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Polytriangulane Wesley D. Allen, Henrik Quanz, and Peter R. Schreiner J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00669 • Publication Date (Web): 01 Aug 2016 Downloaded from http://pubs.acs.org on August 1, 2016

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Journal of Chemical Theory and Computation

Polytriangulane Wesley D. Allen*,1 Henrik Quanz,2 and Peter R. Schreiner*,2 1

Center for Computational Quantum Chemistry and Department of Chemistry, University of Georgia, Athens, Georgia 30602, USA 2

Institute for Organic Chemistry, Justus-Liebig University, Heinrich-Buff-Ring 58, 35392 Giessen, Germany

Abstract The infinite spiro-annelation of cyclopropanes in a non-branched form would produce a σhelicene called polytriangulane, an unknown hydrocarbon with the formula CnHn comprised exclusively of formal C(sp3) atoms. The structure of polytriangulane is elucidated here via a rigorous mathematical analysis of a C85H88 prototype optimized by M06-2X/6-31G(d) density functional theory and an idealized polymer composed of equilateral cyclopropane units. The spiro carbons in polytriangulane form an exact, nonrepeating helix with a steep rise angle near 35°, a radius of only 0.41 Å, and irrational periodicity parameter very close to

τ ideal = 2 [1+ π −1 cos−1 ( 14 )]−1 . A focal point analysis of the ring opening of cyclopropane to propene employing basis sets as large as cc-pCV5Z and correlation treatments as extensive as CCSDT(Q) yields

=17.2 ± 0.1 kcal mol–1.

Subsequent application of

CCSD(T)/CBS theory to a homodesmotic equation for ring aggregation predicts that

=

+16.1 kcal (mol CH)–1 for polytriangulane; hence, this compound is much more stable thermodynamically than acetylene. Similar computations on another hypothetical homodesmotic transformation indicate that the total strain energy in polytriangulane is 42.7 kcal per mole of cyclopropane units.

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Introduction The spiro-annelation of cyclopropanes leads to the family of [n]triangulanes,1-9 whose infinite extension would provide hitherto unknown polytriangulane, a hydrocarbon with the formula CnHn.

In constructing triangulanes beyond [3]triangulane,10-11 linear, helical, and branched

isomers ensue (Scheme 1). The helical [n]triangulanes9 belong to the σ-helicene family,4 which also encompasses the spiro-cyclobutanes,12-13 spiro-cyclopentanes,14-15 oligotwistanes,16-17 and helical diamondoids such as [1(2)3]tetramantane18 (Scheme 2).

The [n]triangulanes,

oligotwistanes, and diamondoids have no internal-rotation degrees of freedom and therefore occur as highly rigid, solitary structures. While the first member of the higher triangulanes, [4]triangulane, was made in racemic form in 1990,9 its first enantioselective synthesis was accomplished by de Meijere et al.5 in 1999. Within a few years de Meijere and co-workers were then able to synthesize enantiomerically pure compounds of this type up to [15]triangulane,1-2 the largest member of all σ-helicenes known to date. Here we go a step further and report on the theoretical and computational exploration of an infinite extension of a C2-symmetric, helical [n]triangulane, named polytriangulane.

Scheme 1. Spiro-annelation of cyclopropane en route to polytriangulane.


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The σ-helicenes are inherently chiral, and their optically rotatory powers mirror their strain energies;19 thus, the helical [n]triangulanes display the largest optical rotations of any alkane of comparable molecular weight.2, 4-5, 20-21 This increase in the Cotton effect has been rationalized on the basis of the Walsh orbital model of the bonding in cyclopropane, in which the strained

carbon-carbon

bonds

exhibit

enhanced

Remarkably, the rotatory power of rigid σ-helicenes2,

Cmethylene 5

p-orbital

contributions.22-23

(as well as π-helicenes24) increases

monotonically with helix length, while it uniformly decreases in this manner for the conformationally flexible polyspirocyclobutanes and polyspirocyclopentanes.13-15

Scheme 2. Other members of the σ-helicene family: flexible trispiranes, rigid tetramantane, and tetratwistane. All structures are depicted in their (M)-helical forms (C2 symmetry).

The question arises whether the helicity of polytriangulane is regular, that is, whether it can be described by a rigorous mathematical formalism, as found for polytwistane.25 As with all increasingly larger, highly strained structures, determination of the incremental thermochemistry and strain energies of polytriangulane relative to its parent cyclopropane provides insight regarding its experimental viability. Here we perform a rigorous mathematical analysis of the structure of polytriangulane by investigating a C85H88 prototype optimized by density functional theory and an idealized polymer composed of equilateral cyclopropane units. We also determine enthalpies of formation of small triangulanes using high-level coupled cluster computations [CCSD(T)] and employ homodesmotic equations to ascertain

and the strain energy of

polytriangulane.


