1. 1 Molecular Hamiltonians and Molecular Spectroscopy

2. A Ze+ B Ze+ 1 e- 2 e- r 2B r 1A r 12 r AB r 2A r 1B where, etc. Solve for allowed quantum states, n = 1, 2, 3, … The Molecular Hamiltonian: 2 nuclei plus 2 electrons

3. 3 Molecular Spectroscopy The Bohr condition: hν = E n - E m EnEn EmEm hν The art of observing transitions between quantum states.

4. 4 The Molecular Hamiltonian α, β label the nuclei; i, j label the electrons Too difficult to solve without further approximations. – Approximate separation of variables in to electronic, vibrational, and rotational degrees of freedom. – Separation also leads to qualitative understanding. Some things are left out of this Hamiltonian: – Intrinsic spin: electronic and nuclear – Relativistic effects: more important for heavy atoms – Add terms to H to include these effects approximately as needed.

5. The Born-Oppenheimer Approximation TNTN V NN HeHe where α, β label the nuclei, and i,j label the electrons. We would like to solve (1) Separation of variables: (2) where we define the electronic wavefunction as solution to (3) when the nuclear coordinates q α are kept fixed. This means that depends parametrically on what fixed nuclear coordinates we have chosen, and so do the resultant energy eigenvalues, U = U(q α ) Then U(q α ) becomes the “effective potential energy” in which the nuclei move, and we can solve for the nuclear motion approximately as (4)

6. Failures of the Born-Oppenheimer Approximation Radiationless transitions: internal conversion and intersystem crossing. Collision-induced curve crossing (Landau-Zener) Conical intersections between electronic surfaces Small corrections to energies and other properties even in ground state molecules Vibrational-rotational-electronic coupling in Rydberg molecules

7. The second step: Separation of Vibration and Rotation The nuclear potential energy U(q α ) for a diatomic molecule

8. 8 Start with product wavefunctions: Electronic, vibrational, rotational, nuclear spins When the coupling terms are inportant, such product wavefunctions can serve as a basis for finding better solutions to the coupled problem. Spacings: – Nuclear spins: MHz (radio frequency) – Rotations: GHz – THz (microwave, mm-wave, THz) – Vibrations: 40 – 4000 cm -1 (1 – 100 THz) – Electronic (valence): 15,000 – 100,000 cm -1 (600 nm – 100 nm) (visible – ultraviolet) – Electronic (core) 100 nm – 0.1 nm (vacuum UV to X-ray)

9. Spectroscopies Rotational spectroscopy in microwave, mm, THz regions – Nuclear spins add hyperfine structure Vibrational spectroscopy in the THz to infrared regions – Add rotational and hyperfine structures Electronic spectroscopy in ultraviolet – Add vibrational, rotational, and hyperfine structures.

10. 10 Rotational and Vibrational Spectroscopies Wavelength 10 cm 1 cm 1 mm 100 μm 10 μm 1 μm Wavenumber 0.1 1 10 100 1,000 10,000 (cm -1 ) Frequency 10 GHz 100 GHz 1 THz 10 THz 100 THz microwave mm-wave THz/far-IR mid-IR near-IR kTkT kTkT (300 K)

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