Density Functional Theory versus Complete Active Space Self

Taras Shevchenko National University of Kyiv, Volodymyrska Street 64, Kyiv ... and Department of Chemistry and Biochemistry, Concordia University, Mon...
0 downloads 0 Views 949KB Size
Download PDF
Article pubs.acs.org/JPCA

Density Functional Theory versus Complete Active Space SelfConsistent Field Investigation of the Half-Metallic Character of Graphite-Like and Amorphous Carbon Nanoparticles Oleksiy V. Khavryuchenko,*,‡,† Volodymyr D. Khavryuchenko,§ and Gilles H. Peslherbe‡ †

Taras Shevchenko National University of Kyiv, Volodymyrska Street 64, Kyiv 01601, Ukraine Centre for Research in Molecular Modeling and Department of Chemistry and Biochemistry, Concordia University, Montréal, Québec, Canada H4B1R6 § Computation Chemistry Group, Revutskogo Street 13, app. 149, Kyiv 02091, Ukraine ‡

S Supporting Information *

ABSTRACT: Model carbon nanoparticles representative of the graphite-like and amorphous domains of active carbon are investigated with density functional theory (DFT) and complete active space self-consistent field (CASSCF) methods. Cyclic carbon clusters containing conjugated carbene groups are found to undergo Jahn− Teller distortion. More importantly, the half-metallicity, that is, the equal or similar stability of various spin states, previously suggested by DFT calculations for both types of nanosized clusters is confirmed by CASSCF calculations. Furthermore, the model carbon clusters are found to possess a multiconfigurational electronic structure dominated by high-spin configurations. When compared to CASSCF results, the single-reference DFT predicts proper electronic structures, characterized by antiferromagnetically coupled electron pairs, at the expense of spin contamination as a reflection of the multiconfigurational character. In fact, spin contamination, which is normally viewed as an error, does not corrupt the energetics of the half-metallic systems and therefore does not preclude the applicability of DFT to such systems.

1. INTRODUCTION Carbon exhibits a large number of structural polymorphs and composites with properties varying from insulating to metalconducting, from mechanically hard to soft, from fragile to elastic,1−4 which warrants the investigation of a wide array of model carbon materials. While carbon may take various forms such as graphite, diamond, fullerenes, or nanotubes with welldefined structures, active carbon is thought to be made up of two major domains, namely, a graphite-like domain and a disordered amorphous one, 3,4 which deserve extensive characterization. A recent quantum chemical investigation has demonstrated that both graphite-like and amorphous model clusters of carbon possess a set of pseudodegenerate spin states, conferring them half-metallic properties.5−14 The number of these pseudodegenerate spin states depends on the quantity of unsaturated carbene-like sites conjugated via graphitic or cumulenic bridges. Such features should have a tremendous effect on the physicochemical properties of carbon nanoparticles.15−19 Previous calculations based on density functional theory (DFT) suggest that the electronic structures of carbon clusters possess significant multiconfigurational character, as reflected in the large extent of spin contamination.6 Indeed, it has been reported that, in some cases, the integer part of the extent of spin contamination bears some information about the distribution of unpaired electrons within degenerate molecular orbitals (MOs), and therefore, the significant spin contamination observed in calculations with single-reference methods © 2014 American Chemical Society

such as DFT would simply reflect the presence of antiferromagnetically coupled electronic pairs,20,21 coinciding with the number of effectively unpaired electrons and characterizing local spin density.22−24 In order to assess the reliability of the previous DFT calculations on carbon clusters and the prediction of their halfmetallic behavior, as well as the multiconfigurational character of their electronic structure, a similar investigation is carried out with the complete active space self-consistent field (CASSCF) approach in this work. Since the model clusters investigated in previous work6 are not suitable for CASSCF calculations because of their large number of atoms and the large active space that the calculations would require, smaller model clusters, still representative of the graphite-like and amorphous domains of active carbon, are designed and investigated, both with DFT and CASSCF methods.

