Molecular Hamiltonians and Molecular Spectroscopy

1. Molecular Hamiltonians and Molecular Spectroscopy

2. The Molecular Hamiltonian:2 nuclei plus 2 electrons 2 e- r12 1 e- r2B r1A r1B r2A A Ze+ rAB B Ze+ where , etc. Solve for allowed quantum states, n = 1, 2, 3, …

3. Molecular Spectroscopy The Bohr condition: hν = En - Em • The art of observing transitions between quantum states. En hν Em

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 TN VNN He • where α, βlabel 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 qα are kept fixed. • This means thatdepends parametrically on what fixed nuclear coordinates we have chosen, and so do the resultant energy eigenvalues, U = U(qα) • Then U(qα) 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 • Separation is only approximate because molecules vibrate and rotate at the same time. • This approximation is generally poorer than the B.O. approx. • Coriolis effects, centrifugal distortion • Rotation is defined relatives to space-fixed coordinates. • Vibration is defined relative to molecule-fixed coordinates(as is electronic motion). The nuclear potential energy U(qα)for a diatomic molecule

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. Rotational and Vibrational Spectroscopies microwave mm-wave THz/far-IR mid-IR near-IR Frequency 10 GHz 100 GHz 1 THz 10 THz 100 THz 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) Vibrational Spectroscopy Rotational Spectroscopy bench-top FTIR FTMW (typical) kT (300 K)