An alternative approach to density functional theory based on self-consistent field theory for ring polymers is applied to neutral atoms hydrogen to neon in their ground states. The spontaneous emergence of atomic shell structure and spherical symmetry-breaking of the total electron density is predicted using ideas of polymer excluded-volume between pairs of electrons to enforce the Pauli-exclusion principle, and an exact electron self-interaction correction. The Pauli potential is approximated and correlations are neglected, leading to comparisons with Hartree-Fock theory. The model shows excellent agreement with Hartree-Fock theory for atomic binding energies and density profiles of the first six elements, providing exact matches for the elements hydrogen and helium. The predicted shell structure starts to deviate significantly past neon and spherical symmetry-breaking is first predicted to occur at carbon instead of boron. The self-consistent field theory energy functional which describes the model is decomposed into thermodynamic components to trace the origin of spherical symmetry-breaking. It is found to arise from the electron density approaching closer to the nucleus in non-spherical distributions, which lowers the energy despite resulting in frustration between the quantum kinetic energy, electron-electron interaction, and the Pauli-exclusion interaction. The symmetry-breaking effect is also found to have minimal impact on the binding energies. The pair density profiles display characteristics of polymer macro-phase separation, where electron pairs occupy lobe-like structures that combined together, resemble traditional electronic orbitals. It is further shown that the predicted densities satisfy known constraints and produce equivalent total electronic density profiles that are predicted by quantum mechanics.

A representation of polymer self-consistent field theory equivalent to quantum density functional theory is given in terms of non-orthogonal basis sets. Molecular integrals and self-consistent equations for spherically symmetric systems using Gaussian basis functions are given, and the binding energies and radial electron densities of neutral atoms hydrogen through krypton are calculated. An exact electron selfinteraction correction is adopted and the Pauli-exclusion principle is enforced through ideas of polymer excluded-volume. The atoms hydrogen through neon are examined without some approximations which permit cancellation of errors and spontaneous shell structure is observed. Correlations are neglected in the interest of simplicity and comparisons are made with Hartree-Fock theory. The implications of the Pauli-exclusion potential and its approximate form are discussed, and the Pauli model is analyzed using scaling theory for the uniform electron density case where the correct form of the Thomas-Fermi quantum kinetic energy and the Dirac exchange correction are recovered.density functional theory, orbital-free, polymer physics, self-consistent field theory | INTRODUCTIONDensity functional theory (DFT) is one of the most widely used and successful methods for calculating structure and properties of many-body quantum systems. In DFT, a one-particle density is the central quantity instead of a many-particle wave function, making DFT computations orders-of-magnitude more tractable than wave function approaches. In particular, Kohn-Sham DFT (KS-DFT), which uses orbital functions as a route to find the density, can achieve chemical accuracy in many cases.It has been recently shown that polymer self-consistent field theory (SCFT) can also be used to study quantum many-body systems [1][2][3][4].Instead of orbitals, SCFT uses propagators which are solutions to modified diffusion equations. It has been shown that these SCFT equations are formally equivalent to KS-DFT [1], and so, through the theorems of , SCFT is guaranteed to make all the same predictions as quantum mechanics [1,2,4]. The SCFT route has several advantages compared to DFT: the propagators are real-valued functions in contrast to the complex-valued orbitals used in KS-DFT, the diffusion equations are initial-value parabolic equations in contrast to the elliptical boundary-value KS equation, the SCFT algorithm can be made parallel in a straightforward way since the propagators do not span entire systems as do KS orbitals, and the classical partition function derivation of the SCFT equations has implications for the foundations of quantum mechanics [4].

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