Bohr model and dimensional scaling analysis of atoms and molecules

Abstract

It is generally believed that the old quantum theory, as presented by Niels Bohr in 1913, fails when applied to few electron systems, such as the H 2 molecule. Here we review recent developments of the Bohr model that connect it with dimensional scaling procedures adapted from quantum chromodynamics. This approach treats electrons as point particles whose positions are determined by optimizing an algebraic energy function derived from the large-dimension limit of the Schro¨dinger equation. The calculations required are simple yet yield useful accuracy for molecular potential curves and bring out appealing heuristic aspects. We first examine the ground electronic states of H 2 , HeH, He 2 , LiH, BeH and Li 2 . Even a rudimentary Bohr model, employing interpolation between large and small internuclear distances, gives good agreement with potential curves obtained from conventional quantum mechanics. An amended Bohr version, augmented by constraints derived from Heitler-London or Hund-Mulliken results, dispenses with interpolation and gives substantial improvement for H 2 and H 3 . The relation to D-scaling is emphasized. A key factor is the angular dependence of the Jacobian volume element, which competes with interelectron repulsion. Another version, incorporating principal quantum numbers in the D-scaling transformation, extends the Bohr model to excited S states of multielectron atoms. We also discuss kindred Bohr-style applications of D-scaling to the H atom subjected to superstrong magnetic fields or to atomic anions subjected to high frequency, superintense laser fields. In conclusion, we note correspondences to the prequantum bonding models of Lewis and Langmuir and to the later resonance theory of Pauling, and discuss prospects for joining D-scaling with other methods to extend its utility and scope.

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

“…The dimensional scaling has been shown to be a powerful tool to describe numerous features of a great variety of (three dimensional) quantum systems and phenomena pertaining to a wide range of fields from atomic and molecular physics [1][2][3][4][5] to general quantum mechanics [6][7][8], quantum field theory and quantum cosmology [9][10][11][12][13] and quantum information [14][15][16][17][18]. It is often possible to approximate the solution of difficult three-dimensional quantum problems by means of a Taylor series development of similar systems with a non-standard dimensionality (i.e., D = 3) in powers of 1/D and then by using an interpolation or extrapolation procedure.…”

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

“…However,for R < 1 ,the orbital energy of 1s g rapidly drops, and tends towards the conjoined nucleus (Mg 1s) limit, while the orbital energy of 2s g exhibits aslight increase for R < 1 , and tends towards the Mg 2s limit. Thus,w hile there may be some mixing of 1s g and 2s g orbitals as SSSW suggest, ar esonance state with the conjoined nucleus like (Mg 1s) state,asdiscussed by Svidzinsky and co-workers, [3] still seems more likely.…”

Interdisciplinary Research—Mathematical Interactions Viewed from Four Portrayals

“…So far, I have coauthored two papers with Dr. Herschbach [5], [18]. I cannot equate the coauthorship with being an intellectual equal of his on the topic of chemical physics.…”