Energy, self-interaction

Do so, we use the formalism of the variational density fitting method [55, 56] where the Coulomb self-interaction energy of the error is minimized ... 

In the Flartree-Fock (FIF) method, the spurious self-interaction energy in the Flartree potential is exactly cancelled by the contributions to the energy from exchange. This would also occur in DFT if we knew the exact Kohn-Sham functional. In any approximate DFT functional, however, a systematic error arises due to incomplete cancellation of the self-interaction energy. 

A problem with DFT that is not restricted to intermolecular complexes is what might be called overdelocalization . In part because of problems in correcting for the classical self-interaction energy, many functionals overstabilize systems having more highly delocalized densities over more localized alternatives. Such an imbalance can lead to erroneous predictions of higher symmetry structures being preferred over lower symmetry ones, as has been observed, for instance, for phosphoranyl radical structures (Lim et al. 1996), transition-state structures for cationic [4-1-3] cycloadditions (Cramer and Barrows 1998), and in the comparison of cumulenes to poly-ynes (Woodcock, Schaefer, and Schreiner 2002). It can... 

If there is no explicit external electromagnetic field, the covariant field equations determine a self-interaction energy that can be interpreted as a dynamical electron mass Sm. Since this turns out to be infinite, renormalization is necessary in order to have a viable physical theory. Field quantization is required for quantitative QED. The classical field equation for the electromagnetic field can be solved explicitly using the Green function or Feynman propagator GPV, whose Fourier transform is —gllv/K2, where k = kp — kq is the 4-momentum transfer. The product of y0 and the field-dependent term in the Dirac Hamiltonian, Eq. (10.3), is... 

The main advantage of this approximation is that it is exact for two-electron systems (if the correct kf(r) = 0 is utilised in (4.15) before performing the functional differentiation (3.17) required for its application) and also correctly accounts for the self-interaction energies of individual closed shells if a shellpartitioning scheme is used [71]. Furthermore, the RWDA reproduces the asymptotic r proportionality of the exact x-only potential (although with the incorrect prefactor of 1/2 [103]). 

The error of 765 mhartree for AEJ dens) reflects the fact that the LDA is missing important nonlocal contributions responsible for the cancellation of the self interaction energy already on the nonrelativistic level. The much larger LDA error for AE ifctl), on the other hand, directly shows that nonlocal corrections are even more relevant for the relativistic correction — [ ]. Fur-... 

We should note for the future that this Coulomb expression contains the electronic self-interaction energies which are (in HF theory) cancelled by the exchange terms. We therefore write... 

There is one very obvious pitfall in all that we have said so far the Coulomb energy of repulsion of a density with itselT includes the interaction of each electron with all the others plus the interaction of that electron with itself. This self-interaction energy is clearly spurious in the MO method it is automatically cancelled by the diagonal exchange terms... 

The Coulombic electron-electron self-interaction energy (J[pl)... 

Levy minimization (p. 679) local density approximation. LDA (p. 687) non-nuclear attractor (p. 670) one-particle density matrix (p. 698) Perdew-Wang functional (p. 688) self-interaction energy (p. 708) spin polarization (p. 687)... 

Note that the electron-electron repulsion term was written by also considering the Fermi-Amaldi N- ) N factor (Parr Yang, 1989 Putz, 2003), which ensures the correct self-interaction behavior when only one electron is considered, the self-interaction energy must be zero, F (N- 1) 0. 

Note that this correction has the problem that the Kohn-Sham equation is not invariant for the unitary transformation of occupied orbitals, even after the correction, differently from the Hartree-Fock equation. In the Hartree-Fock equation, the variations of the Coulomb self-interaction energy and its potential for the unitary transformations of occupied orbitals cancel out with those of the exchange self-interaction, while these are not compensated, even after the correction in the Kohn-Sham equation. Therefore, the effect of the self-interaction correction depends on the difference in occupied orbitals before and after the unitary transformation. For removing this difference, it is usual to localize the orbitals before the self-interaction correction (Johnson et al. 1994). Note, however, that there are various types of orbital localization methods, and the effect of the selfinteraction correction inevitably depends on them. Combining with the optimized effective potential (OEP) method (see Sect. 7.5) may be one of the most efficient ways to solve this problem. This combination enables us to consistently obtain localized potentials with no self-interaction error. 

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