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International Journal of Molecular Sciences 2010

Compactness Aromaticity of Atoms in Molecules

DOI: 10.3390/ijms11041269

Mihai V. Putz

Keywords: chemical reactivity principles, polarizability, electronegativity, chemical hardness, quantum semi-empirical methods, quantum ab initio methods, aromaticity rules

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Abstract:

A new aromaticity definition is advanced as the compactness formulation through the ratio between atoms-in-molecule and orbital molecular facets of the same chemical reactivity property around the pre- and post-bonding stabilization limit, respectively. Geometrical reactivity index of polarizability was assumed as providing the benchmark aromaticity scale, since due to its observable character; with this occasion new Hydrogenic polarizability quantum formula that recovers the exact value of 4.5 a 0 3 for Hydrogen is provided, where a 0 is the Bohr radius; a polarizability based–aromaticity scale enables the introduction of five referential aromatic rules (Aroma 1 to 5 Rules). With the help of these aromatic rules, the aromaticity scales based on energetic reactivity indices of electronegativity and chemical hardness were computed and analyzed within the major semi-empirical and ab initio quantum chemical methods. Results show that chemical hardness based-aromaticity is in better agreement with polarizability based-aromaticity than the electronegativity-based aromaticity scale, while the most favorable computational environment appears to be the quantum semi-empirical for the first and quantum ab initio for the last of them, respectively.

References

[1]	Kekulé, AF. Untersuchungen uber aromatische Verbindungen. Liebigs Ann. Chem？1866, 137, 129–136.
[2]	Thomson, JJ. On the structure of the molecule and chemical combination. Philos. Mag？1921, 41, 510–538.
[3]	Hückel, E. Quantentheoretische Beitr？ge zum Benzolproblem. Z. Physik？1931, 70, 204–286.
[4]	Julg, A; Fran？oise, P. Recherches sur la géométrie de quelques hydrocarbures non-alternants: son influence sur les énergies de transition, une nouvelle définition de l'aromaticité. Theor. Chem. Acta？1967, 8, 249–259.
[5]	Boldyrev, AI; Wang, LS. All-Metal Aromaticity and Antiaromaticity. Chem. Rev？2005, 105, 3716–3757.
[6]	Doering, WV; Detert, F. Cycloheptatrienylium oxide. J. Am. Chem. Soc？1951, 73, 876–877.
[7]	Chattaraj, PK; Sarkar, U; Roy, DR. Electronic structure principles and aromaticity. J. Chem. Edu？2007, 84, 354–358.
[8]	Mandado, M; Moa, MJG; Mosquera, RA. Exploring basic chemical concepts with the quantum theory of atoms in molecules. In Progress in Quantum Chemistry Research; Hoffman, EO, Ed.; Nova Science Publishers: New York, NY, USA, 2007; pp. 1–57.
[9]	Katritzky, AR; Topson, RD. The σ and π inductive effects. J. Chem. Edu？1971, 48, 427–431.
[10]	Schleyer, PVR; Maerker, C; Dransfeld, A; Jiao, H; Eikema Hommes, NJRV. Nucleus-independent chemical shifts: A simple and efficient aromaticity probe. J. Am. Chem. Soc？1996, 118, 6317–6318.
[11]	Chen, Z; Wannere, CS; Corminboeuf, C; Puchta, R; Schleyer, PVR. Nucleus-Independent Chemical Shifts (NICS) as an aromaticity criterion. Chem. Rev？2005, 105, 3842–3888.
[12]	Moran, D; Simmonett, AC; Leach, FE; Allen, WD; Schleyer, PVR; Schaeffer, HF, III. Popular theoretical methods predict benzene and arenes to be nonplanar. J. Am. Chem. Soc？2006, 128, 9342–9343.
[13]	Randi？, M. Aromaticity and conjugation. J. Am. Chem. Soc？1977, 99, 444–450.
[14]	Gutman, I; Milun, M; Trinasti？, N. Graph theory and molecular orbitals. 19. Nonparametric resonance energies of arbitrary conjugated systems. J. Am. Chem. Soc？1977, 99, 1692–1704.
