Valence atom with bohmian quantum potential: the golden ratio approach
© Putz; licensee Chemistry Central Ltd. 2012
Received: 6 September 2012
Accepted: 29 October 2012
Published: 12 November 2012
The alternative quantum mechanical description of total energy given by Bohmian theory was merged with the concept of the golden ratio and its appearance as the Heisenberg imbalance to provide a new density-based description of the valence atomic state and reactivity charge with the aim of clarifying their features with respect to the so-called DFT ground state and critical charge, respectively.
The results, based on the so-called double variational algorithm for chemical spaces of reactivity, are fundamental and, among other issues regarding chemical bonding, solve the existing paradox of using a cubic parabola to describe a quadratic charge dependency.
Overall, the paper provides a qualitative-quantitative explanation of chemical reactivity based on more than half of an electronic pair in bonding, and provide new, more realistic values for the so-called “universal” electronegativity and chemical hardness of atomic systems engaged in reactivity (analogous to the atoms-in-molecules framework).
Recently, the crucial problem regarding whether chemical phenomena are reducible to physical ones has had an increasingly strong impact on the current course of conceptual and theoretical chemistry. For instance, the fact that elements arrange themselves in atomic number (Z) triads in approximately 50% of the periodic system seems to escape custom ordering quantifications [1, 2]. The same applies to the following: the fascinating golden ratio (τ) limit for the periodicity of nuclei beyond any physical first-principle constants, which provides specific periodic laws for the chemical realm [3–6]; the fact that atoms have no definite atomic radii in the sense of a quantum operator, and even the Aufbau principle, which, although chemically workable, seems to violate the Pauli Exclusion Principle ; at the molecular level, the well-celebrated reaction coordinate, which, although formally defined in the projective energy space, does not constitute a variable to drive optimization in the course of chemical reactions, appearing merely as a consequence of such reactions ; the problem of atoms in molecules , i.e., how much of the free atoms enter molecules and how much independency the atoms preserve in bonding; and chemical bonding itself, which ultimately appears to be reinterpreted as a special case of bosonic condensation with the aid of bondons – the quantum bosons of chemical bonding, which, without being elementary, imbue chemical compounds with a specific reality [10, 11].
In the same context, the specific measure of chemical reactivity, electronegativity (χ), which lacks a definite quantum operator but retains an observable character through its formal identity with the macroscopic chemical potential χ=-μ[12, 13], was tasked with carrying quantum information within the entanglement environment of Bohmian mechanics [14–17] and has thus far been identified with the square root of the so-called quantum potential χ = V Q 1/2.
However, the striking difference between an atom as a physical entity, with an equal number of electrons and protons (thus in equilibrium), and the same atom as a chemical object, with incomplete occupancy in its periphery quantum shells (thus attaining equilibrium by changing accepting or releasing electrons), is closely related to the electronegativity phenomenology in modeling chemical reactivity. Moreover, this difference triggers perhaps the most important debate in conceptual chemistry: the ground vs. valence state definition of an atom.
The point is that curve (2) is not chemically minimized, although it is very often assumed to be in the DFT invoked by the chemical reactivity literature [13, 26–29]; however, the curve cannot be considered indicative of a sort of ground state (neither reactive nor critical states of Figure 1). Additionally, by comparing the curves of Figure 1 (a) and (b), the curve of eq. (2) occurs above both the reactive and critical curves of Figure 1; it thus should represent the chemical valence state with which to operate. Therefore, much caution should be taken when working with eq. (2) in assessing the properties of atoms, molecules, atoms in molecules, etc. Nevertheless, this is another case of chemistry not being reducible to physics and should be treated accordingly. It is worth noting that Parr, the “father” of eq. (2) and a true pioneer of conceptual density functional theory [30, 31], had tried to solve this dichotomy by taking the “valence as the ground state of an atom in a perturbed environment”. This statement is not entirely valid because perturbation is not variation such that it may be corrected by applying the variational principle to eq. (2), for example. In fact, using such variation should be considered a double variational technique that is necessary to arrive at the celebrated chemical reactivity principles of electronegativity and chemical hardness, as recently shown .
