above, the summation is over all Kohn–Sham orbitals and is the time-dependent occupation number for orbital . If the potential vs(r,t) can be determined, or at the least well-approximated, then the original Schrödinger equation, a single partial differential equation in 3N variables, has been replaced by N differential equations in 3 dimensions, each differing only in the initial condition. The problem of determining approximations to the Kohn–Sham potential is challenging. Analogously to DFT, the time-dependent KS potential is decomposed to extract the external potential of the system and the time-dependent Coulomb interaction, vJ. The remaining component is the exchange-correlation potential: In their seminal paper, Runge and Gross approached the definition of the KS potential through an action-based argument starting from the Dirac action Treated as a functional of the wave function, A[Ψ], variations of the wave function yield the many-body Schrödinger equation as the stationary point. Given the unique mapping between densities and wave function, Runge and Gross then treated the Dirac action as a density functional, and derived a formal expression for the exchange-correlation component of the action, which determines the exchange-correlation potential by functional differentiation. Later it was observed that an approach based on the Dirac action yields paradoxical conclusions when considering the causality of the response functions it generates. The density response function, the functional derivative of the density with respect to the external potential, should be causal: a change in the potential at a given time can not affect the density at earlier times. The response functions from the Dirac action however are symmetric in time so lack the required causal structure. An approach which does not suffer from this issue was later introduced through an action based on the Keldysh formalism of complex-time path integration. An alternative resolution of the causality paradox through a refinement of the action principle in real time has been recently proposed by Vignale. Linear response TDDFT Linear-response TDDFT can be used if the external perturbation is small in the sense that it does not completely destroy the ground-state structure of the system. In this case one can analyze the linear response of the system. This is a great advantage as, to first order, the variation of the system will depend only on the ground-state wave-function so that we can simply use all the properties of DFT. Consider a small time-dependent external perturbation . This gives and looking at the linear