Theoretical Study on Drum-Shaped Polymers (1,3,5-Triazine)2n Composed of Nonplanar π-Extended Polymerization Units.

A nonplanar extended π-system can be found not only in compounds formed by multiple ortho-fused benzenes, such as helicenes and corannulenes, but also in compounds formed by bonding of atoms on the large π-extended rings. (1,3,5-Triazine)2n (n ≥ 3) are the latter type of compounds that are characterized by monomer units composed entirely of a 1,3,5-triazine core (general formula: C3N3). The first seven polymers (C3N3)2n (n = 3-9) with a drum shape were investigated computationally. Analyses of natural bonding orbitals and atoms in molecules were applied to investigate the bonding properties. In contrast to the planar structure of the 1,3,5-triazine core, the monomer units in (C3N3)2n (n = 3-9) are transformed from their planar π-system to a warped one. Similar to properties of the nonplanar π-system in [n]helicenes and corannulenes, the nonplanar heterocyclic aromatic configuration of the polymerization units is the determinant of the physical and chemical properties of these polymers. The discovery of nonplanar heterocyclic aromatic structures opens up a broad prospect for the study of azacyclic compounds. The results will be the supplement to the study of heterocyclic helicenes and corannulenes.

Introduction

As widely used reactants in syntheses, the heterocyclic aromatic 1,3,5-triazine-based compounds have been subjected to extensive research theoretically and experimentally[1,2] owing to their high stabilities, low solubilities, and little chemical reactivities.[3] Three ortho- and peri-fused pan class="Chemical">1,3,5-triazine rings can form a concentric nitrogen atom, the coplanar π-extended core of molecule C6N7H3, called tri-s-triazine. Planar C6N7H3 and its substitutive derivatives C6N7R3 were investigated experimentally[4−7] and theoretically[8−10] in detail. Three ortho- and peri-fused 1,3,5-triazine rings can also be designed and optimized as a stable molecule of C7N6H6, which is featured to the pyramidal, carbon-centered (or rather, pyramidal vertex) core frame C7N6[11] where the three fused 1,3,5-triazine rings are transformed to a nonplanar π-extended structure. The carbon atom at the pyramidal vertex of the core frame C7N6 is very reactive, so much so that it exhibits strong carbanionoid or radical properties.[11] For the past few years, nonplanar polycyclic aromatic compounds with helical chirality, such as helicenes[12]and corannulenes,[13] have attracted increasing attention for their unique helical structure, physical properties,[14] and wide potential applications in asymmetric synthesis,[15] self-assembly,[16] chiral materials,[17] and optoelectronic materials.[18] In contrast to a large number of studies on nonplanar polycyclic aromatic polymers with a monomer unit of benzene rings, studies on nonplanar heterocyclic aromatic polymers with a monomer unit of 1,3,5-triazine rings are rare. It has been verified that the nonplanar (warped) π-system does exist thermodynamically and makes the compounds containing it exhibit different chemical properties in our previous work.[11] Another type of polycyclic aromatic polymers can be designed by bonding two aromatic rings, such as diphenylene and terphenyl, instead of ortho- and peri-fused rings. Similarly, each 1,3,5-triazine can bond to other three 1,3,5-triazines by its three carbon atoms. Thus, the polycyclic aromatic polymers (C3N3) are formed, where the polymers consist fully of a 1,3,5-triazine core (i.e., C3N3). In (C3N3), the polymerization units no longer retain the planar π-extended configurations but are transformed to nonplanar, convex/concave ones. Given the special properties of azacyclic aromatic compounds such as triazines, combined with their warped π-extended configurations, polymer (C3N3) may be endowed with more unique physicochemical properties and may be a new kind of nanoparticle or nanomaterial.

Results and Discussion

Molecular Structures and Definitions

2,4,6-Tricyano-1,3,5-triazine (denoted as R)[19,20] is a very widely used chemical raw material. Assuming that multiple R can be pan class="Chemical">polymerized (addition polymerization instead of condensation polymerization) by their whole nitrile groups, a class of new compounds (general formula: (C3N3)) containing only triazine nuclei will be formed. Obviously, m must be an even number for a closed-shell system, and subsequently, polymer (C3N3) can be expressed as polymer (C3N3)2, which is denoted as R, where R = (C3N3)2 and subscript is a specific number. Since all nitrile groups participate in polymerization, there is no nitrile group left in R. Each nitrile group is polymerized with two other nitrile groups to form an s-triazine polymerization unit, and the final R are surely three-dimensional structures with a geometric shape of a drum after optimization. The drum-shaped structures and the nonplanar, convex/concave π-system of the 1,3,5-triazene unit are the typical characteristics of this kind of polymer.