[15]	Balaban, AT; Schleyer, PVR; Rzepa, HS. Crocker, not armit and robinson, begat the six aromatic electrons. Chem. Rev？2005, 105, 3436–3447.
[16]	Ciesielski, A; Krygowski, TM; Cyranski, MK; Dobrowolski, MA; Balaban, AT. Are thermodynamic and kinetic stabilities correlated? A topological index of reactivity toward electrophiles used as a criterion of aromaticity of polycyclic benzenoid hydrocarbons. J. Chem. Inf. Model？2009, 49, 369–376.
[17]	Ciesielski, A; Krygowski, TM; Cyrański, MK; Dobrowolski, MA; Aihara, J. Graph–topological approach to magnetic properties of benzenoid hydrocarbons. Phys. Chem. Chem. Phys？2009, 11, 11447–11455.
[18]	Tarko, L. Aromatic molecular zones and fragments. ？2008, 24–45.
[19]	Tarko, L; Putz, MV. On electronegativity and chemical hardness relationships with aromaticity. J. Math. Chem？2010, 47, 487–495.
[20]	Kruszewski, J; Krygowski, TM. Definition of aromaticity basing on the harmonic oscillator model. Tetrahedron Lett？1972, 13, 3839–3842.
[21]	Krygowski, TM. Crystallographic studies of inter- and intramolecular interactions reflected in aromatic character of π-electron systems. ？1993, 33, 70–78.
[22]	Pauling, L; Wheland, GW. The nature of the chemical bond. V. The quantum-mechanical calculation of the resonance energy of benzene and naphthalene and the hydrocarbon free radicals. J. Chem. Phys？1933, 1, 362–375.
[23]	Pauling, L; Sherman, J. The nature of the chemical bond. VI. The calculation from thermochemical data of the energy of resonance of molecules among several electronic structures. J. Chem. Phys？1933, 1, 606–618.
[24]	Wheland, GW. The Theory of Resonance and Its Application to Organic Chemistry; Wiley: New York, NY, USA, 1944.
[25]	Hess, BA; Schaad, LJ. Hückel molecular orbital π resonance energies. Benzenoid hydrocarbons. J. Am. Chem. Soc？1971, 93, 2413–2416.
[26]	Dauben, HJ, Jr; Wilson, JD; Laity, JL. Diamagnetic susceptibility exaltation as a criterion of aromaticity. J. Am. Chem. Soc？1968, 90, 811–813.
[27]	Flygare, WH. Magnetic interactions in molecules and an analysis of molecular electronic charge distribution from magnetic parameters. Chem. Rev？1974, 74, 653–687.
[28]	Schleyer, PvR; Maerker, C; Dransfeld, A; Jiao, H; Hommes, NJRvE. Nucleus-Independent Chemical Shifts: A simple and efficient aromaticity probe. J. Am. Chem. Soc？1996, 118, 6317–6318.
[29]	Chen, Z; Wannere, CS; Corminboeuf, C; Puchta, R; Schleyer, PvR. Nucleus-Independent Chemical Shifts (NICS) as an Aromaticity Criterion. Chem. Rev？2005, 105, 3842–3888.
[30]	Feixas, F; Matito, E; Poater, J; Solà, M. On the performance of some aromaticity indices: A critical assessment using a test set. J. Comput. Chem？2008, 29, 1543–1554.
[31]	Giambiagi, M; de Giambiagi, MS; dos Santos, CD; de Figueiredo, AP. Multicenter bond indices as a measure of aromaticity. Phys. Chem. Chem. Phys？2000, 2, 3381–3392.
[32]	Bultinck, P; Ponec, R; van Damme, S. Multicenter bond indices as a new measure of aromaticity in polycyclic aromatic hydrocarbons. J. Phys. Org. Chem？2005, 18, 706–718.
[33]	Poater, J; Fradera, X; Duran, M; Solà, M. An Insight into the Local Aromaticities of Polycyclic Aromatic Hydrocarbons and Fullerenes. Chem. Eur. J？2003, 9, 1113–1122.
