Abstract
Hückel molecular orbital (HMO) quantities, viz., electron densities, charge densities, bond orders, free valences, total π-electron energies and highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO–LUMO) or band gaps of {X,Y}-cyclacene graphs under next-nearest-neighbour (nnn) approximations are expressed in analytical forms within a certain range of nnn approximation parameter (m). The critical values of m for {X,Y}-cyclacenes with varying X (=C, N, B) and Y (=C, N, B) are calculated. For {X,X}-cyclacenes with a π-electron on each atom, all HMO quantities except total π-electron energies for a given value of m are found to be independent of X. The cyclic dimer (CD) is constructed in obtaining the eigenvalues corresponding to the singular points of the density of states (DOS) of such {X,Y}-cyclacene. Although the HOMO–LUMO gap of the CD differs from that of the cyclacene with a large number of repeating units (i.e. n ⟶ ∞) but becomes the same for m = 0. The analytical expressions can be used for facile computer programming in obtaining the HMO quantities. Such nnn interaction approximations actually release, to some extent, the strain that results in due to the geometrical structures of such cyclacenes, which is evident from the plots of strain energy per segment vs. contribution of such interactions on the total π-electron energy, where the slopes decrease with an increase in m. The vertical absorption energy difference for singlet-triplet states bears excellent linear correlation with the HOMO–LUMO gaps for a certain m value (m = 0.3) in the case of an even n, but for an odd n, such energy difference remains invariant.
1 Introduction
A cyclacene [1], [2], [3], [4], [5], [6], [7], [8] is a cylindrical or belt-like polycyclic aromatic compound composed of only hexagonal rings and is supposed to be obtained by gluing two bonds of the two terminal hexagons of a linear polyacene. Such compounds were proposed as potentially game-changing molecules in chemistry [5], [6] even before the discovery of carbon nanotubes (CNTs) [7] in 1991, although it appeared in the literature in 1954 as a hypothetical molecule for theoretical study [8]. Cyclacenes, being the building blocks of several important carbon nanostructures, like nanotubes and nanotories, have become important materials for experimental and theoretical investigation [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Cyclacenes are found to appear as intervening belt in the elongated fullerenes [1], [2], [3], [18], [19], [20], [21]. Wegner presented the first synthesis of an aromatic nanobelt on the way to carbon nanotubes [22]. The synthesis of one type of cyclacenes through iterative Wittig reactions followed by a nickel-mediated aryl–aryl coupling reaction was reported, and ultimately the cylindrical shape of its belt structure was confirmed by X-ray crystallography [23]. Very recently, the difficulties for the synthesis of the cyclacenes over the linear analogues due to their radicaloid character were addressed, and the potential use of these systems as molecular templates for the growth of well-shaped carbon nanotubes were explored [24] using Finite-Temperature Density Functional Theory (FT-DFT) and restricted-active-space (RAS) spin-flip (SF) methods. Battaglia et al. [25], [26] used the complete active space self-consistent field method and second-order n-electron valence perturbation theory to investigate the radicality and the vertical singlet–triplet energy gap of cyclacenes as a function of the system size and performed a theoretical study for computing the orbital energies and eigenvectors analytically for such molecules at tight-binding level. DFT calculations on heterocyclacenes with respect to geometry and relative energies were made, and thermodynamically stable macrocycles built from linear annelation of conjugated rings to a hoop shape were predicted [27].
Very recently, eigensolutions (eigenvalues and eigenvectors) [1], [2] of {X,Y}-cyclacenes with X (=C, N, B, etc.) and Y (=C, N, B, etc.) were obtained in analytical forms under the nearest neighbour (nn) as well as the next-nearest-neighbour (nnn) approximations. The weight of an edge (bond) between two vertices (X and Y) under nnn approximations is expressed as
2 Methodology
Considering a butadiene unit as the recurring unit n-{X,Y}-cyclacene graph shown in Figure 1, the analytical expressions of eigenvaluues (λt) and eigenvectors or molecular orbitals (
Now the eigenvectors (MOs) in the t-th representation can be expressed [1], [2] as
On the basis of (2), two secular equations are obtained, as given below, that can be used for expressing eigenvalues and eigenvectors or molecular orbitals (MOs) of a cyclacene graph G.
