Exchange-correlation integral kernel

The derivative of the exchange-correlation potential in terms of electron density, /xc, is called the exchange-correlation integral kernel. Define the response function of the electron density, /ks, for the infinitesimal change in the Kohn-Sham potential,... 

For the exchange-correlation integral kernel, /xc, the local form,... 

Fig. 7.5 Energy distributions of the Coulomb and exchange-correlation self-interactions through their integral kernels for (a) the HOMO of the hydrogen atom, (b) the components of (a), (c) the HOMO of the helium atom, and (d) the LUMO of the helium atom. See Tsuneda et al. (2010)...

With eqn (7) the time-dependent Kohn-Sham scheme is an exact many-body theory. But, as in the time-independent case, the exchange-correlation action functional is not known and has to be approximated. The most common approximation is the adiabatic local density approximation (ALDA). Here, the non-local (in time) exchange-correlation kernel, i.e., the action functional, is approximated by a time-independent kernel that is local in time. Thus, it is assumed that the variation of the total electron density in time is slow, and as a consequence it is possible to use a time-independent exchange-correlation potential from a ground-state calculation. Therefore, the functional is written as the integral over time of the exchange-... 

The exchange-correlation part of the action integral (also termed the XC kernel) includes all the quantum effects of the electron-electron time-dependent interaction and is the analogous of the exchange-correlation functional of the standard time-independent Kohn-Sham formalism. Using the action integral (Eq. (4.57)) into the Euler equation (Eq. (4.49)) gives... 

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