Structural Chemistry & Crystallography Communication

Keywords

Energy of hydrogen atom; Additivity of radii; Bond lengths; Bohr radius; Golden ratio; Ionic radii; Hydrogen bond

Introduction

Bohr’s theory of the hydrogen atom [1] celebrated recently its centennial. It gave the correct magnitude for the ground state energy (E_H),

E_H = eI_H = - (1/2)(e²/κa_B) 1

where, I_H is the ionization potential, e is the unit of charge and κ is the electric permittivity constant and a_B is the Bohr radius (distance of the electron from the proton). However, the negative sign has irked the author’s conscience, since this implies that the energy of our Universe, which consists of more than 70% hydrogen, is negative. To quote from Gamov [2] about ‘False sign’: “All the theories expire or bring disappointment, The Sign is forever the fly in your ointment….”.

As regards bond lengths, although Pauling [3] suggested that covalent radii are additive, he could not account for many chemical bonds, especially the partially and completely ionic types and had to postulate many theories, concepts and correction factors. However, as correction factors do not explain exact science, the nature of the chemical bond became a theoretical speculation of many kinds. On re-investigating the Bohr’s equation (1), the author arrived at the surprising result that the Bohr radius is divided at the point of electrical neutrality into two Golden sections pertaining to the electron and proton. For an introduction to the Golden ratio, see [4]. This made a breakthrough in the interpretation of bond lengths as simple sums of the radii of the adjacent atoms and or ions [5-9]. The section below gives a brief account of this finding, and the interpretation over the years of existing bond length data in various molecules are listed in the references [5-19].

Additivity of Atomic and Ionic Radii in Bond Lengths and Structures of Molecules at the Atomic Level

For the same magnitude of energy as given by equation (1) of Bohr’s theory, the author [5,6] proposed a new interpretation of the ionization potential, I_H, of the hydrogen atom as the difference in potential between that of the proton (p⁺), (=e/ka_p) and of the electron (e^-), (=-e/κa_e) at a mutual distance of the Bohr radius,

a_B(=a_p+a_e)=0.529 Å from each other, [20-27],

I_H=(1/2)(e/κa_B)=(1/2)[(e/κa_p)-(e/κa_e)] 2

The ionization energy, E_H=eI_H=(1/2)(e/κa_B) can be interpreted as the electromagnetic energy of an atomic condenser [5-9], with κa_B as the capacitance. As the electron and proton are charged particles with magnetic momenta, this interpretation seems meaningful. From the above equation (2) and the equality, a_B(=a_p+a_e), one obtains for the ratio [28-36],

a_e/a_p=a_B/a_e=(1+5^1/'2)/2=φ 3

where φ, a mathematical constant, is the Golden ratio [4]. This surprising result shows that the Golden ratio which manifests in the geometry of many creations in the Universe [4] is right in core of the atom! Thus, the ratio of the electrostatic radii of the electron and proton in the hydrogen atom is the Golden ratio, and an atom is a unique construction of Nature [37-49]. On finding that the covalent bond length d(HH) between the two hydrogen atoms in the molecule H₂ is the diagonal of a square with the Bohr radius as a side, and since the latter is d ivided into two Golden sections, a_e=a_B/φ and a_p=a_B/φ², pertaining to the electron and proton, d(HH) is actually the sum,

d(HH)=2^1/2(a_B)=d(H^-)+d(H⁺) 4

where d(H^-)=d(HH)/φ and d(H⁺)=d(HH)/φ² are the anionic and cationic radii of H. Note that H^- and H⁺ are the ionic resonance forms of H in the hydrogen molecule, suggested by Pauling [3]. For more details, see [9].

On subtracting the cationic radius, d(H⁺) from the bond length of partially ionic alkali metal hydrides and hydrogen halides, it was found that the alkali metal cations are Golden sections of the respective interatomic distances in the metal lattice and the halogens have their covalent radii (which are half the covalent bond length).

