Integral electron kinetic energy

We now need to discuss how these contributions that are required to construct the Kohn-Sham matrix are determined. The fust two terms in the parenthesis of equation (7-12) describe the electronic kinetic energy and the electron-nuclear interaction, both of which depend on the coordinate of only one electron. They are often combined into a single integral, i. e ... 

Thus we know about atoms in a molecule uniquely defined by the potentials at their nuclei. By the same token, explicit calculations of the electronic kinetic energies and of all two-electron integrals are avoided. 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

The one-electron matrices T, V(H) and V(He) (i.e. Hcore ) follow immediately from the one-electron integrals. The kinetic energy matrix is... 

Fig. 4.29. Normalized integrated intensities (left) of substrate core levels in dependence on deposition time for the spectra shown in Fig. 4.26. The deposition rate is estimated to be 2nmmin 1. The lines in the left graph are obtained by curve fitting of the data to an exponential decay. The derived attenuation times are displayed in the right graph in dependence on electron kinetic energy together with theoretical energy-dependent escape depth calculated using the formula by Tanuma, Powell, and Penn [37] and using a y/ E law [38]...

To obtain the total electronic kinetic energy we simply multiply by the electron density and integrate over the crystal ... 

As illustrated earlier, setting the generator equal to e-p defines the atomic force and the variational principle leads to the integral atomic force law, or the equation of motion for an atom in a molecule. Finally, it was shown that, when F = — sr-p, the commutator defines the electronic kinetic energy and virial for an atom, and the variational principle yields the relationship between these quantities, the atomic virial theorem. These three relationships—the equation of continuity, the equation of motion, and the virial theorem—form the basis for the understanding of the mechanics of an atom in a molecule. 

The term [VrP F( )] /p vi ) is known as the von Weizsacker kinetic energy fw[P i (r)] [4]. Hence, in the other words, Corollary 1.1 tells that square-integrable. Usually, the von Weizsacker term is only a part of the total many-electron kinetic energy [4]. The exception is the Hartree-Fock 2-electron model systems for which fw[p>i (r)] is the exact kinetic energy. We further have... 

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