PCV Project: Excitons in Molecular Spectroscopy

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1 PCV Project: Excitons in Molecular Spectroscopy Introduction The concept of excitons was first introduced by Frenkel (1) in 1931 as a general excitation delocalization mechanism to account for the ability of solids to absorb high energy quanta radiation without local melting. Since then, the concept has been successfully applied to diverse physical systems, such that further classification of different exciton types has been developed, depending on the situation. When the intermolecular interaction is strong, as in the case of ionic crystals, the atomic exciton model is applicable. This is known as the weakly-bound case because the exciton can be treated as a positive centre-electron pair travelling through the crystal, with significant charge separation between the electron and its positive hole. For the purpose of this project, however, we will just consider the case of molecular excitons, which is typified by weak intermolecular electronic interaction (van der Waals), so that the component molecules of the system preserve their individuality. This is known as the tightly-bound case because the excited electron is associated with a specific molecular centre, and does not migrate from one centre to another. The molecular exciton model (2,3) may be defined as describing resonance interaction between excited states of composite molecular systems, in which nondegenerate states of the individual molecular components become split in the composite particle, usually considered to be caused by dipole-dipole interaction. Selection rules apply for the electric dipole transition between the ground state and the exciton band, which can lead to some interesting spectroscopic properties - some of which will be explored later. The precise quantitative characteristics of the exciton band structure will depend on the intermolecular orientation, centre-to-centre distance, and the intensity of the component molecule optical transition involved in the state resonance interaction. Historically, most of the research on molecular exciton theory and its applications was originally applied to molecular crystals. However, Kasha has applied the model very successfully to elucidate the spectral properties of smaller molecular aggregates, such as weakly-bound organic dimers & trimers of various geometries, long-chain molecular polymers, helical molecular arrays such as polypeptides and nucleic acid, and molecular lamellar aggregates. More recently the molecular exciton model has been applied to explain more exotic spectroscopic effects in nanoscale systems such as quantum dots and carbon nanotubes (4). The vibrational exciton model has also been developed and used to predict the IR spectra of aerosol particles (5).

2 In this project, we will begin by developing the necessary theoretical framework of the molecular exciton model, following Kasha s wave-function formalism. It appears rather similar to the familiar molecular orbital theory, but the physical basis and interpretations derived from it are quite different. We will begin with the simple case of a dimer, exploring how the dipole-dipole configuration and intermolecular geometry affect the exciton splitting energy diagrams. We will then extend the model to include long chain linear polymers, and compare our results with the dimer case. Theoretical framework The molecular exciton model is a state interaction theory. If the intermolecular electron overlap is small, so that the molecular units preserve their individuality in the composite particle, the model will satisfy the requirements of a perturbation theory. We may then seek solutions (wave-functions and energies) for the particle in terms of the constituent wave-functions and energies for the electronic states of the components. The ground state wave-function of the dimer has the description: Ψ G = φ u φ v where φ u represents the ground state wave-function of molecule u and φ v the corresponding one for molecule v (all wave-functions are assumed to be real). The Hamiltonian operator for the dimer is: H = H u + H v + V uv where H u and H v are the Hamiltonian operators for the isolated molecules, u and v, and V uv is the intermolecular perturbation potential. The latter is a coulombic potential, which may be approximated as a point-dipole interaction (6). 1. Derive an expression for the energy of the ground state of the dimer using the time-independent Schrödinger equation. Your final expression should contain a term which represents the van der Waals interaction energy (an energy lowering) between the ground states of molecules u and v, as well as terms representing the ground state energies of the isolated molecules. The excited state dimer wave-functions (exciton wave-functions) may be written: Ψ E = rφ u φ v + sφ u φ v where φ u and φ v represent excited state wave-functions for a particular excited state of molecules u and v, with r and s coefficients to be determined.

3 2. Write the Schrödinger equation for the excited state. Assuming u and v are identical, derive two simultaneous equations for H uu and H uv. (Hint: multiply both sides of the Schrödinger equation by φ u φ v and integrate over coordinates for molecules u and v, then repeat for φ u φ v.) 3. Write the determinant for the coefficients r and s, and determine the roots. (Hint: For non-trivial solutions, we must set the determinant = 0). The roots correspond with the energies of the excited state. They should contain four terms one representing the excited state energy of an individual molecule, one for the ground state of the other molecule, one for the van der Waals interaction between the molecules, and finally a term corresponding to the exciton splitting. In the point-dipole approximation, the exciton splitting term becomes: E = M u. M v r 3 3(M u. r)(m v. r) r 5 and represents an interaction energy due to the exchange of excitation energy between molecule u and molecule v. M u is the transition moment in molecule u, and r is the position vector of the v dipole referred to the u dipole as origin. 4. Write a simple expression for the transition energy ΔE for the composite molecule from the ground state to excited state(s), taking the difference of the van der Waals terms as ΔD. This expression is the characteristic form of the transition energy between states of a particle by molecular exciton theory. Energy level diagrams: Dimer Now that we have derived the basis for exciton splitting in molecular systems, we will explore how various changes in relative geometry and transition dipole configurations affect the transition energies, making use of simple energy level diagrams. 5. Draw a qualitative energy level diagram showing how the relative energies change on going from isolated monomers to dimer. Label the energy of the ground state E G and excited states E E and E E, and indicate the van der Waals interaction ΔD and the exciton splitting E. For now we don t need to consider relative geometries or transition dipole configurations.

