Electronic structure of rhombus-shaped nanographenes: system size evolution from closed- to open-shell ground states

Abstract

We theoretically study and characterize a set of rhombus-shaped nanographenes of increasing size, or n-rhombenes, where n = 2–6, displaying zigzag edges leading to an enhancement of the (poly)radicaloid nature and the appearance of intrinsic magnetism as a function of n. Due to that system-dependent radicaloid nature, we employ spin-flip methods able to capture the challenging physics of the problem, thus providing accurate energy differences between high- and low-spin solutions. The theoretical predictions agree with the experimentally available magnetic exchange coupling for the recently synthesized 5-rhombene, as well as with the size at which the transition from a closed-shell to an open-shell ground-state solution occurs. We also investigate if standard DFT methods are able to reproduce the trend disclosed by spin-flip methods and if the results are highly dependent on the functional choice and/or the intrinsic spin contamination.

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1. Introduction

There is an increasing interest^1–7 on open-shell graphene structures (i.e., nanometer-sized flat graphene nanoforms such as nanoflakes, nanoislands, etc.) with tailored edges and sizes, due to several factors, namely: (i) the dependence of total and local magnetism on the particular topology (e.g. shape and/or sublattice imbalance) of the systems, theoretically predicted time ago;^8–10 and (ii) the recent advances in atomically-induced open-shell nanographene synthesis of a variety of these challenging nanoforms.^11–15 As part of these research interests, we will consider in this study a family of graphene nanoforms displaying a rhombus shape arising from fusing two triangular fragments, with the chemical formula C_2n²+4nH_4n+2 and dubbed as n-rhombenes, with n indicating the system size (Fig. 1). More specifically, n is the number of benzenoid rings along any of both edges. These systems are predicted to have a singlet (S = 0) ground state following the Ovchinnikovs rule¹⁶ and its latter generalization known as Lieb's theorem,¹⁷ given that for these molecules an equal number of C atoms holds in the two interprenetating sublattices in which the topological structure can be decomposed. However, the closed- or open-shell nature of that singlet state might be highly dependent on the size, as well as the energy position of other high-lying states, which also prompted previous theoretical investigations.^18–20

Fig. 1 Chemical structures of the n-rhombenes studied (H atoms are not shown).

Going into more details, the smallest member of the family (i.e., 2-rhombene or benzo[def]phenanthrene, simply known as pyrene) has been synthesized time ago,²¹ and it is also found on interstellar searches,²² used as template for metal–organic frameworks,²³ or for fusing other acenes and azaacenes,²⁴ to name just a few examples of its chemical versatility and functionality. Note also that the synthesis of other members of the family had not been reported, possibly due to the increase of their polyradicaloid (or open-shell) character mentioned above, until the emergence of new atomic-scale techniques allowing their experimental realization. Actually, the surface-assisted synthesis of 4-rhombene and 5-rhombene on Au(111) was very recently reported in ref. 25, further characterized by Scanning Tunneling Microscopy (STM) and probed by inelastic electron tunnelling spectroscopy (IETS) and differential conductance spectroscopy measurements. Those experiments revealed a closed-shell singlet (S = 0) ground-state for 4-rhombene but an open-shell singlet for 5-rhombene, separated by 102 meV from a low-lying triplet (S = 1) state, in agreement with various calculations carried out concomitantly to that experimental investigation.²⁶

However, there are still a few questions that can be treated from a computational point of view regarding these systems, as well as the evolution of electronic properties as a function of the system size, namely: the degree of (poly)radicaloid character and its characterization in terms of the number and distribution of unpaired electrons, the impact of spin contamination on the computational results, the evolution of the magnetic exchange coupling, the energy location of other viable high-spin states such as the quintet, the energy needed to oxidize and reduce the molecules, etc. Additionally, the use of standard Density Functional Theory (DFT) on these systems is not exempted from difficulties,^27,28 due to the increasing open-shell character with n, unless using some non-standard extensions to deal with fractional occupation of molecular orbitals.²⁹ The main goal of the present study is to address these open questions by means of electronic structure calculations. Nevertheless, to circumvent these difficulties and obtain highly accurate results to be taken as reference, we will also apply here more time-intensive and sophisticated methods such as the Restricted Active Space Spin-Flip (RAS-SF) method,^30–32 which has been previously demonstrated to be a reliable tool to obtain very accurate results for (poly)radicaloid compounds.³³ The RAS-SF will also be coupled³⁴ with a short-range exchange–correlation energy functional (RAS-srDFT) to incorporate in a balanced way dynamical (or short-range) and non-dynamical (or long-range) correlation effects.^35,36 Therefore, we will study the set of n-rhombenes, with n = 2–6, to address not only the evolution of electronic properties with the system size but also to simultaneously assess the accuracy of the methods chosen.

