Thermochemistry of the Smallest Hyperbolic Paraboloid Hydrocarbon: A High-Level Quantum Chemical Perspective

: [5.5.5.5]hexaene is a [12]annulene ring with a symmetrically bound carbon atom in its center. This is the smallest hydrocarbon with a hyperbolic paraboloid shape. [5.5.5.5]hexaene and related hydrocarbons are important building blocks in organic and materials chemistry. For example, penta-graphene—a puckered 2D allotrope of carbon—is comprised of similar repeating subunits. Here, we investigate the thermochemical and kinetic properties of [5.5.5.5]hexaene at the CCSD(T) level by means of the G4 thermochemical protocol. We ﬁnd that this system is energetically stable relative to its isomeric forms. For example, isomers containing a phenyl ring with one or more acetylenic side chains are higher in energy by ∆ H 298 = 17.5–51.4 kJ mol − 1 . [5.5.5.5]hexaene can undergo skeletal inversion via a completely planar transition structure; however, the activation energy for this process is ∆ H ‡298 = 249.2 kJ mol − 1 at the G4 level. This demonstrates the high conﬁgurational stability of [5.5.5.5]hexaene towards skeletal inversion. [5.5.5.5]hexaene can also undergo a π -bond shift reaction which proceeds via a relatively low-lying transition structure with an activation energy of ∆ H ‡298 = 67.6 kJ mol − 1 . Therefore, this process is expected to proceed rapidly at room temperature.


Introduction
Carbon atoms that are surrounded by an annulene ring have attracted considerable attention over the past 50 years [1][2][3][4][5]. A [12]annulene with a symmetrically bound carbon in its center is of particular importance since it adopts a hyperbolic paraboloidal shape due to a central sp 3 carbon surrounded by 12 sp 2 carbons. Scheme 1 illustrates the hyperbolic paraboloidal structure of the [12]annulene- [5.5.5.5]hexaene (1). In this structure, the central sp 3 carbon is located at the saddle point of the surface created by the four adjacent pentagons. It is well known that hyperbolic paraboloid surfaces of the equation z = axy possess rigidity in the x and y directions [6,7]. Indeed, this may explain the high rigidity of penta-graphene-a two-dimensional allotrope of carbon. Penta-graphene, which consists of repeating hyperbolic paraboloid units, each comprising four pentagon rings sharing a vertex, has been suggested to be more rigid than graphene [8,9].
Computational quantum chemistry is a branch of chemistry that uses computational simulations to study the chemical properties of molecules and materials [10,11]. Computational quantum chemistry provides means for the rational design and development of new molecules and materials with tailored chemical properties. Due to significant advances in quantum theory and supercomputers, computational simulations are capable of unprecedented predictive accuracy. The present work focuses on the structural, thermodynamic, and kinetic properties of [5.5.5.5]hexaene. In particular, we use high-level composite ab initio methods [12,13] to show that [5.5.5.5]hexaene is (i) configurationally stable with respect to structural inversion, (ii) energetically stable relative to its isomeric forms, and (iii) has a relatively low lying π-bond shift transition state. stable with respect to structural inversion, (ii) energetically stable relative to its isomeric forms, and (iii) has a relatively low lying π-bond shift transition state. Scheme 1. (a) Skeletal structure of a hyperbolic paraboloidal shape created by four pentagon rings sharing a common vertex, (b) the optimized structure of [5.5.5.5] hexaene, and (c) a 3D representation of the hyperbolic paraboloidal structure.

Computational Details
In order to obtain accurate thermochemical and kinetic properties for the hydrocarbons considered in the present work, calculations have been carried out using the highlevel, ab initio, G4 and G4(MP2) procedures [14][15][16]. These procedures combine coupledcluster with single, double, and perturbative triple excitation (CCSD(T)) calculations as well as second-and fourth-order Møller-Plesset perturbation theories to approximate the CCSD(T) energy in conjunction with a triple-ζ-quality basis set (CCSD(T)/TZ). Both theories have been found to provide thermochemical and kinetic properties for hydrocarbons with chemical accuracy (arbitrarily defined as 1 kcal mol −1 = 4.2 kJ mol −1 ) [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. G4 theory is computationally more demanding and in most cases more accurate than G4(MP2) theory; therefore, the main text reports the G4 results whilst the G4(MP2) results are provided as Supporting Information. The geometries of all structures have been obtained at the B3LYP/6-31G(2df,p) level of theory [28][29][30] as prescribed in the G4 and G4(MP2) protocols. Harmonic vibrational frequencies have been calculated at the same level of theory to confirm that all stationary points are equilibrium structures (i.e., with all real frequencies) or transition structures (i.e., with one imaginary frequency). The connectivity of the transition structures were confirmed by performing intrinsic reaction coordinate calculations [31,32]. Zero-point vibrational energies and enthalpic temperature corrections have been obtained from the harmonic frequencies and scaled by appropriate scaling factors as recommended in G4 and G4(MP2) theories [15,16]. All DFT and ab initio calculations were performed using the Gaussian 09 program suite [33].

