2.1. Molecular Properties and Solvent Effects
As a first step, we analyze the situation for an isolated ADB molecule. The transition to the lowest excited singlet state, S
1, is optically allowed and the state is primarily described by an excitation from the highest occupied molecular orbital, HOMO, to the lowest unoccupied molecular orbital, LUMO. Both orbitals are largely localized on the central anthracene unit. This is also observed for the transition density associated with the S
0→S
1 transition (see
Figure 2). It can be explained by a nearly complete breaking of conjugation between the anthracene core and the attached phenylenes, which is a consequence of an almost perpendicular arrangement of the π-planes in the respective units (with a twist angle of 84°). Consequently, when comparing the lowest excitation energies in anthracene and in the isolated ADB molecule, one obtains an only rather small red-shift of 0.15 eV (see
Table 1). The reason why there is any shift at all is a minor spread of the excited state onto the phenylenes, as can be inferred from the shapes of the frontier orbitals and the transition density (see
Figure 2). Consistently, the electron withdrawing carboxylic acid substituents have an only very weak impact on the optical properties of ADB. This can be concluded from the data for diphenylanthracene (DPA) (see
Table 1), which essentially coincide with those for ADB. Finally, it should be mentioned that the transition dipole for the lowest excited state in all molecules discussed in this paragraph is parallel to the short molecular axis of the anthracene moiety (cf.
Figure 2).
The strong localization of the S
1 state in the apolar center of the ADB molecule also explains, why including solvents of varying polarity in the calculations has hardly any impact on the lowest excited state (see
Table 1). This theoretical finding is also consistent with experimental studies on DPA, which show that decreasing the polarity of the solvent from ε = 24 (ethanol) to ε = 2.4 (toluene) increases the excitation energy by at most 0.05 eV [
55,
56,
57]. As a consequence, solvent effects are ignored for the present system and will not be accounted for in the following discussion.
As far as the emission properties of ADB are concerned, we calculate an energy of 2.78 eV for the purely electronic S
1→S
0 transition (calculated for the S
1 equilibrium geometry). This corresponds to a rather large shift of ~ 0.55 eV between the absorption and emission maxima. The magnitude of the shift can be explained by a reduced twist angle of the phenylene rings relative to the anthracene in the excited state equilibrium geometry (56° in S
1 vs. 84° in S
0 geometry), which results in an increase of conjugation. Overall, comparing the results in
Table 1 shows that for the isolated molecule the simulations agree exceptionally well with the experiments. Thus, in the following we will focus on understanding the situation, when the ADB linkers are incorporated into the MOF.
2.2. Formation of H- and J-Aggregates in Zn-ADB SURMOF-2: Anthracene Dimers as Model Systems
For understanding the excited states of ADB incorporated into the MOF, as a first step, it is useful to analyze possible exciton couplings between the chromophores in the MOF structure. Depending on the relative alignment of the transition dipoles on adjacent chromophores, H-type or J-type aggregates are formed in a solid-state assembly. In H-type aggregates, the transition dipoles on adjacent chromophores are aligned in parallel, whereas J-type aggregates are formed with a head-to-tail alignment of the transition dipoles. In both aggregate types, the first excited state is shifted to lower energies compared to the isolated chromophore, but only in J-type aggregates this excited state is optically allowed, resulting in a red-shift of the absorption spectrum. In H-type aggregates, the first optically allowed state is typically found at energies higher than the parent state in the isolated chromophore, which results in a blue-shift [
58].
For the following, discussion of the fundamental aspects of exciton coupling in Zn-ADB SURMOF-2 we will replace the ADB chromophores by anthracene units to simplify the interpretation. This does in no way affect the key conclusions regarding exciton coupling that will be discussed in this section and is justified by the strong localization of the excited state on the anthracene moiety in the isolated ADB molecule (vide supra).
