Current-density functional theory

The basic idea of DFT is to deduce a theory that provides quantum mechanical observables solely on the basis of the electron density or, more precisely, on the basis of the 4-current rather than on the relativistic wave function. Naturally, one would like to base this deduction on QED [389]. However, in the spirit of this book, we would be satisfied by a first-quantized semi-classical theory. 

Hohenberg and Kohn [386] derived two theorems in the realm of nonrelativistic quantum mechanics, which state that the energy can be calculated solely from the electron density and that any trial density produces an energy, that is an upper bound to the desired exact energy. The proofe are straightforward and not involved. However, they also provoked scepticism, which inspired rigorous work on the mathematical foundations of DFT, but we cannot delve deeper into these issues here. 

Exc [jf ] denotes an exchange- orrelation current-density functional, which collects all quantum, i.e., exchange and correlation effects. For the sake of simplicity, we assume that no external electromagnetic potentials are present other than the monopole potential of the atomic nuclei. Unfortunately, Eye [/ ] is not known in analytical form, which is the reason why present-day DFT is hampered by more or less accurate approximations to Exc[/ ]- 

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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... 

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. 

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). 

Nevertheless this formalism is usually referred to as relativistic density functional theory, rather than current-density functional theory. 

In the nonrelativistic context current-density functional theory is based on the nonrelativistic limits of the paramagnetic current (87) and/or the magnetization density (89) [128,129]. In the relativistic situation, however, a density functional approach relying on jp or m can only be considered an approximation, as long as the external magnetic field does not vanish. In order to clarify the relation between these two points of view the weakly relativistic limit of RDFT has to be analyzed. The weakly relativistic limit of the Hamiltonian (23) can be derived either by a direct expansion in 1/c or by a low order Foldy-Wouthuysen transformation,... 

A BRIEF INTRODUCTION TO FOUR-CURRENT DENSITY FUNCTIONAL THEORY... 

Applications have been carried out for the correlation energy density functional E(.[p] [22,28,31,32], the kinetic and exchange energy functionals [24], the kinetic component of the correlation energy [32,33], the current-density functional theory [29], the second-order density matrix [30], and the total energy of atoms and molecules [23],... 

If one compares the attempts reviewed in sec. 3.2 to base majiy-electron quantum mechanics on the two-particle density matrix, i.e. a 2-particle density matrix functional theory with the current density functional theory one realizes that for the former the functional is exactly known, while the full n-representability condition is unknown. For DFT on the other hand, the functional is unknown, but the n representability does not cause problems. Why should one take incomplete information on n-representability as more serious as lack of information on the exact functional Possibly there was just more reluctance in the two-particle-density matrix functional community to be satisfied with approximate n-representability conditions than in the density functional community to accept approximate density functionals, and that this different attitude was decisive for the historical development. 

Hohenberg and Kohn were able to give two very simple existence theorems which underpin the whole of current density functional theory. 

Liu, S. (1996). Local-density approximation, hierarchy of equations, functional expansion, and adiabatic connection in current-density-functional theory. Phys. Rev. A 54, 1328-1336. 

By the calculations of the time-dependent current density functional theory using this equation, accurate excitation energies are obtained for some n tt excitations (van Faassen and de Boeij 2004). Meanwhile, however, it has been found that quite poor excitation energies are produced for the excitations of some types of molecules. On the other hand, calculations of the adiabatic excitation energy benchmark set, containing 109 molecules, show that the vector potential correction hardly affects the calculated excitation energies (Bates and Furche 2012). 

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