Density-response function

In the previous section we obtained an expression for the functional derivative of an arbitrary expectation value. We now make a special choice for the operator O and we choose O = h(r ). In that case we obtain from equation (90)  

If we change the potential by a constant then the density does not change. We therefore must have 

Now equation (98) implies that Av(r) = 0 which together with equation (99) implies that 8v(r) = C. We have therefore shown that only constant potentials yield a zero-density variation, and therefore the density response function is invertible up to a constant. One should, however, be careful with what one means with the inverse response function. The response function defines a mapping V - v4 from the set of potential variations from a nondegenerate ground state, which we call 8V and is a subset of L3/2 + L°°, to the set of first order densities variations, which we call 

that are produced by it. We have just shown that the inverse x 8,4 8V is well-defined modulo a constant function. However, there are density variations that can never be produced by a potential variation and which are therefore not in the set 

An example of such a density variation is one which is identically zero on some finite volume. 

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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] ... 

From this equation it follows that dg,A Pa is diagonal in the spin indices. We will therefore in the following put density variation 5p (r) determines the potential variation 5vs,(r) only up to a constant (see also [66] ). To find an explicit expression for the above functional derivative we must find an expression for the inverse density response function i A. In order to do this we make the following approximation to the Greens function (see Sharp and Horton [39], Krieger et al. [21]) ... 

The last term on the right-hand side is readily identified with the inverse Xs1 (r>r ) °f the density response function of a system of non-interacting particles... 

Adiabatic-connection fluctuation-dissipation theorem allows one to express the exchange-correlation energy-functional by means of imaginary-frequency density response function ( A) of the system with the scaled Coulomb potential (A/ r — r )11,13 ... 

To arrive at Eq. (180) we have used the definitions (145), (148), (171) and (175) of the density response functions. Furthermore, we have abbreviated the kernel of the (instantaneous) Coulomb interaction by w(x, x ) = 3(t — t )/ r — r. Finally, by inserting Eq. (180) into (168) one arrives at the time-dependent Kohn-Sham equations for the second-order density response ... 

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 Peierls transition is described in terms of the polarization function (or the density-response function ) xikl)- For the potential V(r) = cos qr a metallic... 

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. 

Fourier transforms of the bare Coulomb interaction and the density-response function, respectively. 

For a periodic crystal, we introduce the following Fourier expansion of the linear density-response function... 

To compute the interacting RPA density-response function of equation (32), we follow the method described in Ref. [66]. We first assume that n(z) vanishes at a distance Zq from either jellium edge [67], and expand the wave functions (<) in a Fourier sine series. We then introduce a double-cosine Fourier representation for the density-response function, and find explicit expressions for the stopping power of equation (36) in terms of the Fourier coefficients of the density-response function [57]. We take the wave functions <)),(7) to be the eigenfunctions of a one-dimensional local-density approximation (LDA) Hamiltonian with use of the Perdew-Zunger parametrization [68] of the Quantum Monte Carlo xc energy of a uniform FEG [69]. 

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]... 

In terms of the set of KS orbitals / (r), 4>a(r) (we use indices i and j for occupied orbitals (occupation/ = 1) and indices a and b for virtual orbitals (occupation f = 0), respectively), the matrix elements for the single particle density response function induced by a perturbation with frequency co become... 

This is true for an arbitrary nonconstant potential variation. We therefore see that the eigenvalues of when we regard x as an integral operator, are negative. Moreover we see that the only potential variation that yields a zero density variation is given by 8v = C where C is a constant. This implies that is invertible. We will go more closely into this matter in a later section where we will prove the same using the explicit form of the density response function. 

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