Coulomb interaction

The matrix elements within H0, H1 and H2 (Eqs. 3.27, 3.28, 3.29) contain all electrostatic in-teraction energies between the individual chromophores, the dendrimer and the surrounding solvent molecules.

In general, the Coulomb interaction between two chromophores may be written as J_mn(ab,cd) =

Z

dr_mdr_nψ^∗_ma(r_m)ψ^∗_nb(r_n)V_mnψ_nc(r_n)ψ_md(r_m)_{. (3.36)} V_mncovers the complete Coulomb interaction between the chromophoresmandn, andψ_ma is the electronic wave function of chromophoremin the electronic statea.

The meaning brackets(ab,cd)in Eq. 3.36 can be explained as follows: if a = bandc = d, this is the Coulomb interaction between two chromophores in the statesaandc; if a 6= b, the transition density of chromophoremwhile it changes its electronic state fromatobis denoted.

The interaction between transition densities will be utilized later to describe electronic state transitions of the single chromophores via excitation energy transfer (cf. Sec. 3.3.3) within the CC.

To compute the couplingsJmn(ab,cd), the electronic charge density ζ_ab^m(x) =eNm

Z

drδ(x−rq)ψ^∗_ma(r)ψ_mb(r). (3.37) is introduced. In the formula for ζ^m_ab, N_m is the number of electrons in chromophore m. In Eq. 3.37, the treatment of the charge density instead of the linear combination of products of wave functions decreases the actual phase space under consideration to the 3-dimensional Cartesian space.

Fitting the electronic charge density to partial charges

The electronic charge density of the chromophores can be fitted to partial charges located on the positions of the nucleiR_i, as shown in [32]. However, the notation of [83] will be used since it is better aligned with the notation of the rest of this thesis. In order to simplify the formulas, newW-potentials are introduced. For the electron-electron interaction one may write

W^el-el(x−x⁰) = ^e

2

4πe₀ 1

|x−x⁰|^{. (3.38)}

TheW-potential for the electron-nucleus coupling is written as W_m^el-nuc(_x) = ^e

The indexmmeans that the nuclei of chromophoremare considered. Z_iis the atomic number andR_ithe position of atomi. According to the definitions given above theW-potential for the nucleus-nucleus interaction is

3.3 The CC Hamiltonian With the definition of the single electron density of chromophorem,ζ^m_ab(x)(Eq. 3.37), the gen-eral interaction between two chromophoresmandncan be written as

J_mn(ab,cd) =

The charge density of the chromophoremis defined as n^m_ab(_x) =ζ^m_ab(_x)−δ_a,b

∑

i∈m

eZ_iδ(_x−_Ri)_{. (3.42)} It consists of the nuclei charge contribution (nucleusicontributes the chargeeZ_iat positionR_i) and the contribution of the electronic charge density ζ^m_ab(x)(Eq. 3.37). Inserting Eq. 3.42 into Eq. 3.41 results in

The interactions between charge densities in Eq. 3.43 can be fitted to respective partial charges [32], as will be shown below.

Partial charges

Quantum-chemical computations may give the charge densityn^m_ab(x)of a system. However, the knowledge of the charge density does not help to compute the Coulomb interaction between two molecules for specific MD snapshots. The solution for this problem is the definition of a set of partial charges{q_mi(ab)}. The partial chargeq_mi(ab)is located at the position of atomiof moleculem,R_mi. Theq_mi(ab)are chosen that way that the potential of the set ofq_mi(ab)equals the potential of the charge densityn^m_ab(x). There exist different methods to derive the partial charges from the charge density. The most common methods are explained and compared in [86].

Interaction between Pheos in the electronic ground state

While treating the Coulomb interaction of chromophores in the electronic ground state, Eq. 3.36 can be rewritten as

J_mn(gg,gg) =

Z

dr_mdr_nψ_mg^∗ (r_m)ψ^∗_ng(r_n)V_mnψ_ng(r_n)ψ_mg(r_m)_{. (3.44)} The computation of the (electrostatic) coupling between chromophores and solvent molecules can be done in a uniform manner. The respective interaction between the electronic ground state charge densities is then given by

J_mn(gg,gg) = ¹

Interaction between Pheos in the excited states

According to Eq. 3.46, the interaction between two Pheos in the statesaandccan be written as Jmn(ac,ca) = ¹

4πe₀

∑

i,j

q_mi(aa)q_nj(cc)

|R_mi−R_nj| ^{, (3.47)}

with a,c ∈ {g,e,f}, the electronic ground stateg, the electronic first excited electronic statee, and the second excited electronic state f. If both chromophores are in the electronic ground state (a= g,c=g), Eq. 3.46 is reproduced.

Excitonic coupling

If Eq. 3.36 is translated to the interaction between two transition densities, it may be rewritten as

Both transition densities are represented by the charge density n^m_ab, since the interaction be-tween those two transition densities leads to the de-excitation (excitation) of chromophore m from the stateato the statec, while chromophorenis excited (de-excited) from the electronic statec to the state a at the same time. In the current consideration, the charge densityn^m_ab is identified with the charge density n^m_eg. Only the excitonic coupling between the Pheos in the electronic ground state and the Pheos in the first excited electronic state is taken into consider-ation.

From the transition densities, the respective transition partial charges may be calculated [32], and the excitonic coupling can be written as a sum of Coulomb interactions between those transition partial charges:

It has to be mentioned that (if the distance between both chromophores is large) this description is equivalent with the description of the excitonic coupling via the dipole-dipole interaction between transition dipole moments d_m andd_n. If the distance between the chromophores is much shorter than 10 Å, the description given above is the much better, since no approximation is included (cf. [75]). In terms of the transition partial charges, the transition dipole momentdm

may be written as

dm =

∑

i

q_mi(ac)_{Rmi. (3.50)}

For this thesis, the Pheo transition dipole moment (as well as the respective partial charges) was normalized to the measured transition dipole moment of chlorophyll a (there exists no measurement for Pheo). Theπ-electron system is the same for both molecules, only one chain is longer for chlorophyll a. Thus, the assumption that the transition dipole moments of both molecules are the same is reasonable. The value of the transition dipole moment is 4.6 D.

Coupling between covalently bound molecules

The electrostatic interaction is interpreted as an intermolecular interaction, for example the in-teraction between chromophoremand chromophorenor the interaction between chromophore mand the dendrimer. This picture is not completely correct, since the dendrimer structure is covalently bound to each chromophore. Thus, the PNcomplex is one molecule. The Coulomb interaction between each chromophore and the dendrimer contributes to the energy gap