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Plasmonics is a rapid growing field, which has enabled both fundamental science and inventions of various quantum optoelectronic devices. An accurate and efficient method to calculate the optical response of metallic structures with feature size in the nanoscale plays an important role. Quantum hydrodynamic theory (QHT) provides an efficient description of the free-electron gas, where quantum effects of nonlocality and spill-out are taken into account. In this work, we introduce a general QHT that includes diffusion to account for the broadening, which is a key problem in practical applications of surface plasmon. We will introduce a density-dependent diffusion coefficient to give very accurate linewidth. It is a self-consistent method, in which both the ground and excited states are solved by using the same energy functional, with the kinetic energy described by the Thomas-Fermi and von Weizs\"{a}cker (vW) formalisms. In addition, our QHT method is stable by introduction of an electron density-dependent damping rate. For sodium nanosphere of various sizes, the plasmon energy and broadening by our QHT method are in excellent agreement with those by density functional theory and Kreibig formula. By applying our QHT method to sodium jellium nanorods, we clearly show that our method enables a parameter-free simulation, i.e. without resorting to any empirical parameter, such as size-dependent damping rate and diffusing coefficient. It is found that there exists a perfect linear relation between the resonance wavelength and aspect radio. The width decreases with increasing aspect ratio and height. The calculations show that our QHT method provides an explicit and unified way to account for size-dependent frequency shifts and broadening of arbitrarily shaped geometries. It is reliable and robust with great predicability, and hence provides a general and efficient platform to study plasmonics.