Both aggregate types discussed above can be found in Zn-ADB SURMOF-2, as schematically shown in
Figure 3: In
a-direction, one observes J-type (green molecules) as well as H-type aggregates (blue molecules). The center-to center distance between the chromophores in both cases amounts to 19.80 Å. This is so large that the effect of exciton coupling becomes vanishingly small, resulting in a negligible splitting between the lowest excited states (see
Table 2). For symmetry reasons, exactly the same situation as in
a-direction is also found in
b-direction (where now neighboring blue molecules form J-aggregates and neighboring green molecules form H-aggregates). The situation is fundamentally different in
c-direction, where all nearest-neighbor molecules form H-aggregates. As in this direction the molecules are rather close, the impact of the coupling becomes noticeable. Thus, in the following we will be exclusively concerned with H-aggregates stacked in
c-direction. Notably, here the center to center distance of 5.81 Å in the equilibrium structure of the MOF corresponds to a distance of 3.78 Å between neighboring π-planes of the anthracenes, as illustrated in
Figure 3d. This is a consequence of a tilting of the anthracenes, which also results in a slip of the centers of neighboring chromophores parallel to the π-planes by 4.42 Å.
For the
c-stacked, H-aggregate-type dimer of anthracene, the splitting between the lowest excited state and the first state with appreciable oscillator strength amounts to 0.39 eV (see
Table 2). Increasing the number of interacting chromophores somewhat increases the splitting (see last entry in
Table 2), but considering the rather short distance between the neighboring π-planes, this splitting still appears rather small. Moreover, in a typical H-aggregate one would expect the oscillator strength for the excitation into the S
1 state to be exactly zero for symmetry reasons.
To rationalize these findings, as a first step the excited state structure of a cofacial (i.e., not slipped) anthracene model dimer shall be discussed: In the dimer, hybrid orbitals are formed from the molecular HOMOs and LUMOs of each of the molecules. In a single-particle picture, four excitations between these orbitals are possible (see
Figure 4). For symmetry reasons, two of these excitations are optically allowed and two of them are forbidden (solid vs. dashed arrows in
Figure 4). In the actual time dependent density functional theory (TD-DFT) calculations the single-particle excitations mix. This yields four excited states, which can be characterized by linear combinations of either the allowed or the forbidden single-particle excitations. This is shown in the first entry of
Table 3 (slip 0.0 Å) for the two cofacial anthracene molecules. In the following, the four excited states will be denoted as S
a, S
b, S
c, and S
d. Here, S
a refers to an excited state dominated by the positive linear combination of the forbidden single-particle transitions OS→UA and OA→US. In this context, O denotes to the highest occupied and U to the lowest unoccupied orbital of the dimer with a specific symmetry, where S and A specify, whether the dimer orbitals are symmetric (i.e., positive) or antisymmetric (i.e., negative) linear combinations of the orbitals of the individual molecules (see
Figure 4). S
b refers to the positive linear combination of the allowed transitions OA→UA and OS→US and S
c and S
d denote the negative linear combinations of the respective transitions (see last column of
Table 3). In line with the involved single-particle excitations, transitions to states S
a and S
d are strictly optically forbidden (see oscillator strengths in
Table 3), while excitations to states S
b and S
c are, in principle, optically allowed, although in our simulations the oscillator strength for excitations into S
b are consistently much smaller than for excitations into S
c. As a consequence, the position of S
c determines the position of the first peak in the absorption spectrum.
For the cofacial anthracene dimer (first entry in
Table 3), the expected situation for a conventional H-aggregate is recovered: The optically forbidden S
a state is lowest in energy and the first state with appreciable oscillator strength (S
c) lies 0.78 eV above the first excited state. To understand the different properties of the anthracene dimer in the H-aggregate configuration adopted in Zn-ADB SURMOF-2 discussed before, one has to consider the slip of the centers of neighboring chromophores in the MOF (
Figure 3d): As discussed by Kazmaier and Hoffmann for simple model systems and for perylene [
59] (and later found for a variety of organic semiconductors [
60,
61,
62]), displacing the centers of coplanar molecules relative to each other results in a periodic variation of the splitting of the respective hybrid orbitals as a function of the displacement. This occurs due to the symmetry of the individual orbitals (see
Figure 5a). Moreover, because of the decrease of the spatial overlap of the molecules with increasing displacement, the amplitude of the oscillations decreases. These changes of the orbital energies cause also variations in the energies of the above-mentioned four excited states, as shown in
Figure 5b. These variations do not directly coincide with the variations for the orbital energies, which has two reasons: First, due to the different nodal patterns of the HOMO and LUMO (compare
Figure 4), the slips at which the splitting between the HOMO and the HOMO − 1 vanishes differs from the slips at which the same occurs for the LUMO and LUMO + 1. Second, the dominant single-particle excitation describing specific excited states changes with the displacement, as shown by the filling of the symbols in
Figure 5 (for details see figure caption).
