MIT5_74s09_lec08

MIT5_74s09_lec08 - MIT OpenCourseWare http:/ocw.mit.edu...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Andrei Tokmakoff, MIT Department of Chemistry, 3/17/08 6- 12 6.4 NUCLEAR MOTION COUPLED TO ELECTRONIC TRANSITION 1 Here we will start with one approach to a class of widely used models for the coupling of nuclear motions to an electronic transition that takes many forms and has many applications. We will look at the specific example of electronic absorption experiments, which leads to insight into the vibronic structure in absorption spectra. Spectroscopically, it is also used to describe wavepacket dynamics; coupling of electronic and vibrational states to intramolecular vibrations or solvent; or coupling of electronic states in solids or semiconductors to phonons. Further extensions of this model can be used to describe fundamental chemical rate processes, interactions of a molecule with a dissipative or fluctuating environment, and Marcus Theory for non-adiabatic electron transfer. Two-electronic state as displaced harmonic oscillators We are interested in describing the electronic absorption spectrum for the case that the electronic energy depends on nuclear configuration. The simplified model for this is two identical harmonic oscillators potentials displaced from one another along a nuclear coordinate, and whose 0-0 energy splitting is E e E g . We will calculate the electronic absorption spectrum in the interaction picture ( HHV = 0 + () ) using the t time-correlation function for the dipole operator. The Hamiltonian for the matter represents two Born- Oppenheimer surfaces = G HG + E HE (6.1) H 0 G E where the Hamiltonian describing the ground and excited states have contributions from the nuclear energy and the electronic energy H G = g + g EH . (6.2) H = + E e e
Background image of page 2
Andrei Tokmakoff, MIT Department of Chemistry, 3/17/08 6- 13 The harmonic vibrational Hamiltonian has the same curvature in the ground and excited states, but the excited state is displaced by d relative to the ground state. H g = p 2 + 1 m ω 0 2 q 2 (6.3) 2 m 2 H e = p 2 + 1 m 0 2 ( q d ) 2 (6.4) 2 m 2 Now we are in a position to evaluate the dipole correlation function e iH t / h μ e iH t / h 0 0 C μμ t p n n . (6.5) () = n = , nEG with the time propagator iH + E t / h − ( e E e ) t / h + iH t / h ( g g ) e 0 = G e G + E e E (6.6) We begin by making two approximations: 1) Born-Oppenheimer Approximation . Although this is implied in eq. (6.2) when we write the electronic energy as independent of q , specifically it means that we can write the state of the system as a product state in the electronic and nuclear configuration: G , = g n (6.7) 2) Condon Approximation . This approximation states that there is no nuclear dependence for the dipole operator. It is only an operator in the electronic states.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 17

MIT5_74s09_lec08 - MIT OpenCourseWare http:/ocw.mit.edu...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online