Energies for Cyclic and Acyclic Aggregations of Adamantane Sharing Six-membered Rings

Tension energies for acyclic and cyclic assemblies of adamantane units sharing hexagons of carbon atoms converge in a size-extensive manner for large numbers n of adamantane units. In most cases the convergence features diagrams of strain energy per adamantane unit E (n–1) versus n–2 for cyclic aggregates and versus n–1 for acyclic aggregates having linear dependence with slopes of opposite signs. However, we found two exceptions which are discussed in the present paper, with convergence involving linear dependence with slopes of the same sign.


INTRODUCTION
[3] They are the most stable among all their valence isomers because all their C-C bonds are in staggered conformation.Further prominence arises because of their exceptional strength and stability, as possibilities to construct robust nanomachines.But especially in contemplated [4] designs involving circular nanostructures (e.g.wheels or gears) there should be notable stresses and strains -so that the investigation of such in polymantane-based materials should be appropriate.The 2016 Chemistry Nobel Prize to Jean-Pierre Sauvage, Sir J. Fraser Stoddart and Bernard L. Feringa for their design and production of molecular machines stresses this statement.And it should be an area of interest to Nenad Trinajstić.
In a previous paper (which will be considered to be Part 1, whereas the present paper can be considered to be Part 2 this series) were presented acyclic and cyclic strings of adamantane or diamantane cells that shared vertices, edges, or 6-membered rings. [5]In order to provide information without needing to consult Part 1, some repetition of text and figures from Part 1 is made.
Hydrogen-depleted formulas are used in the following unless otherwise stated.Diamondoids consist of adamantane units or cells sharing vertices, edges, or 6-membered carbon rings.They are partially characterized by their constitutional (molecular) formulas, and uniquely characterized by their structural formulas.When pairs of adamantane units share a chair-shaped 6-membered "face" they are called polymantanes.By inscribing a virtual vertex in the center of each unit, and connecting vertices corresponding to adjacent cells, one obtains the inner dual (i.e. dualist) which also characterizes uniquely the polymantane.In the present context, the terms "cyclic" and "acyclic" will refer to the dualist.Polymantanes occur in three common kinds: catamantanes when the dualist is acyclic; perimantanes when the dualist contains sixmembered rings; and coronamantanes when the dualist has larger rings which are not peripheries of "internal" sixmembered ring aggregates.To specify polymantane structures, Balaban and Schleyer proposed a concise notation describing the dualist. [6]A similar proposal applied to polycyclic condensed benzenoid hydrocarbons (catafusenes, perifusenes, and coronafusenes) had been formulated earlier by Balaban and Harary. [7]Catamantane dualists may be characterized by a sequence of digits (1, 2, 3 and/or 4) specifying the four possible orientations along S tetrahedral directions around each vertex along the longest path of the dualist, choosing among all possibilities the one corresponding to the smallest number formed by reading sequentially all digits from one end of that path to the other end.Branchings are denoted by brackets.
Returning to the analogy with polycyclic catacondensed benzenoids, when their dualists attain certain dimensions they may connect their endpoints forming cycles and converting catafusenes into perifusenes or coronafusenes.Alternatively, these ends may exit the twodimensional plane to yield tridimensional helicenes, which are well-known stable compounds.By contrast, there are no real four-dimensional catamantanes, although their imaginary counterparts can be conceptually described.A major difference from polycyclic benzenoids, which on following a path in the benzenoid network the path does not return to its starting position despite the corresponding walk on the honeycomb net so returning -because of the ability of the benzenoid net to twist out of the plane of the honeycomb net -a sort of distortion not available to polymantane structures.Catamantanes are of two types: regular C4n+6H4n+12, and irregular if there are fewer hydrogens.If digits 1, 2, 3, and 4 correspond to letters p, q, r, and s, irregular catamantanes have dualists with a sequence of digits of type p-q-r-p; in such cases the ends of the dualist come so close together that in order to avoid hydrogen overlap, the C-C-C bond angles must undergo distortion.Moreover, whenever a sequence p-q-r-p-q would occur by adding successive adamantane cells, a perimantane results automatically with its cyclic dualist which has the digit sequence 12312, equivalent to 123123 because the last edge of the dualist (denoted by digit 3) is inserted to properly accommodate the space available, since there is no fourth dimension!