2. COMPUTATIONAL METHODS DFT calculations are performed with the hybrid threeparameter Becke Lee−Yang−Parr functional (referred to as B3LYP)25 as defined in the TurboMole program,26 together with Ahlrichs’ triple-ζ split-valence basis set augmented by polarization functions (def2-TZVP).27 The unrestricted Kohn− Sham formalism is employed throughout this work; brokenReceived: March 4, 2014 Revised: July 13, 2014 Published: July 25, 2014 7052

dx.doi.org/10.1021/jp502207n | J. Phys. Chem. A 2014, 118, 7052−7057

The Journal of Physical Chemistry A

Article

orbitals are discussed hereafter. Finally, exploratory CASSCF(4,4) calculations were also performed (cf. Supplement A, Supporting Information) and found to provide an inadequate description of the cluster spin states, compared to the larger active space CASSCF(8,8) calculations, essentially lifting their pseudodegeneracy and proving that the large (8,8) active space is indeed needed in CASSCF calculations. In order to construct smaller models of amorphous-domain carbon nanoparticles, the prescriptions previously established for the amorphous C92H18 cluster6 are followed: (i) the H/C atomic ratio must be ∼0.2, as found experimentally for active carbon; (ii) there must be two-coordinated carbene-type carbon sites (potentially spin-active); and (iii) conjugation chains must link the spin sites. Because benzyne-type motifs such as phenyl rings dehydrogenated on one side have been shown to provide spin conductivity,6 along with conformational rigidity and the lowest H/C ratio, a cyclic model of three carbene centers connected by three benzyne motifs (denoted A6H and shown in Figure 1b) was first considered as the smallest possible model (A stands for “amorphous” and “6” denotes the number of H atoms in the model. An analogous notation is used for A16H and A22H further). For this geometry, all spin states up to the septet are pseudodegenerate, but upon geometry relaxation, the initial cluster geometry collapses for all lower-multiplicity states than the septet due to the proximity of unsaturated carbon atoms with oppositely polarized radicals, forming a fullerene-like curved graphene “bowl”. Although this is an interesting finding by itself, which could explain the possible mechanism of fullerene-like motif formation during the carbonization of polymer precursors, it precludes the use of the A6H model as a reduced variant of the truly amorphous domain, which is characterized in the first place by the presence of postpolymeric carbenes or cumulenic chains.6 Therefore, a generic model (denoted A16H and shown in Figure 1c) containing the A6H moiety rigidly connected at the bottom by three aliphatic chains linked to a single CH group in the center (Figure 1c) was constructed. Although deviating from the required H/C ratio overall, this model retains the desired dehydrogenation level for the “upper” three-site carbene-containing cycle. Finally, in order to unveil the properties of cyclic carbene-containing systems, another model (denoted A22H and shown in Figure 1d) was also constructed, with all noncarbene carbon atoms saturated by hydrogens (to the level of benzene). For each target multiplicity, CASSCF calculations were performed for a total of six singlet, six triplet, three quintet, and one septet eigenstates with a (6,6) active space (six electrons in six orbitals) for these model clusters.

symmetry (BS) singlet and triplet states are also characterized for select systems by using a spin−flip procedure and the density matrix of the most stable high-multiplicity state as the initial density matrix of the calculations. CASSCF28,29 calculations are also performed with the def2-TZVP basis set, with the initial wave function taken from the DFT calculations. Energies are evaluated for a number of eigenstates at the cluster geometry optimized for the lowest-energy eigenstate of a given target-multiplicity manifold obtained without state averaging. The ORCA ab initio DFT, and semiempirical self-consistent field MO theory package30 was used for all calculations. Figure 1 displays the carbon structures considered in this work. The G-4H cluster (G stands for “graphite-like” and “-4”

Figure 1. B3LYP/def2-TZVP optimized structures of the carbon clusters: (a) graphite-like G-4H; (b) A6H transforming from the septet-preferred conformation (side and top view, left) to the lowerspin-state-preferred conformation (right); (c) A16H transforming from the septet-preferred conformation (side and top view, left) to the quintet-preferred conformation (right); (d) A22H transforming from the septet-preferred conformation (side and top view, left) to the quintet-preferred conformation (right).

