We have compared the TeV gamma-rays with the new 12CO J=2-1 data toward HESS J1745-303 in the Galactic center and confirmed that the molecular gas MG358.9-0.5 toward (l, b)=(358.9, -0.5 at VLSR=-100-0 km s-1 shows a reasonable positional agreement with the primary peak (northern part) of the gamma-ray source. For the southern part of HESS J1745-303, we see no CO counterpart, whereas the HI gas in the Parkes 21 cm HI dataset shows a possible counterpart to the gamma-ray source. This HI gas may be optically thick as supported by the HI line shape similar to the optically thick 12CO. We estimate the total mass of interstellar protons including both the molecular and atomic gas to be 2x10^6 Mo and the cosmic-ray proton energy to be 6x10^{48} ergs in the hadronic scenario. We discuss possible origins of the cosmic-ray protons including the nearby SNR G359.1-0.5. The SNR may be able to explain the northern gamma-ray source but the southern source seems to be too far to be energized by the SNR. As an alternative, we argue that the second-order Fermi acceleration in the inter-clump space surrounded by randomly moving high-velocity clumps may offer a possible mechanism to accelerate protons. The large turbulent motion with velocity dispersion of ~15 km s-1 has energy density two orders of magnitude higher than in the solar vicinity and is viable as the energy source.
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