Exact exchange orbital functionals

Local Density Approximation, Optimized Effective Potential, and Exact Exchange Orbital Functionals... 

The first term, the kinetic energy, is difficult to calculate directly from the density, and it is for this reason that the molecular orbitals mentioned above are introduced a very good approximation to the kinetic energy corresponding to the density can be calculated from the orbitals as it would be in HF theory. This approach does not however yield the exact kinetic energy because it assumes that the electrons in each orbital do not interact with electrons in other orbitals. The exact exchange-correlation functional must therefore contain a corrective term to incorporate the effect of electronic interactions on their kinetic energy. In practice, such a term is not explicitly included in common functionals. 

The main advantage of such an approach is that it allows for greater flexibility in the choice of appropriate xc functionals. In particular, the OEP method can be used for the treatment of the exact exchange energy functional, defined by inserting KS orbitals in the Fock term, i.e. 

SIC, or self-interaction corrected LDA [21, 22], and more recently the exact exchange (EXX) functionals [23], both of which include a dependence of the exchange functional on the occupied molecular or crystalline orbitals. 

The fourth rung of the ladder in Fig. 10.2 is important because the most common functionals used in quantum chemistry calculations with localized basis sets lie at this level. The exact exchange energy can be derived from the exchange energy density, which can be written in terms of the Kohn-Sham orbitals as... 

If the functional applied does not include any exact exchange then (A-B) will be a diagonal matrix that only depends on the orbital eigenvalues. The working equation then becomes a single standard eigenvalue equation ... 

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