Practical Tight-Binding HÃ¼ckel-like Modeling of Electronic Properties

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Practical Tight-Binding Hückel-like Modeling of Electronic Properties of Saturated Hydrocarbons – on Sigma-Aromaticity of Cyclopropane and Correlations of Alkane Ionization Energies and Enthalpy of Formation Corrected for Protobranching Jerry Ray Dias J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b05467 • Publication Date (Web): 30 Jul 2018 Downloaded from http://pubs.acs.org on August 4, 2018

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The Journal of Physical Chemistry

Practical Tight-Binding Hückel-Like Modeling of Electronic Properties of Saturated Hydrocarbons – on sigma-aromaticity of cyclopropane and correlations of alkane ionization energies and enthalpy of formation corrected for protobranching Jerry Ray Dias* Department of Chemistry University of Missouri Kansas City, MO 64110-2499 [email protected] Abstract This testing of σ-aromaticity in cyclopropane and protobranching in alkanes via a Hückel-Like Model proposed by Klein and Larson is expedited by mirror-plane fragmentation per the method of McClelland originally developed for conjugated polyenes. Alkane carbon sp3 orbitals are represented by edge weighted tetrahedral stellate graphs connected to hydrogen orbitals.

INTRODUCTION Klein and Larson presented a stellation model for determining the Hückel-type molecular orbital eigenvalues of saturated hydrocarbons.1 This model gave two fundamental proofs, namely that alkanes are stable with half their eigenvalues being positive and half negative. We will demonstrate graph theoretical methods for facile calculation eigenvalues of alkanes per this amended model and show that the results nicely correlate their experimental properties. Also the prior work of Estrada who used an atom-bond topological model for alkanes based on a tightbinding Hückel-like Hamiltonian will be compared to our results.2 Preliminaries The stellated graph G* associated with an alkane has a vertex set V* = V(G*) corresponding to four sp3 hybrid orbitals on each carbon and one for each hydrogen with an edge set E* = E(G*) partitioned into internal (E*int) and external (E*ext) edge subsets. The internal edge set consists of 1 ACS Paragon Plus Environment

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six tetrahedrally arranged edges of weight 0.5 linking the four hybrid orbitals on each C atom; these edges are weighted 0.5 because the p-orbitals have contributed only 2 electrons of the carbon 4 electrons. Each external bond edge of normal weight connect an H to a stellated C orbital. Overall, the stellated graph G* associated with an alkane is an orbital network graph of a hydrocarbon. All the substituents to any given carbon have the sp3 angles of 109.5o. The number of stellate edges (qst) is given by qst = 9N + 1 ‒ r (r = No. cycloalkane rings) where N is the generation which is equal to the number of carbons (CNH2N+2‒2r). The number of network orbitals (stellate vertices) is given by η = 6N + 2 ‒2r. The characteristic polynomial P(G*;X) of the stellate graphs of the alkanes gives Hückellike molecular orbital (MO) eigenvalue/eigenvector solution for the alkanes. Each MO will have a specific energy value (eigenvalue) and wave function (eigenvector). The eigenvalue unit is β which has dimension of energy with a negative value. Throughout we will omit this unit and the reader should assume that it is implied. The stellate graphs directly represent the alkane C-H orbitals whereas the Estrada atom based connectivity model treats each carbon atom and its attached hydrogen as a united atom. It considers 1,2-, 1,3-, and 1,4-interactions between the united-atoms in an alkane. Here the nature of the interactions does not matter but only the number of united-atoms of each type interacting to each other.[2] Sachs graph theoretical method The determination of the coefficients an of the characteristic polynomial P(G;X) by the Sachs graph theoretical method is summarized by the following formula

