A Hohenberg-Kohn theorem for non-local potentials

International Journal of Quantum Chemistry, 2003

According to the Hohenberg-Kohn theorem, there is an invertible one-to-one relationship between the HamiltonianĤ 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 HamiltonianĤ(t). In the proof of the theorem, HamiltoniansĤ/Ĥ(t) that differ by an additive constant C/ function C(t) are considered equivalent. Since the constant C/ function C(t) is extrinsically additive, the physical system defined by these differing HamiltoniansĤ/Ĥ(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 H/Ĥ(t). Hohenberg-Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians {Ĥ}/{Ĥ(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)

Journal of Physics and Chemistry of Solids, 2012

The Hohenberg-Kohn theorem is generalized to the case of electrons in the presence of both an external electrostatic EðrÞ ¼ À=vðrÞ and magnetostatic BðrÞ ¼ = Â AðrÞ field. For the non-degenerate ground state, it is established that the basic variables are the ground state density rðrÞ and physical current density jðrÞ by proving the relationship between the densities frðrÞ,jðrÞg and the external potentials fvðrÞ,AðrÞg is one-to-one. A constrained-search proof is also provided. It is further explained why the basic variables cannot be the density rðrÞ and the paramagnetic current density j p ðrÞ as presently thought to be the case.

Quantal Density Functional Theory, 2016

The Hohenberg-Kohn theorems for a system of N electrons in an external electrostatic field are generalized to the added presence of a uniform magnetostatic field. The theorems are proved for Hamiltonians of both spinless electrons and electrons with spin. It is thereby shown that the basic variables in each case are the nondegenerate ground state density ρ(r) and physical current density j(r), i.e. knowledge of {ρ(r), j(r)} uniquely determines the external scalar v(r) and vector A(r) potentials to within a constant and the gradient of a scalar function, respectively. The proofs differ from the original HK proof because the relationship between the potentials {v(r), A(r)} and the nondegenerate ground state wave function is no longer one-to-one but many-to-one. Further, in addition to the constraint in the original HK proof of fixed electron number N, the constraint of fixed canonical orbital angular momentum L (for spin less electrons) and the added constraint of fixed spin angular momentum S (for electrons with spin) is required. The consequence of these proofs to the existing spin and current density functional theories is remarked upon.

1999

A generalised Hohenberg-Kohn theorem is described in terms of the sign of the secondorder energy variation. Independently, it is also corroborated within the perturbation theoretical framework. An alternative formulation of the Hohenberg-Kohn theorem, based on the relationships involving the matrix representations of density functions and the Hamiltonian operator variations, is shown to extend the validity of the theorem to the excited states of the Hamiltonian operators possessing non-degenerate spectra. Finally, a connection with Brillouin's theorem when energy variation becomes stationary is also outlined.

Physics Letters A, 2001

The time evolution of the mean values of the position and momentum operators for a Schrödinger particle in a one-dimensional box is reviewed. The connection of both (d/dt) X and (d/dt) P with local densities and Bohm's quantum potential is pointed out. New boundary non-local terms are obtained.

Physical Review A, 2016

External potentials play a crucial role in modelling quantum systems, since, for a given interparticle interaction, they define the system Hamiltonian. We use the metric space approach to quantum mechanics to derive, from the energy conservation law, two natural metrics for potentials. We show that these metrics are well defined for physical potentials, regardless of whether the system is in an eigenstate or if the potential is bounded. In addition, we discuss the gauge freedom of potentials and how to ensure that the metrics preserve physical relevance. Our metrics for potentials, together with the metrics for wavefunctions and densities from [I. D'Amico, J. P. Coe, V. V. França, and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011)] paves the way for a comprehensive study of the two fundamental theorems of Density Functional Theory. We explore these by analysing two manybody systems for which the related exact Kohn-Sham systems can be derived. First we consider the information provided by each of the metrics, and we find that the density metric performs best in distinguishing two many-body systems. Next we study for the systems at hand the one-to-one relationships among potentials, ground state wavefunctions, and ground state densities defined by the Hohenberg-Kohn theorem as relationships in metric spaces. We find that, in metric space, these relationships are monotonic and incorporate regions of linearity, at least for the systems considered. Finally, we use the metrics for wavefunctions and potentials in order to assess quantitatively how close the many-body and Kohn-Sham systems are: We show that, at least for the systems analysed, both metrics provide a consistent picture, and for large regions of the parameter space the error in approximating the many-body wavefunction with the Kohn-Sham wavefunction lies under a threshold of 10%.

Physical Review Letters, 2004

For a given excited state there exist densities that arise from more than one external potential. This is due to a qualitatively different energy-density relationship from that of the ground state and is related to positive eigenvalues in the nonlocal susceptibility for excited states. Resulting problems with the generalization of the density functional methodology to excited states are discussed.

The time evolution of the mean values of the position and momentum operators for a Schrödinger particle in a one-dimensional box is reviewed. The connection of both (d/dt) X and (d/dt) P with local densities and Bohm's quantum potential is pointed out. New boundary non-local terms are obtained.  (S. De Vincenzo), [email protected] (L. González-Díaz).

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1998

We present a family of birational transformations in CP 2 depending on two, or three, parameters which do not, generically, preserve meromorphic 2-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called the 'post-critical set', we get some new structures, some 'non-analytic' 2-form which reduce to meromorphic 2-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the degrees of the parameters, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in CP 2 is first carried out using the Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, once again, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic 2-form can be preserved for this mapping. These birational transformations in CP 2 , which, generically, do not preserve any meromorphic 2-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic 2-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more 'probabilistic' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic 2-form explains most of the (numerical) discrepancies between the topological and probabilistic approaches of dynamical systems.

Physical Review Letters, 1994

We consider interacting particles in an external harmonic potential. We extend the 8 = 0 case of the generalized Kohn theorem, giving a "harmonic-potential theorem" (HPT), demonstrating rigid, arbitrary-amplitude, time-oscillatory Schrodinger transport of a many-body eigenfunction. We show analytically that the time-dependent local-density approximation (TDLDA) satisfies the HPT exactly. Other approximations, such as linearized TDLDA with frequency-dependent exchange correlation kernel and certain inhomogeneous hydrodynamic formalisms, do not. A simple modification permits such explicitly frequency-dependent local theories to satisfy the HPT, however.