Electronic Properties of Cyclacenes from TAO-DFT

Owing to the presence of strong static correlation effects, accurate prediction of the electronic properties (e.g., the singlet-triplet energy gaps, vertical ionization potentials, vertical electron affinities, fundamental gaps, symmetrized von Neumann entropy, active orbital occupation numbers, and real-space representation of active orbitals) of cyclacenes with n fused benzene rings (n = 4–100) has posed a great challenge to traditional electronic structure methods. To meet the challenge, we study these properties using our newly developed thermally-assisted-occupation density functional theory (TAO-DFT), a very efficient method for the study of large systems with strong static correlation effects. Besides, to examine the role of cyclic topology, the electronic properties of cyclacenes are also compared with those of acenes. Similar to acenes, the ground states of cyclacenes are singlets for all the cases studied. In contrast to acenes, the electronic properties of cyclacenes, however, exhibit oscillatory behavior (for n ≤ 30) in the approach to the corresponding properties of acenes with increasing number of benzene rings. On the basis of the calculated orbitals and their occupation numbers, the larger cyclacenes are shown to exhibit increasing polyradical character in their ground states, with the active orbitals being mainly localized at the peripheral carbon atoms.

cryptoannulenic effect) is expected to be responsible for the properties of n-cyclacene. Besides, the structure of n-cyclacene can also be regarded as two fused trannulenes (i.e., circular, all-trans cyclic polyene ribbons) 19,21 . From the bond length analysis of n-cyclacene, there is bond length alternation in the benzene ring, and the aromaticity is reduced due to the structural strain, which can hence be responsible for the properties of n-cyclacene.
Even though there has been a keen interest in n-cyclacenes, the studies of their electronic properties are scarce. While n-cyclacene may be synthesized via an intramolecular cyclization of n-acene (a chain-like molecule with n linearly fused benzene rings, e.g., see Fig. 1 of ref. 23), the synthetic procedure has been very challenging, and has not succeeded in producing pure n-cyclacene 23,24,26 , possibly due to its highly strained structure and highly reactive nature 12,26 . As the stabilities of annulated polycyclic saturated hydrocarbons decrease rapidly with the number of fused benzene rings 16 , the synthesis of larger n-cyclacenes should be even more difficult.
To circumvent the formidable computational expense of high-level ab initio multi-reference methods, we have recently developed thermally-assisted-occupation density functional theory (TAO-DFT) 54,55 , a very efficient electronic structure method for studying the properties of large ground-state systems (e.g., containing up to a few thousand electrons) with strong static correlation effects [56][57][58] . In contrast to KS-DFT, TAO-DFT is a density functional theory with fractional orbital occupations, wherein strong static correlation is explicitly described by the entropy contribution (see Eq. (26) of ref. 54), a function of the fictitious temperature and orbital occupation numbers. Note that the entropy contribution is completely missing in KS-DFT. Recently, we have studied the electronic properties of zigzag graphene nanoribbons (ZGNRs) using TAO-DFT 56 . The ground states of ZGNRs are found to be singlets for all the widths and lengths studied. The longer ZGNRs should possess increasing polyradical character in their ground states, with the active orbitals being mainly localized at the zigzag edges. Our results are in good agreement with the available experimental and highly accurate ab initio data. Besides, on the basis of our TAO-DFT calculations, the active orbital occupation numbers for the ground states of ZGNRs should exhibit a curve crossing behavior in the approach to unity (singly occupied) with increasing ribbon length. Very recently, the curve crossing behavior has been confirmed by highly accurate ab initio multi-reference methods 53 ! TAO-DFT has similar computational cost as KS-DFT for single-point energy and analytical nuclear gradient calculations, and reduces to KS-DFT in the absence of strong static correlation effects. Besides, existing XC density functionals in KS-DFT may also be adopted in TAO-DFT. Relative to high-level ab initio multi-reference methods, TAO-DFT is computationally efficient, and hence very powerful for the study of large polyradical systems. In addition, the orbital occupation numbers from TAO-DFT, which are intended to simulate the natural orbital occupation numbers (NOONs) [i.e., the eigenvalues of one-electron reduced density matrix] 59 , can be very useful for assessing the possible polyradical character of systems. Recent studies have demonstrated that the orbital occupation numbers from TAO-DFT are qualitatively similar to the NOONs from high-level ab initio multi-reference methods, giving promise for applying TAO-DFT to large polyradical systems 53,54,56,57 .
Due to its computational efficiency and reasonable accuracy for large systems with strong static correlation effects, in this work, TAO-DFT is adopted to study the electronic properties of n-cyclacenes (n = 4-100). As n-cyclacenes have not been successfully synthesized, no experimental data are currently available for comparison. Therefore, our results are compared with the available high-level ab initio data as well as those obtained from various XC density functionals in KS-DFT. In addition, as n-cyclacene can be considered as an interconnection of n-acene, the electronic properties of n-cyclacene are also compared with those of n-acene to assess the role of cyclic topology.

