3.1. Transport Gap
Calculations have been carried out for the systems studied in the previous paper (See reference [3
]), three electron-acceptors: 1,3,5-tris(phenyl-2-pyridylamino)benzene (E1), 2,4,6-tris[di(2-pyridyl)-amino1,3,5-triazine (E2) and 2,4,6-tris(carbazolo)-1,3,5-triazine (TRZ2) (E3), (with numbers 40, 169 and 170, respectively, as assigned in reference [1
]) and two hole-acceptors: N,N′-Bis(3-methylphenyl)-N,N′-diphenylbenzidine (TPD) (H1) and 4,4′-di(N-carbazolyl)biphenyl (CBP) (H2) (molecules 86 and 91 in reference [1
]). (See Figure 1
). Hereafter we first consider how the optimal dielectric constant was determined and, subsequently, the transport gap.
DFT plus PCM calculations were carried out for the IP, EA,
, versus the dielectric constant (
). Figure 2
, Figure 3
and Figure 4
show the results for the molecule E1, although similar results are obtained for the rest of the studied molecules. It is noted that the polarizable character of PCM is manifested clearly in the stronger effects it has on the parameters of the charged system (IP and EA) than on those of the neutral system (
). These calculations have been made using the best basis set (Def2TZVPP) and three functionals, B3LYP (Figure 2
), PBE0 (Figure 3
) and PBEPBE (Figure 4
). The first two hybrid functionals show a crossing of the IP-vs-
curves, and, the EA-vs.-
curves. The crossing of the IP-
curves occurred at very similar values of the dielectric constant, namely,
for B3LYP and
for PBE0 (see Figure 2
and Figure 3
and Table 1
). Although both functionals show crossing, thus allowing us to derive an optimal dielectric constant from this analysis, we chose PBE0 for the majority of the calculations presented here for two reasons; most theoretical analyses support PBE0 as the most suitable functional for organic materials and molecules [4
], and the dielectric constant derived from the use of this functional gives
values well within the range 3–6, which is the expected range for the materials at hand [2
]. Note that the rather popular B3LYP functional gives dielectric constants as high as 9.4 (see molecule E2 in Table 1
). On the other hand, with the PBEPBE functional (Figure 4
) no crossing was found over a rather wide range (1–80). In fact, this is not the only functional that did not show crossing, rather at least two additional functionals were identified (BLYP and
HCTH). For the long range corrected functionals, these crossing occur at values close to
, as it was indicated previously.
The qualitative effect that the choice of the functional may have had on the results for the optimal dielectric constant was not seen when the basis set was changed. The use of a different basis set may have varied the values of
at which the curves IP(EA)-vs.-
cross, albeit for the molecules investigated here crossing occured no matter the basis. This is illustrated by the results reported in Table 1
for the optimal dielectric constant, obtained with the PBE0 functional and the three previously indicated basis sets. It is noted that the maximum difference was always smaller than 15%, being in most cases smaller than 5%. In the following, the dielectric constant introduced in the PCM will be the average of those at which the curves IP(EA)-vs.-
, obtained with the combination PBE0/Def2TZVPP.
3.2. Optical Gap and Exciton Binding Energies
Energies of the excited states for each of the five organic molecular materials investigated here were calculated within TD-DFT framework with the hybrid exchange-correlation functional PBE0 and using the Def2TZVPP basis set and with a procedure similar to that of reference [4
]. The dielectric constant was calculated as explained in the previous subsection and incorporated into the PCM. The calculated energies for the four lowest lying excited states are reported in Table 2
. Their oscillator strengths are also given. It is noted that while the energies of the four excited states of each molecule did not differ much, their oscillator strengths may have been significantly different, determining, for molecules H1 and H2 (hole-acceptors) the main transition and thus, the first excited state. Nevertheless, for molecules E1, E2 and E3 (electron-acceptors), the first excited state was not that one which shows the larger oscillator strength. Then, for the first excited state of H1 and H2 together with the first three excited states of the systems E1, E2 and E3, state-specific (SS) solvation calculations were carried out, within the non-equilibrium solvation-linear response approximation (See references [13
]). The results for the most important implicated excitations and their oscillator strengths are shown in Table S6 of the electronic supporting information (ESI)
. A summary which includes the results of the excited states of less energy having a considerable oscillator strength is shown in Table 3
. As expected, solvation calculations slightly decreased the energy of the excited states, and significantly modified the value of the oscillator strengths.
reports experimental data collected in reference [1
] for the IP, EA and the optical gap (all in eV). Theoretical results (last five columns) were obtained by means of PBE0/Def2TZVPP and incorporating the PCM with the dielectric constant
derived as explained succinctly in Table 1
and extensively in the main text.
stand for transport and optical gap, respectively. It is noted that theoretical results for
, albeit slightly larger than the experimental data, are not that different. Some hints that may help to understand this discrepancy are outlined in the following paragraph. As regards the exciton binding energies, they are of the order of the expected values for these materials. However, it is likely that due to overestimation of
, our calculations underestimated their actual values. As remarked above, solvation calculations reduce in a small amount (less than 0.2 eV) the excited state energy and, thus, the optical gap.
Exciton binding energies in organic materials are usually much larger than those currently observed in inorganic compounds, namely, i.e., meV, as compared to tenths of eV typical of organic materials. However, it is unlikely that this large difference in may be solely understood in terms of differences in the dielectric constants. Instead, it is widely accepted that electron-electron and electron-phonon interactions may also play a substantial role. Both interactions are commonly larger in inorganic materials, contributing appreciably to the reduction of the optical gap and, thus, the exciton binding energy. The no-consideration of electron-phonon interactions in the present calculations, the incomplete treatment of electron-electron interactions, and the different nature of their electronic transition may surely be the reasons of the difference in the results showed above.
In order to attain an understanding of the nature of the electronic transitions in these systems, it is worth drawing iso-contour plots of the difference between the local electron densities in ground- and excited-states (Figure 5
and Figure 6
). Thus, calculations of the GEDT between the ground- and the excited-states have been carried out using the procedure of Le Bahers et al. [66
]. Table 5
shows the results (see full results in Table S7 of ESI
) for the charge transferred (
being the position vector in three-dimensions. In this expression
stand for positive or negative differences between densities of the excited and ground state wave-functions that can be quantified by the spatial distance between the barycenters of these density distributions
and the difference between the dipole moments computed for the ground and the excited states
It is readily noted in Figure 6
that, in hole-acceptor systems, the excitation localizes on the central part of the molecule backbone (see reference [4
]) and the spatial distance between the barycenters of the two density distributions
has values that are far from the range within which GEDT excitations usually lie (
Å) Then, they should be considered as local excitations. The three electron-acceptor systems E1 and E2, in turn, show a second and a third excited state with considerably higher oscillator strengths, such that, albeit having higher energies, may compete with the first excited state. Actually, the very small oscillator strength of the latter makes more probable the excitation to the second excited state.
A consequence of what has been pointed out in the preceding paragraph is that the excited states of systems E1 and E3 can be safely considered as GEDT excited states, with
approximately equal to 1.95 and 2.40 Å. Instead, material E2, albeit having a first excited state with
of 2.25 Å, its small oscillator strength indicates that the intensity of the excitation might be small compared to the transition to the second excited state, that, with
of only 0.57 Å, should be considered as a local excitation. These remarks are better illustrated by Figure 5
that shows iso-contour plots of ground- and excited-state density difference for the three electron-acceptor systems. Figures S1–S5
of the ESI show iso-contour plots of some of the molecular orbitals likely involved in the transitions that define the optical gap.