Electronic Response to Time-Dependent Perturbations
In Chap. 3 we discussed how to determine the ground state of a system of ions and electrons within the DFT-LDA. We have seen that this ground state corresponds to a minimum of the total energy written as a functional of the electronic density. If a weak e
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In Chap. 3 we discussed how to determine the ground state of a system of ions and electrons within the DFT-LDA. We have seen that this ground state corresponds to a minimum of the total energy written as a functional of the electronic density. If a weak external time-dependent perturbation is applied to the electronic system (for instance, a low intensity electromagnetic field of a sufficiently high frequency such that one can neglect the response of the ionic degrees of freedom), the electronic density will start oscillating around the minimum energy configuration. For such weak external fields, one can, to a very good approximation, retain only the linear part of the response. In this approximation, the frequency of the induced oscillations will be determined by the curvature of the energy surface around its minimum. As the density oscillates, the effective potential changes and, in turn, induces a change in the density itself. Therefore, to describe these electron 'vibrations' we must require self-consistency between the variations of the density and those of the effective potential. In this way, a time dependent theory can be obtained starting, e.g., from Hartree, Hartree-Fock or LDA energy functional within the linear response approximation. However, the first two approaches are intrinsically based on a mean-field approximation and neglect correlations beyond those arising from the Pauli principle. On the other hand, the time-dependent extension of the LDA (TDLDA), could also be derived, making again a local approximation, from an exact theorem which is the time-dependent counterpart of the HK one (see e.g. [61]) and which leads to the so-called time-dependent DFT (TDDFT). In the following, we derive explicitly the TDLDA equations from the LDA Hamiltonian without going through the TDDFT formalism. The related framework known as the Random Phase Approximation (RPA)(see e.g. [10]) will be also mentioned.
4.1 Linear Response: RPA and TDLDA Let us denote by H 0 an unperturbed, time-independent self-consistent-field Hamiltonian of the form 1i2 Ho = - 2 --V' 2 + Vetr[Qo], (4.1) me R. A. Broglia et al., Solid State Physics of Finite Systems © Springer-Verlag Berlin Heidelberg 2004
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4. Electronic Response
where the effective potential Veff[Qo] includes the Hartree, ionic, and, in the case of the Kohn-Sham equations of DFT, also an exchange-correlation potential (see e.g. (3.26)). H 0 is evaluated at the self-consistent density of the ground state, Qo. The associated set of time-dependent Schrodinger equations is (4.2) whose solutions 'Ph(t) = exp( -iEht/n)cph(O) are the (stationary) one-particle states of the system, where 'Ph(O) is the solution of (3.25). The label h refers here to occupied states. As a result of a perturbation, electrons can be promoted from occupied states h, which become 'hole' states, to empty (or unoccupied) states p, which we denote as 'particle' states. We specify the perturbation which is added to the Hamiltonian (4.1) by writing it as, Vpert(r, t)
= Vpert(r) exp( -iwt) + h.c.
(4.3)
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