Potential reductio

This equation, together with the variational principle, allowed Hohenberg and Kohn [1] to show by a trivial reductio ad absurdum that there is a one-to-one relationship between the density p and the external potential Oext- This implies that the energy E is a functional of the electronic density of the ground state i.e., for a given external potential u,... 

The form (128) suffers from the nonlinearity of H in the external vector potential The standard reductio ad absurdum of the HK-argument only works if H is linear in the external potentials. Thus no existence theorem can be proven with (128). On the other hand, such a proof is possible for the form (129), using the density n and the gauge dependent current jp - c/e)V x m as basic DFT... 

Proof ([1], p. B865). The proof proceeds by reductio ad absurdum." We assume the existence of two external potentials Vi(r) and V2(r) such that... 

IV) A self-contradiction (ad absurdum) of Eq. (22) might also mean that the to-be-refuted assumptions (i) or/and (ii) of the Hohenberg-Kohn theorem are selfcontradictory with Eq. (19) and this is precisely the case of many-electron Coulomb systems with Coulomb-type class of external potentials. In other words, the original reductio ad absurdum proof of the Hohenberg-Kohn theorem based on the assumption (19) is incompatible with the ad absurdum assumption (ii) since the Kato theorem is valid for such systems [18]. 

Deb [31], Smith [32], and E. Bright Wilson (quoted by Lowdin [33] for the recent applications of the Kato theorem to the Hohenberg-Kohn theorem see also [34—36]). Therefore, if a given pair of iV-electron systems with the Hamiltonians Hi and H2 of the type (1) are characterized by the same groxmd-state one-electron densities (= to-be-refuted assumption (ii)), their external potentials Vi(r) and V2(r) of the form (24) are identical. The latter contradicts (19) and hence, the assumption (ii) cannot be used in the proof via reductio ad absurdum of the Hohenberg-Kohn theorem together with the assumption (19). In other words, they are Kato-type incompatible with each other. 

The first step of any DFT is the proof of a Hohenberg-Kohn type theorem [6]. In its traditional form, this theorem demonstrates that there exists a one-to-one correspondence between the external potential and the (one-body) density. The first implication is clear With the external potential it is alwaj possible (in principle) to solve the many-body Schrodinger equation to obtain the many-body wave-function. From the wave-function we can trivially obtain the density. The second implication, i.e. that the knowledge of the density is sufficient to obtain the external potential, is much harder to prove. In their seminal paper, Hohenberg and Kohn used the variational principle to obtain a proof by reductio ad ahsurdum. Unfortunately, their method cannot be easily generalized to arbitrary DFTs. The Hohenberg-Kohn theorem is a very strong statement From the density, a simple property of the quantum mechanical system, it is possible to obtain the external potential and therefore the many-body wave-function. The wave-function, in turn, determines every observable of the system. This implies that every observable can be written as a junctional of the density. 

The authors demonstrated this first theorem by proving by reductio ad absurdum that there can not be two different Vexti ) that result in the same ground state electron density or what is the same, the ground state density uniquely specifies the external potential Vextir). A direct consequence derived from this first principle is that all the ground state properties of a system are defined by its electron density. 

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