Journal of the American Chemical Society ARTICLE Figure 5. Upper panels: (x, y)-plane-integrated orbital charge-densities for the HOMO and LUMO of 3N, to quantify orbital localization in the direction of the long molecular axis (orange areas). The black curves show the corresponding data for the band derived from those orbitals in the SAM at θ = 1/2. Middle panels: Charge rearrangements ΔFi(z) = FiSAM(z) Fimol(z) for the HOMO (left) and LUMO (right) upon monolayer formation. Red (blue) areas show accumulation (depletion) of electron density. Lowest panels: Cumulative charge rearrangements Qi(z) for these states upon monolayer formation. The molecules in the background serve as guide to the eye. implies that the calculated gap-reduction is not a consequence of dielectric screening by neighboring molecules, an effect that is not captured by hybrid calculations (cf., Determining a Suitable Computational Approach section). The real-space representations in Figure 4 can also be asso- ciated with the collectively induced QCSE: In the case of terphenyl (lower panel), both HOMO and LUMO are deloca- lized over the molecule, and this does not change upon mono- layer formation. In contrast, for terpyrimidine the HOMO and LUMO are to some degree localized on opposite ends of the molecule already in the isolated system (molecular eﬀect). This localization is further enhanced in the monolayer (collective eﬀect). Because the electrostatic energy increases across the SAM (cf., Figure 3a), this corresponds to a localization of the HOMO level at the high-energy side of the SAM as in the QCSE model in Figure 3c (right panel). In other words, the corresponding electron density is shifted along the electric ﬁeld. Note that this is counterintuitive from the perspective of depolarization eﬀects (Figure 3a), as the total electron density is (naturally) shifted opposite to the electric ﬁeld direction as a consequence of the negative charge of electrons. To better quantify this eﬀect, we calculated orbital charge- densities of the frontier states integrated over the (x, y)-plane using the deﬁnition given in the Methods section. They are shown for the isolated 3N molecule as orange areas in the upper panels of Figure 5. The black curves show the corresponding charge densities for the bands derived from these orbitals in the Figure 6. (a) Simpliﬁed sketch of the electron electrostatic energy across a SAM of 3N molecules at low (dashed gray) and high (solid black) packing density. The orange symbols illustrate for two positions how the energy is changed diﬀerently at diﬀerent positions in the well upon increasing the SAM packing density. (b) Energetic shifts of the HOMO (black squares), HOMO 1 (gray circles), and LUMO (black diamonds) derived bands in a SAM of 3N molecules upon increasing the SAM packing density θ, aligned at the average electrostatic energy E = (Elveafct + Erviagcht)/2 (cf. Figure 3a). The energies are determined from the onsets of the respective peaks in the DOS. SAM (θ = 1/2). As inferred already from the contour plots in Figure 4, HOMO contributions on the central ring are shifted to the rightmost pyrimidine and “virtual” charge density corresponding to the molecule’s LUMO accumulates on the leftmost ring at the expense of the rightmost ring. This trend is even more clearly resolved when plotting the respective charge-density diﬀerences, ΔFi(z) = FiSAM(z) Fimol(z), in the central panel of Figure 5, where red (blue) areas show accumulation (depletion) of electron density. In this context it is also useful to plot the cumulative charge rearrangement, Qi(z), deﬁned as Qi(z) = R z0 ΔFi(z0) dz0 (see bottom panel of Figure 5). Qi(z) > 0 gives the number of electrons that have been transferred from the region right of z to its left. For Qi(z) < 0, the direction of the charge transfer is reversed. The latter quantity shows that the eﬀect of self-localization is sizable: for the HOMO, 0.3 electrons are accumulated on the rightmost ring. The reorganization of the LUMO is of the same magnitude, but opposite in sign. Collectively Induced QCSE: Eigenenergies. The above findings lead to the question of how the collectively induced electric field influences the electronic structure of the SAM beyond orbital localization. The simplified potential wells in Figure 6a sketch the electrostatics across the SAM at low (gray) and high (black) packing density. The orange markings illustrate how the energy is changed differently at different positions in the well. As the orbitals are differently localized along the long molecular axis already in the isolated molecule and even more so in the SAM (vide supra), it can be expected that their energies are differently affected by increasing the packing density (and, 18639 dx.doi.org/10.1021/ja203579c |J. Am. Chem. Soc. 2011, 133, 18634–18645