Each atom section 3–5 has four subsections: A for D = 1 B for D → ∞ C for ϵ D ( 1 ), the first-order perturbation terms D for ϵ 3, the ground-state energy at D = 3 is obtained from the interpolation formula. We outline the following sections: in section 2 the interpolation formula in section 3 treat helium in section 4 lithium in section 5 beryllium in section 6 hydrogen molecule. This article exhibits the applicability of an unorthodox formula, a blend of dimensions with first-order perturbations, to more complex many-body systems. The first-order terms actually provide much of the dimension dependence. Then at D = 3, the ground state energy of helium ϵ 3 can be obtained by linking ϵ 1 and ϵ ∞ together with the first-order perturbation coefficients ϵ 1 ( 1 ) and ϵ ∞ ( 1 ) of the 1/ Z expansion. In the D → ∞ limit, the electrons assume positions fixed relative to another and to the nucleus, with wave functions replaced by delta functions.

In the D = 1 limit, the Coulombic potentials are replaced by delta functions in appropriately scaled coordinates. It makes use of only the dimensional dependence of a hydrogen atom, together with the exactly known first-order perturbation terms with λ = 1/ Z for the dimensional limits of the electron-electron 〈1/ r 12〉 interaction. Recently, a simple analytical interpolation formula emerged using both the D = 1 and D → ∞ limits for helium. Other dimensional scaling approaches were extended to N-electron atoms, renormalization with 1/ Z expansions, random walks, interpolation of hard sphere virial coefficients, resonance states, and dynamics of many-body systems in external fields. It was arduous and asymptotic but by summation techniques attained very high accuracy for D = 3.

The approach began with the D → ∞ limit and added terms in powers of δ = 1/ D. Years ago, a D-scaling technique used with quantum chromodynamics was prompted for helium. Taking a spatial dimension other than D = 3 can make a problem much simpler and then use perturbation theory or other techniques to obtain an approximate result for D = 3. Dimensional scaling, as applied to chemical physics, offers promising computational strategies and heuristic perspectives to study electronic structures and obtain energies of atoms, molecules, and extended systems.

Hydrogen molecule