an = ∑ (-1)c(s)2r(s) 2 ACS Paragon Plus Environment

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sεSn where 0 < n < N, s is a Sachs graph, Sn is a set of all Sachs graphs with exactly n vertices, c(s) is the total number of disjoint K2 and Cm components, and r(s) is the number of monocyclic rings (circuits) in s. All combinations of lK2 and kCm such that 2l +km = n are enumerated. N is the number of atoms in the entire pπ system. The coefficients an of P(G;X) of a molecular graph G are determined by enumerating all Sachs graphs s for each value of n. Each Sachs graph s is drawn within parentheses and the set Sn of all such graphs are denoted in brackets. By definition ao = 1. Since only K2 and Cm components are permitted, S1 = ϕ (the empty set) giving a1 = 0. The set S2 of all Sachs graphs on two vertices leads to the value of a2 = ‒ q which is the negative number of graph edges in G. The same Sachs formula applies to the computation of the matching polynomial coefficients but all cyclic components are omitted (i.e, 2r

(s)

= 0) and only disjoint

K2 components are enumerated.3 It needs to be emphasized that the above is for graphs with the weights of all edges and vertices being unity. Modified Sachs graph theoretical method for heteroatomic (edge/vertex weighted) graphs Aihara has fomulated the following modification of Sachs graph theoretical method for graphs having edges and vertices with weights different from unity.4 In a π conjugated hydrocarbon with heteroatoms, the heteroatoms can be depicted by self-loop representation. Each heteroatom in a molecular graph is specified by addition of a self-loop with a weight of h. A self-loop is a hypothetical π bond in which both ends terminate in the same heteroatom. Such a self-loop is counted as one of the components of a Sachs graph in addition to the disjoint pπ bonds (K2) and/or disjoint π cycles (Cm).

an = ∑ {(-1)c(s)2r(s)kx∏hα} 3 ACS Paragon Plus Environment


sεSn

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α

The power value x of the weighted edge k is determined by two rules: 1) when each disjoint heterobond is counted in the Sachs graph s, it must be multiplied by the square of heterobond parameter k for each time it appears in s; 2) when a π cycle with heterobonds is counted in the same Sachs graph s, it must be multiplied by the product of all the heterobond parameters present in the cycle. Figure 1 illustrates this modified Sachs method with the pyrrole heterocycle. Note that the odd coefficients a1 and a3 now arise from the self-loop of weight of h in the set of Sachs graphs S1 and S3, respectively. The stellate graphs of the alkanes will be devoid of self-loops of weight of h, and we will only be concerned with the internal weighed edges which for convenience are assigned k = 0.5 which we will later show is a reasonable value. Eigenvalue determination via mirror-plane fragmentation of larger symmetrical molecular graphs into smaller ones Molecular graphs of conjugated polyenes possessing a plane of symmetry can be decomposed into subgraphs (fragments) having smaller secular determinants. The characteristic polynomials of the mirror-plane fragments are factors of the characteristic polynomial of the original parent molecular graph. The roots of the characteristic polynomials are the eigenvalues that correspond to orbital energies. The program of Balasubramanian is routinely used.5 McClelland has described the following rules for decomposition polyene conjugated hydrocarbons.6 1) The symmetry plane divides the molecular graph into a left-hand and right-hand subgraph. 2) Vertices on the plane of symmetry are included in the left-hand subgraph. 3) The weight of the edge between the plane of symmetry and a vertex not on the plane of symmetry but in the lefthand subgraph is √2. 4) If the symmetry plane bisects an edge originally connecting the left and right subgraphs, the vertex in the left subgraph is weighted + 1 (Coulomb integral = α – β) and 4 ACS Paragon Plus Environment