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Results and Discussion Topology of Polytriangulane While polytriangulane has the simple formula CnHn, the molecule contains two sets of equivalent carbon atoms, designated here as Cspiro and Cmethylene. An expanded color-coded view of a polytriangulane segment is shown in Figure 1. The Cspiro atoms, each bonded to four other carbons, constitute a helical chain that circumscribes a cylinder of inner radius R. Methylene units bond to consecutive spiro carbons, forming three-membered (Cspiro, Cmethylene, Cspiro) rings that project outward from the inner wall of the cylinder. Figure 1 shows the helical chain of Cspiro– Cspiro bonds in red; the Cspiro–Cmethylene and C–H bonds appear in blue and black, respectively. Ha,2j+4

Hb,2j+5

C2j+5 Ha,2j+2

C2j+2 C2j

C2j+6

Hb,2j+7

C2j+4 C2j+3

Hb,2j+3

C2j+1 Hb,2j+1

Ha,2j+6

Ha,2j

Figure 1. Geometric patterns and atomic labeling in a polytriangulane segment. The C–C bonds are color-coded to identify the distances r1(red) and r2(blue) = r3(green); C–H bonds are shown in black. The chain of red bonds forms the helical backbone. The colors of the bonds subtending the inter-ring C–C–C angles are θ11(red, red), θ12(red, blue) = θ13(red, green), and θ23(blue, green). Finite [n]triangulanes can only exhibit approximate helical structures but evolve toward a structural limit with a wealth of possibilities as n grows to infinity (Scheme 1). Among the questions of concern are: (a) the period of repetition within the helix, (b) equivalences in the bond distances and angles, (c) the radius and angle of inclination of the helix, (d) the dihedral angle between consecutive cyclopropane rings, and (e) whether the structure contains exact or merely approximate helices.

We answer these and others questions via a comprehensive

mathematical treatment of the atomic positions and geometric degrees of freedom of infinite polytriangulane in the Supporting Information; key aspects of the analysis are summarized here.


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To formulate the Cartesian coordinates of the atoms in polytriangulane, we choose to align the polymer along the z-axis and define the helical lattice function

X(s; R,α ,ζ , β ,δ ) = R { cos (α s + β ) , sin (α s + β ) , 4ζ s + δ } ,

(1)

in which s is an independent variable describing location along the polymer chain, and all other quantities relate to the internal structure of the molecule. The Cartesian coordinates of the carbon atoms can be represented as

(

)

x 2 j Cspiro = X( j; R, α ,ζ ,0, 0)

(2)

and

(

)

x 2 j+1 Cmethylene = X( j + 12 ; ρ R, α , ζ ρ −1, β , δ ) ,

(3)

where j runs over all the integers, and R is the inner radius of the helix formed by the spiro carbons. The carbon framework involves the six internal parameters (R, ρ, α, ζ, β, δ), the last five of which are dimensionless quantities.

The positions of the hydrogen atoms in

polytriangulane can be expressed as

(

x 2 j (H a ) = X j + 12 ; Rρ H ,α ,ζ ρ H−1 , β H ,δ H a

a

a

a

)

(4)

and

(

x 2 j+1 (H b ) = X j + 12 ; Rρ H ,α ,ζ ρ H−1 , β H ,δ H b

b

b

b

).

(5)

For each of the two groups (Ha, Hb), three independent parameters (ρH, βH, δH) are required. The (α, ζ) parameters in eqs 4 and 5 must be identical to those for the C atoms in order for the structure of the monomer units to be replicated along the chain. Formulas for the bond vectors connecting the various atoms can be obtained immediately from eqs 1–5, allowing internal coordinates to be expressed in terms of the helical parameters. The index j can be eliminated in the computation of internal coordinates by using standard

(

)

trigonometric identities such as cos u ± v = cosucos v sinusin v . In this manner ten essential conditions are derived:

(

r12 = 2R 2 8ζ 2 +1− cos α

)

2 r22 = R 2  ρ 2 + 1+ ( 2ζ + ρδ ) − 2 ρ cos ( 12 α + β )   


(6) (7)

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2 r32 = R 2  ρ 2 + 1+ ( 2ζ − ρδ ) − 2 ρ cos ( 12 α − β )   

(

r12 cosθ11 = 2R 2 −8ζ 2 − cos α + cos 2 α

(8)

)

(9)

r2r3 cosθ 23 = R 2  ρ 2δ 2 − 4ζ 2 + 1+ ρ 2 cos α − 2 ρ cos ( 12 α ) cos β 

(10)

r1r3 cos θ13 = R 2  −4ζ ( 2ζ − ρδ ) +1− cos α − 2ρ sin ( 12 α ) sin (α − β ) 

(11)

r1 r2 cosθ12 = R 2  −4ζ ( 2ζ + ρδ ) + 1− cos α − 2ρ sin ( 12 α ) sin (α + β ) 

(12) .

