In my opinion, your question is too broad. Since the class of core-shell particles is sufficiently wide as I know. I know that this particles are used as catalysts. That is, the catalyst is just on the shell and thus increases the efficiency of catalysis. Also it could be used for targeted transport in biomedical research or as a fluorescent label.
As for the gap, it will be something middle of the distance between of the HOMO-LOMO levels of simple molecule (a molecule may not exist, such as Au2 but the splitting of the levels could be estimated) and the distance between the C.B. and V.B. of the corresponding bulk state, but to be more precise, you need to separately investigate this particle. For example, if we consider a cluster of gold, we'll see what I wrote above. But the picture could be changed much if we cover this particle by some shell - it is usually some organic and additional levels should be investigated for each case.
Perhaps you would be worth to see something about quantum chemistry.
The main advantages of core-shell nanoparticles are (1) because of the shell material coating , properties of the core material such as reactivity can be decreased and thermal stability can be modified, so that the overall particle stability and dispersibility of the core particle increases and (2) the shell material can provide surface chemistry for further modification and functionalization of the nanoparticles.
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