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the vertex in the right subgraph is weighted – 1 (Coulomb integral = α + β). 5) The eigenvalues of the original molecular graph equal the sum of eigenvalues belonging to both subgraphs. Righthand fragments identify the eigenvalues with antisymmetric eigenvectors (wave functions) in regard to the mirror-plane. In general, molecular graphs with greater than two-fold symmetry will possess approximately two-thirds of its HMO eigenvalues doubly degenerate. One set of these degenerate eigenvalues correspond to the eigenvalues belonging to the right-hand mirrorplane fragment. As a typical example, all conjugated polyene cycles, like benzene, have doubly degenerate HMO eigenvalues as prescribed by the Frost circle mnemonic. One set of these doubly degenerate eigenvalues will have antisymmetric eigenvectors and other symmetric eigenvectors. Herein, we show that McClelland rules also apply to the stellate graphs of saturated alkane hydrocarbons. RESULTS AND DISCUSSION Mirror-plane fragmentation of alkane stellate molecular graphs To illustrate mirror-plane fragmentation, consider the stellate graph of methane in Figure 2. The characteristic polynomial of the stellate molecular graph (G*) is to the 8th degree because of the four carbon hybrid orbitals and the four hydrogen s orbitals. Three successive mirror-plane fragmentations leads to three identical fragments, each with the binomial X2 + 0.5X – 1 = 0. A fourth fragment has the binomial X2 – 1.5X – 1 = 0. The roots of these polynomials give all the eigenvalues for the methane stellate graph. In general, a0 = 1 by definition for all stellate graphs and the a2 coefficient of the characteristic polynomial [P(G*;X)] is the sum a2 = ∑� (-1)k2. For the methane stellate graph

a2 = (-1)(6x0.52 + 4x1.02) = – 5.5 where the first term comes from the six weighted 0.5 edges in 5 ACS Paragon Plus Environment


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the tetrahedron and the second from the four normal weighted edges to the hydrogens. The a3 coefficient is a3 = ∑� (-1)2k3. For the methane stellate graph, there are four adjoined C3 cycles giving a3 = ‒ 4x2x0.125 = ‒ 1.0. For all stellate graphs of the acyclic alkanes, the tail coefficient

aN = ± 1 (+ 1 for N = odd and – 1 for N = even molecular systems of size N = number of carbons); for monocyclic alkanes the tail coefficient is aN = ± (1 + k2N + 2kN). While the above gives the coefficients to the characteristic polynomial of alkane stellate graphs, the coefficients to the matching polynomial excludes the set of Sachs graphs with cyclic components. For the use of the characteristic and matching polynomials in the determination of topological resonance energy (TRE) the reader is referred to the recent review by Aihara.3 We now illustrate the Sachs graphical computation of the matching polynomial for methane and its hypothetical TRE. Consider the stellate graph of methane. All alkanes must be by necessity devoid of topological resonance energy (TRE). Thus, let us assume the stellate graph of methane with weighted internal edges of 0.5 does have resonance energy. Per the modified Sachs graph method by Aihara, the matching polynomial of the stellate graph of methane is X8 – (4 +6k2)X6 + (6 +12k2 + 3k4)X4 – (4 + 6k2)X2 + 1 = 0 which upon assigning k = 0.5 gives X8 – 5.5X6 + 9.1875X4 – 5.5X2 + 1 = 0. Note that in the Sachs graph method for determining matching polynomials coefficients all cyclic components are omitted (i.e, 2r

(s)

= 0 in the set of Sachs

graphs) and only disjoint K2 components are enumerated. Solving this matching polynomial gives Eσ(methane)ref = 8.69959. Now TRE = Eσ(methane) ‒ Eσ(methane)ref = 8.68468 – 8.69959 = ‒ 0.015 ≈ 0 which is essentially zero as expected for the TRE of all alkanes in general. This shows that the stellate internal edge weight assignment of 0.5 is both a convenient and 6 ACS Paragon Plus Environment