denotes the number of H atoms abstracted from the edge), shown in Figure 1a, is a reduced version of the C96H18 hexagulene used in previous DFT calculations,6 which can be considered as a model of the graphite-like domain of active carbon. The model comprises two rows of aromatic rings with one dehydrogenated edge, providing conformational rigidity and, as we will see later, additional stabilization of highmultiplicity states due to deep delocalization. For each target multiplicity, CASSCF calculations were performed for a total of six singlet, six triplet, and one quintet eigenstates with a (8,8) active space (eight electrons in eight orbitals) for this model cluster. Two types of initial orbitals were used in the CASSCF calculations; (1) one set of orbitals is that of the model guess orbitals determined by the ORCA program (where the initial electron density is constructed as a superposition of predetermined spherical neutral-atom densities) for the DFToptimized triplet structure; and (2) another is the set of orbitals obtained from “single-point” restricted open-shell Kohn−Sham energy calculations for the DFT-optimized quintet structure. The results obtained with both sets of initial orbitals are qualitatively the same (cf. Supplement A (Supporting Information) for active space MOs and energies), and therefore, only the results obtained with the former set of

3. RESULTS AND DISCUSSION The structure of all model clusters considered was first optimized by DFT. Table 1 collects the cluster relative energies obtained for various multiplicity states. The similar stability of the spin states from singlet to quintet is obvious from this data. This demonstrates that the small models considered in this work exhibit the same half-metallic properties as the larger model clusters considered in previous work,6 that is, they possess a number of pseudodegenerate spin states. The small model clusters thus provide an avenue to benchmark the results of DFT calculations against those of the more rigorous but more computationally intensive CASSCF method, which is the very focus of this work. 7053

dx.doi.org/10.1021/jp502207n | J. Phys. Chem. A 2014, 118, 7052−7057

The Journal of Physical Chemistry A

Article

interconverting between conformations, cannot be observed in active carbon by structural methods such as high-resolution transition electron microscopy (HR-TEM). Also, Jahn−Teller distortion could lead to difficulties in attempts to resolve magnetic interactions in such flexible clusters within the Heisenberg model, where high symmetry and a low level of electron delocalization are assumed. The CASSCF-optimized geometry of all carbon clusters is consistent with that obtained from DFT, although differing in bond lengths for the carbene groups in amorphous cluster models by up to 0.05 Å and in the CCC angle by up to 10°. This deviation is less pronounced for the G-4H cluster, with a standard deviation in the C−C bond lengths of structures optimized with different multiplicities of up to 0.06 Å. As for amorphous clusters, differences in the DFT and CASSCF predictions of the point symmetry of structures are only observed in the case of triplet-optimized geometries, where CASSCF still yields a Cs symmetry structure (a discrepancy possibly due to the limitations of the single-reference nature of DFT). The DFT and CASSCF relative energies of the carbon clusters in various spin states are shown in Figure 3 and in

Table 1. B3LYP/def2-TZVP Select Results for Clusters in Various Spin States multiplicity

magnetic moment, Ms

1b 3 5 7

0 1 2 3

1b 3 5 7 9 11

0 1 2 3 4 5

1b 3 5 7 9 11

0 1 2 3 4 5

extent of spin contamination

relative energya

G-4H 2.02 1.03 0.31 0.08

1.0 1.0 0.0 16.9

2.09 1.12 0.05 0.03 0.04 0.05

0.7 0.0 1.0 27.4 62.3 97.7

1.15 0.02 0.04 0.03 0.04 0.07

0.0 1.8 0.1 22.7 105.7 177.8

A16H

A22H

a

Relative energy (in kcal/mol) with respect to the lowest energy of a given cluster. bBS singlet state.

