A two-level donor-acceptor model

Model Hamiltonian As a first step I discuss the eigenvalues and eigenfunctions of a coupled two-level system consisting of a donor (D) and an acceptor (A). I model this system in the basis of|ADi (ground state),|A^∗Di (only A is excited), and|AD^∗i (only D is excited) by the exciton Hamiltonian

Hˆ^ex =EAD|ADihAD| (5.3)

+E_A^∗_D|A^∗DihA^∗D|+E_AD^∗|AD^∗ihAD^∗| +V|A^∗DihAD^∗|+V|AD^∗ihA^∗D|.

E_AD, E_A^∗_D, and E_AD^∗ are the energies of the antisymmetrized product states|ADi,

|A^∗Di, and|AD^∗i. V is the coupling matrix elementhA^∗D|Vˆ^C|AD^∗i=hAD^∗|Vˆ^C|A^∗Di with the Coulomb interaction Vˆ^C, which mediates between the excited donor and acceptor states. In many typical exciton models [CGK06, §9.2], the ground state is not necessarily part of the model Hamiltonian but the system is only modeled in the vector space that is spanned by |A^∗Di and |AD^∗i. Since I want to model the density oscillation that results from a real-time TDDFT calculation after a boost-like excitation, I include the ground state explicitly into the model.

A resonant coupling in the following means that E_A^∗_D =E_AD^∗. Otherwise, donor and acceptor are non-resonant. I further define

E¯ = E_A^∗_D+E_AD^∗ to the first state |1i and the lower sign belongs to the second state |2i. In the case of resonant coupling with ∆E = 0, the excited eigenstates reduce to a symmetric and an antisymmetric coupling between the single acceptor and donor transitions

|1/2i = ^√1

2(±sgn(V)|A^∗Di+|AD^∗i) with energies E_1/2 = ¯E ± |V|. In the case of vanishing coupling with V = 0 and ∆E > 0, the eigenvectors are |1i = |A^∗Di and

|2i =|AD^∗i with energies E₁ =E_A^∗_D and E₂ =E_AD^∗. For ∆E <0 the eigenvectors are|1i=|AD^∗i and |2i=−|A^∗Diwith energies E1 =EAD^∗ and E2=EA^∗D.

The splitting of the energy levels of the single chromophores into the energy levels of the coupled exciton system is often called Davydov splitting [Dav64]. In a real-time TDDFT simulation, the dipole moment shows oscillations with the excitation energies E_1/2−E₀ as discussed in section 4.2. Therefore, the dipole spectrum shows two peaks that are separated by2√

∆E²+V².

This can be used to investigate coupling strengths between chromophores from a real-time TDDFT calculation. If one of the excited eigenstates is dipole-forbidden, such as the antisymmetric state in the case of resonant coupling between two equal chromophores, one must choose an excitation and an observable that do not show this dipole symmetry in order to see the spectral line. One option is to apply the boost in the donor space only and observe the dipole moment in the donor or acceptor half-spaces separately. I use this to compute the coupling strengths between two resonant sodium dimers and two BChls for different distances in a subsequent section. In the time domain, the dipole moments in the donor and acceptor half-spaces show a beat signal that can be evaluated equivalently for the coupling strength [HKK10].

Boost excitation For simplicity, I assume in the following that the single donor and acceptor states do not overlap and the space can be divided into a donor half-spaceV_D and an acceptor half-space V_A. In order to model the density fluctuation that arises from a real-time TDDFT calculation, I apply a weak boost-like excitation in the donor half-spaceV_D at t= 0. After that, the system is in the state

|ψ(t= 0)i=c_AD|ADi+c_AD^∗|AD^∗i (5.6)

=c₀|0i+c₁|1i+c₂|2i.

