Cage-Shaped Molecules Derived by Applying the Edge Strategy to a Cubane Skeleton

Abstract

The edge strategy of Fujita’s unit-subduced-cycle-index (USCI) approach (S. Fujita, “Symmetry and Combinatorial Enumeration in Chemistry,” Springer, 1991) is shown to be effective for derivation based on a cubane skeleton, where twelve edges accommodate a set of methano-bridges and/or ethano-bridges.

1 INTRODUCTION

Systematic derivation of high symmetry chiral molecules has been discussed qualitatively by Farina and Morandi [1]. A more quantitative approach based on mathematical foundations has been developed by Fujita under the name the edge strategy (cf. Chapter 17 of [2]), which is one embodiment of Fujita’s unit-subduced-cycle-index (USCI) approach [2]. The edge strategy has been applied to tetrahedrane of T_d-symmetry [3,4], cyclobutadiene of D_2d-symmetry [4], dodecahedrane of I_h -symmetry [5], and fullerene of I_h -symmetry [6].

The present article is devoted to the application of the edge strategy to a cubane skeleton of O_h -symmetry, because cubane itself [7] and its derivatives such as basketane [8,9] and D₃ -trishomocubane [10,11,12] have attracted the interest of synthetic organic chemists, as summarized in a review [13]. Although systematic discussions based on D₃-twisted bicyclo[2.2.2]octane molecular framework as a common structural unit have appeared [14], the edge strategy applied to a cubane skeleton aims at providing more quantitative perspectives.

2 THE EDGE STRATEGY APPLIED TOA CUBANE SKELETON

2.1 Theoretical Foundations

According to the edge strategy, an appropriate set of edges (bonds) selected from the twelve edges (bonds) of a cubane skeleton 1 is replaced by a set of m₁ methano-bridges (solid circles) and m₂ ethano-bridges (pairs of solid circles), so as to give a cage-shaped molecule. For example, a solid circle is placed to give 2, which corresponds to the formation of homocubane 3 with a single methano-bridge, as shown in Figure 1. The substitution of a pair of solid circles produces 4, which corresponds to the formation of basketane 5 with a single ethano-bridge.

Figure 1.

 Edge strategy applied to a cubane skeleton. A methano-bridge (a solid circle) or an ethane-bridge (a pair of solid circles) is placed on an edge to give a cage-shaped molecule.

The cubane skeleton 1 belongs to the point group O_h. Its 33 subgroups up to conjugacy construct a non-redundant set of subgroups (SSG) [15]:   

$\begin{array}{c}{\text{SSG}}_{O_h}\\ =\{\underset{\left(1\right)}{C_1},\underset{\left(2\right)}{C_2},\underset{\left(2\right)}{{C^{\prime }}_2},\underset{\left(2\right)}{C_s},\underset{\left(2\right)}{{C^{\prime }}_s},\underset{\left(2\right)}{C_i},\underset{\left(3\right)}{C_3},\\ \underset{\left(4\right)}{C_4},\underset{\left(4\right)}{S_4},\underset{\left(4\right)}{D_2},\underset{\left(4\right)}{{D^{\prime }}_2},\underset{\left(4\right)}{C_{2v}},\underset{\left(4\right)}{{C^{″}}_{2v}},\underset{\left(4\right)}{C_{2h}},\underset{\left(4\right)}{{C^{\prime }}_{2h}},\\ \underset{\left(6\right)}{D_3},\underset{\left(6\right)}{C_{3v}},\underset{\left(6\right)}{C_{3i}},\underset{\left(8\right)}{D_4},\underset{\left(8\right)}{C_{4v}},\underset{\left(8\right)}{C_{4h}},\underset{\left(8\right)}{D_{2d}},\underset{\left(8\right)}{{D^{\prime }}_{2d}},\underset{\left(8\right)}{D_{2h}},\underset{\left(8\right)}{{D^{\prime }}_{2h}},\\ \underset{\left(12\right)}{T_{}},\underset{\left(12\right)}{D_{3d}},\underset{\left(16\right)}{D_{4d}},\underset{\left(24\right)}{O_{}},\underset{\left(24\right)}{T_h},\underset{\left(24\right)}{T_d},\underset{\left(48\right)}{O_h}\},\end{array}$(1)

where the subgroups are aligned in the ascending order of their orders, as listed in pairs of parentheses.