2 rH2 = R 2  ρ 2 + ρ H2 + ( ρ Hδ H − ρδ ) − 2 ρρ H cos ( β − β H )   

(13)

 ρ 2 − ( ρδ + 2ζ ) ( ρ Hδ H − ρδ ) − ρρ H cos ( β − β H )   r2 rH cos θ H 2 = R   – ρ cos ( 12 α + β ) + ρ H cos ( 12 α + β H ) 

(14)

 ρ 2 − ( ρδ − 2ζ ) ( ρ Hδ H − ρδ ) − ρρ H cos ( β − β H )   r3rH cosθ H3 = R   – ρ cos ( 12 α − β ) + ρ H cos ( 12 α − β H ) 

(15)

2

2

Here the Cspiro–Cspiro distance is signified by r1, while r2 and r3 denote the Cspiro–Cmethylene bond lengths; moreover, inter-ring C–C–C angles between bonds of length ri and rj are labeled as θij. Finally, rH represents a C–H bond distance, and (θH2, θH3) are the corresponding H–C–C angles involving the C–C bonds of lengths (r2, r3). The key conclusion from this stage of the analysis is that if a solution to eqs 6-12 exists, then the carbon framework of polytriangulane forms a true σhelicene whose parameters (R, ρ, α, ζ, β, δ) are manifestly determined by the internal coordinates (r1, r2, r3, θ11, θ23, θ13, θ12). Likewise, if a solution to eqs 13–15 can be found for each of the two groups of hydrogen atoms (Ha, Hb), then the helical structure of the entire molecule is established. Because a nonlinear redundancy condition exists for the six bond angles around a tetravalent atom,26 eqs 6-12 contain only six independent degrees of freedom, precisely the number required to fix the carbon helical parameters (R, ρ, α, ζ, β, δ). Instead of the redundant set of inter-ring angles (θ11, θ23, θ13, θ12), only the wag (χw), rock (χr), and twist (χt) angle deformations for the spiro linkages are needed to specify the carbon framework:


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χ w = 12 (θ11 + θ13 − θ12 − θ 23 )

(16)

χ r = 12 (θ11 − θ13 + θ12 − θ 23 )

(17)

χ t = 12 (θ11 − θ13 − θ12 + θ 23 ) .

(18)

Novel formulas presented in the SI show how to determine all the inter-ring angles from the set of independent internal variables (r1, r2, r3, χw, χr, χt). Remarkably, the system of eqs 6-12 and 13-15 can be solved in closed form, even in the general case that no bond distances or angles are equivalent.

The derivation given in SI

culminates in eqs E33–E38 for the carbon parameters and eqs G19, G5, G20 for the hydrogen counterparts. For infinite polytriangulane the Cspiro–Cmethylene bonds are equivalent, so that r2 = r3 and θ12 = θ13. With inclusion of this symmetry, simplified formulas are obtained for the carbon quantities:

2 µ cosθ − 2cosθ − 1) − 2 (1+ cosθ )  2 µ (1− cosθ ) + cosθ ( cos α = (1+ 2µ cosθ ) − 2 (1+ cosθ ) 1− 2µ (1− cosθ )  2

12

2

11

11

23

2

2

12

11

1+ cos α + 2 µ cosθ12 ( cos α − 1) + 2cosθ11  

ρ=− ζ=

2cos ( 12 α ) (1+ cosθ11 )

sin ( 12 α ) − cos α − cosθ11 8 cos ( 12 θ11 )

11

 ,

(19)

23

,

(20)

,

(21)

β =π ,

(22)

δ =0 ,

(23)

and

R=

{

r1 (1+ 2 µ cosθ12 ) + 2 (1+ cosθ11 )  2 µ 2 (1− cosθ 23 ) − 1 2

}

2cos ( 12 θ11 ) 8µ cosθ12 + 8µ 2 (1− cosθ 23 ) − 2 (1+ cosθ11 ) 

,

(24)

where µ = r2/r1 = r3/r1. The instantaneous lead angle γ of the spiraling curves connecting the Cspiro atoms, representing inclination relative to the plane perpendicular to the axis of the helix, satisfies sin γ =

4ζ

α + 16ζ 2

.

(25)

2


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The rise of the Cspiro helix is also measured by the C–C–C–C torsion angle (ω) between consecutive spiro carbons, as given by

cos ω = 2csc 2 θ11 (1+ cos α ) (1+ cosθ11 ) − 1 .

