Hohenberg-Kohn theorems in electrostatic and uniform magnetostatic fields

Abstract

The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby, a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle in terms of these densities is also developed. These theorems are further generalized to electrons with spin by imposing the added constraint of fixed canonical orbital and spin angular momenta. The proofs differ from the original HK proof and explicitly account for the many-to-one relationship between the potentials and the nondegenerate ground state wave function. A Percus-Levy-Lieb constrained-search proof expanding the domain of validity to N-representable functions, and to degenerate states, again for fixed electron number and angular momentum, is also provided.

Uniform magnetic fields in density-functional theory

“…and that j m;ψ k is not independent of k. The theorem above is the point of departure for the density-functional formalism proposed in Ref. 9. In this setting, the magnetic vector potentials are restricted to a three-dimensional family, whereas the physical current densities belong to an infinite-dimensional function space, hinting at considerable redundancy of the formalism.…”

“…To date, no physical LDFT has been proposed or analysed in the literature. A hybrid formalism, based on the triple (ρ, L, j), has been proposed [9]. We now compare this approach with LDFT formalism.…”

Electron Correlations in Local Effective Potential Theory

“…, x N ; x = rσ; (rσ) the spatial and spin coordinates of each electron. In recent work [12], we have proved that the basic variables for the physical system described above, in which the interaction of the magnetic field is solely with the orbital angular momentum, are the nondegenerate ground state density ρ(r) and the physical current density j(r). The proof is for uniform magnetic fields and for fixed electron number N and canonical angular momentum L. The proof is rigorous in the original HK sense in that knowledge of {ρ(r), j(r)} uniquely determines the potentials {v(r), A(r)} to within a constant and the gradient of a scalar function, respectively.…”

“…The mapping to the model system possessing the same basic variables {ρ(r), j(r)} ensure the constancy [12] of both the electron number N and orbital angular momentum L.…”

“…Finally, consider the case of N electrons in both an external electrostatic E (r) = −∇v(r) and magnetostatic B(r) = ∇ × A(r) field (This case, and the corresponding basic variables [12], is discussed in greater detail in the following section. Here, we focus on the correlations within the corresponding LEPT).…”