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

## Abstract

**:**

_{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.

## 1. Introduction

^{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].

## 2. Computational Details

^{−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].

_{13}H

_{8}isomers considered in the present work relative to the bottom-of-the-well G4 reference values. The considered DFT methods, ordered by their rung on Jacob’s ladder [34], are the generalized gradient approximation (GGA) functionals BLYP [28,35], PBE [36], BPBE [35,36] and BP86 [35,37] the meta-GGA functionals TPSS [38] and MN15-L [39] the hybrid-GGAs B3LYP [28,29,30] B3PW91 [27,40], PBE0 [41] and the range-separated CAM-B3LYP [42] the hybrid-meta GGAs M05-2X [43], M06-2X [44], MN15 [39], BMK [45], and PW6B95 [46]. All calculations have been performed in conjunction with the def2-QZVPP basis set [47]. With the exception of the Minnesota functionals, which account for dispersion in the functional form, the empirical D3 dispersion correction has been used in conjunction with the Becke–Johnson damping potential (denoted by D3BJ) [48,49,50].

## 3. Results and Discussion

**1**, Scheme 1) has the molecular formula of C

_{13}H

_{8}, Scheme 2 depicts key structural C

_{13}H

_{8}isomers

**2**–

**14**that have been identified using the ChemSpider database [51]. [5.5.5.5]hexaene and the C

_{13}H

_{8}isomers in Scheme 2 are, by definition, highly unsaturated. As such, most of the C

_{13}H

_{8}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 C

_{13}H

_{8}isomers are synthetically accessible, for example, isomers

**2**and

**3**[52,53,54]. 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**.

_{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 (

**11**–

**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**.

**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.

^{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°.

_{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].

^{‡}

_{298}= 67.6 kJ mol

^{−1}at the G4 level. Therefore, this process is expected to proceed rapidly at room temperature.

^{−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}).

## 4. Conclusions

**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

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**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.

**Figure 1.**Top and side views of the optimized B3LYP/6-31G(2df,p) structures of (

**a**) 1 ([5.5.5.5]hexaene, D

_{2}symmetry), (

**b**) planar transition structure for structural inversion of

**1-TS**(D

_{inv}_{2h}symmetry), and (

**c**) transition structure for the double-bond shift in

**1-TS**(C

_{shift}_{2v}symmetry); bond lengths are given in Å (black font), and angles are given in degrees (green and red font). See Table 1 for the relative G4 enthalpies at 298 K.

**Scheme 3.**Schematic illustration of the (

**a**) skeletal inversion (

**1-TS**) and (

_{inv}**b**) π-bond shift (

**1-TS**) reactions in [5.5.5.5]hexaene.

_{shift}Struct. | ∆E_{e} | ∆H_{0} | ∆H_{298} | Struct. | ∆E_{e} | ∆H_{0} | ∆H_{298} |
---|---|---|---|---|---|---|---|

1 | 0.0 | 0.0 | 0.0 | 10 | 52.3 | 43.1 | 51.4 |

2 | 15.3 | 8.1 | 17.5 | 9 | 61.2 | 42.0 | 52.8 |

3 | 25.8 | 15.2 | 24.3 | 11 | 281.8 | 263.2 | 277.3 |

5 | 29.5 | 18.3 | 28.0 | 12 | 283.6 | 264.3 | 278.7 |

4 | 29.8 | 18.7 | 28.3 | 13 | 363.6 | 359.9 | 361.4 |

6 | 54.9 | 36.3 | 46.8 | 14 | 498.7 | 472.8 | 488.2 |

7 | 54.8 | 36.7 | 46.9 | 1-TS_{inv} ^{a} | 262.7 | 247.5 | 249.2 |

8 | 57.4 | 38.5 | 48.9 | 1-TS_{shift} ^{b} | 74.6 | 68.4 | 67.6 |

**Table 2.**Statistical analysis for the performance of selected DFT procedures for calculating the relative energies of the isomers in Scheme 2 relative to the electronic ∆E

_{e}reference values from G4 theory (in kJ mol

^{−1})

^{a,b}.

RMSD | MAD | MSD | LD | |
---|---|---|---|---|

BLYP-D3BJ | 87.7 | 79.8 | −76.2 | −143.4 |

BP86-D3BJ | 27.8 | 24.3 | −11.0 | −52.2 |

PBE-D3BJ | 27.4 | 24.1 | −2.2 | −42.5 |

BPBE-D3BJ | 23.6 | 19.9 | 1.8 | −34.2 |

TPSS-D3BJ | 38.4 | 32.8 | −26.3 | −69.6 |

MN15-L | 30.6 | 27.9 | −16.9 | −58.7 |

B3LYP-D3BJ | 62.0 | 57.9 | −54.1 | −91.8 |

CAM-B3LYP-D3BJ | 38.5 | 37.0 | −35.1 | −50.2 |

PBE0-D3BJ | 23.1 | 17.8 | 17.4 | 38.8 |

B3PW91-D3BJ | 14.9 | 13.3 | 2.7 | 22.0 |

M06-2X | 47.9 | 45.7 | −45.7 | −61.3 |

MN15 | 24.7 | 22.6 | −22.6 | −36.5 |

PW6B95-D3BJ | 28.8 | 22.8 | −22.8 | −46.8 |

M05-2X | 22.0 | 20.7 | −19.8 | −34.0 |

BMK-D3BJ | 16.2 | 15.2 | 11.5 | −22.2 |

^{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.

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Karton, A.
Thermochemistry of the Smallest Hyperbolic Paraboloid Hydrocarbon: A High-Level Quantum Chemical Perspective. *C* **2023**, *9*, 41.
https://doi.org/10.3390/c9020041

**AMA Style**

Karton A.
Thermochemistry of the Smallest Hyperbolic Paraboloid Hydrocarbon: A High-Level Quantum Chemical Perspective. *C*. 2023; 9(2):41.
https://doi.org/10.3390/c9020041

**Chicago/Turabian Style**

Karton, Amir.
2023. "Thermochemistry of the Smallest Hyperbolic Paraboloid Hydrocarbon: A High-Level Quantum Chemical Perspective" *C* 9, no. 2: 41.
https://doi.org/10.3390/c9020041