Introduction to Real Time propagation in Yambo

Time Dependent Effective Schrödinger Equation

The Yambo code implements a Time Dependent Effective Schrödinger Equation(TD-ESE), where the coupling between electrons and the external field is described by means of the Modern Theory of Polarization:

Where [math]\displaystyle{ | v_{i \mathbf{k}} \rangle }[/math] are the time-dependent valence bands, Hsysk is the Hamiltonian, [math]\displaystyle{ \mathcal E(t) }[/math] is the external field and the term [math]\displaystyle{ i e \partial k }[/math] corresponds to the dipole operator in periodic systems. For more details on this last term see Ref. and . Differently from other codes, the equations of motion of Yambo are in the length gauge and not in the velocity one. If you want to know more about the advantages and disadvantages of the two gauges read section 2.7 of Ref. . The Hamiltonian and the initial wave-functions are obtained from DFT. Then in order to obtain linear response, we probe the system with a delta function field, that excites all the frequencies at the same footing, and Fourier transforms. We calculate the Berry polarization

and from its Fourier transform we get the linear response. Indeed

[math]\displaystyle{ \chi(\omega) = \frac{P(\omega)}{E(\omega)} }[/math]

that is related to the dielectric constant through the relation [math]\displaystyle{ \epsilon(\omega) = 1 + 4 \pi \chi(\omega) }[/math]. See also Ref.