The relations given in (3) result in the following eigenvalue equations [1], [2]
The solutions of (4) results in the eigenvalue expression [1], [2] as follows
Using (5) along with the required normalization conditions for such cases, (2) can be found to be written as
where
and
Now, to express the Hückel’s quantities [1], [2], [3], [4], [14], [15], [16], [17], [29], [30], [31], we need to find out the eigenvectors (MOs) that are occupied, i.e. to find out the half of the total eigenvalues that have larger values in the β unit than the rest. In the case of m = 0, as we see from the eigenvalue expressions given in (5) that the eigenvalues in the β unit follow the decreasing order as
2.1 Critical Values of m for {X,Y}-Cyclacenes With Even n
For an even n,
Case I: If the value of hX exceeds (
Now putting
whose solution gives the critical value of m (taking the positive root) as
below which the eigenvalues obey the relation
Case II: If (
As for {B,N}-cyclacenes
i.e. for such cyclacenes, frontier orbitals do not change with the change in m values within the specified range.
For other cyclacenes, equating
whose acceptable root is the positive as m is positive and is given as
We can see from (14) that the critical m does not depend on the number of rings of cyclacene. The values of m were calculated for different {X,Y}-cyclacenes with X, Y = C, N, B and are given in Table 1. The values of hX, hY, kXX,
{X,Y}-Cyclacene | Critical values of m | |
---|---|---|
n-even | n-odd | |
{C,C} | 0.434 | 0.420 |
{N,N} | 0.434 | 0.420 |
{B,B} | 0.434 | 0.420 |
{C,N} | 0.693 | 0.452 |
{N,C} | 0.370 | 0.415 |
{C,B} | 0.386 | 0.416 |
{B,C} | 0.575 | 0.436 |
{N,B} | 0.543 | 0.432 |
2.2 Critical Values of m for {X,Y}-Cyclacenes With Odd n
For an odd n, two eigenvalues,
For {B,N}-cyclacenes,
which is always positive for all values of m (
For all other cyclacenes, the critical value of m, below which
that ultimately results in m in following form
As for n = 3, the coefficient of m shown in (16) is negative, the value of m in (17) would be taken as
whereas for all other values of n, the coefficient of m given in (16) is positive; thus, the value of m would be taken as
From (16), it reveals that the critical value of m for odd cyclacenes depends on n, i.e. the number of rings in a given cyclacene and approximate values of m in varying n can be calculated with the help of (18) and (19).
For a large n, the value of m as given by the (19) reaches at
The value of m given in (20) is the value up to which one can ascertain about the validity of
Thus, to calculate the Hückel MO (HMO) quantities [1], [2], [3], [4], [14], [15], [16], [17], [29], [30], [31], [32], [33], [34], [35], [36] under nnn approximations [2], [28] within the range of m given in Table 1, we need to consider the MOs with eigenvalues
Using (5), (21)–(24), the HMO quantities, viz., electron density, charge density, bond orders, free valences, total π-electron energy, etc., were derived in analytical forms and are given in the following sections.
3 Hückel Molecular Orbital (HMO) Quantities
From the symmetry of a cyclacene graph, we can see that there are two different sets of vertices (1 and 2) and only two different sets of symmetry equivalent bonds (1–2 and 2–3); thus, for such systems, there are two sets of charge densities, two sets of bond orders, and hence, two sets of free valences that were derived in analytical forms as discussed in the following sections.
3.1 Electron Density and Charge Density
Electron density (
Thus, for the odd n with the help of (21), (22), and (25), the electron and charge densities can be expressed as
In the case of the odd cyclacenes with X = Y, the above (26) and (27) reduce to the following equations on substitution of (5)
and
respectively.
