  We recently showed that the Dephasing Representation (DR) provides an
efficient tool for computing ultrafast electronic spectra and that further
acceleration is possible with cellularization [M. \v{S}ulc and J.
Van\'i\v{c}ek, Mol. Phys. 110, 945 (2012)]. Here we focus on increasing the
accuracy of this approximation by first implementing an exact Gaussian basis
method, which benefits from the accuracy of quantum dynamics and efficiency of
classical dynamics. Starting from this exact method, the DR is derived together
with ten other methods for computing time-resolved spectra with intermediate
accuracy and efficiency. These methods include the Gaussian DR, an exact
generalization of the DR, in which trajectories are replaced by communicating
frozen Gaussian basis functions evolving classically with an average
Hamiltonian. The newly obtained methods are tested numerically on time
correlation functions and time-resolved stimulated emission spectra in the
harmonic potential, pyrazine S0/S1 model, and quartic oscillator. Numerical
results confirm that both the Gaussian basis method and the Gaussian DR
increase the accuracy of the DR. Surprisingly, in chaotic systems the Gaussian
DR can outperform the presumably more accurate Gaussian basis method, in which
the two bases are evolved separately.
