A semiclassical initial value representation (SC-IVR) for centroid dynamics (CD) is derived to provide a more accurate description of quantum time correlation functions (TCFs). The time-dependent quasi-density operator (QDO) in SC-IVR-CD is shown to exhibit more exact behavior than that of the CMD approximation, thereby improving the calculated quantum TCFs. Three typical one-dimensional model potentials (weak anharmonic, double well, and quartic) were studied as examples in calculating TCFs with the SC-IVR-CD method. Importantly, for intermediate temperatures, SC-IVR-CD gives a more exact description of the coherent behavior of the quantum TCFs than does centroid molecular dynamics (CMD).