### Abstract

The density matrix of a spherically symmetric system can be expanded as a Fourier–Legendre series of Legendre polynomials. Application is here made to harmonically trapped electron pairs (i.e., Moshinsky’s and Hooke’s atoms), for which exact wavefunctions are known, and to the helium atom, using a near-exact wavefunction. In the present approach, generic closed form expressions are derived for the series coefficients of . The series expansions are shown to converge rapidly in each case, with respect to both the electron number and the kinetic energy. In practice, a two-term expansion accounts for most of the correlation effects, so that the correlated density matrices of the atoms at issue are essentially a linear functions of . For example, in the case of Hooke’s atom, a two-term expansion takes in 99.9% of the electrons and 99.6% of the kinetic energy. The correlated density matrices obtained are finally compared to their determinantal counterparts, using a simplified representation of the density matrix , suggested by the Legendre expansion. Interestingly, two-particle correlation is shown to impact the angular delocalization of each electron, in the one-particle space spanned by the and variables.

Received 23 May 2008
Accepted 20 August 2008
Published online 30 September 2008

Acknowledgments:
The present authors were not aware of Ref. 14. They hereby thank the referee for drawing its content to their attention during the referring process.

Article outline:

I. INTRODUCTION
II. FOURIER–LEGENDRE EXPANSION OF THE “SPINLESS” DENSITY MATRIX
A. One-electron functions
B. Fourier–Legendre series expansion
III. EXPANSION OF THE DENSITY MATRIX IN THE REPRESENTATION
A. Moshinsky’s atom
B. Hookium
C. Helium atom
IV. CONVERGENCE
A. Moshinsky’s atom
B. Hookium
C. Helium atom
V. COMPARISON WITH UNCORRELATED MATRICES
VI. CONCLUSIONS

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