Stack bonding in pentacene and its derivatives

Abstract

Understanding the nature of π-stacking interactions is important to molecular recognition, self-assembly, and organic semiconductors. The stack bond order (SBO) model of π-stacking has shown that the conformations of dimers are found at orientations where the combinations of monomer MOs are overall bonding within the stack. DFT calculations show that parallel displaced minima found on the potential energy surface for the π-stacked dimers of pentacene and perfluoropentacene occur when the dimer MOs are constructed from combinations of monomer MOs with an allowed SBO. An examination of the MOs of π-stacked dimers extracted from X-ray structures of alkynyl derivatives like TIPS-pentacene pack at one or more of the minima expected to show similar MO patterns. The π-stacking variability within these materials can be attributed to a balance between the minima allowed by SBO theory and steric effects within the lattice. The offset orientation of the pentacene cores observed in packing of these materials is attributed to the increased overlap of monomer lobes in the dimer and a reduction in two-orbital-four-electron repulsions. Charge mobility estimated from the frontier MOs of the dimer is related to the MO structure that favors PD conformations.

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Introduction

π-Stacking interactions contribute to charge mobility in organic electronic materials (OEMs).^1–7 Pentacene derivatives are utilized in a number of OEMs, including organic light-emitting diodes (OLEDs), organic field-effect transistors (OFETs), and organic solar cells (OSCs).^8,9 However, the herringbone packing of pentacene (1) and perfluoropentacene (2) lacks π–π interactions, which limits their transport properties.¹⁰ Substitutions with bulky alkynyl substitutions (i.e., 3–7, Fig. 1A) were included to improve processing and charge mobility within organic electronics derived from these materials. These derivatives pack in π-stacked “brick-and-mortar” arrangements which vary in the face-to-face overlap of the pentacene cores.^10–15

Fig. 1 (A) Pentacene (1), perfluoropentacene (2), 6,13-TIPS-pentacene (3, R = iPr, TIPS = triisopropylsilylethynyl), 6,13-t-butylethynylpentacene (4, R = tBu) [10], 5,14-TIPS-pentacene (5), 4-CF3-6,13-TIPS-pentacene (6), octafluoro 6,13-TIPS-pentac TIPS-pentacene (7). (B) Example of SBO in a noncovalent π-stacking interaction of a homodimer for a (+) and (−) combination of two monomer π-type MOs. At the S conformation, these combinations for stack antibonding (SA) and stack bonding (SB) combinations, respectively (SBO = 0). Parallel displacement shifts the character of the (+)-type MO from SA to SB for a net increase in SBO that increases the interpenetration of the dimer MOs and lowers the interaction energy of the stack by reducing the 2o4e repulsions.

Various groups have used theoretical approaches to understand the electronic structure of OEMs.^16–18 Our group has introduced the concept of the stack bond order (SBO) as an intuitive molecular-orbital (MO) based means for interpreting the conformational preferences of π-stacking interactions.^19–21 These nonbonded interactions universally prefer parallel-displaced (PD) and twisted (TW) conformations over the fully eclipsed sandwich (S) conformation. The SBO approach explains this preference similarly to the bond order in conventional MO theory. Within a π-stacked dimer, the MOs of monomers A and B of the same irreducible representation combine into additive (+) and subtractive (−) linear combinations (eqn (1) and (2)).

ϕe_irrep(+) = ϕe^A_irrep + ϕe^B_irrep (1)

ϕe_irrep(−) = ϕe^A_irrep − ϕe^B_irrep (2)

At the fully eclipsed sandwich (S) conformation, these pairs of π-type dimer MOs adopt stack antibonding (SA) character for the (+) linear combination and stack bonding (SB) character for the (−) combination (Fig. 1B). Given that the SBO is defined as the difference in the number of SB and SA-type MOs, the S conformation has an SBO of zero, which is interpreted in the same way as a zero bond order in conventional MO theory. Therefore, in analogy to the He dimer, the S conformation is predicted by the SBO to be unbound as a repulsive two-orbital-four-electron (2o4e) interaction, where the (+) and (−) linear combinations are well separated energetically (Fig. 1B). Likewise, fully eclipsed π-stacking in any aromatic system would be “forbidden”.²¹ However, PD of the stack stabilizes the (+) combination because the lobes are able to constructively overlap while the (−) combination destabilizes as the overlap decreases. At the minimum PD point in the benzene dimer, both of these MOs have stack bonding character (nonzero SBO) and are near degenerate such that the interaction between them is no longer repulsive.¹⁹ Therefore, the distortion by PD along the slip coordinate in formally non-covalent π-stacking interactions occurs to minimize the repulsive nature of the 2o4e interaction.

