D.1.1 The Förster-type potential expansion in Donor-Acceptor systems To investigate the coupling mechanism and check for Förster-type coupling behavior, I sug-gested in Pub1 a dipole coupling scheme that can be applied in the TDDFT context. In the following, I outline some of the theoretical background of this concept that is needed to explain the PARSEC implementation. For including a dipole-dipole coupling approximation in the KS framework, there are two valid starting points: the Hartree energy and the Hartree potential. Both routes involve a multipole expansion of the 1/|r−r0|factor in the Hartree integral. Starting from the Hartree energy, one rst performs the multipole expansion and then draws the functional derivative to obtain the corresponding potential (see Pub1 and Ref. [Hof08] for details). Along this route, one obtains the potential in dipole-dipole coupling approximation (superscript dd and superscript index E)
vHddE[nD, nA] = vH[nD] +vddHE,D[nA]
Θ(−x)+ vH[nA] +vHddE,A[nD]
Θ(x). (D.1) 85
The Heaviside function splits the entire real space into a D (Θ(−x)) and an A (Θ(x)) part (see Fig. 1 of Pub1). Hence, the superscripts D and A indicate properties that are computed only on the D or A side, respectively. Based on this notation, vH[nD(A)]is the Hartree potential of D(A) and vHddE,D(A)[nA(D)] is the potential resulting from the interaction between D and A in the D(A) half space. With the help of the multipole moments of the D(A) density,
Ni=
wherei= D,A, the Förster potential that the D density generates in the half space of A is vddHE,A[nD](rA) =ND The corresponding potentialvHddE,D[nA](rD)is obtained from equation (D.3) by interchanging D and A superscripts and replacing R by −R.
Alternatively, the multipole expansion can be inserted directly into the Hartree potential and one obtains the potential
vHddv[nD, nA] = vH[nD] +vHddv,D[nA]
Θ(−x)+ vH[nA] +vHddv,A[nD]
Θ(x) (D.4)
with the superscript index v that denotes the potential route, where vHddv,A[nD] = ND Pub1 demonstrates that these two approaches result in potentials that dier qualitatively outside the regions where the density is high. From a comparison of the two potentials with the true Hartree potential, Pub1 concludes thatvHddv[nD, nA]is more appropriate as it exhibits the asymptotic decay to zero expected naturally from a physical potential.
Both ways of using Förster-type potentials are implemented in the PARSEC code.
The Förster-type potential expansion feature can be (de)activated using the boolean ag Use_foerster. The input parameter foerster_type allows for the choice of the potential-determination route along the previous lines: foerster_type = 1 is the default and uses Eq.
(D.4), whereas foerster_type = 2 uses Eq. (D.1). The separation of the grid into two half
D.1. FÖRSTER-TYPE POTENTIALS AND GRID PARTITIONING 87 spaces is always performed by the planes of the coordinate system that are perpendicular to the coordinate axes. To this end, one needs to indicate the desired coordinate axis x, y, or z via the input parameter foerster_half_space (string) and the partitioning is performed accordingly. To guarantee for consistent calculations, the selected scheme is used for the ground state and during real-time propagation throughout.
D.1.2 Using Förster-type potentials in supermolecular systems
For the investigations on supermolecular systems of Chap. 5, I extended the Förster-type potential evaluation of the previous section to deal with more complicated molecular arrange-ments. This functionality may be (de)activated by the boolean ag Use_multi_foerster when Use_foerster = true. The statements of the previous section concerning foerster_type ap-ply accordingly.
Two dierent types of arrangements, linear and circular ones, may be used to separate the entire grid into partitions, according to which one may place single molecules. These setups can be addressed by the integer input parameter Foerster_neighbor_type: numbers 1 to 3 correspond to linear arrangements, whereas numbers 10 and 11 imply circular setups.
