Helium atom Coulomb hole

A Hartree-Fock SCF wave function takes into account the interactions between electrons only in an average way. Actually, we must consider the instantaneous interactions between electrons. Since electrons repel each other, they tend to keep out of each other s way. For example, in helium, if one electron is close to the nucleus at a given instant, it is energetically more favorable for the other electron to be far from the nucleus at that instant. One sometimes speaks of a Coulomb hole surrounding each electron in an atom. This is a region in which the probability of finding another electron is small. The motions of electrons are correlated with each other, and we speak of electron correlation. We must find a way to introduce the instantaneous electron correlation into the wave function. 

The mutual avoidance of electrons in the helium atom or in the hydrogen molecule is caused by Coulombic repulsion of electrons (described in the previous subsection). As we have shown in this chapter, in the Haitree-Fock method the Coulomb hole is absent, whereas methods that account for electron correlation generate sueh a hole. However, electrons avoid each other also for reasons other than their charge. The Pauli principle is another reason this occurs. One of the consequences is the fact that electrons with the same spin coordinate cannot reside in the same place see p. 34. The continuity of the wave function implies that the probability density of them staying in the vicinity of each other is small i.e.. 

Figure 2 The wave function of helium atom in its electronic ground state. The upper part (a) represents the difference between the Hartree-Fock and Hylleraas wave functions in a plane that contains the nucleus and fixed electron, called in the literature the Coulomb hole [5]. In the middle part (b) the F12 geminal function with 7 = 1.0 is plotted. The bottom part (c) represents the difference between (a) and (b).

DFT efforts are directed towards elaborating such a potential, and the only criterion of whether a model is any good, is comparison with experiment. However, it turned out that there is a system for which every detail of the DFT can be verified. Uniquely, the dragon may be driven out the hole and we may fearlessly and with impunity analyze all the details of its anatomy. The system is a bit artificial, it is the harmonic helium atom (harmonium) discussed on p. 185, in which the two electrons attract the nucleus by a harmonic force, while repelling each other by Coulombic interaction. For some selected force constants k, e.g., for A =, the Schrodinger equation can be solved analytically. The wave function is extremely simple, see p. 507. The electron density (normalized to 2) is computed as... 

In order better to bring out the details of the electron-electron interaction in the ground-state helium atom, we have in Figure 7.3 plotted the difference between the exact wave function and the Hartree-Fock wave function. In the exact wave function, the probability amplitude for the free electron is shifted away from the fixed electron, creating a Coulomb hole around this electron. The Coulomb hole is wide but rather shallow, with a minimum of —0.068 (relative to the Hartree-Fock description) close to the fixed electron (more precisely, at a distance of 0.49 o Ifom the nucleus). Since the free electron is pushed away from the fixed electron, its amplitude accumulates elsewhere and has a maximum of 0.012 relative to the Hartree-Fock description on the opposite side of the nucleus - at a distance of 0.79uo ffum the nucleus and 1.29no from the fixed electron. 

Fig. 13. The Coulomb hole in the ground-state helium atom. The plotted function represents the difference between the exact wave function and the Hartree-Fock wave function in a plane that contains the nucleus and the fixed electron, the positions of which are indicated by vertical bars (atomic units).

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