Hartree-Fock exchange

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

The exchange part, ex, which represents the exchange energy of an electron in a uniform electron gas of a particular density is, apart from the pre-factor, equal to the form found by Slater in his approximation of the Hartree-Fock exchange (Section 3.3) and was originally derived by Bloch and Dirac in the late 1920 s ... 

Barone, V., 1994, Inclusion of Hartree-Fock Exchange in Density Functional Methods. Hyperfine Structure of Second Row Atoms and Hydrides , J. Chem. Phys., 101, 6834. 

Buhl, M., 1997, Density Functional Calculations of Transition Metal NMR Chemical Shifts Dramatic Effects of Hartree-Fock Exchange , Chem. Phys. Lett., 267, 251. 

Challacombe, M., E. Schwegler, and J. Almlof. 1995. Linear scaling computations of the Hartree-Fock exchange matrix. J. Chem. Phys. 105, 2726. 

Laming, G. J., N. C. Handy, and W. H. Miller. 1995. Comparison of the Gaussian and Bessel Function Exchange Functionals with the Hartree-Fock Exchange for Molecules. J. Phys. Chem. 99, 1880. 

Lelj, F., C. Adamo, and V. Barone. 1994. Role of Hartree-Fock exchange in density functional theory. Some aspects of the conformational potential energy surface of glycine in the gas phase. Chem. Phys. Lett. 230, 189. 

Three types of exchange/correlation functionals are presently in use (i) functionals based on the local spin density approximation, (ii) functionals based on the generalized gradient approximation, and (iii) functionals which employ the exact Hartree-Fock exchange as a component. The first of these are referred to as local density models, while the second two are collectively referred to as non-local models or alternatively as gradient-corrected models. 

It was this observation which gave rise to so-called hybrid derrsity ftmctional models, such as the B3LYP model. Here, the Hartree-Fock exchange energy is added to the exchange energy from a partictrlar density functional model with one or more adjustable parameters. 

Kohn-Sham Equations. The set of equations obtained by applying the Local Density Approximation to a general multi-electron system. An Exchange/Correlation Functional which depends on the electron density has replaced the Exchange Energy expression used in the Hartree-Fock Equations. The Kohn-Sham equations become the Roothaan-Hall Equations if this functional is set equal to the Hartree-Fock Exchange Energy expression. 

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