Functional Derivative of the Zero Point Energy Functional from the Strong Interaction Limit of Density Functional Theory

We derive an explicit expression for the functional derivative of the subleading term in the strong interaction limit expansion of the generalized Levy--Lieb functional for the special case of two electrons in one dimension. The expression is derived from the zero point energy (ZPE) functional, which is valid if the quantum state reduces to strongly correlated electrons in the strong coupling limit. The explicit expression is confirmed numerically and respects the relevant sum-rule. We also show that the ZPE potential is able to generate a bond mid-point peak for homo-nuclear dissociation and is properly of purely kinetic origin. Unfortunately, the ZPE diverges for Coulomb systems, whereas the exact peaks should be finite.Comment: 12 pages, 7 figure

Kohn-Sham equations with functionals from the strictly-correlated regime: Investigation with a spectral renormalization method

We re-adapt a spectral renormalization method, introduced in nonlinear optics, to solve the Kohn-Sham (KS) equations of density functional theory (DFT), with a focus on functionals based on the strictly-correlated electrons (SCE) regime, which are particularly challenging to converge. Important aspects of the method are: (i) the eigenvalues and the density are computed simultaneously; (ii) it converges using randomized initial guesses; (iii) easy to implement. Using this method we could converge for the first time the Kohn-Sham equations with functionals that include the next leading term in the strong-interaction limit of density functional theory, the so-called zero-point energy (ZPE) functional as well as with an interaction-strength-interpolation (ISI) functional that includes both the exact SCE and ZPE terms. This work is the first building block for future studies on quantum systems confined in low dimensions with different statistics and long-range repulsions, such as localization properties of fermions and bosons with strong long-range repulsive interactions in the presence of a random external potential

Fermionic statistics in the strongly correlated limit of Density Functional Theory

Exact pieces of information on the adiabatic connection integrand $W_{\lambda}[\rho]$, which allows to evaluate the exchange-correlation energy of Kohn-Sham density functional theory, can be extracted from the leading terms in the strong coupling limit ($\lambda\to\infty$, where $\lambda$ is the strength of the electron-electron interaction). In this work, we first compare the theoretical prediction for the two leading terms in the strong coupling limit with data obtained via numerical implementation of the exact Levy functional in the simple case of two electrons confined in one dimension, confirming the asymptotic exactness of these two terms. We then carry out a first study on the incorporation of the fermionic statistics at large coupling $\lambda$, both numerical and theoretical, confirming that spin effects enter at orders $\sim e^{-\sqrt{\lambda}}$

Locating ship strike risk hotspots for fin whale (Balaenoptera physalus) and sperm whale (Physeter macrocephalus) along main shipping lanes in the North-Western Mediterranean Sea

Abstract The Pelagos Sanctuary is the only pelagic marine protected area in the Mediterranean Sea, instituted for the conservation of cetaceans. Considering the number and size of commercial and touristic ports located along its coasts, this protected area is highly impacted by human activities, and especially marine traffic. Fin whales and sperm whales are regularly sighted in the Pelagos Sanctuary, and ship strikes with large vessels are one of the main threats affecting these two species. Mapping hotspots of distribution along main shipping lanes could be an effective conservation tool, as they directly allow locating high risk areas. In this work, we used data collected during summer from 2009 to 2019, along main marine traffic corridors in the central region of Pelagos Sanctuary. Ship strike hotspots have been identified, considering the persistence of distribution hotspots over the 11 years period. Hotspots occurrence has then been predicted over the entire Pelagos Sanctuary area applying Generalized Additive Models, allowing for ship strike risk assessment over the marine protected area. Our results highlighted the recurrence of important areas for both species along shipping lanes characterized by high vessel traffic, identifying regions where to address conservation measures