Results and Discussion
Scheme 1. (a) Skeletal structure of a hyperbolic paraboloidal shape created by four pentagon rings sharing a common vertex, (b) the optimized structure of [5.5.5.5] hexaene, and (c) a 3D representation of the hyperbolic paraboloidal structure.

Computational Details
In order to obtain accurate thermochemical and kinetic properties for the hydrocarbons considered in the present work, calculations have been carried out using the high-level, ab initio, G4 and G4(MP2) procedures [14][15][16]. These procedures combine coupled-cluster with single, double, and perturbative triple excitation (CCSD(T)) calculations as well as second-and fourth-order Møller-Plesset perturbation theories to approximate the CCSD(T) energy in conjunction with a triple-ζ-quality basis set (CCSD(T)/TZ). Both theories have been found to provide thermochemical and kinetic properties for hydrocarbons with chemical accuracy (arbitrarily defined as 1 kcal mol −1 = 4.2 kJ mol −1 ) [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. G4 theory is computationally more demanding and in most cases more accurate than G4(MP2) theory; therefore, the main text reports the G4 results whilst the G4(MP2) results are provided as Supporting Information. The geometries of all structures have been obtained at the B3LYP/6-31G(2df,p) level of theory [28][29][30] as prescribed in the G4 and G4(MP2) protocols. Harmonic vibrational frequencies have been calculated at the same level of theory to confirm that all stationary points are equilibrium structures (i.e., with all real frequencies) or transition structures (i.e., with one imaginary frequency). The connectivity of the transition structures were confirmed by performing intrinsic reaction coordinate calculations [31,32]. Zero-point vibrational energies and enthalpic temperature corrections have been obtained from the harmonic frequencies and scaled by appropriate scaling factors as recommended in G4 and G4(MP2) theories [15,16]. All DFT and ab initio calculations were performed using the Gaussian 09 program suite [33].
In the present work, we examine the relative stability of the C 13 H 8 isomers and show that [5.5.5.5]hexaene (1) is the energetically most stable isomer within this space of highly unsaturated structures. This stability may be attributed to the conjugated [12]annulene ring. We also show that, with the exception of isomer 13, which contains a highly strained cyclobutene ring, all of the isomers that contain a phenyl ring are relatively stable with isomerization energies of ∆H 298 = 17.5-52.8 kJ mol −1 relative to isomer 1. For example, isomers 2 and 3 are associated with isomerization energies of 17.5 and 24.3 kJ mol −1 , respectively, relative to isomer 1.
