Fundamental Kinetic-Energy Integral

On the other hand, ab initio (meaning from the beginning in Latin) methods use a correct Hamiltonian operator, which includes kinetic energy of the electrons, attractions between electrons and nuclei, and repulsions between electrons and those between nuclei, to calculate all integrals without making use of any experimental data other than the values of the fundamental constants. An example of these methods is the self-consistent field (SCF) method first introduced by D. R. Hartree and V. Fock in the 1920s. This method was briefly described in Chapter 2, in connection with the atomic structure calculations. Before proceeding further, it should be mentioned that ab initio does not mean exact or totally correct. This is because, as we have seen in the SCF treatment, approximations are still made in ab initio methods. 

Our evaluation of the Fundamental Integral cannot proceed further than (18) unless we now specify the two-electron function f(x). We are now in a position to consider some of the integral types which arise in quantum chemical calculations overlap, kinetic-energy, electron-repulsion, nuclear-attraction and anti-coulomb. 

A fundamental problem in density-functional theory (DFT) is that of how to calculate the exact total energy of a Coulombic system starting from the mere knowledge of the exact density. Of course, this problem could be trivially solved if one were able to express the exact energy as an explicit functional of the density. However, this would imply that we have at our disposal explicit expressions for both the kinetic and the exchange-correlation functionals. As this is not yet possible, one must rely on round-about methods such as the line-integral method recently advanced by van Leeuwen and Baerends. ... 

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