Supercell convergence of charge-transfer energies in pentacene molecular crystals from constrained DFT

Singlet fission (SF) is a multiexciton generation process that could be harnessed to improve the efficiency of photovoltaic devices. Experimentally, systems derived from the pentacene molecule have been shown to exhibit ultrafast SF with high yields. Charge-transfer (CT) configurations are likely to play an important role as intermediates in the SF process in these systems. In molecular crystals, electrostatic screening effects and band formation can be significant in lowering the energy of CT states, enhancing their potential to effectively participate in SF. In order to simulate these, it desirable to adopt a computational approach which is acceptably accurate, relatively inexpensive, and which scales well to larger systems, thus enabling the study of screening effects. We propose an electrostatically corrected constrained density functional theory (cDFT) approach as a low-cost solution to the calculation of CT energies in molecular crystals such as pentacene. Here we consider an implementation in the context of the onetep linear-scaling DFT code, but our electrostatic correction method is in principle applicable in combination with any constrained DFT implementation, also outside the linear-scaling framework. Our newly developed method allows us to estimate CT energies in the infinite crystal limit, and with these to validate the accuracy of the cluster approximation.

Figure 1

The pentacene (${\text{C}}_{22}{\text{H}}_{14}$) single molecule and molecular crystal ($S$ phase [23]). The unit cell contains two molecules. The third lattice vector $\mathbf{c}$ points out of the page.

Figure 2

(a) Schematic of the cDFT scheme used in this work. A nonlocal constraining potential (illustrated by a two-dimensional potential energy surface) constructed from atom-centered functions is applied to the single-electron density matrix. This causes charge to redistribute to obey chosen population constraints and allows the description of CT excitations within the framework of standard DFT. (b) Block scheme of truncated NGWF overlap matrix to ensure site localization of contravariant duals. Blue and red denote the constrained sites, gray the remaining system.

Figure 3

Dimer geometries and CT excitation energies from cDFT (quoted in electronvolts). The significant energy gap between the two CT states in the herringbone dimer can be rationalized by considering the different charge distributions of electron and hole, and the geometry. The hole orbital corresponds to the pentacene highest occupied molecular orbital (HOMO) which has a node on the long axis of the molecule. The electron orbital (lowest unoccupied molecular orbital, LUMO), on the other hand, does not feature such a node. The partial alignment of the upper molecule with the dipole vector means that the bimodal charge distribution on the upper molecule has a lower Coulomb energy in configuration 1 as compared to configuration 2. In the parallel case the two CT states are related by inversion symmetry and the energies are degenerate.

Figure 4

(Top) Herringbone 1, (center) herringbone 2, and (bottom) parallel. Uncorrected energies (blue) and dipole corrections (red) for CT states across a range of supercell embeddings. Dashed lines indicate corrected mean energies. Note that the dipole correction is negative for the 531 supercell for both herringbone configurations.