C 2023, 9, x FOR PEER REVIEW 3 of 9 [5.5.5.5]hexaene (1, Scheme 1) has the molecular formula of C13H8, Scheme 2 depicts key structural C13H8 isomers 2-14 that have been identified using the ChemSpider database [51]. [5.5.5.5]hexaene and the C13H8 isomers in Scheme 2 are, by definition, highly unsaturated. As such, most of the C13H8 isomers are polyynes (isomers 2-12, Scheme 2), and many isomers include a phenyl ring (isomers 2-10 and 13, Scheme 2). Isomer 1 does not contain a phenyl ring; however, it contains a nonplanar conjugated π-system. Therefore, [5.5.5.5]hexaene is expected to be relatively stable compared to the isomers in Scheme 2. Yet, to the best of our knowledge, [5.5.5.5]hexaene (1) has not been synthesized; however, several of the C13H8 isomers are synthetically accessible, for example, isomers 2 and 3 [52][53][54]. In the present work, we examine the relative stability of the C13H8 isomers and show that [5.5.5.5]hexaene (1) is the energetically most stable isomer within this space of highly unsaturated structures. This stability may be attributed to the conjugated [12]annulene ring. We also show that, with the exception of isomer 13, which contains a highly strained cyclobutene ring, all of the isomers that contain a phenyl ring are relatively stable with isomerization energies of ∆H298 = 17.5-52.8 kJ mol −1 relative to isomer 1. For example, isomers 2 and 3 are associated with isomerization energies of 17.5 and 24.3 kJ mol −1 , respectively, relative to isomer 1. Scheme 2. Illustration of key C13H8 isomers taken from the ChemSpider database. Table 1 gives the G4 isomerization energies on the electronic (∆Ee), enthalpic at 0 K (∆H0), and enthalpic at 298 K (∆H298) potential energy surfaces (PESs). Remarkably, our G4 calculations show that 1 is the energetically most stable isomer on the ∆Ee, ∆H0, and ∆H298 PESs. From here onwards, we will focus on the ∆H298 values; however, we note that the same trends are observed on the ∆Ee and ∆H0 PESs. We also note that there is a reasonably Scheme 2. Illustration of key C 13 H 8 isomers taken from the ChemSpider database. Table 1 gives the G4 isomerization energies on the electronic (∆E e ), enthalpic at 0 K (∆H 0 ), and enthalpic at 298 K (∆H 298 ) potential energy surfaces (PESs). Remarkably, our G4 calculations show that 1 is the energetically most stable isomer on the ∆E e , ∆H 0 , and ∆H 298 PESs. From here onwards, we will focus on the ∆H 298 values; however, we note that the same trends are observed on the ∆E e and ∆H 0 PESs. We also note that there is a reasonably good agreement between the G4 and G4(MP2) values. For example, the largest deviation of 12.7 kJ mol −1 between the two theories is obtained for isomer 14. All the G4(MP2) isomerization energies are given in Table S1 of the Supporting Information. All the isomers 2-10 contain an aromatic phenyl ring with one (or more) linear acetylenic carbon chain. Isomer 2, which contains a single (−C≡C−) 3 chain terminated with a methyl group, is less stable than [5.5.5.5]hexaene (1) by 17.5 kJ mol −1 . Moving the methyl group to the Ph ring in the ortho, meta, and para positions relative to the acetylenic chain (isomers 3-5) destabilizes the isomers. In particular, we obtain isomerization energies of 24.3 (3), 28.0 (5), and 28.3 (4) kJ mol −1 relative to isomer 1. Isomers 6-9 all involve three acetylene substituents and one methyl substituent. These isomers, along with isomer 10, are energetically less stable than isomer 1 by 46.8-52.8 kJ mol −1 . All the other isomers C 2023, 9, 41 4 of 9 (11)(12)(13)(14) involve long acetylenic or cumulenic chains or a highly strained cyclobutene ring. These isomers are highly energetic and lie as much as 277.3-488.2 kJ mol −1 above isomer 1. Table 1. CCSD(T) energies relative to isomer 1 (in kJ mol −1 ) obtained from G4 theory for the fourteen C 13 H 8 isomers shown in Schemes 1 and 2 and transition structures in Figure 1. a transition structure of 1 for skeletal inversion (see Figure 1b and Scheme 3a). b transition structure of 1 for π-bond shift (see Figure 1c and Scheme 3b).

