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New perspectives on the fundamental theorem of density functional theory



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 equation image/equation image(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