Kohn-Sham response function

One uses a simple CG model of the linear responses (n= 1) of a molecule in a uniform electric field E in order to illustrate the physical meaning of the screened electric field and of the bare and screened polarizabilities. The screened nonlocal CG polarizability is analogous to the exact screened Kohn-Sham response function x (Equation 24.74). Similarly, the bare CG polarizability can be deduced from the nonlocal polarizability kernel xi (Equation 24.4). In DFT, xi and Xs are related to each other through another potential response function (PRF) (Equation 24.36). The latter is represented by a dielectric matrix in the CG model. 

The Kohn-Sham response function Xs is readily expressed in terms of the static unperturbed Kohn-Sham orbitals... 

In this section we are going to develop a different approach to the calculation of excitation energies which is based on TDDFT [69, 84, 152]. Similar ideas were recently proposed by Casida [223] on the basis of the one-particle density matrix. To extract excitation energies from TDDFT we exploit the fact that the frequency-dependent linear density response of a finite system has discrete poles at the excitation energies of the unperturbed system. The idea is to use the formally exact representation (156) of the linear density response n j (r, cu), to calculate the shift of the Kohn-Sham orbital energy differences coj (which are the poles of the Kohn-Sham response function) towards the true excitation energies Sl in a systematic fashion. 

Fig. 4.2 Excitation spectra of the ethane molecule, which were calculated directly from the Kohn-Sham response functions, /ks> of LDA and PBE exchange-correlation functionals, compared to the experimental one (Exp.) (Marques et al. 2001)...

The first factor is evaluated directly from the explicit functional form, the second follows from the linear response limit of the Kohn-Sham equations as does the last one (the inverse Kohn-Sham response function). 

Here Xs is the Kohn-Sham response function, constructed from KS energies and orbitals ... 

We call the Fukui function / (r) the HOMO response. Equation 24.39 is demonstrated as follows. The PhomoW is the so-called Kohn-Sham Fukui function denoted as f (r) [32]. According to the first-order perturbation theory, one has... 

The frontier orbitals responses (or bare Fukui functions) f (r) and the Kohn-Sham Fukui functions (or screened Fukui functions)/, (r) are related by Dyson equations obtained by using the PRF and its inverse [32]. Indeed, by using Equation 24.57 and the chain rule for functional derivatives in Equation 24.36, one obtains... 

Abstract We review Kohn-Sham density-functional theory for time-dependent response func-... 

Jonsson et al. review tlie Kohn-Sham density functional theory (DFT) for time-dependent (TD) response functions. They describe the derivation of the working expressions. They also review recent progress in the application of TD-DFT to open shell systems. They reported results on several properties (i) hyperpolarizabilities (e.g. para-nitroaniline, benzene, Cgg fullerene), (ii) excited state polarizabilities (e.g. pyrimidine), (iii) three-photon absorption and (iv) EPR spin Hamiltonian parameters. 

Let be the ground-state Kohn-Sham wave-functions for the system under study. We prepare the initial state for the time propagation by exciting the electrons with the electric field v r,t) = —kox S t), where = x,y,z. The amplitude ko must be small in order to keep the response of the system linear and dipolar. Through this prescription all frequencies of the system are excited with equal weight. At t = 0+ the initial state for the time evolution reads... 

The ESR hyperfine coupling is determined by triplet perturbations. Thus, in principle one should use an unrestricted wave function to describe the reference state. However, it is also possible to use a spin-restricted wave function (Fernandez et al. 1992) and take into account the triplet nature of the perturbation in the definition of the response. Within such a (e.g., SCF or MCSCF) restricted-unrestricted approach, first-order properties are given as the sum of the usual expectation value term and a response correction that takes into account the change of the wave function induced by the perturbation (of the type (0 H° 0)). This restricted-unrestricted approach has also been extended to restricted Kohn-Sham density functional theory (Rinkevicius et al. 2004). 

The calculation of the induced electron density may be done in the context of the Kohn-Sham approach to density functional theory, because the response of a KS system to a change in the one particle effective potential (r) corresponds to that of a system of non-interacting electrons. 

Here, the frequency-dependent response functions y(r, r oj) and y0(r, oj) correspond, respectively, to the actual interacting system and the equivalent Kohn-Sham noninteracting system. Using the expression of the effective potential, one can write... 

Thus, the response kernel for the interacting system can be obtained from that of the noninteracting system if one has a suitable functional form for the XC energy density functional for TD systems. The standard form for the kernel yo(r, r" Kohn Sham orbitals (/ (r), their energy eigenvalues sk, and the occupation numbers nk, is given [17,19] by... 

Fukui functions and other response properties can also be derived from the one-electron Kohn-Sham orbitals of the unperturbed system [14]. Following Equation 12.9, Fukui functions can be connected and estimated within the molecular orbital picture as well. Under frozen orbital approximation (FOA of Fukui) and neglecting the second-order variations in the electron density, the Fukui function can be approximated as follows [15] ... 

It is worth noting that screened response x/C r ) can be computed from the Kohn-Sham orbital wave functions and energies using standard first-order perturbation theory [3]... 

Similar relations can be obtained for the nonlinear/ functions. Kohn-Sham orbital formulations of these nonlinear responses can be constructed along the lines described previously [32] and will be presented elsewhere. 

The paper is organized as follows. In Section 2, derivation of the the SRPA formalism is done. Relations of SRPA with other alternative approaches are commented. In Sec. 3, the method to calculate SRPA strength function (counterpart of the linear response theory) is outlined. In Section 4, the particular SRPA versions for the electronic Kohn-Sham and nuclear Skyrme functionals are specified and the origin and role of time-odd currents in functionals are scrutinized. In Sec. 5, the practical SRPA realization is discussed. Some examples demonstrating accuracy of the method in atomic clusters and nuclei are presented. The summary is done in Sec. 6. In Appendix A, densities and currents for Skyrme functional are listed. In Appendix B, the optimal ways to calculate SRPA basic values are discussed. 

The time-dependent density functional theory [38] for electronic systems is usually implemented at adiabatic local density approximation (ALDA) when density and single-particle potential are supposed to vary slowly both in time and space. Last years, the current-dependent Kohn-Sham functionals with a current density as a basic variable were introduced to treat the collective motion beyond ALDA (see e.g. [13]). These functionals are robust for a time-dependent linear response problem where the ordinary density functionals become strongly nonlocal. The theory is reformulated in terms of a vector potential for exchange and correlations, depending on the induced current density. So, T-odd variables appear in electronic functionals as well. 

The kernel K(r,r ) is the transpose of Jf(r,r ), the Kohn-Sham potential-response function (KSPRF),... 

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