In this work, the varieties of molecular structural parameters (bond lengths, bond angles, and dihedral angles) are the focus as the number of polymerization units increases. Obviously, it is difficult to compare all the structural parameters of these pan class="Chemical">polymers because the total atoms contained in each polymer of R are different and the serial numbers of bonding atoms (C–C bonds) between the polymerization (heteroaromatic s-triazine core) units are not exactly the same among R. As mentioned above, the molecules in R are in the shape of a drum, with the z axis coincident with the rotating principal axis. There is some symmetry between the polymerization units (see Table S1 in the Supporting Information). This means that small part of a geometric structure can carry the required information of a molecule in R. The part of the first polymerization unit (formed by N1, C2, N3, C4, N5, and C6 of s-triazine ring) and its three bonding units is the most representative. To facilitate the comparison of the structural parameters of each polymer, it needs to renumber the atoms on the three s-triazine units, which are bonded to the first polymerization unit. Fortunately, the atom C6 is always bonded to the atom C8 on another polymerization unit in all polymers. Therefore, only the atoms on the two polymerization units, which are bonded to atoms C2 and C4, respectively, need to be renumbered. In fact, it just needs to renumber the two carbon atoms that are bonded to atoms C2 and C4 and the two adjacent nitrogen atoms of both bonding carbon atoms. The carbon atom that is bonded to C2 is renumbered to 8′ (C8′). The smaller serial number of the nitrogen atom that is bonded to C8′ is renumbered as 7′ (N7′), and another nitrogen atom that is bonded to C8′ is renumbered as 9′ (N9′). Similarly, the carbon atom that is bonded to atom C4 is renumbered to 8″ (C8″). The smaller serial number of nitrogen atom that is bonded to C8″ is renumbered as 7″ (N7″), and another nitrogen atom that is bonded to C8″ is renumbered as 9″ (N9″) (see Figure ). The renumbered atoms vs their original serial numbers in R are listed in Table .

Figure 1

Renumbered atoms and their corresponding original serial numbers in the 3D view of R (the groups of atoms surrounded by lines).

Table 1

Renumbered Atoms and Their Corresponding Original Serial Numbers in R (R = (C3N3)2, = 3–9)

polymer	N7′	C8′	N9′	N7″	C8″	N9″
R3	21	22	23	25	26	27
R4	21	22	23	25	26	27
R5	27	28	29	31	32	33
R6	31	36	35	37	38	39
R7	67	72	71	31	32	33
R8	67	72	71	31	32	33
R9	67	72	71	31	32	33

The optimized geometries are shown in Figure (top views or z-axis views). It can be found that all the polymerization pan class="Chemical">(s-triazine) units present a double (resonant ) pattern. The other views (either x-axis or y-axis view of each polymer of R because they are very similar) are shown in Figure S1 (Supporting Information).

Figure 2

z-axis views of optimized structures of R at the B3LYP/cc-pvDZ level (R = (C3N3)2, = 3–9).

In the molecule (C3N3)2 sequence, only the molecules whose n is less than 10 can be optimized to the drum type (x-axis or y-axis view) at the B3LYP/cc-pvDZ theory level. The molecule (pan class="Chemical">C3N3)2×10 (i.e., n = 10) is optimized to a shape of binoculars (bicylinder-type z-axis view, Figure S2 in the Supporting Information), the molecule (C3N3)2×14 (n = 14) is optimized into a trinocular tube viewed from the z axis (see Figure S3 in the Supporting Information) at the B3LYP/cc-pvDZ theory level, and so on. This is the reason why only the molecules (C3N3)2 (n = 3–9) are investigated in this work.

As mentioned above, the polymerization (heteroaromatic pan class="Chemical">s-triazine core) units in (C3N3)2 are warped to a nonplanar, convex/concave π-system by the steric repulsions of adjacent units and bond tensions. To evaluate these effects, the R (Figure ) optimized at the same level (i.e., B3LYP/cc-pvDZ) was used as the reference molecule to compare the varieties of the molecular structure parameters.

Figure 3

Optimized structure parameters of 2,4,6-tricyano-1,3,5-triazine (R) at the B3LYP/cc-pvDZ level.

The seven polymers R (R = (pan class="Chemical">C3N3)2, = 3–9) and the reference molecule of R were successfully optimized at the restricted B3LYP/cc-pvDZ theory levels. The real lowest vibrational frequencies (see Table S2 in the Supporting Information) of R and R indicate that the polymers and the R have stable equilibrium configurations. The optimized geometric structures of R and R are shown in Figures and 3, respectively.

Bond Lengths

According to the classification criteria for structural parameters set out in the previous section, only the optimized bond lengths within a polymerization unit (the first s-triazine core, all of which are carbon–nitrogen bonds) and the bonds (all of which are C–C bonds) to the three adjacent polymerization units are listed in Table . It can be found that the carbon–nitrogen bonds in the polymerization unit (s-triazine ring core) range from 1.3145 Å (C4=N5 of R) to 1.3585 Å (C2=N3 of R), which are typical lengths of the carbon–nitrogen double bond (1.32 ± 0.02 Å) or carbon–nitrogen resonant bond (, 1.36 ± 0.02 Å). From the point of view of bond length, it can be judged that the electron configuration of triazine, i.e., π-extended configuration, is still reserved in each polymerization unit because all the atoms in the ring are connected to their adjacent atoms by alternate single–double bonds or a resonant bond. However, the bonds on the polymerization unit have different lengths. Specifically, the carbon–nitrogen double bonds on either side of atoms C2, C4, and C6 are in groups on the first polymerization unit of high-symmetry polymers R (D3), R (D5), and R (D6), except that of R (O of symmetry, the bond lengths on the ring are exactly the same). Each bond is different within one polymerization unit for low-symmetry polymers R (C2), R (D8), and R (C1). The drum-type polymers can be divided into two parts, top and bottom drumheads (P and P in Figure of y-axis view of polymer R), by the coordinate plane xOy. For convenience, the hexagon composed of N1, C2, N3, C4, N5, and C6 is defined as U1. U2, U3, and U4 are the three polymerization (s-triazine) units that are bonded to U1. U1, U2, and U4 are located at the top part of a drum (P), and U3 is located at the bottom part of a drum (P). According to the symmetry principle, the effect of U2 and U4 on U1 is similar or the same, while the effect of U3 on U1 is different from that of U2 and U4 on U1. This results in the lengths of N1C2 and C2N3 (i.e., the carbon–nitrogen double bond at both sides of atom C2) being similar to those of N1C6 and C6N5 but different from those of N3C4 and C4N5 (see Table ). The lengths of N1C2 (or C2N3) and N1C6 (or N5C6) are slightly longer than that of the carbon–nitrogen double bonds in R (1.3395 Å), while the length of N3C4 (or C4N5) is slightly shorter than those of the carbon–nitrogen double bonds in R. That is, the s-triazine cores of the polymers are transformed to nonregular hexagons because some bonds are lengthened and some bonds are shortened. C4–C8″ between U1 and U3 is located between P and P. Both C2–C8′ (between U1 and U2) and C6–C8 (between U1 and U4) are the bonds in P. The lengths of the three carbon–carbon bonds are very close to the length of the C–C single bond (about 1.54 Å) except the lengths (1.5914 Å) of C2–C8′ and C6–C8 in R. The two extraordinary bond lengths (1.5914 Å) are caused by the huge bond tension, which can be offset partly by lengthening bond lengths. The bond tensions are relieved, of course, not only by the lengthening of the bond between polymerization units but also by the transverse deformation (from regular to irregular hexagons) and longitudinal deformation (from planar to wavy undulating or concave/convex rings) of the s-triazine unit.