For the even n, as we see from the expressions of the MOs that at t = n/2, the MO
Using (5), the above (30) and (31) for the {X,Y}-cyclacenes with X = Y are converted to
and
From (28), (29), (32), and (33), we can see that for the atom-homogeneity (X = Y) of cyclacenes, the electron and, hence, the charge densities are independent on the atom made up of the cyclacene if each atom of the molecule carries a π-electron.
From the above (26), (27), (30), and (31), we have
For {X,Y}-cyclacenes with no nnn interactions and with X = Y, i.e. m = 0, (28), (29), (32), and (33) reduce to
which is consistent with (34). Electron densities and charge densities for {X,X}-cyclacenes with X = C, B, N are calculated using (28), (29), (32), and (33) and given in Table 2.
{X,Y}-Cyclacene | n | ED | CD | BO | FV | Eπ in β unit | ||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 1 | 2 | 1–2 | 2–3 | 1 | 2 | {C,C}/{N,N}/{B,B} | ||
{C,C} {N,N} {B,B} m = 0.0 | 3 | 1.000 | 1.000 | 0.000 | 0.000 | 0.622 | 0.379 | 0.489 | 0.110 | 17.190/33.287/9.556 |
4 | 1.000 | 1.000 | 0.000 | 0.000 | 0.576 | 0.477 | 0.580 | 0.103 | 22.246/43.721/12.154 | |
5 | 1.000 | 1.000 | 0.000 | 0.000 | 0.599 | 0.418 | 0.533 | 0.115 | 28.154/54.991/15.494 | |
6 | 1.000 | 1.000 | 0.000 | 0.000 | 0.588 | 0.449 | 0.556 | 0.107 | 33.613/65.820/18.443 | |
7 | 1.000 | 1.000 | 0.000 | 0.000 | 0.594 | 0.431 | 0.544 | 0.113 | 39.309/76.883/21.599 | |
8 | 1.000 | 1.000 | 0.000 | 0.000 | 0.591 | 0.441 | 0.550 | 0.109 | 44.874/87.816/24.640 | |
{C,C} {N,N} {B,B} m = 0.1 | 3 | 1.032 | 0.968 | −0.032 | 0.032 | 0.621 | 0.381 | 0.491 | 0.110 | 17.200/33.296/9.564 |
4 | 1.021 | 0.979 | −0.021 | 0.021 | 0.575 | 0.479 | 0.582 | 0.103 | 22.254/43.729/12.161 | |
5 | 1.031 | 0.969 | −0.031 | 0.031 | 0.598 | 0.420 | 0.535 | 0.115 | 28.169/55.006/15.507 | |
6 | 1.024 | 0.976 | −0.024 | 0.024 | 0.587 | 0.451 | 0.557 | 0.107 | 33.627/65.835/18.456 | |
7 | 1.029 | 0.971 | −0.029 | 0.029 | 0.593 | 0.433 | 0.545 | 0.113 | 39.330/76.903/21.617 | |
8 | 1.026 | 0.974 | −0.026 | 0.026 | 0.590 | 0.443 | 0.552 | 0.109 | 44.894/87.837/24.658 | |
{C,C} {N,N} {B,B} m = 0.2 | 3 | 1.064 | 0.936 | −0.064 | 0.064 | 0.617 | 0.389 | 0.497 | 0.109 | 17.229/33.325/9.590 |
4 | 1.042 | 0.958 | −0.042 | 0.042 | 0.573 | 0.484 | 0.586 | 0.102 | 22.280/43.755/12.184 | |
5 | 1.062 | 0.938 | −0.062 | 0.062 | 0.596 | 0.426 | 0.541 | 0.115 | 28.217/55.052/15.549 | |
6 | 1.050 | 0.950 | −0.050 | 0.050 | 0.585 | 0.457 | 0.562 | 0.106 | 33.673/65.879/18.495 | |
7 | 1.059 | 0.941 | −0.059 | 0.059 | 0.591 | 0.438 | 0.550 | 0.112 | 39.392/76.964/21.671 | |
8 | 1.053 | 0.947 | −0.053 | 0.053 | 0.588 | 0.449 | 0.557 | 0.108 | 44.959/87.900/24.714 | |
{C,C} {N,N} {B,B} m = 0.3 | 3 | 1.099 | 0.901 | −0.099 | 0.099 | 0.612 | 0.401 | 0.508 | 0.108 | 17.281/33.375/9.634 |
4 | 1.066 | 0.934 | −0.066 | 0.066 | 0.569 | 0.494 | 0.594 | 0.100 | 22.326/43.799/12.223 | |
