Coulomb hole illustration

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

Figure 1. Illustration of the copper-oxygen singlet formation on the square lattice. The additional doped hole cannot go to the copper site due to strong on-site Coulomb repulsion and is distributed among four oxygen sites forming together with the copper spin a Zhang-Rice singlet.

Figure 5.1 Sketch of the spectrum of the Dirac Hamiltonian for a free electron (a) and a bound electron in a Coulombic potential attractive for electrons [see chapter 6] (b). In (c) creation of an electron-positron pair is illustrated according to Dirac s hole theory, where all negative-energy states are assumed to be occupied for the vacuum state.

This relation illustrates the basic feature that the ionic part of the pseudopotential and the orthogonalisation hole are screened by the induced charge density. However, since the orthogonalisation hole consists of electrons, its potential also includes an exchange and correlation effect, apart from the Coulomb interaction. 

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