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reasonable value. Also, refer to the Supporting Information regarding the stellate model and the weighting factor based on counting Sachs graphs. Eigenvalue degeneracy All the stellate graphs of the alkanes are multiply degenerate in the eigenvalues of 0.78078 and ‒ 1.28078. The stellate graph of methane is triply degenerate with these eigenvalues and this degeneracy increases by one for each additional carbon in going from ethane to pentane and beyond (Figure 3). The HOMO-LUMO gap progressively decreases in going from methane to pentane while the eigenvalue range increases. The stellate graphs of methane, cyclopropane, and isobutane all have greater than 2-fold symmetry and should have more than two-thirds of the eigenvalues degenerate. One set of the degenerate eigenvalues correspond to the eigenvalues of the right-hand mirror-plane fragment. Figure 2 shows that multiple fragmentation of the stellate graph of methane leads to four ultimate subgraphs, three of which are identical thereby explaining its triple degeneracy in the eigenvalues of 0.78078 and ‒ 1.28078. If one considers ethane in Figure 4, it should be evident that interaction between neighbor carbons is directional from tetrahedral vertex to tetrahedral vertex. In fact, this model allows eclipsing and staggered conformations to occur which because they can freely rotate are averaged in this model as it is done in standard thermal chemical values. The two successive mirror-plane fragmentation of ethane (Figure 4) reveals that the origin of the multiple degeneracy of the eigenvalues of 0.78078 and ‒ 1.28078 can be attributed to the frequent occurrence of the same subgraph. Mirror-plane fragmentation of cyclopropane stellate graph give a right-hand fragment which gives one set of the doubly degenerate eigenvalues and this is corroborated by its irreducible subgraph as shown in Figure 5. This same irreducible subgraph is obtained by

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rotational symmetry (Cn = 2π/n) for all stellate graphs of cycloalkanes, depending upon the value of n. This same irreducible graph was used to obtain the eigenvalues of cyclobutane listed in Figure 3 by setting n = 4. Mirror-plane fragmentation of the stellate graphs of propane and isobutane give the same right-hand fragment which explains why both are subspectral in the eigenvalues of 1.98044, 0.57846, ‒ 0.59846, and – 1.46172 in addition to the eigenvalues of 0.78078 and ‒ 1.28078 (Figures 6 and 7). Because the stellate graph of isobutane has 3-fold symmetry, it is doubly degenerate in these common eigenvalues. Alkane enthalpy of formation (∆Hfo) and ionization energy (IE) correlations Figure 8 summarizes the stellate σ-energy (Eσ ) summed from data in Figure 3 along with the HOMO/LUMO energy and ionization energy of methane to pentane. A plot of experimental standard enthalpy of formation (∆Hfo) corrected for protobranching [∆Hfo(corr)]of the alkane against their stellate sigma energy (Eσ) is given in Figure 9. The enthalpy data corrected for protobranching was obtained from the Supporting Information by Schleyer and coworkers.7-9 Protobranching stabilization is due to electron correlation which is absent in the tight-binding HMO.10 The highly correlative plot (R2 = 1) in Figure 9 is very linear further supporting the idea that protobranching is another parameter needed in thermodynamic additive evaluations. The concept of protobranching by Schleyer and coworkers was foreshadowed by the work of Randić11 and Bertz12. Randić used a branching index based on adjacency matrix with the smallest associated binary numbers and showed a unique correlation with experimental enthalpies (see Figure 3 in ref. 11) which displays a clear protobranching trend. Bertz used the term “branching” based on pairs of adjacent lines, i.e., propane subunits.12 The Estrada atombond connectivity model was also shown to support the enthalpy stabilizing effect of protobranching in alkanes.2 8 ACS Paragon Plus Environment