The twelve edges of the cubane skeleton 1 construct an orbit governed by the coset representation $O_h(/{C^{″}}_{2v}),$ where the size of the orbit is equal to the degree of the coset representation, i.e., $\left|O_h\right|/\left|{C^{″}}_{2v}\right|=48/4=12.$

2.2 The Fixed-Point-Matrix (FPM) Method of Fujita’s USCI Approach

Fujita’s USCI approach supports four methods of symmetry-itemized enumeration [2]. Among them, the present article applies the fixed-point-matrix (FPM) method to count cubane derivatives. It is because the FPM method gives results in succinct tabular forms, while the other methods provide equivalent results in more complex forms (e.g., in the form of generating functions). A set of unit subduced cycle indices with chirality fittingness (USCI-CFs) for the $O_h(/{C^{″}}_{2v})$-orbit is cited from Tables 4 and 5 of [16] as follows:   

$\begin{array}{c}(b_1^{12},b_2^6,b_1^2{b}_2^5,a_1^4{c}_2^4,a_1^2{c}_2^5,c_2^6,b_3^4,\\ b_4^3,c_4^3,b_4^3,b_2^2{b}_4^2,a_2^4{c}_4,a_2^2{c}_4^2,a_1^2{a}_2{c}_4^2,a_2^2{c}_4^2,a_2{c}_2{c}_4^2,\\ b_3^2{b}_6,a_3^2{c}_6,c_6^2,b_4{b}_8,a_4^3,a_4{c}_8,a_4{c}_8,a_4^2{c}_4,a_4^3,a_2^2{c}_8,\\ b_{12},a_6{c}_6,a_4{a}_8,b_{12},a_{12},a_{12},a_{12}),\end{array}$(2)

where the USCI-CFs are aligned in accord with the SSG (Eq. 1).

Suppose that each edge is substituted by a methano-bridge, an ethano-bridge, or a null unit (no substitution). Dummy variables x, y, and _z are respectively used for null units (no substitution), for methano-brides, and for ethano-bridges. Then, a single inventory function is used as follows:   

$a_d=b_d=c_d=x^d+y^d+z^d.$(3)

The inventory function (Eq. 3) is introduced into the USCI-CF of each subgroup listed in equation. 2. The resulting equation is expanded to give a generating function for the subgroup at issue, where the coefficient of each term $x^{m_0}{y}^{m_1}{z}^{m_2}$ represents the number of fixed derivatives with m₀ null units (no substitution), m₁ methano-bridges (solid circles), and m₂ ethano-bridges (pairs of solid circles), where m₀+m₁+m₂ = 12. A fixed-point vector (FPV) is obtained by aligning the coefficients of $x^{m_0}{y}^{m_1}{z}^{m_2}$ in the order of the SSG (Eq. 1). Then, the corresponding fixed-point matrix (FPM) is obtained by moving m₀, m₁, and m₂ to satisfy m₀+m₁+m₂ = 12, where the FPV for $x^{m_0}{y}^{m_1}{z}^{m_2}$ appears as a row vector of the FPM.

The resulting FPM is multiplied by the inverse mark table $M_{O_h}^{-1}$ (e.g., Table 2 of [16]) to give an isomer-counting matrix (ICM):   

$\text{ICM}=\text{FPM}\times M_{O_h}^{-1},$(4)

where each row shows the numbers of cage-shaped derivatives with $x^{m_0}{y}^{m_1}{z}^{m_2}$ in the alignment of the SSG (Eq. 1). The procedure for calculating the ICM has been programmed by means of the Maple system in a similar way to the source code described in [16].

Table 1.  Symmetry-Itemized Numbers of Cage-Shaped Molecules (Part I)

Table 2.  Symmetry-Itemized Numbers of Cage-Shaped Molecules (Part II)

Because the term $x^{m_0}{y}^{m_1}{z}^{m_2}$ appears symmetrically in the generating function, the partition [m₀, m₁, m₂] (m₀ ≥ m₁ ≥ m₂) is used to represent terms $x^{m_0}{y}^{m_1}{z}^{m_2},$$x^{m_0}{y}^{m_2}{z}^{m_1},$$x^{m_1}{y}^{m_0}{z}^{m_2},$$x^{m_1}{y}^{m_2}{z}^{m_0},$ and so on.

