Hartree term

The external potential Vext(r) contains the nuclear or ionic contributions and possible external field contributions. The Hartree term Vuif) is the classic electrostatic potential of the electronic cloud... 

The constant C, however, depends on the nature of the interaction between a pair of electrons. If it is mainly of exchange type then C is positive and of order unity, as we have seen. There is also a term of Hartree type—namely that due to direct interaction between the electrons. This has the opposite sign. Finkelstein (1983) and Altshuler and Aronov (1983) found that when the Hartree term is included C should be multiplied by... 

There is no Hartree term for lesser function, because the times t and t2 are always at the different branches of the Keldysh contour, and the <5-function <5(ti — T2) is zero. 

We note the presence of the direct or Hartree term and the exchange term, third term inside the brackets and first term outside, respectively. The term i — j can be included in the direct term since it is canceled by the corresponding inclusion into the exchange term. Therefore, we have the well-known result that the HF approximation does not include self-interaction terms. 

Time-dependent density functional theory (TDDFT) as a complete formalism [7] is a more recent development, although the historical roots date back to the time-dependent Thomas-Fermi model proposed by Bloch [8] as early as 1933. The first and rather successful steps towards a time-dependent Kohn-Sham (TDKS) scheme were taken by Peuckert [9] and by Zangwill and Soven [10]. These authors treated the linear density response of rare-gas atoms to a time-dependent external potential as the response of non-interacting electrons to an effective time-dependent potential. In analogy to stationary KS theory, this effective potential was assumed to contain an exchange-correlation (xc) part, r,c(r, t), in addition to the time-dependent external and Hartree terms ... 

The construction of FKS and calculation of the energy in DFT is akin to the PS version of the HF treatment in that both combine numerical and analytical techniques. For the coulomb (Hartree) term, which is identical in either case, exactly the same procedure could be followed. However, in DFT it is usual to replace the matrix k a(rg) with a vector A,(r ) obtained by fitting the density function p using an auxiliary set of basis functions fi. The contribution to p from the effects of inter-fragment overlap and electron redistribution may be found using (15). Multiplication of the typical term in ARMV — see expression (15) — by 0ju(r) [Pg.154]

In Eq. (32), we have split the external potential into a static nuclear potential and an explicitly time-dependent perturbation. The exchange-correlation functional Exc[p] in Eq. (31) contains all two-electron interactions except the Hartree term 7[p]—that is, it includes the effects of exchange and correlation. In addition, it corrects for the error made in the evaluation of the kinetic energy according to Eq. (32). The last term in Eq. (31) represents the classical nuclear-nuclear repulsion energy. 

Hartree term E describing the Coulomb interaction of the electron density with itself whereas the last term is the HF-exchange energy E . In practical calculations, the occupied orbitals defining 4 are written as a linear combination of known atomic orbitals (LCAO) also referred to as basis functions... 

This external PP is also called the unscreened PP, and the subtraction of >//[ ] and from vfp[n t is called the unscreening of the atomic PP . It can only be done exactly for the Hartree term, because the contributions of valence and core densities are not additive in the xc potential (which is a nonlinear functional of the total density). 

In the atomic-sphere approximation the electrostatic interactions of (7.15), i.e. the Hartree term and (8.31,32), reduce to the interaction with the field -2Zt/r inside the sphere. In addition, the spheres interact via the Madelung term... 

The first two terms are straightforward and are equal to the core contribution, The Coulomb repulsion contribution (the Hartree term) can be expanded in terms of the basis functions and the density matrix, P ... 

In the absence of external potentials, the electrostatic potential energy of nuclei and electrons can be represented by the Coulombic interactions among the electrons and nuclei. There are three groups of electrostatic interactions interactions between nuclei, interactions between electrons and nuclei, and interactions between electrons. Following the Born-Oppenheimer approximation, we neglect nuclei interactions in our DG-based model. Using Coulomb s law, the repulsive interaction between electrons can be expressed as the Hartree term ... 

Note that in the electrostatic or Hartree term (0), the density... 

The starting point in this chain of equations is the Hartree (H) approximation, in which case E = 0. An approximate "Hartree" calculation is shown in Fig.4 for Si. This calculation is "approximate" in that it is not a self-consistent H calculation, but in-stead2is based on a self-consistent pseudopotential-LDA calculation. After achieving selfconsistency the LDA exchange-correlation V was dropped in the extraction of the eigenvalues E and only tile Hartree term retained. We note that in this approximation Si is a semi-metal with partial band-overlap. 

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