^{1,a)}and Russell J. Boyd

^{1,b)}

### Abstract

The local quantum theory is applied to the study of the momentum operator in atomic systems. Consequently, a quantum-based local momentum expression in terms of the single-electron density is determined. The limiting values of this function correctly obey two fundamental theorems: Kato’s cusp condition and the Hoffmann-Ostenhof and Hoffmann-Ostenhof exponential decay. The local momentum also depicts the electron shell structure in atoms as given by its local maxima and inflection points. The integration of the electron density in a shell gives electron populations that are in agreement with the ones expected from the Periodic Table of the elements. The shell structure obtained is in agreement with the higher level of theory computations, which include the Kohn–Sham kinetic energy density. The average of the local kinetic energy associated with the local momentum is the Weizsäcker kinetic energy. In conclusion, the local representation of the momentum operator provides relevant information about the electronic properties of the atom at any distance from the nucleus.

Erin R. Johnson, Axel D. Becke, Chérif F. Matta, Jason K. Pearson, and Dan Matusek are gratefully acknowledged for helpful discussions and suggestions.

Thanks to the Natural Sciences and Engineering Research Council of Canada for financial support. Computational facilities are provided in part by ACEnet, the regional high performance computing consortium for universities in Atlantic Canada. ACEnet is funded by the Canada Foundation for Innovation (CFI), the Atlantic Canada Opportunities Agency (ACOA), and the provinces of Newfoundland and Labrador, Nova Scotia, and New Brunswick.

I. INTRODUCTION

II. QUANTUM-STATISTICAL ESTIMATION OF LOCAL OBSERVABLES

III. LOCAL ESTIMATE OF MOMENTUM OPERATORIN COORDINATE REPRESENTATION

A. Local kinetic energy theorem

IV. RESULTS

A. Bohr atomic model

B. The local momentum at the origin is given by Kato’s cusp condition

C. The local momentum in the middle range describes the shell structure of atoms

V. CONCLUSIONS

### Key Topics

- Quantum mechanics
- 14.0
- Density functional theory
- 9.0
- Kinetic theory
- 9.0
- Ionization potentials
- 6.0
- Particle distribution functions
- 6.0

## Figures

Radial distribution function and local momentum for H and He at numerical Hartree–Fock level.

Radial distribution function and local momentum for H and He at numerical Hartree–Fock level.

Ionization potential as derived from local theory , Koopmans’ theorem , and experiment .

Ionization potential as derived from local theory , Koopmans’ theorem , and experiment .

Local electron momentum (logarithmic scale) vs the radial distribution function (linear scale) for the initial and final elements of the fifth row of the Periodic Table (Ref. 28).

Local electron momentum (logarithmic scale) vs the radial distribution function (linear scale) for the initial and final elements of the fifth row of the Periodic Table (Ref. 28).

The shell structure of the calcium atom as given by the local momentum. The shells are denoted by capital letters and the corresponding number of electrons are given below the electronic population curve, (in gray). The inflection points that separate each shell are given by the circles.

The shell structure of the calcium atom as given by the local momentum. The shells are denoted by capital letters and the corresponding number of electrons are given below the electronic population curve, (in gray). The inflection points that separate each shell are given by the circles.

Local electron momentum for closed shells.

Local electron momentum for closed shells.

## Tables

Valence local kinetic energies and the ionization energies from Koopmans’ theorem, , and experimental ionization energies (Ref. 46).

Valence local kinetic energies and the ionization energies from Koopmans’ theorem, , and experimental ionization energies (Ref. 46).

Shell radii and electron populations at the HF level as given by the inflection points of the local momentum, as given by Eq. (9) (first row), from the local minima in ELF (second row; Ref. 53) and KSKE density by Navarrete-Lopez *et al.* (third row; Ref. 54), and by Schmider and Becke (fourth row; Ref. 55). is the electron density enclosed at a distance of 10 a.u.

Shell radii and electron populations at the HF level as given by the inflection points of the local momentum, as given by Eq. (9) (first row), from the local minima in ELF (second row; Ref. 53) and KSKE density by Navarrete-Lopez *et al.* (third row; Ref. 54), and by Schmider and Becke (fourth row; Ref. 55). is the electron density enclosed at a distance of 10 a.u.

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