Let us first turn our attention to the DFT predictions of cluster structures and spin properties. The structure of the G4H cluster does not change much upon optimization with different target multiplicities (with a standard deviation in the C−C bond lengths of all structures below 0.004 Å). Figure 2

Figure 2. B3LYP/def2-TZVP spin density distributions in the quintetstate carbon clusters: (a) G-4H and (b) A22H. Spin density distributions were visualized with the UCSF Chimera package.31

Figure 3. Relative energies of the G-4H, A16H, and A22H clusters in various target-multiplicity states (and geometries; see the text): (a) B3LYP/def2-TZVP and (b) CASSCF/def2-TZVP. The energy reference is that of the singlet state.

displays spin density distributions for the G-4H and A16H clusters. The quintet structure of G-4H, with four singly occupied sites, is obvious from Figure 2a. However, one would expect a septet structure for the amorphous-domain models, with three biradical carbenes and thus six unpaired electrons for the A16H and A22H clusters. This is due to Jahn−Teller distortions of the cluster from C3v to Cs symmetry for the quintet and singlet states (quintet structures shown in Figure 1c and d) and to C1 symmetry for the triplet state, which provides an additional stability ranging from 22 to 27 kcal/mol. Localization of the spin density on one of the three edge carbons in the quintet A22H, shown in Figure 2b, suggests that the distortion could occur equivalently in all three directions, leading to a triply conformationally degenerate distorted state (with relatively small barriers for interconversion). This might explain why cyclic postpolymeric motifs, being floppy and easily

Supplement A (cf. Tables S1 and S2, Supporting Information). The CASSCF lowest-energy structures of A16H and G-4H correspond to a quintet state, precisely as predicted by DFT (Table 1). More importantly, inspection of Figure 3 immediately reveals that the half-metallicity of the carbon clusters, that is, the existence of multiple pseudodegenerate spin states, predicted by DFT is confirmed by CASSCF calculations. A more detailed examination of the CASSCF relative energies versus target multiplicity shown in Figure 4 and Tables S1 and S2 (Supplement A, Supporting Information) further reveals that all carbon clusters considered, both graphite-like and amorphous and for each target multiplicity and corresponding geometry, possess a number of pseudodegenerate eigenstates. For instance, the quintet, lowest triplet, second triplet, and first singlet states of the G-4H cluster lie within 1 kcal/mol, while all 7054

dx.doi.org/10.1021/jp502207n | J. Phys. Chem. A 2014, 118, 7052−7057

The Journal of Physical Chemistry A

Article

Scheme 1. Structure of the Lowest-Lying Triplet Eigenstate of the A16H Cluster Optimized in the Triplet Statea

a

The spin assignment being arbitrary, only one possible electronic configuration illustrative of the configuration state function is represented in this scheme.

carbon materials is not so critical to their spin properties. The G-4H cluster has a simpler electronic structure with respect to the number of major contributions but still possesses singlet and triplet eigenstates with important open-shell contributions. Interestingly, for G-4H optimized with a septet target multiplicity (an electronically excited state), the lower-multiplicity eigenstates remain multiconfigurational and made up of the higher-lying second singlet, third triplet, and so forth states. Therefore, all carbon clusters examined with the CASSCF approach have half-metallic character, as previously shown by DFT for larger model systems, and this property is retained upon excitation to higher electronic levels (Figure S6, Supporting Information). The DFT and CASSCF relative energies as a function of multiplicity (Figures 3 and 4c) exhibit a similar behavior and demonstrate the similar stability of states with multiplicities from 1 to 5 for A16H (we note that CASSCF calculations with DFT-optimized geometries lift the pseudodegeneracy, with the singlet state rising in energy probably due to the increased diamagnetic character, that is, the tendency to doubly occupy orbitals). Thus, the actual pseudodegeneracy of the spin states is consistent with a single-determinant approach like DFT predicting open-shell BS singlet and triplet states (with four SOMOs), with antiferromagnetically coupled pairs of electrons that provide the same polyradical character as a quintet state (differing only by magnetization or spin flip). From the data in Table 1, there appears to be a correlation between the magnetic moment of the system (Ms) and the integer part of the extent of spin contamination (N) within the range of pseudodegenerate states. As a rule of thumb, the sum of Ms and N is constant and equal to the magnetic moment Mmax of the highest-multiplicity pseudodegenerate state (e.g., 2 for the singlet, triplet, and quintet states of G-4H and A16H). The integer part of spin contamination in the case of half-metallic systems should not be viewed as an error but as and intrinsic property of the system, namely, a measure of antiferromagnetically coupled electron pairs (where each unit of N represents one antiferromagnetically coupled electron pair), as previously reported in ref 20 and nicely illustrated with the example of dinuclear complexes of transition metals.

