of a system. Such calculations require, in addition to the exchange-correlation potential, the exchange-correlation kernel – the functional derivative of the exchange-correlation potential with respect to the density. Formalism Runge–Gross theorem The approach of Runge and Gross considers a single-component system in the presence of a time-dependent scalar field for which the Hamiltonian takes the form where T is the kinetic energy operator, W the electron-electron interaction, and Vext(t) the external potential which along with the number of electrons defines the system. Nominally, the external potential contains the electrons' interaction with the nuclei of the system. For non-trivial time-dependence, an additional explicitly time-dependent potential is present which can arise, for example, from a time-dependent electric or magnetic field. The many-body wavefunction evolves according to the time-dependent Schrödinger equation under a single initial condition, Employing the Schrödinger equation as its starting point, the Runge–Gross theorem shows that at any time, the density uniquely determines the external potential. This is done in two steps: Assuming that the external potential can be expanded in a Taylor series about a given time, it is shown that two external potentials differing by more than an additive constant generate different current densities. Employing the continuity equation, it is then shown that for finite systems, different current densities correspond to different electron densities. Time-dependent Kohn–Sham system For a given interaction potential, the RG theorem shows that the external potential uniquely determines the density. The Kohn–Sham approaches chooses a non-interacting system (that for which the interaction potential is zero) in which to form the density that is equal to the interacting system. The advantage of doing so lies in the ease in which non-interacting systems can be solved – the wave function of a non-interacting system can be represented as a Slater determinant of single-particle orbitals, each of which are determined by a single partial differential equation in three variable – and that the kinetic energy of a non-interacting system can be expressed exactly in terms of those orbitals. The problem is thus to determine a potential, denoted as vs(r,t) or vKS(r,t), that determines a non-interacting Hamiltonian, Hs, which in turn determines a determinantal wave function which is constructed in terms of a set of N orbitals which obey the equation, and generate a time-dependent density such that ρs is equal to the density of the interacting system at all times: Note that in the expression of density