Extended cusp condition

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. 

However, we also found, by explicit calculation, that the extended cusp condition was violated by the uniform electron gas of high density. To understand why, first note that, for a uniform system, Eq. (94) becomes... 

In the high density limit, gr(u) dominates. Now gr(u) varies between 1/2 and I on a length scale of order l/(2kF), and so g x(u) 0(2kF). We also know g x(u) must be positive for some value of u. Thus, as kF —> oo, g j (u) becomes very large and positive, while gx(u) 1, so that the extended cusp condition is violated. It is straightforward to check explicitly the well-known formula for gx u) for the uniform gas[33] to see that it has these properties. 

These two results suggest a possible contradiction. On the one hand, we showed that, at high densities, the uniform gas violates the extended cusp condition, and... 

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... 

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. 

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... 

This simple rationalisation of the Kato cusp conditions in 3 spatial dimensions makes it obvious how to extend them to higher dimensions, for a Hamiltonian in which the interparticle potentials retain their 1/r character. All the above analysis goes through unchanged, save that in eq. (5) we make the replacement... 

Modern implementations of the MP2-F12 method combine the CABS approximation ]20] with robust density fitting techniques [21, 22] and local approaches ]23]. The coefficients are usually constrained at the values predetermined from the cusp conditions, as one half for singlet pairs and one quarter for triplet pairs in the spin-adapted formalism [24, 25]. The MP2-F12 methods have been extended to treat open-shell systems with unrestricted [26, 27, 28], restricted [29, 30] and multireference [29] formalisms. 

W. Kutzehrigg. Generalization of Kato s Cusp Conditions to the Relativistic Case. In D. Mukherjee, Ed., Aspects ofMany-Body Effects in Molecules and Extended Systems, Volume 50 of Lecture Notes in Chemistry, p. 353-366, Berlin, Heidelberg, 1989. Springer-Verlag. 

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