[Paper Review] Alchemical perturbation density functional theory (APDFT)

von Rudorff, Guido Falk, von Lilienfeld, O. Anatole|arXiv (Cornell University)|Sep 5, 2018

Advanced Chemical Physics Studies32 citations

TL;DR

This paper introduces Alchemical Perturbation Density Functional Theory (APDFT), an orbital-free method that uses Taylor-series expansions of electron density derivatives under alchemical perturbations (e.g., nuclear charge changes) to predict electronic properties of iso-electronic molecules. It achieves accuracy comparable to high-level methods like PBE0 at negligible cost, using only a single reference density and its perturbations.

ABSTRACT

We introduce an orbital free electron density functional approximation based on alchemical perturbation theory. Given convergent perturbations of a suitable reference system, the accuracy of popular self-consistent Kohn-Sham density functional estimates of properties of new molecules can be systematically surpassed---at negligible cost. The associated energy functional is an approximation to the integrated energy derivative, requiring only perturbed reference electron densities: No self-consistent field equations are necessary to estimate energies and electron densities. Electronic ground state properties considered include covalent bonding potentials, atomic forces, as well as dipole and quadropole moments.

Motivation & Objective

• To develop a computationally efficient method for predicting electronic properties of iso-electronic molecules.
• To surpass standard DFT accuracy using only reference electron density and its derivatives.
• To eliminate the need for self-consistent field iterations in property prediction.
• To enable systematic accuracy improvement via higher-order perturbation terms.
• To allow large-scale screening of chemical space using one high-accuracy reference calculation.

Proposed method

• Uses alchemical perturbation theory to model changes in nuclear charge as a continuous parameter λ.
• Expands the energy and electron density in a Taylor series around the reference system (λ = 0).
• Applies the Hellmann-Feynman theorem to express energy derivatives as integrals over the external potential difference and density derivatives.
• Constructs a lambda-averaged electron density ˜ρ via summation of density derivatives, enabling energy prediction via ∫∆v·˜ρ dr.
• Derives density and energy derivatives using finite differences or analytical derivatives of the reference density.
• Truncates the series after few terms, achieving convergence for small ΔZ (e.g., ΔZ = 1) systems with fixed geometry.

Experimental results

Research questions

• RQ1 Can alchemical perturbation theory be used to predict electronic properties without self-consistent field iterations?
• RQ2 Can the accuracy of DFT-like properties be systematically improved using only reference density derivatives?
• RQ3 Does the lambda-averaged density ˜ρ converge and yield accurate energies and properties for iso-electronic systems?
• RQ4 Can this method predict forces, dipole, and quadrupole moments with high fidelity?
• RQ5 Can one high-accuracy reference calculation enable reliable predictions across a broad chemical space?

Key findings

• APDFT achieves energy, force, and electrostatic property predictions comparable to PBE0 at perturbation orders 3–4 for CO and BF relative to N2 reference.
• For He, energy errors drop below 1 mH with n = 4 expansion order using HF or DFT functionals.
• The method predicts dipole and quadrupole moments with accuracy matching the reference level.
• Convergence is observed for hydrogenic atoms and free atoms, with numerical evidence supporting convergence for small ΔZ systems.
• The approach enables geometry optimization and chemical composition scanning via alchemical gradients, with higher-order terms extending applicability to larger chemical space regions.
• Only the reference electron density and its derivatives are needed, allowing application to any reference method that outputs densities.