Article pubs.acs.org/JPCA
Electron Localization Function Study on the Chemical Bonding in a Real Space for Tetrahedrane, Cubane, Adamantane, and Dodecahedrane and Their Perfluorinated Derivatives and Radical Anions Slawomir Berski,* Agnieszka J. Gordon, and Zdzislaw Latajka Faculty of Chemistry, University of Wroclaw, F. Joliot-Curie 14, 50-383 Wroclaw, Poland S Supporting Information *
ABSTRACT: The nature of chemical bonding in caged cycloalkanes CnXn, CnFn−•, (n = 4, 8, 20; X = H, F), and C10X16, C10F16−•, (X = H, F) has been investigated using topological analysis of the ELF function, electron density, and the Laplacian of electron density at density functional theory (DFT) level. The bonding analysis performed for the perfluorinated radical anion of dodecahedrane (C20F20−•), bestowing an additional electron, shows an unexpected local maximum of the ELF inside the carbon cage. The presence of such an attractor confirms the sigma stellation concept presented by Irikura (J. Phys. Chem. A 2008, 112, 983) and essential change of the electron localization inside the cage. The basin belongs to the rare asynaptic type, V(asyn), and its mean electron population is 0.26 (0.36e). The value of the integrated spin density, 0.13e, shows that both spin-up and spindown electrons reside in the vicinity of the cage center. A similar attractor has been found for perfluorinated radical anion of adamantane (C10F16−•). However, the saturation of the basis set suggests that such an attractor may be an artifact. For both caged perfluorinated tetrahedrane and cubane (CnFn −•, n = 4, 8), no valence attractors are present inside the cage. Unpaired electron density is concentrated mainly on the C−C bonding basins. The results obtained in this study are complementary to those based on the molecular orbital theory presented by Irikura.
I. INTRODUCTION The concept of a sigma stellation has been described by Irikura1 as a possible tool for the design of electron boxes. Although fluorine is the most electronegative element, fluorocarbons (e.g., CF4) do not express large electron affinities. The solution for constructing negative ions, through fluorinated carbon cages with enhanced electron affinity, is therefore a starlike arrangement of vacant molecular orbitals. When empty σ* orbitals are present in the space of the fluorinated carbon cage and arranged in such a way that they overlap (Figure 1, ref 1), the lowest of the resulting molecular orbitals are delocalized and expected to show an unusual stability. In the case of perfluorinated tetrahedrane (C4F4), cubane (C8F8), adamantane (C10F16), and dodecahedrane (C20F20), an additional
electron occupies a delocalized and totally symmetrical orbital, confined to the inside of the carbon cage. The electron affinity (EA) results confirm its expected value increase. Going from tetrahedrane (C4H4) to tetrafluorotetrahedrane (C4F4), the EA increases from −1.37 to 0.15 eV. Similarly from cubane (C8H8) to octafluorocubane (C8F8) there is an increase from −1.14 to 1.63 eV. The EA values also change when comparing the results for adamantane (C10H16) and hexadecafluoroadamantane (C10F16) (from −1.00 to 1.46 eV), and dodecahedrane (C20H20) and eicosafluorododecahedrane (C20F20) (from −0.70 to 3.73 eV1). Positive values of EAs mean that the molecule binds an electron.1 From the point of view of the size of the molecular system investigated, an extension of Irikura’s is study by Wang et al. on the excess electron encapsulated inside the C60F60 cage. The process of constructing new radical anions via sigma stellation has been investigated through the molecular orbital concept (MO). However, the electronic structure of a molecule can also be described in real space terms using topological analysis of the following scalar fields: the electron density, ρ(r),3 electrostatic potential, ε(r),4,5 electron localization function, ELF, η(r),6−8 or electron localizability indicator,
Figure 1. Starlike arrangement of the carbon-fluoribe antibonding σ* orbitals, where overlapping causes unusually high electron stabilities of the fluorocarbon cages as suggested by Irikura.1 © XXXX American Chemical Society
Received: February 21, 2014 Revised: May 20, 2014
A
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ELI-D, YDσ(r).9 Both the η(r) and YDσ(r) functions enable easy description of the chemical bonding in terms of atomic cores, lone pairs, and chemical bonds, thus having a clear chemical meaning and therefore are most frequently used. The topological analysis of the electron localization function or electron localizability indicator provides a mathematical model of the Lewis’s valence theory. The molecular position space is divided into basins of attractors corresponding to a one-to-one representation of the expected chemical objects. It is rooted in the chemical interpretation of the η(r) and YDσ(r) functions. The gradient dynamical system of the studied function f is characterized by its critical points at given position rC, that is the points at which