5 | 1.095 | 0.905 | −0.095 | 0.095 | 0.591 | 0.436 | 0.551 | 0.114 | 28.299/55.133/15.620 | |
6 | 1.077 | 0.923 | −0.077 | 0.077 | 0.580 | 0.467 | 0.572 | 0.105 | 33.753/65.958/18.565 | |
7 | 1.090 | 0.910 | −0.090 | 0.090 | 0.586 | 0.448 | 0.560 | 0.112 | 39.502/77.072/21.766 | |
8 | 1.082 | 0.918 | −0.082 | 0.082 | 0.583 | 0.459 | 0.566 | 0.107 | 45072/88.011/24.813 | |
{C,C} {N,N} {B,B} m = 0.4 | 3 | 1.136 | 0.864 | −0.136 | 0.136 | 0.603 | 0.419 | 0.526 | 0.107 | 17.359/33.452/9.702 |
4 | 1.092 | 0.908 | −0.092 | 0.092 | 0.562 | 0.508 | 0.608 | 0.100 | 22.396/43.868/12.285 | |
5 | 1.130 | 0.870 | −0.130 | 0.130 | 0.583 | 0.451 | 0.566 | 0.115 | 28.423/55.255/15.728 | |
6 | 1.107 | 0.893 | −0.107 | 0.107 | 0.573 | 0.482 | 0.586 | 0.104 | 33.876/66.078/18.672 | |
7 | 1.124 | 0.876 | −0.124 | 0.124 | 0.579 | 0.462 | 0.574 | 0.113 | 39.667/77.233/21.910 | |
8 | 1.113 | 0.887 | −0.113 | 0.113 | 0.575 | 0.474 | 0.581 | 0.107 | 45.245/88.180/24.963 |
3.2 Bond Orders
The bond order prs of the bond (r-s) between the r and s atoms is defined [4], [16], [17], [29], [30], [31], [33], [34], [35], [36], [37], [38], [39], [40] as
where crj is the jth MO coefficient at the rth atom of cyclacene, and Nj is the population of the jth MO.
In the case of the odd n, from (5), we can see that for t = n, such cyclacene graph has nondegenerate eigenvalues, but for t ≠ n, eigenvalues are doubly degenerate as the tth and (n-t)th representations are equivalent. For this case, the expressions of bond orders given in (36) take the form
Taking r-s as 1–2 we have
Thus, putting the values of the coefficients in (38), the final form of bond order of the 1–2 bond is as follows
Similarly, the bond order of the 2–3 bond can be expressed as
As
Equation (41) finally takes the following form. Following substitution, the coefficients form (21) and (22)
In the case of {X,X}-cyclacenes with odd number of vertices, the above (39) and (42) for bond orders get reduced to
In the case of the even cyclacene graphs, the representations corresponding to t = 0 and n/2 are nondegenerates, whereas the rest of the representations are doubly degenerate. For such cyclacene graphs, the bond orders can be expressed as
Thus,
For t = n/2 from the expressions of the MOs, we can see that the coefficients of
which takes the final form as below
For such cyclacenes, the bond order of the 2–3 bond can be expressed as
Putting the values of
which is finally reduced after incorporation of the values of the coefficients from (21) and (22) to the following form
For {X,X}-cyclacenes with even n, (47) and (50) are reduced to
and
Thus, (43), (44), (51), and (52) show that the {X,Y}-cyclacenes with atom-homogeneity (i.e. X = Y), and the bond orders do not depend on the atom made up of the cyclacene if each atom of the molecule carries an π-electron. The bond orders of the 1–2 and 2–3 bonds for the {X, X}-cyclacenes with X = C, B, N were calculated using (43), (44), (51), and (52) and are given in Table 2.
