# The frequency-dependent xc kernel

# The frequency-dependent xc kernel

The key quantity in the formulation of linear-response theory within time-dependent density-functional theory is the so-called exchange-correlation (xc) kernel. This chapter discusses the properties of the xc kernel. In general, it is a complex and frequency-dependent object, and a functional of the ground-state density. Several exact properties such as the zero-force theorem, static and high-frequency limit, and the long-range behaviour are discussed. The xc kernel can be related to a time-dependent variational principle provided one pays attention to certain time-dependent boundary conditions to ensure causality. The adiabatic approximation of the xc kernel is discussed. Finally, the third section of the chapter reviews the xc kernel of the homogeneous electron liquid. Many of its exact properties are covered, such as the high- and low-frequency limit and the long-wavevector limit. Three widely used parametrizations are presented and the analytical behaviour in the complex frequency plane is investigated.

*Keywords:*
exchange-correlation kernel, zero-force theorem, boundary conditions, adiabatic approximation, homogeneous electron liquid, long-wavevector limit, parametrizations, complex frequency plane

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