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 exchange-correlation energy is not always a differentiable functional of the spin density. As a consequence, various types of applications of SDFT must be critically reexamined.
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 exchange-correlation energy is not always a differentiable functional of the spin density. As a consequence, various types of applications of SDFT must be critically reexamined.
Recently, nonuniqueness of external electrostatic and magnetic fields yielding a given many-electron ground state has been pointed out [K. Capelle and G. Vignale, Phys. Rev. Lett. 86, 5546 (2001); H. Eschrig and W. E. Pickett, Solid State Commun. 118, 123 (2001)], implying the nondifferentiability of the ground-state energy functional of spin-density-functional theory (SDFT), on the basis of which the applicability of widely used DFT methods in SDFT has been put into question and the need for a critical reexamination of those applications has been concluded. Here it is shown, for collinear magnetic fields, that the nonuniqueness of the external potentials in SDFT does not imply the nonexistence of number-conserving functional derivatives as well, with the use of which therefore problems arising from the nondifferentiability are avoided.
It has recently been shown how to replace spin-wave functions with purely spatial wave functions in the formulation of spin-density functional theory (SDFT), without any change to the resultant density functional expressions [F. Zahariev and M. Levy, Phys. Rev. A 100, 062507 (2019)]. The purely spatial wave functions that are obtained are much more convenient to use and thus allow for a relatively easy decomposition and manipulation of terms in the constrained-search formulation of SDFT. All the essential ingredients of SDFT, including the kinetic, exchange, and correlation contributions, are explicitly defined in terms of the universal functional expressed in this spin-free manner. Constrained-search derivations of the Oliver-Perdew relations for the kinetic and exchange terms are presented. The up-down spin component of the correlation term is found to contain the up-down spin component of the Hartree term. A spin-dependent generalization of the adiabatic connection is put forth and a connection with spin-dependent coordinate scaling is established for the correlation energy.
Standard approximations for the exchange-correlation functional are known to deviate from linear dependence of the energy on the electron and spin numbers (in -space). Violation of this flat-plane condition underlies the failure of all known approximate functionals to describe band gaps in strongly correlated systems. Then crucial for further development of functionals is recognition of the behavior of the energy as a function of, and its derivatives (derivatives discontinuity pattern at 0K limit). In this Letter, the energy derivatives are analyzed thoroughly. It is shown that apart from the well-known type of discontinuity pattern, three other patterns are possible in the vicinity of the singlet state point. The zero temperature limits of energy derivatives at this point are derived, found different for various directions. Existence of all discontinuity patterns is illustrated on the example of diatomic molecules set.
The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of quantum mechanics, and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems. Here we show that in formulations of density-functional theory (DFT) that employ more than one density variable, applied to systems with a degenerate ground state, there is a subtle loophole in the HK theorem, as all mappings between densities, wave functions, and potentials can break down. Two weaker theorems which we prove here, the joint-degeneracy theorem and the internal-energy theorem, restore the internal, total, and exchange-correlation energy functionals to the extent needed in applications of DFT to atoms, molecules, and solids. The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems.
It has been known for some time that the exchange-correlation potential in time-dependent density functional theory is an intrinsically nonlocal functional of the density as soon as one goes beyond the adiabatic approximation. In this paper we show that a much more severe nonlocality problem, with a completely different physical origin, plagues the exchange-correlation potentials in time-dependent spin density functional theory. We show how the use of the spin current density as the basic variable solves this problem, and we provide an explicit local expression for the exchange-correlation fields as functionals of the spin currents.