 + m 2 ) r = internuclear distance m1m1 m2m2 r

Rigid Rotor Model In wavenumbers (cm -1 ): Separation between adjacent levels: F(J) – F(J-1) = 2BJ

Rotational Energy Levels Selection Rules: Molecule must have a permanent dipole.  J =  1 J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.

Rotational Spectra J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, J” → J’F(J’)-F(J”) 3 → 42(1.91)(4)15.3 cm -1 4 → 52(1.91)(5)19.1 cm -1 5 → 62(1.91)(6)22.9 cm -1 6 → 72(1.91)(7)26.7 cm -1 7 → 82(1.91)(8)30.6 cm -1 8 → 92(1.91)(9)34.4 cm -1 9 → 102(1.91)(10)38.2 cm -1

Intensity of Transitions J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, %T cm -1

Are you getting the concept? Calculate the most intense line in the CO rotational spectrum at room temperature and at 300 ° C. The rigid rotor rotational constant is 1.91 cm -1. Recall: k = 1.38 x J/K h = x Js c = 3.00 x 10 8 m/s

The Non-Rigid Rotor : Account for the dynamic nature of the chemical bond:  J = 0,  1 D is the centrifugal distortion constant (D is large when a bond is easily stretched) Typically, D < *B and B = 0.1 – 10 cm -1

More Complicated Molecules Still must have a permanent dipole  J = 0,  1 K is a second rotational quantum number accounting for rotation around a secondary axis A.

Vibrational Transitions Simplest Case: Diatomic Molecule Harmonic Oscillator Model: Two atoms connected by a spring. v = vibrational quantum number (v = 0, 1, 2, …) = classical vibrational frequency k = force constant (related to the bond order). in Joules in cm -1

Vibrational Energy Levels J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, Selection Rules: 1)Must have a change in dipole moment (for IR). 2)  v =  1

Anharmonicity Ingle and Crouch, Spectrochemical Analysis Selection Rules:  v =  1,  2,  3, …  v = 2, 3, … are called overtones. Overtones are often weak because anharmonicity at low v is small.

Rotation – Vibration Transitions The rotational selection rule during a vibrational transition is:  J =  1 Unless the molecule has an odd number of electrons (e.g. NO). Then,  J = 0,  1  J = 0,  1 B v signifies the dependence of B on vibrational level

Rotation – Vibration Transitions Ingle and Crouch, Spectrochemical Analysis If  J = -1  P – Branch If  J = 0  Q – Branch If  J = +1  R – Branch

Rotation – Vibrational Spectra Why are the intensities different? J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.

Are you getting the concept? In an infrared absorption spectrum collected from a mixture of HCl and DCl, there are eight vibrational bands (with rotational structure) centered at the values listed below. Identify the cause (species and transition) for each band. Band LocationSpecies/Transition 2096 cm cm cm cm cm cm cm cm -1 Atomic masses H → amu D → amu 35 Cl → amu 37 Cl → amu

Raman Spectra J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, Selection Rule:  J = 0,  2

Polyatomics If linear  (3N – 5) vibrational modes (N is the # of atoms) (N is the # of atoms) If non-linear  (3N – 6) vibrational modes Only those that have a change in dipole moment are seen in IR.

Linear Polyatomic J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, How many vibrational bands do we expect to see?

Nonlinear Polyatomic (Ethylene) J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.

Infrared Spectroscopy Near Infrared: 770 to 2500 nm Near Infrared: 770 to 2500 nm 12,900 to 4000 cm -1 Mid Infrared: 2500 to 50,000 nm (2.5 to 50  m) Mid Infrared: 2500 to 50,000 nm (2.5 to 50  m) 4000 to 200 cm -1 Far Infrared: 50 to 1000  m Far Infrared: 50 to 1000  m 200 to 10 cm -1

Infrared Spectroscopy: Vibrational Modes Ingle and Crouch, Spectrochemical Analysis

Pretsch/Buhlmann/Affolter/ Badertscher, Structure Determination of Organic Compounds Group Frequencies Estimate band location:

Are you getting the concept? Estimate the stretching vibrational frequency for a carbonyl group with a force constant, k, of 12 N/cm. If a C=S bond had the same force constant, where would its stretching band appear in the infrared absorption spectrum? Recall: 1 amu = x kg 1N = 1 kg*m*s -2 Atomic masses C → amu O → amu S → amu

Infrared Spectroscopy Near Infrared: 770 to 2500 nm Near Infrared: 770 to 2500 nm 12,900 to 4000 cm -1 * Overtones * Combination tones * Useful for quantitative measurements Mid Infrared: 2500 to 50,000 nm (2.5 to 50 um) Mid Infrared: 2500 to 50,000 nm (2.5 to 50 um) 4000 to 200 cm -1 * Fundamental vibrations * Fingerprint region 1300 to 400 cm -1 (characteristic for molecule as a whole) (characteristic for molecule as a whole) Far Infrared: 2.5 to 1000 um Far Infrared: 2.5 to 1000 um 200 to 10 cm -1 * Fundamental vibrations of bonds with heavy atoms (useful, e.g., for organometallics) atoms (useful, e.g., for organometallics)

Example of an Overtone Wagging vibration at 920 cm -1. Overtone at approximately 2 x 920 cm -1 = 1840 cm -1.

N.B. Colthup et al., Introduction to Infrared and Raman Spectroscopy, Academic Press, Boston, Fermi Resonance

Example of a Fermi Resonance Stretching vibration of C-C=(O) at 875 cm -1. Overtone at approximately 2 x 875 cm -1 = 1750 cm -1 coincides with C=O stretch