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Time-Dependent Variational Principle in Density Functional Theory

Elsevier Science & Technology
DOI: 10.1016/s0065-3276(08)60462-1


Publisher Summary This chapter discusses mixed state time-dependent variational principle, time-independent density functional theory (DFT), and time-dependent DFT. It presents the formulation of exact equations for the evolution of electronic space-spin densities, which are equivalent to the Heisenberg equation of motion for the electrons in the system. Using the observation that densities can be expressed as quadratic expansions of functions, the chapter also formulates exact equations for these one-particle functions. Contemporary versions of DFT are presented in two separate conceptual and logical frameworks, one each for time-independent and time-dependent DFT. Even within the constrained search formulation of time-independent DFT, there are several rather deep and interconnected questions. Commonly these are discussed in terms of symmetry (and symmetry breaking), occupation number distributions, and functional differentiability. Their resolution is important for both fundamental reasons and to provide pathways to more powerful and reliable DFT approximations. Implicit in them is the issue of mixed states. Because customary formulations of time-dependent DFT are separate, it is not evident how resolution of those issues in the time-independent case would carry over. Further, the importance of parameter-space metrics in the electron nuclear dynamics work of Yngve Öhrn versus the absence of such metrics in conventional time-dependent DFT suggests strongly that a formulation directly from the time-dependent variational principle would be beneficial and clarifying, in that it would provide a rigorous basis for the use of dynamics in parameter space, including mixed states right from the start and would provide a significantly enhanced foundation for constructing approximations.

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