Kato cusp conditions

Beyond this it is also clear that trial functions must satisfy the Kato cusp conditions explicitly or implicitly, or convergence will remain too slow for high-precision results. Morgan[5] has emphasized that satisfying these cusp conditions is of greater importance than building in the correct asymptotic... 

The behavior of relativistic wave functions at the Coulomb singularities of the Hamiltonian have been studied [84]. The nuclear attraction potentials don t cause any problem. There are weak singularities of the type r with p slightly smaller than 0, as they are familiar for the H-like ions. The limits r —> 0 and oo commute, and the Kato cusp conditions [85] arise in the nrl. For the coalescence of two electrons the two limits do not commute. An expansion in powers of c is possible to the lowest orders and leads to results consistent with those reported above. 

If we now perform an average about an infinitesimally small sphere centered at r,j = 0, this will eliminate from ip all except the s-wave component, which is itself annihilated by JD -. Assuming that d ip/dr j is bounded, as it is for eigenfunctions of the hydrogen atom, then the coefficients of the l/r,j singularities in eqq. (4) and (5) should cancel, which leads to the Kato cusp condition given by eq. (2). 

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... 

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

This important property has been proven by Ayers in two steps. The fact that nuclear positions, Ra, and the atomic numbers of the nuclei, Za, can be determined from the cusp conditions... 

For the cusp condition, we proposed the method [141] using the Slater-type orbitals (STOs) by the STO-GTO expansion method [143,144]. Following Kato [145], the exact... 

The cusp values of the wave function are also the necessary conditions of the exact wave function. Kato [34] rigorously derived the cusp conditions for many-electron systems as... 

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),... 

Myers CR, Umrigar CJ, Sethna JP, Morgan JD III (1991) Fock s expansion, Kato s cusp conditions, and the exponential ansatz. Phys Rev A 44 5537-5546... 

This ratio is comparable to Kato s nonrelativistic cusp condition [482], which is the ratio of the derivative of the nonrelativistic radial function to itself (note... 

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. 

Strictly speaking the nuclear attractors do not represent critical points, because of the cusp condition (Kato 1957). 

Figure 4 indicates that the electron-electron cusp value c(s) for the highly accurate N ax = 9 wave function is a constant close to 5 in the interval 1 < j < 8. In 1957, Kato proved (for r > 0) that the exact He ground state wave function must satisfy the electron-electron cusp condition " ... 

Although the majority of electronic quantum chemistry uses gaussian expansions of atomic orbitals [10,11], the present work uses exponential type orbital (ETO) basis sets which satisfy Kato s conditions for atomic orbitals they possess a cusp at the nucleus and decay exponentially at long distances from it [39]. It updates work since 1970 and detailed elsewhere [3,6,15,18,28,31,42,51,52,57,60,62]. 

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