^{1,a)}

### Abstract

The Shannon information entropy of 1-normalized electron density in position and momentum space and , and the sum , respectively, are reported for the ground-state H, , , , He, , Li, and B atoms confined inside an impenetrable spherical boundary defined by radius . We find new characteristic features in denoted by well-defined minimum and maximum as a function of confinement. The results are analyzed in the background of the irreducible lower bound stipulated by the entropyuncertainty principle [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys.44, 129 (1975)]. The spherical confinement model leads to the values which satisfy the lower bound up to the limits of extreme confinements with the interesting new result displaying regions over which a set of upper and lower bounds to the information entropy sum can be locally prescribed. Similar calculations on the H atom in excited states are presented and their novel characteristics are discussed.

The author (K.D.S.) is indebted to Dr. H. E. Montgomery, Jr. for the constant motivation and encouragement throughout the course of this work and especially checking out the calculations at using accurate variational wave function. The financial support received from the Department of Science and Technology, New Delhi is gratefully acknowledged. This work is dedicated to Professor R.G. Parr on the occasion of his 84th birthday.

I. INTRODUCTION

II. SHANNON ENTROPY OF FREE ATOMS

III. SPHERICALLY CONFINED ATOM MODEL

IV. METHOD OF CALCULATIONS

V. RESULTS AND CONCLUSIONS

A. Confined -like atoms

B. Confined -like atoms

C. Confined and atoms

D. Confined H-excited state

E. Shell-confined and states

### Key Topics

- Entropy
- 54.0
- Ground states
- 15.0
- Wave functions
- 11.0
- Boundary value problems
- 7.0
- Numerical modeling
- 7.0

## Figures

The model of shell confinement within the concentric spherical shell of impenetrable boundaries. The radius of inner and outer boundaries are the variables which define the confinement.

The model of shell confinement within the concentric spherical shell of impenetrable boundaries. The radius of inner and outer boundaries are the variables which define the confinement.

(a) The variation with the radius of confinement of Shannon information entropy in position space and Shannon information entropy in momentum space for the H-like atoms in the ground state. (b) The variation with the radius of confinement of Shannon information entropy sum for the H-like atoms in the ground state.

(a) The variation with the radius of confinement of Shannon information entropy in position space and Shannon information entropy in momentum space for the H-like atoms in the ground state. (b) The variation with the radius of confinement of Shannon information entropy sum for the H-like atoms in the ground state.

The variation with the radius of confinement of Shannon information entropy sum for the He-like atoms in the ground state. The results are derived from the BLYP functional.

The variation with the radius of confinement of Shannon information entropy sum for the He-like atoms in the ground state. The results are derived from the BLYP functional.

The variation with the radius of confinement of Shannon information entropy sum for the ground-state Li and B atoms. The results are derived from the KLI functional.

The variation with the radius of confinement of Shannon information entropy sum for the ground-state Li and B atoms. The results are derived from the KLI functional.

A comparison of the variation with the radius of confinement of Shannon information entropy sum for the H atom in the excited state with that in the ground state.

A comparison of the variation with the radius of confinement of Shannon information entropy sum for the H atom in the excited state with that in the ground state.

The variation with the radius of confinement of Shannon information entropy sum for the and atom in the *shell*-confined state (see Fig. 1). The inner radius is fixed at for both atoms. The results for He are derived using the BLYP functional.

The variation with the radius of confinement of Shannon information entropy sum for the and atom in the *shell*-confined state (see Fig. 1). The inner radius is fixed at for both atoms. The results for He are derived using the BLYP functional.

## Tables

The numerical results of total energy, the Shannon entropy in position space , in momentum space , and the Shannon entropy sum , for the ground-state H-like atoms: H, , and . All values are in a.u. The last entry under each atom refers to the free state.

The numerical results of total energy, the Shannon entropy in position space , in momentum space , and the Shannon entropy sum , for the ground-state H-like atoms: H, , and . All values are in a.u. The last entry under each atom refers to the free state.

The numerical values of total energy, the Shannon entropy in position space , in momentum space , and the Shannon entropy sum , for the ground-state He-like atoms: , He, and , using the BLYP functional. All values are in a.u.

The numerical values of total energy, the Shannon entropy in position space , in momentum space , and the Shannon entropy sum , for the ground-state He-like atoms: , He, and , using the BLYP functional. All values are in a.u.

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