Theoretical Principles of Fluorescence Spectroscopy

Abstract

The chapter outlines general principles of fluorescence spectroscopy. Basic principles of radiative and nonradiative transitions (including the Jablonski diagram and Franck–Condon principle) are described and explained. The fundamentals of important fluorescence techniques, such as the steady-state and time-resolved measurements, fluorescence anisotropy, solvent relaxation method, fluorescence quenching, and nonradiative energy transfer, are discussed in detail. Special attention is devoted to the fast dynamics of individual transitions and processes influencing them at the molecular level. The end of the chapter focuses on excimers and exciplexes and mainly on the weakly bound complexes (so-called J and H aggregates), because the literature describing their behavior is relatively rare and pertinent pieces of information are not easy to find.

Appendix: Simple Quantum Mechanics Explanation of Nondegenerate Transitions Between Energy Levels in “J” and “H” Dimers

The ground-state wave function of a dimer composed of molecules A and B, Ψ _G = Φ = ψ _A ψ _B, is a totally symmetrical product with respect to all the symmetry operations of the dimer AB. The first excited state can be described by two equivalent wave functions, Φ ₁ = ψ _A ψ _B ^*·and·Φ ₂ = ψ _A ^* ψ _B. Their energies are degenerate. The delocalized stationary states corresponding to the “exciton,” i.e., to the state in which the excited electron is not localized in any of them, are described by a symmetrical and antisymmetrical combination of the two above functions:

$$ {\varPsi}_{+}=\left(1/\surd 2\right) \left({\varPhi}_1+{\varPhi}_2\right)=\left(1/\surd 2\right) \left({\psi}_{\mathrm{A}}{\psi}_{\mathrm{B}}^{*}+{\psi}_{\mathrm{A}}^{*}{\psi}_{\mathrm{B}}\right) $$

(43)

$$ {\varPsi}_{-}=\left(1/\surd 2\right) \left({\varPhi}_1-{\varPhi}_2\right)=\left(1/\surd 2\right) \left({\psi}_{\mathrm{A}}{\psi}_{\mathrm{B}}^{*}-{\psi}_{\mathrm{A}}^{*}{\psi}_{\mathrm{B}}\right) $$

(44)

The node of the wave function does not correspond to a change in the sign of the wave function, but to a change in the orientation of the dipole moment. The energies of states Ψ ₊ and Ψ ₋ are E ± = ΔE ± E′, where ΔE is the energy difference between the excited and ground states of the monomer and E′ is the perturbation (energy splitting) due to interaction of the excited and ground-state dipoles. The value + E′ corresponds to Ψ ₊ and similarly for –E′. This value can be calculated using the perturbation Hamiltonian and the wave functions of the unperturbed degenerate states Φ ₁ and Φ ₂. The perturbation Hamiltonian can be expressed as the classic expression for the energy of interacting dipoles. If we take into account only the changes in the dipoles in one dimension (which is the case for most fluorophore dimers), we can write

$$ {H}_{\mathrm{pert}}=\frac{e^2}{4\pi {\varepsilon}_0{r}^3}{\displaystyle \sum_{ij}{x}_{\mathrm{A}}^i{x}_{\mathrm{B}}^j} $$

(45)

where e is the elementary charge, ε ₀ is the dielectric permittivity of vacuum, and x ⁱ describes the positions of the electrons in molecule A (x ^j in molecule B).

After insertion in (45), we get

$$ {E}^{\prime }={\displaystyle \iint {\varPhi}_1{\widehat{H}}_{\mathrm{pert}}{\varPhi}_2d{\tau}_{\mathrm{A}}d{\tau}_{\mathrm{B}}}=\frac{e^2}{4\pi {\varepsilon}_0{r}_{\mathrm{A}\mathrm{B}}^3}{\displaystyle \iint {\psi}_{\mathrm{A}}{\psi}_{\mathrm{B}}^{\ast }}{\displaystyle \sum_{\mathrm{ij}}{x}_{\mathrm{A}}^i{x}_{\mathrm{B}}^j}{\psi}_{\mathrm{A}}^{\ast }{\psi}_{\mathrm{B}}d{\tau}_{\mathrm{A}}d{\tau}_{\mathrm{B}} $$

