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International Journal of Quantum Chemistry

New perspectives on the fundamental theorem of density functional theory

Xiao‐Yin Pan, Viraht Sahni

Published Jan 1, 2008• 17 citations
Publisher

Abstract

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

DOI
10.1002/qua.21826

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Abstract According to the Hohenberg–Kohn theorem, there is an invertible one‐to‐one relationship between the Hamiltonian of a system and the corresponding ground‐state density ρ(r). The extension of the theorem to the time‐dependent case by Runge and Gross states that there is an invertible one‐to‐one relationship between the density ρ(rt) and the Hamiltonian (t). In the proof of the theorem, Hamiltonians / (t) that differ by an additive constant C/function C(t) are considered equivalent. Because the constant C/function C(t) is extrinsically additive, the physical system defined by these differing Hamiltonians / (t) is the same. Thus, according to the theorem, the density ρ(r)/ρ(rt) uniquely determines the physical system as defined by its Hamiltonian / (t). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians { }/{ (t)} that differ by an intrinsic constant C/function C(t) but which represent different physical systems and yet possess the same density ρ(r)/ρ(rt). The intrinsic constant C/function C(t) contains information about the different physical systems and helps differentiate between them. In such a case, the density ρ(r)/ρ(rt) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians { }/{ (t)}. Thus, although the proof of the Hohenberg–Kohn theorem is independent of whether the constant C/function C(t) is additive or intrinsic, the applicability of the theorem is restricted to excluding the case of the latter. The corollary is as follows: degenerate Hamiltonians { }/{ (t)} that represent different physical systems, but differ by a constant C/function C(t), and yet possess the same density ρ(r)/ρ(rt), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

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arXiv:1412.7052•Journal of Physics Condensed Matter•Apr 29, 2015•73 citations
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
Abstract According to the Hohenberg–Kohn theorem, there is an invertible one‐to‐one relationship between the Hamiltonian of a system and the corresponding ground‐state density ρ(r). The extension of the theorem to the time‐dependent case by Runge and Gross states that there is an invertible one‐to‐one relationship between the density ρ(rt) and the Hamiltonian (t). In the proof of the theorem, Hamiltonians / (t) that differ by an additive constant C/function C(t) are considered equivalent. Because the constant C/function C(t) is extrinsically additive, the physical system defined by these differing Hamiltonians / (t) is the same. Thus, according to the theorem, the density ρ(r)/ρ(rt) uniquely determines the physical system as defined by its Hamiltonian / (t). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians { }/{ (t)} that differ by an intrinsic constant C/function C(t) but which represent different physical systems and yet possess the same density ρ(r)/ρ(rt). The intrinsic constant C/function C(t) contains information about the different physical systems and helps differentiate between them. In such a case, the density ρ(r)/ρ(rt) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians { }/{ (t)}. Thus, although the proof of the Hohenberg–Kohn theorem is independent of whether the constant C/function C(t) is additive or intrinsic, the applicability of the theorem is restricted to excluding the case of the latter. The corollary is as follows: degenerate Hamiltonians { }/{ (t)} that represent different physical systems, but differ by a constant C/function C(t), and yet possess the same density ρ(r)/ρ(rt), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
The general expectation that, in principle, the time-dependent density-functional theory (TDDFT) is an exact formulation of the time evolution of an interacting NNN-electron system is critically reexamined. It is demonstrated that the previous TDDFT foundation, resting on four theorems by Runge and Gross (RG) [Phys. Rev. Lett. 52, 997 (1984)], is invalid because undefined phase factors corrupt the RG action integral functionals. Our finding confirms much of a previous analysis by van Leeuwen [Int. J. Mod. Phys. B 15, 1969 (2001)]. To analyze the RG theorems and other aspects of TDDFT, an utmost simplification of the Kohn-Sham (KS) concept has been introduced, in which the ground-state density is obtained from a single KS equation for one spatial (spinless) orbital. The time-dependent (TD) form of this radical Kohn-Sham (rKS) scheme, which has the same validity status as the ordinary KS version, has proved to be a valuable tool for analysis. The rKS concept is used to clarify also the alternative nonvariational formulation of TD KS theory. We argue that it is just a formal theory, allowing one to reproduce but not predict the time development of the exact density of the interacting NNN-electron system. Besides the issue of the formal exactness of TDDFT, it is shown that both the static and time-dependent KS linear response equations neglect the particle-particle (ppp-ppp) and hole-hole (hhh-hhh) matrix elements of the perturbing operator. For a local (multiplicative) operator this does not lead to a loss of information due to a remarkable general property of local operators. Accordingly, no logical inconsistency arises with respect to DFT, because DFT requires any external potential to be local. For a general nonlocal operator the error resulting from the neglected matrix elements is of second order in the electronic repulsion.
The premise of density-functional theory is that knowledge of the ground-state density uniquely determines the Hamiltonian, and thereby, via solution of the corresponding time-independent Schr\"odinger equation, all the properties of the system. The density therefore constitutes a basic variable of quantum mechanics. There are at present two paths from the density to the Hamiltonian: the Hohenberg and Kohn proof of the bijectivity between the external potential and the basic variable, and the Percus, Levy, and Lieb constrained-search proof. We argue the Hohenberg- and Kohn-type proof to be the more fundamental, and that this is the case in general when both external electrostatic and magnetostatic fields are present, and the basic variables are the ground-state density and physical current density.
Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schrödinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain Sturm-Liouville problem, which we discuss for different situations. Based on these considerations we then present a discussion of the famous Runge-Gross theorem which provides a density-potential mapping for time-analytic potentials. Further we give conditions such that the general fixed-point approach is well-defined and converges under certain assumptions. Then the application of such a fixed-point procedure to lattice Hamiltonians is discussed and the numerical realization of the density-potential mapping is shown. We conclude by presenting an extension of the density-potential mapping to include vector-potentials and photons.