Description of excitation-energy transfer

The amplitude of the density fluctuation or the induced dipole moment oscillates between donor and acceptor with the same frequency ~˜ω=√

∆E²+V² as the occu-pation of the single donor and acceptor excited states from equations (5.11) and (5.12).

It can therefore be used as an indicator for a total excitation energy per chromophore within the scope of the model.

I exemplify this by means of two sodium dimers that are aligned parallel to each other with an inter-dimer distance of 16 a₀. The single dimers are aligned with their symmetry axis in z-direction and are separated from each other in x-direction. In one system the sodium dimers are resonant with an intra-dimer bond length of5.78 a0 and a coupling strength⁴¹ ofV = 0.0695 eV. In a second system the bond of the acceptor dimer is shortened by 0.5 a0 as in [Hof+13]. The latter results in an off-resonance⁴²of

|∆E|= 0.0628 eVand a coupling strength of V = 0.0661 eV with very similar values

|∆E| ≈V.

A sodium dimer is very well described by the two-level model since it shows an isolated, strong excitation with transition dipole along its symmetry axis. I performed a real-time TDDFT simulation of the combined donor-acceptor systems with a boost in the donor half-space parallel to the symmetry axis of the sodium dimer with energy E^boost ≈6.8·10⁻⁴eV. Figures 5.6 (a1) and (a2) show the resulting absolute density fluctuation⁴³ of acceptor and donor

δN_A/D(t) = Z

V_A/D

|δn(r, t)|d³r . (5.20)

41The computation of coupling strengths is explained below.

42I computed∆E from the excitation energies of the single dimers.

43Note that the integral over a transition density without taking its absolute value is always zero Rρijd³r= 0 since the underlying eigenstates|iiand|jiare orthogonal.

0

Resonant ( E=0) Non-resonant ( E V)

(a1) (a2)

(b2) (b1)

(c1) (c2)

Linear fit Linear fit

Figure 5.6: Absolute density fluctuation (a1)+(a2), squared absolute density fluctua-tion (b1)+(b2), and energy according to equafluctua-tion (5.2) (c1)+(c2) of donor and acceptor in a resonant (a1)-(c1) and a non-resonant (a2)-(c2) donor-acceptor system consisting of two sodium dimers. In figures (a1)+(a2) a linear fit to the envelope of the acceptor density fluctuation is displayed.

The envelope of δN_A/D is a measure for the strength of the density oscillation in the respective half-space. It can be estimated numerically by time-integration over the periodic time of the fast oscillation ¯τ = ^2π_ω¯ , i.e., The latter is usually fulfilled as long as the two chromophores can be described by the exciton Hamiltonian model and do not build a single molecule.

The envelopes of δN_A/D in figures 5.6 (a1) and (a2) are fits to the estimated en-velopes from equation (5.21) with the assumed shape according to (5.15). In the resonant case in figure 5.6 (a1), δN_AandδN_Dshow the time dependence as predicted from the donor-acceptor model with the predicted beat. The envelope of the acceptor is proportional to |sin (˜ωt)|, the envelope of the donor is proportional to |cos (˜ωt)| according to equation (5.16). In the non-resonant case in figure 5.6 (a2), the envelope of the acceptor shows the same behaviour proportional to|sin (˜ωt)|as in the resonant case, just scaled by ^{√ |V|}

The initial slope of the acceptor envelope is proportional to|V|in both cases since in equation (5.15)

^V

~ω˜ sin(˜ωt)

−→t→0 ^|V|

~ t. A linear fit to the acceptor envelope is also indicated in figures 5.6 (a1) and (a2). The initial slopes of the acceptor density fluctuation of both systems are, as expected, very similar since the coupling strength is almost identical. However, the beat frequency ω˜ is larger by a factor of about√

2 in the non-resonant case.

The envelope of δN_A/D already shows the correct frequency of the energy transfer between donor and acceptor. One expects the energy to behave in the way of the coefficients |c_A^∗_D|² and |c_AD^∗|² from equations (5.11) and (5.12). This is perfectly reproduced by the energy per chromophore as defined through equations (5.1) and (5.2) and shown in figures 5.6 (c1) and (c2). As expected, the energy is completely transferred between donor and acceptor in the resonant case and only partially with amplitude _∆E^V2+V^{2 2} ≈ ¹₂ in the non-resonant case. The total energy is indicated by the black line.⁴⁴

The oscillations of the total energy mark the numerical accuracy. The faster oscil-lations that modulate the energies in the single donor and acceptor half-spaces cancel each other. They are no noise but probably the effect of slightly overlapping densities of donor and acceptor.

Considering the density fluctuations, the correct time dependence with the correct dependence on V and ∆E for the acceptor is achieved by the envelope of(δN_A)². In the resonant case, it is easily seen that this holds true for the donor. In the general case, the time-dependence of (δND)² according to (5.15) is more involved. However, one can show that

44Note that the single contributions of the energy density as defined in equation (5.1) show rapid oscillations [Sol16] that cancel each other when summed up.

where h(t) is a fast oscillating function with a time-dependent frequency but unity which is the shape expected from the coefficient|c_AD^∗|². This also explains the shape of the envelope ofδNDin figure 5.6 (a2). The proof of equation (5.22) is presented in appendix D.

(δN_A)² and(δN_D)²are shown in figures 5.6 (b1) and (b2) for the resonant and non-resonant system. In the non-resonant case, both show exactly the behaviour one expects for an energy per chromophore, such asδE_A/D in figure 5.6 (c1). In the non-resonant case, the time-dependence for both, the donor and acceptor separately, is correct as well. However, the sum of the two envelops, indicated by the black line, is not constant.

The reason lies in the different amplitudes of the donor and acceptor oscillations since R

|ρ_AA^∗|d³r and R

|ρ_DD^∗|d³r in equation (5.15) are not necessarily equal. Since the non-resonant system still consists of two molecules that differ only slightly from each other, their transition densities are still similar. The interpretation of(δN_A)²and (δN_D)² as energy is therefore still valid. Yet, if the chromophores are very different, it is, to my knowledge, not guaranteed thatR

|ρ_AA^∗|d³r and R

|ρ_DD^∗|d³r are similar. In this case,(δN_A)² and (δN_D)² cannot be analyzed quantitatively in this simple way.

Finally, I want to point out that in figures 5.6 (b1) and (b2) I showR

|δn_A/D|d³r2

, not R

|δn_A/D|²d³r. Still, I do not state that this is necessarily the best measure for a pragmatically defined energy per chromophore. There could be a possibility to relate the amplitude of the density fluctuation directly to an energy such that energy conservation is reproduced correctly. The numerical details of the presented calculations are listed in appendix E.2.3.