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The experimental bond dissociation energy (BDE) for methane and the C-C bond in ethane are BDE(methane) = 104.8 kcal/mol (CH4 → CH3• + H•) and BDE(ethane) = 89.9 kcal/mol (CH3-CH3 → 2CH3•) with the ratio BDE(ethane)/ BDE(methane) = 89.9/104.8 = 0.857. The stellate energy of methane radical is Eσ = 6.95154. Using the data in Figure 8 for methane and ethane, we get (15.38757 – 2x6.95154)/(8.68468 – 6.95154) = 1.48449/1.73314 = 0.857 which has the same ratio as the respective BDE above. This example again show that there is an excellent correlation the between bond dissociation enthalpies and the alkane stellate electronic energy. Figure 10 gives a plot of experimental ionization energies (IE) of the alkanes (ethane to pentane) against their stellate graph HOMO values (β). With inclusion of methane, this plot would not be linear. The HOMO value for isobutane (0.57846) is identical to that of propane but is doubly degenerate with an occupancy of 4 electrons. This explains why its ionization energy (10.57 eV) is lower than that of propane (10.95 eV) because of Jahn-Teller effects would remove this degeneracy in isobutane by elevating one energy level and lowering the other. Cyclopropane and σ-aromaticity Despite the much greater angle distortion, cyclopropane has a strain energy only slightly larger than cyclobutane. This problem of the nearly equivalent strain energies was readdressed, leading to new estimates for the stabilization of cyclopropane due to CH bond strengthening (11.7 kcal/mol) and to σ-aromaticity (11.3 kcal/ mol).13 Our evaluation of the σ-aromaticity in cyclopropane is summarized in Figure 11. We model this evaluation by disregarding the localized contribution of the three pendant groups that are the origin of the triply degenerate eigenvalues of 0.78078 and – 1.28078. Thus, we only consider the bond-alternant resonant inner stellate orbitals of cyclopropane. This is similar to what Klein and Larson did in their proof that 9 ACS Paragon Plus Environment


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cyclobutane has half-positive and half-negative eigenvalues (Fig. 5 in ref. 1). Aihara and Ishida showed that bond-alternant benzene possessed unusually high aromaticity14 and we used their method for this bond-alternant cyclopropane unit which is an analog of bond-alternant of benzene. The characteristic polynomial of the edge weighted stellate subgraph of cyclopropane is P(G*;X) = X6 – (3 + 3k2)X4 + (3 + 3k2 + 3k4)X2 – (1 + k6 + 2k3) = X6 – (3 + 0.75)X4 + (3 + 0.75 + 0.1875)X2 – (1 + 0.015625 + 0.25) = X6 – 3.75 X4 + 3.9375X2 – 1.265625. The corresponding matching (or acyclic) polynomial is given by P(G*;X)ac = X6 – (3 + 3k2)X4 + (3 + 3k2 + 3k4)X2 – (1 + k6) = X6 – 3.75 X4 + 3.9375X2 – 1.015625. Using the roots of these equations to determine the energies [Eσ and Eσ(ref)] of the bond-alternant cyclopropane and its hypothetical reference, respectively, we can determine its topological resonance energy [TRE = Eσ ‒ Eσ(ref) = 6.46409 ‒ 6.38470 = 0.0794 β]. The reader is referred to the recent Aihara review for the role of topological resonance energy (TRE) in evaluation of aromaticity.3 If this idealized cyclopropane had all equal stellate bonds of weight one rather than the bond-alternant of 0.5 and 1.0 weights, it becomes identical to benzene (TRE = 0.273 β). Using these values, we estimate that cyclopropane has (0.0794/0.273)100% = 29.1% the aromaticity of benzene. This aromaticity percentage can be compared to [11.3/28.8]x100% = 39.2% determined using Schleyer and coworkers values 11.3 kcal/mol for the σ-aromaticity of cyclopropane12 and benzene aromaticity of 28.8 kcal/mol.7 The ionization energy of cyclopropane (9.86 eV) is lower than expected because of the doubly degeneracy its HOMO. Cyclobutane and σ-antiaromaticity Using the irreducible subgraph in Figure 5 and its corresponding equations the eigenvalues of cyclobutane can be obtained by assigning n = 4. These eigenvalues are summarized in Figure 3. Our evaluation of the σ-antiaromaticity in cyclobutane is summarized in Figure 12 that follows 10 ACS Paragon Plus Environment

