Schrödinger Equation – Model Systems: We have carefully considered the development of the Schrödinger equation for important model systems – the one, two and three dimensional PIAB (particle in a box) and the POAR (particle moving on a planar ring - rotation about a fixed point/axis).

Schrödinger Equation – Model Systems: For each of the model systems looked at we have (a) looked at the classical energy expression, (b) constructed a Hamiltonian operator using the classical energy expression as a guide, (c) set up the Schrödinger equation and (d) solved the Schrödinger equation to determine (1) the form of the wave functions (eventually normalized) and (2) the expressions for energies (eigenvalues).

Atomic and Molecular Energies: Hamiltonians and energies for atoms and molecules are more important than those of model systems – although real and model systems have similarities. We will look at rotational energies, vibrational energies and electronic energies in this course. In the first two cases the wave functions will not be fully developed.

Rotational Energies of Molecules: In condensed phases (solid and gas) individual molecules cannot rotate freely. Why? We say that rotational motion is quenched. In the gas phase, where molecules are not in contact, free rotation is possible. It turns out that, in the gas phase, rotational energies are quantized – only particular frequencies are observed in the rotational spectrum of a molecule.

Rotational Energies of Diatomics: Diatomic molecules have very simple rotational spectra. A list of some of the strongest low frequency lines in the rotational spectra of carbon monoxide ( 12 C 16 O) and carbon monosulfide ( 12 C 32 S) are presented on the next slide. The data are taken form the NIST database

Rotational Spectra of 12 C 16 O and 12 C 32 S: Line Number 12 C 16 O Frequency (MHz) 12 C 32 S Frequency (MHz)

Rotational Spectra of 12 C 16 O and 12 C 32 S: Any spectroscopic experiment gives us energy level spacings – if we remember Planck’s constant and ΔE = hν. The data on the previous slide tell us that the energy level spacings are larger for 12 C 16 O than for 12 C 32 S. This is reminiscent of what we saw earlier for our model systems – the PIAB and the POAR.

PIAB and POAR Energies Revisited:

Rotational Energies – Diatomic Molecules: We might expect that the energy expression for a freely rotating molecule would be similar to the energy expression for the POAR. This turns out to be true. The solution of the Schrӧdinger equation in this case is both time consuming and mathematically formidable. We’ll content ourselves with the result in this course.

Rotational Energies – Diatomics:

Rotational Energies – Diatomics:

Spectroscopic Constants: The rotational constant, B, is the quantity which is determined by experiment when spectroscopy is used to study a rotational spectrum. The fundamental unit of B is s -1 or Hz (although MHz and GHz are seen). In infrared work one sees “rotational constants” reported with the unit cm -1.

Rotational Energy Level Patterns:

Rotational Energies – Linear Molecules: J Value (Rotational Quantum Number) J(J+1) ValueRotational Energy hB 266hB 31212hB 42020hB 53030hB 64242hB

Linear Molecules – Selection Rules: In any spectroscopic experiment we need to know what transitions between energy levels are possible. Here, we need to know how J can change when a transition from one rotational energy level to another occurs. Class Example: Determine the selection rule for rotational spectroscopy (absorption experiment) using experimental data for CO.

Rotational Constant Calculations: Class Examples: Use structural data and atomic masses (for specific isotopes) to determine rotational constants. We will explore the use of both symmetry and center of mass calculations in calculating values for (a) moments of inertia and (b) rotational constants for simple molecules.

Rotational Spectra: Class Example: Use our “derived” selection rule for rotational spectra of linear molecules to obtain closed form expressions for (a) the rotational energy differences corresponding to observable rotational transitions and (b) a closed form expression for observed frequencies in linear molecule rotational spectra.