Struct
Having established that [5.5.5.5]hexaene (1) is a relatively stable C 13 H 8 isomer, it is of interest to examine its structure in more detail. This is a symmetric molecule of D 2 symmetry. The bond lengths and angles are shown in Figure 1a. We begin by noting that even though [5.5.5.5]hexaene is highly nonplanar, the double and single bonds of the outer [12]annulene ring exhibit bond length alternation, indicating a partly delocalized π-system. The double C=C bonds are of similar lengths to those of typical C=C bonds, namely, 1.360 and 1.343 Å, and the single C-C bonds are shorter than typical single C-C bonds, namely, 1.441 and 1.468 Å (Figure 1). For comparison, the length of the conjugated double bonds in cyclopentadiene is 1.346 Å, i.e., in between the lengths of the double bonds in 1, and the length of the single bond sandwiched between the two double bonds in cyclopentadiene is 1.468 Å. The above geometrical parameters show that the central carbon in [5.5.5.5]hexaene closer to planarity than a tetrahedral carbon; it is therefore of interest to examine the re tion barrier height for the skeletal inversion in this structure. The inversion transiti structure is completely planar and has a D2h symmetry (see Figure 1b and Scheme 3a). T four bonds between the central carbon and the carbons of the [12]annulene ring are s  Table 1 for the relative G4 enthalpies at 298 K. C 2023, 9, 41 5 of 9 [5.5.5.5]hexaene and related systems have been initially proposed in the quest to form a tetracoordinate carbon with a planar configuration [1][2][3][4][55][56][57]. Indeed, the outer conjugated annulene imposes, to some extent, a planar configuration around the central sp 3 carbon. This is demonstrated by bond angles around this carbon being smaller than the ideal tetrahedral angle. The two bond angles around the central carbon are 96.4 • and 98.1 • . The planarity of the central carbon can be assessed by examining the angle between the planes of the two opposite cyclopentadiene rings, which is 56.0 • (cfr. to an angle of 90 • in a perfect tetrahedral configuration). We note that the angle between the planes of the two opposite cyclopentene rings is very similar and amounts to 56.9 • .
The above geometrical parameters show that the central carbon in [5.5.5.5]hexaene is closer to planarity than a tetrahedral carbon; it is therefore of interest to examine the reaction barrier height for the skeletal inversion in this structure. The inversion transition structure is completely planar and has a D 2h symmetry (see Figure 1b and Scheme 3a). The four bonds between the central carbon and the carbons of the [12]annulene ring are significantly elongated relative to the equilibrium structure (1, Figure 1a). All of the bonds of the [12]annulene ring are slightly shortened relative to the equilibrium structure; therefore, the bond length alternation in the [12]annulene ring is maintained. As expected, the skeletal inversion has a high barrier height of ∆H ‡ 298 = 249.2 kJ mol −1 at the G4 level, demonstrating the high configurational stability of [5.5.5.5]hexaene towards skeletal inversion. The high energy of the planar TS for the skeletal inversion may be attributed to two factors: (i) the strain energy associated with a planar structure of four fused five-membered rings sharing a central carbon atom, and (ii) the antiaromaticity of this planar TS, which involves 12 (4n) π-electrons [5,58]. It is of interest to examine whether the conjugated [12]annulene ring can undergo a π-bond shift reaction (illustrated schematically in Scheme 3). Figure 1c shows the optimized transition structure for this reaction. The geometric configuration of the central carbon atom in this transition structure is very similar to that in the equilibrium structure. The bond length alternation in this TS is significantly reduced, with the difference in length between the two types of bonds being merely 0.039 Å. The activation energy for the π-bond shift reaction is ∆H ‡ 298 = 67.6 kJ mol −1 at the G4 level. Therefore, this process is expected to proceed rapidly at room temperature.
Finally, it is useful to examine the performance of a representative selection of DFT methods across the rungs of Jacob's Ladder in order to accommodate future computational investigations of larger fragments of penta-graphene and related structures. Table  2 gives the root mean square deviations (RMSDs), mean absolute deviations (MADs), mean signed deviations (MSDs), and largest deviations (LDs) for the considered DFT methods. We begin by noting that highly unsaturated hydrocarbons have been shown to be an extremely challenging target for most DFT functionals [20,59]. Let us begin with examining the performance of the pure GGA methods. BLYP-D3BJ results in an unacceptably large RMSD of 87.7 kJ mol −1 , which is mostly attributed to the poor performance of the LYP correlation functional. Replacing the LYP functional with either the P86 or PBE It is of interest to examine whether the conjugated [12]annulene ring can undergo a π-bond shift reaction (illustrated schematically in Scheme 3). Figure 1c shows the optimized transition structure for this reaction. The geometric configuration of the central carbon atom in this transition structure is very similar to that in the equilibrium structure. The bond length alternation in this TS is significantly reduced, with the difference in length between the two types of bonds being merely 0.039 Å. The activation energy for the π-bond shift reaction is ∆H ‡ 298 = 67.6 kJ mol −1 at the G4 level. Therefore, this process is expected to proceed rapidly at room temperature.