Irikura’s discovery1 of the existence of an additional electron inside the carbon cage with a possibility of modifying the electronic structure of the whole molecule inspired our research into the nature of bonding in selected cycloalkanes and perfluorinated cycloalkanes. We have chosen tetrahedrane, cubane, adamantane, and dodecahedrane and their pefluorinated derivatives and radical anions as model compounds. Through this study we would like to be able to answer the following questions: What are the differences in electronic structures of the caged cycloalkanes (CnHn, n = 4, 6, 20; C10H16), perfluorinated cycloalkanes (CnFn, C10F16), and perfluorinated cycloalkanes with an additional electron (CnFn−•, C10F16−•)? Where exactly, on which particular bond or lone pair, represented by localization basins of the η(r) field, is the unpaired electron localized? If sigma-stellation concept is true and an additional electron indeed occupies the central point of the carbon cage in perfluorinated cycloalkanes (CnFn−.,C10F16−•), where the carbon−fluorine antibonding orbitals overlap, how does this reflect the topology of the ρ(r) and η(r) functions? If a new valence attractor in the central region of carbon cage represents the perturbation of the electronic structure, caused by electron attachment, what is the spin composition of localization basins for this region? What is the synaptic type8,14 of the localization basin associated with the valence attractor localized in the center of the carbon cage?
∇f (rC; {α}) = 0
The number of positive eigenvalues of the Hessian (second derivatives) matrix at rC is called the index of the critical point, I(P). In R3, I(P) ranges from 0 to 3. The local maxima called attractors, are critical points with I(P) = 0. The stable manifold, i.e., the set of points defining all the trajectories ending at a given critical point, of an attractor is named a basin. The separatrices, which are the bounding surfaces, lines, and single points between basins, are the stable manifolds of critical points having at least one strictly positive index. There are basically two kinds of basins: the core basins, C(A), encompassing the nuclei with Z > 2 and of atomic symbol A, and the valence basins, V(A, B, ...), the union of which constitute the valence shell of the molecule. The valence basins are characterized by their synaptic order, the number of core basins with which common boundary is shared.8 When proton is located within a valence basin, it is counted as formal core. Monosynaptic basins are associated with lone pairs and disynaptic ones with twocenter. Polysynaptic basins are characteristic of the multicentric bonds. Quantitatively, basin properties are calculated by integrating the relevant density of the property, e.g., the basin population, which for the open shell systems can be written as the sum of its spin contributions: N̅ (Ωi) =
II. COMPUTATIONAL DETAILS The CnXn, (n = 4, 8, 20; X = H, F) and C10X16 (X = H, F) molecules have been studied in singlet state (M = 1) with the charge of 0. The perfluorinated radical anions (CnFn−•, C10F16) are studied in a doublet state (M = 2) with the charge of −1. The optimization of geometrical structures has been performed using the Gaussian 0915 program with the unrestricted DFT wave functions. The B3LYP functional16−18 in conjunction with 6-31++G(d,p)19,20 and aug-cc-pVTZ21,22 basis sets has been used as implemented in the G09 program. The def2-TZVPPD basis set23,24 has been obtained from the Basis Set Exchange software and the EMSL Basis Set Library.25,26 The localization of the minima on potential energy surfaces (PES) has been verified by nonimaginary harmonic frequencies. The stability of the wave function has been confirmed for each case. Topological analysis of ELF function was carried out using DGrid-4.6,27 TopMod,28,29 and TopMod09 programs. The “spin-polarized” formula has been used within the Dgrid-4.6 program. The parallelepipedic grid of points, surrounding each molecule within the radius of 5 bohrs with a step of 0.05 bohr has been adopted. The topological analysis of electron density field was carried out using AIMAll program.30 The graphical representation of molecules and ELF-basins has been obtained using JMol31 and VMD32 programs. The manual search for the critical points in the perfluorinated adamantane radical anion has been performed starting with the points on the straight line joining the attractor in the center of molecule, V(asyn), to the C(C) attractors and scanned with a step of 0.1 bohr using the search09 program from TopMod09 package. The gradient path joining attractors through (3,−1) CP has been traced using the search09 program.