Table 2

Optional List of Bond Lengths (in Å) of R at the B3LYP/cc-pvDZ Theory Level (R = (C3N3)2, = 3–9)

Figure 4

y view of the drum-shaped polymer R. P is the mirror unit of P with respect to the plane xOy. U2, U3, and U4 are the three polymerization (s-triazine) units that are bonded to the first unit U1.

y view of the drum-shaped polymer R. P is the mirror unit of P with respect to the plane xOy. U2, U3, and U4 are the three pan class="Chemical">polymerization (s-triazine) units that are bonded to the first unit U1.

Bond Angles

Based on the same rule as the optional list of bond lengths, only the bond angles of which vertices are atoms on U1 are listed (Table ). There are six ∠N–C–N and ∠C–N–C angles (bond angles inside of U1, denoted as BAset1) and six ∠N–C–C (bond angle outside of U1, denoted as BAset2) of each polymer. The bond angle values of ∠N–C–N and ∠C–N–C are listed in the first six rows (rows 1 to 6) and the bond angle values of ∠N–C–C are listed in the last six rows (rows 7 to 12). Compared to the ∠C–N–C angle of the planar pan class="Chemical">s-triazine core in R (113.7°, see Figure ), the ∠C–N–C angles of U1 are enlarged (greater than 113.7°) generally except that of the ∠C2–N1–C6 angle (111.9°) of U1 in R and ∠C2–N3–C4 angle (111.1°) of U1 in R. The ∠N–C–N angles are reduced (less than 126.3°, see Figure ). The sum of angles inside of the planar R core is 126.3° × 3 + 113.7° × 3 = 720° (see the bond angle values in Figure ), which is, as well known, the maximum value for any hexagon. The sums of BAset1 (∠N–C–N and ∠C–N–C angles) for R to R are 715.6°, 719.1°, 718.1°, 716.4°, 714.1°, 712.5°, and 713.9° (see row 13 of Table ), respectively. The sums of BAset1 are less than 720°, indicating that the s-triazine units are distorted longitudinally from planar hexagons to spatial ones due to the bond tensions and weak intramolecular interactions. These results also help us confirm that the s-triazine units in R are transformed into wavy undulating or concave/convex rings. All the ∠N–C–C angles (BAset2) are generally decreasing (greater than 116.8°) compared to that of R (116.8°, see Figure ) except the bond angles of ∠N1–C2–C8′ and ∠N1–C6–C8 in R to R. It can be seen that the sums (rows 14 to 16 in Table ) of bond angles with the same carbon vertex are less than 360°, indicating that the three bonds on a carbon atom are also not coplanar.

Table 3

Optional List of Bond Angles (in Degrees) of R at the B3LYP/cc-pvDZ Theory Level (R = (C3N3)2, = 3–9)

no.	bond angle	R3 (D3h)	R4 (Oh)	R5 (D5h)	R6 (D6h)	R7 (C2)	R8 (D8)	R9 (C1)
1	∠C2–N1–C6	111.9	114.7	114.6	113.9	113.4	113.2	113.3
2	∠C2–N3–C4	115.0	114.7	114.5	114.6	117.1	118.2	111.1
3	∠C4–N5–C6	115.0	114.7	114.5	114.6	112.2	111.4	119.6
4	∠N1–C2–N3	125.0	125.0	124.9	124.7	122.6	121.5	126.6
5	∠N3–C4–N5	123.7	125.0	124.7	123.9	123.1	122.4	122.6
6	∠N1–C6–N5	125.0	125.0	124.9	124.7	125.7	125.8	120.7
7	∠N1–C2–C8′	109.7	113.5	116.7	120.1	124.7	128.0	123.2
8	∠N3–C2–C8′	112.6	113.5	112.7	110.8	109.1	107.5	106.5
9	∠N3–C4–C8″	114.1	113.5	113.5	113.7	111.8	111.3	120.4
10	∠N5–C4–C8″	114.1	113.5	113.5	113.7	116.4	117.8	108.0
11	∠N1–C6–C8	109.7	113.5	116.7	120.1	122.1	124.2	131.0
12	∠N5–C6–C8	112.6	113.5	112.7	110.8	108.5	106.7	106.2
13	sum of inner angles	715.6	719.1	718.1	716.4	714.1	712.5	713.9
14	sum around C2	347.3	352.0	354.3	355.6	356.4	357.0	356.3
15	sum around C4	351.9	352.0	351.7	351.3	351.3	351.5	351.0
16	sum around C6	347.3	352.0	354.3	355.6	356.3	356.7	357.9

