Helium harmonic atom

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

Table 11.1. Harmonium (harmonic helium atom). Comparison of the components (a.u.) of the total energy E[po calculated by the HF, BLYP, and BP methods with the exact values (row KS exact Kohn-Sham solution). ...

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.

Harmonic helium atom. In this peculiar helium atom the electrons are attracted to the nucleus by harmonic springs (of equal strength) of equilibrium length equal to zero. For A- = I an exact analytical solution exists. The exact wave function is a product of two... 

Harmonic helium atom. In this system the electrons ... 

Hylleraas function (p. 506) harmonic helium atom (p. 507) James-Coolidge function (p. 508) Kolos-Wolniewicz function (p. 508) geminal (p. 513)... 

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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