Journal of the American Chemical Society thus, increasing the internal field). To illustrate that such shifts are actually observed, Figure 6b shows the evolution of the HOMO, HOMO 1, and LUMO derived states of 3N as a function of the packing density relative to E = (Elveafct + Erviagcht)/2 (cf., Figure 3a). The HOMO shifts up by up to 0.9 eV, while the LUMO is destabilized by only 0.1 eV, consistent with the above- discussed gap-reduction by 0.8 eV. The observation that the LUMO is destabilized instead of stabilized (as one might expect from its localization) is most likely due to the somewhat arbitrary choice of the average vacuum level as energy reference, but could also be an indication that, while the model in Figure 6a provides a consistent qualitative picture of the situation, it does not reproduce all quantitative details. Furthermore, the energetic distance between HOMO and HOMO 1 is increased by nearly 0.4 eV (cf., Figure 4) because of the very diﬀerent localization of those states. As a consequence, the π π* gap (i.e., the gap between the HOMO 1 and the LUMO) is reduced by only 0.4 eV. An interesting observation concerns the signiﬁcant shifts already at very low packing densities. The energy diﬀerence between HOMO and HOMO 1, for example, is increased by 0.2 eV already between θ = 1/64 (considered as the isolated molecule) and θ = 1/8, and the gap is reduced even more. This clearly shows that the observed orbital- energy shifts are primarily of electrostatic origin, fully consistent with the occurrence of a collectively induced QCSE. QCSE and Depolarization. Finally, it needs to be understood how the fact that the HOMO is shifted toward the “right” side of the SAM upon increasing the packing density (which corre- sponds to increasing the molecular dipole) can be reconciled with the generally observed depolarization effects in polar SAMs. That the latter occur also in oligopyrimidine SAMs can be inferred, for example, from the sublinear increase of ΔEvac with coverage in Figure 3b. This can only be explained by the total charge density reacting in a qualitatively different manner to the electric field than the charge density associated with the HOMO-derived band. To explain that, it is useful to identify “related” molecular orbitals. This can be done by applying the quasiband picture outlined earlier, in which each orbital of an oligomer is interpreted as the product of a quantum-well eigenfunction (the envelope function) and a function derived from an orbital of a single repeat unit (e.g., the HOMO or LUMO of pyrimidine).102 111 The latter then determines which quasiband a given orbital of the oligomer belongs to, with the number of states in a quasiband equaling the number of repeat units. Figure 7a shows the orbitals related in this way to the HOMOs and LUMOs of 3N and 3P. As terphenyl is an apolar molecule, sine functions are suitable envelope functions (cf., middle panel of Figure 3c).109 In contrast, signiﬁcant asymmetries are found in the orbitals of terpyrimidine, which is due to the envelope functions in this polar molecule being better described by linear combinations of Airy functions (cf., left panel of Figure 3c). Note that, although this very simple model does not quantitatively reproduce the localization of each orbital, its value is evident as it perfectly reproduces the changes of sign of the wave functions along the backbone. Bearing in mind that the QCSE “picture” in the rightmost panel of Figure 3c is derived from the envelope-function approximation, it is useful to consider all orbitals belonging to a quasiband “simultaneously”. The corresponding ΔFi(z)- and Q i(z)-plots for the HOMO-related states (the HOMO 2 and the HOMO 3) in 3N are, therefore, included in Figure 7b. For the lowest-lying orbital (HOMO 3) the opposite trend than for ARTICLE Figure 7. (a) Isodensity plots for related occupied (lower plots) and unoccupied (upper plots) molecular orbitals of 3N (left) and 3P (right).116 Also shown are schematic standing-wave envelope functions. (b) Upper panels: Charge rearrangements ΔFi(z) = FiSAM(z) Fimol(z) for the HOMO 2 (left) and HOMO 3 (right) upon monolayer formation. Red (blue) areas show accumulation (depletion) of electron density. For the equivalent plot for the HOMO, see Figure 5. Lower panels: Cumulative charge rearrangements upon SAM formation. Left panel: Qi(z) for the three related states HOMO (black), HOMO 2 (crossed gray), and HOMO 3 (light gray) of 3N. Right panel: Sum over the three Qi-curves shown in the left panel (black color) and sum over the Qi of all occupied states (gray color). the HOMO is observed (cf., Figure 5); i.e., electron density is shifted to the left part of the molecule. The HOMO 2, on the other hand, experiences a shift of charge density from the perimeters to the center of the molecule, almost perfectly canceling the combined eﬀect of HOMO and HOMO 3. Indeed, the sum of 18640 dx.doi.org/10.1021/ja203579c |J. Am. Chem. Soc. 2011, 133, 18634–18645