Dihedral Angles

In this paper, only two types of dihedral angles (Table ) related to U1 are considered: one is that the dihedral angles of all four vertices on the U1 are marked by ∠N–CN–C or ∠C–NC–N (denoted as DAset1), and another is that the dihedral angles with the C–C bond as the edge are marked by ∠N–CC–N (denoted as DAset2). A dihedral angle is commonly denoted as θ and has a range of 0° ≤ θ ≤ 180° mathematically. The minus sign (or nothing) ahead of a value in Table indicates a dihedral vector because the angle (θ) or supplementary angle (π–θ) of the normal vector between a pair of half-planes is processed into a dihedral angle chemically. Therefore, the minus sign ahead of the dihedral angle should be ignored when only the size of the dihedral angle is involved. In this work, the supplementary angle (180° – θ) of a dihedral angle is considered as its dihedral angle (acute angle) for the following discussion when a dihedral angle is obtuse (θ > 90°). For example, ∠N1–C2C8′–N9′ in R is taken as 36.0° (the calculated θ = 144.0°, see row 8 in Table ). For an individual polymer, the six dihedral angles in DAset1 are variant except that of R, in which the six dihedral angles are all 7.9°. Since the four vertices that make up the dihedral angle are also the vertices of the same hexagon (U1), the size of the dihedral angle represents the fact that four atoms are not coplanar, and hence, the bonds between them fluctuate. The greater the size, the greater the degree of deviation from the bisector of the dihedral angle constituted by the four vertices. The maximum dihedral angle ∠N3–C4N5–C6 (θ =29.1° in R) indicates maximum fluctuation of the three continuous bonds of N3–C4, C4–N5, and N5–C6. It can be also concluded that the pan class="Chemical">s-triazine units are all undulating rings in R. The dihedral angle, such as ∠N3–C4N5–C6, can also be taken as the inclination angle of N3–C4 to the plane ΔC4N5C6 or N5–C6 to the plane ΔC3N4C5. Surprisingly, N3–C4, C4–N5, and N5–C6 are resonant bonds (, see Figure ) with a large inclination angle (29.1°) of N3C4 to the plane ΔC4N5C6 or N5C6 to the plane ΔC3N4C5. The studies on the presence of large dihedral angles of inner aromatic rings, such as inside of U1, have not been reported. Although Itami et al. recently reported a large dihedral angle (34.5°) in corannulenes in their work,[13f] the four vertices (∠b–c–d–e = 34.5° in Table 1 of ref (13f)) of this large dihedral angle are respectively located at three ortho-fused benzene rings instead of an individual benzene ring.

Table 4

Optional List of Dihedral Angles (in Degrees) of R at the B3LYP/cc-pvDZ Theory Level (R = (C3N3)2, = 3–9)

no.	dihedral angle	R3 (D3h)	R4 (Oh)	R5 (D5h)	R6 (D6h)	R7 (C2)	R8 (D8)	R9 (C1)
1	∠C6–N1C2–N3	–23.8	7.9	–1.9	–6.4	–13.9	–16.5	1.4
2	∠C2–N1C6–N5	23.8	–7.9	1.9	6.4	0.7	–1.8	15.0
3	∠N1–C2N3–C4	12.1	–7.9	–6.3	–6.1	6.4	11.5	–20.9
4	∠C2–N3C4–N5	2.0	7.9	16.0	20.8	16.0	13.5	26.0
5	∠N3–C4N5–C6	–2.0	–7.9	–16.0	–20.8	–27.1	–29.1	–12.2
6	∠C4–N5C6–N1	–12.1	7.9	6.3	6.1	18.5	23.4	–10.4
7	∠N1–C2C8′–N7′	0.0	0.0	154.8	0.0	9.2	13.8	–34.1
8	∠N1–C2C8′–N9′	144.0	150.5	0.0	157.7	168.9	174.3	132.6
9	∠N3–C2C8′–N7′	144.0	150.5	0.0	157.7	149.7	146.7	166.2
10	∠N3–C2C8′–N9′	0.0	0.0	154.8	0.0	9.9	13.7	–27.1
11	∠N3–C4C8″–N7″	149.8	150.5	150.0	149.1	135.4	130.3	173.1
12	∠N3–C4C8″–N9″	0.0	0.0	0.0	0.0	–13.4	–18.7	25.3
13	∠N5–C4C8″–N7″	0.0	0.0	0.0	0.0	–13.5	–18.7	25.1
14	∠N5–C4C8″–N9″	149.8	150.5	150.0	149.1	162.3	167.7	–122.6
15	∠N1–C6C8–N7	144.0	0.0	0.0	0.0	8.9	13.8	–18.8
16	∠N1–C6C8–N9	0.0	150.5	154.8	157.7	150.0	146.7	–176.3
17	∠N5–C6C8–N7	0.0	150.5	154.8	157.7	168.7	174.3	144.2
18	∠N5–C6C8–N9	144.0	0.0	0.0	0.0	9.7	13.7	–13.3