2. Theoretical framework

The characterization of the (poly)radicaloid nature of these compounds will be first estimated by the fractional occupation of molecular orbitals induced by (near-)degeneracy effects, which in open-shell systems can be simulated by minimizing the Gibbs electronic free energy (G_el = E_el − T_elS_el) of the system, at a fictitious pseudo-temperature (i.e., electronic) called T_el and imposing a Fermi–Dirac distribution around the Fermi level E_F:with f_i being the fractional occupation numbers, and θ = k_BT_el. This kind of methods are collectively denoted as Finite–Temperature DFT (FT-DFT).^37–39

Reference values for energy differences between singlet (S), triplet (T), and quintet (Q) spin states (vide infra) will be obtained by the Restricted Active Space Spin-Flip (RAS-SF) method, allowing multiple spin-flip excitations⁴⁰ within the active space chosen of doubly occupied (RAS1), singly occupied (RAS2), and virtual (RAS3) subspaces. The RAS-SF Hamiltonian is defined as that of n-spin-flip excitations expanded in terms of the number of holes (particles) in the doubly occupied (virtual) space:

R̂^nSF = r̂^nSF₀ + r̂^nSF_h + r̂^nSF_p + r̂^nSF_hp + r̂^nSF_2h + r̂^nSF_2p+ …, (2)

where r̂^nSF₀ performs all possible spin-flip excitations within RAS2 and h and p subindices indicate the number of holes and particles in RAS1 and RAS3, respectively. In the approximation used here p = h = 1; i.e., the right side of eqn (2) is truncated at the third term. The eigenvalues are obtained from a wavefunction expanded over a high-spin single reference restricted set of Slater determinants, , where |Φ_R〉 are the allowed electronic configurations. These calculations will be also coupled with the PBE⁴¹ short-range exchange–correlation energy functional, to introduce both types (long- and short-range) of electron correlation effects, briefly denoted as RAS-srPBE in the following, since this functional is chosen for this RAS-srDFT extension.

The relative energies between those low-lying and different spin states will also be obtained by Density Functional Theory (DFT), employing semi-local and both Global–Hybrid (GH) and Double–Hybrid (DH) density functionals, to bracket if the accuracy of the calculations depends on the choice of the exchange–correlation functional. The general expression for a GH functional is given by:

E^GH_xc[ρ] = c_xE^EXX_x + (1 − c_x)E_x[ρ] + E_c[ρ], (3)

with the exchange contribution being a linear function of the EXact-eXchange (EXX) energy, E^EXX_x, and the exchange functional, E_x[ρ], weighted by a coefficient c_x (note that for a semi-local functional c_x = 0). This expression can be extended to a DH form as:

E^DH_xc[ρ] = c_xE^EXX_x + (1 − c_x)E_x[ρ] + c_cE^PT2_c + (1 − c_c)E_c[ρ], (4)

that is, adding the 2nd-order Perturbation Theory energy, E^PT2_c, to the correlation functional, E_c[ρ], weighted now by the coefficient c_c. The expressions for:with 〈ab||ij〉 = 〈ab|ij〉 − 〈ab|ji〉, rely both on the set of occupied ({ϕ_i}) and virtual ({ϕ_a}) orbitals self-consistently generated by eqn (3) and used hereinafter. We will select for these calculations the semi-local PBE⁴¹ (c_x = 0), the hybrid PBE0⁴² (c_x = 1/4) and PBEHH (c_x = 1/2), and the double-hybrid PBE0-DH⁴³ (c_x = 1/2 and c_c = 1/8) exchange–correlation functionals, all of them belonging to the same non-empirical family of expressions but differing in their hybridization.^44,45 Furthermore, a Broken–Symmetry (BS) solution will also be carefully obtained at all those levels for the corresponding open-shell singlet states.

