DFT addresses systems under external conditions that are constant in time, essentially focusing on the ground state of the system (although extensions to excited states exist).

Naturally, one raises the question: what about a system exposed to a time-varying external potential, \(v_{\mathrm{ext}}(\vec r,t)\), for example, a molecule irradiated by a laser pulse? Again, as in ground-state DFT, the formal framework for the problem is the time-dependent Schrödinger equation, and again, the enormous difficulty of directly solving a system with more than a few electrons arises. Clearly, performing the time evolution of a many-electron system does not make the task easier.

In 1984 Runge and Gross have shown the one-to-one correspondence between the time-dependent potential and the time-dependent electron density, establishing time-dependent density functional theory  (TDDFT). 

Given the initial state, the time-dependent density \(n(\vec r,t)\) entirely describes the system at any further moment in time.  Of course, as in ground-state DFT, development of approximations for exchange and correlation is required. In the TD-context, this task is even more difficult: for example, the xc potential for some point \(\vec r\) and time \(t\) has to take into account not only the density at that very time, but also at all previous times (memory effect).

In addition to full time propagation, the establishment of TDDFT allows to examine the reaction of a system to a small external perturbation, namely, how the density at point \(\vec r\) and time \(t\)  changes due to a small variation of the external potential at point \(\vec r'\) and time \(t'\). Such an investigation bears the name of linear response. Within the linear response regime, one aims at finding e.g. the absoption spectrum of a material, evaluate the fundamental and optical gaps, describe excitons and more. 

References

1. E. Runge and E. K. U. Gross, Density-functional theory for time-dependent systems, Phys. Rev. Lett. 52, 997 (1984)

2. M. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E. K. U. Gross, eds., Time-Dependent Density Functional Theory, Vol. 706, Lectures in Physics (Springer, 2006)

3. C. A. Ullrich, Time-Dependent Density-Functional Theory: Concepts and Applications (Oxford University Press, 2012)

4. M. A. L. Marques, N. T. Maitra, F. M. S. Nogueira, E. K. U. Gross, and A. Rubio, eds., Fundamentals of Time-Dependent Density Functional Theory (Springer, 2012).

5. N. T. Maitra, Perspective: fundamental aspects of time-dependent density functional theory, J. Chem. Phys. 144, 220901 (2016)

6. S. Sharma, J. K. Dewhurst, and E. K. U. Gross, Optical Response of Extended Systems Using Time-Dependent Density Functional Theory, Top. Curr. Chem. 347, 235–258 (2014)

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