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The average energy curvature as a function of the particle number is amolecule-specific quantity, which measures the deviation of a given functionalfrom the exact conditions of density functional theory (DFT). Related to thelack of derivative discontinuity in approximate exchange-correlationpotentials, the information about the curvature has been successfully used torestore the physical meaning of Kohn-Sham orbital eigenvalues and to developnon-empirical tuning and correction schemes for density functionalapproximations. In this work, we propose the construction of a machine-learningframework targeting the average energy curvature between the neutral and theradical cation state of thousands of small organic molecules (QM7 database).The applicability of the model is demonstrated in the context ofsystem-specific gamma-tuning of the LC-$\omega$PBE functional and validatedagainst the molecular first ionization potentials at equation-of-motion (EOM)coupled-cluster references. In addition, we propose a local version of thenon-linear regression model and demonstrate its transferability and predictivepower by determining the optimal range-separation parameter for two largemolecules relevant to the field of hole-transporting materials. Finally, weexplore the underlying structure of the QM7 database with the t-SNEdimensionality-reduction algorithm and identify structural and compositionalpatterns that promote the deviation from the piecewise linearity condition.

Learning the energy curvature versus particle number in approximate density functionals