The spin-contamination associated to DFT calculations can be reduced to better estimate the singlet–triplet and singlet–quintet energy differences, or ΔE_ST and ΔE_SQ, respectively. Denoting the low-spin (M_s = 0) and high-spin (e.g. M_s = 1 for a triplet) solutions as LS and HS, respectively, one of the simplest forms has been proposed by Yamaguchi et al.^46–48 exploiting the difference between the calculated mean values of the Ŝ² spin operator for the two solutions involved, such as:

ΔE(HS − LS) = δ·[E(HS) − E(LS)], (7)

with:where n_S corresponds to the 〈Ŝ²〉 difference between the ideal spin eigenvalues, e.g., n_S = 2 for a singlet–triplet. In the case of n-rhombenes, we will calculate ΔE_ST and ΔE_SQ values, with S, T, and Q corresponding to the lowest-energy solutions fixing M_s = 0 (singlet), M_s = 1 (triplet), and M_s = 2 (quintet) or, in other words, the S₀ − T₁ and S₀ − Q₁ energy difference. In the case of the singlet states, an open-shell solution will also be imposed which will be used to calculate those ΔE_ST and ΔE_SQ values and thus infer for which value of n the closed- and open-shell solutions might differ.

2.1 Computational details

The geometries of the compounds are initially optimized by the B97-3c method⁴⁹ leading to planar structures identified by all-real 3N − 6 vibrational frequencies. The FT-DFT calculations are done with the recommended TPSS⁵⁰ functional and the default temperature of 5000 K, together with the def2-TZVP basis set,⁵¹ employing the ORCA 5.0 package.⁵² The spatial plots of ρ_U(r) are done with the AVOGADRO 1.97 visualizer.⁵³ The same ORCA 5.0 package is also used for the DFT calculations with different exchange–correlation functionals and the def2-SVP and def2-TZVP basis sets. The RIJCOSX technique,⁵⁴ together with the auxiliary def2/JK and def2-nVP/C basis sets,⁵⁵ is fixed for the latter calculations to alleviate the computational cost. The cc-pVTZ basis set is also used for some test calculations, employing again the corresponding auxiliary basis sets. For the RAS-SF calculations, done here with the Q-CHEM 6.0 package,⁵⁶ we also assess the dependence of the results with respect to different basis sets (6-31G*, 6-311G*, def2-SVP, and def2-TZVP). For the RAS-based calculations, the number of electrons (N_e) in the number of active orbitals (N_o), or briefly (N_e,N_o), is chosen consistently with the fractional occupation disclosed at the FT-DFT level, that is, (4,4) for 2-rhombene, (6,6) for 3-rhombene, and (10,10) for the rest of n-rhombenes (4- to 6-rhombene). A triplet state was used as the high-spin reference for RAS-SF, except for n = 5–6 for which a quintet state was instead used.

3. Results and discussion

3.1 Geometrical features

The B97-3c optimized geometry for the 2-rhombene molecule, the only system for which X-ray data exists,⁵⁷ are in close agreement with the experimental results (averaged error of 0.0080 Å) and very close to those calculated for consistency at the PBE0/def2-TZVP and PBE0-DH/def2-TZPV levels (averaged errors of 0.0065 and 0.0062 Å, respectively) which are known to be very accurate methods for geometry optimizations.⁵⁸ The Bond Length Alternation (BLA) can be calculated by B97-3c to estimate the average difference in lengths between single and double bonds along the zigzag edges:⁵⁹where d_i,j is the distance between atoms i and j out of the N total C atoms. BLA values (in Å, as calculated at the B97-3c level) of 0.029, 0.033, 0.031, 0.027, and 0.023, are found from 2-rhombene to 6-rhombene, respectively, indicating only slight variations of aromaticity as a function of the system size n. Note also that these values are lower than those found before⁶⁰ for oligoacetylenes of similar length, as it is also expected from the extended conjugation of the (fully planar) n-rhombenes. The same trend for BLA values is reproduced by PBE0/def2-SVP or PBE0/def2-TZVP methods, with the latter giving slightly higher values (in Å) of 0.034, 0.041, 0.040, and 0.036 for 2- to 5-rhombene, respectively. The InfraRed (IR) spectra of the set of n-rhombenes, at the B97-3c level, shows (see the ESI†) pronounced peaks at ≈850 and ≈3130 cm⁻¹ for the CH out-of-plane bending and CH stretching, respectively, whose intensity progressively increases with the system size and thus with the number of CH bonds, in agreement with the expected results for polycyclic aromatic hydrocarbons.⁶¹