In the following we discuss three classes of nonbranched caramantanes and perimantanes sharing hexagons.The notation for long acyclic catamatanes contains, in square brackets, the repeating unit followed by three dots.To start with catamatanes, their dualists are continuous sequences of straight lines with equal length.It is known that a non-planar sequence of three lines on the diamond lattice is chiral. [8,9]Indeed, taking as an example the tetramantanes, and ignoring the branched [1(2)3]tetramantane, there are two isomeric regular non-branched tetramantanes: the achiral zigzag [121]tetramantane whose dualist is a z-like coplanar sequence of three equal lines, and the chiral [123]tetramantane with a nonplanar dualist In the former case one may continue lengthening the dualist in a self-similar manner resulting in a quasilinear zigzag poly[12…]catamantane which is examined in detail in the following.However, in the latter case, there are two possibilities: (i) lengthening the dualist by repeating the 123 triplet, which leads to cancellation of chirality yielding an irregular poly[123…]catamantane; and (ii) adding a fourth direction in a self-similar manner, which leads to a regular, helical and chiral poly[1234…]catamantane.The latter aggregate had been discussed in Part 1, and will again be examined in detail below.The present paper also includes a new regular catamantane and a perimantane.Figure 1 displays for each type of acyclic polymantane several views of their dualists.As in Part 1, in all following figures, virtual vertices of dualists are colored in olive; quaternary (C), tertiary (CH) and secondary (CH2) carbon atoms are presented in red, black, and light blue, respectively.
The last aggregate in Figure 1 has a dualist (whose vertices mimic carbon atoms of all-trans-perhydro-acene) whose dualist mimics the carbon atoms of the zigzag poly[12…]catamantane, the first aggregate in Figure 1.We refer to Schleyer-Williams-Blanchard "strain" energy as the SWB-tension energy [10] delivered by the most recent CambridgeSoft MM2 package that we use.In adamantane this SWB-tension energy amounts to about 6 kcal/mol, despite there being very little identifiable "strain" in terms of anomalous bond lengths or bond angles.Indeed rather than such geometric strain, SWB ascribe [9] their so-called "strain" energies due to interactions between nonneighbor atoms and bonds.In homologous diamondoids this tension energy increases with the number of adamantane units, and varies among isomeric diamonoids as shown in the following examples.On comparing the three regular isomeric tetramantanes C22H28, SWB-tension It can be seen that a zigzag two-dimensional conformation of the dualist plays the dominant favorable role leading to compactness of the diamondoid, and that branching of dualists (which changes the partition formula) is relatively unimportant, with [1234]-, and [12(3)4]-pentamantane having practically equal strain energies.An unfavorable aspect leading to high strain is the association of large topological distances with low geometrical distances; an extreme example is provided by irregular catamantanes.
Results for various adamantane polymers can be viewed to have total energies or SWB-tension energies approximated in an additive manner.This presumes an energy εint for an internal adamantyl unit and another energy εend for an end (terminal) adamantyl unit.For an open-chain polymer or aggregate, this means that the (SWB-tension) energy for the whole polymer of adamantyl units is: Or equivalently so that a plot of En/n versus 1/n is expected to be asymptotically linear, with an intercept equal to in ε and a slope which is either positive or negative as end in . Indeed, there is evidence [11][12][13][14] that such a functional form is highly accurate.
For all cases examined in Part 1, it was found that acyclic aggregates yielded SWB plots with negative slope, meaning that an adamantane end-unit has less net tension than internal units.However, we have continued searching and although most aggregates behaved similarly, we find two counterexamples discussed in the present communication, with plots for both cyclic and acyclic aggregates with positive slope, but still converging for large numbers of adamantane units.
Continuing our "additive" analysis, we see that for n -fold cyclically symmetric structures, there are n equivalent uints, each with a common contribution n ε .But since the units must be geometrically distorted (relative to the open-chain units), there must be some strain, so that n ε should generally be n ε  .The distortion (or strain) in the bond length and/or bond angles naturally is proportional to 1 / n .If the linear-chain experiences no screw-like torsion along the chain, which is to say it approaches a simple translational symmetry in the interior, then the curvatureinduced stress energy should be proportional to the square of this geometric strain.Thus for each of the n units of the polymer there should be an additional contribution so that a plot of / n is anticipated to be linear.With the (geometric) strain 1 / n  measuring deviations from the ideal unstressed case, there can also be higher order curvature-strain corrections in 1 / n (say ).If there were to be a screw-like torsion in the open-chain case, and this is "straightened out" in going to the cyclic chain, then there would be an additional correction  to n ε , so that Here / γ n is a genuine net strain-energy contribution to the overall tension energy n E .The following separate sections deal with three classes of polymantanes sharing hexagons of carbon atoms.The 1 st section repeats, in abbreviated form, data from Part 1 but without repeating the corresponding tables which appeared in Part 1.

