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How do you compare the strength of H bonds when it's the same atom on both sides of the H? Like $\ce{N-H\bond{...}N}$ vs $\ce{F-H\bond{...}F}$? Is one still stronger than the other, or are they both the same since in both cases the pull on both sides is equal?

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In some cases, such as [F-H-F]- (biflouride), there is a symmetric hydrogen bond, with both H-F bonds equally strong.

However, just because the two atoms bonding to the H are the same doesn't mean it is a symmetric hydrogen bond. For example water dimer does not have a symmetrical hydrogen bond, the H is closer to one O atom than the other.

There can be cases where the hydrogen being exactly in the middle is a potential energy minimum, and other cases where the middle is not a local minimum but there are two minima offset from the middle.

See Symmetry of hydrogen bonds in solution for more information.

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