Analytic Structure of Many-Body Coulombic Wave Functions

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

doi:10.1007/s00220-008-0664-5

Cited by 56 publications

(86 citation statements)

References 28 publications

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“…However, it has been pointed out by Fournais et al that higher order regularity can be expected for a more general additive ansatz. 13 This is consistent with the early work of Kutzelnigg and Morgen, 14 where, based on such an additive ansatz, different cusp conditions are applied to different types of electron pairs, and thus a higher order convergence (such as M −7/3 ) is achieved. Unfortunately such kind of additive ansatz is not size consistent.…”

Section: Introductionsupporting

confidence: 88%

Combining the Transcorrelated Method with Full Configuration Interaction Quantum Monte Carlo: Application to the Homogeneous Electron Gas

Luo

Alavi

2018

J. Chem. Theory Comput.

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We suggest an efficient method to resolve electronic cusps in electronic structure calculations, through the use of an effective transcorrelated Hamiltonian. This effective Hamiltonian takes a simple form for plane wave bases, containing up to two-body operators only, and its use incurs almost no additional computational overhead compared to that of the original Hamiltonian. We apply this method in combination with the full configuration interaction quantum Monte Carlo (FCIQMC) method to the homogeneous electron gas. As a projection technique, the non-Hermitian nature of the [physics.comp-ph]

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Section: Introductionsupporting

confidence: 88%

Combining the Transcorrelated Method with Full Configuration Interaction Quantum Monte Carlo: Application to the Homogeneous Electron Gas

Luo

Alavi

2018

J. Chem. Theory Comput.

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“…Our technique of proof can be directly extended to this situation. This observation might be helpful in the approximation of the wave functions in view of their analytic structure outside the coalescence points of more than two particles [5].…”

Section: Introductionmentioning

confidence: 90%

“…This can be deduced without much effort from the results in [5]. The regularizing factor thus covers the singularities of the eigenfunctions in the neighborhood of such points completely.…”

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confidence: 93%

The mixed regularity of electronic wave functions multiplied by explicit correlation factors

Yserentant¹

2011

ESAIM: M2AN

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Abstract. The electronic Schrödinger equation describes the motion of N electrons under Coulombinteraction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics 2000, Springer (2010], the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The present paper complements this work. It is shown that one can reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.

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“…It often serves as a model for electronic wave functions. Its singular behavior at the diagonal x 1 = x 2 is the same as that of the solutions of the electronic Schrödinger equation at the positions where two electrons of distinct spin meet [7]. The function represents at the same time the ground state of the so-called hookium or harmonium atom [11], an artificial two-electron system with the Hamiltonian…”

Section: A Counterexamplementioning

confidence: 87%

The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces

Kreusler

Yserentant

2012

Numer. Math.

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We continue the study of the regularity of electronic wave functions in Hilbert spaces of mixed derivatives. It is shown that the eigenfunctions of electronic Schrödinger operators and their exponentially weighted counterparts possess, roughly speaking, square integrable mixed weak derivatives of fractional order ϑ for ϑ < 3/4. The bound 3/4 is best possible and can neither be reached nor surpassed. Such results are important for the study of sparse grid-like expansions of the wave functions and show that their asymptotic convergence rate measured in terms of the number of ansatz functions involved does not deteriorate with the number of electrons. Mathematics Subject Classification (2000)35J10 · 35B65 · 41A25 · 41A63

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Analytic Structure of Many-Body Coulombic Wave Functions

Cited by 56 publications

References 28 publications

Combining the Transcorrelated Method with Full Configuration Interaction Quantum Monte Carlo: Application to the Homogeneous Electron Gas

Combining the Transcorrelated Method with Full Configuration Interaction Quantum Monte Carlo: Application to the Homogeneous Electron Gas

The mixed regularity of electronic wave functions multiplied by explicit correlation factors

The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces

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