  In this work we present a derivation of the real-time time-dependent
orbital-optimized M{\o}ller-Plesser TDOMP2 and its biorthogonal companion,
time-dependent non-orthogonal OMP2 (TDNOMP2), theory starting from the
time-dependent bivariational principle and a parametrization based on the
exponential orbital-rotation operator formulation commonly used in
time-independent molecular electronic-structure theory. We apply the TDOMP2
method to extract absorption spectra and frequency-dependent polarizabilities
and first hyperpolarizabilities from real-time simulations, comparing the
results with those obtained from conventional time-dependent coupled-cluster
singles and doubles (TDCCSD) simulations and from its second-order
approximation TDCC2. We also compare with results from CCSD and CC2 linear and
quadratic response theory. Our results indicate that while TDOMP2 absorption
spectra are of the same quality as TDCC2 spectra, frequency-dependent
polarizabilities and hyperpolarizabilties from TDOMP2 simulations are
significantly closer to TDCCSD results than those from TDCC2 simulations.
