The harmonic helium atom

Woznidd, in Theory of Electronic Shells in Atoms and Molecules (ed. A. Yutsis), Mintis, Vilnius, 1971, p. 103. 

Note that the exact wave function (its spatial part ) is a geminal (Le. two-electron function). 

Let me be naive. Do we have two harmonic springs here Yes, we do. Then, let us treat them first as independent oscillators and take the product of the ground-state functions of both oscillators exp[—-I- /- )]. Well, it would be good to 

From (10.10) we see that for ri = rz = const (in such a case both electrons move on the surface of the sphere), the larger value of the function (and eo ipso of the probability) is obtained for larger ryi- This means that, it is most probable that the electrons prefer to occupy opposite sides of a nucleus. This is a practical manifestation of the existence of the Coulomb hole around electrons, i.e. the region of the reduced probability of finding a second electron the electrons simply repel each other. They cannot move apart to infinity since both are held by the nucleus. The only thing they ean do is to be close to the nucleus and to avoid each other - this is what we observe in (10.10). 

One-electron problems are the simplest. For systems with two electrons we can apply certain mathematical tricks which allow very accurate results. We are going to talk about such calculations in a moment. 

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Already the Morse potential looks very difficult to manage, to say nothing of the harmonic helium atom. 

Fig. 11.12. The long-chased electron correlation dragon is finally found in its correlation hole, and we have an ratceptional opportunity to see what it looks like. Correlation potential-efficiency analysis of various DFT methods and cranpariscxi with the exact theory for the harmonic helium atom (with the force constant k = ) according to Kais et al. Panel (a) shows correlation potential Vc (which is less important than the exchange potential) as a function of the radius r (a) and of density p (b). The same notation is used as in Fig. 11.10. The DFT potentials produce plots that differ widely from the exact correlation potential...

The harmonic helium atom represents an instructive example that pertains to medium electronic densities. It seems that the dragon of the correlation energy does not have hundreds of heads and is of quite good character, although it remains a bit unpredictable. 

Fig. 4.23. The harmonic helium atom. The electrons repel by Coulombic forces and are attracted by the nucleus by a harmonic (non-CoulombicJ force.

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... 

The Hohenberg-Kohn theorem can be proved for an arbitrary external potential-this property of the density is called the v-representability. The arbitrariness mentioned above is necessary in order to define in future the functionals for more general densities (than for isolated molecules). We will need that generality when introducing the functional derivatives (p. 584) in which p(r) has to result from any external potential (or to be a v-representable density). Also, we will be interested in a non-Coulombic potential corresponding to the harmonic helium atom (cf. harmonium, p. 589) to see how exact the DFT method is. We may imagine p, which is not u-representable e.g., discontinuous (in one, two, or even in every point like the Dirichlet function). The density distributions that are not u-representable are out of our field of interest. 

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