Finally, it is useful to examine the performance of a representative selection of DFT methods across the rungs of Jacob's Ladder in order to accommodate future computational investigations of larger fragments of penta-graphene and related structures. Table 2 gives the root mean square deviations (RMSDs), mean absolute deviations (MADs), mean signed deviations (MSDs), and largest deviations (LDs) for the considered DFT methods. We begin by noting that highly unsaturated hydrocarbons have been shown to be an extremely C 2023, 9, 41 6 of 9 challenging target for most DFT functionals [20,59]. Let us begin with examining the performance of the pure GGA methods. BLYP-D3BJ results in an unacceptably large RMSD of 87.7 kJ mol −1 , which is mostly attributed to the poor performance of the LYP correlation functional. Replacing the LYP functional with either the P86 or PBE correlation functionals results in significant improvements in performance. Namely, the BP86-D3BJ and BPBE-D3BJ methods result in RMSDs of 27.8 and 23.6 kJ mol −1 , respectively. We note that PBE-D3BJ results in a similar performance to BP86-D3BJ, albeit PBE-D3BJ is associated with a much smaller MSD of merely −2.2 kJ mol −1 ( Table 2). The considered meta-GGA methods (TPSS-D3BJ and MN15-L) do not offer an improvement over the best-performing GGA methods. Moving on to the hybrid-GGA methods, B3LYP-D3BJ shows poor performance, which is considerably improved by the range-separated CAM-B3LYP method. PBE0-D3BJ results in an RMSD of 23.1 kJ mol −1 , which is a noticeable improvement over the pure PBE-D3BJ method. Overall, the hybrid-GGA B3PW91-D3BJ provides the best performance of all the considered DFT methods with an RMSD of 14.9 kJ mol −1 . Moving on to the hybrid meta-GGA methods, M06-2X provides poor performance with an RMSD of 47.9 kJ mol −1 . Interestingly, its predecessor M05-2X results in an RMSD of just 22.0 kJ mol −1 . We note that both M05-2X and M06-2X include similar amounts of exact Hartree-Fock exchange, namely, 56% and 54%, respectively. However, M06-2X is more heavily parametrized than M05-2X, which seems to work less well for the highly challenging isomerization energies at hand. Both MN15 and PW6B95-D3BJ do not offer an improvement over the performance of M05-2X. However, BMK-D3BJ results in a significantly lower RMSD of 16.2 kJ mol −1 (which is similar to that of B3PW91-D3BJ for which we obtain an RMSD of 14.9 kJ mol −1 ). a RMSD = root mean square deviation, MAD = mean absolute deviation, MSD = mean signed deviation, and LD = largest deviation. b the cumulenic isomer 14 is highly challenging for DFT methods and is excluded from the evaluation dataset.

Conclusions
In this work, we use the high-level G4 composite ab initio method to investigate the thermodynamic and kinetic properties of the smallest prototypical hyperbolic paraboloidal hydrocarbon-[5.5.5.5]hexaene (1). We find that this system is energetically stable relative to its isomeric forms. For example, isomers containing a phenyl ring with one or more acetylenic side chains are higher in energy on the enthalpic potential energy surface at 298 K by ∆H 298 = 17.5-51.4 kJ mol −1 , whereas long acetylenic carbon chains are higher in energy by as much as ∆H 298 = 277.3-488.2 kJ mol −1 . [5.5.5.5]hexaene can undergo skeletal inversion via a completely planar transition structure; however, the activation energy for this process is relatively high, being ∆H ‡ 298 = 249.2 kJ mol −1 at the G4 level. This demonstrates the high structural stability of [5.5.5.5]hexaene. Finally, we find that the transition structure for the π-bond shift reaction of the conjugated [12]annulene ring is relatively low-lying with an activation energy of ∆H ‡ 298 = 67.6 kJ mol −1 . Therefore, this process is expected to proceed rapidly at room temperature. We hope that these accurate theoretical results will inspire further experimental explorations of [5.5.5.5]hexaene.
Supplementary Materials: The following supporting information can be downloaded at https:// www.mdpi.com/article/10.3390/c9020041/s1: G4(MP2) reaction energies and barrier heights for all the reactions considered in the present work (Table S1); optimized geometries for all the species considered in this work (Table S2).
Funding: This research received no external funding.

Data Availability Statement:
The data that support the findings of this study are available in the supplementary material of this article and from the corresponding author upon reasonable request.