Abstract
We present a simple interpolation formula using dimensional limits $D=1$ and $D=\infty$ to obtain the $D=3$ ground-state energies of atoms and molecules. For atoms, these limits are linked by first-order perturbation terms of electron-electron interactions. This unorthodox approach is illustrated by ground-states for two, three, and four electron atoms, with modest effort to obtain fairly accurate results. Also, we treat the ground-state of H$_2$ over a wide range of the internuclear distance R, and compares well with the standard exact results from the Full Configuration Interaction method. Similar dimensional interpolations may be useful for complex many-body systems.
Highlights
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 [1,2,3,4]
Other dimensional scaling approaches were extended to N-electron atoms [7], renormalization with 1/Z expansions [8], random walks [9], interpolation of hard sphere virial coefficients [10], resonance states [11], and dynamics of many-body systems in external fields [12, 13]
The ground-state energy for the Be-atom has been calculated by applying various methods for e.g., the Configuration Interaction (CI) method with Slater-type orbitals (STOs) [29], the Hylleraas method (Hy) [30], the Hylleraas-Configuration Interaction method (Hy-CI) [31], and the Exponential Correlated Gaussian (ECG) method [32, 33]
Summary
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 [1,2,3,4]. Other dimensional scaling approaches were extended to N-electron atoms [7], renormalization with 1/Z expansions [8], random walks [9], interpolation of hard sphere virial coefficients [10], resonance states [11], and dynamics of many-body systems in external fields [12, 13]. A simple analytical interpolation formula emerged using both the D = 1 and D → ∞ limits for helium [14] 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/r12 interaction.
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