Notably, the slip-induced changes in excitation energies are large enough for the order of the states to change as a function of the displacement. In fact, for a slip of 4.42 Å (the value obtained in Zn-ADB SURMOF-2, indicated by a vertical line in
Figure 5) the optically weakly allowed S
b state comes to lie lowest in energy instead of the strictly symmetry-forbidden S
a state. This is reminiscent of the situation in crystals of dicyanodistyrylbenzene based molecules, where static symmetry-breaking renders the lowest excited state of H-aggregate coupled chromophores optically allowed [
63]. Moreover, the splitting between the lowest excited state and the first state with appreciable oscillator strength (S
c) decreases by essentially a factor of two between the cofacial dimer (slip 0.0 Å) and the dimer in the Zn-ADB SURMOF-2 configuration (4.42 Å, see
Table 3). This explains the somewhat unexpected excited state properties of the anthracene-dimer extracted from the Zn-ADB SURMOF-2 structure (vide supra). In passing we note that especially the change of the order of the states due to the slip will become relevant later, when discussing the emission properties of Zn-ADB SURMOF-2.
Still, there is one aspect of the calculations on the anthracene H-aggregates which is at variance with the experimental observation for Zn-ADB SURMOF-2: The simulations predict a blue shift of the absorption maximum (by 0.12 eV for the tetramer in
Table 2 compared to the isolated molecule), while in the experiments a minor red shift of the absorption peak by 0.05 eV is observed (see
Table 1). To understand that, one has to go beyond representing the chromophores in the simulations by anthracene units. In particular, one has to study to what degree the conformations of the actual ADB units change upon incorporation into Zn-SURMOF2.
2.3. Impact of the Chemical Linkages and The Solid-State Conformation on the Optical Properties of ADB in Zn-ADB SURMOF-2
As a first step towards answering that question, the impact of the bonding of the ADB chromophore to the metal nodes is assessed. For this purpose, we optimized a single ADB molecule suspended between two Zn-paddle wheels (pw) with the Zn and O atoms fixed to the positions they adopt in the periodic structure of the MOF discussed below. The Zn-nodes are additionally saturated by three acetate groups per paddlewheel (see
Figure 6a). This structure in the following will be referred to as
opt(pw-ADB-pw)
1, where the subscript denotes that only a single ADB unit is considered and the superscript refers to a full geometry optimization (where only the Zn and O atoms are fixed). This geometry optimization yields a structure very similar to the isolated ADB molecule with bond angles within 0.5 degrees and bond lengths within 0.02 Å (cf.
Supplementary Material, SI.2). The changes in bond-lengths are primarily triggered by fixing the Zn-Zn distance between the paddle wheels. Also, the nature and energy of the lowest excited state are very similar to those in the isolated molecule (cf.,
Table 1 and
Table 4). This supports the finding from above that terminal substituents have essentially no impact on the lowest excited states of the chromophores. It also implies that for the present combination of chromophore and metal node, there is no “through-bond” electronic coupling (like in certain electrically conductive MOFs [
14]) between adjacent chromophores in
a- and
b-direction.
