Exchange operator wave function calculations, functionals

It is evidently additive the additive operator V is calculated over unperturbed wave functions, which means that at this approximation the interacting charges are rigid. The exchange energy, 1 nonadditive for all exchanges mixing electrons of three or more atoms. 

Functionals. The difference between the Fock operator, in wave function based calculations and the analogous operator in DFT calculations is that the Coulomb and exchange operators in T are replaced in DFT by a functional of the electron density. In principle, this functional should provide an exact formula for computing the Coulombic interactions between an electron in a KS orbital and all the other electrons in a molecule. To be exact, this functional must include corrections to the Coulombic repulsion energy, computed directly from the electron density, for exchange between electrons of the same spin and correlation between electrons of opposite spin. 

One guesses at an initial set of wave functions, , and constructs the Hartree-Fock Hamilton S which depends on the through the definitions of the Coulomb and exchange operators, (/ and One then calculates the new set of , and compares it (or the energy or the density matrix) to the input set (or to the energy or density matrix computed from the input set). This procedure is continued until the appropriate self-consistency is obtained. 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

For ionized helium3, on the other hand, the experimental value of the hyperfine structure of the metastable 22S level, 1083 360 0 020 Mc/s [98] differs from the uncorrected calculated value by 182 22 parts in 108. This calculated value includes all the terms included in equation 11.3 except Gr and Ct. The values of the nuclear structure and exchange current corrections depend on the choice of the nuclear wave functions and exchange current operators. Foley and Nessler [47] find a structure correction of 138 parts in 106, and exchange current corrections of 2 parts in 108 for one type of interaction current, and 230 parts in 108 for another. The measured hyperfine structure therefore discriminates against the latter. 

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. 

So far we have made no approximations to the Coulomb and exchange terms in the potential, which are still expressed in terms of the large and small components. In an actual calculation we would not have the large and small components, only the approximate transformed wave function. The potentials must therefore be expressed in terms of the transformed wave functions and the transformed operators, taking into account that these are not exact. 

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