# Bohr model and dimensional scaling analysis of atoms and molecules

## Anatoly Svidzinsky, Goong Chen, Siu Chin, Moochan Kim, Dongxia Ma,
Robert Murawski, Alexei Sergeev, Marlan Scully, and Dudley Herschbach

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 SchrÃ¶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.

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