Harmonic helium atom harmonium

T vo-electron systems already represent a serious problem for quantum chemistry, because the mutual correlation of electron motions must be carefully taken into account. As we will see in further chapters, such calculations are feasible, but the 

Unfortunately, this wonderful two-electron system is (at least partially) nonphysical. It represents a strange helium atom, in which the two electrons (their distance denoted by r 2) interact through the Coulombic potential, but each is at- 

The Hamiltonian of this problem (the adiabatic approximation and atomic units are used) has the form  

It is amazing in itself that the Schrodinger equation for this system has an analytical solution (for k = ), but it could be an ratremety complicated ana cal formula. It is a sensation that the solution is dazzlingly beautiful and simple 

The wave function represents the product of the two harmonic oscillator wave functions (Gaussian functions), but also an additional extremely simple correlation factor (1 + r 2). As we will see in Chapter 13, exactly such a factor is required for the ideal solution. In this exad function there is nothing else, just what is needed.  

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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... 

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). ...

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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