The Electron Density is Smooth Away from the Nuclei

Fournais, Søren; Hoffmann‐Ostenhof, M.; Hoffmann‐Ostenhof, Thomas; Sørensen, Thomas Østergaard

doi:10.1007/s002200200668

Cited by 58 publications

(80 citation statements)

References 18 publications

(22 reference statements)

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“…If R 0 ∈ Σ n , then this set contains at least a segment outside Σ. This contradicts the fact, (21), that Ψ has isolated zeroes outside Σ. On the other hand since we do not know if Ψ has usual derivatives everywhere we cannot reasonably assume that the factors ϕ and χ are everywhere regular.…”

Section: The Coulomb Hamiltonianmentioning

confidence: 88%

“…Whether the inclusion of such cusps by means of trial wavefunctions involving r ee ij in electronic structure calculations could improve their accuracy has been quite widely studied; an example can be found in 15 . We shall consider the matter further from a mathematical standpoint 16 when examining the proposed factorisations.…”

Section: The Coulomb Hamiltonianmentioning

confidence: 99%

“…The best known regularity of an eigenfunction Ψ of the Coulomb Hamiltonian H ′ is only that its first distributional derivatives are bounded 12,16 ; in particular, we do not know if one can differentiate Ψ everywhere in the usual sense. Fortunately, we have some further information on Ψ; we know by elliptic regularity 17 (see Appendix VIII G), that the following two statements are valid:…”

Section: The Coulomb Hamiltonianmentioning

confidence: 99%

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On factorization of molecular wavefunctions

Jecko¹,

Sutcliffe²,

Woolley³

2015

J. Phys. A: Math. Theor.

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Recently there has been a renewed interest in the chemical physics literature of factorization of the position representation eigenfunctions {Φ} of the molecular Schrödinger equation as originally proposed by Hunter in the 1970s. The idea is to represent Φ in the form ϕχ where χ is purely a function of the nuclear coordinates, while ϕ must depend on both electron and nuclear position variables in the problem. This is a generalization of the approximate factorization originally proposed by Born and Oppenheimer, the hope being that an 'exact' representation of Φ can be achieved in this form with ϕ and χ interpretable as 'electronic' and 'nuclear' wavefunctions respectively. We offer a mathematical analysis of these proposals that identifies ambiguities stemming mainly from the singularities in the Coulomb potential energy.

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Section: The Coulomb Hamiltonianmentioning

confidence: 88%

Section: The Coulomb Hamiltonianmentioning

confidence: 99%

Section: The Coulomb Hamiltonianmentioning

confidence: 99%

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On factorization of molecular wavefunctions

Jecko¹,

Sutcliffe²,

Woolley³

2015

J. Phys. A: Math. Theor.

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“…Note that the maximization gives the same order for the estimates as in Lemma 4. However in certain cases a slightly smaller maximum is achieved due to the constraint in (30).…”

Section: Ktmentioning

confidence: 99%

Sparse grids for the Schrödinger equation

Griebel¹,

Hamaekers²

2007

ESAIM: M2AN

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Abstract. We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system. Mathematics Subject Classification. 35J10, 65N25, 65N30, 65T40, 65Z05.

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“…Earlier the same authors had proven that ρ ψ is C ∞ [154]. Jecko [284] has an alternate proof of Theorem 19.9.…”

Section: As Withmentioning

confidence: 96%