Electrostatic potential spatial minima

In order to stably levitate an object, the net force on it must be zero, and the forces on the body, if it is perturbed, must act to return it to its original position. The object must be at a local potential minimum that is, the second derivatives with respect to all spatial coordinates of the potential must be positive. This may seem, at first sight, to be trivial to arrange. However, any system whose potential is a solution to Laplace s equation is automatically unstable A statement in words of Laplace s equation is that the sum of the second partial derivatives of the potential is zero, and so not all can be simultaneously positive. This has long been known for electrostatic potentials, having been stated by Earnshaw(n) Millikan s scheme for suspending charged particles is thus only neutrally stable, since the fields within a Millikan capacitor provide no lateral constraint. 

It has been traditional to define a van der Waals potential (which combines Coulomb s law and the Lennard-Jones 6-12 potential function) and thereby subsume electronic shell repulsion, London forces, and electrostatic interactions under the term van der Waals interaction. Unfortunately, the resulting expression is an oversimplified treatment of the electrostatic interactions, which are only calculated between close neighbors and are considered to be spatially isotropic. Both of these implicit assumptions are untrue and do not represent physically realistic approximations. We prefer to use the term van der Waals distance for the intemuclear separation at which the 6-12 potential function is a minimum (see Fig. 6), the van der Waals radius being one-half this value when the two interacting atoms are identical, and explicitly treat the Lennard-Jones and electrostatic terms separately. While the term van der Waals interaction may have some value as a shorthand in structure description, it should be avoided when energetics are treated quantitatively. 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

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