Further, on subtracting the Golden ratio based cationic radius of the alkali metals from the interionic distance in alkali halides, the Golden ratio based radii of the halogen ions were obtained. See for all details, Ref. [50-59]. Thus, it could be generalized that the covalent bond length between two atoms of the same kind, d(AA)=d(A^-)+d(A⁺), the sum of their Golden ratio based cationic and anionic radii (which are similar to the ionic resonance forms of hydrogen in H₂). This finding paved the way to interpret the experimental bond lengths as simple sums of the appropriate radii of the adjacent atoms and or ions and thereby establish the precise structures at the atomic level of simple as well as complex molecules. References [50-59] below give the work from 2004 -2016. For a chapter in a book, a pdf of full poster and a review talk in power point, see Ref. [28,31,52].

References

1. Gamow G (1966) Thirty years that shook physics.
2.  Pauling L (1960) The nature of the chemical bond and the structure of molecules and crystals: an introduction to modern structural chemistry. Cornell University Press.
3. Livio M (2008) The golden ratio: The story of phi. The world's most astonishing number. Broadway Books.
4. Heyrovska R (2004) Hydrogen as AN Atomic Condenser. In: APS Division of Atomic. Molecular and Optical Physics Meeting Abstracts.
5. Heyrovska R (2004) The Decisive Role of the Golden Ratio in Atomic Dimensions. In: APS Division of Atomic. Molecular and Optical Physics Meeting Abstracts.
6.  Heyrovska R (2004) The Golden Ratio, Atomic, Ionic and Molecular Capacities and Bonding Distances in Hydrides. In: International Joint meeting of ECS, USA and Japanese, Korean and Australian Societies, Honolulu, Hawaii.
7. Working Group (2005) Pracovního Setkání Fyzikálních Chemiků a Elektrochemiků, Masarykova Univerzita v Brně, Unor. Vth Working Meeting of Physical Chemists and Electrochemists, Masaryk University, Brno, pp: 32-33.
8. Heyrovska R (2005) The Golden ratio, ionic and atomic radii and bond lengths. Molecular Physics 103: 877-882.
9. Heyrovska R, Narayan S (2005) A New Contribution for WYP 2005: The Golden Ratio, Bohr Radius, Planck’s Constant, Fine‐Structure Constant and g‐Factors. In: AIP Conference Preceedings 795: 205-205.
10. Heyrovska R, Narayan S (2005) Fine-structure Constant, Anomalous Magnetic Moment, Relativity Factor and the Golden Ratio that Divides the Bohr Radius, p: 24.
11. Heyrovska R (2005) Linear relation between the ionic radii of alkali and halogen ions in the crystal (based on the Golden ratio and in aqueous solutions).
12. Working Group (2006) Pracovního Setkání Fyzikálních Chemiků a Elektrochemiků, Masarykova Univerzita v Brně, Unor. VIth Working Meeting of Physical Chemists and Electrochemists, Masaryk University, Brno, pp: 38-39.
13. Heyrovská R (2009) The Golden ratio in the creations of Nature arises in the architecture of atoms and ions. Innovations in Chemical Biology, pp: 133-139.
14. Heyrovska R (2006) Dependence of the length of the hydrogen bond on the covalent and cationic radii of hydrogen, and additivity of bonding distances. Chem Phys Lett 432: 348-351.
15. Heyrovska R (2007) Dependences of molar volumes in solids, partial molal and hydrated ionic volumes of alkali halides on covalent and ionic radii and the golden ratio. Chemical Physics Letters 436: 287-293.
16. Heyrovska R (2007) Linear Dependencies of Van Der Waals. Covalent and Valence Shell Radii of Atoms of Groups 1a - 8a on their Bohr Radii.