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the same method used for the σ-aromaticity in cyclopropane. We model this evaluation by disregarding the localized contribution of the four pendant groups that are the origin of the quadruply degenerate eigenvalues of 0.78078 and – 1.28078. Thus, we only consider the bondalternant resonant inner stellate orbitals of cyclobutane. This is identical to what Klein and Larson did in their proof that cyclobutane has half-positive and half-negative eigenvalues (Fig. 5 in ref. 1). Aihara and Ishida showed that bond-alternant monoenes possessed unusually high aromaticity/antiaromaticity14 and we used their method for this bond-alternant cyclobutane unit which is an analog of bond-alternant of cyclooctatetraene. The characteristic polynomial of the edge weighted stellate subgraph of cyclobutane is P(G*;X) = X8 – (4 + 4k2)X6 + (6 + 8k2 + 6k4)X4 – (4 + 4k2 + 4k4 + 4k6)X2 + (1 + k8 ‒ 2k4) = X8 – 5X6 + 8.375X4 ‒ 5.3125X2 + 0.87890625 for k = 0.5. The corresponding matching (or acyclic) polynomial is given by P(G*;X)ac = X8 – (4 + 4k2)X6 + (6 + 8k2 + 6k4)X4 – (4 + 4k2 + 4k4 + 4k6)X2 + (1 + k8) = X8 – 5X6 + 8.375X4 ‒ 5.3125X2 + 1.00390625 for k = 0.5. Using the roots of these equations to determine the energies [Eσ and Eσ(ref)] of the stellate bond-alternant cyclobutane and its hypothetical reference, respectively, we can determine its topological resonance energy [TRE = Eσ ‒ Eσ(ref) = 8.472136 ‒ 8.509082 = ‒ 0.03695 β]. If this idealized cyclobutane had all equal stellate bonds of weight one rather than the bond-alternant of 0.5 and 1.0 weights, it becomes identical to cyclooctatetrene (TRE = ‒ 0.595 β). Using these values, we estimate that cyclobutane has (‒ 0.03695/‒0.595)100% = 6.2% the antiaromaticity of cyclooctatetrene. Overall, strain energy differences between cyclopropane and cyclobutane needs to take into consideration the stabilizing effect of σ–aromaticity of cyclopropane and the σ–antiaromaticity of cyclobutane. CONCLUSION



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This work demonstrates that the McClelland mirror-plane method for factorization secular determinants of Hückel molecular orbitals developed originally for conjugated polyenes is applicable for simplifying alkane stellate Hückel-like calculations. With the Fermi energy being located at E = 0, we argue that any reasonable quantum mechanical energy model should have an equal number antibonding and bonding eigenvalues and that these should sum to zero. The stellate model for tight-binding calculation of the electronic properties of the alkanes meets these criteria and gives results that nicely correlate with their experimental standard enthalpy of formation (∆Hfo) after correcting for protobranching and ionization energies (IE). These linear plots obtained support the protobranching stabilization concept associated with ∆Hfo for the alkanes. The result of the Estrada atom-bond connectivity model also supports the stabilizing effect of protobranching in alkanes.2 This stellate model reveals that cyclopropane exhibits an electron delocalization not present in acyclic alkanes. The irreducible subgraph shown in Figure 5 for cyclopropane and its associated equations can be used to solve all cycloalkane eigenvalues by changing the value of n. How this irreducible subgraph and its associated equations were derived, the reader should refer to pages 47-49 in ref 15.15 Acknowledgement This work was supported in part by a grant from by the UM Board of Curators (K0906077).

REFERENCES [1] Klein, D.J.; Larson, C.E. Eigenvalues of saturated hydrocarbons. J. Math. Chem. 2013, 51, 1608-1618.