[34]	Matito, E; Poater, J; Duran, M; Solà, M. An analysis of the changes in aromaticity and planarity along the reaction path of the simplest Diels–Alder reaction. Exploring the validity of different indicators of aromaticity. J. Mol. Struct. (Theochem)？2005, 727, 165–171.
[35]	Matito, E; Duran, M; Solà, M. The aromatic fluctuation index (FLU): A new aromaticity index based on electron delocalization. J. Chem. Phys？2005, 122, 014109.
[36]	Matito, E; Salvador, P; Duran, M; Solà, M. Aromaticity measures from Fuzzy-Atom Bond Orders (FBO). The Aromatic fluctuation (FLU) and the para-delocalization (PDI) indexes. J. Phys. Chem. A？2006, 110, 5108–5113.
[37]	Cioslowski, J; Matito, E; Solà, M. Properties of aromaticity indices based on the one-electron density matrix. J. Phys. Chem. A？2007, 111, 6521–6525.
[38]	Putz, MV. On absolute aromaticity within electronegativity and chemical hardness reactivity pictures. MATCH Commun. Math. Comput. Chem？2010, 64, 391–418.
[39]	Brinck, T; Murray, JS; Politzer, P. Polarizability and volume. J. Chem. Phys？1993, 98, 4305–4307.
[40]	Hati, S; Datta, D. Hardness and electric dipole polarizability. Atoms and clusters. J. Phys. Chem？1994, 98, 10451–10454.
[41]	Waller, I. Stark's effect of the second order with hydrogen and the Rydberg correction of the spectrum of He and Li. Z. Physik？1926, 38, 635–646.
[42]	Epstein, PS. The Stark effect from the point of view of Schr？dinger’s quantum theory. Phys. Rev？1926, 28, 695–710.
[43]	Hassé, HR. The polarizability of the helium atom and the lithium ion. Proc. Cambridge Phil. Soc？1930, 26, 542–555.
[44]	McDowell, HK; Porter, RN. Reduced free-particle Green's functions in quantum-mechanical perturbation calculations. J. Chem. Phys？1976, 65, 658–672.
[45]	McDowell, HK. Exact static dipole polarizabilities for the excited S states of the hydrogen atom. J. Chem. Phys？1976, 65, 2518–2522.
[46]	Delone, NB; Krainov, VP. Multiphoton Processes in Atoms; Springer: Berlin, Germany, 1994.
[47]	Krylovetsky, AA; Manakov, NL; Marmo, SI. Quadratic Stark effect and dipole dynamic polarizabilities of Hydrogen-like levels. Laser Phys？1997, 7, 781–796.
[48]	Bratsch, SG. Electronegativity equalization with Pauling units. J. Chem. Edu？1984, 61, 588–590.
[49]	Bratsch, SG. A group electronegativity method with Pauling units. J. Chem. Edu？1985, 62, 101–104.
[50]	Parr, RG; Donnelly, RA; Levy, M; Palke, WE. Electronegativity: The density functional viewpoint. J. Chem. Phys？1978, 68, 3801–3808.
[51]	Sen, KD; J？rgenson, CD. Structure and Bonding; Springer: Berlin, Germany, 1987; Volume 66.
[52]	Putz, MV. Contributions within Density Functional Theory with Applications to Chemical Reactivity Theory and Electronegativity; Dissertation Com.: Parkland, FL, USA, 2003.
[53]	Putz, MV. Systematic formulation for electronegativity and hardness and their atomic scales within density functional softness theory. Int. J. Quantum Chem？2006, 106, 361–386.
[54]	Putz, MV. Absolute and Chemical Electronegativity and Hardness; Nova Science Publishers: New York, NY, USA, 2008.
[55]	Putz, MV. Quantum and electrodynamic versatility of electronegativity and chemical hardness. In Quantum Frontiers of Atoms and Molecules; Putz, MV, Ed.; Nova Science Publishers: New York, NY, USA, 2010.
[56]	Koopmans, T. Ordering of wave functions and eigenvalues to the individual electrons of an atom. Physica？1934, 1, 104–110.
