Hartree interaction term

The last term is introduced within the self-consistent Hartree approximation (within the functional up to one vertex), //// = 10/9 accounts different coefficients in functional for the self-interaction terms (ri4 - for the given field d and dla)2 d )2 terms), cf [20], We presented da = 2k d t>ke lkfix, ... 

C 1/2 contains a factor gkpay1, which should be about 1, and an unknown term from the Hartree interaction. 

Model Hartree-Fock calculations which include only the electrostatic interaction in terms of the Slater integrals F0, F2, F and F6, and the spin-orbit interaction , result in differences between calculated and experimentally observed levels596 which can be more than 500 cm-1 even for the f2 ion Pr3. However, inclusion of configuration interaction terms, either two-particle or three-particle, considerably improves the correlations.597,598 In this way, an ion such as Nd3+ can be described in terms of 18 parameters (including crystal field... 

The main advantage of this method is automatic orthogonality of all functions P,(r). Its principal disadvantage is an inaccurate account of the energy of self-interaction of the ith electron. We can avoid this by employing instead of (28.19) the Hartree-type term... 

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. 

Table 3. Core-valence electron coulomb interaction terms, evaluated from relativistic Hartree-Fock-Slater wave functions...

In molecules and clusters, genuine exchange (as well as correlation) among identical nuclei is very small because, at typical internuclear separations, the overlap of nuclear wave functions is rather small. However, the exact xc functional also contains self-exchange contributions which are not small and which cancel the self-interaction terms contained in the Hartree potentials in Eqs. (71) and (72). Hence it will be a very good approximation to represent Fjc by the self-exchange terms alone. This leads to... 

From the examination of AE decompositions performed in our group in Pisa (of which we quote the first papers Bonaccorsi et al., 1971 Alagona et al., 1972, 1973) we learned that solute-water interactions can be reduced to the interaction terms of electrostatic origin, i.e. those described by a simple Hartree product wavefunction E sjthat the validity... 

Hence, the electronic energy is a functional ofthe density E i = rf[/>(r)] instead of a functional of the orbitals E = d[ ( / ( r )] (see Eq. (12.6) for the explicit expression in Hartree-Fock theory). However, the formally exact principles of DFT are connected with the drawback that the energy expression as a function of the density is not known. It can be approximated and is usually divided into different physically meaningful contributions (i.e. an electron-electron interaction energy functional is formulated, which is decomposed in a classical Poisson-type electrostatic interaction term and a rest term - the exchange-correlation functional -which is not known). 

Note that here and later on r denotes the single-particle coordinate whereas R is still used as abbreviation for all nuclear positions as in Eq. (1). The potential (5) consists, on one hand, of an external potential V(r,R), which in our case is time-dependent owing to the atomic motion R( ). On the other hand, there are electron-electron interaction terms, namely the Hartree and the exchange-correlation term, which depend both via the density p on the functions tpj. The exchange-correlation potential VIC is defined within the so-called adiabatic local density approximation [25] which is the natural extension of the lda from stationary dpt. It is assumed to give reliable results for problems where the time scale of the external potential (in our case typical collision times) is larger than the electronic time scale. 

Another recent development is the implementation of DK Hamiltonians which include spin-orbit interaction. An early implementation shared the restriction of the relativistic transformation to the kinetic energy and the nuclear potential with the efficient scalar relativistic variant electron-electron interaction terms were treated in nonrelativistic fashion. Further development of the DKH approach succeeded in including also the Hartree potential in the relativistic treatment. This resulted in considerable improvements for spin-orbit splitting, g tensors and molecular binding energies of small molecules of heavy main group and transition elements. Application of Hamiltonians which include spin-orbit interaction is still computationally demanding. On the other hand, the SNSO method is an approximation which seems to afford a satisfactory level of accuracy for a rather limited computational effort. 

Only one solution is found if < 1. This dehnes the restricted Hartree-Fock solution. If 1, three solutions are found, the restricted Hartree-Fock solution, but also two solutions with x 0. So if the effective one-center repulsion energy is large relative to the covalent interaction term, the line-width function F, in the ground state a different electron density for spin ground state now becomes paramagnetic. In calculations of the total energy one has to compute Eq.(2.229) separately for spin states a and o and to correct it for double counting of the one-center electron-electron interaction ... 

In the case of Hartree-Fock calculations or for self-interaction free methods in general, each interaction term has the form 2SiSjJy and the expressions for the energy differences, obtained in a similar way, are those given in Eqs. (7) ... 

We have already seen in section 8.1 that (i) a Dirac electron with electromagnetic potentials created by all other electrons [cf. Eq. (8.2)] cannot be solved analytically, which is the reason why the total wave function as given in Eq. (8.4) cannot be calculated, and also that (ii) the electromagnetic interactions may be conveniently expressed through the 4-currents of the electrons as given in Eq. (8.31) for the two-electron case. Now, we seek a one-electron Dirac equation, which can be solved exactly so that a Hartree-type product becomes the exact wave function of this system. Such a separation, in order to be exact (after what has been said in section 8.5), requires a Hamiltonian, which is a sum of strictly local operators. The local interaction terms may be extracted from a 4-current based interaction energy such as that in Eq. (8.31). Of course, we need to take into account Pauli exchange effects that were omitted in section 8.1.4, and we also need to take account of electron correlation effects. This leads us to the Kohn-Sham (KS) model of DFT. 

In Section 3.2.1, we discussed the two main sets of approximations available to us when considering the interaction terms present in the Hamiltonian operator. The first is ab initio theory, which has as its basis Hartree-Fock theory the second is density functional theory, which recasts the basic equations in terms of the electron density rather than the wavefunction directly. 

There is increasing interest in the relativistic treatment of atoms/ molecules/ and solids. A relativistic Hartree-Fock scheme [Hartree-Fock-Dirac (HFD) method] based on the variation in the total energy obtained with a single Slater determinant (in which the one-electron orbitals are four-component Dirac spinors), using a Dirac-type Hamiltonian for each electron and including Coulomb interaction, was developed some time ago.< For the remaining interaction terms the first-order perturbation of the Breit interaction operato reduced to large components (Pauli approximation) is usually taken into account (see, however, the work of Mann and Johnson ). 

The first term on the right-hand side may be recognized as the Hartree term (or direct interaction term) J and the other is a nonclassical term containing the effects of exchange and correlation. The same formal rewriting may, of course, be done for the current density term... 

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