3.2 The (poly)radicaloid character of the systems

A simple yet effective estimate of the radicaloid character is obtained by the FT-DFT method, providing the energy distribution of electrons around the Fermi level, which can also be represented as a spatial density:with f_i being the fractional occupation number (0 < f_i < 2) corresponding to that subset of orbitals ({φ_i}_U) holding a fractional occupation. The integration of ρ_U(r) directly leads to N_U, a global measure of the number of strongly correlated electrons. The N_U values for the (singlet) ground-state of n-rhombenes are 0.19, 0.81, 1.85, 3.14, and 4.60 for 2-rhombene, 3-rhombene, 4-rhombene, 5-rhombene, and 6-rhombene, respectively, increasing significantly with the system size as it does also the length of the zigzag edges (see Fig. 2). This relationship can be easily understood by the real-space distribution of ρ_U(r), see the ESI,† which is strongly localized at those C atoms belonging to the zigzag edges, taking also into account that ρ_U(r) (and thus N_U) is a size-extensive magnitude. This topology of ρ_U(r) agrees with the distribution found before for other graphene nanoribbons⁶² or linear and cyclic oligoacenes.⁶³

Fig. 2 Evolution of the N_U values of n-rhombenes, as calculated by the FT-TPSS/def2-TZVP method, as a function of the zigzag length (in Å) measured as the sum of the (2n) CC bond lengths along one of the four (symmetrical) edges of the systems.

The evolution of the ground-state fractional occupation numbers through the n-rhombenes family is presented in Fig. 3, which includes now a large manifold of orbitals to better appreciate how the (poly)radicaloid nature evolves with the system size. It is immediately apparent that the number of orbitals with fractional occupation increases with the system size, with the ground-state of smaller n-rhombenes (n ≤ 3) displaying a vanishing or small diradicaloid character, consistent with the information available in the literature for e.g. pyrene.⁶⁴ On the other hand, the larger n-rhombenes tackled here (i.e., 4-rhombene, 5-rhombene, and 6-rhombene) exhibited a more marked diradicaloid character, with a larger number of orbitals with occupation f_i ≈ 1, which is also in agreement with the larger N_U values found from ρ_U(r). Overall, the largest n-rhombenes should be considered as of a significantly (poly)radicaloid nature.

Fig. 3 Orbital occupation numbers (f_i) of n-rhombenes, as calculated by the FT-TPSS/def2-TZVP method.

Additionally, the degree of the (poly)radicaloid character of the compounds can also be estimated by the radical indexes Y_i number⁶⁵ (i = 0, 1, 2) generally obtained as Y_i = f_LUNO+i, with f_LUNO being the fractional occupation number of the Lowest Unoccupied Natural Orbital. For a N-electron system, the Highest Occupied Natural Orbital (HONO) is defined as the (N/2)th orbital while the Lowest Unoccupied Natural Orbital (LUNO) is defined as the (N/2 + 1)th orbital, with the rest of orbitals defined correspondingly. At the RAS-SF/6-31G* level, we obtain values for each of the Y₀, Y₁, and Y₂ index, which are respectively related to the di-, tetra-, and hexa-radicaloid character. Note that Y_i ≈ 1 suggests a high radical character (e.g. Y₀ = 0 for a closed-shell system and Y₀ = 1 for a diradical). Fig. 4 shows the Y_i values for the set of n-rhombenes, indicating a low radicaloid character for 2- and 3-rhombene, a moderate diradicaloid character for 4-rhombene, and a low to moderate tetraradicaloid character for the largest 5- and 6-rhombene. This conclusion is in qualitative agreement with previous and simplified topological approaches for the treatment of conjugated hydrocarbons.⁶⁶

Fig. 4 Radical index (Y_i) of n-rhombenes, as calculated by the RAS-SF/6-31G* method.

