Cubane belongs to a typical class of molecular solids such as carbon clusters (C60, C70). Due to its peculiar structure (i.e., the C-C-C bond angle is 90o rather than the customary 109.5o), cubane stores a great deal of energy, which implies its explosiveness. Cubane was firstly synthesized in 1964  but its remarkable properties, such as potential pharmaceutical applications, have been realized only recently. One of its derivatives, tetranitrocubane, is a very powerful explosive, yet it is extremely stable, a great advantage from a safety standpoint. Another derivative, the dipivaloylcubane compound, exhibits moderate activity against human immunodeficiency virus without impairing healthy cells. This greatly motivates extensive studies both experimentally and theoretically. For example, ab initio calculations provide valuable information on its bond structure and electronic properties. The numerical computations reproduce the experimental bond lengths quite satisfactorily . Spectroscopic properties of cubane have also been explored recently. Miaskiewicz et al.  calculated the vibrational spectra of cubane by the density-functional theory. The infrared and Raman spectra have been independently studied by Vlahacos et al.  and Jursic . Zakrzewski and Ortiz  reported the vertical ionization energies of cubane. On the other hand, the chemical stability of cubane structure is certainly interesting from an application point of view.
The remarkable structural properties of cubane greatly motivate us to investigate its underlying driving force. We employ an ab initio method to calculate its bond structure for a pure cubane. All the calculations are performed at the Hartree-Fock level by theMolpro package . The basis set used for hydrogen is (4S 1P)/[2S 1P], whereas for carbon (9S 4P 1D)/[3S 2P 1D] is used. The basis functions are all valence triple zeta correlated basis sets.
Until Eaton and Cole  carried out the first synthesis of cubane, scientists did not believe it could be made. It was thought that even though one made cubane, it would explode spontaneously because a tremendous amount of energy is stored inside the molecule. But when Eaton and Cole made the compound, it turned out to be very stable. Previous calculations showed that the stable structure is cubic (see Figure 10.5) .
There are two kinds of bond lengths, namely the length RC-C between two neighboring carbon atoms and the length RC-H between neighboring hydrogen and carbon atoms. In the present study, the structure of cubane is fully optimized
Figure 10.5. A typical cubane structure. The origin of the coordinate system is at the center.
without any constraints, which yields a cubic structure with only two kinds of bond lengths, RC—C = 1.56175 A, RC—H = 1.08774 A, in good agreement with both experimental  and theoretical results . Experimentally, Hedberg et al.  found RC _C = 1.5618 A. Theoretical calculations using different basis sets show that RC—C = 1.557—1.580 A. Note that hereafter all the calculations are based on this fully-optimized structure. In order to see how the total energy changes as a function of one specific bond length, we fix one bond length while variating the other. In Figure 10.6, we plot the total energy as a function of RC—C, where RC—H = 1.08774 A. One sees that with an increase in RC—C, the energy decreases first and then increases. A minimum appears at 1.56175 A. The estimated dissociation energy is 2.2 Hartree, which indicates that cubane is very stable. A polynomial fit shows that the potential energy changes as
Figure 10.6. The potential energy first increases as the helium atom moves towards cubane and then decreases. A peak appears at about z = — 1.86 A. The vertical dashed line refers to the z—component of the neighboring hydrogen atom's position. Note that the cubane framework is fixed.
Figure 10.7. Analogously the total energy of cubane changes as a function of Rc-H whereas Rc-c is fixed at 1.56175 A.
10. Theoretical Investigations in Retinal and Cubane -303 -304 '
where A0 = -305.216, A1 = 25.8292, A2 = 2.20239, A3 = -71.5941, and A4 = 1. 14699. The units of the As are chosen such that the overall unit is Hartree. (The same is true for those Bs below). We notice that the potential form is very different from those listed in Table I. We in fact tried to use those potentials but none of them is able to reproduce our potential. In some sense, our potential does not follow Morse and Lennard-Jones potentials, but behaves more as the Buckingham potential.
Next we fix Rc-c = 1.56175 A and change Rc-H from 0.6 A to 1.4 A. The total energy is plotted as a function of Rc-H in Figure 10.7. The minimum is around 1.08774 A, which reasonably agrees with the previous results. The experimental length is 1.098 A and the theoretical results range from 1.075 to 1.106 A. The potential can be fitted as
Rc-h where B0 = -310.33, B1 = 0.948781, B2 = 3.36077, and B3 = 0.717603. comparing V(Rc-c) with V(Rc-h), one finds that V(Rc-H) changes more smoothly. Similarly, the phenomenological potential  has a different form from our ab initio potentials, which reflects that cubane does not perfectly fall into the usual hydrocarbon category. This is a direct consequence of its unique structure as aforementioned. The dissociation energy for the hydrogen atoms is found to be 0.9 Hartree, corresponding to 105 K. Thus at room temperature, the framework of cubane is very stable.
These two potential forms suggest the complication of the potential in a real simulation. Those simple model potentials in Table 10.1 may have limited applications particularly for nanosystems. This is important for future investigations in other systems.
Acknowledgment. This work was supported by the U.S. Army Research Office under contract W911NF041038. The content of the information does not necessarily reflect the position or the policy of the federal government, and no official endorsement should be inferred.
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