Regular Helix Poly[1234…]catamantanes
Diamantane results from two adamantane units sharing a "face" of chair-shaped hexagon of carbon atoms, and higher catamantanes proceed with further face-sharings.In this section, we study linear-chain and cyclic helical catacondensed aggregates of adamantane units sharing hexagons, but with the building units based on (facesharing) fusions of diamantane units, conserving their reciprocal orientation.
Figure 3 shows a small portion of such an acyclic helix (from front and side views).In the middle front view of Figure 3, one sees that four adamantane units, or two diamantane units, make up one turn of the helix.The IUPAC (von Baeyer) nomenclature of such aggregates was discussed earlier, though we use the Balaban-Schleyer notation for catamantanes (as is based on dualists, with digits 1,2,3,4 identifying the four possible directions of bonds around sp 3 hybridized carbon atoms).One of the three isomeric tetramantanes, the chiral [123]tetraman-tane, is the precursor of the chiral helical systems discussed here.
In Figures 4 and 5 one can see the variation of the   SWB-tension and total energy per diamantane unit versus 1/n 2 for cyclic aggregates, and versus 1/n for acyclic aggregates, respectively.
As seen in Tables 1 and 2, for each new C8H8 diamantane unit the PM6-computed energy increases by 1090 kcal/mol, and the MM2-calculated strain energy for acyclic chains by 9.1 kcal/mol.
Unlike the aggregation examined in the preceding section, the straight line representing the ratio between  tension energies of acyclic zz-cata[n]diamantanes and n values exhibits a positive slope when plotted versus n -1 as seen in Figure 6.A tendency to relax end ε to be less than in ε is seen in that an internal unit is under tension from neighbor units on both sides (rather than just a single unit neighboring to an end unit).An energy cost for end ε over in ε is identifiable when the end unit involves a greater number of atoms and bonds to undergo interactions than appear in the (then smaller) internal units.Thus the sign of the slope for / n E n  in our plots depends on the balance between these two counter-vailing tendencies.As indicated earlier, zigzag dualists give rise to the most compact arrangement of adamantane units, having their carbon atoms almost as in the diamond lattice.This may be the reason why in this case we have εin < εend.
Interestingly, with increasing n values, the so-called "non-1,4-Van der Waals" SWB-stress energies of cyclic zzcata[n]diamantanes decrease and then change sign around n = 40 becoming negative at higher n values.The minimal values of 1,4-VDW and SWB tension energies correspond to n = 37 and 58, respectively.

Blade-perimantane
If the dualist has vertices corresponding to the carbon atoms in an all-antiperhydro-k-acene, then the diamondoid is a "blade perimantane".The dualist of its dualist leads to the zig-zag regular poly[12…]catamantane examined in the preceding section.
Acenes such as anthracene, (k = 3 benzenoid rings), tetracene, (k = 4 benzenoid rings), etc. have n= 4k + 2 carbon atoms, and therefore for this class of perimantanes we will discuss only integer k values.This same number   Like the preceding zz-catamantanes, blade perimantanes contain most of their carbon atoms in a compact arrangement, close to that present in the diamond net, and this is probably the explanation for the difference of SWB strain between Figures 6 and 9 on one hand (slopes with the same sign), and Figure 4 on the other hand (slops with opposite signs) (Figure 11, Tables 3-4).

CONCLUSIONS
In Part 1 we presented one class each of cyclic and acyclic strings of regular catamantanes sharing hexagons of n

Figure 2 .
Figure 2. Dualists of pentamantanes with their notation and strain energies: upper row, the three isomeric regular and the unique irregular [1231]-pentamantane.Lower row, the three iisomeric branched pentamantanes.

Figure 3 .
Figure 3. Three views of a fragment with 4 diamantane and 8 adamantane cells for the acyclic regular acyclic helix poly[1234…]catamantane without its dualist.

Figure 6 .
Figure 6.zz-Cata-[n]diamantanes.Upper panel in red: cyclo-zz-cata[10]diamantane, and the plot of the strain energy per diamantane unit versus n -2 .Lower panel in blue: zz-cata[28]14amantane (front and side views with dualists), and the plot of the strain energy per unit versus n -1 .For structures, the c

Figure 9 .
Figure 9. Two views of a cyclic blade polymantane.

Figure 10 .
Figure 10.Upper panel in red: cyclic blade-perimantane and the plot of the Total energy per adamantane unit En -1 versus n -2 .Lower panel in blue: acyclic blade-perimantane and the plot of the total energy per unit versus n -1 .

Figure 11 .
Figure 11.SWB-Tension energies (in kcal/mol) for bladeperimantane.Upper plot in red: cyclic blade-perimantane.Lower plot in blue: acyclic blade-perimantane, and the plot of the SWB-tension energy per adamantane unit versus n -2 .