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From densities to Hamiltonians: the main idea of DFT

Carsten A. Ullrich, Adam Wasserman

Published Aug 20, 2026• 0 citations
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Abstract

Abstract The usual way to proceed in quantum mechanics is to solve Schrödinger’s equation and obtain all observables of interest from the resulting eigenstates. Schrödinger’s equation depends on a given potential, which establishes a map from potentials to observables, one of them being the probability density. This chapter introduces the main idea of DFT, namely, that this map can be inverted. An example, for a one-electron system, shows how it is possible to construct, from a given density, that potential which produces that density as its ground-state density. This then establishes a formal map from ground-state densities to Hamiltonians and, hence, to observables. It is then discussed what these findings imply, and how they need to be generalized to establish the DFT formal framework.

DOI
10.1093/oso/9780198910664.003.0003

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Abstract The usual way to proceed in quantum mechanics is to solve Schrödinger’s equation and obtain all observables of interest from the resulting eigenstates. Schrödinger’s equation depends on a given potential, which establishes a map from potentials to observables, one of them being the probability density. This chapter introduces the main idea of DFT, namely, that this map can be inverted. An example, for a one-electron system, shows how it is possible to construct, from a given density, that potential which produces that density as its ground-state density. This then establishes a formal map from ground-state densities to Hamiltonians and, hence, to observables. It is then discussed what these findings imply, and how they need to be generalized to establish the DFT formal framework.
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.
This work studies in detail the possibility of defining a one-to-one mapping from charge densities as obtained by the time-dependent Schrödinger equation to external potentials. Such a mapping is provided by the Runge-Gross theorem and lies at the very core of time-dependent density functional theory. After introducing the necessary mathematical concepts, the usual mapping "there" - from potentials to wave functions as solutions to the Schrödinger equation - is revisited paying special attention to Sobolev regularity. This is scrutinised further when the question of functional differentiability of the solution with respect to the potential arises, a concept related to linear response theory. Finally, after a brief introduction to general density functional theory, the mapping "back again" - from densities to potentials thereby inverting the Schrödinger equation for a fixed initial state - is defined. Apart from utilising the original Runge-Gross proof this is achieved through a fixed-point procedure. Both approaches give rise to mathematical issues, previously unresolved, which however could be dealt with to some extent within the framework at hand.
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.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.
The advent of the Hohenberg-Kohn theorem in 1964, its extension to finite- T , Kohn-Sham theory, and relativistic extensions provide the well-established formalism of density-functional theory (DFT). This theory enables the calculation of all static properties of quantum systems without the need for an n -body wavefunction ψ. DFT uses the one-body density distribution instead of ψ. The more recent time-dependent formulations of DFT attempt to describe the time evolution of quantum systems without using the time-dependent wavefunction. Although DFT has become the standard tool of condensed-matter computational quantum mechanics, its foundational implications have remained largely unexplored. While all systems require quantum mechanics (QM) at T =0, the pair-distribution functions (PDFs) of such quantum systems have been accurately mapped into classical models at effective finite- T , and using suitable non-local quantum potentials (e.g., to mimic Pauli exclusion effects). These approaches shed light on the quantum → hybrid → classical models, and provide a new way of looking at the existence of non- local correlations without appealing to Bell's theorem. They also provide insights regarding Bohmian mechanics. Furthermore, macroscopic systems even at 1 Kelvin have de Broglie wavelengths in the micro-femtometer range, thereby eliminating macroscopic cat states, and avoiding the need for ad hoc decoherence models.