  A non-linear conjugate gradient optimization scheme is used to obtain
excitation energies within the Random Phase Approximation (RPA). The solutions
to the RPA eigenvalue equation are located through a variational
characterization using a modified Thouless functional, which is based upon an
asymmetric Rayleigh quotient, in an orthogonalized atomic orbital
representation. In this way, the computational bottleneck of calculating
molecular orbitals is avoided. The variational space is reduced to the
physically-relevant transitions by projections. The feasibility of an RPA
implementation scaling linearly with system size, N, is investigated by
monitoring convergence behavior with respect to the quality of initial guess
and sensitivity to noise under thresholding, both for well- and ill-conditioned
problems. The molecular- orbital-free algorithm is found to be robust and
computationally efficient providing a first step toward a large-scale, reduced
complexity calculation of time-dependent optical properties and linear
response. The algorithm is extensible to other forms of time-dependent
perturbation theory including, but not limited to, time-dependent Density
Functional theory.
