In quantum mechanics, specifically time-dependent density functional theory, the Runge–Gross theorem (RG theorem) shows that for a many-body system evolving from a given initial wavefunction, there exists a one-to-one mapping between the potential (or potentials) in which the system evolves and the density (or densities) of the system. The potentials under which the theorem holds are defined up to an additive purely time-dependent function: such functions only change the phase of the wavefunction and leave the density invariant. Most often the RG theorem is applied to molecular systems where the electronic density, ρ(r,t) changes in response to an external scalar potential, v(r,t), such as a time-varying electric field. The Runge–Gross theorem provides the formal foundation of time-dependent density functional theory. It shows that the density can be used as the fundamental variable in describing quantum many-body systems in place of the wavefunction, and that all properties of the system are functionals of the density. The theorem was published by Erich Runge and Eberhard K. U. Gross in 1984. As of September 2021, the original paper has been cited over 5,700 times.
Overview The Runge–Gross theorem was originally derived for electrons moving in a scalar external field. Given such a field denoted by v and the number of electron, N, which together determine a Hamiltonian Hv, and an initial condition on the wavefunction Ψ(t = t0) = Ψ0, the evolution of the wavefunction is determined by the Schrödinger equation (written in atomic units)
H ^ v ( t ) | Ψ ( t ) ⟩ = i ∂ ∂ t | Ψ ( t ) ⟩ . {\displaystyle {\hat {H}}_{v}(t)|\Psi (t)\rangle =i{\frac {\partial }{\partial t}}|\Psi (t)\rangle .}
At any given time, the N-electron wavefunction, which depends upon 3N spatial and N spin coordinates, determines the electronic density through integration as
ρ ( r , t ) = N ∑ s 1 ⋯ ∑ s N ∫ d r 2 ⋯ ∫ d r N | Ψ ( r , s 1 , r 2 , s 2 , . . . , r N , s N , t ) | 2 . {\displaystyle \rho (\mathbf {r} ,t)=N\sum _{s_{1}}\cdots \sum _{s_{N}}\int \ \mathrm {d} \mathbf {r} _{2}\ \cdots \int \ \mathrm {d} \mathbf {r} _{N}\ |\Psi (\mathbf {r} ,s_{1},\mathbf {r} _{2},s_{2},...,\mathbf {r} _{N},s_{N},t)|^{2}.}
Two external potentials differing only by an additive time-dependent, spatially independent, function, c(t), give rise to wavefunctions differing only by a phase factor exp(-i α(t)), with dα(t)/dt = c(t), and therefore the same electronic density. These constructions provide a mapping from an external potential to the electronic density:
v ( r , t ) + c ( t ) → e − i c ( t ) | Ψ ( t ) ⟩ → ρ ( r , t ) . {\displaystyle v(\mathbf {r} ,t)+c(t)\rightarrow e^{-ic(t)}|\Psi (t)\rangle \rightarrow \rho (\mathbf {r} ,t).}
The Runge–Gross theorem shows that this mapping is invertible, modulo c(t). Equivalently, that the density is a functional of the external potential and of the initial wavefunction on the space of potentials differing by more than the addition of c(t):
ρ ( r , t ) = ρ [ v , Ψ 0 ] ( r , t ) ↔ v ( r , t ) = v [ ρ , Ψ 0 ] ( r , t ) {\displaystyle \rho (\mathbf {r} ,t)=\rho [v,\Psi _{0}](\mathbf {r} ,t)\leftrightarrow v(\mathbf {r} ,t)=v[\rho ,\Psi _{0}](\mathbf {r} ,t)}
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