3.3 Free Valence
Free valence [33], [34], [35], [41], [42], [43], [44], [45], [46] is considered as one of the criteria to measure the reactivity of an atom in a molecule. This concept can be considered as the quantum mechanical expression of ideas advanced at the beginning of the century by Flürscheim (affinity demand), Werner (residual affinity), Thiele (partial valence) according to which the atoms in a molecule may not have used up their capacity to combine [47]. Each atom in a conjugated system (molecule) can exert only a certain π-bonding power (real), although it has a maximum capacity (theoretical, threshold) for combining (bonding) in the molecule. The difference between the theoretical capacity and real capacity of bonding is always positive. As this difference goes up, the surplus valence increases at the position for accommodating another group or atom at that position; thus, reactivity increases as free valence increases at a position. Free valence was originally defined by Linus Pauling in terms of bond orders calculated from the resonance valence bond theory. It was shown later on that a similar conclusion of free valence can be reached from the HMO theory, and it was C.A. Coulson who first proposed the free valence (Fr) in the HMO formalism [41], [42] as
where Nmax is the maximum bond order at atom r, and prs is the bond order of the bond between the r and s atoms. Here, Nmax is the sum of the bond orders at the central vertex (atom) of a hypothetical molecule or molecular species where the central vertex is bonded to three peripheral vertices each with one electron including the central vertex.
The free valence of vertex 1 or 4 can be expressed with the help of (53) as
For an odd n, putting the bond orders value from (39) in (54), we have
Similarly, for the even n, inserting the bond order value from (47) in (54), we have
The free valence at atoms 1 or 4 for {X,X}-cyclacenes with odd and even n can be obtained as
and
respectively.
The free valence of the vertex 2 or 3 for such cyclacenes can be expressed as
Now, for the cyclacenes with an odd n, the above expression (59) is reduced in taking the values of bond orders from (39) and (42) as follows
Similarly, for an even n, using (47) and (50), (58) is reduced to
n | Strain energy per fragment ( | |||||||
---|---|---|---|---|---|---|---|---|
M06-2X | M06-2X D3(BJ) | m = 0.0 | m = 0.1 | m = 0.2 | m = 0.3 | m = 0.4 | m = 0.43 | |
6 | 157.7 | 157.6 | 0.0 | 0.0146 | 0.0599 | 0.1401 | 0.2629 | 0.31 |
7 | 122.2 | 121.6 | 0.0 | 0.0203 | 0.0829 | 0.1923 | 0.3573 | 0.4197 |
8 | 93.6 | 93.0 | 0.0 | 0.0209 | 0.0853 | 0.1987 | 0.3717 | 0.4376 |
9 | 81.2 | 80.7 | 0.0 | 0.0253 | 0.1032 | 0.2397 | 0.4462 | 0.5245 |
10 | 65.5 | 65.0 | 0.0 | 0.0269 | 0.1095 | 0.2548 | 0.4758 | 0.5598 |
11 | 60.2 | 59.8 | 0.0 | 0.0305 | 0.1243 | 0.2887 | 0.538 | 0.6326 |
As given earlier, with the free valence at atom 2 for {X,X}-cyclacenes with odd and even n, (60) and (61) change to the following forms
and
Similar to the cases of electron (and, hence, charge) densities and bond orders, the free valences are independent of the atom the cyclacene is made up if each atom of cyclacene with atom-homogeneity caries a π-electron as is evident from (57), (58), (62), and (63). Free valences of vertices 1 and 2 for {X,X}-cyclacenes with X = C, B, N are calculated using (62) and (63) and given in Table 2.
3.4 Total π-Electron Energy
Assuming that each atom in the {X,Y}-cyclacene molecule possesses one π-electron, then, the expression of total π-electron energy [1], [2], [3], [4], [14], [15], [17], [29], [30], [31], [34], [35], [36], [48], [49], [50] can be derived with the help of (5) as
For the {X,Y}-cyclacenes with X = Y, i.e.
