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Why is the chemical shift for hydrides in TM complexes so strongly shielded, to the point where it's negative?

Initially, I thought it was because hydrides are $\ce{H-}$, but isn't the charge meaningless once it coordinates to a metal centre?

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You essentially answer this question yourself. The chemical shifts of transition metal hydrides are often largely negative because they are highly shielded. That's what chemical shifts are a measure of - the amount of shielding from the external magnetic field. A very small hydride nucleus buried in a great big fluffy cloud of d-shell electrons of a transition metal is incredibly highly shielded from the base magnetic field.

The negative number of the chemical shift is arbitrary; what is relevant is that chemical shifts to the right are more shielded, and those to the left are less shielded.

Read the information about chemical shifts on this great online resource.

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