Static density response function

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. 

Moreover, Berkowitz and Parr [39] have shown that the static density response may be expressed in terms of the softness kernel s(f,r/), the global softness S and the Fukui functions /(r) as follows ... 

The first term is the expectation value and from the fact that the density change is the static linear response function... 

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). 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

Owing to the static HK theorem, the initial potential Pq = ext[ o] is a functional of the unperturbed ground-state density Mq, so that the response function x, by Eq. (145), is a functional of Mq as well. 

The linear photoresponse of metal clusters was successfully calculated for spherical [158-160, 163] as well as for spheroidal clusters [164] within the jellium model [188] using the LDA. The results are improved considerably by the use of self-interaction corrected functionals. In the context of response calculations, self-interaction effects occur at three different levels First of all, the static KS orbitals, which enter the response function, have a self-interaction error if calculated within LDA. This is because the LDA xc potential of finite systems shows an exponential rather than the correct — 1/r behaviour in the asymptotic region. As a consequence, the valence electrons of finite systems are too weakly bound and the effective (ground-state) potential does not support high-lying unoccupied states. Apart from the response function Xs, the xc kernel /xc[ o] no matter which approximation is used for it, also has a self-interaction error. This is because /ic[no] is evaluated at the unperturbed ground-state density no(r), and this density exhibits self-interaction errors if the KS orbitals were calculated in LDA. Finally the ALDA form of /,c itself carries another self-interaction error. 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

Analytic response theory, which represents a particular formulation of time-dependent perturbation theory, has constituted a core technology in much of the this development. Response functions provide a universal representation of the response of a system to perturbations, and are applicable to all computational models, density-functional as well as wave-function models, and to all kinds of perturbations, dynamic as well as static, internal as well as external perturbations. The analytical character of the theory with properties evaluated from analytically derived expressions at finite frequencies, makes it applicable for a large range of experimental conditions. The theory is also model transferable in that, once the computational model has been defined, all properties are obtained on an equal footing, without further approximations. 

We therefore find that A < 0. However, we already know that A = 0 is only possible if/(r) is constant. We have therefore obtained the result that if a nonzero density variation is proportional to the potential that generates it, i.e., 8n = A 8v, then the constant of proportionality A is negative. This is exactly what one would expect on the basis of physical considerations. In actual calculations one indeed finds that the eigenvalues of x are negative. Moreover, it is found that there is a infinite number of negative eigenvalues arbitrarily close to zero, which causes considerable numerical difficulties when one tries to obtain the potential variation that is responsible for a given density variation. We finally note the invertability proof for the static response function can be extended to the time-dependent case. For a recent review we refer to Ref. [15]. 

Champagne et al.206 have studied solvent effects, through a continuum model, on the a and response functions of polyacetylene chains in the TDHF approximation. They find large increases in the values which they relate to the solvatochromic shifts in the lowest optically allowed transition. Density functional theory has also been assessed207 in connection with the calculation of the same response functions, but has been found to be inadequate due to the inability of the exchange/correlation potentials to satisfactorily represent the effects of the ends of the polymer. Schmidt and Springborg208 have calculated the static hyperpolarizability of polyacetylene and polycarbonitrile in DFT in the presence of external fields. 

This energy expression can be used to build up the respective variational functional to get the molecular orbitals [above]. A crucial step in the general self-consistent reaction field procedure is the estimation of the solvent charge density needed to obtain the response function G(r,r ) and the reaction potential. The use of Monte Carlo or molecular dynamics simulations of the system consisting the solute and surrounding solvent molecules has been proposed to find the respective solvent static and polarization densities. 

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