3.3 Singlet–triplet and singlet–quintet energy gap

3.3.1 Results from wavefunction methods. First of all, we will assess for 2- and 3-rhombene RAS-based methodological aspects to better bracket the accuracy of the RAS-SF results for the rest of the largest systems. The use of a large (def2-TZVP) or a cost-effective (6-31G*) basis set brings only a small difference for the target ΔE(HS − LS) values, that is ΔE_ST and ΔE_SQ energy differences: Going from def2-TZVP to 6-31G* modifies for 2-rhombene (3-rhombene) the ΔE_ST value by 0.05 (0.08) eV, while ΔE_SQ was affected by 0.14 (−0.01) eV. In this comparison, we will also include next the RAS-srPBE/6-31G* results for which, similarly, the ΔE_ST and ΔE_SQ values are only slightly affected, as can be seen in Fig. 5 summarizing the impact of these technical aspects on the results. Therefore, we will rely in the following on the cost-effective RAS-SF/6-31G* results for the largest (and more challenging) n-rhombenes as a good compromise between accuracy and computational cost.

Fig. 5 Energy difference (in eV) between the low- and high-spin states of 2- and 3-rhombene, and N_U, as calculated at various RAS-based theoretical levels.

The evolution of ΔE_ST and ΔE_SQ values with the system size is presented in Fig. 6, with both energy differences smoothly decreasing as a function of n (for completeness, the figure also includes the S₁ state). The smaller systems, 2- to 4-rhombene, keep a substantial energy difference between low-spin and high-spin (be T₁ or Q₁) states, which is fully consistent with their low radicaloid character. For the longest rhombenes, the ΔE_ST and ΔE_SQ values are considerably lower, with the T₁ state being always the lowest one. Actually, for the case of 5-rhombene, the T₁ state is energetically predicted (RAS-SF/6-31G* level) at 139 meV above the (open-shell) S₀ state, in close agreement with the experimentally determined energy difference between those states of 102 meV²⁵ (note that the surface is known to play a negligible influence to adequatly compare experimental and calculated values according to recent studies⁶⁷). For 6-rhombene, that ΔE_ST values slightly decreases to 56 meV, a value still few times larger than the Landauer limit⁶⁸ [i.e., k_BT ln(2) ≈ 18 meV, with k_B the Boltzmann constant and T the temperature] for energy dissipation at room temperature, a key magnitude for having stable spintronics devices. Note that for 6-rhombene, the ΔE_ST and the ΔE_SQ values are energetically close just separated by less than 0.2 eV.

Fig. 6 Energy difference (in eV) between the low- and high-spin states of n-rhombenes, and N_U, as calculated by the RAS-SF/6-31G* method.

The results by RAS-SF/6-31G* will also be compared for 4- to 6-rhombenes with those obtained using (Strongly-Contracted) N-electron Valence second-order Perturbation Theory or (SC-)NEVPT2,^69–71 as a sanity check given the high accuracy of the latter method for related systems,⁷² employing the same active space (10,10) and basis set than for RAS-SF. Interestingly, the T₁ state for 5-rhombene (6-rhombene) is also found lying 139 (75) meV above the S₀ state, in perfect agreement with the RAS-SF results. Finally, other results from literature are compared here for the sake of completeness: (i) NEVPT2/STO-3G calculations^73,74 led to negligible ΔE_ST values of 16 and 3 meV for 5- and 6-rhombene, respectively, and too low ΔE_SQ values too, thus indicating the need of more realistic basis sets; and (ii) TAO-LDA calculations⁶⁴ reported ΔE_ST values for 4- to 6-rhombene of 440, 204, and 116 meV, closely following the trend of the values calculated here although slightly overestimating the experimental result for 5-rhombene.