While the intermonomer mixing of p_π AOs has been assumed to be greatest at S due to the eclipsing of the ring atoms, this applies to the SB-type MOs as the cancelation by the SA-type MOs character depletes the overall inter-ring electron density. PD and/or TW converts one or more dimer MO with SA character at the S conformation to SB for a non-zero SBO (Fig. 1B) which increases the overlap between the monomer lobes as well as the inter-ring density, a major component of the dispersion. These conformations also decrease the exchange-repulsion term in the interaction energy²² which, with the increase in the interpenetration of the electron densities, allows closer stacking of the rings than in the S orientation. The most favourable overlap between the p_π-type AOs tends to occur in those conformations that stack similar to graphite-like AB (i.e., Bernal-type²³) packing.²¹ While the favourable overlap may not be evident from the atomic positions in the packing, it can be understood through visual inspection of the MOs in combination with SBO considerations. The number of possible π-stacking conformations increases with the size of the aromatic system as the frontier π-type monomer MOs are divided into more lobes which can form SB-type MOs at varying PD slip distances. The potential for multiple conformations with non-zero SBO is consistent with the polymorphism observed in some polyaromatic hydrocarbons (PAHs).²¹ From these observations, a set of stacking bonding rules has been proposed for aromatic homodimers:²¹

(a) Sandwich and other π-stacked conformations that eclipse the ring atoms are forbidden because they have equal numbers of SB- and SA-type dimer MOs (SBO = 0, maximum 2o4e repulsion).

(b) Parallel displaced π-stacked conformations maximize constructive interactions between the monomers because one or more SA-type MO at the S conformation will be converted to SB (SBO > 0) to minimize 2o4e repulsions.

(c) Substituted or polyaromatic molecules may have multiple observable π-stacked conformations due to the nodal structure of their extended π-systems.

(d) Twisted conformations are less effective for conversion of SA-type MOs to SB and may be preferred in cases where steric interactions prohibit PD.

In this contribution, the SBO model is used to show how orbital interactions lead to the multiple π-stacking motifs observed in pentacene derivatives. The factors that favor Bernal-type packing are also shown to be linked to MO patterns previously discussed in estimates of the charge mobility.

Methods

Calculations were performed using the Gaussian 09 quantum chemistry package.²⁴ The M06-2X exchange correlation functional²⁵ was used with and without Grimme's D3 empirical dispersion correction (M06-2X-D3).²⁶ These functionals give interaction energies of comparable accuracy to modern advanced functionals for noncovalent dimer test sets.²⁷ See ref. 21 for a more detailed explanation of the choice of functional for these calculations. The basis sets were triple-ζ quality with polarization functions on all atoms.²⁸ An ‘ultrafine’ grid was used for all calculations. Relative energies are not corrected for basis set superposition error (BSSE) because the goal of the study is to compare trends in π-stacking interactions, noting that a previous study of coronene dimers demonstrated that the shape of the potential energy curve did not change when BSSE was included.²⁹ Dimer structures and potential energy surfaces were generated with monomers fixed at their optimized structures and held parallel. Full optimization of the PD-OV^xy minima either led to a slightly more stable structure (<1.0 kcal mol⁻¹) with minimal distortions from the reported conformation or to a lower energy PD-OV^xy conformation. The MOs of these fully optimized structures are not significantly different from those shown below.

