### Existence of minimizers for Kohn-Sham models in quantum chemistry

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In this article, we provide error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the electronic...

The purpose of the present article is to compare different phase-space sampling methods, such as purely stochastic methods (Rejection method, Metropolized independence sampler, Importance Sampling), stochastically perturbed Molecular Dynamics methods (Hybrid Monte Carlo, Langevin Dynamics, Biased Random Walk), and purely deterministic methods (Nosé-Hoover chains, Nosé-Poincaré and Recursive Multiple Thermostats (RMT) methods). After recalling some theoretical convergence properties for the...

The present work is a mathematical analysis of two algorithms, namely the Roothaan and the level-shifting algorithms, commonly used in practice to solve the Hartree-Fock equations. The level-shifting algorithm is proved to be well-posed and to converge provided the shift parameter is large enough. On the contrary, cases when the Roothaan algorithm is not well defined or fails in converging are exhibited. These mathematical results are confronted to numerical experiments performed by chemists.

The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock model as well as in some instances of the density functional theory. In a number of applications, it is convenient to approximate this integral operator by a multiplication operator, i.e. by a local potential. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator including the Slater potential, the optimized effective...

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