3.3.2 Assessment of DFT methods. A simple yet useful estimate of the performance of the PBE-based functionals (all calculations are done with the def2-TZVP basis set unless otherwise noticed) will be the comparison of the results with the experimentally available HOMO–LUMO gap (0.73 eV) of 4-rhombene. Due to the fact that the ground-state of 4-rhombene is experimentally predicted to be a closed-shell system, the closest estimate is provided by PBE (0.66 eV) followed by PBE0 (1.36 eV) and PBEHH (2.13 eV), thus indicating that a relatively low c_x weight [see eqn (3)] is favoured in this case. Actually, a calculation with an ad hoc PBEh hybrid functional⁷⁵ (for which c_x = 1/10) leads to an intermediate value of 0.93 eV between those calculated by PBE and PBE0, showing how (linearly) dependent the HOMO–LUMO gap with respect to c_x is. This finding prompts us to discard PBEHH (the functional with the highest c_x weight) for the rest of the calculations shown here.

The FT-DFT method will be first applied for a fast estimate of the ΔE_ST (ΔE_SQ) energy differences, with calculated values (all in eV) of 1.864, 0.869, 0.430, 0.255, and 0.174 (5.239, 3.115, 1.792, 1.095, 0.741) and thus decreasing with the system size, in agreement with marked radicaloid character of the largest systems and with the T₁ state lower in energy with respect to the Q₁ state, as also obtained by the RAS-SF calculations. Since the relative stabilization of the HS with respect to the LS state is known to be strongly affected by the functional choice,^76–78 or more specifically by the fraction of exact-exchange,⁷⁹ we will next assess the performance of the PBE and PBE0 models, with the results shown in Fig. 7. When PBE is applied, the ground-state open-shell (singlet) solution collapsed to the closed-shell form for compounds 2- to 4-rhombene, but not for 5- and 6-rhombene. Thus, for 5- and 6-rhombene, the closed-shell singlet lies at 40 and 176 meV above in energy, respectively, with respect to the open-shell singlet state. This functional also provides for 5-rhombene the T₁ state separated by 174 meV from the open-shell state, an energy gap slightly overestimated with respect to both experimental (102 meV) and RAS-SF (139 meV) values. The Q₁ state is found much higher at 846 meV, close to the value obtained at the RAS-SF level (868 meV). In the case of 6-rhombene, the triplet lies above the singlet by 235 meV, which roughly represents a 3-fold overestimation of the RAS-SF result and evolves contrarily to the trend provided by RAS-SF and NEVPT2 results since ΔE_ST is found lower for 6-rhombene than for 5-rhombene. Note also that the ΔE_ST and ΔE_SQ energy differences are known to decrease for increasing radicaloid nature, as it has been previously shown for e.g. linear and cyclic acenes.⁶³

Fig. 7 Energy difference (in eV) between the low- and high-spin states of n-rhombenes, as calculated by DFT/def2-TZVP methods and spin-decontaminated as commented in the text.

On the other hand, the PBE0 model predicted an open-shell ground-state for 4-rhombene, contrarily to experimental results, with the corresponding closed-shell state at 170 meV higher in energy. For the case of 5- and 6-rhombene, the closed-shell singlet lies as high as 630 and 1214 meV above in energy, respectively, with respect to the open-shell singlet. Additionally, the large spin contamination (see Table 1) is identified as the responsible factor for predicting the Q₁ state significantly more stable than the T₁ for 5-rhombene, as already anticipated by the RAS-SF results since the manifold of states was found very close in energy for the largest systems: ΔE_ST and ΔE_SQ are predicted to be 255 and 654 meV before applying the spin correction given by eqn (8), which modifies the values to 1403 and 882 meV, respectively. A similar situation is found for 6-rhombene, with values for ΔE_ST and ΔE_SQ of 1181 and 474 meV, respectively, once the spin correction is applied, see eqn (7) and (8). Finally, the evolution with the system size of the ΔE_ST and ΔE_SQ calculated values is presented in Fig. 7 taking into account the spin correction, with all the individual values for n-rhombenes at both PBE and PBE0 methods given as part of the ESI.†

Table 1 〈Ŝ²〉 calculated values for S₁, T₁, and Q₁ states of n-rhombenes, as calculated by DFT/def2-TZVP methods

[n]	PBE			PBE0
	S1	T1	Q1	S1	T1	Q1
2	0.000	2.014	6.005	0.000	2.045	6.009
3	0.000	2.017	6.030	0.000	2.072	6.111
4	0.000	2.022	6.037	0.970	2.102	6.162
5	0.709	2.026	6.046	1.767	2.130	6.217
6	1.368	2.029	6.055	2.540	3.213	6.271