The systems that belong to such {X,Y}-cyclacene systems are {C,C}, {C,N}2n+, {N,C}2n+, {N,N}4n+, {C,B}2n−, {B,C}2n−, {B,B}4n−, etc. The systems {C,N}2n+ and {N,C}2n+ are generated on removal of one electron from each N-atom of {N,N}-cyclacene; the systems {C,B}2n− and {B,C}2n− are realized after adding one electron to each B-atom of {B,B}. Similarly, the system {N,N}4n+ comes into the picture on removal of one electron from each N-atom of {N,N}-cyclacene system, whereas the {B,B}4n− appears when one electron is added to each B-atom of {B,B}-cyclacene.
Putting (65) under the condition of non-nnn interaction (i.e. m = 0), we have
The systems that belong to such {X,Y}-cyclacene systems are {C,C}, {N,N}4n+, {B,B}4n-, etc. The total π-electron energies for the {X,X}-cyclacenes with X = C, B, N were calculated using (65) and are given in Table 2.
3.4.1 Strain Energies of {C,C}-cyclacene and Their Total π-Electron Energies Under nnn Interactions
Strain energies [51] per fragment (EStrain) for the {C,C}-cyclacenes were obtained from the TD-DFT level calculation with dispersion-uncorrected (M06-2X) and dispersion-corrected (M06-2X-D3(BJ)) methods and with the 6-31+G∗ basis set (given in Table 3). The formula used for such calculations is,
where
One such plot with m = 0.3 is shown in Figure 2.
3.5 HOMO–LUMO or Band Gap (Δ)
HOMO–LUMO or band gap (Δ) is an important quantity to predict reactivity and stability of a molecule [2], [29], [30], [31], [48]. To find out this quantity for {X,Y}-cyclacenes, we need to figure out first the HOMO and LUMO energy levels from the eigenvalue expressions given in (20). As eigenvalues follow the order
Case-I: Cyclacenes with {X,Y} = {C,C}, {B,B}, {N.N}, {C,B}, and {B,C}
HOMO–LUMO expression for n-even
HOMO–LUMO expression for n-odd
n-{C,C}-cyclacenes | HOMO–LUMO gap (Δ) in β unit | ||||||
---|---|---|---|---|---|---|---|
m = 0.0 | m = 0.1 | m = 0.2 | m = 0.3 | m = 0.4 | m = 0.43 | ||
6 | 1.24 | 0.62 | 0.61 | 0.60 | 0.59 | 0.20 | 0.02 |
7 | 0.66 | 0.34 | 0.34 | 0.35 | 0.35 | 0.37 | 0.37 |
8 | 0.98 | 0.41 | 0.42 | 0.43 | 0.44 | 0.20 | 0.02 |
9 | 0.80 | 0.28 | 0.22 | 0.22 | 0.23 | 0.24 | 0.24 |
10 | 0.80 | 0.30 | 0.31 | 0.32 | 0.33 | 0.20 | 0.02 |
11 | 0.92 | 0.15 | 0.15 | 0.15 | 0.16 | 0.17 | 0.17 |
Thus, the HOMO–LUMO or band gaps (Δ) for the even and odd n (some higher values) in the β unit can be expressed from (67) and (68) as
for the even n and
for odd-n, respectively.
Case-II: Cyclacenes with {X,Y} = {B,N}, {C,N}, {N,B}, and {N,C}
For the even-n, the expressions of HOMO and LUMO energies for such cyclacenes are as follows:
whereas for the odd-n, the HOMO–LUMO energies are the same as given in (68) and HOMO–LUMO gap is same as given in (70) above.
Thus, the expression of the HOMO–LUMO or band gap (Δ) for such systems with the even-n in the β unit would look like
Here, we can see that for the {X,Y}-cyclacenes under the approximations of a large number of repeating units (i.e. n → ∞), all the expressions of the HOMO–LUMO or band gap (Δ) given in (69), (70), and (72) reduce to the same expression as shown by (72).