(26)

The dihedral angle ψ between adjacent cyclopropane rings in polytriangulane is related to the C– C–C bond angles by the formula

cosψ =

cosθ12 cosθ13 − cosθ11 cosθ 23 , sin φ13 sin φ45

(27)

where φij denotes an intra-ring angle between bonds of lengths ri and rj. For infinite polytriangulane the Cmethylene–H bonds are equivalent, but the H–C–C angles

θH2 and θH3 are not equal for the individual hydrogens. In this case the simplified βH equation for each hydrogen atom is cos ( 2β H ) =

(D

2

− D1 ) ( D1 + D2 + 2D0 ) ε ε 2 − 4  D02 + D0 ( D1 + D2 ) + D1 D2 

(D − D ) 1

2

2

+ ε2

,

(28)

where D0 =

D1 =

4 ρζ 2 (1− λ ) sin ( 12 α )

,

(29)

16ζ 2  ρ 2 ( 2λ − 1) − rH2 R −2  + ε 2 4 ρ (1− λ ) sin ( 12 α )

,

D2 = ρ (1− λ ) sin ( 12 α ) ,

λ=

(30) (31)

µr1rH ( cosθ H3 + cosθ H 2 )

,

(32)

− cosθ H 2 ) .

(33)

2 ρ R 2  ρ + cos ( 12 α ) 

and

ε=

µr1rH R2

( cosθ

H3

For chemically reasonable structures, ε will be a very small quantity, and βH can be approximated as 1

 D2 + D1 + 2D0   . D −D

β H ≈ cos −1  2 

2

(34)

1


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Once βH is known from eq 28, the formulas for the other helical parameters of hydrogen are more compact; specifically,

ρH =

ρ ( λ − 1)

(35)

cos ( β H )

and

δ H = ( 2ζρ H )

−1

 ρ 2 − ρ cos ( 12 α + β ) − µ r1rH R −2 cos θ H 2   .  + ρ H cos ( 12 α + β H ) + ρ ρ H cos ( β H ) 

(36)

The mathematical system defined by eqs 6-15 exhibits a four-fold degeneracy of polytriangulane solutions for any specified set of bond distances and angles: {(α, ζ, β, δ, βH, δH), (–α, –ζ, –β, –δ, –βH, –δH), (–α, ζ, –β, δ, –βH, δH),

(α, –ζ, β, –δ, βH, –δH)}, with (helicity, direction of

propagation) = {(P, +z), (P, –z), (M, +z), (M, –z)}, in order. Thus, right- and left-handed helices are possible in equal numbers.

Computational Analysis of Polytriangulane Quantum chemical computations were employed to analyze the three-dimensional structure of polytriangulane by optimizing the geometry of closed-shell prototypes using the M06-2X functional combination with a 6-31G(d) basis set. Cutting out n cyclopropane rings from a polytriangulane strand yields n+1 Cspiro, n Cmethylene, and 2n H atoms, along with two severed C– C bonds at each terminus that can be capped with H atoms to produce a closed-shell C2n+1H2n+4 species. Our prototypes of this kind contained up to n = 42 rings, giving C85H88 as the largest species studied and the one on which we focus here. This C85H88 strand exhibited C2 symmetry, and the Cartesian coordinates of the optimized geometry are provided in the SI. To probe characteristics of infinite polytriangulane, edge effects were eliminated by removing the five rings at each end of the C85H88 molecule from the mathematical analysis and the plots displayed here. Technical challenges were encountered in exactly converging on the equilibrium geometry of C85H88, causing minute but inconsequential variations in the optimized geometric parameters. Table 1 presents parameters for an infinite polytriangulane helix derived from the average (r1, r2, r3, χw, χr, χt, rH, θH2, θH3) values of the central 32 rings of C85H88.


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Table 1. Geometric parameters of polytriangulane derived from M06-2X/6-31G(d) optimization of a C85H88 prototype and an idealized structure with equilateral cyclopropane rings. C85H88

idealized

C85H88

idealized

r1

1.458

1.503

ψ

90.5

90

r2 = r3

1.498

1.503

R

0.409

0.425

χw=χr

0.4

0

ρ

3.805

3.674

χt

0.4

0

α

4.450

4.460

θ11

138.6

138.6

ζ

0.797

0.791

θ23

137.8

138.6

β

π

π

θ12 = θ13

137.8

138.6

δ

0

0

rH,a = rH,b

1.087

1.079

ρH,a = ρH,b

5.597

5.389

θH2,a = θH3,b

117.7

117.7

βH,a

2.776

2.779

θH3,a = θH2,b

118.1

117.7

βH,b

3.507

3.505

ω

98.8

98.2

δH,a = –δH,b

–0.181

–0.178

γ

35.6

35.3

τ = 2πα–1

1.4120

1.409

Distances ri, rH, R in Å; angles θij, θi ω,γ, ψ in deg; all other parameters are dimensionless. R is the radius of the Cspiro helix. The bond distances (ri, rH) are depicted in Figure 1; (θ11, θ12, θ13, θ23) denote angles between [(red, red), (red, blue), (red, green), (blue, green)] C–C bonds therein.