[57]	Sanderson, RT. Principles of electronegativity Part I. General nature. J. Chem. Edu？1988, 65, 112–119.
[58]	Mortier, WJ; Genechten, KV; Gasteiger, J. Electronegativity equalization: application and parametrization. J. Am. Chem. Soc？1985, 107, 829–835.
[59]	Putz, MV. Maximum hardness index of quantum acid-base bonding. MATCH Commun. Math. Comput. Chem？2008, 60, 845–868.
[60]	Pearson, RG. Hard and Soft Acids and Bases; Dowden, Hutchinson & Ross: Stroudsberg, PA, USA, 1973.
[61]	Pearson, RG. Hard and soft acids and bases—the evolution of a chemical concept. Coord. Chem. Rev？1990, 100, 403–425.
[62]	Putz, MV; Russo, N; Sicilia, E. On the application of the HSAB principle through the use of improved computational schemes for chemical hardness evaluation. J. Comput. Chem？2004, 25, 994–1003.
[63]	Pearson, RG. Chemical Hardness: Applications from Molecules to Solids; Wiley-VCH: Weinheim, Germany, 1997.
[64]	Sen, KD; Mingos, DMP. Structure and Bonding; Springer: Berlin, Gernay, 1993; Volume 80.
[65]	Parr, RG; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, NY, USA, 1989.
[66]	Putz, MV. Density functionals of chemical bonding. Int. J. Mol. Sci？2008, 9, 1050–1095.
[67]	Parr, RG; Chattaraj, PK. Principle of maximum hardness. J. Am. Chem. Soc？1991, 113, 1854–1855.
[68]	Putz, MV. Semiclassical electronegativity and chemical hardness. J. Theor. Comput. Chem？2007, 6, 33–47.
[69]	Dirac, PAM. Quantum mechanics of many-electron systems. Proc. Roy. Soc. (London)？1929, A123, 714–733.
[70]	L？wdin, P-O. On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals. J. Chem. Phys？1950, 18, 365–376.
[71]	L？wdin, P-O. Some remarks on the resemblance theorems associated with various orthonormalization procedures. Int. J. Quantum Chem？1993, 48, 225–232.
[72]	Hoffmann, R. An extended Hückel theory. I. Hydrocarbons. J. Chem. Phys？1963, 39, 1397–1412.
[73]	Roothaan, C. C.J. New developments in molecular orbital theory. Rev. Mod. Phys？1951, 23, 69–89.
[74]	Pople, JA; Nesbet, RK. Self-consistent orbitals for radicals. J. Chem. Phys？1954, 22, 571–572.
[75]	Pople, JA; Beveridge, DV. Approximate Molecular Orbital Theory; McGraw-Hill: New York, NY, USA, 1970.
[76]	Pople, JA; Santry, DP; Segal, GA. Approximate self-consistent molecular orbital theory. I. Invariant procedures. J. Chem. Phys？1965, 43, S129–S135.
[77]	Pople, JA; Segal, GA. Approximate self-consistent molecular orbital theory. II. Calculations with complete neglect of differential overlap. J. Chem. Phys？1965, 43, S136–S151.
[78]	Pople, JA; Segal, GA. Approximate self-consistent molecular orbital theory. III. CNDO results for AB2 and AB3 systems. J. Chem.Phys？1966, 44, 3289–3297.
[79]	Oleari, L; DiSipio, L; De Michelis, G. The evaluation of the one-centre integrals in the semi-empirical molecular orbital theory. Mol. Phys？1966, 10, 97–109.
[80]	Slater, JI. Quantum Theory of Atomic Structure; McGraw-Hill Book Company: New York, NY, USA, 1960.
[81]	Baird, NC; Dewar, MJS. Ground states of σ-bonded molecules. IV. The MINDO method and its application to hydrocarbons. J. Chem. Phys？1969, 50, 1262–1275.
[82]	Dewar, MJS; Hasselbach, E. Ground states of .sigma.-bonded molecules. IX. MINDO [modified intermediate neglect of differential overlap]/2 method. J. Am. Chem. Soc？1970, 92, 590–598.