The bonds between the polymerization units are optimized to single C–C bonds at the B3LYP/cc-pvDZ level. There are four independent dihedral angles around each C–C bond that act as the edge of dihedral angles. This kind of dihedral angle (DAset2) varies greatly. Some of them are zero (0°), such as ∠N3–C4C8″–N9″, ∠N5–C4C8″–N7″, etc., in R, R, R, and R, indicating that the atoms N3, C4, C8″, N9″, and N7″ are coplanar or indicating that the pairs of half-planes ΔN3C4C8″ and ΔC4C8″N9″ and those of half-planes ΔN5C4C8″ and ΔC4C8″N7″ have the same twisting angle and twisting orientation. Some of them are very large (for ∠N5–C4C8″–N9″, θ = 180°–122.6° = 57.4°), indicating that the two bonds N5–C4 and C8″–N9″, which should have been in the same plane, are severely distorted by bond tensions and steric repulsions. pan class="Chemical">Polymers with high symmetry (R, R, R, and R) have four coplanar vertices of dihedral angles (∠N–CC–N = 0°) with C–C edges, while polymers with low symmetry or no symmetry (R, R, and R) do not (∠N–CC–N ≠ 0° in DAset2).

NBO Analyses

NBO analysis indicates that all the atoms in the polymers are sp2-hybridized at the B3LYP/cc-pvDZ theory level and two kinds of orbitals (σ and π) are obtained. For comparison, NBO analysis of the reference molecule R is also carried out at the same theory level. The numbers of σ- and π-orbitals of R are listed in Table . It should be noted that some π-orbitals (5, 1, and 1 in R, R, and R, respectively) are determined by NBO to be antibonding lone-pair orbitals (LP*) of pan class="Chemical">carbon atoms. Due to the resonance effect, these electron pairs are mainly concentrated on the side of the C atom in the resonance structure of NC and thus allocated to the C atom by NBO as a lone pair. This lone pair can only occupy LP* orbitals because the valence bonds of these carbon atoms are already fulfilled and the LP* energies of the carbon atoms are closer to the π-orbitals. Whether the electron pair of the resonance structure of NC is partitioned by NBO into π electrons or into lone-pair electrons does not affect the recognition of the carbon–nitrogen as double (resonant) bonds. Based on this reasoning, the numbers of the σ-bond obtained from NBO are equal to the total numbers of σC–C and resonant covalent bonds of NC (see Figure and Table ) obtained from structure optimizations.

Table 5

Numbers of σ and π Bonding Orbitals of R at the B3LYP/cc-pvDZ Theory Level (R = (C3N3)2, = 3–9)

bonding orbital	R3 (D3h)	R4 (Oh)	R5 (D5h)	R6 (D6h)	R7 (C2)	R8 (D8)	R9 (C1)
σ	45	60	75	90	105	120	135
π	13 + 5a	24	29 + 1a	36	42	48	53 + 1a

The resonance structure of NC is partitioned by NBO into lone-pair electrons of the C atom instead of π electrons.

In NBO analyses, the delocalized resonant double NC bonds are translated into alternate single–double bonds, which are characterized by localized Lewis’ configurations of a shared electronic pair. The localized double bonds (N=C) are only distributed in the pan class="Chemical">polymerization units like U1, U2, U3, and so on. Each unit has three localized N=C double bonds (as an example, all the 126 bonding orbitals are listed in Table S3 of the Supporting Information for reference). The σC–N and πC–N bonds can be clearly distinguished by the values of electron occupancies (Table ) and the energies (Table ) of bonding orbitals. It is well known that the energy of a σ-orbital (here, about −0.86 au for σC–N, see Table ) is lower than that of its π-orbital (all about −0.36 au for πC–N). The maximum electron occupancy of any bonding orbital is 2.0e. A perfectly localized orbital would have an electron occupancy of 2.0e. The number of delocalized electrons can be measured simply by 2.0e minus the orbital occupancy. It is common sense that the delocalization of a π-orbital is greater than that of a σ-orbital. This means that the occupancy of a π-orbital is lower than that of a σ-orbital. Taking the two molecular orbitals between atoms N1 and C6 in R as an example, the one with an electron occupancy close to 2.0e (1.978e) is defined as a σ-orbital (i.e., σN1–C6), while the one with a small electron occupancy (1.655e) is defined as a π-orbital (i.e., πC–N) by NBO. The delocalized electron is filled in its own antibonding orbital. For example, the occupancy of antibonding π-orbital (i.e., π*C–N) is about 0.365e (see Table ). The occupancies of all σC–N and σC–C orbitals are greater than 1.96e, while those of all πC–N orbitals are less than 1.69e. Comparing the orbital occupancies of σC–N (1.970e–1.983e) in U1 and that (1.982e) in the s-triazine core of R and the orbital occupancies of πC–N (1.625e–1.697e) in U1 and that (1.657e) in the s-triazine core of R, it is found that the occupancy differences between the same orbital type (σC–N, σC–C, and πC–N) are very small. However, there exist large difference values (0.045–0.065 au) of orbital energies between the orbitals in U1 (−0.89505 to −0.84614 au for σ-orbitals and −0.36051 to −0.34649 au for π-orbitals) and the s-striazine core of R (−0.92662 au for σ-orbitals and −0.39605 au for π-orbitals). Whether it is a σ-orbital or π-orbital, the orbital energies of U1 are always greater than those of the corresponding orbitals in the s-striazine core of R. The increase in orbital energy of the polymerization unit is undoubtedly caused by bond tensions, which also led to the distortion of the rigid plane of the s-triazine monomer into undulating configration of the polymerization unit.

