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International Journal of Quantum Chemistry

Density functional theory and multicomponent wave functions

Jacek Karwowski

Published Mar 19, 2012• 5 citations
Publisher

Abstract

Abstract Some consequences of using multicomponent wave functions are analyzed. In particular, paradoxical properties of a density functional derived from Dirac and Lévy‐Leblond equations are discussed. The application of a recently formulated N‐electron “relativistic Schrödinger theory,” based on a 4N‐component wave function, to the construction of a Lorentz and gauge covariant density functional theory is suggested. © 2012 Wiley Periodicals, Inc.

DOI
10.1002/qua.24037

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Abstract Some consequences of using multicomponent wave functions are analyzed. In particular, paradoxical properties of a density functional derived from Dirac and Lévy‐Leblond equations are discussed. The application of a recently formulated N‐electron “relativistic Schrödinger theory,” based on a 4N‐component wave function, to the construction of a Lorentz and gauge covariant density functional theory is suggested. © 2012 Wiley Periodicals, Inc.
An observation on a pathological behavior of an exact density functional derived from either relativistic (Dirac) or nonrelativistic (L\'evy-Leblond) quantum-mechanical equation is reported. As expected, in the case of a one-electron atom the variational minimum of this functional is equal to the exact ground-state energy. However, apart from the correct density, this minimum is reached also by an infinite set of densities which do not correspond to the exact wave function. This paradoxical property of the functional is related to the multicomponent structure of both Dirac and L\'evy-Leblond wave functions. In particular, imposing the correct boundary conditions upon the trial densities removes only a part of the fake solutions. The results of this study demonstrate that in density-functional theories derived from models based on multicomponent wave functions, one should not expect any simple relation between the accuracy of the energy and the correctness of the corresponding density.
Abstract Spin‐density functional theory (SDFT) is the most widely implemented and applied formulation of density functional theory. However, it is still finding novel applications, and occasionally encounters unexpected problems. In this article, we first briefly describe a few of the latter, related to issues such as nonuniqueness, noncollinearity, and currents. We then turn to an example of the former, namely SDFT for the Heisenberg model. It is shown that time‐honored concepts of Coulomb DFT, such as the local‐density approximation, can be applied to this (and other) model Hamiltonians, as well, once the concept of “density” has been suitably reinterpreted. Local‐density‐type approximations for the inhomogeneous Heisenberg model are constructed. Numerical applications to finite‐size and impurity systems demonstrate that DFT is a computationally efficient and reasonably accurate alternative to conventional methods of statistical mechanics for the Heisenberg model. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
Abstract Attempts to put classical and quantum mechanics on an equal footing gave rise to establishing of many known phase‐space representations of quantum mechanics. The class of self‐dual representations occupies a distinguished position among them, as they could be endowed with a Hilbert space structure closely resembling the standard picture of quantum mechanics. Using a phase‐space variational principle and the concept of a phase‐space wave function, we established a phase‐space version of the density functional theory parallel to the original Hohenberg–Kohn formalism and the constrained‐search approach by Levy and Lieb. © 1995 John Wiley & Sons, Inc.
Abstract The fundamental theorem of time‐independent/time‐dependent density functional theory due to Hohenberg–Kohn (HK)/Runge–Gross (RG) proves the bijectivity between the density ρ(r)/ρ(rt) and the Hamiltonian / (t) to within a constant C/function C(t), and wave function Ψ/Ψ (t). The theorems are each proved for scalar external potential energy operators. By a unitary or equivalently a gauge transformation that preserves the density, we generalize the realm of validity of each theorem to Hamiltonians, which additionally include the momentum operator and a curl‐free vector potential energy operator defined in terms of a gauge function α (R)/α (Rt). The original HK/RG theorems then each constitute a special case of this generalization. Thereby, a fourfold hierarchy of such theorems is established. As a consequence of the generalization, the wave function Ψ/Ψ (t) is shown to be a functional of both the density ρ(r)/ρ(rt), which is a gauge‐invariant property, and a gauge function α(R)/α(Rt). The functional dependence on the gauge function ensures that as required by quantum mechanics, the wave function written as a functional is gauge variant. The hierarchy and the dependence of the wave function functional on the gauge function thus enhance the significance of the phase factor in density functional theory in a manner similar to that of quantum mechanics. Various additional perspectives on the theorem are arrived at. These understandings also address past critiques of time‐dependent theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
It is not possible to use the Schrödinger equation to describe the motion of a particle with relativistic effects since it is not invariant relative to the Lorentz transformation. However, the attempt to generalize quantum mechanics in the relativistic domain on the basis of the one particle theory is somehow difficult. So we find the appropriate equations on the basis of the relativistic invariance. To obtain the non-relativistic equations, we take the limit as the speed of light tends to infinity. This obviously recovers the Schrödinger equation. Under this heading, we extend the non-relativistic Schrödinger equation in the description of particles with speeds comparable with that of light. This extension should be consistent with the Lorentz transformation [ 12 ]. The characteristic feature of the relativistic wave equations is the spin which provides a useful gauge in the applicability of particular equations that describe particular kinds of particles. We have two types of relativistic equations, i.e., the equation with spin-zero due to Schrödinger and spin-half due to Dirac that describes the electron.
Density functional theory is a rapidly developing branch of many-particle physics that has found applications in atomic, molecular, solid state and nuclear physics. This text describes the conceptual framework of density functional theory and discusses in detail the derivation of explicit functionals from first principles as well as their application to Coulomb systems. Both non-relativistic and relativistic systems are treated. The connection of density functional theory with other many-body methods is highlighted. The presentation is self-contained; the book is thus suited for a graduate course on density functional theory.