Results and discussion

Dimers of pentacene (1_dimer) and perfluoropentacene (2_dimer)

1D and 2D potential energy curves and surfaces for 1_dimer and 2_dimer. From the previous study of coronene and other large PAHs,²¹ two types of PD minima were allowed by the SBO model: those which appear slipped over the bonds of the fused rings (PD-OB) and those which are slipped over the vertices of the rings (PD-OV, i.e., Bernal-like packing). For the oblong pentacenes 1 and 2, displacement along only the R_x,slip (long axis of pentacene) or R_y,slip (short axis of pentacene) coordinates gives PD-OB^x and PD-OV^y conformations, respectively (Fig. 2). Multiple PD-OB^x minima in the 1D potential curve are found along the long R_x,slip coordinate at increments of one half-ring distance from the maxima at S or eclipsed (E) conformations (Fig. 3 and Table 1). These minima occur at similar R_x,slip distances to the PD-OB^x minima of coronene and other PAHs.²¹ One PD-OV^y minimum is found at an R_y,slip distance of roughly half the distance to travel across the ring, consistent with minima for the benzene and the lowest energy PAH dimer conformations.^19–21 Each of these conformations corresponds to a minimum in the inter-ring distance R_vert.

Fig. 2 Possible PD conformations of pentacene-type dimers for translation along the R_x,slip-only (PD-OB^x), R_y,slip-only (PD-OVy), and both R_x,slip and R_y,slip coordinates (PD-OV^xy_I and PD-OV^xy_II). Eclipsed conformations along the over-the-bond R_x,slip coordinate (E-OB) are maxima and are predicted to have zero SBO. Not all conformations will necessarily be observed.

Fig. 3 Potential energy curves with respect to the eclipsed S conformation (E_int,S = E(R_x,slip) − E(S)) along the R_x,slip coordinate only (PD-OB^x, orange) and the R_x,slip coordinate allowing R_y,slip to optimize (PD-OV^xy, blue) for 1_dimer (A) and 2_dimer (B). Variation in the inter-ring distance R_vert for 1_dimer (C) and 2_dimer (D). Variation in R_y,slip for 1_dimer (E) and 2_dimer (F).

Table 1 DFT(M06-2X) interaction energies E_int with and without empirical dispersion corrections calculated relative to separated monomers and, in parentheses, the eclipsed S conformation. Values in italics are minima using the default integration grid and are unstable on the larger “ultrafine” grid (see text)

