[Paper Review] Node-less atomic wave functions, Pauli repulsion and systematic projector augmentation

Peter E. Blöchl, Clemens J. Först|arXiv (Cornell University)|Oct 22, 2012

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper introduces a novel method to construct node-less atomic wave functions and energy-dependent, node-reduced partial waves that retain full information about atomic eigenstates. By inverting the Schrödinger equation, it derives an effective potential describing Pauli repulsion from core electrons, enabling a systematic projector augmentation scheme in the PAW method and offering a transparent, physically motivated alternative to conventional pseudopotentials with improved accuracy for correlated systems.

ABSTRACT

A construction of node-less atomic orbitals and energy-dependent, node-reduced partial waves is presented, that contains the full information of the atomic eigenstates and that allows to represent the scattering properties in a transparent manner. By inverting the defining Schrödinger equation, the Pauli repulsion by the core electrons can be represented as effective potential. This construction also provides a description of the Pauli repulsion by an environment. Furthermore, the representation leads to a new systematic scheme for a projector augmentation. The relation to Slater orbitals is discussed.

Motivation & Objective

• To develop a physically transparent representation of atomic wave functions that eliminates nodes while preserving full information about eigenstates.
• To describe Pauli repulsion from core electrons as an effective, energy-dependent, semi-local potential.
• To enable a systematic, physically grounded scheme for projector augmentation in the Projector Augmented Wave (PAW) method.
• To provide a framework for constructing tight-binding orbitals and incorporating strong electron correlations into DFT-based simulations.

Proposed method

• Constructs node-less wave functions recursively via the relation $(\hat{H} - E_n)|u_n\rangle = -|u_{n-1}\rangle$, starting from the ground state.
• Derives a transformation between standard bound states $|\psi_n\rangle$ and node-less functions $|u_n\rangle$ using energy differences: $|\psi_n\rangle = \sum_{m=1}^n |u_m\rangle \prod_{j=1}^{m-1}(E_j - E_n)$.
• Represents the Pauli repulsion as an effective potential by inverting the Schrödinger equation, yielding a semi-local pseudo-potential description.
• Uses energy-dependent node-reduced wave functions $|q_n(E)\rangle$ defined by $|q_n(E)\rangle = \frac{|q_{n-1}(E)\rangle - |q_{n-1}(E_{n-1})\rangle}{E - E_{n-1}}$, ensuring correct origin behavior.
• Applies a power-series expansion in $r$ to solve the inhomogeneous radial Schrödinger equation, with recursion relations for coefficients $a_j$.
• Derives energy-derivative relations showing $|q_{n+j}(E_n)\rangle = \frac{1}{j!}|q_n^{(j)}(E_n)\rangle$, enabling systematic construction of higher-order projectors.

Experimental results

Research questions

• RQ1 How can node-less atomic wave functions be systematically constructed to retain full information about atomic eigenstates while eliminating radial nodes?
• RQ2 Can Pauli repulsion from core electrons be represented as an effective, energy-dependent, semi-local potential through inversion of the Schrödinger equation?
• RQ3 How can the resulting wave functions enable a systematic and physically transparent projector augmentation scheme in the PAW method?
• RQ4 What is the mathematical and physical connection between these node-less wave functions and Slater-type orbitals or conventional pseudopotentials?

Key findings

• The node-less wave functions $|u_n\rangle$ are constructed recursively from bound states $|\psi_n\rangle$ using energy differences, ensuring exact reconstruction of the original eigenstates.
• The leading-order behavior of $|q_n(E)\rangle$ at the origin scales as $r^{\ell + 2(n-1)}$, with energy-independent prefactors, confirming the increasing repulsion with quantum number.
• The energy-derivative of $|q_n(E)\rangle$ at $E_n$ yields $|q_{n+j}(E_n)\rangle = \frac{1}{j!}|q_n^{(j)}(E_n)\rangle$, enabling a systematic, recursive projector construction.
• The effective potential describing Pauli repulsion is derived directly from the Schrödinger equation and is semi-local, avoiding divergences common in early pseudopotential approaches.
• The method provides a natural basis for tight-binding orbitals and enables the incorporation of strong electron correlations into DFT frameworks via a physically grounded augmentation scheme.
• The formalism is extendable to relativistic systems and maintains consistency with the PAW method, offering a transparent and systematic alternative to standard pseudopotential construction.