The C–C bond distances of C85H88 are plotted in Figure 2. A strong clustering is seen, as each ring displays two distinct bond distances, r1 = 1.4581 ± 0.0001 Å and r2 = r3 = 1.4978 ± 0.0001 Å. Both of these values are considerably smaller than the C–C distance (1.5087 Å) exhibited by cyclopropane at the same level of theory [M06-2X/6-31G(d)]. Moreover, r1 lies roughly halfway between the prototypical distances for carbon-carbon single and double bonds, indicating a Cspiro–Cspiro bond order substantially greater than 1 along the backbone of polytriangulane. The small r1 distance is consistent with the expectation that the carbons in triangulanes are not sp3-hybridized. Indeed, the predicted 1.458 Å distance compares favorably with the length of the central C(sp2)–C(sp2) bond in trans-1,3-butadiene (1.461 Å M06-2X/631G(d), 1.468 Å from electron diffraction27). Figure 3 shows that the C–C–C inter-ring bond angles of C85H88 also group into narrow ranges: θ11 = 138.63 ± 0.07°, θ12 = θ13 = 137.82 ± 0.08°, and θ23 = 137.80 ± 0.04°. The effective equivalence of θ23 with (θ12, θ13) is accidental and not a consequence of the C2 symmetry of the molecule. Nonetheless, the central part of the C85H88


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strand clearly exhibits spiro connections very close to those of an idealized structure in which all C–C–C inter-ring bond angles are equal.

Figure 2. Evidence of two distinct C–C bond distances in polytriangulane: a plot of the lengths of the C–C bonds made by carbon atom j along the chain of the C85H88 fragment. The color code is the same as in Figure 1.

Figure 3. Evidence of three distinct C–C–C bond angles in polytriangulane: a plot of the angles formed by carbon atom j along the chain of the C85H88 fragment. Each angle is shown with the color complementary to those of the two included bonds, using the coloring scheme of Figure 1.


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Figure 4. Minute variations of the dihedral angle ψ between adjacent cyclopropane rings and the ω(C–C–C–C) torsion angle between consecutive spiro carbons along the chain of the C85H88 fragment. The ω(C–C–C–C) torsion angle between consecutive spiro carbons is a measure of the angle of inclination of the polytriangulane helix. Figure 4 reveals that for the C85H88 fragment the ω angles lie in the narrow range 98.83°±0.07°. The corresponding instantaneous lead angle γ of the spiraling curves connecting the Cspiro atoms is γ = 35.6° (Table 1). Thus, the rise of the polytriangulane helix is very steep; in comparison, the corresponding lead angle in polytwistane is γ0 = 15.7°.25 A consequence of this steep rise is that the radius (R) of the polytriangulane Cspiro helix is only 0.409 Å (Table 1). The dihedral angles ψ between adjacent cyclopropane rings along the chain of C85H88 are also plotted in Figure 4, where only minute variations are seen. The tight range of ψ = 90.54° ± 0.03° indicates the constancy of orientation of the cyclopropane rings, which are very nearly perpendicular to each other. The C–H bonds of polytriangulane are equivalent, as shown by the fact that C85H88 exhibits rH bond lengths of 1.0874 Å with a range less than 0.0001 Å. Figure 5 plots the H–C–C bond angles along the chain of the prototype. Two distinct values are apparent, θH2,a = θH3,b = 117.70° ± 0.01° and θH3,a = θH2,b = 118.06° ± 0.02°, which are indicative of a slight twisting of the methylene group in the cyclopropane rings that nonetheless maintains the equivalence of the two hydrogen atoms.


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Figure 5. Plots of two distinct H–C–C bond angles along the chain of the C85H88 fragment. As shown in Figure 1, θHi,j denotes the angle between the ri(C–C) (i = 2, 3) bond and the C–Hj bond (j = a, b). An idealized reference structure of polytriangulane can be constructed by assembling identical cyclopropane units composed of equilateral triangles. In this case, we select µ = r2/r1 = r3/r1 = 1 in the above equations. Moreover, vector analysis shows that in order for all inter-ring C–C–C bond angles (θ11, θ12, θ13, θ23) to be equivalent, these angles must assume the value

θ 0 = cos −1 ( − 34 ) ≈ 138.6°. Placing these values of µ and θ0 into eqs 19–26, we obtain elegant

analytic solutions for the helical parameters of idealized polytriangulane: α = π + cos −1

ρ=

27 2

,ζ=

5 8

, β = π, δ = 0, R =

2 5

( )

( ), 1 4

−1 r , γ = sin −1[(1+ 101 α 2 ) −1/2 ] , and ω = cos − 14 , whose

numerical values are listed in Table 1. Note that the radius of the idealized Cspiro helix (R = 0.425 Å) is determined solely by applying the scale factor

2 5

to the common C–C bond length

of the cyclopropane monomer (r =1.503 Å). To complete the idealized model, we select the monomer parameters rH = 1.079 Å and θH2,a = θH2,b = θH3,a = θH3,b = 117.7° for the hydrogen atoms;28 eqs 28, 35, and 36 then yield ρH,a = ρH,b = 5.389, βH,a = 2.779, βH,b = 3.505, and δH,a = –δH,b = –0.178. If the molecular axis of idealized polytriangulane is placed in the z direction, then the Cartesian coordinates of carbon atoms k = 0, 1, 2, ... are given by


13


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Page 14 of 22

,

(37)

where the mk and nk quantities are irregular integers, and

denotes the greatest integer less

than or equal to j. In particular, for k = 0, 1, 2, ...,16,

m = {4,9,−2,−27,−14,−9,22,117,34,−81,−122,−387,−14,711,502,837 }

(38)

n = {0,−3,−2,−3,2,15,6,−3,−14,−57,−10,69,66,159,−26,−435} .