(46)

Because x ⁱ describes the positions in A only and x ^j in B only, expression (44) can be rewritten

$$ {E}^{\prime }=\frac{1}{4\pi {\varepsilon}_0{r}_{\mathrm{A}\mathrm{B}}^3}\left[{\displaystyle \int {\psi}_{\mathrm{A}}{\displaystyle \sum e{x}_{\mathrm{A}}^i{\psi}_{\mathrm{A}}^{\ast }d{\tau}_{\mathrm{A}}}}\right]\left[{\displaystyle \int {\psi}_{\mathrm{B}}{\displaystyle \sum e{x}_{\mathrm{B}}^j{\psi}_{\mathrm{B}}^{\ast }d{\tau}_{\mathrm{B}}}}\right]=\frac{1}{4\pi {\varepsilon}_0{r}_{\mathrm{A}\mathrm{B}}^3}{\overrightarrow{\mu}}_{\mathrm{A}} {\overrightarrow{\mu}}_{\mathrm{B}} $$

(47)

where ${\overrightarrow{\mu}}_{\mathrm{A}}$ and ${\overrightarrow{\mu}}_{\mathrm{B}}$ are the transition moments of the individual molecules. The transition moments of the dimer are

$$ {\overrightarrow{\mu}}_{+}={\displaystyle \int \int {\varPsi}_{\mathrm{G}}}\left({\overrightarrow{\mu}}_{\mathrm{A}}+{\overrightarrow{\mu}}_{\mathrm{B}}\right){\varPsi}_{+}d{\tau}_{\mathrm{A}}d{\tau}_{\mathrm{B}} $$

(48)

$$ {\overrightarrow{\mu}}_{-}={\displaystyle \int \int {\varPsi}_{\mathrm{G}}}\left({\overrightarrow{\mu}}_{\mathrm{A}}+{\overrightarrow{\mu}}_{\mathrm{B}}\right){\varPsi}_{-}d{\tau}_{\mathrm{A}}d{\tau}_{\mathrm{B}} $$

(49)

After the insertion of the expressions for the wave functions and application of the orthogonality properties of the wave functions of different states of the same molecule, we get

$$ {\overrightarrow{\mu}}_{+}=\left({\scriptscriptstyle \frac{1}{\sqrt{2}}}\right)\left({\overrightarrow{\mu}}_{\mathrm{A}}+{\overrightarrow{\mu}}_{\mathrm{B}}\right) $$

(50)

$$ {\overrightarrow{\mu}}_{-}=\left({\scriptscriptstyle \frac{1}{\sqrt{2}}}\right)\left({\overrightarrow{\mu}}_{\mathrm{A}}-{\overrightarrow{\mu}}_{\mathrm{B}}\right) $$

(51)

The above-outlined simple theoretical description provides a clue to deciding which transition is allowed and which is forbidden. For a coplanar arrangement of two aromatic rings with both dipole moments oriented in the same direction, energy contribution E´ is positive, Eq. (47). State Ψ ₊ has higher energy than Ψ ₋ and also than the excited state of the monomer. The transition moment for transition Ψ _G → Ψ ₊ is ${\overrightarrow{\mu}}_{+}=\left({\scriptscriptstyle \frac{1}{\sqrt{2}}}\right)\left(2{\overrightarrow{\mu}}_{\mathrm{A}}\right)\ne 0$ and this transition is allowed. Transition Ψ _G → Ψ ₋ is forbidden because ${\overrightarrow{\mu}}_{-}=0$. If the dipole moments are antiparallel, E´ is negative. This means that Ψ ₊ has lower energy and that transition Ψ _G → Ψ ₊ is forbidden because the two contributions to the final dipole moment cancel each other. It follows that the absorption spectra are identical in the two cases. Using analogous qualitative analysis for the orientation of aromatic rings in one plane, we can find that, for the “head-to-tail” as well as the “head-to-head” orientation of the dipole moments, the allowed transition will be the excitation to the lower excited state. The energy of the lower state will be the same in both cases and the dimers will be strongly fluorescent species.