Figure 4. CASSCF/def2-TZVP relative energies of geometryoptimized carbon clusters as a function of target multiplicity: (a) G4H, (b) A16H, and (c) A22H. The set of lowest-lying eigenstates for a given target-multiplicity geometry is given as S1−first singlet, S2− second singlet, T1−first triplet, T2−second triplet, T3−third triplet, Q−quintet. The energy reference is that of Q (a) and S1 (b,c) in triplet-optimized clusters.

six lowest states lie within 2 kcal/mol (Figure 4a). CASSCF calculations with DFT-optimized geometries lead to the same outcome (results not shown), except that the composition of the eigenstates contains contributions from a larger number of configurations. Therefore, the CASSCF calculations and the resulting pseudodegeneracy of the spin states are fairly insensitive to the choice of optimized geometry employed. Analysis of the electronic configurations of the most stable pseudodegenerate states (cf. Supplement B, Supporting Information) shows that in the majority of cases, the eigenstates with multiplicity equal to or lower than that of the targetmultiplicity one are multiconfigurational, with the same orbital makeup and occupancy, and only differing by magnetization. For instance, the lowest-energy triplet eigenstate of A16H optimized in the triplet state (one of the pseudodegenerate states) is made up of a fair number of triplet configurations, to some extent some with high open-shell character (four SOMOs), as illustrated in Scheme 1. In contrast, the next triplet eigenstate (only 0.08 kcal/mol higher in energy) is overwhelmingly dominated by an open-shell configuration (with four SOMOs, 57%). The quintet eigenstates generally implicate fewer configurations, with one highly dominant contribution. These features are also found for cluster structures optimized in other target-multiplicity states. The A22H cluster exhibits a similar behavior as A16H, and thus, benzyne links provide no enhancement or decrease of electronic delocalization in comparison to phenyl rings, suggesting that the high level of dehydrogenation observed in

4. CONCLUSIONS In this work, we report a comparative investigation of small models of graphite-like and amorphous nanoparticles by DFT and CASSCF calculations, examining their structural and 7055