The current line of work takes a step forward by employing the double variation of the parabolic energy curve of type (2) to provide the quantum (DFT) valence charge of an atom (say, N ** ) and to compare it either quantitatively and qualitatively with the chemical critical charge N * . The goal of these efforts is to gain new insight into the valence state and chemical reactivity at the quantum level. To this end, the relation of Bohmian mechanics to the concept of the golden ratio will be essential and will be introduced in the following.
The consequences of the joint consideration of Bohmian mechanics and the golden ratio for the main atomic systems will be explored, and the quantum chemical valence state will be accordingly described alongside the so-called universal electronegativity and chemical hardness, refining the work of Parr and Bartolotti  as well as generalizing the previous Bohmian-Boeyens approach [3, 4].
Two apparently disjoint theories of matter will be employed to characterize the quantum valence of an atom: Bohmian mechanics – furnishing the main equation for total energy – and the fundamental quantum mechanics through the Heisenberg combined with de Broglie principles providing the wave-particle indeterminacy framework in which the golden ration dependency of Z/N naturally appears as quantifying the valence states of atoms considered the “ground state” of the atomic chemical reactivity.
Because of the need to reduce Copenhagen’s indeterminacy for quantum phenomena, i.e., by associating it the quantum description of “Newtonian” forms of motion, though by preserving probability densities, quantum averages, etc., the so-called “minimalist” quantum theory may be formulated following the Bohm quantum mechanical program as follows.
which is the variation in the quantum potential with electron exchange under a constant classical or external potential.
when the above relations (6) and (10) are substituted into eq. (15).
It is worth noting that although we obtained the total energy (17) in the Bohmian mechanics context, it showcases a clear electronic density dependency, not under a density functional (as DFT would require) but merely as a spatial function, which is a direct reflection of the entanglement behavior of Bohmian theory through the involvement of a quantum potential. However, in most cases, and especially for atomic systems, eq. (17) will yield numerical values under custom density function realizations.
Golden ratio imbalance for valence states of atoms
Worth remarking the results of type (20) and (22), here based on chemical reactivity specialization of Heisenberg type equations (18a) and/or (18b), were previously obtained at the level of neutron-protonic imbalance, inside the atomic nuclei, based on well-founded empirical observations . The present golden ratio appearance is ultimately sustained also by the deviation from the N=Z condition for so-called “quark atoms” (as another way in considering the atoms in a quantum valence state), earlier identified as true matter’s entities responsible for matter’s reactivity at the atomic level .
Therefore the atomic structure branching (22) can be regarded as the present golden ratio extension to valence atom and as such employed; actually, its consequences regarding the characterization of the quantum valence states of atoms within the Bohmian quantum potential are the main aims of the present endeavor and will be discussed next.
Atomic implementation and discussion
On Slater density for valence atoms
The identity between eqs. (25b) and (25c) gives sufficient support to the present Slater density approach eq. (23a) in modeling the valence atoms or the atoms at their frontiers approaching reactivity (i.e. atoms-in-molecules complexes by chemical reactions).
Quantum chemical bonding and reactivity indices
• kinetic energy(31a)
• nucleus-electronic interaction(31b)
• inter-electronic interaction (see also Appendix)(31c)
Having the completely analytical density in terms of number of reactive electrons as in eq. (33), worth pointing here on the so called sign problem relating with its variation, e.g., its gradient, the gradient of its square root, etc. Although this problem usually arises in density functional theory when specific energy functionals are considered in gradient forms, see for instance ref. , there is quite instructive discussing the present behavior and its consequences.