The PARSEC input block Foerster_centers was implemented to dene the position of the grid segments via the coordinates of their centers. For the sake of clearness, I use the term hub for these points in the following. Note that the hubs always need to be listed in consecutive order. In case of linear setups one additionally needs to specify the orientation of the alignment in terms of the coordinate axis using the parameter foerster_half_space.
Then, the grid slicing is performed along each center line of neighboring hubs where center lines are perpendicular to the specied coordinate axis.
Circular setups are supposed to be arranged on a circle around the origin in the x-y -plane. The slicing is performed in terms of angles. If hubs are given via the input block Foerster_centers, PARSEC computes the angles corresponding to the center lines that run through the origin and through the middle of all neighboring hubs. Here, the number of hubs should equal the number of wedges specied via the integer input parameter Foer-ster_wedge_number. The grid partitioning is performed along planes given by these center lines and the z-axis. When no hubs are dened, PARSEC performs an equidistant circular slicing while it assumes that the rst molecule is positioned on the x-axis. In this case, one needs to give the number of wedges using the parameter Foerster_wedge_number.
Within each of these arrangements, one may choose between dierent ways of how the interaction between the subsystems is implemented. I start with the options that are avail-able in the cases of linear arrangements. Foerster_neighbor_type = 1 corresponds to an interaction between next neighbors only for all moments of the multipole series except for the monopole part. The latter contribution is by denition the most long-range one of the mul-tipole series. It is important for reasonable absolute values of the potential and, therefore, needs to be applied also beyond next neighbors. In case of Foerster_neighbor_type = 2, PARSEC computes the pairwise interaction between all grid parts. Foerster_neighbor_type
= 3 models periodic boundary conditions: It computes the multipole moments of the last (rst) molecule of linear arrangements and places a ctitious molecule with the same prop-erties in front of the rst (behind the last) molecule. The intermolecular interaction works only between next neighbors and assumes equidistant molecular spacing.
As this proceeding is just a crude approximation to periodic systems, the idea with
circular arrangements is to design closed systems where each molecule naturally has next neighbors on both sides and no explicit periodic boundary conditions need to be applied. In the circular setup of Foerster_neighbor_type = 10, the interaction contributions beyond the monopole term work only between next neighbors whereas monopole interaction applies between all molecules. Foerster_neighbor_type = 11 uses pairwise interaction between all molecules of the setup.
D.1.3 Partition-selective excitation and observation
For TD investigations, it may be relevant to excite only specic subsystems instead of the entire system. In cases where the previously explained partitioning is used, such subsystems can be related to partitions of the grid. In D-A systems where the separation is performed into two half spaces, such a feature can be controlled by the boolean ag Use_sel_boost. If Use_sel_boost = true, one may indicate via boost_half_space in which of the half spaces the excitation is supposed to work. This parameter reads a string with two positions where the rst position discriminates between the negative (n) and the positive (p) side of the coordinate axis and the second position determines the axis (x, y, or z) along which the separation occurs.
A similar feature is implemented in case of more complex supermolecuar systems by the integer input parameter Excite_center. If Excite_center = 0, the excitation is per-formed only in one of the half spaces of the grid, thus one additionally needs to specify boost_half_space. Alternatively, one may address the grid partition where excitation is per-formed via the hubs given by Foerster_centers. To this end, one needs to set Excite_center to the number corresponding to the position of the hub in the Foerster_centers input list.
Finally, the input option Excite_center = -1 deactivates selective excitation and the excita-tion process is applied in all partiexcita-tions of the grid. Note that both space-selective excitaexcita-tion features may also be used independent of the Förster-type potential expansion.
In PARSEC calculations with multiple centers, some important observables are calculated not only in the entire system but also for each system partition separately. In cases of Förster-type potential approximations or selective excitation, the dipole moment and the integral over the density corresponding to each grid partition are written to the le multi_center.out during time propagation. Here, the TD dipole moment is calculated relative to the centers of mass at the initial time t= 0 corresponding to each grid partition.