Coulomb cusp/hole

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. 

Clearly, with this ansatz, very high accuracy is attained with only a few terms in the expansion. Indeed, the plots of the Coulomb hole, the kinetic energy and the Coulomb cusp in this chapter... 

Another more subtle condition is the electron-electron cusp condition. As two electrons approach each other, their Coulomb interaction dominates, and this leads to a cusp in the exchange-correlation hole at zero separation[15]. It is most simply expressed in terms of the pair distribution function. We define its spherically-averaged derivative at zero separation as... 

Physically, C, which controls the large wavevector decay of ( / i, depends on the behavior of the system at small interelectronic separations. In fact, C is proportional to the system-average of the cusp in the exchange-correlation hole at zero separation. If smooth function of r - r, then g x(r, r) = 0, and C would vanish, as it does at the exchange-only level (i.e., to first order in e2). However, as we saw in section 2.2, the singular nature of the Coulomb interaction between the electrons leads to the electron-coalescence cusp condition, Eq. (31). For the present purposes, we wish to keep track explicitly of powers of the coupling constant, so we rewrite Eq. (31) as... 

Bingel i ) has investigated the consequence of Kato s theorems for the pair density. It seems that generally the pair density has a cusp like that shown in Fig. 3. The Coulomb hole is not as deep as the Fermi hole and it has a cusp. 

The Coulomb hole that wavefunctions are predicted to have for close anti-parallel-spin electrons is also called a correlation hole. As a condition for wavefunctions containing correlation holes, Kato proposed a correlation cusp condition (Kato 1957),... 

The Hartree-Fock wavefunction violates this condition, because it gives zero for the left-hand side of this equation. As shown in Fig. 3.1, a wavefunction satisfying this condition contains a correlation hole, which contains a sharp dip, called a cusp, near ri2 = 0. This correlation hole causes anti-parallel-spin electrons to be further apart, and therefore reduces Coulomb interactions, thus lowering the total electronic energies. Sinanoglu named this electron correlation in the correlation cusp condition as dynamical correlation (Sinanoglu 1964). 

Electrons with different spins are thus correlated in their motion. This correlation cannot be accounted for by a wave function in the form of a single Slater determinant, since Equation 1.91 cannot be written as a product of two orbitals. The exact properties of the Coulomb hole can be do ived from the Hamiltonian operator. One may show that there is cusp of the following type when two electrons come close to each other ... 

The problem with conventional configuration interaction or coupled-cluster wavefunctions is that the cusp conditions cannot be fulfilled, or more precisely that the overall shape of the wavefunction in the vicinity of electron coalescence, the Coulomb hole, converges extremely slowly with the size of the underlying basis set expansion. Figure 1 demonstrates that the expansion in orbital products does not provide terms which are linear in While at large separation, the wavefunction shape is well described even by the shortest Cl expansion, even the largest Cl expansion is far off at rj2 0. 

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