Electronic density response function

Van der Waals (dispersion) functionals have been developed to reduce the enormous computational time required in the AC/FDT method while maintaining accuracy and ease of use, as in the London classical potential. Lundqvist and coworkers proposed a dispersion functional, called the Andersson-Langreth-Lundqvist (ALL) functional by using a local density approximation for the electron density response function of the AC/FDT method (Andersson et al. 1996),... 

Introduction of the static density response function for a system with a constant number of electrons yields the RF - DFT model. This second approach is expected to be more useful in the analysis of chemical reactivity in condensed phases. 

Local density functional theory may be introduced within the RF model of solvent effects thorugh the induced electron density. The basic quantity for such a development is the linear density response function [39] ... 

Here Xa(r, r, m) and Xt(r, f, are the exact density-density response functions (157) of each separate system in the absence of the other. Xa is defined by the linear density response nia(r) exp(wt) of the electrons in system a to an externally applied electron potential energy perturbation Ff (r)e ... 

The calculated quantities include charge density wave (CDW), spin density wave, antiferro-orbital ordering, and electron-pairing response functions. The corresponding operators are the density operator A , the spin density operator the antiferro-orbital operator A°, and the singlet pairing operator , all of which are defined in terms of the original orbitals of e(= and 9 = d 2, .2) as... 

The same procedure can be applied for derivatives from other representations. Electronic properties obtained by differentiation are usually classified by its dependence on the position. Global properties have the same value everywhere, such as the chemical potential, hardness and softness. Electron density, Fukui function and local softness change throughout the molecule, and they are called local properties. Finally, kernel properties depend on two or more position vectors, like the density response and softness kernels. Global parameters describe molecular reactivity, local properties provide information on site selectivity, while kernels can be used to understand site activation. 

We compute the one-electron eigenfunctions involved in the evaluation of the independent particle density response function [Pg.230]

The many-body ground and excited states of a many-electron system are unknown hence, the exact linear and quadratic density-response functions are difficult to calculate. In the framework of time-dependent density functional theory (TDDFT) [46], the exact density-response functions are obtained from the knowledge of their noninteracting counterparts and the exchange-correlation (xc) kernel /xcCf, which equals the second functional derivative of the unknown xc energy functional ExcL i]- In the so-called time-dependent Hartree approximation or RPA, the xc kernel is simply taken to be zero. 

A second and more widely used approach for the computation of excitation energies within DFT is based on the linear-response formulation of the time-dependent perturbation of the electronic density. The basic quantity in linear response TDDFT (LR-TDDFT) is the time-dependent density-density response function [33]... 

Fig. 9.10 The density-response function x-x q) describes the redistribution Sn q) of the electron density by a periodic modulation Vq q) of the potential as a function of the wavevector q (cf. also Fig. 9.8). In a 1-d system, x has a singularity atq = 2kf. This singularity does not exist in 2-d- and 3-d systems. From [9] and [M4], Chap. 2.

The origin of the distortion is the reaction (the response ) of the conduction electrons in the 1-d metal to a periodic modulation of the periodic potential. The amplitude n of the electron density in the 1-d metal exhibits an increasing and divergent component when the wavevector of the potential Vq(q) of the periodic modulation of the lattice potential (which is due to the phonons), i.e. the wavevector q of the periodic perturbation, has the value q = 2kp. Figure 9.10 shows the so-called polarisation function or density-response function x( ). It describes the redistribution 8n(q) of the electron density n(q) in the presence of this periodic potential Vq(q) ... 

The divergence of the density-response function x q) occurs only in a 1-d metal [M4, M2, Chap. 17]. It gives rise to the collective state of the electrons (the charge-density wave) and the static lattice distortion with the same period q = 2kp, as well as the opening of an energy gap at the Fermi energy Ep (Fig. 9.8c). 

Which fundamental properties X could we be interested in Realizing that the electron density distribution function contains all the information about the system in the ground state (Hohenberg and Kohn theorems), its response to several perturbations is certainly of fundamental importance. Other properties also provide valuable information, such as the energy and the electronic chemical potential of the system. We will consider all of these and try to find analytical expressions for their response to, or resistance against, changes in N or v(r). 

The present authors have used the dielectric screening method for all their ab-initio computations of the macroscopic dielectric constant and phonon frequencies of Si and Ge. For the calculation of the electron energies and wave functions needed in the expression of the electron density response matrix they apply the local density approximation in the Hamiltonian. The advantage and shortcomings of this approximation are treated at length in the papers by J.T. Devreese, R. Martin, K. Kune, S. Louie, A. Baldereschi and R. Resta in these proceedings. 

In other words, the density response function of the interacting uniform electron gas is... 

The response function that enters (4.62), xkSj is the density-density response function of a system of non-interacting electrons and is, consequently, much easier to calculate than the full interacting x- In terms of the unperturbed stationary Kohn-Sham orbitals it reads... 

Here, xa (i ) is the causal density response function at imaginary frequencies of a system in which the electrons interact through a modified Coulomb potential w (r) = A/r, and whose ground state density is equal to the actual one. Xa (i ) is related to the polarisation function P (ia ) through the equality ... 

Frequency-dependent response properties from one-electron density matrix functionals have been reviewed by Pernal and Cioslowski, stressing on the point that the accuracy of the computed data is limited by both errors... 

A final set of reactivity indices constitute the so-called nuclear reactivity indices. As can be seen from eqn (26), the Fukui function measures the electron density response due to the change of the number of electrons of the system. However, despite the fact that the electron density determines all ground state properties of an atomic or a molecular system, the response of the nuelei due to this perturbation remains unknown a reponse kernel is needed to translate electron density changes in external potential changes. Cohen et at. circumvented this problem by introducing a nuclear Fukui function, which they defined as the Hellmann-Feynman force F acting on the nuclei due to a perturbation in the number of electrons at a constant external potential ... 

As the nuclear charge increases for a same number of electrons, the system become harder and more polarizable. A many particle system is completely characterized by total number of electrons (AO and the chemical potential v(r) while % and r] describe the response of the system when N changes at fixed v(r) the linear density response function R r, r ) depicts the same for the variation of v(r) for constant N [76] ... 

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