4 The excited state resonance interaction may be approximated via the quasi-classical vector model, where we consider the interaction of transition moment dipoles electrostatically. Consider three cases: (i) parallel (ii) in-line and (iii) oblique transition dipoles (shown below): (i) (ii) (iii) In each case, the polarization axis for the electronic transition in the unit molecule is shown aligned with the long axis of the molecule represented in the oval profile. 6. Draw qualitative energy level diagrams for the three cases shown above (i-iii). Considering the dipole phase relations (in-phase/out-of-phase), for each case indicate and explain which transitions are allowed and which are forbidden. The exciton splitting energy, corresponding to the separation ΔE = E E E E, is given by: E = 2 M 2 3 (cosα + 3cos 2 θ) r uv where M is the transition moment for the singlet-singlet transition in the monomer, r uv is the centre to centre distance between molecules u and v, α is the angle between polarization axes for the component absorbing units and θ is the angle made by the polarization axes of the unit molecule with the line of molecular centres. The transition moments to the exciton states E E and E E are given by: M = 2Mcosθ M = 2Msinθ 7. Using the mathematical software of your choice, plot the exciton splitting energy (in arbitrary units) as a function of θ, α, r and M respectively, in each case keeping all other parameters constant. Explain your observations. How would you expect the absorption spectrum of these three dimers to differ from the monomer? Now consider the case of co-planar inclined transition dipoles (iv):

5 The exciton band splitting in this case is given by the formula: E = 2 M 2 3 (1 3cos 2 θ) r uv The transition moments for this case are given by M = 0 and M = 2M. 8. Once again, plot the exciton splitting as a function of θ. At what angle is ΔE = 0? And lastly for the dimer we consider the case of non-planar transition dipoles, as shown in (v): The exciton splitting in this case is given by: E = 2 M 2 3 (cosα 3cos 2 θ) r uv where α is the angle between the two molecular planes defined by the diagram above, and θ is the angle between the polarization axes and the line of molecular centres. The transition moments for electric dipole transitions from the ground state to the two exciton states E E and E E vary continuously with angle α. 9. Plot the exciton splitting as a function of angles θ and α in this final example for the dimer and describe your observations. Comment on the allowed/disallowed character of the transitions as a function of the two angles. Linear chain polymers Using the same theoretical framework outlined previously, we can extend the model to linear chain molecular aggregates. 10. Assuming N identical molecules (N very large), construct the ground and excited state wave-functions as the product of independent molecular functions. Derive the corresponding energies of the ground and excited states.

6 Consider the two geometries I and II shown below: For case I the exciton splitting is given by: E = 4 N 1 N M 2 r (1 3 3cos2 α) The transition moments for this case are given by: M = 0 and M = (N 1/2 )M when π/2 α > arc cos (1/3 1/2 ) M = (N 1/2 )M and M = 0 when 0 α < arc cos (1/3 1/2 ) For case II the exciton splitting is given by: E = 4 N 1 N M 2 r (1 3 + cos2 α) The transition moments for this case are given by: M = (N 1/2 )Mcosα M = (N 1/2 )Msinα 11. Plot the exciton splitting for the two cases as a function of N and α. What do you observe? How does this compare with what we saw for the dimer? 12. Do you think there are any deficiencies in the molecular exciton model? Are all of the assumptions valid? How might we improve the model?

7 13. (Optional) Extend the molecular exciton model to helical particles composed of N identical molecules (of the form shown below), and derive the exciton splitting energies. Draw corresponding energy-level diagrams for the cases of parallel and perpendicular polarization of the transition moment vectors with respect to the helical axis. References (1) J. Frenkel, Phys. Rev. 37, 7, 1276 (1931). (2) Kasha, M., Rawls, H. R., & Ashraf El-Bayoumi, M. (1965). The exciton model in molecular spectroscopy. Pure and Applied Chemistry, 11(3-4), (3) Kasha, M. (1976). Molecular excitons in small aggregates. Spectroscopy of the excited state, 12, (4) Scholes, G. D., & Rumbles, G. (2006). Excitons in nanoscale systems. Nature materials, 5(9), (5) Sigurbjörnsson, O. F., Firanescu, G., & Signorell, R. (2009). Intrinsic particle properties from vibrational spectra of aerosols. Annual review of physical chemistry, 60, (6) E. G. McRae and M. Kasha. Physical Processes in Radiation Biology, p. 23. Academic Press, New York (1964).