Computational Details
All calculations are performed with a development version of Q-Chem 4.0 60 , using the 6-31 G(d) basis set with the fine grid EML (75, 302), consisting of 75 Euler-Maclaurin radial grid points and 302 Lebedev angular grid points. Results are calculated using KS-LDA (i.e., KS-DFT with the LDA XC density functional 29,30 ) and TAO-LDA (i.e., TAO-DFT with the LDA XC density functional and the LDA θ-dependent density functional θ E LDA (see Eq. (41) of ref. 54) with the fictitious temperature θ = 7 mhartree (as defined in ref. 54). Note that KS-LDA is simply TAO-LDA with θ = 0, and hence it is important to assess the performance of KS-LDA here to assess the significance of TAO-LDA.
The ground state of n-cyclacene/n-acene (n = 4-100) is obtained by performing spin-unrestricted KS-LDA and TAO-LDA calculations for the lowest singlet and triplet energies of n-cyclacene/n-acene on the respective geometries that were fully optimized at the same level of theory. The singlet-triplet energy (ST) gap of n-cyclacene/n-acene is calculated as (E T − E S ), the energy difference between the lowest triplet (T) and singlet (S) states of n-cyclacene/n-acene.

Results and Discussion
Singlet-Triplet Energy Gap. Figure 2 shows the ST gap of n-cyclacene as a function of the number of benzene rings, calculated using spin-unrestricted KS-LDA and TAO-LDA. The results are compared with the available data 25 , calculated using the complete-active-space second-order perturbation theory (CASPT2) 46 (a high-level ab initio multi-reference method) as well as the M06L functional 32 (a popular semilocal XC density functional) and the B3LYP functional 33,34 (a popular hybrid XC density functional) in KS-DFT (see Table S1 in Supplementary Information).
As can be seen, the anticipated even-odd oscillations in the ST gaps may be attributed to the cryptoannulenic effects of n-cyclacenes 17,18,20,23 . However, the amplitudes of the even-odd oscillations are considerably larger for KS-DFT with the XC functionals, which are closely related to the degree of spin contamination (as discussed in ref. 25). In general, the larger fraction of Hartree-Fock (HF) exchange adopted in the XC functional in KS-DFT, the higher the degree of spin contamination for systems with strong static correlation effects. For example, the ST gap obtained with KS-B3LYP is unexpectedly large at n = 10, due to the high degree of spin contamination 25 .
On the other hand, as commented in ref. 25, the ST gaps obtained with CASPT2 are rather sensitive to the choice of active space. Since the complete π-valence space was not selected as the active space (due to the prohibitively high cost), the CASPT2 results here should be taken with caution. Recent studies have shown that a sufficiently large active space should be adopted in high-level ab initio multi-reference calculations 47,49,53 for accurate prediction of the electronic properties of systems with strong static correlation effects, which can, however, be prohibitively expensive for large systems. Note that the ST gap obtained with CASPT2 unexpectedly increases at n = 12, possibly due to the insufficiently large active space adopted in the calculations 25 .
To assess the role of cyclic topology, Figs 3 and 4 show the ST gap of n-cyclacene/n-acene as a function of the number of benzene rings, calculated with spin-unrestricted TAO-LDA (see Table S2 in Supplementary Information). Similar to n-acenes, the ground states of n-cyclacenes remain singlets for all the cases investigated. In contrast to n-acene, the ST gap of n-cyclacene, however, displays oscillatory behavior for small n, and the oscillation vanishes gradually with increasing n. For small n, n-cyclacene with even-number benzene rings exhibits a larger ST gap (i.e., greater stability) than that with odd-number benzene rings. For sufficiently large n (n > 30), the ST gap of n-cyclacene converges monotonically from below to the ST gap of n-acene (which monotonically decreases with increasing n). At the level of TAO-LDA, the ST gaps of the largest n-cyclacene and n-acene studied (i.e., n = 100) Due to the symmetry constraint, the spin-restricted and spin-unrestricted energies for the lowest singlet state of n-cyclacene/n-acene, calculated using the exact theory, should be identical 50,54-56 . Recent studies have shown that KS-DFT with conventional XC density functionals cannot satisfy this condition for the larger n-cyclacene/n-acene, due to the aforementioned spin contamination 25,47,49,50,[54][55][56] . To assess the possible symmetry-breaking effects, spin-restricted TAO-LDA calculations are also performed for the lowest singlet energies on the respective optimized geometries. Within the numerical accuracy of our calculations, the spin-restricted and spin-unrestricted TAO-LDA energies for the lowest singlet state of n-cyclacene/n-acene are essentially the same (i.e., essentially no unphysical symmetry-breaking effects occur in our spin-unrestricted TAO-LDA calculations).

