A conceptually correct model for covalent bonding, suitable for teaching at the undergraduate general chemistry level, is presented. The quantum mechanical basis for the stability of the covalent bond can be derived from a qualitative discussion of the uncertainty principle, which relates the electron’s spatial confinement to its kinetic energy. The qualitative discussion is backed by quantitative data obtained from the analytical quantum mechanical solution of the H2+ molecule. In addition, the stability of molecules is used as a tangible background against which the inferences emerging from the philosophical debate between classical determinism and the Copenhagen interpretation can be tested.
A conceptually correct model for covalent bonding, suitable for teaching at the undergraduate general chemistry level, is presented. The quantum mechanical basis for the stability of the covalent bond can be derived from a qualitative discussion of the uncertainty principle, which relates the electron’s spatial confinement to its kinetic energy. The qualitative discussion is backed by quantitative data obtained from the analytical quantum mechanical solution of the H2+ molecule. In addition, the stability of molecules is used as a tangible background against which the inferences emerging from the philosophical debate between classical determinism and the Copenhagen interpretation can be tested.
The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.
The chemical bond is a central organizing concept in chemistry, yet it is absent from the molecular Hamiltonian and no "bond operator" exists. Bonding is therefore not a primitive physical entity but a derived descriptor emerging from the quantum state. The logical consequences of this observation are revisited. Statements such as "bonding stabilizes structure" when taken literally risk circular reasoning (petitio principii), whereby bonding is inferred from a stationary structure and then invoked as its cause. The same caution applies to concepts such as steric repulsion, which is also a derived descriptor. Bonding accompanies stable or metastable states and correlates with their properties without constituting their cause. Illustrative examples are drawn from QTAIM, non-covalent interaction (NCI) approach, protein structure, and hydrogen-hydrogen bonding. Causation, language, and the autonomy of chemistry are also briefly discussed. The aim is not at all to diminish the role of bonding, but to place it at the correct logical level, that is, as a powerful, state-dependent descriptor that organizes, classifies, and predicts chemical behavior without serving as its fundamental cause.
Abstract The experimental trends in bonding and structure which we have discussed in the previous chapter cannot be understood within a classical framework. None of the elements and only very few of the thousand or more binary AB compounds are ionic in the sense that the electrostatic Madelung energy controls their bonding. And even for ionic systems, it is a quantum mechanical concept that stops the lattice from collapsing under the resultant attractive electrostatic forces: the strong repulsion that arises as the ion cores start to overlap is direct evidence that Pauli’s exclusion principle is alive and well and hard at work! In this chapter we will briefly outline the key experiments which led to the breakdown of the classical world view and the birth of quantum mechanics. The concept of a wave packet will be used to resolve the problem of wave-particle duality and to underpin Heisenberg’s uncertainty principle. A plausibility argument will then be given for the derivation of the Schrodinger equation. It will be solved both for the free-electron gas and free atoms as the former is the starting point for our treatment of the bonding and structure of sp-valent metals, the latter for our treatment of covalent systems. Although this chapter is primarily concerned with introducing the concepts behind quantum mechanics, we will end with a short section on the recent predictive power of first principles calculations. It is these computationally intensive calculations that have given credence to the simple models which are developed and presented in subsequent chapters.
With local realism quantum mechanics established, we can simply describe an extranuclear electron as a large-scale elastic ring with an elastic phase trajectory. Several small molecules can thus be strictly calculated through the logical method of establishing an accurate mechanical equilibrium equation describing the molecular structure, then solving the strict solutions of this mechanical equation and the corresponding wave equation. The results (bond length and dissociation energy) are in good agreement with observed results—i.e. if it is only coincidence, there should not be such a high probability of agreement between calculated and observed results. The method of local realism quantum mechanics is no longer the semi-empirical method. The method to calculate the electron pairing energy uses a linear regression of the ionization energy obtained through experiment. Nonetheless, it is exciting that there are diatomic molecules such as Na2, K2 and asymmetric HF molecules that possess a non-zero non-bonding electron number in the calculation examples. Moreover, the molecular structures are very intuitive, and the calculation method is much simpler than existing methods.