17. Heyrovska R (2007) Atomic Structures of the Molecular Components in DNA and RNA based on Bond Lengths as Sums of Atomic Radii.
18. Heyrovska R, Narayan S (2008) Structures of molecules at the atomic level: Caffeine and related compounds. 10th Eurasia Conference on Chemical Sciences, Manila, Phillipines, p: 330.
19. Heyrovska R (2008) Direct Dependence of Covalent, Van Der Waals, and Valence Shell Radii of Atoms on their Bohr Radii for the Main Group Elements. Philippine Journal of Science 137: 133-139.
20. Heyrovska R (2008) Structures of the Molecular Components in DNA and RNA with Bond Lengths Interpreted as Sums of Atomic Covalent Radii. The Open Structural Biology Journal.
21. Heyrovska R (2008) Atomic Structures of all the Twenty Essential Amino Acids and a Tripeptide, with Bond Lengths as Sums of Atomic Covalent Radii.
22. Heyrovska R (2008) Atomic structures of graphene, benzene and methane with bond lengths as sums of the single, double and resonance bond radii of carbon.
23. Heyrovska R (2008) Atomic structures of graphene, benzene and methane with bond lengths as sums of the single, double and resonance bond radii of carbon. arXiv Preprint arXiv: 0804.4086.
24. Heyrovska R (2008) Atomic Structure of Benzene which Accounts for Resonance Energy.
25. Heyrovska R (2010) Bond Energy Sums in Benzene, Cyclohexatriene and Cyclohexane Prove Resonance Unnecessary. arXiv Preprint arXiv: 0807.4140.
26. Heyrovska R (2008) Various Carbon to Carbon Bond Lengths Inter-related via the Golden Ratio, and their Linear Dependence on Bond Energies. arXiv Preprint arXiv: 0809.1957.
27. Heyrovska R (2008) Direct dependence of covalent, van der Waals and valence shell radii of atoms on their Bohr radii for elements of Groups 1A-8A. Philippine Journal of Science 137: 133-139.
28. Heyrovska R (2009) The Golden ratio in the creations of Nature arises in the architecture of atoms and ions. Innovations in Chemical Biology, Proceedings of the 9th Eurasia Conference on Chemical Sciences, Antalya, Turkey.
29. Heyrovska R (2009) Golden Sections of Interatomic Distances as Exact Ionic Radii and Additivity of Atomic and Ionic Radii in Chemical Bonds, p: 1184.
30. Heyrovska R (2009) Golden sections of inter-atomic distances as exact ionic radii of atoms. Nature Precedings.
31. Heyrovska R, Narayan S (2009) Atomic Structures of Molecules Based on Additivity of Atomic and/or Ionic Radii. Nature Precedings.
32. Heyrovska R (2009) Aqueous Redox Potentials Found to be Inversely Proportional to the Bohr Radius 216th ECS Meeting, Vienna, Austria. Electrochem Soc Trans 25: 159-163.
33. Heyrovska R (2010) Radii of redox components from absolute redox potentials compared with covalent and aqueous ionic radii. Electroanalysis 22: 903-907.
34. Heyrovska R (2010) Bonding distances as Exact Sums of the Radii of the Constituent Atoms in Nanomaterials-Boron Nitride and Coronene.
35. Heyrovska R, Atchison L, Narayan S (2010) Precise Atomic Structures of Three Novel Nanomaterials in Nanotechnology, Biomedicine and Cosmology: Graphene, Boron Nitride and Coronene. Nature Precedings.
36. Heyrovska R (2011) Gender discrimination is a drawback for the progress of science. Symposium: International Women in Science, Challenges and Triumphs, International Year of Chemistry.
37. Heyrovska R, Atchison L, Narayan S (2011) Precise atomic structures of two important molecules in biochemistry: Ascorbic acid (vitamin C) and aspirin (acetyl salicylic acid). Nature Precedings.
