On the gradient expansion of the exchange energy within linear response theory and beyond

Abstract

-In the present work, we reexamined the gradient expansion of the exchange energy of an electron gas with a slowly varying charge density. We stay within the exchange-only approximation of Sharp, Horton, Talman, and Shadwick but go to second order in the deviation from the homogeneous limit. The coefficient of the lowest-order gradient correction is obtained analytically both for a bare and a screened Coulomb interaction-the former yielding the value previously obtained by Kleinman numerically and by Engel and Vosko analytically. A screened Coulomb interaction gives Sham's coefficient in the limit of infinite screening length. The cause of the difference between the coefficients of Kleinman and Sham is clearly exhibited. The coefficients of the two next highest-order gradient corrections, one of which originates in second-order response theory, is shown to diverge as the screening length becomes large. The bare Coulomb interaction gives finite coefficients of which the one originating from linear response is obtained analytically and differs from the presumably correct result obtained by Engel and Vosko. This discrepancy demonstrates the extreme sensitivity of the analytical expressions to different regularization procedures. We suggest that coefficients should rather be chosen according to the performance of the resulting gradient approximations in weakly perturbed electron gases.

Density gradient expansion of correlation functions

“…The problem does, for instance, not appear when Yukawa-screened Coulomb interactions are used. 38 The conclusion is that there is nothing wrong with the Kirzhnits method, or the nonlinear response theory derivation of the gradient expansion of the exchange hole presented here, but one has be aware that for Coulomb interactions the procedures of expanding correlation functions in terms of density gradients followed by integrations involving Coulomb potentials may not yield the same result as directly expanding the integrated quantity in terms of density gradients. For this reason the original GGA of Perdew and Wang 10 based on the gradient expansion of the exchange hole was later reparameterized 16,40 to accommodate the correct gradient coefficient for the exchange energy.…”

“…The coefficient B x is the same gradient coefficient as obtained by Gross and Dreizler 31 and earlier by Sham. 34 However, the correct analytic exchange gradient coefficient is known [35][36][37][38][39] to be a factor 10/7 larger. The reason for this discrepancy is clearly described by Svendsen and von Barth 38 who showed that the Coulomb interaction is too singular to allow for the interchange of the operations of integration and the expansion in wave vectors.…”

Straightforward gradient approximation for the exchange energy ofs−pbonded solids

“…By a simultaneous study of the slowly-varying limit and the limit of low-order response, both the forms and the coefficients in front of different gradient corrections can be determined. 6 We will now investigate the possibility of modeling the exact exchange functional by retaining these forms, while using the coefficients as adjustable parameters.…”

“…In previous work 6,7 we have argued that the electronic density of an atom is certainly not a linear perturbation of the homogeneous electron gas. We have also suggested that higher-degree gradients might be useful in attempts to model the true exchangecorrelation or exchange functionals.…”

A study of accurate exchange-correlation functionals through adiabatic connection

“…Accurate functionals were later constructed by parameterizing the Monte-Carlo calculations of Ceperley and Alder 16 and making sure that the high density limit of the correlation functional was reproduced. Further work on correlation functionals is that given by Langreth As is clear, the approach to developing functionals has been to start with the simplest functional given by the LDA, making gradient correction to it up the second order or the fourth order 21,22 and then demanding that the resulting functional satisfy exact conditions for them. The results have been that highly accurate energies can now be calculated using density functionals developed over the past two decades.…”