14: Time-dependent Quantum Dynamics
- Page ID
- 60571
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The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions.
- 14.1: Time-Dependent Vector Potentials
- Modifying the nuclear and electronic momenta of the full electronic and nuclear-motion Hamiltonian to address interactions with light results in "interaction perturbations" that induce transitions among the various electronic/vibrational/rotational states of a molecule. The one-electron additive nature of these perturbation plays an important role in determining the kind of transitions that can be induce.
- 14.2: Time-Dependent Perturbation Theory
- The mathematical machinery needed to compute the rates of transitions among molecular states induced by such a time-dependent perturbation is contained in time dependent perturbation theory (TDPT).
- 14.3: Application to Electromagnetic Perturbations
- Light-induced transitions between states can be treated with first-order time-dependent perturbation theory. One results of this is the first-order Fermi-Wentzel "golden rule" expression that gives the rate as the square of a transition matrix element between the two states involved, of the first order perturbation multiplied by the light source function evaluated at the transition frequency.
- 14.4: The "Long-Wavelength" Approximation
- To make progress in further analyzing the first-order results obtained above, it is useful to consider the wavelength λλ of the light used in most visible/ultraviolet, infrared, or microwave spectroscopic experiments. Even the shortest such wavelengths (ultraviolet) are considerably longer than the spatial extent of all, but the largest molecules (i.e., polymers and biomolecules for which the approximations we introduce next are not appropriate).
Thumbnail: Energy diagram illustrating the Franck–Condon principle applied to the solvation of chromophores. The parabolic potential curves symbolize the interaction energy between the chromophores and the solvent. The Gaussian curves represent the distribution of this interaction energy. (CC -BY-SA 3.0; Mark Somoza)