The ICM of the partition [m₀, m₁, 0] is shown in Table 1, where each row contains the symmetry-itemized values in the alignment of SSG (Eq. 1). The partition [m₀, m₁, 0] (corresponding to $x^{m_0}{y}^{m_1}$ or $x^{m_0}{z}^{m_2}$) means the single substitution of methano-bridges (or ethano-bridges) on edges. Each value of Table 1 represents the number of pairs of (self-) enantiomers, where a pair of self-enantiomers means an achiral derivative. Note that the values for [11,0,1] are equal to the values collected in the [11,1,0] -row; the values for [5,7,0] (or for [5,0,7]) are equal to the values collected in the [7,5,0]-row; and so on.

Table 2 collects symmetry-itemized numbers of cage-shaped molecules, each of which exhibits mixed appearance of m₁ methano-bridges (m₁ > 0) and m₂ ethano-bridges (m₂ > 0).

3 CAGE-SHAPED MOLECULES DERIVED BYTHE EDGE STRATEGY

3.1 Cage-Shaped Molecules with Either Methano- or Ethano-Bridges

Table 1 summarizes the symmetry-itemized enumeration of cage-shaped molecules with methano- or ethano-bridges.

The value 1 at the intersection between the [12,0,0]-row and the O_h-column (the 33rd column) in Table 1 indicates the existence of one O_h-molecule, which corresponds to cubane 1 itself, dodecakismethanocubane ([0,12,0]), or dodecakisethanocubane ([0,0,12]).

The [11,1,0]-row of Table 1 (at the 14th-column) indicates that there appears one ${C^{″}}_{2v}$-molecule, which corresponds to a cage-shaped molecule with one methano-bridge 2 (homocubane 3) or a cage-shaped molecule with one ethano-bridge 4 (bascketane 5, strictly speaking [11,0,1]), as depicted in Figure 1.

The [10,2,0]-row of Table 1 indicates that there are one pair of enantiomeric ${C^{\prime }}_2$-molecules, one ${C^{\prime }}_s$-molecule, one C_2v-molecule, and one ${D^{\prime }}_{2h}$-molecule. These bismethanocubanes are depicted in Figure 2.

Figure 2.

 Cage-shaped molecules (bismethanocubanes) by placing two methano-bridges on a cubane skeleton.

The ${C^{\prime }}_2$-molecule 6(or its enantiomer $\overline{6}$), which is called C₂-bishomocubane, possesses a two-fold rotation axis running through the midpoints of the edges 2–6 and 4–8. Synthesis and absolute configuration of optically active C₂-bishomocubane $\overline{6}$ have been reported [17]. The ${C^{\prime }}_s$-molecule 7 possesses a mirror plane containing vertices 2, 4, 8, 6. The C_2v-molecule 8 possesses a two-fold axis running through the centers of the top and bottom faces. The ${D^{\prime }}_{2h}$-molecule 9 possesses a vertical two-fold axis running through the centers of the top and bottom faces as well as two two-fold axes perpendicular to the vertical axis.

The [9,3,0]-row of Table 1 indicates that there are two pairs of enantiomeric C₁-molecules ($10/\overline{10}$ and $11/\overline{11}$), one pair of enantiomeric ${C^{\prime }}_2$-molecules ($12/\overline{12}$), two C_s-molecules (13 and 14), one ${C^{\prime }}_s$-molecule (15), one ${C^{″}}_{2v}$-molecule (16), one pair of enantiomeric D₃-molecules ($17/\overline{17}$), and one C_3v-molecule (18). These trismethanocubanes are depicted in Figure 3.

Figure 3.

 Cage-shaped molecules (trismethanocubanes) by placing three methano-bridges on a cubane skeleton.