[83]	Dewar, MJS; Lo, DH. Ground states of sigma-bonded molecules. XVII. Fluorine compounds. J. Am. Chem. Soc？1972, 94, 5296–5303.
[84]	Bingham, RC; Dewar, MJS; Lo, DH. Ground states of molecules. XXV. MINDO/3. Improved version of the MINDO semiempirical SCF-MO method. J. Am. Chem. Soc？1975, 97, 1285–1293.
[85]	Bingham, RC; Dewar, MJS; Lo, DH. Ground states of molecules. XXVI. MINDO/3 calculations for hydrocarbons. J. Am. Chem. Soc？1975, 97, 1294–1301.
[86]	Bingham, RC; Dewar, MJS; Lo, DH. Ground states of molecules. XXVII. MINDO/3 calculations for carbon, hydrogen, oxygen, and nitrogen species. J. Am. Chem. Soc？1975, 97, 1302–1306.
[87]	Bingham, RC; Dewar, MJS; Lo, DH. Ground states of molecules. XXVIII. MINDO/3 calculations for compounds containing carbon, hydrogen, fluorine, and chlorine. J. Am. Chem. Soc？1975, 97, 1307–1311.
[88]	Dewar, MJS; Lo, DH; Ramsden, CA. Ground states of molecules. XXIX. MINDO/3 calculations of compounds containing third row elements. J. Am. Chem. Soc？1975, 97, 1311–1318.
[89]	Murrell, JN; Harget, AJ. Semi-empirical Self-consistent-field Molecular Orbital Theory of Molecules; Wiley Interscience: New York, NY, USA, 1971.
[90]	Ohno, K. Some remarks on the Pariser-Parr-Pople method. Theor. Chim. Acta？1964, 2, 219–227.
[91]	Klopman, G. A semiempirical treatment of molecular structures. II. Molecular terms and application to diatomic molecules. J. Am. Chem. Soc？1964, 86, 4550–4557.
[92]	Pople, JA; Beveridge, DL; Dobosh, PA. Approximate self-consistent molecular-orbital theory. V. Intermediate neglect of differential overlap. J. Chem. Phys？1967, 47, 2026–2034.
[93]	Dewar, MJS; Thiel, W. Ground states of molecules. 38. The MNDO method. Approximations and parameters. J. Am. Chem. Soc？1977, 99, 4899–4907.
[94]	Dewar, MJS; McKee, ML. Ground states of molecules. 41. MNDO results for molecules containing boron. J. Am. Chem. Soc？1977, 99, 5231–5241.
[95]	Dewar, MJS; Rzepa, HS. Ground states of molecules. 40. MNDO results for molecules containing fluorine. J. Am. Chem. Soc？1978, 100, 58–67.
[96]	Davis, LP; Guidry, RM; Williams, JR; Dewar, MJS; Rzepa, HS. MNDO calculations for compounds containing aluminum and boron. J. Comp. Chem？1981, 2, 433–445.
[97]	Dewar, MJS; Storch, DM. Development and use of quantum molecular models. 75. Comparative tests of theoretical procedures for studying chemical reactions. J. Am. Chem. Soc？1985, 107, 3898–3902.
[98]	Thiel, W. Semiempirical methods: current status and perspectives. Tetrahedron？1988, 44, 7393–7408.
[99]	Clark, TA. Handbook of Computational Chemistry; John Wiley and Sons: New York, NY, USA, 1985.
[100]	Dewar, MJS; Zoebisch, EG; Healy, EF; Stewart, JJP. Development and use of quantum mechanical molecular models. 76. AM1: A new general purpose quantum mechanical molecular model. J. Am. Chem. Soc？1985, 107, 3902–3909.
[101]	Dewar, MJS; Dieter, KM. Evaluation of AM1 calculated proton affinities and deprotonation enthalpies. J. Am. Chem. Soc？1986, 108, 8075–8086.
[102]	Stewart, JJP. MOPAC: A semiempirical molecular orbital program. J. Comp. Aided Mol. Design？1990, 4, 1–103.
