Coulomb cusp illustration

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

To illustrate the nuclear and electronic Coulomb cusp conditions, we have in figure 7.5 plotted the ground-state helium wave function with one electron fixed at a point 0.5ao from the nucleus. On the left, the wave function is plotted with the free electron restricted to a circle of radius 0.5ao centred at the nucleus (with the fixed electron at the origin of the plot) on the ri t, the wave function is plotted on the straight line through the nucleus and the fixed elearon. The wave function is differentiable everywhere except at the points where the particles coincide. 

The usual Hilbert-space requirement of continuous gradients is not appropriate to Coulombic point-singularities of the potential function u(r) [ 196]. This is illustrated by the cusp behavior of hydrogenic bound-state wave functions, for which the Hamiltonian operator is... 

Here R/ denotes the positions of the nuclei, Zk their atomic number, and a0 = h2/me2 is the Bohr radius. For a Coulomb system one can thus, in principle, read off all information necessary for completely specifying the Hamiltonian directly from examining the density distribution the integral over n(r) yields N, the total particle number the position of the cusps of n(r) are the positions of the nuclei, R. and the derivative of n(r) at these positions yields Zf, by means of Eq. (19). This is all one needs to specify the complete Hamiltonian of Eq. (2) (and thus implicitly all its eigenstates). In practice one almost never knows the density distribution sufficiently well to implement the search for the cusps and calculate the local derivatives. Still, Kato s theorem provides a vivid illustration of how the density can indeed contain sufficient information to completely specify a nontrivial Hamiltonian.11... 

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