Journal of the American Chemical Society part of the Fock exchange is absent by construction. As discussed in ref 81, the absence of the long-range Fock exchange is not detrimental to the molecular electronic structure, as it hardly inﬂuences the highly localized orbitals which exhibit a large SIE. The open black squares in Figure 2a depict the correction to the LDA orbital energies as calculated with HSE, again relative to the shift of the LDA-HOMO, ΔεiHSE:= (εiHSE εiLDA) (εHHSOEMO εLHDOAMO).75 Clearly, HSE and GKLI corrections compare very well, and both result in a strong modiﬁcation of the order of the occupied frontier orbitals compared to the LDA (and also GGA) calculations (cf. Figure 2b). This way the two approaches mutually support each other, and we conclude that HSE is well-suited for describing the systems at hand. Certain shortcomings, however, remain: Hybrid functionals do not account for the narrowing of the fundamental gap by dielectric screening in the SAM, as this requires nonlocal correlation.82 84 Such eﬀects should, however, have no qualita- tive impact on the SAM’s electronic structure, although they will need to be revisited in a later chapter, when the interaction of the SAM with the metal is discussed. Another potential pitfall linked to DFT is the reliable calcula- tion of molecular polarizabilities. For example, Champagne et al.85 have argued that typical exchange-correlation functionals tend to overestimate polarizabilities. Some of us have shown that this is not always the case86,87 and that it depends on the particular electronic structure of the system in question.88 An (at least partial) remedy is typically the inclusion of exact exchange. As molecular dipole moments and polarizabilities are absolutely crucial parameters for the present paper, we have performed extensive test calculations employing a number of diﬀerent DFT functionals, Hartree Fock (HF) and second- order Møller Plesset perturbation theory (MP2). These tests (which are discussed in considerable detail in the Supporting Information) show that for the present system all methods give consistent results for the polarizabilities as well as for the dipole moments; the hybrid functional calculations yield a HOMO LUMO gap that is somewhat larger than the optical gap calculated by time-dependent density-functional theory (in con- trast to the LDA and GGA functionals, which severely under- estimate the gap). This further conﬁrms the suitability of HSE for describing the problems at hand. Molecule-to-Monolayer Transition. With the appropriate tools in hand, we can now turn to first calculating the molecular and then the monolayer properties. Figure 2c shows the density of states (DOS) of 3Nup and 3Ndown molecules as calculated with HSE. This plot illustrates that the position of the thiol docking group has only minor impact on the molecular electronic structure. The ordering of the frontier orbitals is the same with all frontier orbitals having π-character (see insets), and also the eigenvalues are comparable; the gap differs by only 0.2 eV. Assembling the polar molecules into a monolayer creates a two-dimensional array of dipoles, μ, which give rise to a change in the electron electrostatic energy between the docking- and the tail-group sides of the SAM. The net eﬀect is described via the Helmholtz solution to the Poisson equation (ΔEvac = eμ/ε0A) and is proportional to the areal dipole density.89,90 In densely packed SAMs, in which ΔEvac originates from polar tail-group substituents (the commonly studied situation), the correspond- ing shift in the potential landscape is strongly conﬁned to the region of the tail-group.33 This can be explained by purely electrostatic arguments, as it can be shown that for a square two-dimensional array of point dipoles the decay-length of the ﬁeld ARTICLE Figure 3. (a) (x, y)-averaged electrostatic energy E of an electron across a hypothetical free-standing 3Nup-SAM at θ = 1 /2 (gray curve; for details on the further averaged red curve, see text). The molecule in the background and the “box-potential” serve as guide to the eye. The zero of the energy axis is set to the average of the left and the right vacuum energy, E = (Elveafct + Erviagcht)/2 . (b) Step in the electrostatic energy across the SAM, ΔEvac = Erviagcht Elveafct, as a function of the SAM packing density θ for 3Nup (black squares), 3Ndown (orange disks), 3N (green diamonds). The ﬂat evolution for the apolar 3P SAM (gray triangles) is shown for comparison (see Figure 1b for the molecular structures). Note that the energy scale is reversed for 3Ndown. (c) Sketches of the potential and the lowest eigenstates of 1D potential wells with inﬁnitely high barriers for a particle in a box exposed to an electric ﬁeld (left panel) and without electric ﬁeld (middle panel). The rightmost panel sketches the situation encountered for the envelope functions in the valence and conduction bands that are responsible for the quantum-conﬁned Stark eﬀect (QCSE) in a semiconductor quantum-well structure. is equivalent to the interdipole distance divided by 2π.42 In SAMs consisting of polarizable molecules, this conﬁnement is further enhanced by depolarization eﬀects.91,92,31,93,30,59,56,57,41,43 The situation is markedly diﬀerent for oligopyrimidines as shown for a 3Nup SAM at the highest investigated packing density, θ = 1/2, in Figure 3a. As the total molecular dipole originates from each of the dipolar pyrimidine rings, ΔEvac is established across the SAM in a more or less continuous way.44 Even though the plane-averaged electrostatic energy displayed in Figure 3a oscillates due to the nuclei, one can clearly see that it is superimposed with an approximately linear potential gradient, which is due to the combination of the ﬁelds generated by the 18637 dx.doi.org/10.1021/ja203579c |J. Am. Chem. Soc. 2011, 133, 18634–18645