One of the crucial aspects not captured by the
opt(pw-ADB-pw)
1 model system is, how the conformation of the chromophore is changed by the neighboring linkers in
c- direction. To capture the influence of the neighbors, as a first step we optimized the structure of the 3D MOF employing periodic boundary conditions. For reasons explained in the Methods Section, this has been done using the PBE functional [
64]. From the periodic structure we extracted a tetramer cluster repeated in
c-direction (consisting of four ADB molecules bonded to two saturated Zn-paddle wheels). This cluster was then further optimized with the PBE0 functional [
65] (like in the molecule-based simulations), fixing the positions of the Zn atoms. This is done to deal with geometries (in particular bond lengths) obtained at a level of theory consistent with the previously discussed simulations. The structure of that tetramer cluster,
opt(pw-ADB-pw)
4, is shown in
Figure 6c. In this cluster, the geometries of the outermost (pw-ADB-pw) units are impacted by edge effects, but comparing the structures shown in
Figure 6b,c, one sees that the two central units adopt an arrangement fully consistent with the periodic geometry optimizations. A more quantitative analysis of the results shows that all twist angles agree to within less than 1° and that bond lengths are within 0.004 Å compared to the values from the periodic simulations (for more details see
Supporting Information SI.4). Thus, we used one of these units (viz tet2 in
Figure 6c) as the basic building block for the model systems used in the following. The structures constructed using this “cut-out” monomer will be denoted as
cut(pw-ADB-pw)
n, where the index n denotes the number of repeating units.
The geometric changes of all “cut” structures compared to optimized monomer structure,
opt(pw-ADB-pw)
1, arise from the impact of the neighboring ADB chromophores inside the MOF. These changes primarily concern the twist of the anthracene and phenylene moieties of ADB relative to the plane in which the neighboring Zn atoms are arranged (compare
Tables S3 and S4 in the
Supplementary Materials). In the optimized monomer,
opt(pw-ADB-pw)
1, the phenylenes are essentially in the plane of the Zn atoms (see
Figure 6a). Such a conformation is prevented in the 3D periodic structure by steric constraints, as can be inferred from the structure shown in
Figure 6b. In fact, for phenylenes in the plane of the Zn atoms, the H atoms on neighboring rings would come much too close to each other. Consequently, the phenylenes are twisted by 24° relative to the Zn plane in the periodic conformation. Also, the orientation of the anthracene units is significantly modified. While the anthracene plane is nearly perpendicular to the Zn-plane in
opt(pw-ADB-pw)
1 (at 82°), the twist between the two planes is reduced to 42° in the periodic structure. This reduced anthracene-Zn plane twist is primarily a result of van der Waals interactions between neighboring anthracenes trying to reduce the distance between the π-planes of the molecules (as shown in
SI.3 contained in the
Supplementary Materials). Most importantly, as a consequence of that also the angle between the phenylene and anthracene planes is reduced from 81° in
opt(pw-ADB-pw)
1 to 66° in the periodic structure.
To study the impact of the changes in twist angles, we first discuss the properties of a single “cut” chromophore,
cut(pw-ADB-pw)
1, (see highlight in
Figure 6c). The energy of the lowest excited state of
cut(pw-ADB-pw)
1 is distinctly red-shifted (by 0.17 eV) compared to the fully optimized monomer,
opt(pw-ADB-pw)
1, and the oscillator strength is significantly increased (see
Table 4). This can be attributed to the change of the conformation of the chromophore when incorporated into the MOF, where the main aspect is that the reduced twist between the phenylenes and the anthracene in the ADB unit results in an increased conjugation (see
Supplementary Material SI.5).
