Rayleigh scattering correlation microspectroscopy is developed and applied to study diffusion dynamics of some nanospheres in water. It was clearly found that the diffusion constant of gold nanoparticles decreased with increasing excitation laser power at the excitation wavelength of higher absorption cross section. This behavior was explained in terms of a coupling between laser trapping by the scattering excitation laser itself and laser heating of the particle. In the case of non-absorbing nanospheres such as silica and polystyrene, the excitation power dependence can be ascribed only to the laser trapping. Experimental setup is introduced, theoretical formulation is described, and future development of this measurement is considered.