(39)

and

The geometric parameters of the idealized polytriangulane model are compared with those of the C85H88 prototype in Table 1. The agreement of the helical parameters ω, γ, ψ, α, ζ,

βH, and δH is remarkable. For example, ω = (98.8°, 98.2°) and ζ = (0.797, 0.791) in the (C85H88, idealized) cases. Therefore, the idealized model recovers the essential aspects of the helical structure of polytriangulane. Of greatest significance, the model yields

τ ideal =

2π

α ideal

=

2π = 1.40887795... , π + cos −1 ( 14 )

(40)

showing that the periodicity of the helix is an irrational number. The C85H88 prototype has a very similar value, τ = 1.4120. The actual α parameter for a non-idealized, infinite polytriangulane helix would satisfy eq 19 and thus τ would also be irrational in this case.

In brief,

polytriangulane is a nonrepeating σ-helicene, in which the azimuthal angles of the carbon and hydrogen atoms never exactly repeat as the chain is traversed. From a practical perspective the periodicity parameters can provide the number of Cspiro atoms (nε) that must be traversed to achieve an azimuthal coincidence within a specified threshold (ε). For the idealized model (ε, nε) = (1.2°, 31) and (0.1°, 603), while for the C85H88 prototype (ε, nε) = (1.2°, 24) and (0.1°, 425). Polytriangulane Thermochemistry The thermochemistry of infinite polytriangulane is analyzed here by employing a homodesmotic reaction29 for extending [n]polytriangulane by one cyclopropane unit:

[n]polytriangulane + cyclopropane + neopentane → [n + 1]polytriangulane + 2 propane

(41)

This equation is homodesmotic because both reactants and products have the following structural elements: 3n+7 formal C(sp3)–C(sp3) bonds; and n+1 C(sp3)H0, n+5 C(sp3)H2, and 4 C(sp3)H3


14

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units. The similarity of the bonding on both sides of eq 41 leads to a high degree of error cancellation in quantum chemical computations, thus providing an effective approach for accurate thermochemistry. In explicit form eq 41 becomes

C2n+1H 2n+4 + C3H 6 + C ( CH 3 )4 → C2n+3H 2n+6 + 2 CH 3CH 2CH 3 .

(42)

The enthalpy change at 0 K for this aggregation reaction is given by ,

(43)

in which . Selecting the established values

(44) = –19.68±0.06 kcal mol–1[30,31] and

= –31.45 kcal mol–1 fixes qagg as –7.91 kcal mol–1. In the limit n → ∞ , eq 43 gives the enthalpy of formation for polytriangulane per mol of CH units as .

(45)

A highly accurate enthalpy of formation of cyclopropane can be computed from , where

(46)

is the enthalpy change for the ring opening of cyclopropane to

form propene. In Table 2, a focal-point analysis is laid out for

based on all-electron

computations through the CCSDT(Q) level and extrapolation of cc-pCVXZ (X = 3, 4, 5) energies to the CBS limit. Impressive convergence is seen for the entries in both the columns and rows, and the final prediction of

= –8.76 kcal mol–1 should be accurate to ca. 0.1 kcal mol–1.

= 8.43±0.07 kcal mol–1,31 we arrive at

Adopting –1

0.1 kcal mol .

=17.2 ±

This value compares favorably with earlier high-level computational

extrapolations (17.4 ± 1.5 kcal mol–1)32 and the experimental value (16.9 ± 0.1 kcal mol–1).16 To find ∆ agg H 0° (∞) , the M06-2X/6-31G(d) level of theory was employed to compute the enthalpy change ∆ agg H 0° (n) of eq 43 for the series n = 1–23. The reaction energies were quite stable, rapidly converging to a mean value of 6.08 kcal mol−1 with a standard deviation of only 0.13 kcal mol−1. Alternative B3LYP/6-31G(d) computations were performed for the series n = 1–8, which converged to 1.96 ± 0.08 kcal mol–1 for the reaction energy. With inclusion of both


15


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Page 16 of 22

dispersion (+D3)33 and geometrical counterpoise (gCP) corrections,34 the B3LYP/6-31G(d) reaction energy increased to 2.26 kcal mol–1. Other DFT results are given in the SI. Overall, a disturbing lack of consensus is found in the DFT computations. Although the DFT methods are not capable of delivering reliable final values for ∆ agg H 0° (∞) , we can conclude from the results that this quantity is sufficiently converged with the n = 3 reaction of eq 42 and that extrapolation to infinite n is not necessary. Table 2. All-electron focal-point analysisa of cyclopropane to propene.