38. Heyrovska R (2011) Structures at the Atomic Level of Cobalt, Zinc and Lead Niobates (with an Appendix: Atomic structure of cobalt niobate crystal).
39. Heyrovska R (2011) Atomic and molecular structures of positronium, dipositronium and positronium hydride. Nature Precedings.
40. Heyrovska R, Narayan S (2008) Structures of molecules at the atomic level: Caffeine and related compounds. Philippine Journal of Science 140: 119-124.
41. Heyrovska R (2011) Precise Molecular Structures of Cysteine, Cystine, Hydrogen - Bonded Dicysteine, Cysteine Dipeptide, Glutathione and Acetyl Cysteine Based on Additivity of Atomic Radii. Nature Precedings.
42. Heyrovska R (2012) New insight into DNA damage by cisplatin at the atomic scale. Nature Precedings, pp: 16-21.
43. Heyrovska R (2012) Precise Atomic Structures of L-DOPA, Dopamine, Noradrenaline, Adrenaline, Isoprenaline, 5-HTP, Serotonin and Histamine with Bond Lengths as Exact Sums of Adjacent Atomic Radii. International J Sci 1: 1-9.
44. Heyrovska R (2013) Atomic and Ionic Radi of Elements and Bohr Radii from Ionization Potentials are Linked Through the Golden Ratio. International J Sci 2: 82-92.
45. Heyrovska R (2013) Bond Lengths, Bond Angles and Bohr Radii from Ionization Potentials Related via the Golden Ratio for H2+, O2, O3, H2O, SO2, NO2 and CO2. International J Sci 2: 1-4.
46. Heyrovska R (2013) Atomic, Ionic and Bohr Radii Linked via the Golden Ratio for Elements Including Lanthanides and Actinides. International J Sci 2: 63-68.
47. Heyrovska R (2013) Atomic, Ionic and Bohr Radii Linked via the Golden Ratio for Elements of Groups 1-8 including Lanthanides and Actinides. International Journal of Sciences 2: 63-68.
48. Heyrovska R (2013) Golden ratio based fine structure constant and rydberg constant for hydrogen spectra. International Journal of Sciences 2: 28-31.
49. Heyrovska R (2015) The Golden ratio, a key geometrical constant in atomic architecture. In: Proceedings of the 113th Statistical Mechanics Conference.
50. Heyrovska R (2014) Bond Lengths as Exact Sums of the Radii of Adjacent Atoms and or Ions in the Structures of Molecules (Lead Lecture). 13th Eurasia Conference on Chemical Sciences, p: 165.
51. Heyrovska R (2014) New interpretation of the structure and formation of ozone based on the atomic and Golden ratio based ionic radii of oxygen.
52. Heyrovska R (2015) The Golden ratio, a key geometrical constant in atomic architecture. In: Proceedings of the 113th Statistical Mechanics Conference, p: 22.
53. Heyrovska R (2015) Structural insights at the atomic level of important materials.
54. Heyrovska R (2016) Sorry Bohr, Ground State Energy of Hydrogen Atom is Not Negative.
55. Heyrovska R (2016) Simple Interpretation of the Bond Lengths and Bond Angles in Stratospheric Chlorine Monoxide and Peroxide Based on Atomic and Ionic Radii.
56. Heyrovska R (2016) The Coulombic Nature of the van der Waals Bond Connecting Conducting Graphene Layers in Graphite. Graphene 5: 35.
57. Heyrovska R (2016) A Simple and Precise Interpretation of the Bond Lengths and Angles in Diborane in Terms of Atomic and Ionic Radii.
58. Heyrovska R (2016) The Coulombic Nature of the van der Waals Bond Connecting Conducting Graphene Layers in Graphite. Dedicated to geo-carbon expert, Prof. Gustaf Arrhenius, of Scripps Institution of Oceanography, CA. Graphene 5: 35-38.
59. Heyrovska R (2014) A Simple and Exact Interpretation of the Bond Lengths and Stacking Distances in Benzene and its Dimers in Terms of Atomic Covalent Radii.