Table 6

Occupancies (in e) of σ-, π-, and π*-Orbitals of R at the B3LYP/cc-pvDZ Theory Level (R = (C3N3), = 3–9)a

orbital	R3 (D3h)	R4 (Oh)	R5 (D5h)	R6 (D6h)	R7 (C2)	R8 (D8)	R9 (C1)	RM
σN1–C2	1.978	1.982	1.983	1.984	1.984	1.984	1.983	1.982
σN1–C6	1.983	1.983	1.984	1.984	1.984	1.983	1.985	1.982
σC2–N3	1.983	1.983	1.983	1.981	1.982	1.982	1.977	1.982
σN3–C4	1.981	1.982	1.982	1.981	1.979	1.977	1.984	1.982
σC4–N5	1.983	1.983	1.983	1.983	1.984	1.983	1.976	1.982
σN5–C6	1.970	1.982	1.982	1.981	1.978	1.977	1.981	1.982
πN1–C6	1.655	1.647	1.655	1.661	1.685	1.697	1.695	1.657
πC2–N3	1.655	1.647	1.654	1.661	1.687	1.693	1.625	1.657
πC4–N5	1.655	1.647	1.637	1.631	1.627	1.628	1.686	1.657
σC2–C8′	1.963	1.969	1.969	1.968	1.968	1.968	1.967	1.979
σC4–C8″	1.962	1.969	1.967	1.967	1.965	1.964	1.964	1.979
σC6–C8	1.963	1.969	1.969	1.969	1.968	1.968	1.970	1.979
π*N1–C6	0.367	0.354	0.364	0.372	0.357	0.353	0.354	0.371
π*C2–N3	0.366	0.354	0.354	0.355	0.355	0.360	0.365	0.371
π*C4–N5	0.376	0.354	0.352	0.353	0.354	0.360	0.364	0.371
Σπ	4.965	4.941	4.946	4.953	4.999	5.018	5.006	4.971
Σπ*	1.109	1.062	1.070	1.08	1.066	1.073	1.083	1.113
Σπ + Σπ*	6.074	6.003	6.016	6.033	6.065	6.091	6.089	6.084

R is the reference molecule 2,4,6-tricyano-1,3,5-triazine.

Table 7

Energies (in au) of R at the B3LYP/cc-pvDZ Theory Level (R = (C3N3)2, = 3–9)a

orbital	R3 (D3h)	R4 (Oh)	R5 (D5h)	R6 (D6h)	R7 (C2)	R8 (D8)	R9 (C1)	RM
σN1–C2	–0.87244	–0.87731	–0.87623	–0.87429	–0.86862	–0.86681	–0.87590	–0.92662
σN1–C6	–0.86551	–0.87585	–0.87560	–0.87395	–0.87871	–0.87989	–0.86954	–0.92662
σC2–N3	–0.88784	–0.87585	–0.86649	–0.85983	–0.86517	–0.86783	–0.84614	–0.92662
σN3–C4	–0.87244	–0.87731	–0.88388	–0.88916	–0.89070	–0.89160	–0.88000	–0.92662
σC4–N5	–0.86551	–0.87585	–0.88312	–0.88871	–0.88541	–0.88261	–0.89505	–0.92662
σN5–C6	–0.88784	–0.87731	–0.86852	–0.86225	–0.85149	–0.84664	–0.86994	–0.92662
πN1–C6	–0.35254	–0.35302	–0.34896	–0.34681	–0.34790	–0.34895	–0.34785	–0.39605
πC2–N3	–0.35254	–0.35302	–0.35554	–0.35747	–0.36051	–0.36001	–0.34649	–0.39605
πC4–N5	–0.35254	–0.35302	–0.35087	–0.34981	–0.34882	–0.34805	–0.35792	–0.39605
σC2–C8′	–0.63801	–0.66467	–0.67861	–0.68035	–0.68246	–0.68077	–0.68013	–0.79818
σC4–C8″	–0.67602	–0.66467	–0.66016	–0.65727	–0.66009	–0.66388	–0.66176	–0.79818
σC6–C8	–0.63801	–0.66467	–0.67936	–0.68121	–0.68245	–0.68081	–0.67132	–0.79818
π*N1–C6	–0.05067	–0.04146	–0.04247	–0.04399	–0.04707	–0.04874	–0.04874	–0.09069
π*C2–N3	–0.04158	–0.04146	–0.03599	–0.04437	–0.03016	–0.03138	–0.03850	–0.09069
π*C4–N5	–0.03590	–0.04146	–0.04435	–0.03269	–0.04093	–0.03928	–0.03265	–0.09069

R is the reference molecule 2,4,6-tricyano-1,3,5-triazine.

U1 is a nonplanar π-extended ring because it is characterized by resonance structures and alternate single–double bonds like the pan class="Chemical">s-triazine core of R. Σπ is the sum of occupancies of π-orbitals and Σπ* is the sum of occupancies of π*-orbitals. At a macrolevel, the term Σπ + Σπ* is the conjugate electron number of U1. The value of the term Σπ + Σπ* can be expressed as 6 + x, where x is a very small number and mainly from the contribution of lone pairs whether it is for nonlanar U1 or for the rigid-planar s-triazine core of R. Macroscopically, the conjugate electron number of U1 is surely going to be 6, which satisfies Huckel’s rule (i.e., with aromaticity). That is, U1 is a heterocyclic aromatic unit like the s-triazine core of R. By that analogy, each polymerization unit of R is a nonplanar s-triazine core.