[103]	Stewart, JJP. Optimization of parameters for semiempirical methods. I. Method. J. Comput. Chem？1989, 10, 209–220.
[104]	Stewart, JJP. Optimization of parameters for semiempirical methods. II. Applications. J. Comput. Chem？1989, 10, 221–264.
[105]	Del Bene, J; Jaffé, HH. Use of the CNDO method in spectroscopy. I. Benzene, pyridine, and the diazines. J. Chem. Phys？1968, 48, 1807–1814.
[106]	Del Bene, J; Jaffé, HH. Use of the CNDO method in spectroscopy. II. Five-membered rings. J. Chem. Phys？1968, 48, 4050–4056.
[107]	Del Bene, J; Jaffé, HH. Use of the CNDO method in spectroscopy. III. Monosubstituted benzenes and pyridines. J. Chem. Phys？1968, 49, 1221–1229.
[108]	Ridley, JE; Zerner, MC. Triplet states via intermediate neglect of differential overlap: Benzene, Pyridine and the Diazines. Theor. Chim. Acta？1976, 42, 223–236.
[109]	Bacon, AD; Zerner, MC. An intermediate neglect of differential overlap theory for transition metal complexes: Fe, Co and Cu chlorides. Theor. Chim. Acta？1979, 53, 21–54.
[110]	Stavrev, KK; Zerner, MC; Meyer, TJ. Outer-sphere charge-transfer effects on the spectroscopy of the [Ru(NH3)5(py)]2+ Complex. J. Am. Chem. Soc？1995, 117, 8684–8685.
[111]	Stavrev, KK; Zerner, MC. On the Jahn–Teller effect on Mn2+ in zinc-blende ZnS crystal. J. Chem. Phys？1995, 102, 34–39.
[112]	Cory, MG; Stavrev, KK; Zerner, MC. An examination of the electronic structure and spectroscopy of high- and low-spin model ferredoxin via several SCF and CI techniques. Int. J. Quant. Chem？1997, 63, 781–795.
[113]	Anderson, WP; Edwards, WD; Zerner, MC. Calculated spectra of hydrated ions of the first transition-metal series. Inorg. Chem？1986, 25, 2728–2732.
[114]	Anderson, WP; Cundari, TR; Zerner, MC. An intermediate neglect of differential overlap model for second-row transition metal species. Int. J. Quantum Chem？1991, 39, 31–45.
[115]	Boys, SF. Electronic wavefunctions. I. A general method of calculation for stationary states of any molecular system. Proc. Roy. Soc？1950, A200, 542–554.
[116]	Szabo, A; Ostlund, NS. Modern Quantum Chemistry - Introduction to Advanced Electronic Structure Theory; Dover Publications Inc.: New York, NY, USA, 1996.
[117]	Clementi, E; Roetti, C. Roothaan-Hartree-Fock atomic wavefunctions: Basis functions and their coefficients for ground and certain excited states of neutral and ionized atoms, Z ≤ 54. At. Data Nucl. Data Tables？1974, 14, 177–478.
[118]	Hehre, WJ; Stewart, RF; Pople, JA. Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J. Chem. Phys？1969, 51, 2657–2665.
[119]	Collins, JB; Schleyer, PvR; Binkley, JS; Pople, JA. Self-consistent molecular orbital methods. XVII. Geometries and binding energies of second-row molecules. A comparison of three basis sets. J. Chem. Phys？1976, 64, 5142–5152.
[120]	Stewart, RF. Small Gaussian expansions of Slater-type orbitals. J. Chem. Phys？1970, 52, 431–439.
[121]	Hartree, DR. The wave mechanics of an atom with a non-Coulomb central field. Proc. Cam. Phil. Soc？1928, 24, 89–111.
[122]	Hartree, DR. The wave mechanics of an atom with a noncoulomb central field. Part I. Theory and methods. Part II. Some results and discussions. Proc. Cam. Phil. Soc？1928, 24, 111–132.
