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Register a new userLearning the energy curvature versus particle number in approximate density functionals
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The average energy curvature as a function of the particle number is a molecule-specific quantity, which measures the deviation of a given functional from the exact conditions of density functional theory (DFT). Related to the lack of derivative discontinuity in approximate exchange-correlation potentials, the information about the curvature has been successfully used to restore the physical meaning of Kohn-Sham orbital eigenvalues and to develop non-empirical tuning and correction schemes for density functional approximations. In this work, we propose the construction of a machine-learning framework targeting the average energy curvature between the neutral and the radical cation state of thousands of small organic molecules (QM7 database). The applicability of the model is demonstrated in the context of system-specific gamma-tuning of the LC-$omega$PBE functional and validated against the molecular first ionization potentials at equation-of-motion (EOM) coupled-cluster references. In addition, we propose a local version of the non-linear regression model and demonstrate its transferability and predictive power by determining the optimal range-separation parameter for two large molecules relevant to the field of hole-transporting materials. Finally, we explore the underlying structure of the QM7 database with the t-SNE dimensionality-reduction algorithm and identify structural and compositional patterns that promote the deviation from the piecewise linearity condition.
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We develop relativistic short-range exchange energy functionals for four-component relativistic range-separated density-functional theory using a Dirac-Coulomb Hamiltonian in the no-pair approximation. We show how to improve the short-range local-density approximation exchange functional for large range-separation parameters by using the on-top exchange pair density as a new variable. We also develop a relativistic short-range generalized-gradient approximation exchange functional which further increases the accuracy for small range-separation parameters. Tests on the helium, beryllium, neon, and argon isoelectronic series up to high nuclear charges show that this latter functional gives exchange energies with a maximal relative percentage error of 3 %. The development of this exchange functional represents a step forward for the application of four-component relativistic range-separated density-functional theory to chemical compounds with heavy elements.
Machine learning is employed to build an energy density functional for self-bound nuclear systems for the first time. By learning the kinetic energy as a functional of the nucleon density alone, a robust and accurate orbital-free density functional for nuclei is established. Self-consistent calculations that bypass the Kohn-Sham equations provide the ground-state densities, total energies, and root-mean-square radii with a high accuracy in comparison with the Kohn-Sham solutions. No existing orbital-free density functional theory comes close to this performance for nuclei. Therefore, it provides a new promising way for future developments of nuclear energy density functionals for the whole nuclear chart.
Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and highly accurate energies are achieved. Accurate {em constrained optimal densities} are found via a modified Euler-Lagrange constrained minimization of the total energy. A projected gradient descent algorithm is derived using local principal component analysis. Additionally, a sparse grid representation of the density can be used without degrading the performance of the methods. The implications for machine-learned density functional approximations are discussed.
Empirical fitting of parameters in approximate density functionals is common. Such fits conflate errors in the self-consistent density with errors in the energy functional, but density-corrected DFT (DC-DFT) separates these two. We illustrate with catastrophic failures of a toy functional applied to $H_2^+$ at varying bond lengths, where the standard fitting procedure misses the exact functional; Grimmes D3 fit to noncovalent interactions, which can be contaminated by large density errors such as in the WATER27 and B30 datasets; and double-hybrids trained on self-consistent densities, which can perform poorly on systems with density-driven errors. In these cases, more accurate results are found at no additional cost, by using Hartree-Fock (HF) densities instead of self-consistent densities. For binding energies of small water clusters, errors are greatly reduced. Range-separated hybrids with 100% HF at large distances suffer much less from this effect.
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