dx.doi.org/10.1021/jp502207n | J. Phys. Chem. A 2014, 118, 7052−7057

The Journal of Physical Chemistry A

Article

Electronic Structure of the Active Carbon Graphite-Like and Amorphous Domains. Chem. Phys. Lett. 2011, 513, 261−266. (7) Khavryuchenko, V. D.; Tarasenko, Y. A.; Strelko, V. V.; Khavryuchenko, O. V.; Lisnyak, V. V. Quantum Chemical Study of Polyaromatic Hydrocarbons in High Multiplicity States. Int. J. Mod. Phys. B 2007, 21, 4507−4515. (8) Khavryuchenko, V. D.; Khavryuchenko, O. V.; Tarasenko, Y. A.; Lisnyak, V. V. Computer Simulation of N-Doped Polyaromatic Hydrocarbons Clusters. Chem. Phys. 2008, 352, 231−234. (9) Philpott, M. R.; Cimpoesu, F.; Kawazoe, Y. Geometry, Bonding and Magnetism in Planar Triangulene Graphene Molecules with D3h Symmetry: Zigzag Cm442+4m+1H3m+3 (m: 2, .. 15). Chem. Phys. 2008, 354, 1−15. (10) Philpott, M. R.; Cimpoesu, F.; Kawazoe, Y. Bonding and Magnetism in High Symmetry Nano-Sized Graphene Molecules: Linear Acenes C 4m+2 H2m+4 (m=2,...25); Zigzag Hexangulenes C6m**2H6m (m=2,...10); Crenelated Hexangulenes C 6 ( 3 m * * 2 − 3 m + 1 ) H 6 ( 2 m − 1 ) (m=2,...6); Zigzag Triangulenes Cm**2+4m+1H6m (m=2,...15). Mater. Trans. 2008, 49, 2448−2456. (11) Khavryuchenko, V. D.; Khavryuchenko, O. V.; Tarasenko, Yu. A.; Shkilnyy, A.; Stratiichuk, D.; Lisnyak, V. V. Nanostructurisation in the SKS Active Carbon, Characterised by SEM, TEM, EDX and Quantum-Chemical Simulations. Int. J. Mod. Phys. B 2010, 24, 1449− 1462. (12) Kan, E.; Hu, W.; Xiao, C.; Lu, R.; Deng, K.; Yang, J.; Su, H. Half-Metallicity in Organic Single Porous Sheets. J. Am. Chem. Soc. 2012, 134, 5718−5721. (13) Lee, K. W.; Lee, C. E. Half-Metallic Carbon Nanotubes. Adv. Mater. 2012, 24, 2019−2023. (14) Fong, C. Y.; Pask, J. E.; Yang, L. H. Materials for Engineering: Vol. 2. Half-Metallic Materials and Their Properties; Imperial College Press: London, 2013; p 304. (15) Khavryuchenko, V. D.; Khavryuchenko, O. V.; Lisnyak, V. V. Effect of Spin Catalysis in H2S Oxidation: A Quantum Chemical Insight. Catal. Commun. 2010, 11, 340−345. (16) Buchachenko, A. L.; Berdinsky, V. L. Electron Spin Catalysis. Chem. Rev. 2002, 102, 603−612. (17) Minaev, B. F. The Role of Exchange Interaction in Mechanisms of Spin-Catalysis. Theor. Exp. Chem. 1996, 32, 1−12. (18) Buchachenko, A. L.; Berdinsky, V. L. Spin Catalysis as a New Type of Catalysis in Chemistry. Russ. Chem. Rev. 2004, 73, 1033− 1039. (19) Schwarz, H. On the Spin-Forbiddeness of Gas-Phase Ion− Molecule Reactions: A Fruitful Intersection of Experimental and Computational Studies. Int. J. Mass Spectrom. 2004, 237, 75−105. (20) Zilberberg, I.; Ruzankin, S.Ph. Expansion of the Unrestricted Determinant in the Basis of Paired Orbitals. Chem. Phys. Lett. 2004, 394, 165−170. (21) Rodriguez, J. H.; McCusker, J. K. Density Functional Theory of Spin-Coupled Models for Diiron-Oxo Proteins: Effects of Oxo and Hydroxo Bridging on Geometry, Electronic Structure, and Magnetism. J. Chem. Phys. 2002, 116, 6253−6270. (22) Ramos-Cordoba, E.; Matito, E.; Mayer, I.; Salvador, P. Toward a Unique Definition of the Local Spin. J. Chem. Theory Comput. 2012, 8, 1270−1279. (23) Mayer, I. Local Spins: Improving the Treatment for Single Determinant Wave Functions. Chem. Phys. Lett. 2012, 539−540, 172− 174. (24) Ramos-Cordoba, E.; Matito, E.; Salvador, P.; Mayer, I. Local Spins: Improved Hilbert-Space Analysis. Phys. Chem. Chem. Phys. 2012, 14, 15291−15298. (25) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648. (26) von Arnim, M.; Ahlrichs, R. Performance of Parallel TURBOMOLE for Density Functional Calculations. J. Comput. Chem. 1998, 19, 1746−1757. (27) Schäfer, A.; Horn, H.; Ahlrichs, R. Fully Optimized Contracted Gaussian Basis Sets for Atoms Li to Kr. J. Chem. Phys. 1992, 97, 2571.

