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( r t ) and the Hamiltonian $\hat{H}$ / $\hat{H}$ ( 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 ( R t ). 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( r t ), which is a gauge‐invariant property, and a gauge function α( R )/α( R t ). 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