∫Ω ρ(r)dr = N̅ α(Ωi) + N̅ β(Ωi) i
The variance of the basin population is calculated as σ 2(N̅ , Ωi) =
∫Ω dr1 ∫Ω π(r1, r2)dr2 − [N̅ (Ωi)]2 + N̅ (Ωi) i
i
where π(r1,r2) is the spinless pair function. The variance is a measure of the quantum mechanical uncertainty of the basin population, which can be interpreted as a consequence of the electron delocalization.8 One of the most intersting results obtained from topological analysis of η(r) or YDσ(r) fields is much smaller basin population (less than 1e) obtained for some formally single covalent bonds. Such bonds have been named “electrondepleted”.10 Sini et al.11 showed that among electron-depleted bonds (A−B) a special class of bonds exists, distinct from the covalent and ionic bonds. These bonds are characterized by the resonance forms of the A+B− and A−B+. From topological point of view the localization basin, V(A,B), characterizing such bond possesses a population smaller than 1e, a large variance, and a dominant role of the fluctuation of the electron pair density.12 Electron-depleted bonds were described on the basis of the valence bond theory (VB) and the name “the charge-shift bonds” (CS) was suggested.13 B
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(C4F4−•) does not perturb the topological structure of the carbon cage. In order to check that the topology of the η(r) field in C4F4−• is not dependent on the basis set choice, the geometrical optimizations have been performed at the UB3LYP/aug-ccpVTZ and UB3LYP/def2-TZVPPD computational levels. The results show the same topology of ELF as the one obtained using a smaller basis set. Furthermore, no valence attractor inside the carbon cage has been found. The basin populations (N̅ ) for the attractors are collected in Table 2. The values obtained for the H−C and C−C bonds in both C4H4 and C4F4−• molecules are close to the formal value of 2e, expected for the single bond. It is worth noting that, in perfluorinated tetrahedrane radical anion, the V(C,C) basin population is 0.39e larger, although the length of the C−C bond has not been practically changed (Δr = 0.003 Å). For a longer bond, a smaller basin population should be expected. The population of the V(C,F) basin representing the C−F bond is much smaller than 1e (0.73e). The electron depleted character of the C−F bonds has been recently reported by Shaik et al.13 for the CH3F molecule. The small population of the V(C,F) basin (0.86e) and the large population variance, σ2 (0.64) is thought to be a confirmation of the covalent−ionic fluctuation mechanism for the charge-shift bonds.12 The small basin population obtained for the C−F bond in C4F4−• also confirms its character as a charge-shift (CS) type bond. Partition of the molecular electron density into the ELFbasins, corresponding to the cores, bonds, and lone pairs, can also help to locate the position of the additional electron. The integrated spin density, ρspin, for the V(C,C) basin is 0.1e, about five times more than the value of 0.02e, obtained for the V(C,F) basin. The unpaired electron occupies mainly the C−C bonds because all six V(C,C) basins are represented by 0.6e of ρspin. Irikura1 has shown using the MO approach that some part of the unpaired electron density is delocalized inside the carbon cage. His calculations yielded the ρspin value of 0.026e. Our results are in agreement with the MO results since the six V(C,C) basins cover the entire space inside carbon skeleton. The separatrix of each V(C,C) valence basin “touch” each other in the C4F4−• center of the mass, where the repellor, the (3,+3) critical point of η(r) and ρ(r) fields, (η(3,+3)(r) = 0.347, ρ(3,+3)(r) = 0.184 e/bohr3) is found. Hence, a part of 0.6e (ρspin) should be localized within the limits of integration used by Irikura.1 III.2. C8H8, C8F8, C8F8−•. Topological analysis of all the derivatives of cubane, (C8H8), has been performed using both DGrid-4.6 and TopMod09 programs. With the exception of the nonbonding attractors Vi(F) corresponding to formal lone electron pairs of the F atoms, the same topological structure has been obtained. The valence bonding attractors, V(C,C) and V(C,F), are localized for each C−C bond in C8H8 and C−C and C−F bonds in C8F8 and C8F8−•. Topological analysis of η(r) function does not show any valence attractor inside the carbon cage, where only the V(C,C) basins are observed for the C−C bonds. Thus, similarly to the case of perfluorinated tetrahedrane radical anion, the attachment of electron to perfluorinated cubane does not change its η(r)-topological (electronic) structure. In a search for critical point around the C8F8−• center of mass, only a repellor, (3,+3) critical point (η(3,+3)(r) = 0.052, ρ(3,+3)(r) = 0.036 e/bohr3) has been found, for both η(r) and ρ(r) fields. It is worth noting that both values are much smaller than those found for the perfluorinated tetrahedrane radical