(in kcal mol–1) for the ring opening of

∆Ee(RHF)

+δ [MP2]

+δ [CCSD]

cc-pCVDZ cc-pCVTZ cc-pCVQZ cc-pCV5Z

–9.48 –9.46 –9.59 –9.67

+3.77 +4.54 +4.68 +4.80

–2.28 –2.31 –2.25 –2.22

–0.48 –0.34 –0.35 –0.34

–0.16 [–0.16] [–0.16] [–0.16]

–8.64 [–7.73] [–7.68] [–7.60]

CBS LIMIT

[–9.70]

[+4.93]

[–2.20]

[–0.34]

[–0.16]

[–7.48]

FUNCTION X (fit points)

–cX

a+be (3,4,5)

–3

a+bX (4,5)

–3

a+bX (4,5)

+δ [CCSD(T)] +δ [CCSDT(Q)]

–3

a+bX (4,5)

NET

additivity

Reference geometry, ∆(ZPVE) , and ∆(rel): AE-CCSD(T)/cc-pCVTZ

∆ open H 0° = ∆Ee(FPA) + ∆(ZPVE) + ∆(rel) + ∆(DBOC) = –7.4 –1.28 –0.059 –0.015 = –8.76 kcal mol–1 The symbol δ denotes the increment in the energy difference (∆Ee) with respect to the previous level of theory in the hierarchy RHF → MP2 → CCSD → CCSD(T) → CCSDT(Q). Bracketed numbers result from basis set extrapolations (using the specified functions and fit points) or additivity approximations, while unbracketed numbers were explicitly computed. The focal-point tables target ∆Ee[CCSDT(Q)] in the complete basis set limit (NET/CBS LIMIT). a

Our rigorous evaluation of ∆ agg H 0° (3) is detailed in Table 2, wherein a focal-point analysis is shown involving explicit computations through the CCSDT(Q)/cc-pVQZ level and extrapolation to the CBS limit. The error-canceling properties of the homodesmotic equation are evident in the very strong convergence of both the electron correlation increments and the basis set variations. For example, the +δ [CCSD(T)] increment amounts to less than 0.3 kcal mol–1, and the cc-pVQZ basis set provides results within 0.2 kcal mol–1 of the CBS limit. With the auxiliary corrections for ZPVE, core electron correlation, special relativity, and DBOC, we arrive at ∆ agg H 0° (3) =7.11 kcal mol–1. Inserting this result into eq 45 along with our qagg and values, we obtain


= +16.11 kcal (mol CH)–1. By

16

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comparison,

= +1.28 kcal (mol CH)–1 for polytwistane, another compound with the

formula CnHn.25 Even though polytriangulane is thus substantially less stable thermodynamically than polytwistane, it could still be formed in a highly exothermic, hypothetical polymerization of = +27.35 kcal (mol CH)–1.

acetylene, for which

Table 3. Valence focal-point analysisa of ∆ agg H 0° (3) (in kcal mol–1) for forming [4]triangulane from [3]triangulane. ∆Ee(RHF) cc-pVDZ cc-pVTZ cc-pVQZ CBS LIMIT

+δ [MP2]

+δ [CCSD]

+δ [CCSD(T)]

NET

4.20 4.48 4.53

+2.26 +2.82 +3.01

–0.37 –0.66 –0.74

+0.04 +0.17 +0.24

+6.13 +6.81 +7.04

[4.54]

[+3.14]

[–0.79]

[+0.29]

[+7.18]

FUNCTION a+be–cX a+bX–3 a+bX–3 X (Fit points) (3, 4, 5) (4, 5) (4, 5) Reference geometry: AE–MP2/cc–pCVTZ ∆H agg (3) = ∆Ee(FPA) + ∆(ZPVE) + ∆(core) + ∆(rel) + ∆(DBOC)

a+bX–3 (4, 5)

= 7.18 –0.049 –0.050 +0.000 +0.027 = 7.11 kcal mol–1 a

For notation see footnote of Table 2.

