Current density functionals

G. Vignale, in Current Density Functional Theory and Orbital Magnetism, Vol. 337 of Nato ASI Series, Series B, edited by E. K. U. Gross and R. M. Dreizler (Plenum Press, NewYork, 1995), p. 485. 

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

Berger JA, Snijders JG (2002) Ultranonlocality in time-dependent current-density-functional theory Application to conjugated polymers, Phys. Rev. Lett, 83 694-697... 

Further, there are asymptotically corrected XC kernels available, and other variants (for instance kernels based on current-density functionals, or for range-separated hybrid functionals) with varying degrees of improvements over adiabatic LDA, GGA, or commonly used hybrid DFT XC kernels [45]. The approximations in the XC response kernel, in the XC potential used to determine the unperturbed MOs, and the size of the one-particle basis set, are the main factors that determine the quality of the solutions obtained from (13), and thus the accuracy of the calculated molecular response properties. Beyond these factors, the quality of the... 

It is directly possible to prove a HK-theorem for the form (3.55) using the density n and the gauge-dependent current jp — (c/e)V x m as basic DFT variables, but not for the form (3.54) which would suggest to use n and the full current j. One is thus led to the statement that the first set of variables can legitimately be used to set up nonrelativistic current density functional theory, indicating at first glance a conflict with the fully relativistic DFT approach. 

A density (or current density) functional representation of the relativistic noninteracting kinetic energy can either be obtained by the (linear) response technique discussed in Appendix D or by a direct gradient expansion (GE) on the basis of (2.38), whose kinetic contribution is given by... 

This equation defines the TDKS potentials Ajo implicitly in terms of the functionals A[j] and A,[j]. Clearly, Eq. (137) is rather complicated. The external-potential terms 5 and J are simple functionals of the density and the paramagnetic current density. The complexity of Eq. (137) arises from the fact that the density, Eq. (128), and the paramagnetic currents, Eqs. (129), (134), are complicated functionals of j. Hence a formulation directly in terms of the density and the paramagnetic current density would be desirable. For electrons in static electromagnetic fields, Vignale and Rasolt [61-63] have formulated a current-density functional theory in terms of the density and the paramagnetic current density which has been successfully applied to a variety of systems [63]. A time-dependent HKS formalism in terms of the density and the paramagnetic current density, however, has not been achieved so far. 

The exchange-correlation scalar potential of the current-density functional theory of Vignale and Rasolt [101,102] is then derived. Again as in the zero... 

Since the current density in the bulk measures the surface charges, the time-dependent current-density functional theory (CDFT) appears to be a way to investigate this problem. At least the results presented by de Boeij et al. [171] for the bulk susceptibility and by van Faassen et al. [172] for the polarizability of linear chains are encouraging, although this may not be the case for second-and third-order effects. 

Calculating xw within the framework of plain spin density functional theory (SDFT), there is no modification of the electronic potential due to the induced orbital magnetization. Working instead within the more appropriate current density functional theory, however, there would be a correction to the exchange correlation potential just as in the case of the spin susceptibility giving rise to a Stoner-like enhancement. Alternatively, this effect can be accounted for by adopting Brooks s orbital polarization formalism (Brooks 1985). 

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