	E int (M06-2X/(TZVP))	R vert, Å	R x,slip, Å	R y,slip, Å	WBIAB	E int (M06-2X-D3/(TZVP))	R vert, Å	R x,slip, Å	R y,slip, Å
1dimer
S	−6.5 (0.0)	3.66	0.00	0.00	0.032	−9.8 (0.0)	3.67	0.00	0.00
PD-OB^xA	−13.3 (−6.8)	3.38	1.31	0.00	0.164	−16.5 (−6.7)	3.38	1.31	0.00
PD-OB^xB	−11.3 (−4.8)	3.38	3.70	0.00	0.126	−14.3 (−4.4)	3.38	3.69	0.00
PD-OB^xC	−8.3 (−1.8)	3.39	6.11	0.00	0.083	−10.7 (−0.9)	3.39	6.08	0.00
PD-OB^xD	−5.5 (+1.0)	3.32	8.37	0.00	0.049	−7.3 (+2.5)	3.42	8.27	0.00
PD-OV^y	−14.1 (−7.6)	3.32	0.00	1.56	0.078	−17.2 (−7.4)	3.33	0.00	1.54
PD-OV^xyIA	−17.3 (−10.8)	3.28	1.25	1.27	0.157	−20 (−10.5)	3.28	1.12	1.27
PD-OV^xyIB	−14.9 (−8.4)	3.27	3.46	1.17	0.121	−17.8 (−8.0)	3.27	3.45	1.17
PD-OB^xyC	−11.8 (−5.3)	3.27	5.50	1.36	0.089	−14.4 (−4.4)	3.27	5.45	1.37
PD-OV^xyIIC	−9.0 (−2.4)	3.22	7.41	1.43	0.078	−10.8 (−1.0)	3.19	7.29	1.40
PD-OV^xyIID	−5.6 (+0.9)	3.19	9.78	1.36	0.055	—	—	—	—
2dimer
S	−11.9 (0.0)	3.44	0.00	0.00	0.084	−15.7 (0.0)	3.45	0.00	0.00
PD-OB^xA	−20.9 (−9.0)	3.24	1.23	0.00	0.230	−24.6 (−8.9)	3.24	1.23	0.00
PD-OB^xB	−18.2 (−6.3)	3.23	3.63	0.00	0.176	−21.6 (−5.9)	3.23	3.61	0.00
PD-OB^xC	−14.0 (−2.1)	3.24	6.03	0.00	0.122	−16.8 (−1.1)	3.25	6.01	0.00
PD-OB^xD	−9.8 (+2.1)	3.26	8.39	0.00	0.075	−12.3 (+3.4)	3.26	8.32	0.00
PD-OV^y	−22.9 (−11.0)	3.18	0.00	1.29	0.165	−26.6 (−10.9)	3.18	0.00	1.29
PD-OV^xyIA	−26.3 (−14.4)	3.14	1.19	0.90	0.248	−29.9 (−14.2)	3.15	1.19	0.89
PD-OV^xyIB	−22.8 (−10.9)	3.13	3.44	0.96	0.148	−26.2 (−10.5)	3.14	3.42	0.96
PD-OV^xyIIB	−18.4 (−6.5)	3.14	5.11	1.23	0.130	−21.5 (−5.7)	3.15	5.02	1.25
PD-OV^xyIIC	−14.5 (−2.6)	3.12	7.50	1.23	0.105	−16.9 (−1.1)	3.13	7.44	1.23
PD-OV^xyIID	−10.3 (+1.6)	3.9	9.81	1.21	0.082	−12.1 (+3.7)	3.10	9.74	1.21

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Variation along both R_x,slip and R_y,slip coordinates generates one type of PD-OB^xy and two types of PD-OV^xy π-stacking motifs (Fig. 2). PD-OV^xy_I conformations orient the 2-carbon over the center of a ring of the other monomer and occur when one unit slips half the diameter of the ring in the R_y,slip direction (∼1.4 Å) and increments of n + 0.5 ring distances in the R_x,slip direction. PD-OV^xy_II conformations place the 2-carbon over a tertiary carbon of a fused ring and occur when one unit slips by one C–C bond length in the R_y,slip direction and n ring distances in the R_x,slip direction. These conformations are consistent with Bernal-type stacking and analogous to the two PD-OV^xy conformations previously discussed for PAHs.²¹ PD-OB^xy conformations occur with the slippage of a half-ring distance over the bond from a fully eclipsed conformation (i.e., S or E-OB^x) and were less energetically favorable than those with PD-OV stacking in unsubstituted PAHs.²¹ The alkynyl substituted pentacenes adopt various conformations in their brick-and-mortar packing¹⁰ (e.g., 3, PD-OV^xy_IID; 4, PD-OV^xy_IIB; 5, PD-OV^xy_IA; 6, PD-OV^xy_IIC; 7, PD-OV^xy_IC). A recently synthesized tethered pentacene dimer arranges into the PD-OV^xy_IB conformation.³⁰ Derivatives with substitutions to the pentacene core such as nitriles,^31,32 ethers,³³ thioethers,³⁴ and thiophenes,³⁵ as well as tethered anthracenes and substituted hexacenes and heptacenes³⁶ also stack in similar Bernal-type or PD-OB orientations. Because the π-systems of the alkyne spacers are conjugated with the pentacene core, these groups could potentially participate in stack bonding.