2.4. The Final Absorption Spectrum of Zn-ADB SURMOF-2: Combining Conformational Changes and Aggregate Formation
In the previous sections we have shown that when assembling ADB chromophores into Zn-SURMOF-2, there are two competing effects regarding the change in the absorption spectrum: On the one hand, there is a red-shift of the lowest excited state in the MOF due to the conformational changes triggered by inter-linker interactions and a related increase of conjugation (see
Section 2.3 and horizontal arrow in
Figure 7). On the other hand, H-aggregate formation in
c-direction causes a blue-shift of the first strongly allowed state, as discussed in
Section 2.2 (see
Table 2). This raises the question, how the combination plays out in the actual MOF, in which exciton coupling and conformational changes happen simultaneously. To address that, we constructed a dimer and a tetramer by assembling the “cut” monomer entities described above. This yielded
cut(pw-ADB-pw)
2 and
cut(pw-ADB-pw)
4, where the former system has the advantage that its properties in terms of the nature of orbitals and excited states can be discussed in analogy to the situation of the anthracene dimers from
Section 2.2. Indeed, it turns out that the nature and order of the excited states in
cut(pw-ADB-pw)
2 are equivalent to those of the anthracene dimer with a slip of 4.42 Å (cf.,
Table 3 and
Table 4): In the TD-DFT simulations on
cut(pw-ADB-pw)
2, the lowest excited state displays S
b character (the negative linear combination of OA→UA and OS→US transitions), while the state with the largest oscillator strength is the third excited state possessing S
c character (the respective positive linear combination). Compared to the slipped anthracene dimer, the oscillator strengths of both states are increased, which is due to the spreading of the transition density onto the phenylene units (cf.,
Supplementary Material SI.5; see also comparison of the properties of anthracene and ADB in
Table 1). This effect is a consequence of the reduced twist between the anthracene and the phenylene unit, when the ADB chromophores are incorporated into the MOF. As the oscillator strength associated with the S
c state is by a factor of more than five higher than that of the S
b state, the energy of S
c determines the position of the first absorption peak. Due to the dimer formation, in
cut(pw-ADB-pw)
2 this state is slightly blue-shifted by 0.08 eV compared to the corresponding monomer
cut(pw-ADB-pw)
1 (see blue arrow
Figure 7 and
Table 4). Combining this blue shift by 0.08 eV with the red-shift by 0.17 eV between
opt(pw-ADB-pw)
1 and
cut(pw-ADB-pw)
1 due to conformational changes yields the overall red-shift of 0.09 eV between an isolated ADB molecule and
cut(pw-ADB-pw)
2. This is schematically shown by the black arrow in
Figure 7.
A similar situation is obtained when calculating the excited states of the tetramer,
cut(pw-ADB-pw)
4. (see
Table 4). Again, the lowest exited state is red-shifted compared to the monomer and states at higher energies dominate the absorption spectrum due to their larger oscillator strengths. The state with the highest oscillator strength amongst the first 20 excited states (S
14) is even somewhat further blue-shifted than in
cut(pw-ADB-pw)
2. Calculating a theoretical absorption spectrum from a superposition of Gaussian peaks with full widths at half maximum (FWHM) of 0.30 eV centered at the energies of the excited states of the tetramer and scaled by their oscillator strengths yields an absorption maximum at 3.29 eV (cf.
Supplementary Material SI.7). This is only slightly higher than the experimental absorption maximum at 3.27 eV (see
Table 1). In passing we note that an unambiguous signature of the weakly allowed S1 state in
cut(pw-ADB-pw)
2 and
cut(pw-ADB-pw)
4 cannot be identified in the experimental spectra, as discussed in more detail in the
Supplementary Material (SI.9).
Overall, the above considerations explain, why the first absorption peak in Zn-ADB SURMOF-2 is not blue-shifted but rather red-shifted compared to the isolated ADB chromophore in solution in spite of the formation of H-aggregates. What remains to be explained is the significant shift of 0.64 eV between the absorption and emission maxima in Zn-ADB SURMOF-2.
2.5. Explaining the Red-Shifted Emission of ADB Molecules Incorporated into Zn-ADB SURMOF-2
For the isolated ADB molecule in solution, the rather large shift of 0.45 eV between absorption and emission maxima could be explained by a reduction of the twist angle between anthracene and phenylene units from 84° to 56° in the excited state equilibrium conformation (see
Section 2.1). In the MOF that angle is already decreased to 66° in the ground state, primarily due to the van der Waals interaction between neighboring chromophores (see
Section 2.3). Moreover, a further planarization of the ADB linkers incorporated into Zn-ADB SURMOF-2 in the excited state is prevented by steric constraints due to the already tight packing of the ADB chromophores in
c-direction in the ground state. Indeed, when optimizing the geometry of the (pw-ADB-pw)
4 tetramer in the S
1 electronic configuration, yielding
S1(pw-ADB-pw)
4, the changes in tilt angles are only very minor (see
Supporting Information SI.7). This applies in particular to the two central pw-ADB-pw units, where the S
1 state is primarily localized, as can be inferred from the excitation-induced changes in bond lengths and from the transition density shown in
Figure 8.
