Hartree-Fock theory electronic gradient

DFT has come to the fore in molecular calculations as providing a relatively cheap and effective method for including important correlation effects in the initial and final states. ADFT methods have been used, but by far the most popular approach is that based on Slater s half electron transition state theory [73] and its developments. Unlike Hartree-Fock theory, DFT has no Koopmans theorem that relates the orbital energies to an ionisation potential, instead it has been shown that the orbital energy (e,) is related to the gradient of the total energy E(N) of an N-electron system, with respect to the occupation number of the 2th orbital ( , ) [74],... 

The expressions for the electronic gradient and Hessian derived in this subsection play an important role in Hartree-Fock theory and will be used repeatedly in the remainder of this chapter. Note carefully, however, that these expressions are valid only at the expansion point ic = 0. More general expressions, valid for any k, are considered in Exercise 10.1 but are not needed here. Expressions for higher-order derivatives of the energy may also be derived by the techniques of this subsection but are not considered here since they are not required for the optimization and the characterization of the wave function. 

THE ELECTRONIC GRADIENT IN ORBITAL-BASED HARTREE-FOCK THEORY... 

As we said in the introduction, the only consistent framework for a relativistic many-electron system is QED. By means of the Hartree-Fock limit of this theory, after renormalization, and using gradient techniques, Engel and Dreizler [22] found a complete energy functional where both terms of the two previous sections appear naturally. 

Values of R for free atoms have been calculated in Table 3 from the restricted Hartree-Fock functions by perturbation theory and are usually in the range 0.0-0.5. For molecules, however, the only satisfactory procedure is to introduce the dependence on m right from the start and indeed to include the m dependence as one of the parameters to be minimised in the var-ation method. In H - Cl the inner shell polarisation accounts for 17% of the electronic field-gradient l2)... 

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