Bohr model and dimensional scaling analysis of atoms and molecules

Contactium: A strongly correlated model system

At the limit of an infinite confinement strength ω, the ground state of a system that comprises two fermions or bosons in harmonic confinement interacting through the Fermi-Huang pseudopotential remains strongly correlated. A detailed analysis of the one-particle description of this "contactium" reveals several peculiarities that are not encountered in conventional model systems (such as the two-electron harmonium atom, ballium, and spherium) involving Coulombic interparticle interactions. First of all, none of the natural orbitals (NOs) {ψn(ω;r)} of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their non-ascendingly ordered occupation numbers {νn} turn out to be related to the eigenfunctions and eigenvalues of a zero-energy Schrödinger equation with an attractive Gaussian potential. This observation enables the derivation of their properties, such as the n-4/3 asymptotic decay of νn at the n→∞ limit (which differs from that of n-8/3 in the Coulombic systems), the independence of the confinement energy vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩ of n, and the n-2/3 asymptotic decay of the respective contribution νntn to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical to those of the two-electron harmonium atom at the ω → ∞ limit, despite the entirely different interparticle interactions in these systems.

High Dimensional Atomic States of Hydrogenic Type: Heisenberg-like and Entropic Uncertainty Measures

High dimensional atomic states play a relevant role in a broad range of quantum fields, ranging from atomic and molecular physics to quantum technologies. The D-dimensional hydrogenic system (i.e., a negatively-charged particle moving around a positively charged core under a Coulomb-like potential) is the main prototype of the physics of multidimensional quantum systems. In this work, we review the leading terms of the Heisenberg-like (radial expectation values) and entropy-like (Rényi, Shannon) uncertainty measures of this system at the limit of high D. They are given in a simple compact way in terms of the space dimensionality, the Coulomb strength and the state’s hyperquantum numbers. The associated multidimensional position–momentum uncertainty relations are also revised and compared with those of other relevant systems.

Physical origin of chemical periodicities in the system of elements

The Periodic Law, one of the great discoveries in human history, is magnificent in the art of chemistry. Different arrangements of chemical elements in differently shaped Periodic Tables serve for different purposes. “Can this Periodic Table be derived from quantum chemistry or physics?” can only be answered positively, if the internal structure of the Periodic Table is explicitly connected to facts and data from chemistry. Quantum chemical rationalization of such a Periodic Tables is achieved by explaining the details of energies and radii of atomic core and valence orbitals in the leading electron configurations of chemically bonded atoms. The coarse horizontal pseudo-periodicity in seven rows of 2, 8, 8, 18, 18, 32, 32 members is triggered by the low energy of and large gap above the 1s and nsp valence shells (2 ≤ n ≤ 6 !). The pseudo-periodicity, in particular the wavy variation of the elemental properties in the four longer rows, is due to the different behaviors of the s and p vs. d and f pairs of atomic valence shells along the ordered array of elements. The so-called secondary or vertical periodicity is related to pseudo-periodic changes of the atomic core shells. The Periodic Law of the naturally given System of Elements describes the trends of the many chemical properties displayed inside the Chemical Periodic Tables. While the general physical laws of quantum mechanics form a simple network, their application to the unlimited field of chemical materials under ambient ‘human’ conditions results in a complex and somewhat accidental structure inside the Table that fits to some more or less symmetric outer shape. Periodic Tables designed after some creative concept for the overall appearance are of interest in non-chemical fields of wisdom and art.

Reply to Correspondence on "Core Electron Topologies in Chemical Compounds: Case Study of Carbon versus Silicon"

In their Correspondence, von Szentpály, Schwarz, Stoll, and Werner claim that the main conclusions of our Communication previously published in this journal are based on computational artifacts and oversimplified models. We clarify the justification of our simple one-electron model to describe one-electron physics, and refute their criticism based on what they call "computational artifacts." We remind that our main conclusion on the crucial role of qualitative changes in core electron wavefunctions is evidenced not only by wavefunction topologies the complainants cling to, but also by several other physical observables, which remain unrefuted. Hence, the conclusions of our original Communication stand.

Rotation-vibrational energies for some diatomic molecules with improved Rosen-Morse potential in D-dimensions

By using the Nikiforov-Uvarov method, we solve the Schrödinger equation for the improved Rosen-Morse potential model in D spatial dimensions. We obtained the rotation-vibrational energies and the wave function, respectively. The ro-vibrational energies spectral of NO(a⁴π_i) and [Formula: see text] in D-dimensions have been computed by using the rotation-vibrational energy eigenvalues equation.

Core Electron Topologies in Chemical Compounds: Case Study of Carbon versus Silicon

The similarities and differences between carbon and silicon have attracted the curiosity of chemists for centuries. Similarities and analogies can be found in their saturated compounds, but carbon exhibits a cornucopia of unsaturated compounds that silicon (and most other elements) cannot replicate. While this qualitative difference is empirically well known, quantum chemistry has previously only described quantitative differences related to orbital overlap, steric effects, or orbital energies. We study C₂ and Si₂ and their hydrides X₂ H_2n (X=C, Si; n=1, 2, 3) by first-principles quantum chemical calculation, and find a qualitative difference in the topologies of the core electrons: carbon has the propensity to alter its core electron topology when forming unsaturated compounds, and silicon has not. We draw a connection between the core electron topologies and ionization energies, and identify other elements we expect to have similarly flexible core topologies as carbon.

Exploring Unorthodox Dimensions for Two-Electron Atoms

Melding quantum and classical mechanics is an abiding quest of physical chemists who strive for heuristic insights and useful tools. We present a surprisingly simple and accurate treatment of ground-state two-electron atoms. It makes use of only the dimensional dependence of a hydrogen atom, together with the exactly known first-order perturbation value of the electron-electron interaction, both quintessentially quantum, and the D → ∞ limit, entirely classical. The result is an analytic formula for D-dimensional two-electron atoms with Z ≥ 2. For D = 3 helium, it gives accuracy better than 2 millihartrees, which is better than current density functional theory. A kindred explicit formula for correlation energy exploits interpolation between D → ∞ and D = 1 or 2; when added to the Hartree-Fock energy, it improves accuracy for D = 3 helium to better than 0.1 millihartrees.

D-dimensional energies for cesium and sodium dimers

We solve the Schrödinger equation with the improved Rosen−Morse potential energy model in D spatial dimensions. The D-dimensional rotation-vibrational energy spectra have been obtained by using the supersymmetric shape invariance approach. The energies for the 33[Formula: see text]g+ state of the Cs2 molecule and the 51Δg state of the Na2 molecule increase as D increases in the presence of fixed vibrational quantum number and various rotational quantum numbers. We observe that the change in behavior of the vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

100th anniversary of Bohr's model of the atom

In the fall of 1913 Niels Bohr formulated his atomic models at the age of 27. This Essay traces Bohr's fundamental reasoning regarding atomic structure and spectra, the periodic table of the elements, and chemical bonding. His enduring insights and superseded suppositions are also discussed.