3.5.1 Vertical Absorption Energy Differences Between Singlet–Triplet States for {C,C}-Cyclacenes and Their HOMO–LUMO Gaps (Δ)
The HOMO–LUMO gap of a molecule could be a reasonable factor in determining the singlet–triplet splitting behaviour. This led us to search the correlations of the HOMO–LUMO gaps (Δ) with that of the vertical absorption energy differences
3.6 Density of States (DOS)
The density of states (DOS) [11], [48], [52], [53], [54] of a system (molecular network or solid) gives the idea of the number of states per unit energy range at a given energy level that are supposed to be occupied, and it is an important theoretical quantity used in the description of electronic states of the system concerned. The cyclic dimer (CD) is a dimer obtained from a system with repeating monomers under cyclic boundary conditions, and its importance in the routine derivation of the DOS is now well recognized [11], [53], [54], [55]. From the HMO method, it has been found that almost all molecular orbital energies of the CD of a periodic polyhex network coincide [11], [53] with the singular points in its DOS, which has been reconfirmed by the extensive variable-β,γ version of PPP MO-type calculations. Thus, the secrecy of the electronic properties of the infinitely large periodic polyhex networks is shown to be hidden in their respective CD graphs that are called either polyomino or kegome graphs [11].
The CD graph (GCD) of the {X,Y}-cyclacene is shown in Figure 4 whose vertex weights are shown on the loops, the edge weights on the edges marked as a broken line given below the graph GCD in Figure 4, the edge weights on the two edges (marked as solid lines) that are parallel but perpendicular to the bottom line of the page are of weight kYY each, whereas all the solid edges are of weight kXY each. The graph GCD possesses a plane of symmetry and is factored into two graphs
The HOMO–LUMO gap or band gap (Δ) of the cyclic dimer in the β unit can be obtained from (74) as
As we see from (72) and (75) above, in presence of the nnn interactions, the expression of the density of states obtained from the HOMO–LUMO or band gap (Δ) under the approximations of n → ∞ differs from that obtained from the CD. However, in absence of the nnn interaction (i.e. for m = 0), both the approaches end up with the same expression as given in (72).
4 Conclusion
An {X,Y}-cyclacene graph with varying X (=C, N, B) and Y (=C, N, B) as well as a number of hexagonal rings represents a variety of cyclacenes. The HMO quantities, viz., electron and charge densities, bond orders, free valences, total π-electron energies, HOMO–LUMO or band gaps, etc., under the nnn approximations for such systems were expressed in compact analytical forms that are very much useful in calculating the values of the respective quantities in a very easy and convenient manner. Although the HMO quantities for the {X,X}-cyclacenes were calculated and are shown in Table 2, the said quantities for the {X,Y}-cyclacenes can also be calculated with the use of the analytical expressions given in (26), (27), (30), (31), (39), (42), (47), (50), (55), (56), (60), (61), and (64). Facile computer programs can be made with the analytical expressions for making the calculations further simple and very less time consuming. Furthermore, the HMO quantities are very useful in judging several properties of the concerned molecule, for example, the relative charge density (electron density) on an atom indicates the vulnerability on its position for an electrophile (nucleophile), the bond order of a bond dictates its relative strength, the free valence of a atom measures how much its position is susceptible to an electrophilic attack compared to other atomic positions, the total π-electronic energy give the relative stability, the HOMO–LUMO or band gap dictates the stability/reactivity, π-electron transitions, conducting properties of the concerned cyclacenes, etc. Such nnn interaction approximations actually release, to some extent, the strain that results due to the geometrical structures of such cyclacenes. The vertical absorption energy difference for singlet–triplet states bears excellent linear correlation with the HOMO–LUMO gaps for a certain m value (m = 0.3) in case of an even n, but for an odd n, such energy difference remains invariant. The eigenvalues of the cyclic dimer of the cyclacene represents the singular point of the density of states, which is an important quantity for exploring the electronic behaviour of the material. Thus, considering all these, the present study has some important relevance.
Acknowledgement
One of the authors, Mr. Swapnadeep Mondal, thanks the Council of Scientific and Industrial Research (CSIR), New Delhi, for financial assistance under the Junior Research Fellowship. Authors are thankful to the learned Reviewers for their valuable suggestions in improving the article.
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