The two-fold axis of the ${C^{\prime }}_2$-molecule 12 (or $\overline{12}$) runs through the midpoints of edges 1–6 and 4–8. The mirror plane of the C_s-molecule 13 (or 14) is perpendicular to the front and back faces and bisects the cube vertically. The mirror plane of the ${C^{\prime }}_s$-molecule 15 is contained in the plane 2–6–4–8. The two-fold axis of the ${C^{″}}_{2v}$-molecule 16 runs through the midpoints of the edges 2–6 and 4–8. The three-fold axis of the D₃-molecule 17 (or $\overline{17}$) runs through the vertices 6 and 4. The three-fold axis of the C_3v-molecule 18 runs through the vertices 6 and 4.

Among the trismethanocubanes depicted in Figure 3, the D₃-molecule 17 or $\overline{17}$ is called D₃-trishomocubane. The syntheses of $17/\overline{17}$ have been reported by Underwood [18] and by Kent et al. [10]. The absolute configuration of (−)D₃-trishomocubane 17 has been determined by Helmchen and Staiger [11], while the absolute configuration of (+)D₃-trishomocubane $\overline{17}$ has been determined by Nakazaki et al. [12].

3.2 Cage-Shaped Molecules with Both Methano- and Ethano-Bridges

Let us examine the ${C^{″}}_{2v}$-column (the 14th column) of Table 2. The value of each row shows the number of cage-shaped molecules, as collected in Figure 4.

Figure 4.

 Cage-shaped molecules of ${C^{″}}_{2v}$-symmetry by placing methano- and ethano-bridges on a cubane skeleton.

For the purpose of depicting cage-shaped molecules of ${C^{″}}_{2v}$-symmetry, we presume that the two-fold axis of ${C^{″}}_{2v}$ runs through the midpoints of the edges of 2–6 and 4–8. According to Fujita’s USCI approach [2], the derivation of such ${C^{″}}_{2v}$-molecules is controlled by the following subduction:   

$\begin{array}{c}{O}_h(/{C^{″}}_{2v})\downarrow {C^{″}}_{2v}\\ =2{C^{″}}_{2v}(/C_1)+{C^{″}}_{2v}(/C_s)+2{C^{″}}_{2v}(/{C^{″}}_{2v}).\end{array}$(5)

This subduction corresponds to the USCI-CF $a_1^2{a}_2{c}_4^2$, which appears as the 14th element of equation. 2.

The twelve edges of a ${C^{″}}_{2v}$-molecule are divided into the following orbits:   

$\left\{2-6\right\}{\Delta }_1{C^{″}}_{2v}(/{C^{″}}_{2v})$(6)

  

$\left\{4-8\right\}{\Delta }_2{C^{″}}_{2v}(/{C^{″}}_{2v})$(7)

  

$\left\{1-5,3-7\right\}{\Delta }_3{C^{″}}_{2v}(/C_s)$(8)

  

$\left\{1-2,2-3,5-6,6-7\right\}{\Delta }_4{C^{″}}_{2v}(/C_1)$(9)

  

$\left\{1-4,3-4,5-8,7-8\right\}{\Delta }_5{C^{″}}_{2v}(/C_1)$(10)

Each orbit accommodates methano-bridges (or ethano-bridges) in accord with its size.

For example, one ${C^{″}}_{2v}$-molecule 19, which corresponds to the value appearing at the intersection between the [10,1,1]-row and the ${C^{″}}_{2v}$-column (the 14th column) of Table 2, is depicted by placing an ethano-bridge on the orbit Δ₁, a methano-bridge on the orbit Δ₂, and null units on the remaining orbits Δ₃, Δ₄, and Δ₅.

The derivation of 26 and 27 illustrates two different modes of packing of ethano-bridges. Two ethano-bridges separately occupy two one-membered orbits Δ₁ and Δ₂ in 26, so that they are inequivalent to each other. On the other hand, two ethano-bridges occupy a two-membered orbit Δ₃ of 27, so that they are equivalent to each other.

CONCLUSION

The edge strategy proposed in Chapter 17 of [2] is effective for derivation based on a cubane skeleton, where twelve edges accommodate a set of methano-bridges and/or ethano-bridges. The fixed-point-matrix (FPM) method of Fujita’s USCI approach is applied to symmetry-itemized enumeration of cage-shaped molecules by starting from the cubane skeleton of O_h.

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