The boost is applied in the way of section 4.2. Hence, c_AD^∗ = ik·r_DD^∗ with the transition dipole −er_DD^∗ of the D→ D^∗ excitation and cAD =p

1− |c_AD^∗|² due to the normalization of the states. The coefficients cj = hj|ψ(t = 0)i, which relate to the eigenvectors in the second line of equation (5.6), are calculated by expressing the states |AD^∗i and|ADi through the eigenvectors. The coefficients are

c0 =cAD =p Note that the coefficients c0,c1, and c2 are time-independent and given by equation (5.7) since they measure the occupation of the eigenstates |0i, |1i, and |2i after the boost. Since the Hamiltonian remains time-independent for t >0, the occupation of the eigenstates remains constant as well. The coefficientscA^∗DandcAD^∗, however, are time-dependent. If donor and acceptor states are non-overlapping, the coefficients are given through Hence, the probability of finding the system in one of the states |A^∗Di or |AD^∗i is

|c_A^∗_D(t)|² =|k·rDD^∗|² V² the total excitation energy between donor and acceptor with frequencyω. In the case˜ of resonant coupling, this is a complete beat signal. Without coupling the energy remains at the donor. Finally, in the general case of V 6= 0 and ∆E 6= 0, the beat is incomplete and only a part of the energy is transferred between donor and acceptor.

Density fluctuation By applying the same approximations for a weak boost as in section 4.2, one can write the density fluctuation as

δn(r, t) =hψ(t)|ˆn|ψ(t)i −n0(r) =−2i X

j=1,2

cjsin(ω0jt)ρ0j(r) . (5.13)

Here, the ground-state density is n₀ = hAD|ˆn|ADi. Following the same notation as in the preceding sections, ~ω0j = Ej −E0 is the excitation energy for the |0i → |ji transition with transition densityρ0j =h0|ˆn|ji.

One can express the transition densities in equation (5.13) through the basis vectors {|ADi,|A^∗Di,|AD^∗i} by using equation (5.5). In the case of non-overlapping donor and acceptor states, hAD|ˆn|A^∗Di = ρ_AA^∗ and hAD|ˆn|AD^∗i = ρ_DD^∗ are the transi-tion densities of the singleA → A^∗ and D → D^∗ transitions of acceptor and donor, respectively. This results in By inserting equation (5.14) into equation (5.13), the density fluctuation can be ex-pressed in terms of the donor and acceptor transition densities

δn(r, t) = 2k·rDD^∗

The density fluctuation in equation (5.15) has contributions from the single donor and acceptor transition densities, which can be assigned to density fluctuations in their respective half-spaces VD and VA. In the case of resonant coupling, it follows that ^|V|

~ω˜ = 1 and ^∆E

~ω˜ = 0. The respective expression for the density fluctuation δn^∆E=0(r, t) = 2k·rD{sgn(V) [sin (˜ωt) cos (¯ωt)]ρAA^∗(r) (5.16)

+ [cos (˜ωt) sin (¯ωt)]ρ_DD^∗(r)}

has a symmetric form for acceptor and donor. On both sides the density has a rapidly fluctuating contribution ω¯ that is enveloped by a sine or cosine with frequency ω.˜ Again, the slow oscillation withω˜ can be identified with the energy transfer between donor and acceptor. The fast oscillation with ω¯ is the result of the ground state

|0i=|ADi being still occupied after the boost excitation.

In the case of no coupling with V = 0, the situation is reversed and the density fluctuation reduces to

As expected, only the initially excited donor density oscillates with its related excita-tion energyEAD^∗−EAD.

In the general case for ∆E 6= 0 andV 6= 0 in equation (5.15), the time dependence on the acceptor side remains simple with the amplitude of the fluctuation scaled down

by a factor of ^{√ |V|}

∆E²+V². On the donor side the time dependence gets more involved and is discussed in a subsequent section in more detail.

The time-dependent induced dipole moments of acceptor and donor,δµ_Aandδµ_D, follow directly from equation (5.15) The dipole moments of donor and acceptor show the same time dependence as the induced density.