Paramagnetic current

A crude physical interpretation of Eq, (2) is that the factor (pBIAE) represents the applied field B is the magnetic fiux density and fi the Bohr magneton) driving the angular momentum and inducing a paramagnetic current, and the factor represents the... 

Direct application of Ax for the quantitative evaluation of aromaticity is, however, not practicable since its magnitude is not determined by ring currents only. Quite substantial effects may be played by a local contribution by the 7r-bond anisotropy and the anisotropy of CC and CH (r-bond magnetic susceptibilities as well as by the anisotropy due to local paramagnetic currents (for more detail, see, e.g., 66MI1). 

This expression indicates that it is the sum of the paramagnetic current operator and the curl of the magnetisation density,... 

If A(r, to) = A (r, to), then the statement of the theorem is trivially true tecause the initial paramagnetic currents and the initial densities are identical so that... 

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. 

Paramagnetic currents on the atom in question. This allows for the fact that the electron cloud does not have spherical symmetry. This term is insignificant for H because the 2p and other states lie at such high energies, but is important in cobalt magnetic resonance. 

Diamagnetic and paramagnetic currents on neighboring atoms. These effects may be significant (particularly the paramagnetic currents on cobalt) and may upset any simple correlation between the observed chemical shift and the effects mentioned in paragraph (1) above. 

The gauge-invariant physical current j(r) is related to the paramagnetic current jp(r) entering the above integral virial theorem by... 

So it is that the two cyclazines, 33 and 34, formally similar in structure, have n.m.r. spectra dominated by the ring currents which are surely present in their peripheral K-systems a diamagnetic current in the case of the [10]annulene-like cycl[3,2,2]azine and a paramagnetic current in the [12]annulene-like cyd[3,3,3]azine. 

To study the magnetic properties of matter one would often like to be able to obtain information on the currents in the system and their coupling to possible external magnetic fields. Important classes of experiments for which this information is relevant are nuclear magnetic resonance and the quantum Hall effects. SDFT does not provide explicit information on the currents. RDFT in principle does, but standard implementations of it are formulated in a spin-only version, which prohibits extraction of information on the currents. Furthermore, the formalism of RDFT is considerably more complicated than that of SDFT. In this situation the formulation of nonrelativistic current-DFT (CDFT), accomplished by Vignale and Rasolt [140, 141], was a major step forward. CDFT is formulated explicitly in terms of the (spin) density and the nonrelativistic paramagnetic current density vector jp(r). Some recent applications of CDFT are Refs. [142, 143, 144, 145]. E. K. U. Gross and the author have shown that the existence of spin currents implies the existence of a link between the xc functionals of SDFT and those of CDFT [146], Conceptually, this link is similar to the one of Eq. (99) between functionals of DFT and SDFT, but the details are quite different. Some approximations for xc functionals of CDFT are discussed in Refs. [146, 147, 148]. 

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,... 

DFT [95] was generalized to include relativistic effects by MacDonald and Vosko [96] and by Ramana and Rajagopal.[97] By omitting small diamagnetic effects one-electron equations were obtained that contained a scalar effective potential as well as an effective magnetic field caused by the polarization. In cases where the paramagnetic currents cannot be neglected, it is necessary to resort to a current-density formalism.[98,99]... 

Another ingredient whose importance is being increasingly recognized [62-67] is the paramagnetic current density, defined in atomic units as... 

For usual nonrelativistic wavefunctions represented by the Slater determinant, the right-hand side of Eq. (6.104) is derived using the paramagnetic current density. 

In this equation, the exchange-correlation energy functional, xc, is assumed to be a functional of the electron density and paramagnetic current density, and the corresponding exchange-correlation potential, and vector potential, A c, are represented as... 

Vignale and Kohn proved that this time-dependent vector potential is Fourier-transformed (t u>) using the current density j, which is usually the paramagnetic current density jp, to... 

Current Density Functional Theory (CDFT). - The Kohn-Sham density functional expression has to be modified when magnetic fields are present so that the density functional depends on the paramagnetic current density. A local exchange-correlation term then has the form... 

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