Molecular Orbital Diagram

The deformation of the s-triazine unit in R can be observed by comparing their molecular orbital diagrams with the monomer pan class="Chemical">s-triazine or the planar benzene (introduced here for reference). There are many types of orbitals, including bonding orbitals, antibonding orbitals, core orbitals, lone-pair orbitals, vacant orbitals, and multiple orbitals for each type of polymer. In view of this, only the diagrams of the lowest-energy σ-orbital (denoted as σLEO) in R (Figure ) and R/benzene (Figure ) are provided. Since there are only carbon and nitrogen atoms in R, their σLEO number can be easily determined by adding the number of atoms to one. The σLEO numbers are 37, 49, 61, 73, 85, 97, and 109 for R to R, respectively. As can be seen in Figure , the σLEO of R is mainly composed of the σ-orbital in each s-triazine unit (here, it is denoted as σring) except that of R, in which some σring’s contribute little to the σLEO. In terms of the pattern of σring, it is different from the σLEO of R/benzene. The σLEO’s of R/benzene are pie-shaped, while the σring is double sunken pie-shaped. Whether it is R or R/benzene, all the atoms on the six-membered ring are sp2-hybridized. The difference in orbital patterns is because the overlap direction of sp2 hybrid orbitals has been changed in R. On six-membered rings of R/benzene, sp2 hybrid orbitals of adjacent atoms lie in the same plane and overlap in the head-to-head mode (and the largest overlapping mode) to form a bond. In R, due to the tension of the polymerization units, the sp2 hybrid orbitals of adjacent atoms do not overlap to form a bond in the same plane but with an offset angle in the normal direction of the s-triazine plane. On the one hand, the undulating overlapping mode of the sp2 hybrid orbital twists the pattern of σring into a double sunken pie shape. On the other hand, the overlapping degree of the sp2 hybrid orbitals in σLEO is reduced, resulting in a decrease in bond strength. This is consistent with the change in orbital energy.

Figure 5

z view diagrams of the lowest-energy σ-orbital (labeled by MO and number) in R at the B3LYP/cc-pvDZ theory level (R = (C3N3)2, = 3–9).

Figure 6

x view and z view diagrams of the lowest-energy σ-orbital (labeled by MO and number) in R/benzene at the B3LYP/cc-pvDZ theory level.

Unlike σLEO, the lowest π-orbital is difficult to be located. Any molecular orbital, including the π-orbital, is the superposition of orbitals with similar energies within the molecule. Just as the σ-orbital of the molecule discussed above being not the superposition of two hybrid orbitals in the head-to-head mode, two p-orbitals of adjacent atoms do not overlap in the shoulder-to-shoulder (parallel) mode but overlap at a certain angle (or called dislocation superposition). This overlapping mode results in the positive phase (plus isovalue) and negative phase (negative isovalue) canceling each other out and a lot of nodes in the molecular orbital diagram (Figure ). In the polymer, there is no π-extended orbital shaped as sandwich biscuits (the lowest-energy π-orbital, denoted πLEO, Figure ) similar to the R/pan class="Chemical">benzene. Even as part of the π-orbital (denoted as πring), the patterns of πring are quite different from the πLEO of R/benzene. The orbital diagram shown in Figure is the representative orbital, which is not necessarily the πLEO of the polymers, selected from the various orbitals of a polymer.

Figure 7

Diagrams of the selected (not necessarily the lowest-energy) π-orbital (labeled by MO and number) in R at the B3LYP/cc-pvDZ theory level (R = (C3N3)2, = 3–9).

Figure 8

x view and z view diagrams of the lowest-energy orbital (labeled by MO and number) in R/benzene at the B3LYP/cc-pvDZ theory level.

As can be seen from the selected molecular orbital diagrams, all the polymerization units have a large π-extended structure like the pan class="Chemical">s-triazine core of R; the orbitals (both σ-orbital and π-orbital) of all the polymerization units are wavy undulating configurations. From the molecular orbital diagram, it is confirmed again that the polymerization unit is a nonplanar (warped) s-triazine unit.

AIM Bonding Analysis

AIM analysis shows that the number of the BCPs with ∇2ρ < 0 is equal to that of chemical bonds obtained from structural optimization and the σ-orbitals obtained from the NBO. Only a small amount of BCPs with ∇2ρ > 0 (i.e., van der Waals force or dispersion force) was found in the three polymers R, R, and R (see Table S4 of the Supporting Information). The van der Waals forces occur between two nitrogen atoms at the side face of a drum (Figure S4 of the Supporting Information). Take R as an example, the eight van der Waals forces (dispersions) are labeled as “V”. The lengths of bond paths (approximately the distance between the atoms) range from 2.9 to 3.2 Å except that of N5···N9 (2.316 Å) and N49···N57 (2.315 Å) in R. The electron densities at this kind of BCP range from 0.008e/bohr[2] to 0.014e/bohr[2] except those of N5···N9 (0.034e/bohr[2]) and N49···N57 (0.035e/bohr[2]) in R. Such long bond paths and small electron densities mean that the interaction (van der Waals force) between the atoms is very weak. These results indicate that weak intramolecular interactions contributed little to the stability of the molecule at the B3LYP/cc-pvDZ level.