[123]	Hartree, DR. The Calculation of Atomic Structures; John Wiley and Sons: New York, NY, USA, 1957.
[124]	Fock, V. N？herungsmethode zur L？sung des quantenmechanischen Mehrk？rperproblems. Z. Physik？1930, 61, 126–140.
[125]	Kohn, W; Sham, LJ. Self-consistent equations including exchange and correlation effects. Phys. Rev？1965, 140, A1133–A1138.
[126]	Johnson, BG; Gill, PMW; Pople, JA. The performance of a family of density functional methods. J. Chem. Phys？1993, 98, 5612–5627. Erratum: Johnson, B.G. , , 9202.
[127]	Slater, JC. Quantum Theory of Molecules and Solids; McGraw-Hill: New York, NY, USA, 1974; Volume 4.
[128]	Becke, AD. Density-functional exchange energy approximation with correct asymptotic behaviour. Phys. Rev. A？1988, 38, 3098–3100.
[129]	Perdew, JP; Chevary, JA; Vosko, SH; Jackson, KA; Pederson, MR; Sing, DJ; Fiolhais, C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B？1992, 46, 667–6687.
[130]	Senatore, G; March, NH. Recent progress in the field of electron correlation. Rev. Mod. Phys？1994, 66, 445–479.
[131]	Lee, C; Yang, W; Parr, RG. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B？1988, 37, 785–789.
[132]	Miehlich, B; Savin, A; Stoll, H; Preuss, H. Results obtained with the correlation energy density functionals of Becke and Lee, Yang and Parr. Chem. Phys. Lett？1989, 157, 200–206.
[133]	Gill, PMW; Johnson, BG; Pople, JA; Frisch, MJ. The performance of the Becke-Lee-Yang-Parr (B-LYP) density functional theory with various basis-sets. Chem. Phys. Lett？1992, 197, 499–505.
[134]	Vosko, SH; Wilk, L; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys？1980, 58, 1200–1211.
[135]	Becke, AD. Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys？1993, 98, 5648–5653.
[136]	Becke, AD. Density-functional thermochemistry. V. Systematic optimization of exchange-correlation functionals. J. Chem. Phys？1997, 107, 8554–8561.
[137]	Putz, MV; Russo, N; Sicilia, E. Atomic radii scale and related size properties from density functional electronegativity formulation. J. Phys. Chem？2003, 107, 5461–5465.
[138]	Program Package, Single Point, Minimal Basis Set-for Ab Initio Methods; HyperChem 7.01; Hypercube, Inc.: Gainesville, FL, USA, 2002.
[139]	Putz, MV. Electronegativity: quantum observable. Int. J. Quantum Chem？2009, 109, 733–738.
[140]	Bethe, H; Jackiw, R. Intermediate Quantum Mechanics, 2nd ed ed.; Benjamin: New York, NY, USA, 1968.
[141]	Jackiw, R. Quantum mechanical sum rules. Phys. Rev？1967, 157, 1220–1225.
[142]	Thomas, W. Uber die Zahl der Dispersionselectronen, die einem starion？ren Zustande zugeordnet sind. Naturwissenschaftern？1925, 13, 510–525.
[143]	Kuhn, W. Regarding the total strength of a condition from outgoing absorption lines. Z. Phys？1925, 33, 408–412.
[144]	Reiche, F; Thomas, W. Uber die Zahl der dispersionselektronen, die einem station？ren Zustand zugeordnet sind. Z. Phys？1925, 34, 510–525.
[145]	Mehra, J; Rechenberg, H. The Historical Development of Quantum Theory: The Formulation of Matrix Mechanics and its Modifications 1925–1926; Springer-Verlag: New York, NY, USA, 1982. Chapter IV.
[146]	Bethe, H. Theory of the passage of fast corpuscular rays through matter (Translated). In (World Scientific Series in 20th Century Physics); Bethe, H, Ed.; World Scientific: Singapore, 1997; Volume 18, pp. 77–154.
[147]	Morse, PM; Feshbach, H. Methods of Theoretical Physics; McGraw-Hill: New York, NY, USA, 1953.

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