New Complete Orthonormal Sets of Exponential-Type Orbitals in Standard Convention and Their Origin

In standard convention, the new complete orthonormal sets of ψ(α*)-exponential type orbitals (ψ(α*)-ETOs) are introduced as functions of the complex or real spherical harmonics and (α★)-modified and L(p★)-generalized Laguerre polynomials ( (α★)-MLPs and L(p★)-GLPs), where, 0 < ζ < ∞, x = 2ζr, p* = 2l + 2 − α*, q* = n + l + 1 − α* and α* is the noninteger or integer (−∞ < α ≤ 2 for α* = α) frictional quantum number (−∞ < α*< 3 for α* ≠ α). It is shown that the origin of the ψ(α*)-ETOs, (α★)-MLPs and L(p★)-GLPs is the self-frictional quantum forces which are analog of radiation damping or self-frictional forces introduced by Lorentz in classical electrodynamics. The relations for the quantum self-frictional potentials in terms of ψ(α*)-ETOs, (α★)-MLPs, and L(p★)-GLPs, respectively, are established. We note that, in the case of disappearing frictional forces (for α* = 1 and ζ = Z/n), the ψ(α*)-ETOs and eigenvalue are reduced to the Schrödinger’s results in nonstandard convention for the hydrogen-like atom and, therefore, the ψ(α*)-ETOs become the noncomplete, i.e., ψnlm(1) ≡ ψnlm. Here, the ψnlm is Schrödinger’s wave function in nonstandard convention. As examples of an application of ψ(α*)-ETOs as basis sets, the calculations have been performed for the total energies of some atomic and ionic systems.