  The density-matrix rho(r, r') of a spherically symmetric system can be
expanded as a Fourier-Legendre series of Legendre polynomials Pl(cos(theta) =
r.r'/rr'). 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 rho(r, r'). 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 P1(cos(theta)) = cos(theta). For example, in the case of
the 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 rho(r, r'), 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 r and r' variables.
