The　elastic　properties　of　solid　solution　structure　of　the　transition　metal　atomistic　Ti,　Zr,　Hf,　V,　Nb　and　Ta　in　CeO2crystal　are　studied　with　using　first　principle　method,　which　is　based　on　the　density　functional　theory　(DFT).　The　investigation　presents　some　results.　The　elastic　constants　(B,　G,　E,　C11and　C12)　of　CeO2and　solid　solution　structure　of　CeO2-M　system　have　increased,　after　which　the　mechanical　properties　of　CeO2-Nb　better　other　solid　solution　structure.　The　elastic　constants　of　C44of　CeO2-Ti　and　CeO2-V　marked　decline　phenomenon.　The　elastic　modulus　of　solid　solution　CeO2-Nb　significantly　increased,　and　anisotropy　is　not　significant　difference　similar　as　CeO2.　Also,　the　solid　solution　of　Ti,　V,　Ta　in　CeO2crystal　of　elastic　modulus　anisotropy　has　a　significant　difference.　The　solid　solution　of　Nb,　Zr,　Hf　in　CeO2crystal　anisotropy　of　the　elastic　modulus　are　not　prominent.　By　calculating　the　energy　band　and　density　of　states　shows　that　CeO2is　an　insulator,　and　there　are　two　band　gap.　©　2015,　Editorial　Office　of　＇Chinese　Rare　Earths＇.　All　right　reserved.