The United Nations General Assembly has officially designated 2025 as the International Year of Quantum Science and Technology (IYQ). This global initiative, supported by UNESCO and co-sponsored by several countries, marks the centenary of quantum sciences, which encompass quantum mechanics, quantum chemistry, and related fields. These disciplines serve as the foundation for fundamental research across the natural sciences. Quantum mechanics provides the theoretical framework for understanding matter and energy at the microscopic scale. While public celebrations of the International Year of Quantum Science and Technology often showcase quantum technologies to engage a broader audience, it is crucial to remember that the emergence of quantum theory primarily represented a ground-breaking shift in the basic sciences. This article explores the complex relationship between quantum mechanics and concepts of traditional chemistry, particularly concerning the microscopic world of atoms and molecules. It highlights how quantum mechanics, a holistic and nonlocal theory, features concepts such as entanglement and the indistinguishability of components that challenge classical chemical notions of distinct interatomic interactions and well-defined molecular structures. While quantum mechanics excels at determining the overall stability and dynamics of the system, the article explains that quantum chemistry relies on the classical Born-Oppenheimer approximation to maintain the view of local interactions and identifiable atoms within a molecule. Then, the article discusses the historical development and limitations of transition state theory in explaining chemical reactions, highlighting chemistry’s specific focus on bond breaking and bond making as a fundamental explanatory principle.
Abstract This paper challenges the conventional probabilistic frameworks of quantum mechanics by introducing the Aktug Geometric Model (AGM), a paradigm rooted in deterministic physics and exact boundary-value solutions. Standard chemical methodologies frequently rely on the mechanical halving of total bond lengths to derive atomic radii—a practice that creates profound structural inconsistencies and gives rise to the covalent tangency paradox. By employing a rigorous energy-density function subject to relativistic invariance, the AGM demonstrates that atoms maintain their intrinsic volumes rather than undergoing arbitrary cloud deformations during molecular synthesis. Through spectroscopic energy mapping, this study models the exact spatial boundaries of the hydrogen atom utilizing a spherical gradient governed by the Sinc function and the root-mean-square (RMS) metrics of wave propagation, accounting for structural sharp-cutoff effects via the Gibbs phenomenon. Quantitatively, while mainstream models enforce an artificial wave-function field contraction to explain the transition from an isolated Hydrogen radius (0.53 Å) to the H2 and H-O bond lengths (0.74 Å and 0.96 Å, respectively), the AGM unveils an elegant atomic velocity locking mechanism operating under a volumetric compression regime. The model proves that the bond length does not compress; instead, it locks precisely at a 0.70 Å threshold for both H2 and H-O systems, representing a true geometric penetration ratio. This constant distance reveals that hydrogen undergoes an expansion rather than a shrinkage when bonding with oxygen to form water, establishing a universal bond constant born out of a specific hydrogen-oxygen resonance. By eliminating the absurdity of the halving approach, the AGM provides a seamless, predictive, and geometrically stable architecture for multiple bond systems, realigning molecular chemistry with the principles of deterministic causality. In standard quantum mechanics and wave theory, the calculability (convergence) of wave packets propagating in free space presents severe mathematical challenges. Wave functions defined over an unbounded space-time continuum disperse over time and yield infinities upon integration, forcing a reliance on probabilistic interpretations or abstract concepts such as quantum tunneling. This paper demonstrates that a strict geometric boundary condition is required for the complex temporal frequency (ω) and spatial frequency (k) to yield physically meaningful results. The Aktug Geometric Model (AGM) confines subatomic energy density within a closed geometric manifold (M), reinterpreting the Heisenberg Uncertainty Principle (Δx⋅Δp≥ℏ/2) not as an intrinsic probabilistic limitation, but as a direct mathematical consequence of bounded geometry.