III. RESULTS AND DISCUSSION III.1. C4H4, C4F4, C4F4−•. In order to examine the electronic structure of the simplest caged cycloalkanes, tetrahedrane (C4H4) and the radical anion of perfluorinated tehrahedrane (C4F4−•) have been chosen. As the full optimization of the geometrical structure of the C4F4 molecule proved to be impossible, we did not include this molecule in the discussion. The optimized bond lengths are collected in Table 1. The topological structures represented by attractors of the η(r) field for studied molecules are compared in Figure 2. Table 1. . Comparison of the Optimized Bond Lengths in Tetrahedrane, Cubane, Adamantane, and Dodecahedrane and Their Perfluorinated Derivatives and Radical Anions of Perfluorinated Derivatives; All Calculations Performed Using 6-31++G(d,p) Basis Seta n=4
C4H4
C4F4
C4F4−•
C−C [Å] C−H(F) [Å] n=8
1.481 1.072 C8 F 8
b C8F8
1.478 1.389 C8F8−•
C−C [Å] C−H(F) [Å] n = 10
1.571 1.092 C10H16
1.579 1.344 C10F16
1.558 1.394 C10F16 −•
C−C [Å] Cs−H(F) [Å] Ct−H(F) [Å] n = 20
1.544 1.099 1.098 C20H20
1.566 1.352 1.362 C20F20
1.548 1.367 1.456 C20F20−•
C−C [Å] C−H(F) [Å]
1.556 1.096
1.565 1.358
1.555 1.379
a
n = number of carbon atoms in molecule. bOptimization of geometrical structure indicated two imaginary frequencies.
Topological analysis of all tetrahedrane derivatives have been performed using both DGrid-4.6 and TopMod09 programs. The same types of topological structures, in terms of the number of local maxima (attractors) and their spatial positions, have been obtained for the carbon C(C) and fluorine C(F) cores and the carbon−carbon and carbon−fluorine bonds (V(C,C) and V(C,F)), respectively. Furthermore, very similar values of the basin populations (N̅ ) have been obtained. Nonbonding electron density of fluorine has been described using a set of point attractors, represented in TopMod09 program by three Vi(F) attractors for each fluorine atom. The carbon skeletons in the C4H4 and C4F4−• molecules are characterized by similar sets of the core C(C) and bonding attractors V(C,C), positioned outside the straight line joining the C(C) core attractors (Figure 2). Such topological feature of η(r) field undoubtedly is associated with the strained character of these molecules. The strain in carbon compounds (cyclotetrahedrane and cyclobutane) has been described for the first time within the ELF framework by Chevreau and Sevin.33 The carbon−fluorine bond is characterized by a single attractor, V(C,F), positioned 0.5 Å from the C(F) core attractor and 1.58 Å from the C(C) core attractor. For the bond of a covalentpolarized character, the presence of the V(C,F) attractor in the vicinity of more electronegative atom supports such classification. In the case of the C−C bonds, the V(C,C) attractor is situated 0.88 Å from each C(C) core attractor, confirming the covalent character of the bond. The most important observation is that the presence of an additional electron C
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Figure 2. Core and valence attractors of η(r) field for tetrahedrane, cubane, and their fluorinated derivatives.
III.3. C10H16, C10F16, C10F16−•. Optimized bond lengths for adamantane (C10H16), perfluorinated adamantane (C10F16), and the radical anion of perfluorinated adamantane (C10F16) are compared in Table 1. Calculated bond lengths are very similar to those published by Kovacs and Szabo.34 The adamantane contains four tertiary (Ct) and six secondary (Cs) carbon centers; therefore, the difference between the basin population of the respective Cs−H(F) and Ct−H(F) bonds can be expected. Once an additional electron is attached, the carbon−fluorine bonds formed through the tertiary carbon atom (Ct−F) undergo a large elongation by 0.094 Å. The (Cs− F) bond formed through the secondary carbon atom becomes only slightly longer (by 0.015 Å). Dramatic structural changes in the length of the Ct−F bonds is likely to effect the values of the basin populations significantly. Topological analysis of all the derivatives of adamantane has been performed using both DGrid-4.6 and TopMod09 programs. In C10H16, each carbon−carbon bond of the four connected cyclohexane rings is represented by the single disynaptic attractor, V(Cs,Ct), with the basin population of 1.87e (see Table 2). Similarly to previously studied cycloalkanes, this population corresponds well with the results expected for the single carbon−carbon bond (2e). Each of the carbon−hydrogen bonds is represented by two protonated attractors, V(H,Cs) and V(H,Ct), with very similar values of the basin population (2.03e and 2.06e, respectively).
