  We present a novel theoretical formulation for performing quantum dynamics in
terms of moments within the single-particle description. By expressing the
quantum dynamics in terms of increasing orders of moments, instead of
single-particle wave functions as generally done in time-dependent density
functional theory, we describe an approach for reducing the high computational
cost of simulating the quantum dynamics. The equation of motion is given for
the moments by deriving analytical expressions for the first-order and
second-order time derivatives of the moments, and a numerical scheme is
developed for performing quantum dynamics by expanding the moments in the
Taylor series as done in classical molecular dynamics simulation. We propose a
few numerical approaches using this theoretical formalism on a simple
one-dimensional model system, for which an analytically exact solution can be
derived. Application of the approaches to an anharmonic system is also
discussed to illustrate their generality. We also discuss the use of an
artificial neural network model to circumvent the numerical evaluation of the
second-order time derivatives of the moments, as analogously done in the
context of classical molecular dynamics simulations.
