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Concept# Time-dependent density functional theory

Summary

Time-dependent density-functional theory (TDDFT) is a quantum mechanical theory used in physics and chemistry to investigate the properties and dynamics of many-body systems in the presence of time-dependent potentials, such as electric or magnetic fields. The effect of such fields on molecules and solids can be studied with TDDFT to extract features like excitation energies, frequency-dependent response properties, and photoabsorption spectra.
TDDFT is an extension of density-functional theory (DFT), and the conceptual and computational foundations are analogous – to show that the (time-dependent) wave function is equivalent to the (time-dependent) electronic density, and then to derive the effective potential of a fictitious non-interacting system which returns the same density as any given interacting system. The issue of constructing such a system is more complex for TDDFT, most notably because the time-dependent effective potential at any given instant depends on the value of the density at all previous times. Consequently, the development of time-dependent approximations for the implementation of TDDFT is behind that of DFT, with applications routinely ignoring this memory requirement.
The formal foundation of TDDFT is the Runge–Gross (RG) theorem (1984) – the time-dependent analogue of the Hohenberg–Kohn (HK) theorem (1964). The RG theorem shows that, for a given initial wavefunction, there is a unique mapping between the time-dependent external potential of a system and its time-dependent density. This implies that the many-body wavefunction, depending upon 3N variables, is equivalent to the density, which depends upon only 3, and that all properties of a system can thus be determined from knowledge of the density alone. Unlike in DFT, there is no general minimization principle in time-dependent quantum mechanics. Consequently, the proof of the RG theorem is more involved than the HK theorem.
Given the RG theorem, the next step in developing a computationally useful method is to determine the fictitious non-interacting system which has the same density as the physical (interacting) system of interest.

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Related courses (1)

CH-452: Computational methods in molecular quantum mechanics

This course will discuss the main methods for the simulation of quantum time dependent properties for molecular systems. Basic notions of density functional theory will be covered. An introduction to