As a consequence, one would expect a smaller shift between the absorption and emission maxima in the MOF. Indeed, when calculating the excited state properties of one of the two central pw-ADB-pw units cut from the optimized S
1 tetramer,
S1,cut(pw-ADB-pw)
1, one observes an only rather moderately shift of ~0.3 eV (see monomer values in
Table 4 and
Table 5). A similar shift is actually observed when comparing the energies of the lowest excited states for
cut(pw-ADB-pw)
4 and
S1(pw-ADB-pw)
4. The respective energies amount to 2.86 eV for ground-state conformation (see
Table 4) and to 2.49 eV for the excited state conformation (see
Table 5), where it should be stressed that both states have the same nature, being dominated by an excitation between equivalent orbitals (see
Supplementary Material SI.8). This clearly shows that geometric relaxations in the excited state in Zn-ADB SURMOF-2 would result in a shift between absorption and emission maxima of only half the experimentally observed value of 0.64 eV.
This suggests that the main features in the emission and absorption spectra are dominated by different electronically excited states: The key difference between them is that the absorption spectrum is influenced by all excited states and dominated by the state(s) with the highest oscillator strength (S
14 in the case of
opt(pw-ADB-pw)
4). Conversely, what counts for the emission characteristics according to Kasha’s rule are the properties of the lowest excited state, i.e., S
1. This is in particular the case here, as due to the slip of the anthracene molecules, transitions between this state and the ground state are not optically forbidden in the H-aggregates of Zn-ADB SURMOF-2 (see
Section 2.2) [
63].
Therefore, to assess the shift between the maxima of the absorption and emission spectra, one has to compare the absorption maximum of
cut(pw-ADB-pw)
4 (which we find at 3.29 eV as discussed in
Section 2.4) and the S
1 energy of
S1(pw-ADB-pw)
4 (of 2.49 eV). This, indeed, yields a red-shift of 0.80 eV, which is consistent with the experimentally observed shift of 0.64 eV. The somewhat larger shift in the calculations occurs not only for Zn-ADB SURMOF-2, but also for ADB in solution (see
Table 1). It is mostly due to a minor underestimation of the emission energy, as becomes evident, e.g., from the comparison between the calculated excited state properties and the experimental spectra in the
Supplementary Material (SI.9).
In passing we note that even if the lowest excited state was forbidden in absorption, dynamic symmetry breaking due to geometry relaxations in the excited state could relax symmetry-selection rules [
63,
68]. This is, however, not the case in our TD-DFT simulations, as can be inferred from the reduced oscillator strength of the lowest excited state for the relaxed geometry of
S1(pw-ADB-pw)
4 compared to the ground-state conformation in
cut(pw-ADB-pw)
4. This is potentially a consequence of the delocalization of the exciton over two chromophores.
The above considerations show that the red-shifted emission of Zn-ADB SURMOF-2 is indeed a direct consequence of inter-chromophore interactions, where the subtleties of exciton coupling in slipped chromophores are crucial, while massive, excitation-induced conformational changes, as one would expect in classical excimers, do not play a role.
A final aspect that should be discussed is the possible consequence of the particularly low oscillator strength of 0.06 associated with the S
1→S
0 emission transition of
S1(pw-ADB-pw)
4. It implies that the radiative lifetime of Zn-ADB SURMOF-2 should be particularly long, in fact, much longer than the measured overall excited state lifetime of ~4 ns [
51] (which would then be determined by the non-radiative lifetime). It also suggests that in case there is some inhomogeneity in the sample including some less well-ordered regions, where the chromophores have a more isolated character, these regions will dominate the emission spectrum immediately after excitations due to the much larger oscillator strengths of isolated chromophores (see
Table 1). Only, when the excited state populations in these regions have decayed or when the excitons have migrated to the fully crystalline parts of the samples, the red-shifted emission of the S
1 state in
S1(pw-ADB-pw)
4 will dominate. Such a red-shift of the emission with time has, indeed, been observed in the experiments on Zn-ADB SURMOF-2 [
51].