• the fact that the (covalent) bond length is proportional to the atomic radii and in inverse correlation with bonding order is well known , and this it is also nicely reflected in eq. (25d); however, changing the sign to negative radii as surpassing the threshold 21/5 and fixing in fact the limit N bonding =4, is consistent with maximum bond order met in Nature; it is also not surprising this self-released limit connects with golden ratio by the golden-spiral optimization of bond-order ; more subtle, it connects also with the 4π symmetry of two spherical valence atoms making a chemical bond (Figure 2, inset): such “spinning” reminds of the graviton symmetry  (the highest spherical symmetry in Nature, with spin equal 2) and justifies the recent treatments of chemical bonding by means of the quasi-particles known as bondons [10, 53], as well as the use of the 4D complex projective geometry in modeling the chemical space as a non-Euclidian one, eventually with a time-space metrics including specific “gravitational effects” describing the bonding ;
• the “gap” between the atomic systems contributing 2 to 3 electrons to produce chemical bond is about double of the golden ratio, ; therefore, this gap marks the passage from the space occupied by a pair of electrons and that required when the third electron is added on the same bonding state: it means that the third electron practically needs one golden measure (τ) to (covalently) share with each of the existing pairing electrons, while increasing the bond order to the level of three; it is therefore a space representation of the Pauli exclusion principle itself, an idea also earlier found in relation with dimensionless representation of a diatomic bonding energy (2τ) at its equilibrium bonding distance τ; when the fourth electron is coming into the previous system, in order the maximum fourth order of bonding to be reach the chemical bonding space is inflating about five times more, yet forbidding further forced incoming electrons into the same space of bonding state as the bonding radius becomes negative in sign.
which is not appropriate for describing the valence state of an atom, as eq. (2) prescribes, despite being similar in form to the Bohmian-based result of eq. (34a). Thus, the previous limitation of the Parr-Bartolotti conclusion  and the paradox raised in describing the valence (parabolically) state with the optimized atomic density (33) are here solved by the double (or the orthogonal) variational implementation, as recently proved to be customary for chemical spaces . In the light of this remark one may explain also the sign difference between the “physical” energy (34b) and that obtained for the “chemical” situation (34a): through simple variational procedure for “physical” energy (30) the result (34b) is inherently negative – modeling systems stability in agreement with the upper branch of eq. (22), whereas the double variational algorithm employing optimized density (33) into the Bohmian shaped energy (17) it produces the positive output (34a) associated with activation energy characteristic for chemical reactivity corresponding to the lower branch of eq. (22).
Synopsis of the critical charges in the physical ground state (N * ) as well as for chemical reactive (valence) state (N ** ) for atoms of the first four periods of the periodic table of elements, as computed from the minimum point of associated interpolations of ionization and electronic affinitiesand of eq. ( 38 ), respectively
Nevertheless, the energetic analysis also reveals the atomic systems Be, B and C to be situated over the corresponding physical stable states; this may explain why boron and carbon present special chemical phenomenology (e.g., triple electronic bonds and nanosystems with long C-bindings, respectively), which is not entirely explained by ordinary physical atomic paradigms [55–60].
The result, nevertheless, appears to be an unusually higher increase in chemical hardness than in electronegativity, which certainly cannot be used to model a reactive-engaged tendency because it is more stable (by chemical hardness) than reactive (by electronegativity); it is, however, consistent with the physical stability of the system, provided by the single variational procedure through which eq. (34b) was produced.
Remarkably, the actual electronegativity of (42) obtained by the quantum Bohm and golden ratio double procedure yields sensible results similar to those of the single variational approach (40); however, the chemical hardness of (43) is approximately 5-fold lower than its “stable” counterpart (41), affirming therefore the manifestly reactive framework it produces – one described by a quadratic equation (34a) instead of a cubic one (34b).
Charge waves in gauge chemical reactivity
along the reaction path accounting for the acidic (electron accepting, 0 ≤ N ≤ +1) and basic (electron donating, –1 ≤ N ≤ 0) chemical behaviors.
In any case, the present analysis provides the qualitative result that the difference between the critical ground state and optimal valence charges is more than half of an electronic pair, giving rise to the significant notion that chemical reactivity is not necessarily governed by a pair of electrons but governed by no less than half of a pair and is related to the golden ratio (τ > 0.5).
