Minimizers for open-shell, spin-polarised Kohn-Sham equations for non-relativistic and quasi-relativistic molecular systems
This interesting paper is devoted to the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic \(N\)-electron Coulomb systems. Note that nowadays the Density Functional Theory (DFT) of \textit{W. Kohn} and \textit{L. J. Sham} [``Self-consistent equations including exchange and correlation effects'', Phys. Rev. A, 140, No. 4, 1133--1138 (1965; \url{doi:10.1103/PhysRev.140.A1133})] is one of the most applicable to the electronic structure calculations in modern quantum chemistry [\textit{A. Szabo} and \textit{N. S. Ostlund}, Modern quantum chemistry: an introduction to advanced electronic structure theory. New York, NY: Palgrave Macmillan (1982)], and also in the condensed matter theory. Here it is established existence of a ground state (minimizer) to the non-relativistic Kohn-Sham problem as well as to the quasi-relativistic one, that is shown for the spin-polarized molecular systems with \(N^{\uparrow }\) electrons (with spin up), and \(N^{\downarrow }\) electrons (with spin down). The open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic \(N\)-electron Coulomb systems are characterized with the properties that the kinetic energy of the electrons is given by either the non-relativistic operator \(-\triangle_{x_n}\) or the quasi-relativistic operator \(\sqrt{-\alpha^{-2}\triangle_{x_n}+\alpha^{-4}}-\alpha^{-2}\). Here \(\alpha \) is the Sommerfeld's fine structure constant. For the standard and extended Kohn-Sham models in the local density approximation the existence of a ground state would be of interest. Note that it is required that the total charge \(Z_{\text{tot}}\) of \(K\) nuclei must be greater than \(N - 1\). Also it is considered the quasi-relativistic setting, where it is required that \(Z_{\text{tot}}\) to be smaller than a critical charge \(Z_c = 2\alpha^{-1}\pi^{-1}\). To show the main result it is required the total electron density \(\rho \) to be defined, and the exchange-correlation functional to be chosen as \[ E_{xc}(\rho )=\int_{\mathbb{R}^3}g(\rho (r))dr,\leqno{(*)} \] yielding the Local Density Approximation (LDA). Here the authors suppose the function \(g\) to be such that the above stated equality (\(\ast \)) incorporates all approximate LDA functionals used in practical implementations. So assume that \(g\) is a twice differentiable function which satisfies \(g\in C^1(\mathbb{R}_{+},\mathbb{R})\), \(g(0)=0\), \(g^{'}\leq 0\), and there exists \(0<\beta_{-}\leq \beta_{+}<\nu \) such that \(\sup\limits_{\mathbb{R}_{+}}\frac{|g^{'}(\rho )|} {\rho^{\beta_{-}}+\rho^{\beta_{+}}}<\infty \). Also there exists \( 1\leq \gamma <3/2 \) such that \(\lim\limits_{\rho\to0^{+}}\sup (g(\rho )/\rho^{\gamma })<0\), where \(\nu =2/3 \) or the non-relativistic case, and \(\nu =1/3\) for the quasi-relativistic case. Thus the main result for the quasi-relativistic, spin-polarized Kohn-Sham problem is the following: If \(N=N^{\uparrow }+N^{\downarrow }\) is a positive integer satisfying \(N\leq Z_{\text{tot}}< Z_{c}=2\alpha^{-1}\pi^{-1}\), then the spin-polarized Kohn-Sham LDA problem has a minimizer satisfying a certain algebraic condition.
scientific article; zbMATH DE number 6670052