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arXiv:cond-mat/0209688 · International Journal of Quantum Chemistry

Corollary to the Hohenberg–Kohn theorem

Xiao‐Yin Pan, Viraht Sahni

Published Jan 1, 2003• 14 citations
arXivHTMLPDF

Abstract

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

DOI
10.1002/qua.10595

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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
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 The Hohenberg–Kohn (HK) theorem for interacting electrons is a cornerstone of modern electronic structure calculations. For a general quantum system, a HK‐type Hamiltonian in the form of can always be defined, by grouping those terms with fixed or preknown coefficients into an internal part of the Hamiltonian , and factorizing the remaining as the superposition of a set of Hermitian operators . It is asked whether the HK theorem can be extended to such a general setting, so that the ground‐state expectation values of as the generalized density can in principle be used as the fundamental variables determining all the properties of the system. It is shown that the question can be addressed by the invertibility of generalized density correlation matrix (GDCM) defined with respect to the operators. This criterion is applied to several representative examples, including the quantum Ising dimer, frustration‐free systems, N ‐level quantum systems and a fermionic Hubbard chain. It is suggested that for a finite‐size system, finding an invertible GDCM under one single configuration is typically sufficient to establish the generic extensibility of the HK theorem in the entire parameter space.
Abstract In this chapter, the Hohenberg-Kohn theorem of DFT is formulated and proved. This theorem says that there is a unique one-to-one correspondence between potentials and ground-state densities in electronic many-body systems. The original proof of Hohenberg and Kohn is presented and discussed, which proceeds in two steps: first, the mapping from potentials to wave functions is shown to be unique, followed by the mapping from wave functions to densities. Two corollaries of the theorem are then discussed: the variational principle, whereby the exact ground-state energy can be found by variation of the energy functional, and then the universality of parts of this energy functional. The so-called constrained search formalism is also briefly discussed.
The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for NNN-representable density kernels, that any finite non-negative differentiable function is an NNN-representable particle density.
The formalism of Kohn and Sham uses a specific (model) hamiltonian which highly simplifies the many-electron problem to that of noninteracting fermions. The theorem of Hohenberg and Kohn tells us that, for a given ground state density, this hamiltonian is unique. In principle, this density can be chosen as that of the real, interacting system. To obtain the energy, or other properties of the real system, approximations are needed. Working with non interacting fermions is an important simplification, but it may be easier to produce approximations with different choices of the model hamiltonian. The feature that the exact density is (ideally) reproduced can be kept in the newly defined fictitious systems. Using model hamiltonians having the same form as the physical one, that is, being built of one- and two-body operators, allows to approach the physical hamiltonian arbitrarily close, and thus a systematic reduction of the approximations.
A density-functional formalism comparable to the Hohenberg-Kohn-Sham theory of the ground state is developed for arbitrary time-dependent systems. It is proven that the single-particle potential $v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ leading to a given vvv-representable density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}t)$ is uniquely determined so that the corresponding map $v\ensuremath{\rightarrow}n$ is invertible. On the basis of this theorem, three schemes are derived to calculate the density: a set of hydrodynamical equations, a stationary action principle, and an effective single-particle Schr\"odinger equation.