Vertical Ionization Potential, Vertical Electron Affinity, and Fundamental Gap.
At the lowest singlet state (i.e., the ground-state) geometry of n-cyclacene/n-acene (containing N electrons), TAO-LDA is adopted to calculate the vertical ionization potential using multiple energy-difference methods, with E N being the total energy of the N-electron system.
With increasing number of benzene rings in n-cyclacene, IP v oscillatorily decreases (see Fig. 5), EA v oscillatorily increases (see Fig. 6), and hence E g oscillatorily decreases (see Fig. 7). However, these oscillations are damped and eventually disappear with increasing n (see Tables S3 to S5 in Supplementary Information). For sufficiently large n (n > 30), the IP v and E g values of n-cyclacene converge monotonically from above to those of n-acene (which monotonically decrease with increasing n), while the EA v value of n-cyclacene converges monotonically from below to that of n-acene (which monotonically increases with increasing n). Note also that the E g value of n-cyclacene (n = 13-54) is within the most interesting range (1 to 3 eV), giving promise for applications of n-cyclacenes in nanophotonics.      for the lowest singlet state of n-cyclacene/n-acene as a function of the number of benzene rings, using TAO-LDA.
Here, f i the occupation number of the i th orbital obtained with TAO-LDA, which ranges from 0 to 1, is approximately the same as the occupation number of the i th natural orbital 53-58 . For a system without strong static correlation ({ f i } are close to either 0 or 1), S vN provides insignificant contributions, while for a system with strong static correlation ({ f i } are fractional for active orbitals and are close to either 0 or 1 for others), S vN increases with the number of active orbitals. As shown in Fig. 8, the S vN value of n-cyclacene oscillatorily increases with increasing number of benzene rings (see Table S6 in Supplementary Information). Nonetheless, the oscillation is damped and eventually disappears with the increase of n. For sufficiently large n (n > 30), the S vN value of n-cyclacene converges monotonically from above to that of n-acene (which monotonically increases with increasing n). Therefore, similar to n-acenes 47,[49][50][51][53][54][55][56][57][58] , the larger n-cyclacenes should possess increasing polyradical character.
Active Orbital Occupation Numbers. To illustrate the causes of the increase of S vN with n, we plot the active orbital occupation numbers for the lowest singlet state of n-cyclacene as a function of the number of benzene rings, calculated using TAO-LDA. Here, the highest occupied molecular orbital (HOMO) is the (N/2) th orbital, and the lowest unoccupied molecular orbital (LUMO) is the (N/2 + 1) th orbital, where N is the number of electrons in n-cyclacene. For brevity, HOMO, HOMO − 1, … , and HOMO − 15, are denoted as H, H − 1, … , and H − 15, respectively, while LUMO, LUMO + 1, … , and LUMO + 15, are denoted as L, L + 1, … , and L + 15, respectively.
As presented in Fig. 9, the number of fractionally occupied orbitals increases with increasing cyclacene size, clearly indicating that the polyradical character of n-cyclacene indeed increases with the cyclacene size. Similar to the previously discussed properties, the active orbital occupation numbers of n-cyclacene also exhibit oscillatory behavior, showing wave-packet oscillations.  Real-Space Representation of Active Orbitals. For the lowest singlet states of some representative n-cyclacenes (n = 4-7), we explore the real-space representation of active orbitals (e.g., HOMOs and LUMOs), obtained with TAO-LDA. Similar to previous findings for n-acenes 47,49,50,56 , the HOMOs and LUMOs of n-cyclacenes are mainly localized at the peripheral carbon atoms (see Fig. 10).

Conclusions
In conclusion, we have studied the electronic properties of n-cyclacenes (n = 4-100), including the ST gaps, vertical ionization potentials, vertical electron affinities, fundamental gaps, symmetrized von Neumann entropy, active orbital occupation numbers, and real-space representation of active orbitals, using our newly developed TAO-DFT, a very efficient electronic structure method for the study of large systems with strong static correlation effects. To assess the effects of cyclic nature, the electronic properties of n-cyclacenes have also been compared with those of n-acenes. Similar to n-acenes, the ground states of n-cyclacenes are singlets for all the cases investigated. In contrast to n-acenes, the electronic properties of n-cyclacenes, however, display oscillatory behavior for small n (n ≤ 30) in the approach to the corresponding properties of n-acenes with increasing number of benzene rings, which to the best of our knowledge have never been addressed in the literature. The oscillatory behavior may be related to the cryptoannulenic effect and structural strain of n-cyclacene, which have been shown to be important for small n, and unimportant for sufficiently large n. On the basis of several measures (e.g., the smaller ST gap, the smaller E g , and the larger S vN ), for small n, n-cyclacene with odd-number benzene rings should possess stronger radical character than that with even-number benzene rings. In addition, based on the calculated orbitals and their occupation numbers, the larger n-cyclacenes are expected to possess increasing polyradical character in their ground states, where the active orbitals are mainly localized at the peripheral carbon atoms.
Since TAO-DFT is computationally efficient, it appears to be a promising method for studying the electronic properties of large systems with strong static correlation effects. Nevertheless, as with all approximate electronic structure methods, a few limitations remain. Relative to the exact full configuration interaction (FCI) method 61 , TAO-LDA (with θ = 7 mhartree) is not variationally correct (i.e., overcorrelation can occur), and hence, the orbital occupation numbers from TAO-LDA may not be the same as the NOONs from the FCI method. To assess the accuracy of our TAO-LDA results, as the computational cost of the FCI method is prohibitive, the electronic properties of n-cyclacenes from relatively affordable ab initio multi-reference methods are called for. The orbital occupation numbers are given in parentheses.