Nonuniqueness of the Potentials of Spin-Density-Functional Theory

Abstract

It is shown that, contrary to widely held beliefs, the potentials of spin-density-functional theory (SDFT) are not unique functionals of the spin densities. Explicit examples of distinct sets of potentials with the same ground-state densities are constructed. These findings imply that the zero-temperature exchangecorrelation energy is not always a differentiable functional of the spin density. As a consequence, various types of applications of SDFT must be critically reexamined. DOI: 10.1103/PhysRevLett.86.5546 PACS numbers: 71.15.Mb, 31.15.Ew, 75.10.Lp The Hohenberg-Kohn (HK) theorem [1], which guarantees that the ground-state single-particle density uniquely determines all observables of a many-body system, is one of the most remarkable theorems of quantum mechanics. It is also at the heart of density-functional theory (DFT), one of the most popular many-body methods [2,3]. In the case of the original formulation of DFT, in which the basic variable is the particle density n͑r͒, the HK theorem can be cast in the form of two logically independent one-to-one maps [2]. Quantum mechanics guarantees that, for a given interaction and particle number, the potential in which the particles move determines the ground-state many-body wave function (via solution of Schrödinger's equation), which in turn determines the ground-state density (by simple integration). The essence of the original HK theorem is that both of these maps are invertible: the ground-state density n͑r͒ uniquely determines the groundstate wave function C͑r 1 , . . . , r N ͒, which in turn determines, up to an additive constant, the potential [1-4],These abstract maps have found extremely important practical applications in the form of the Kohn-Sham (KS) formulation of DFT [5], which is used for almost all band-structure calculations in solid-state physics, and a rapidly increasing number of electronic-structure calculations in quantum chemistry. Many of these applications, however, do not employ the original formulation of DFT, but spin-density-functional theory (SDFT) [6], in which the fundamental variables are the spin-resolved particle densities n s ͑r͒. In SDFT the map from spin densities to wave functions is easily established, but that from wave functions to potentials could not, in spite of considerable effort [2,6,7] be proven and remains an, albeit popular, conjecture. In the early days of SDFT, von Barth and Hedin [6] already pointed out that the uniqueness of the spin-dependent potentials is not guaranteed, and explicitly constructed two potentials which, when used in a one-body Hamiltonian, have common eigenstates. The question of whether a similar construction is possible in the many-body case, however, remained open, and even in the one-body case it was objected that these common eigenstates were not necessarily common ground states [2,7].In this Letter, we settle these questions by first deriving a general equation [Eq. (2), below] whose solutions are the nonunique pieces of the potentials. This equation has two types of solutions, ...

Nonuniqueness and derivative discontinuities in density-functional theories for current-carrying and superconducting systems

“…This is the same prescription employed in Ref. 6 to construct examples for nonuniqueness in SDFT. In the terminology of that reference nonuniqueness arising from such constants of motion is referred to as systematic nonuniqueness.…”

“…In the present work we extend the analysis of Ref. 6 to two other generalizations of DFT, namely, current-DFT 7,8 ͑CDFT͒ and DFT for superconductors. [9][10][11][12] The discovery of nonuniqueness in these generalized DFTs deepens our understanding of the respective XC functionals and flags a warning signal to alltoo-immediate generalizations of the original HK theorem to more complex situations.…”

“…Following original ideas of von Barth and Hedin, 4 it has recently been shown by Eschrig and Pickett 5 and by the present authors 6 that in spin-DFT ͑SDFT͒ the situation is not that simple: while the wave function is still uniquely determined by the spin densities n ↑ (r) and n ↓ (r), the external potentials v ↑ (r) and v ↓ (r) ͓or v(r) and B(r)͔ are not. This implies that SDFT functionals are not always differentiable, and has far-reaching consequences for the construction of better exchange-correlation ͑XC͒ functionals, and for applications to systems such as half-metallic ferromagnets.…”

“…This implies that SDFT functionals are not always differentiable, and has far-reaching consequences for the construction of better exchange-correlation ͑XC͒ functionals, and for applications to systems such as half-metallic ferromagnets. 5,6 SDFT is not the only instance at which the original HK theorem has been generalized. In the present work we extend the analysis of Ref.…”

Spin in density‐functional theory

“…[91,92] Similar to the external potential, which is only known up to a constant, it is also possible to add a constant shift to the external magnetic field B z (r) without changing the wavefunction or the (spin-)density. [93,94] This leads to a number of peculiarities related to the differentiability of the spin-dependent energy functional. [95,96] However, most of these issues do not appear if the treatment is restricted to eigenfunctions of S z (the case of interest here) [97] or can be addressed by constraining M S to a fixed value (as we are always requiring here).…”