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Is there an online database having information on molecules with a certain number of atoms? For example, properties of tri-atomic molecules?

There is this table that I got from NIST which gives a few, but it would be nice to know if more such lists exist.

It would be even better if these molecules could be further categorized according to their type of geometry (for example, not just "triatomic molecules" but "linear triatomic molecules") but the example that I gave from NIST doesn't go this far.

Thanks.

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CRC Handbook of Chemistry and Physics

The example you gave from NIST gives about 55 geometries for triatomic molecules. The CRC Handbook gives a plethora of properties for diatomics (just search "diatomic" in a PDF version of the book) and also gives electron affinities (at least) for triatomics. In the 97th Edition (2016), which is the newest version for which I immediately have a PDF available, the data is organized as follows for electron affinities, with properties for atoms in alphabetical order first, followed by the diatomic molecules in alphabetical order next:

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After the last diatomics that they have alphabetically are finished, triatomics are listed in alphabetical order:

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After the last triatomics that they have alphabetically are finished, polyatomics are listed in alphabetical order:

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The triatomics list here is significantly longer than the one in your example from NIST.

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