electronic properties in various multiplicity states. The present CASSCF calculations confirm our previous DFT-based prediction of the half-metallicity of both the graphite-like and amorphous domains of active carbon. Systems comprising carbene groups conjugated by either graphitic moieties, cumulenic chains, or stressed phenyl rings possess a set of half-occupied MOs that confer them multiconfigurational character. Interpolating this finding to bulk active carbon, which contains both domains, suggests that its physicochemical properties may arise from its polyradical character and collectivization of electrons. To the best of our knowledge, while a number of compounds have been reported to exhibit half-metallic behavior,14 the half-metallicity of pure carbon systems (or, in general, sp elements) has not been considered before. On the more theoretical side, it was shown that DFT is applicable to multiconfigurational systems where the nonreducible part of the spin contamination is not an error but an intrinsic property of the half-metallic system. While the reliability and applicability of DFT for spin-degenerate systems has been widely supported by CASSCF calculations for polynuclear complexes of transition metals32 and compounds with metal−metal multiple bonds,33 the present work demonstrates that DFT can also be employed for half-metallic systems.

■

ASSOCIATED CONTENT

S Supporting Information *

Additional information related to CASSCF calculations and clusters construction (Supplement A) and Cartesian coordinates and computational protocols (Supplement B). This material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. ca. Tel: (044) 2393306. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Computational resources were provided by the Centre for Research in Molecular Modeling (CERMM) and Calcul Québec. This work was funded in part by the Ministére de l’éducation, du Loisir et du Sport du Québec and Concordia University.

■

REFERENCES

(1) Marsh, H.; Rodríguez-Reinoso, F. Activated Carbon; Elsevier Ltd.: London, U.K., 2006; p 536. (2) Polyynes: Synthesis, Properties, and Applications; Cataldo, F., Ed.; CRC Press: Boca Raton, FL, 2006; p 528. (3) Harris, P. J. F. The Encyclopedia of Materials: Science and Technology; Buschow, K. H. J., Cahn, R. W., Flemings, M. C., Ilschner, B., Kramer, E. J., Mahajan, S., Eds.; Elsevier: Amsterdam, The Netherlands; New York, 2001; Vol. 6, p 6197. (4) Ergun, S. Structure in Carbon. Carbon 1968, 6, 141−159. (5) Khavryuchenko, V. D.; Khavryuchenko, O. V.; Lisnyak, V. V. High Multiplicity States in Disordered Carbon Systems: Ab Initio and Semiempirical Study. Chem. Phys. 2010, 368, 83−86. (6) Khavryuchenko, O. V.; Khavryuchenko, V. D.; Lisnyak, V. V.; Peslherbe, G. H. A Density-Functional Theory Investigation of the 7056

dx.doi.org/10.1021/jp502207n | J. Phys. Chem. A 2014, 118, 7052−7057

The Journal of Physical Chemistry A

Article

(28) Jensen, F. Introduction to Computational Chemistry; Wiley: New York, 2006; p 624. (29) Olsen, J. The CASSCF Method: A Perspective and Commentary. Int. J. Quantum Chem. 2011, 111, 3267−3272. (30) Neese, F. The ORCA Program System. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2012, 2, 73−78. (31) Pettersen, E. F.; Goddard, T. D.; Huang, C. C.; Couch, G. S.; Greenblatt, D. M.; Meng, E. C.; Ferrin, T. E. UCSF Chimera  A Visualization System for Exploratory Research and Analysis. J. Comput. Chem. 2004, 25, 1605−1612. (32) Vogiatzis, K. D.; Klopper, W.; Mavrandonakis, A.; Fink, K. Theoretical Study on the Magnetic Properties of Paddlewheels and Trinuclear Clusters with Exposed Metal Sites. ChemPhysChem 2011, 12, 3307−3319. (33) Nishino, M.; Yamanaka, S.; Yoshioka, Y.; Yamaguchi, K. Theoretical Approaches to Direct Exchange Couplings Between Divalent Chromium Ions in Naked Dimers, Tetramers, and Clusters. J. Phys. Chem. A 1997, 101, 705−712.

7057

dx.doi.org/10.1021/jp502207n | J. Phys. Chem. A 2014, 118, 7052−7057