The potential energy surface for simultaneous slipping along both the R_x,slip and R_y,slip coordinates results in more stable minima than the PD-OB^x and PD-OV^y conformations which appear as saddle points on the 2D surface (Table 1 and Fig. 4). The lowest energy PD-OV^xy_IA conformation has the most π-surface overlap of the minima found on the 2D surface. As the stack slips further from the S conformation along R_x,slip, additional shallow minima appear on the surface with R_y,slip varying between 1.1 and 1.5 Å (0.9–1.3 Å for 2_dimer) and R_vert between 3.2 and 3.3 Å (3.1–3.2 Å for 2_dimer) as the dimer adjusts to maximize the interaction between the monomers. These minima are incrementally 3–4 kcal mol⁻¹ less stable than PD-OV^xy_IA consistent with the decreased overlap between the π-systems. The minima nearest S are PD-OV^xy_I-type conformations; the dimers switch to PD-OV^xy_II-type once only three of the monomers’ fused rings are interacting. The minima occur at similar R_x,slip and R_y,slip distances for 1_dimer and 2_dimer, but at shorter inter-ring distances for the latter due to the depletion of π-electron density induced by the electron-withdrawing fluorines.

Fig. 4 Potential energy surfaces for parallel-displaced (PD) along both the R_x,slip and R_x,slip coordinates for 1_dimer (A) and 2_dimer (B). The monomer structures are fixed in this PES. Full optimization of the minima either leads to minimal distortion from the fixed dimer and a slightly lower E_int (<1.0 kcal mol⁻¹) or a lower energy conformation.

The minima listed in Table 1 are 2–4 kcal mol⁻¹ more stable relative to separate monomers when the D3 empirical dispersion correction is included.²⁶ The PD-OV^xy_IIB and PD-OV^xy_IID conformations are only found for 2_dimer where the E_int are generally stronger than for 1_dimer. In addition, the interactions are sensitive to the number of integration lattice points and the inclusion of the D3 empirical dispersion correction. Use of the default grid settings in Gaussian09 finds these conformations as minima where they appear as shoulders on the PES when grids with more integration points are used. The PD-OV^xy_IID conformation of 1 is not a minimum when the D3 empirical dispersion correction is included.

SBO analysis of 1_dimer

To understand the influence of orbital interactions on the conformational preferences of 1_dimer and 2_dimer, the frontier bands of dimer MOs were inspected along each of the 1D and 2D PD coordinates. Minima along each coordinate are consistent with the conversion of at least one (+)-type dimer MO (eqn (1)) with SA character in the S conformation to SB at a PD conformation, as expected by the SBO model. Considerations for favorable MO interactions are simplified by focusing on the occupied frontier MOs. Lower-lying MOs show similar patterns of interconversion between SB and SA as the frontier band along each of the PD coordinates. While these MOs can influence the conformation locations, the frontier MOs are assumed to contribute most significantly to the observed minima in analogy to the Walsh rules.^37,38

For 1_dimer, mixing of the three monomer frontier MOs (ϕ_3b2g HOMO, ϕ_2au HOMO−1, ϕ_3b1g HOMO−2) was considered. At the S conformation, the (+) and (−) linear combinations of these MOs form SA- and SB-type dimer MOs, respectively, which cancel for an SBO of zero (Fig. 5). PD along the dimer short axis (R_y,slip) converts the ϕ_3b2g(+) and ϕ_2au(+) dimer MOs to SB at the PD-OV^y minimum while ϕ_3b1g(+) and the (−)-type MOs remain SB for a net positive SBO. ϕ_3b2g(+) has no nodes along the PD coordinate and maintains its SA character along the PD coordinate. These shifts in MO character are similar to the benzene dimer where there is also only one node perpendicular to the PD coordinate.¹⁹

Fig. 5 Frontier dimer MOs for the sandwich (S) and parallel-displaced (PD) conformations of 1_dimer: (A) dimer MOs for conformations along the y- and x-slip coordinates (PD-OV^y and PD-OB^x, resp.), (B) dimer MOs for conformations found varying the slip coordinate in both the x- and y-directions (PD-OV^xy, PD-OB^xy). MOs were plotted with an isosurface value of 0.01 using Chem3D. The frontier MOs for 2_dimer are similar in character and are found in the Fig. S1, ESI.†

For the PD-OB^x minima along the long axis (R_x,slip) (Fig. 5), all frontier dimer MOs have SB character only for the lowest-energy PD-OB^x_A conformation. The next three PD-OB minima each have one SA-type dimer MO (ϕ_2au(+), ϕ_3b2g(+), and ϕ_3b2g(−), resp.) which, with the reduced overall co-facial overlap of the monomer lobes, contributes to higher E_int for these conformations. In addition, the equal number of SA and SB MOs (SBO = 0) found at the E-OB^x_A conformation is consistent with SBO analysis, which predicts that E conformations will be unbound.