Heats of Formation (HOF)

Heat of formation (HOF, commonly denoted as ΔHf) is an important measure of molecular stability. The HOF values of the polymers are obtained by our previously developed method[21] for estimating the approximate HOF of large-sized compounds at the level of B3LYP/cc-pvDZ (Table ). As far as the HOF criterion is concepan class="Chemical">rned, the relative stabilities of the polymers cannot be estimated because each polymer molecule contains a different number of atoms. It is assumed that nR can self-polymerize to form R by the following reactionwhere

Table 8

Total Energies E0 (in au, Including Zero-Point Energy Corrections), the Enthalpies Hf (in au, Including Zero-Point Energy Corrections), and the ΔHf (in kcal mol–1) of R and R at the B3LYP/cc-pvDZ Theory Level (R = (C3N3)2, = 3–9)a

molecule	Eo	H298	ΔHf	ΔΔHf
C6N6 (RM)	–557.028290	–557.017860	165.7
R3	–1670.737502	–1670.717901	707.8	210.7
R4	–2227.924965	–2227.898752	771.2	108.4
R5	–2785.009094	–2784.975426	900.1	71.6
R6	–3342.004024	–3341.962935	1084.8	90.6
R7	–3898.936043	–3898.887923	1308.8	148.9
R8	–4455.836558	–4455.781344	1552.6	227.0
R9	–5012.720499	–5012.657440	1807.3	316.0

R is the reference molecule 2,4,6-tricyano-1,3,5-triazine.

ΔΔHf is the change in enthalpy from the reactant R to the product R. Obviously, given the same number of atoms, the smaller the value of ΔΔHf, the smaller the HOF per product R. According to this rule, the order of stability of the polymer is R > R > R > R > R > R > R. This order is consistent with the conclusion that the bond tensions in R and R are low and the bond tensions in R and R are high.

Conclusions

The computational and theoretical results of molecules (C3N3)2 (n = 3–9) indicate the following: (1) The geometric structures are cages and drum-shaped. The total number of bonds obtained from structure optimizations, the σ-orbitals obtained from NBO analysis, and the BCPs with ∇2ρ < 0 obtained from AIM analysis are equal. (2) All the atoms in pan class="Chemical">polymers are sp2-hybridized and the polymerization units maintain the resonance (π-extended) structure of s-triazine. The bond between polymerization units is a carbon–carbon single bond (C–C), which is not in the plane of NCN, although all the atoms are sp2-hybridized. (3) The out-of-plane tensions of the C–C bonds cause the core of triazine to be deformed to a wavy undulating (nonplanar π-extended) ring. (4) The twist angle, i.e., dihedral angle of two planes consisting of four consecutive atoms in a resonant structure, can reach up to 29.1°. (5) The warped π-system can be found not only in helicenes and corannulenes, in which the benzene units are ortho-fused, but also in polymers like (C3N3)2 with heterocyclic aromatic hydrocarbon polymerization units bonding by two carbon atoms. (6) The results of this study broaden the research field of helicenes and corannulenes and help organic chemists and material chemists synthesize this kind of compound with structural characteristics.

Computational Methods

All calculations were carried out using the GAUSSIAN09 version C.01 program package.[22] The geometry structures and harmonic vibrational frequencies of R were mainly optimized using the B3LYP method with Dunning’s cc-pvDZ basis set. The convergence criterion of the SCF iteration algorithm was set to 10–8 during geometry optimization. No imaginary frequency (see Table S2 in the Supporting Information) was found for all the structures at the B3LYP/cc-pvDZ theory level, indicating that all of the structures optimized are local minima on the potential energy surface. The optimized geometries are shown in Figure (top views or z-axis views). It can be found that all the polymerization pan class="Chemical">(s-triazine) units present a double (resonant ) pattern. The other views (either x-axis or y-axis view of each polymer of R because they are very similar) are shown in Figure S1 (Supporting Information).

One of the main reasons for choosing the B3LYP method with Dunning’s cc-pvDZ basis set as the computational level of the molecules in this sequence (for example, n = 7, 8, 9, 10, 14, 18 of R) is that the structures of the larger molecules can also be optimized successfully at this computational level. In R, the s-triazine units are distorted (with a warped π-system) by extepan class="Chemical">rnal forces. These forces may be bond tensions or intramolecular dispersion forces. The role of bond tension is discussed in the next section. Quantitative or qualitative analysis of intramolecular dispersion forces needs to use methods with functionals of long-range correction (such as Head–Gorden’s functionals wB97XD) and basis sets with diffuse functions (such as Dunning’s aug-cc-pvDZ) since the B3LYP functional does not address weak intermolecular interactions well.[23] However, using MP2 or wB97XD methods with the aug-cc-pvDZ basis set, the optimizations are either extremely time-consuming or a convergence failure (see Table S5 in the Supporting Information). Even using the B3LYP method with the aug-cc-pvDZ basis set, only molecules R and R in this sequence can be successfully optimized. This means that it is impossible to optimize all of R by using any method with diffuse functions’ basis sets (including the common basis set of 6-31+G(d)). Popelier in his work[24] pointed out that the model used to obtain the electronic density and, hence, the dispersion interaction results, Atoms In Molecule (AIM),[25] is relatively independent of the choice of the functionals as well as the basis sets. The interpretation of the charge density toward chemical concepts is independent of the method by which it has been acquired.[26] In our previous work,[11,21] the B3LYP/cc-pvDZ theory level is chosen to optimize the relative larger-sized molecules and the results are reliable. This is the reason why the theory level of B3LYP/cc-pvDZ in combination with AIM and NBO[27] is used to study this series of molecules in this work.

An attempt is made to find out whether there are weak interactions (van der Waals forces or dispersion forces) in the seven cage polymers. The AIM 2000 program package, based on AIM theory of Bader et al.,[24,25] is designed to yield valuable information such as the energy of an atom,[27,28] the electronic density (ρ(r)) of the critical points, etc. Each point lying between bonded pair atoms is called a bond critical point (BCP) where the Laplace of ∇2ρ is used to determine whether the charge of the region is locally depleted (∇2ρ > 0) or concentrated (∇2ρ < 0). The former is typically relevant to weak interactions between atoms, whereas the latter is used to characterize covalent bonds.