Euler Equation for Density

Euler Equation for Density.—A pillar of the TF theory is that the density-potential relationship (3) can be obtained as the Euler equation of the variation problem posed by minimizing E in equation (20) with respect to p r) subject to the normalization requirement 

Thus we introduce a Lagrange multiplier p and minimize according to 

The principle underlying the whole of the density theory of atoms and molecules has essentially been exposed by the argument leading to equation (24). We shall see below, however, that there is a need to refine the elementary approximation (18) for the kinetic energy density, in order to transcend TF statistical theory. Nevertheless, even without refinement, certain useful relations follow from this simplest form of density theory and we will discuss these now for positive atomic ions. 

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