Also a local analysis of the type of charge that is dominant in atomic stability, i.e., the critical physical ground state or the chemical valence reactive state based on eqs. (45a) and (45b), respectively, may be of considerable utility in refined inorganic chemistry structure-reactivity analysis. To the same extent, it depends on the degree of the polynomials used to interpolate the critical and valence charges over the concerned systems; however, through the present endeavor, we may assert that the analysis should be of the type (48), which in turn remains a sort of integral version of the imbalance equation (20a), in this case for the ground-valence charge gap states of a chemical system.
Aiming to hint at the solution to the current debate regarding the physical vs. chemical definition of an atom and as a special stage of a larger project regarding quantum chemical orthogonal spaces, the present work addresses the challenging problem of defining and characterizing valence states with respect to the ground state within conceptual density functional theory. We are aware of the earlier warnings raised by Parr and Bartolotti and others [18, 33, 63] regarding the limits of density functional theory and of the total energy of atomic systems combined with a Slater-based working density to provide a quadratic form in terms of system charge, as required by the general theory of chemical reactivity of atoms and molecules in terms of electronegativity and chemical hardness. Fortunately, we discovered that the Bohmian form of the total energy of such atomic systems provides, instead, the correct behavior, although it is only density-function-dependent and not a functional expression. Moreover, this finding was reached through the so-called double variational procedure, which, as emphasized earlier, was likely to reproduce the chemical reactivity principles of electronegativity and chemical hardness in an analytical manner; however, such a double analytical variational approach is consistent with the recent advanced chemical orthogonal spaces approaches of chemical phenomenology  as being at least complementary to the physical description of many-electronic systems when they are engaging in reactivity or equilibrium as the atoms-in-molecules Bader theory prescribes [9, 42]. With the present Bohmian approach, the total energy is in fact identified with the quantum potential, thus inherently possessing non-locality and appropriate reactivity features, which are manifested even over long distances [10, 11, 53]; this also generalizes the previous Boeyens electronegativity formulation of electronegativity [5, 6] from the direct relationship between a quantum potential and its charge derivative. The double algorithm was also implemented to discriminate the valence from the ground state charges, this time by using the golden ratio imbalance equation as provided by adaption of the Heisenberg type relationship to chemical reactivity for atoms. This corresponds to an analytical unfolding of the physical and chemical imbalance of the electronic charge stability of atomic systems, paralleling the deviation from the equal electron-to-proton occupancy in physical systems toward electron deficiency in the valence states of chemical systems. This dichotomy was implemented by the golden ratio presented in eq. (22). As a consequence, the difference between valence and ground state charge systems is naturally revealed and allows for the explanation of chemical reactivity and bonding in terms of fractional electron pairs, althrough driven by the golden ratio under the so-called physical-to-chemical charge difference wave function and associated normalizations, all of which represent elaborated or integral forms of the basic imbalance atomic equation. The present results are based on 10th-order polynomial fitted over 32 elements from the first 54 elements of the first four periods of periodic table of elements and can be further pursued by performing such systematic interpolations that preserve the golden ratio relationships, as advanced herein; they may also provide a comprehensive picture of how valence electrons may always be projected/equalized/transposed into ground state electrons within the perspective of further modeling chemical reactions when chemical reactivity negotiates the physical molecular stabilization of atoms in molecules.
a For circular orbits, the lowest ones in each atomic shells – including the valence ones, one has ΔO=Δr=2πr, with r the orbital radii thereof, while O=p is the fixed particle’s momentum on that orbit; therefore, when combined into eq. (18b) they provide the celebrated Bohr-de Broglie relationship rp=nħ solving the atomic spectra of Hydrogen atom in principal quantum numbers (n).
Appendix: Semi-classical inter-electronic energy
For the inter-electronic interaction, see Figure 8; in evaluating V ee [ξ] of eq. (31c), the two-electronic density is approximated by the Coulombic two mono-electronic density product, thus neglecting the second-order density matrix effects associated with the exchange-correlation density.
This work was supported by CNCS-UEFISCDI agency through the research project TE16/2010-2013 within the PN II-RU-TE-2009-1 framework. Inspiring discussions with Profs. Boeyens (University of Pretoria, South Africa) and von Szentpaly (Stuttgart University, Germany) are kindly acknowledged. Constructive referees’ comments and stimulus is also sincerely thanked. This paper is dedicated to Prof. Robert G. Parr for his pioneering quantum work on atomic valence states.
