Electronic cusp

The differences between the single-configuration wavefunctions are more clearly illustrated by comparing their plots of the intracule function h(ri2), also shown in Fig. 1. This plot reveals the absence of an electron-electron cusp for both the closed and split-shell functions, but shows that the inclusion of exp( —yri2) causes the distribution to have a minimum at ri2=0, forming a cusp (of the correct sign) at that point. This feature will be important for the description of phenomena that depend upon the coincidence probability. 

The Z-scaled electron-electron distribution shows more pronounced differences for species of different Z. Values of and h(ri2) are given in Fig. 4. The Z-scaled intercepts (values of ( (ri2))) change appreciably with Z, and even at Z= 10 the electron-electron cusp is clearly visible. None of these graphs exhibit... 

Unlike the Cl basis, however, the radial coordinates in these functions are r< =min(r1, r2) and r> =max(ri,r2). These functions yield improved radial convergence since the basis is better able to represent the electron-electron cusp. The angular functions,... 

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... 

This is borne out by the lack of any cusp in Figure 3. The electron-electron cusp condition is not obeyed by some popular approximations, e.g., the random phase approximation[17,18]. 

Another possible universal condition on non-uniform electron gases is the extended electron-electron cusp condition. To state this condition precisely, we define the electron pair (or intracule) density in terms of the second-order density matrix, Eq. (14), as... 

Finally, we examine the cusp values for the helium atom. Table 3 shows the nuclear-electron and electron-electron cusp values at the distance r = 1.0 a.u., which was explained below Eq. (24). Both nuclear-electron and electron-electron cusp values approach the exact values of —2.0 and 0.5, respectively, as the order n of the FC calculation increases. At = 27, the cusp values are correct to 22 digits, which is about a half of the correct digits of the variational energy, 41 digits, given in Table 1. This result is natural from a theoretical point of view. 

Table 3 Electron-nucleus and electron-electron cusp values of the FC wave fimctions of heUum...

The functional forms of the one- and two-electrons terms x and u are chosen such that they model the nuclear-electron and electron-electron cusp conditions, respectively, and the parameters inherent in these functions are variationally optimized by the QMC procedure. 

Clearly it requires an enormous number of basis functions to achieve sub-microhartree accuracy if one is using a partial wave expansion. The origin of the slow convergence is the slow representation of the electron-electron cusp at ri2 = 0 by the partial wave expansion, which for any finite L yields an approximate wavefunction analytic in Cartesian coordinates at ri2 = 0. Much faster convergence can be obtained by using a basis which explicitly incorporates linear terms (and higher odd powers) in ri2. Such a basis, for example, is provided by the Pekeris functions... 

One might think that correlation energies would be even easier to calculate in a universe with D > 4, and indeed the error in a truncated partial wave expansion due to the inexact representation of the electron-electron cusps becomes much less important, falling off as ( - -1)-. However, as D increases another problem becomes of increasing importance the phenomenon of localisation about the minimum, or minima, of the effective potential. 

To see how electron localisation for large D would effect the partial wave expansions for many-electron atoms, we may for simplicity consider the helium atom, which is described by three internal coordinates, ri, T2, and. The localisation in the variables ri and V2 can easily be accommodated by using basis functions with flexible length scales, but the localisation in 6 would be very slowly described by truncated partial wave expansions, which contain no internal angular scaling parameter. Thus as D increases, the partial wave expansion has less trouble describing the electron-electron cusp, but more and more trouble describing the localisation of electrons. 

Nagy, A. AmoviUi, C. Electron-electron cusp condition and asymptotic behavior for the Pauli potential in pair density functional theory. J. Chem. Phys. 2008,128, 114115. 

CS then used a model for the correlation functions y>(rj,rj) which satisfies the electron-electron cusp condition at r = Vj [75]. The free parameter in... 

The solid line represents the exact result, obtained from the most accurate variational wavefunction of Kinoshita [99]. One can see that the two electrons preferably move on opposite sides of the nucleus The likelihood to come close to each other is only half as large as that of remaining on opposite sides. Moreover, the 2-particle density clearly shows the electron-electron cusp at ri = V2- Figure 2.21 also provides the x-only result, which corresponds to the ground-state KS Slater determinant. As this determinant only contains Pauli, but not Coulomb correlation, the electrons move independently in the x-only approximation. 

However, when the overlap of the charge densities of the interacting monomers becomes significant (typically at around the equilibrium separation) the basis set must be flexible enough to describe the intermolecular electron-electron cusp as well as the intermolecular charge-transfer (CT). The former effect is manifested as part of and the latter as... 

Simple correlation functions of the form found in equation (6) were employed in early QMC computations. These functions were quite useful because the electron-electron cusp conditions were easily satisfied, giving a large reduction in the variance of the local energy. An improved correlation function for atoms was introduced by Unuigar et al. in their treatment of atoms with 2, 4, and 10 electrons. This correlation function is of the form... 

A Cl expansion in orbital products cannot reproduce the electron-electron cusp. This is the origin of its slow convergence, and terms that explicitly depend on r,2 are required to take account of the electron-electron cusp and to overcome the convergence problem. Unfortunately, the straightforward inclusion of these terms in variational many-electron calculations, especially for molecules, leads to very complicated three- and four-electron integrals over operator products of the type, for example. 

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