In Walsh-like diagrams (Fig. 6), the PD-OB^x minima of 1_dimer generally occur at the crossing point of the ϕ_3b1g(+/−) MOs. Crossing points occur when the (+) and (−) linear combinations of monomer MOs are overall SB in character and degenerate with no 2o4e repulsions between the dimer MOs, contributing to the minimum in E_int. Because the sinusoidal behavior of the dimer ε_MOs, where minima correspond to the largest overlap of the monomer MO lobes, depends upon the nodal structure of the monomer MOs (i.e., more nodes lead to shorter wavelengths in the oscillation), the crossing points of different (+)/(−) pairs do not occur at the same R_y,slip distances. For example, PD-OB^x_A appears closest to the crossing points of all three (+)/(−) pairs of dimer MOs, but the other minima have at least one MO that is near maximum SA character due to a node along the xz plane (Fig. 5).

Fig. 6 Walsh-like diagrams for PD along the R_x,slip coordinate for the frontier MOs of 1_dimer (A) and 2_dimer (B). Walsh-like diagram for PD along the R_x,slip coordinate allowing R_y,slip to vary for the frontier MOs of 1_dimer (C) and 2_dimer (D).

PD translation along both R_x,slip and R_y,slip coordinates (Fig. 5) breaks the xz symmetry plane of the dimer to allow lobes that were inaccessible for mixing in PD-OB^x to combine between monomer MOs. This mixing introduces SB character analogous to that found along the PD-OV^y-coordinate. As a result, the PD-OV^xy minimum energy conformations in the 2D PES are located at roughly a half-ring distance slip in the R_y,slip direction. These results are consistent with the conformations observed in X-ray structures of substituted pentacenes.¹⁰ These derivatives do not pack with a mirror plane in the pentacene units, a result that was not expected by the synthetic group, but is explained through SBO analysis. The increase in orbital mixing enhances the inter-ring density and favors shorter R_vert distances and the stronger E_int relative to the PD-OB^x conformations (Table 1). However, not all potential π-stacking conformations (Fig. 2) are found on the PES for 1_dimer due to other contributions to the overall E_int.

The mixing of SB character along both PD coordinates complicates the interpretation of the ε_MO trends in the PD-OV^xy Walsh-type diagrams (Fig. 6). Pairs of dimer MOs maintain the roughly sinusoidal patterns of the PD-OB^x MO energies, where the maxima do not necessarily correlate with the least monomer MO overlap due to the mixing of PD-OV^y SB character. Irregularities also appear in the patterns due to variations in R_y,slip to maximize overlap of the lobes across the xz nodal plane in the monomer MOs. The inflection in the HOMO/HOMO−1 dimer pair near R_y,slip = 2.75 Å can be attributed to an PD-OB-like conformation. The more regular behavior of HOMO−4/5 is due to the lack of nodes in the xz plane that could limit overlap between the broad monomer lobes.

SBO analysis of 2_dimer

The dimer MOs constructed from the four monomer perfluoropentacene frontier MOs were considered in the SBO analysis of 2_dimer. The (+)/(−) pairs of the HOMO (ϕ_7b2g) and HOMO−1 (ϕ_5au) monomer MOs form well-separated individual bands (Fig. 6, Fig. S1, ESI†). Along the PD-OB^x and PD-OV^y coordinates, these dimer MOs have similar character to 1_dimer with some minor mixing of the carbon and fluorine 2p_π AOs. However, the stabilization of monomer HOMO−2 (ϕ_6b1g) by the electron-withdrawing fluorines allows its linear combinations to mix with HOMO−3 (ϕ_6b2g) into a band of four dimer MOs. Similar to 1_dimer, only the PD-OB_A minimum along the R_x,slip coordinate has SB character for all frontier dimer MOs with conformations at longer R_x,slip having at least one of the MOs with a node in the xz plane have SA character (PD-OB^x_B: ϕ_5au(−); PD-OB^x_C: ϕ_6b2g(−) and ϕ_7b2g(+); PD-OB^x_D: ϕ_5au(+), ϕ_6b2g(+), ϕ_7b2g(−)) to decrease the overall SBO of the respective PD-OB^x conformations.