anion. This can be attributed to the larger distance between the carbon atoms. The results for the basin populations, collected in Table 2, show similar values to those calculated for tetrahedrane and its derivatives. The increase in the carbon cage size from C4H4 to C8H8 results in a small electron density redistribution from the C−H to C−C bonds (0.03e). However, the substitution of the hydrogen atoms (C8H8) by fluorine atoms (C8F8) leads to significant saturation of the C−C bonds and consequently to the basin population increase by 0.23e. An additional electron causes further electron density concentration on the C−C bonds (by 0.08e) while the population of the C−F bonds diminishes by 0.15e. The values of the integrated spin densities for single C−C and C−F bonds are two times smaller than in the C4F4−• molecule and seem to be associated with a larger number of the C−C and C−F bonds. The unpaired electron is mainly localized in the C−C bonds, and both C4F4−• and C8F8−• radical anions display very similar distribution of unpaired electron density. The integrated spin density (ρspin) inside the carbon cage in C8F8−•, calculated by Irikura1 is 0.11e. Applying similar analysis as in the case of the C4F4−• molecule, we can conclude that our results are in an agreement with the MO analysis. A fraction of 0.6e (ρspin) from 12 V(C,C) basins will fill the inside of the carbon cage. D
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Table 2. Mean Electron Population (N̅ ) for Selected Localization Basins in Tetrahedrane, Cubane, Adamantane, and Dodecahedrane and Their Perfluorinated Derivatives and Radical Anions of the Perfluorinated Derivatives; the Basin Population of the Core C(C), C(F) and Non-Bonding Basins Vi(F) Have Been Omitted and the Integrated Spin Densities (ρspin) for Radical Anions Are Presented in Parenthesesa N̅ [e]
basin/molecule n=4
C4H4
V(H,C) V(C,C) V(C,F) n=8
2.17 1.82
V(H,C) V(C,C) V(C,F) n = 10 V(H,Cs) V(H,Ct) V(Cs,Ct) V(Cs,F) V(Ct,F) V(asyn) n = 20 V(H,C) V(C,C) V(C,F) V(asyn) a
In C10F16, two bonding disynaptic attractors, V(Cs,F) and V(Ct,F), represent two types of the carbon−fluorine bonds. The lone electron pairs are characterized by a set of nonbonding attractors, Vi(F). Fluorination leads to the saturation of the C−C bonds by 0.26e, although the bonds in C10F16 are 0.022 Å longer than in C10H16. This effect is similar to the one observed for perfluorinated tetrahedrane and perfluorinated cubane. Despite the large withdrawing character of the fluorine atom (inductive effect), both the mesomeric effect and the repulsion between the lone pairs play the crucial role here. The Ct−F and Cs−F basin population vary significantly (0.92e and 1.00e, respectively). Such difference corresponds well to the difference in values of the bond lengths. For the longer Ct−F bonds (1.362 Å), a smaller amount of the electron density is obtained (ΔN̅ = 0.08e). The C−F bonds can be assigned to the electron-depleted type of bonds. The electronic structure of the C10F −• radical anion, represented by core and valence attractors (Figure 3a) is very similar to the one observed in a neutral form of perfluoroadamantane. The number of core attractors, (C(Cs), C(Ct)), and valence nonbonding attractors,Vi(F), remains unchanged. Similarly, for the attractors and synaptic orders related to the carbon−fluorine bonds (V(Cs,F), V(Ct,F)) and the carbon−carbon bonds (V(Cs,Ct)) not much change is observed. However, our results show the presence of a single valence attractor inside the carbon cage region. The attractor is situated at the crossover of the C3 and C2 axis and lies 1.80 Å from the secondary carbon atoms and 1.56 Å from the tertiary carbon atoms. Finding such an attractor is an unusual discovery and is bound to be a consequence of high concentration of both spin-up and spin-down electron density inside the carbon cage. It is also worth noting that the attractor is characterized by a small value of η(3,‑3)(r) (0.176), essentially smaller than usual
C8H8 2.14 1.85 C10H16 2.03 2.06 1.87
C20H20 2.08 1.88
C4F4
C4F4−•
C8F8
2.21 (0.10) 0.73 (0.02) C8F8−•
2.08 0.89 C10F16
2.16 (0.05) 0.74 (0.01) C10F16−•
2.13 1.00 0.92
2.21 (0.04) 0.90 (
