Solving the Ultranonlocality Problem in Time-Dependent Spin-Density-Functional Theory

Abstract

It has been known for some time that the exchange-correlation potential in time-dependent densityfunctional theory is an intrinsically nonlocal functional of the density as soon as one goes beyond the adiabatic approximation. In this paper we show that a much more severe nonlocality problem, with a completely different physical origin, plagues the exchange-correlation potentials in time-dependent spin-density functional theory. We show how the use of the spin current density as a basic variable solves this problem, and we provide an explicit local expression for the exchange-correlation fields as functionals of the spin currents. DOI: 10.1103/PhysRevLett.90.066402 PACS numbers: 71.15.Mb, 71.10.Ca, 71.45.Gm For many years the local density approximation (LDA) has provided the much needed handle on the difficult problem of approximating the density dependence of the exchange-correlation (xc) potential-the single particle potential that incorporates the many-body effects in the Kohn-Sham equation for the ground state density [1]. In LDA, the xc potential V xc r r is simply a function of the local density nr r . This approximation is not unreasonable as long as the functional derivative of V xc r r with respect to nr r 0 -the so-called exchange-correlation kernel f xc r r;r r 0 V xc r r = nr r 0 -is a sufficiently shortranged function of the distance jr r ÿr r 0 j [2].However, much recent work [3][4][5][6][7] has demonstrated that the requirement of short rangedness is not always fulfilled in physical systems, and when this happens the local density approximation is flawed. This does not mean that a local description of exchange and correlation is absolutely impossible, only that such a description cannot be achieved in terms of the particle density.For example, in the density-functional theory of crystalline insulators it has been found [4-6] that the xc potential has an ''ultranonlocal'' dependence on the density, due to the fact that the Fourier transform of the xc kernel f xc k k;k k diverges as 1=k 2 for k ! 0. But, the ultranonlocality disappears if one reformulates the theory in terms of the electric polarizationP Pr r and the exchangecorrelation electric fieldẼ E xc r r associated with it. Another instance of the ultranonlocality problem was discovered in the time-dependent density-functional theory (TDDFT) [8] following the realization that the frequency-dependent LDA [9] fails to satisfy Kohn's theorem [10,11]. The pathology was traced to a singularity of the formk kk k 0 k 2 in the xc kernel f xc k k;k k 0 ; ! for k ! 0 at finitek 0 k 0 and !. The ensuing nonlocality problem was solved by upgrading to time-dependent current-density functional theory (TDCDFT), where the basic variable is the current density, and its conjugate field is a vector potential [3]. TDCDFT has since been applied to the calculation of the optical spectra of solids [12] and the polarizability of long polymer chains [13] with considerable success.In this Letter we show that the ultranonlocality problem occurs in an aggravated for...

Time-dependent exchange-correlation current density functionals with memory

“…Yet, it is inadequate in processes that involve motion of electrons over long distances [22,28] or in cases of double excitations [29]. Thus, it is important to develop methods beyond the adiabatic assumption and several attempts in this direction have been made [4,10,30,31].…”

Dynamical exchange-correlation potentials for the electron liquid in the spin channel

“…7: we supply detailed derivations of the results presented there without proof, and derive an exact relationship between the low-frequency behavior of Re A() and the Landau parameters. We also evaluate the low frequency behavior of Im A() within the mode-decoupling approximation, 13,14 which is supposed to be exact in the high-density limit.…”

“…The small-q divergence cancels out in the density channel, i.e., when one sums over the spin indices, but not in any other combination of the matrix elements f xc, Ј (q,). It has recently been pointed out, 7 that this difficulty can be avoided by switching to a more complete description in which the basic variables are the longitudinal and transverse components of the spin-resolved current density. In the spincurrent density functional theory the xc potential is a vector potential A xc, , which is related to the spin-resolved current density by the following relation:…”

Exchange and correlation effects on plasmon dispersions and Coulomb drag in low-density electron bilayers

“…6 On the other hand, the spin-spin dynamic XC kernel is strongly dependent on q and diverges as q −2 at small wave vectors. 36 Thus, the intra-and interlayer dynamic XC kernels in the long-wavelength limit are represented as…”