Linear Response TD-DFT

To obtain excitation energies and properties within the time-dependent Kohn-Sham framework, it is possible to propagate in time the time-dependent electron density, through the solution of Eq. (4.60), and then extract energies and oscillator strengths from a Fourier analysis of the results [98-102]. Alternatively, the excited-state properties can be determined through the linear response theory. This is an efficient approach which avoids the direct solution of the time-dependent Kohn-Sham equations and is often used in practical applications. 

As shown in Sec. 1.2.3, if we consider a small time-dependent external perturbation Vgxt(r, we can write the linear variation of the electron density as (in the Fourier domain)  

Equations (4.66) and (4.67) can be finally written in the more compact form  

Equation (4.73) has the same form as that of the random-phase approximation (or TD-HF) problem and its solutions were extensively studied in literature [103], 

This work was funded by the ERG Starting Grant FP7 Project DEDOM, Grant Agreement No. 207441. 

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