- Putz MV: Big chemical ideas in the context: the periodic law and the Scerri’s Periodic Table. Int J Chem Model. 2011, 3: 15-22.Google Scholar
- Scerri ER: The Periodic Table – Its Story and Its Significance. 2007, Oxford-New York: Oxford University PressGoogle Scholar
- Boeyens JCA: Emergent Properties in Bohmian Chemistry. Quantum Frontiers of Atoms and Molecules. Edited by: Putz MV. 2011, New York: Nova Publishers Inc, 191-Google Scholar
- Boeyens JCA: New Theories for Chemistry. 2005, Amsterdam: ElsevierGoogle Scholar
- Boeyens JCA: Chemistry from First Principles. 2008, Heidelberg-Berlin: SpringerView ArticleGoogle Scholar
- Boeyens JCA, Levendis DC: Number Theory and the Periodicity of Matter. 2008, Heidelberg-Berlin: SpringerView ArticleGoogle Scholar
- Kaplan IG: Is the Pauli exclusive principle an independent quantum mechanical postulate?. Int J Quantum Chem. 2002, 89: 268-276. 10.1002/qua.10276.View ArticleGoogle Scholar
- Scerri ER: Just how ab initio is ab initio quantum chemistry?. Found Chem. 2004, 6: 93-116.View ArticleGoogle Scholar
- Bader RFW: Atoms in Molecules - A Quantum Theory. 1990, Oxford: Oxford University PressGoogle Scholar
- Putz MV: The bondons: The quantum particles of the chemical bond. Int J Mol Sci. 2010, 11: 4227-4256. 10.3390/ijms11114227.View ArticleGoogle Scholar
- Putz MV: Quantum Theory: Density, Condensation, and Bonding. 2012, Toronto: Apple Academics & CRC PressGoogle Scholar
- Parr RG, Donnelly RA, Levy M, Palke WE: Electronegativity: the density functional viewpoint. J Chem Phys. 1978, 68: 3801-3808. 10.1063/1.436185.View ArticleGoogle Scholar
- Parr RG, Yang W: Density Functional Theory of Atoms and Molecules. 1989, New York: Oxford University PressGoogle Scholar
- Bohm D: A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys Rev. 1952, 85: 166-179. 10.1103/PhysRev.85.166.View ArticleGoogle Scholar
- Bohm D: A suggested interpretation of the quantum theory in terms of “hidden” variables. II. Phys Rev. 1952, 85: 180-193. 10.1103/PhysRev.85.180.View ArticleGoogle Scholar
- Bohm D, Vigier JP: Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations. Phys Rev. 1954, 96: 208-216. 10.1103/PhysRev.96.208.View ArticleGoogle Scholar
- Cushing JT: Quantum Mechanics – Historical Contingency and the Copenhagen Hegemony. 1994, Chicago & London: The University of Chicago PressGoogle Scholar
- Von Szentpály L: Modeling the charge dependence of total energy and its relevance to electrophilicity. Int J Quant Chem. 2000, 76: 222-234. 10.1002/(SICI)1097-461X(2000)76:2<222::AID-QUA11>3.0.CO;2-6.View ArticleGoogle Scholar
- Ayers PW, Parr RG: Variational principles for describing chemical reactions: the Fukui function and chemical hardness revisited. J Am Chem Soc. 2000, 122: 2010-2018. 10.1021/ja9924039.View ArticleGoogle Scholar
- Geerlings P, De Proft F, Langenaeker W: Conceptual density functional theory. Chem Rev. 2003, 103: 1793-1874. 10.1021/cr990029p.View ArticleGoogle Scholar
- Putz MV: Contributions within Density Functional Theory with Applications in Chemical Reactivity Theory and Electronegativity. 2003, Parkland: Dissertation.comGoogle Scholar
- Parr RG: Density functional theory. Annu Rev Phys Chem. 1983, 34: 631-656. 10.1146/annurev.pc.34.100183.003215.View ArticleGoogle Scholar