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[2] Estrada, E. Atom-bond connectivity and energetics of branched alkanes. Chem. Phys. Lett. 2008, 463, 422-425. [3] Aihara, J. Graph Theory of Aromatic Stabilization. Bull. Chem. Soc. Jpn. 2016, 89, 14251454. [4] Aihara, J. General Rules for Constructing Hückel Molecular Orbital Characteristic Polynomials. J. Amer. Chem. Soc. 1976, 98, 6840-6844. [5] Balasubramanian, K. J. Computer Generation of Characteristic Polynomials of EdgeWeighted Graphs, Heterographs, and Directed Graphs. J. Comput. Chem. 1988, 9, 204-211. [6] McClelland, B.J. Graphical Method for Factorization Secular Determinants of Hückel Molecular Orbital Theory. J.C.S. Faraday Soc. Trans. II 1974, 70, 1453-1456. [7] Wodrich, M.D.; Wannere, C.S.; Mo, Y.; Jarowski, P.D.; Houk, K.N.; Schleyer, P. von R. The Concept of Protobranching and Its Many Paradigm Shifting Implications for Energy Evaluation. Chem. Eur. J. 2007, 13, 7731-7744. [8] Fishtik, I. Comment on “The Concept of Protobranching and Its Many Paradigm Shifting Implications for Energy Evaluations.” J. Phys. Chem. A 2010, 114, 3731-3736. [9] Schleyer, P. von R.; McKee, W.C. Reply to the “Comment on ‘The Concept of Protobranching and Its Many Paradigm Shifting Implications for Energy Evaluations.’” J. Phys. Chem. A 2010, 114, 3737-3740. [10] McKee, W.C.; Schleyer, P. von R. Correlation Effects on the Relative Stabilities of Alkanes, J. Am. Chem. Soc. 2013, 135, 13008-13014.



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[11] Randić, M. On Characterization of Molecular Branching. J. Am. Chem. Soc. 1975, 97, 66096615. [12] Bertz, S.H. Branching in Graphs and Molecules. Discrete Applied Mathematics 1988, 19, 65-83. [13] Exner, K.; Schleyer, P. von R. Theoretical Bond Energies: A Critical Evaluation. J. Phys. Chem. A 2001, 105, 3407-3416. [14] Aihara, J.; Ishida, T. Unusuallly High Aromaticity and Diatropicity of Bond-alternant Benzene. J. Phys. Chem. A 2010, 114, 1093-1097. [15] Dias, J.R. Molecular Orbital Calculations Using Chemical Graph Theory; Springer-Verlag: Berlin, 1993.

List of Figures Figure 1. The application of the modified Sachs equation on the molecular graph of the heterocyclic, pyrrole. Figure 2. Successive mirror-plane fragmentation of the stellate graph of methane leads to four ultimate subgraphs, three of them being identical accounting for it triple degeneracy in the eigenvalues of 0.78078 and - 1.28078. Figure 3. Eigenvalue summary of the stellate graphs corresponding to the initial members of the alkane series. Figure 4. Successive mirror-plane fragmentation of the stellate graph of ethane. Figure 5. Two different methods that give solutions of the doubly degenerate eigenvalues of cyclopropane. The same irreducible graph at the bottom can be used to obtain the eigenvalues of all cycloalkanes (CnH2n, n > 2) by changing the value of n in its associated equation. Figure 6. Successive mirror-plane fragmentation of the stellate graph of propane. Note that the right-hand fragment is identical to the right-hand fragment of stellate graph of isobutane in Figure 7. Figure 7. Fragmentation of the stellate graph of isobutane gives the same right-hand fragment present in fragmentation of the stellate graph of propane.


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Figure 8. Summary of the σ-energy (Eσ) summed from data in Figure 3 along with the HOMO/LUMO energy and ionization energy of the alkanes. Note that all the tetrahedron inner edges have weight of 0.5. Figure 9. Plot of ∆Hfo (kcal/mol, corrected for protobranching) versus Eσ (β). Figure 10. HOMO (β) versus IE (eV). Figure 11. Evaluation of σ-aromaticity in carbon skeleton of cyclopropane where stabilizing delocalization is occurring. Figure 12. Evaluation of σ-antiaromaticity in carbon skeleton of cyclobutane where destabilizing delocalization is occurring.



pyrrole

molecular graph k of pyrrole

N H

S1 =

P(G;X) = X5

= a1 = ( 1)12oh =

k

G

h

hX4

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(3+2k2)X3 + 3hX2 + (1 + 4k2)X

(h + 2k2) = 0

h

h

S2 =

= k

a2 = 3( 1)12o + 2( 1)12ok2 = 3

k

2k2

= a3 = 3( 1)22oh = h

S3 = h

h

h

S4 =

= k

k

k

k

a4 = ( 1)22o + 4( 1)22ok2 =1 + 4k2

= ( 1)32oh + ( 1)121k2 =

S5 = h

k

h 2k2

k

Figure 1. The application of the modified Sachs equation on the molecular graph of the heterocyclic, pyrrole.