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3.4 Charged systems

We also investigated the energy associated to charge the systems, known as adiabatic ionization potential (AIP) or electron affinity (AEA) and calculated as energy differences between charged and neutral systems all at their respective optimized geometries: AIP = E_M˙^₊ − E_M and AEA = E_M − E_M˙^₋, with E_M being the energy of the neutral molecule and E_M˙^₊ (E_M˙^₋) the energy of the positively (negatively) charged molecule. The quasiparticle energy gap (QEG) is readily obtained as QEG = AIP − AEA. The corresponding vertical magnitudes are VIP (vertical ionization potential), VEA (vertical electron affinity), and VQEG (vertical quasiparticle energy gap) and are obtained at the ground-state geometries. We will employ the PBE0 and the PBE0-DH models to bracket the dependence of the values with the functional choice. Experimentally available AIP⁸⁰ and AEA⁸¹ values for 2-rhombene (7.426 and 0.406 eV, respectively) will be used as test case to assess the accuracy of the DFT results. Whereas PBE0/def2-SVP gives values of 7.175 and 0.368 eV, respectively, the use of the larger def2-TZVP basis set produced values of 7.148 and 0.394 eV, showing a not very marked dependence of the results on the basis set size (furthermore PBE0/cc-pVTZ led to values of 7.135 and 0.356 eV, respectively). The use of PBE0-DH instead leads to values of 7.271 and 0.284 eV (def2-SVP) and 7.256 and 0.334 eV (def2-TZVP) for AIP and AEA, respectively, slightly more influenced by basis sets effects due to the inclusion of the PT2 term in eqn (4). Choosing the PBE0 functional for its good trade-off between accuracy and computational cost, and looking at the evolution of these values with the system size, see Table 2, we observe a initial decreasing values as a function of n for all the energy magnitudes collected, up to some saturation for the 4- to 6-rhombene. Additionally, the difference between adiabatic and vertical values is reduced with the system size, which is rationalized by an enhanced rigidity of the molecular backbone for the largest systems studied. For the sake of completeness, we also used the FT-DFT method to observe a qualitatively similar evolution of the values with the system size.

Table 2 Energy differences (in eV) between positively or negatively charged and neutral n-rhombenes, as calculated by PBE0/def2-TZVP

[n]	PBE0						FT-DFT
	AIP	VIP	AEA	VEA	AQEG	VQEG	AIP	VIP	AEA	VEA	AQEG	VQEG
2	7.148	7.231	0.394	0.280	6.754	6.950	6.749	6.768	0.652	0.613	6.097	6.156
3	6.076	6.127	1.559	1.502	4.517	4.625	5.834	5.842	1.638	1.621	4.197	4.221
4	5.604	5.615	2.137	2.125	3.467	3.491	5.371	5.374	2.177	2.170	3.194	3.205
5	5.679	5.726	2.166	2.101	3.513	3.625	5.113	5.115	2.497	2.493	2.616	2.622
6	5.568	5.585	2.276	2.237	3.292	3.349	4.945	4.947	2.711	2.709	2.233	2.238

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The inner reorganization energy (Λ) for holes (Λ_h˙^₊) or electrons (Λ_e˙^₋), related to charge migration and transfer between neighbouring molecules^82,83 can also be obtained (in the hopping regime) from the following set of calculations:

Λ_h(e)˙^± = [E_M˙_^±//M − E_M˙^_±] + [E_M//M˙^_± − E_M], (11)

