Variational theorems for the single-particle probability density and density matrix in momentum space

Henderson, George A.

doi:10.1103/physreva.23.19

Cited by 61 publications

(28 citation statements)

References 3 publications

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“…The so-called frequency moments of the density play a relevant role within such a density functional theory framework [3,4]. Much effort has been made to obtain a similar formulation of this theory in the conjugated space, i.e., in terms of the momentum one-particle density γ ( p), with many successful results [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

Angulo

2011

Phys. Rev. A

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Rigorous and universal relationships among radial expectation values of any D-dimensional quantummechanical system are obtained, using Rényi-like position-momentum inequalities in an information-theoretical framework. Although the results are expressed in terms of four moments (two in position space and two in the momentum one), especially interesting are the cases that provide expressions of uncertainty in terms of products r a 1/a p b 1/b , widely considered in the literature, including the famous Heisenberg relationship r 2 p 2 D 2 /4. Improved bounds for these products have recently been provided, but are always restricted to positive orders a,b > 0. The interesting part of this work are the inequalities for negative orders. A study of these relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility, and the height of the peak of the Compton profile, among others.

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Section: Introductionmentioning

confidence: 99%

Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

Angulo

2011

Phys. Rev. A

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“…Position and momentum densities are not equivalent, but complementary, i.e., it is not possible to convert one into the other without additional assumptions. Although in principle, either one of them determines all groundstate properties ͑see the famous Hohenberg-Kohn theorem 52 and its momentum-space equivalent by Henderson 53 ͒, the functional form of this relationship is unknown, and the arising questions form the subject of a whole branch of quantum chemistry, densityfunctional theory ͑DFT͒.…”

Section: Discussionmentioning

confidence: 99%

‘‘Low-momentum electrons’’ and the electronic structure of small molecules

Schmider

1996

The Journal of Chemical Physics

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The electronic Husimi distribution (r ជ , p ជ ) is a ''fuzzy'' density in phase space. Sections through this function with a zero momentum variable (p ជ ϭ0), are shown to be indicative of the spatial locations of chemical bonds and ''free electron pairs'' in molecules. The distribution (r ជ ;0) tends to focus on the inter-nuclear regions in position space. The Laplacian ٌ r 2 (r ជ ;0), of the function may be used to enhance its diffuse features. The argument is made that the momentum-space Hessian of the Husimi function at the momentum-origin ( p ជ ϭ0), includes information about the ''flexibility'' of the electrons and the anisotropy of the latter. The diagonalization of this tensor supplies a pictorial map of preferred directions of electrons in the low-momentum, i.e., ''valence'' region of momentum space. Examples studied in this paper include the H 2 , N 2 , CH 4 , H 2 O, C 2 H 4 and C 6 H 6 systems in their Hartree-Fock approximation.

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“…The knowledge on the spherically-averaged momentum density γ(p) is very limited, even for ground-state neutral atoms, in spite of the recent progress in the experimental access to it by different techniques [1][2][3][4][5][6][7] and its demands as the basic quantity in the development of momentum-space density functional theories [8,9]. This is more striking, when we find in position space that numerical data abound for the spherically-averaged electron density ρ(r), especially at the Hartree-Fock level, as recently reviewed by Angulo et al [10].…”

Section: Introductionmentioning

confidence: 99%

Structure of the electron momentum density of atomic systems

Romera

Koga

Dehesa

1997

Z Phys D - Atoms, Molecules and Clusters

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The present paper addresses the controversial problem on the nonmonotonic behavior of the sphericallyaveraged momentum density γ(p) observed previously for some ground-state atoms based on the Roothaan-HartreeFock (RHF) wave functions of Clementi and Roetti. Highly accurate RHF wave functions of Koga et al. are used to study the existence of extrema in the momentum density γ(p) of all the neutral atoms from hydrogen to xenon. Three groups of atoms are clearly identified according to the nonmonotonicity parameter µ, whose value is either equal to, larger, or smaller than unity. Additionally, it is found that the function p −α γ(p) is (i) monotonically decreasing from the origin for α ≥ 0.75, (ii) convex for α ≥ 1.35, and (iii) logarithmically convex for α ≥ 3.64 for all the neutral atoms with nuclear charges Z = 1-54. Finally, these monotonicity properties are applied to derive simple yet general inequalities which involve three momentum moments p t . These inequalities not only generalize similar inequalities reported so far but also allow us to correlate some fundamental atomic quantities, such as the electron-electron repulsion energy and the peak height of Compton profile, in a simple manner.

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Variational theorems for the single-particle probability density and density matrix in momentum space

Cited by 61 publications

References 3 publications

Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

Generalized position-momentum uncertainty products: Inclusion of moments with negative order and application to atoms

‘‘Low-momentum electrons’’ and the electronic structure of small molecules

Structure of the electron momentum density of atomic systems

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