- Parr RG, Pearson RG: Absolute hardness: companion parameter to absolute electronegativity. J Am Chem Soc. 1983, 105: 7512-7516. 10.1021/ja00364a005.View ArticleGoogle Scholar
- Putz MV: Absolute and Chemical Electronegativity and Hardness. 2008, New York: Nova Publishers Inc.Google Scholar
- Putz MV: Systematic formulation for electronegativity and hardness and their atomic scales within density functional softness theory. Int J Quantum Chem. 2006, 106: 361-389. 10.1002/qua.20787.View ArticleGoogle Scholar
- Chattaraj PK, Parr RG: Density functional theory of chemical hardness. Struct Bond. 1993, 80: 11-25. 10.1007/BFb0036797.View ArticleGoogle Scholar
- Chattaraj PK, Sengupta S: Popular electronic structure principles in a dynamical context. J Phys Chem. 1996, 100: 16129-16130.View ArticleGoogle Scholar
- Chattaraj PK, Maiti B: HSAB principle applied to the time evolution of chemical reactions. J Am Chem Soc. 2003, 125: 2705-2710. 10.1021/ja0276063.View ArticleGoogle Scholar
- Chattaraj PK, Duley S: Electron affinity, electronegativity, and electrophilicity of atoms and ions. J Chem Eng Data. 2010, 55: 1882-1886. 10.1021/je900892p.View ArticleGoogle Scholar
- Ayers PW, Parr RG: Variational principles for describing chemical reactions: reactivity indices based on the external potential. J Am Chem Soc. 2001, 123: 2007-2017. 10.1021/ja002966g.View ArticleGoogle Scholar
- Kohn W, Becke AD, Parr RG: Density functional theory of electronic structure. J Phys Chem. 1996, 100: 12974-12980. 10.1021/jp960669l.View ArticleGoogle Scholar
- Putz MV: Chemical action concept and principle. MATCH Commun Math Comput Chem. 2011, 66: 35-63.Google Scholar
- Parr RG, Bartolotti LJ: On the geometric mean principle of electronegativity equalization. J Am Chem Soc. 1982, 104: 3801-3803. 10.1021/ja00378a004.View ArticleGoogle Scholar
- Kleinert H: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets. 2004, Singapore: World Scientific, 3View ArticleGoogle Scholar
- Guantes R, Sanz AS, Margalef-Roig J, Miret-Artés S: Atom–surface diffraction: a trajectory description. Surf Sci Rep. 2004, 53: 199-330. 10.1016/j.surfrep.2004.02.001.View ArticleGoogle Scholar
- Putz MV: On Heisenberg uncertainty relationship, its extension, and the quantum issue of wave-particle duality. Int J Mol Sci. 2010, 11: 4124-4139. 10.3390/ijms11104124.View ArticleGoogle Scholar
- Pauling L, Wilson EB: Introduction to Quantum Mechanics with Applications to Chemistry. 1985, New York: Dover PublicationsGoogle Scholar
- Lackner KS, Zweig G: Introduction to the chemistry of fractionally charged atoms: electronegativity. Phys Rev D. 1983, 28: 1671-1691. 10.1103/PhysRevD.28.1671.View ArticleGoogle Scholar
- Hohenberg P, Kohn W: Inhomogeneous electron gas. Phys Rev. 1964, 136: B864-B871. 10.1103/PhysRev.136.B864.View ArticleGoogle Scholar
- Putz MV: Density functionals of chemical bonding. Int J Mol Sci. 2008, 9: 1050-1095. 10.3390/ijms9061050.View ArticleGoogle Scholar
- Ghosh DC, Biswas R: Theoretical calculation of absolute radii of atoms and ions. Part 1. The atomic radii. Int J Mol Sci. 2002, 3: 87-113. 10.3390/i3020087.View ArticleGoogle Scholar
- Bader RFW: Definition of molecular structure: by choice or by appeal to observation?. J Phys Chem A. 2010, 114: 7431-7444. 10.1021/jp102748b.View ArticleGoogle Scholar