where E_M˙_^±//M and E_M//M˙^_± are now the energies of the charged molecule at the neutral geometry and that of the neutral at the charged geometry, respectively, with E_M˙^_± being the energy of the charged structures.^84,85 We will employ the PBE0 functional also here due to its accurate results for other conjugated (semiconducting) systems.^86–88 At the PBE0/def2-TZVP level, Λ_h˙^₊ values of 157 and 103 meV are obtained for the smallest 2- and 3-rhombene members of the set of molecules, respectively, showing the typical evolution of these values with the system size (inversely proportional to the number of C atoms⁸⁹) and favourably comparing with for instance circum(oligo)acenes of increasing size, such as circumbenzene or coronene (C₂₄H₁₂) and circumtetracene (C₄₈H₁₈) with values of 127 and 57 meV, respectively, calculated before at the B3LYP/cc-pCVDZ level.⁹⁰ For the case of Λ_e˙^₋, values of 225 and 147 meV are again calculated at the PBE0/def2-TZVP level, larger than those found before for Λ_h˙^₊. The application of the PBE0-DH model led to Λ_h˙^₊ (Λ_e˙^₋) values of 182 and 114 meV (250 and 153 meV), thus confirming the trend found before with PBE0. Note that the reorganization energies of larger systems, following the expected evolution with the system size, would preclude the use of eqn (11) due to the gradual transition between the incoherent (hopping) and coherent (ballistic) transport regimes.⁹¹ As a matter of illustration, for 6-rhombene values of Λ_h⁺ and Λ_e⁻ as low as 36 and 58 meV, respectively, are found.

4. Conclusions

We have systematically studied the electronic structure of increasingly larger rhombus-shaped molecules, or n-rhombenes, ranging from 2-rhombene (C₁₆H₁₀) to 6-rhombene (C₉₆H₂₆), due to the markedly increasing (poly)radicaloid character with the system size. The rationalization of that character is extensively done through the number and spatial extension of unpaired electrons, the fractional occupation of molecular orbitals, the frontier molecular orbital energy gap, and the total energy difference between singlet–triplet and singlet–quintet electronic states. The challenging electronic structure of these systems has prompted the use of multiconfigurational methods, such as RAS-srDFT, RAS-SF, or NEVPT2, leading to: (i) a qualitative agreement between available experimental and theoretical results for the system size at which the transition from a closed-shell to an open-shell ground-state happens, and (ii) a quantitative and close agreement between theoretical and experimental results for the singlet–triplet energy difference for 5-rhombene. This agreement facilitates us to rely on the calculated singlet–triplet energy difference for 6-rhombene (not yet synthesized) which is lower than for 5-rhombene but still larger than the Landauer limit.

The quality of those wavefunction based results also served us to assess different DFT flavors (semi-local, hybrid, and double-hybrid functionals) for estimating the energy difference between different spin states (profoundly influenced by the spin contamination). For that situation, the spin contamination is alleviated by the use of the Yamaguchi's correction, which led to qualitatively correct results for 5-rhombene but not for 6-rhombene, disagreeing thus with results from wavefunction methods. The study of charged (monocationic and monoanionic) systems showed a decreasing (increasing) evolution of ionization potentials (electron affinity) with the system size, closing thus the quasiparticle energy gap as it was also expected due to their increasing (poly)radicaloid nature. The reorganization energy for hole or electron charge carrier transport in the hopping regime, for the smallest members of the family, show reasonably low values for those molecules to act as organic molecular semiconductors. On the other hand, the low-cost FT-DFT method can be considered as an alternative for dealing with (poly)radicaloids systems, providing fast and qualitatively correct results.

Data availability

The data that supports the findings of this study are available within the article [and its ESI†] or are available from the corresponding authors upon reasonable request.

Author contributions

M. E. S.-S. and R. B.-C. contributed equally to this work.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by projects PID2019-106114GB-I00 (“Ministerio de Ciencia e Innovación”), AICO/2021/093 and PROMETEO/2021/017 (“Generalitat Valenciana”).

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Footnote

† Electronic supplementary information (ESI) available: (i) IR spectra of n-rhombenes; (ii) plots of the ρ_U(r) spatial distribution of n-rhombenes; (iii) N_U and Y_i values calculated at the RAS-SF/6-31G* level for n-rhombenes; (iv) HOMO and LUMO orbital energies as calculated at various DFT/def2-TZVP levels; (v) vertical singlet–triplet and singlet–quintet energies calculated at the RAS-SF/6-31G* and DFT/def2-TZVP levels for n-rhombenes; (vi) 〈Ŝ2〉 values for singlet, triplet, and quintet states at the DFT level; (vii) plots of the (triplet and open-shell singlet) spin density spatial distribution of n-rhombenes; (viii) cartesian coordinates of n-rhombenes optimized at the B97-3c level. See DOI: https://doi.org/10.1039/d3cp01103h

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