Hooke s atom

Appendix Exchange-Correlation Potential in Exactly Solvable Hooke s Atom... 

Qian and Sahni [111] have used the Holas-March route to the exchange-correlation potential Vxc r) to construct this quantity in an exactly solvable model. This is the so-called Hooke s atom, in which two electrons, mutually repelling Coulombically, move in an external harmonic potential. For a particular choice of spring constant, the ground-state wave function is analytically known. In the study of Qian and Sahni (see their Fig. 12), the exchange-correlation potential Vxc is calculated exactly from the Holas-March [52] approach. In particular, Qian and Sahni make a full study of the contribution to VXc from the correlation kinetic energy. 

In this section, we discuss first the basic formalism of density functional theory, and the Kohn-Sham equations whose solution yields the ground state density and energy of a system. We next describe a two-electron system, called Hooke s atom, which has the highly unusual feature of an analytic ground state, and is therefore of considerable pedagogical value. Lastly we discuss the notation and organization of this article. 

It is not easy to find a quantum many-body system for which the Schrodinger equation may be solved analytically. However, a useful example is provided by the problem of two electrons in an external harmonic-oscillator potential, called Hooke s atom. The Hamiltonian for this system is... 

Throughout this paper, we use Hooke s atom to illustrate our points. We often use results calculated for the k = 1/4 case, for which the exact density is plotted in Figure 1. We think of this as a sort of poor man s Helium," although we note... 

Figure 1. Ground state density of Hooke s atom for k = 1/4.

Figure 2. Exchange-correlation hole around an electron at the origin of Hooke s atom.

To understand these holes in more detail, we also consider their spin decomposition. Because the Hooke s atom is unpolarized, it has only two distinct spin combinations parallel (TT) and antiparallel (Tl). In Figures 3 and 4 we decompose the spin-averaged hole into these separate contributions. In fact the anti-parallel... 

Also, for a spin-unpolarized two-electron system like the Hooke s atom, the exact parallel hole is made up entirely of exchange. This can be seen most easily in the spin-decomposed second-order density matrix. Since the ground state wavefunction is a spin singlet, it contains no contribution in which both electrons have the same spin. Therefore... 

We note a very important point in density functional theory and the construction of approximate functionals. It is the hole itself which can be well-approximated by, e.g., a local approximation. This is because it is the hole which obeys the exact conditions we have been discussing. To illustrate this point, Figure 6 is a plot of the pair distribution function around the origin in Hooke s atom, both exactly and within the LSD approximation. We see that the two functions are quite different. In particular, the exact pair distribution function has not saturated even far from the center. The corresponding holes of Figure 2, on the other hand, are much more similar. 

We have also calculated this constant for Hooke s atom[18]. We use elementary perturbation theory, treating the Coulomb repulsion as weak. In the high density (non-interacting) limit, we find C(2> in agreement with Eq. (81). We then calculate the leading corrections to C, and find, after tedious calculations[65],... 

In recent work, two of us (Burke and Perdew), with Juan Carlos Angulo[77] were able to prove analytically that this extended cusp condition was true for the ground state of Hooke s atom for all values of the spring constant. We also showed that h(u) was always uni modal for this system, just as had been found for the two electron ions. 

Clearly, if h(u) then obeys the extended cusp condition, so also does hir(u) but hsr u) — h ST(u) < 0 for all u. In the particular case of the Hooke s atom, the high density (or non-interacting) limit is just a pair of three-dimensional harmonic oscillators. The ground state wavefunctions are simple Gaussians, yielding a density... 

Figure 19. h (u) for the high-density limit of Hooke s atom. 

The above analysis explains why the uniform gas behaves so differently from the Hooke s atom in the high density limit. In both systems, h,r(u) behaves very similarly, developing a large, positive derivative for finite u as n —> oo. However, the Hooke s atom h (u) also contains contributions from h lr(u) due to the density gradient, which have no analog in the uniform electron gas. These are sufficient to cancel the contributions from h sr(u), so that the extended cusp condition remains valid for this system. 

The real atom results are from Ref.[78] the Hooke s atom results are from Ref. [11]. 

Kohout, M., Pemal, K., Wagner, F. R., Grin, Y. (2004). Electron locali2ability indicator for correlated wave functions. I. Parallel-spin pairs. Theor. Chem. Acc. 112,453 59. Lam, K. C., Cruz, F. G., Burke, K. (1998). Viral exchange-correlation energy density in Hooke s atom. Int. J. Quantum Chem. 69, 533-540. 

Lam, K. C., Cruz, F. G., Burke, K. Viral (1998). Exchange-correlation energy density in Hooke s atom. Int. J. Quantum Chem. 69,533-540. 

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