- Dreizler RM, Gross EKU: Density Functional Theory. 1990, Heidelberg: Springer VerlagView ArticleGoogle Scholar
- Kryachko ES, Ludena EV: Energy Density Functional Theory of Many Electron Systems. 1990, Dordrecht: Kluwer Academic PublishersView ArticleGoogle Scholar
- Cramer CJ: Essentials of Computational Chemistry. 2002, Chichester: WileyGoogle Scholar
- Capelle K: A bird's-eye view of density-functional theory. Braz J Phys. 2006, 36: 1318-1343. 10.1590/S0103-97332006000700035.View ArticleGoogle Scholar
- Jensen F: Introduction to Computational Chemistry. 2007, Chichester: John Wiley & SonsGoogle Scholar
- Parr RG: The Quantum Theory of Molecular Electronic Structure. 1972, Reading-Massachusetts: WA Benjamin, Inc.Google Scholar
- Cohen AJ, Mori-Sánchez P, Yang W: Challenges for density functional theory. Chem Rev. 2012, 112: 289-320. 10.1021/cr200107z.View ArticleGoogle Scholar
- Petrucci RH, Harwood WS, Herring FG, Madura JD: General Chemistry: Principles & Modern Applications. 2007, New Jersey: Pearson Education, Inc., 9Google Scholar
- Boeyens JC, Levendis DC: The structure lacuna. Int J Mol Sci. 2012, 13: 9081-9096. 10.3390/ijms13079081.View ArticleGoogle Scholar
- Hawking S: The Universe in a Nutshell. 2001, New York: Bantam BooksGoogle Scholar
- Putz MV, Ori O: Bondonic characterization of extended nanosystems: application to graphene's nanoribbons. Chem Phys Lett. 2012, 548: 95-100.View ArticleGoogle Scholar
- Boeyens JCA: A molecular–structure hypothesis. Int J Mol Sci. 2010, 11: 4267-4284. 10.3390/ijms11114267.View ArticleGoogle Scholar
- March NH: Electron Density Theory of Many-Electron Systems. 1991, New York: AcademicGoogle Scholar
- Wentorf RH: Boron: another form. Science. 1965, 147: 49-50. 10.1126/science.147.3653.49.View ArticleGoogle Scholar
- Eremets MI, Struzhkin VV, Mao H, Hemley RJ: Superconductivity in Boron. Science. 2001, 293: 272-274. 10.1126/science.1062286.View ArticleGoogle Scholar
- van Setten MJ, Uijttewaal MA, de Wijs GA, de Groot RA: Thermodynamic stability of boron: the role of defects and zero point motion. J Am Chem Soc. 2007, 129: 2458-2465. 10.1021/ja0631246.View ArticleGoogle Scholar
- Widom M, Mihalkovic M: Symmetry-broken crystal structure of elemental boron at low temperature. Phys Rev B. 2008, 77: 064113-View ArticleGoogle Scholar
- Carbon Bonding and Structures: Advances in Physics and Chemistry. Edited by: Putz MV. 2011, Dordrecht-London: Springer Verlag, Cataldo F, Milani P (Series Editors): Carbon Materials: Chemistry and Physics, Vol. 5.Google Scholar
- Struct Bond. Applications of Density Functional Theory to Chemical Reactivity. Edited by: Putz MV, Mingos DMP. 2012, Berlin-Heidelberg: Springer VerlagGoogle Scholar
- Ferreira R: Is one Electron less than half what an electron pair is?. J Chem Phys. 1968, 49: 2456-2457. 10.1063/1.1670430.View ArticleGoogle Scholar
- Bergmann D, Hinze J: Electronegativity and charge distribution. Struct Bond. 1987, 66: 145-190. 10.1007/BFb0029840.View ArticleGoogle Scholar
- Putz MV: Chemical Orthogonal Spaces. 2012, Kragujevac: Kragujevac University Press, [Gutman I (Series Editor): Mathematical Chemistry Monographs, Vol. 14].Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.