The additional strain introduced to polytriangulane via the spiro connections can be deduced using the following homodesmotic reaction:

[n]polytriangulane + 2 ( n −1) propane → n cyclopropane + ( n −1) neopentane Applying eq 47 through n = 9, the M06-2X/6-31G(d),

(47)

B3LYP/6-31G(d), and

B3LYP+D3+gCP/6-31G(d) density functional methods give 13.8, 9.7, and 10.0 kcal mol–1, respectively, as the extra strain per spiro connection. As in the case of ∆H agg (n) , the disparity in these values is disappointing, but the DFT methods do show that the strain energy increment does converge rapidly with n. Therefore, we executed single-point computations at the AECCSD(T)//AE-MP2/cc-pCVTZ level of theory for the n = 3 and n = 4 reactions in eq 47 and found a mean strain energy increment of 14.56 kcal mol–1 per spiro carbon. If the strain energy of cyclopropane itself is taken as 28.13 kcal mol–1,35 then the total strain energy (Estrain) of the [n]polytriangulanes can be represented as

Estrain = 28.13n + 14.56 ( n − 1)


(48)

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Page 18 of 22

for sizable values of n. This equation corresponds to total strain of 42.69 kcal per mol of cyclopropane units. Summary We have executed a detailed topological and quantum chemical investigation of the σ-helicene polytriangulane, whose structure is a twisted, linear chain of spiro-linked cyclopropane rings. Both an idealized model and a C85H88 prototype optimized by M06-2X/6-31G(d) theory show that the primary Cspiro chain of polytriangulane comprises an exact, nonrepeating helix with an irrational period of repetition (τ) very close to 2 [1+ π −1 cos −1 ( 14 )]−1 . accidental that τ also happens to be near

2.

It is intriguing but

The cyclopropane rings in polytriangulane are

isosceles triangles with (Cspiro–Cspiro, Cspiro–Cmethylene) distances about (0.05, 0.01) Å shorter than in cyclopropane itself. The hydrogens in polytriangulane are equivalent, but a slight twist of the CH2 group relative to the plane of the cyclopropane ring engenders two distinct H–C–C angles separated by 0.4°. The C85H88 structure indicates that the (Cspiro, Cmethylene) atoms lie on a helical strand that winds on the surface of a cylinder whose radius is (R, ρ R) = (0.409, 3.805) Å, in order. The same prototype exhibits a dihedral angle (ψ) of 90.5° between consecutive cyclopropane rings. The idealized model predicts that the helical radius R is equal to

2 5

r , where

r is the side-length of the equilateral triangles describing the cyclopropane rings. The accuracy with which the idealized geometric model predicts this and other structural aspects of the

σ-helicene demonstrates that the root word “triangulane” is especially apt for this compound. Thermochemical computations with a highly convergent FPA scheme provide the most definitive enthalpy of formation to date for cyclopropane:

=17.2 ± 0.1 kcal mol–1.

Subsequent FPA computations applied to a homodesmotic chemical equation for ring aggregation predict that

= +16.1 kcal (mol CH)–1 for polytriangulane. While this enthalpy

is substantially lower than that of acetylene, it does reveal that the isomeric (CH)n compound polytwistane is much more stable thermodynamically. The energetic disfavor of polytriangulane in this comparison is largely due to its total strain energy, which is about 50% greater per mol of CH than in the cyclopropane monomer.


18

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Computational Methods The polytriangulane structures were optimized with the M06-2X36-37 functional in conjunction with a 6–31G(d) basis set using the Gaussian 09 program.38 The "ultrafinegrid" keyword for the density was used throughout except for the [42]triangulane were a "superfinegrid" was employed; the geometry convergence criteria were set to "very tight". For the focal point analysis, the geometries were optimized at the AE–MP239-40 and AE–CCSD(T)41-45 levels of theory in the highest available symmetries using a cc–pCVTZ46 basis set as implemented in the MOLPRO47 program. The final single-point energy computations were executed at the HF, AE– MP2, AE–CCSD, AE–CCSD(T) and AE–CCSDT(Q) levels of theory using cc-pVXZ (X = D, T, Q) basis sets as implemented in MOLPRO.

Acknowledgments This paper is dedicated to Armin de Meijere for his seminal contributions to small ring chemistry. The research at the University of Georgia was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Combustion Program, Grant No. DE-SC0015512.

Supporting Information DFT reaction energies, xyz-coordinates of all optimized structures, and full references for electronic structure codes.

This information is available free of charge via the Internet at

http://pubs.acs.org

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(41) Čížek, J. On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell ‐ Type Expansion Using Quantum ‐ Field Theoretical Methods. J. Phys. Chem. 1966, 45, 4256-4266. (42) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett. 1989, 157, 479-483. (43) Bartlett, R. J.; Watts, J. D.; Kucharski, S. A.; Noga, J. Non-iterative fifth-order triple and quadruple excitation energy corrections in correlated methods. Chem. Phys. Lett. 1990, 165, 513522. (44) Stanton, J. F. Why CCSD(T) works: a different perspective. Chem. Phys. Lett. 1997, 281, 130-134. (45) Razavy, M., Quantum Theory of Tunneling World Scientific Publishing Co Pte Ltd Singapore, 2002; p 540. (46) Dunning, J. T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007-1023. (47) MOLPRO, version 2010.1, a package of ab initio programs ; (see Supporting Information for full references).

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