Electron Density and the External Potentials

We begin by giving a discussion of the properties of the two key objects in density functional theory, the density n and the external potential v. The density has the obvious properties 

These properties follow directly from the definition of the density and the usual normalization condition on the wavefunction. If we take into account that the density is obtained as the density of a bound eigenstate of Hamiltonian (1) we can derive further conditions. For this we put the physical constraint on the many-body system that it has a finite expectation value of the kinetic energy, i.e., 

At this point it is useful to introduce a new space of functions. We say that a function / is in Hl(TZ ) (7Z denotes the real numbers) if 

The space of functions // (lZn) is called a Sobolev space. The supindex 1 refers to the fact that the definition of the norm contains only first order derivatives. 

We therefore see that finiteness of the kinetic energy implies that XP is an element of the function space Hl R N). Differentiation of equation (9) and use of the Cauchy-Schwarz inequality then leads to [1] 

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