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.
[2] 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) At any given time, the N-electron wavefunction, which depends upon 3N spatial and N spin coordinates, determines the electronic density through integration as 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.
The proof relies heavily on the assumption that the external potential can be expanded in a Taylor series about the initial time.
The assumption that the two potentials differ by more than an additive spatially independent term, and are expandable in a Taylor series, means that there exists an integer k ≥ 0, such that is not constant in space.
The first extension of the RG theorem was to time-dependent ensembles, which employed the Liouville equation to relate the Hamiltonian and density matrix.
[5] A proof of the RG theorem for multicomponent systems—where more than one type of particle is treated within the full quantum theory—was introduced in 1986.