L10 - Biophysical Chemistry Chemistry 24a Winter Term...

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Biophysical Chemistry Chemistry 24a Winter Term 2009-10 Instructor: Sunney I. Chan Lecture 10 February 1, 2010 Exciton Interaction. Hypo- and Hyper-chromism. Interacting Chromophores The Quantum Mechanics of Interacting Chromophores Let’s begin with a dimer of two non-interacting chromophores. For simplicity, we will consider a two-level system, namely, chromophores with ground and an excited singlet state only. a ───────── ε a , Φ a o ───────── ε o , Φ o where Φ o , Φ a , ε o , ε a are the eigenfunctions and energies of the one-chromophore hamiltonian, namely, h Φ o = ε o Φ o , h Φ a = ε a Φ a . For a dimer of two non-interacting identical chromophores, H dimer = h (1) + h (2). The Hamiltonian for the dimer is just the sum of one-chromophore hamiltonians. Therefore the wavefunction for the dimer is just the product of the one chromophore eigenfunctions. Ground state : Ψ 0 (1,2) = Φ o (1) Φ o (2), W = 2 ε o
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Excited states : Ψ 1a (1,2) = Φ a (1) Φ o (2), W = ε o + ε a Ψ 2a (1,2) = Φ o (1) Φ a (2), W = ε o + ε a Ψ aa (1,2) = Φ a (1) Φ a (2), W = 2 ε a Energy levels aa ─────── Φ a (1) Φ a (2) 2 ε a 1a ─────── Φ a (1) Φ o (2), Φ o (1) Φ a (2) ε o + ε a 0 ─────── Φ o (1) Φ o (2) 2 ε o 2a Transitions Only 0 1a or 2a allowed! Dipole operator of dimer is μ 1 + μ 2 So, dipole strength for 0 1a transition = | Ψ 1a *(1, 2) ( μ 1 + μ 2 ) Ψ o (1, 2) dr 1 dr 2 = | Φ a *(1) Φ o *(2) ( μ 1 + μ 2 ) Φ o (1) Φ o (2) dr 1 dr 2 = | Φ a *(1) ( μ 1 ) Φ o (1) dr 1 = D oa for monomer Same result for 0 2a transition Spectrum Dipole strength D oa ───────────── ν ν oa
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A dimer of interacting chromophores Now, allow the two chromophores to interact. The Hamiltonian for the system is now H dimer = h (1) + h (2) + V(1,2) where V describes the interaction between the two monomers. The most important interaction is the electrostatic interaction between the two chromophores. At sufficiently close inter- chromophore separation, overlap of the charge clouds of the two molecules may need to be taken into consideration. For neutral molecules, the leading term in V is the electric dipole-dipole interaction, if we expand V in a multipole expansion: ( μ 1 μ 2 / R 12 ³ - 3 ( μ 1 •R 12 )(R 12 μ 2 )/R 12 5 μ 2 μ 1 R 12 Wavefunctions and energy levels Ground level or ground state : Ψ
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