  For inhomogeneous interacting electronic systems under a time-dependent
electromagnetic perturbation, we derive the linear equation for response
functions in a quantum mechanical manner. It is a natural extension of the
original semi-classical Singwi-Tosi-Land-Sjoelander (STLS) approach for an
electron gas. The factorization ansatz for the two-particle distribution is an
indispensable ingredient in the STLS approaches for determination of the
response function and the pair correlation function. In this study, we choose
an analytically solvable interacting two-electron system as the target for
which we examine the validity of the approximation. It is demonstrated that the
STLS response function reproduces well the exact one for low-energy
excitations. The interaction energy contributed from the STLS response function
is also discussed.