As for 1_dimer, PD in both R_x,slip and R_y,slip directions allows for the SA-type MOs at the PD-OB^x conformations to mix in the SB character found in the PD-OV^y conformation. The net increase in the overall SBO relative to the PD-OB^x conformations contributes to lower E_int and shorter inter-ring distances indicative of increased inter-ring density. Patterns in the Walsh diagram (Fig. 6) are challenging to interpret due to the avoided crossings of the a_g and a_u irreducible representations of the MOs in the lower band. However, most minima occur near the crossing point of the highest two pairs of dimer MOs ((+)/(−)ϕ_7b2g(+)/(−)ϕ_5au) with PD-OV^xy_IA shifted closer to the crossing point of (+)ϕ_6b2g and (−)ϕ_6b1g in the lower band. In contrast, PD-OV^xy_IB occurs near the crossing point of three MOs in the lower band.

Trends in electronic properties

The most stable energy π-stacked conformations of 1_dimer and 2_dimer are found at the PD conformation closest to S (PD-OV^xy_IA) which has the most overlap between the lobes of the monomer MOs within the dimer. The trends in E_int are consistent with the inter-ring electron densities as estimated by the sum of the inter-ring Wiberg bond indices (WBI_AB, eqn (3) and Table 1).³⁹

The electron densities are greatest at the first minimum along the PD-OB^x and PD-OV^xy coordinates and decreases as fewer monomer lobes are able to mix. As a result, the inter-ring R_vert distances at the PD-OV^xy minima become significantly shorter than S as a result of the enhanced mixing between monomer MOs. Even though fewer monomer MO lobes can interact at long R_x,slip distances, the R_vert distances are similar due to the distance needed for optimal overlap of the carbon p_π lobes.

The trends in charge mobility were estimated along the PD-OV(xy) coordinate as the ε_MO difference (Δε_MO) for the HOMO and HOMO−1 (electron transfer) and LUMO and LUMO+1 (electron hole) pairs.⁴⁰ For the frontier HOMOs, the difference is minimized at the crossing points and maximized when the pair has maximum SB (HOMO−1) and SA (HOMO) character, (i.e., roughly halfway between the PD-OV^xy minima (Fig. 7)). The extent of overlap of monomer MOs is expected to be related to the magnitude of electronic coupling.¹ In 1_dimer and 2_dimer, the dimer LUMO and LUMO+1 pair is constructed from the (+)/(−) combinations of the π*-type ϕ_b3u MOs which alternate between combinations of SB and SA character. LUMO combinations are generally SB at the minimum energy conformations (Fig. 7). While these unoccupied MOs do not affect the conformation of the π-stack, they influence the electronic properties. This lobal structure of π-systems, previously used to interpret the patterns in the transfer integrals,^40,41 is connected to the structural preferences interpreted through SBO analysis. Minima in the Δε_MO for these unoccupied frontier MOs are also found at the crossing points of the (+) and (−) combinations of the monomer LUMO. Due to the nodal structure of the monomer HOMO and LUMO of 1_dimer and 2_dimer, the extrema in Δε_MO occur in different locations with high hole mobility and variable electron mobility at the PD-OV^xy minima. The HOMO–LUMO gap is largest near the PD-OV^xy minimum structures where LUMO and LUMO+1 are most similar in energy (i.e., when both have SB character and enhanced overlap between the unoccupied monomer MOs).