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X2 + 0.5X 1 = 0 0.78078

2

0.5 0.5 0.5

X + 0.5X 1 = 0 0.78078

0.5

X8

1.5 5.5X6

X5 + 8.8125X4 + X3

vertical mirror-plane fragmentation

X6

0.5X5

X2 1.5X 2.0

1=0

X2 + 0.5X 1 = 0 0.78078

mirror-plane fragmentation of lower left-hand graph X2 + 0.5X 1 = 0 0.78078

horizontal mirror-plane fragmentation

0.70711

0.5

5.5X2 + 1 = 0

2.0 0.78078 0.78078 0.78078 0.5

methane internal edges are weighted 0.5

0.5 0.70711

0.5

0.5

0.5 0.5

0.5

0.5

0.5 0.5

X2 + 0.5X 1 = 0 0.78078

4.25X4 + 0.625X3 + 4.25X2

0.5X

1=0

0.5 4

X

3

2

X 2.75X 2.0 0.78078 0.5

X

1=0

0.5

X2 + 0.5X 1 = 0 0.78078

Figure 2. Successive mirror-plane fragmentation of the stellate graph of methane leads to four ultimate subgraphs, three of them being identical accounting for it triple degeneracy in the eigenvalues of 0.78078 and



Page 18 of 26


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Page 20 of 26


Page 21 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60




mirror-plane fragmentation

Page 22 of 26

0.70711

0.5

0.5 0.70711 0.70711 0.5

3-fold symmetry fragmentation

E = 28.11926 0.57846 LUMO = IP = 10.57 eV 0.70711 0.5

0.5 0.70711

0.86603

0.70711

mirror-plane fragmentation

0.86603

0.5

right-hand fragments

2x

0.5

0.5

0.5 X2 + 0.5X 1 0.78078, 1.28078 X8 X7 5.75X6 3.625X5 + 10.6251X4 3.7187X3 6.5X2 1.375X + 1 2.09567 1.51831 0.78078 0.64317 1.45304

0.5

Note that all the tetrahedron inner edges have weight of 0.5.

X8 0.5X7 5.5X6 3.25X5 + 8.3125X4 3.53125X3 5.125X2 0.875X + 1 1.98044 0.59846 0.78078 0.78078 0.57846

Figure 7. Fragmentation of the stellate graph of isobutane gives the same right-hand fragment present in fragmentation of the stellate graph of propane.


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Figure 9. Plot of ∆Hfo (kcal/mol, corrected for protobranching) versus Eσ (β) 40 35 30 25 20 15 Eσ =

-3.1179ΔHfo

- 47.095

10

R² = 1 5 0 -30

-25

-20

-15

-10

-5

0



Page 24 of 26

Figure 10. HOMO(β) versus IE(eV) 0.63 0.62 0.61 0.6 0.59 0.58 0.57 0.56 0.55 0.54 0.53

HOMO = 0.0695(IE) - 0.1794 R² = 0.9964

10.2

10.4

10.6

10.8

11

11.2

11.4

11.6


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TOC Graphic



0.5 0.5 0.5

0.5 0.5 0.5

0.5 0.5

0.5 0.5 0.5

0.5

ethane stellate graph with internal edges are weighted 0.5 vertical mirror-plane fragmentation

0.5 0.5 0.5

0.5 0.5 0.5

0.5 0.5

0.5 0.5

0.5

0.5

X7

X6

4.5X5 + 2.75X4 + 5.3125X3

3X2

X7

X6

4.5X5 4.75X4 + 4.8125X3

4.5X2

1.75X + 1 1.75X

1

ACS Paragon Plus Environment

Page 26 of 26

Practical Tight-Binding HÃ¼ckel-like Modeling of Electronic Properties

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