Fig. 7 Trends in the HOMO–LUMO gap of the 1_dimer (A) and 2_dimer (B) dimer along the PD-OV^xy coordinate. Minima along the PD-OV^xy PES are indicated with vertical arrows (see Fig. 3 and 6). Trends in the estimates of the electron and hole charge mobilities calculated as the ε_MO difference (Δε_MO) between the HOMO and HOMO−1 and LUMO and LUMO+1 pairs, respectively, of the 1_dimer (C) and 2_dimer (D) dimer over the PD-OV^xy coordinate. LUMO and LUMO+1 for the 1_dimer (E) and 2_dimer (F) dimers along the PD-OV^xy coordinate. MOs were plotted with an isosurface value of 0.01 using Chem3D.

SBO analysis of substituted pentacenes

For a selection of substituted pentacene derivatives 3–7, π-stacked dimers were extracted from their X-ray structures and single point calculations were performed to visualize the MOs (Fig. 8). These molecules incorporate alkyne groups with bulky substituents that favor π-stacking between the pentacene cores through various “brick-and-mortar” packing arrangements.¹⁰ The MOs for these extracted dimers are comparable to those found in 1_dimer and 2_dimer. Due to the longer inter-ring distance in the X-ray structures (∼3.4 Å versus 3.1–3.3 Å in the optimized dimers), some of the MOs do not have effective overlap of lobes, reducing the overall SBO (i.e., HOMO−4 in 3_dimer and 4_dimer, HOMO−5 in 6_dimer, and HOMO and HOMO−1 in 7_dimer). Nonetheless, the overall character is consistent with a non-zero SBO and the interpenetration of the monomer densities. Only in 5_dimer with the most stable PD-OV_IA conformation do all dimer MO have SB character. The close overlap of the monomer MOs along with the high electron/hole mobilities determined in Section 2.1.4 may facilitate the rapid quenching of these materials experimentally.⁴² Although the distances between the monomers in the X-ray structures are longer than the optimized structures, the dimer MOs are analogous to those of 1_dimer (Fig. 8). Note that optimization of these dimers can result in a different conformation indicating that other factors in the solid state play an important role in the packing orientation in addition to the SBO considerations.

Fig. 8 Frontier MOs for dimer pairs extracted from X-ray crystal structures of π-stacked pentacenes arranged from HOMO (top) to HOMO-5 (bottom). SA MOs are indicated with a (*). The overall SBO for the frontier dimer MOs is greater than zero. MOs were generated from an M06-2X/TZVP single point calculation from the extracted experimental structure and plotted with an isosurface value of 0.01 using Chem3D.

Conclusions

Stack bond order (SBO) analysis is an intuitive means of understanding the conformational preferences and variability in π-stacking motifs.^19–21 The observed packing in various pentacene derivatives can be attributed to the manner in which the monomer MOs combine in dimers. As established by the SBO rules for homodimers,²¹ minima of dimers along the R_x,slip and R_y,slip coordinates are found when the number of MOs classified as SB is greater than the number of SA-type MOs. These minima generally occur where at least one pair of dimer MOs consisting of (+)/(−) pair of linear combinations of monomer MOs is near-degenerate and does not contribute to two-orbital-four-electron repulsions. The offset, graphite-like AB (or Bernal-type) π-stacking²³ observed in alkynyl substituted derivatives is attributed to the greater mixing of monomer lobes when the mirror symmetry of the dimer is broken. The offset π-stacking of TIPS-pentacene and related derivatives was not expected by the synthetic group,¹⁰ but is readily explained by SBO analysis. Given the nodal structure of the frontier monomer MOs, multiple minima are found which are consistent with the variable π-stacking arrangements found in X-ray structures of pentacene derivatives. These conformations occur at similar PD distances to other aromatic systems due to the overlap of p_π-type MOs within the constraints of a non-zero SBO. The patterns in stack bonding character that contribute to favourable π-stacking conformations have also been linked to the charge transport properties of these materials.^40,41 The synergy between the structure and charge transport ensures that π-stacked materials pack in a manner to conducive to acting as an organic semiconductor. Future studies will examine how the SBO can be used to interpret the π-stacking within additional classes of aromatic systems important to organic electronic materials.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Calculations were performed on high performance maintained by ODU Information Technology Services.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp03970j

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