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Is there a hydrogen bond between the $\ce{H}$ atom of $\ce{-OH}$ group and the $\ce{O}$ atom of the $\ce{-OCH3}$ group in ortho-methoxyphenol? I was told no. But I want to know why.

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According to a quick calculation on the DF-BP86/def2-SVP level of theory, there is an intramolecular hydrogen bond with a strength of about $\ce{-26.6 kJ/mol}$ according to a quantum theory of atoms in molecule (QTAIM) analysis.

laplacian plot

For an explanation of the evaluation of the bond strength see: E. Espinosa, E. Molins, C. Lecomte. Chem. Phys. Lett. 1998, 285, 170-173.
Analysis done with MultiWFN 3.3.8. Calculations with Gaussian 09 D.01.

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Ortho-methoxyphenol actually can form an intramolecular hydrogen bond, but this can be altered by protic hydrogen acceptor solvents such as DMSO. See https://www.ncbi.nlm.nih.gov/pubmed/19473035 (abstract available in front of the paywall).

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I recommend looking up the relevant bond lengths and seeing if the H and the O "match up" at the right distance to form a hydrogen bond, as dictated by the geometry of the molecule. H-bonds tend between 1.6 and 2 angstroms.

Also note that the most effective hydrogen bonds have all three atoms co-linear (X-H~X); the hydrogen bond becomes weaker as the angle deviates from 180°. You could find the theoretical angle your three H-bonding atoms will share (assuming the distance is reasonable for an H-bond to occur in the first place), and consider how much this angle deviates from 180°.

This brief article mentions models that include the H-bond energy's dependence on the H-bond angle, where the H-bond energy is proportional to some power of the cosine of the angle:

$$U \propto \cos^n(\theta)$$

By this model, when the angle becomes a right angle (a "side-on" H-bond), the potential energy rises to zero. An H-bond in this perpendicular orientation is no longer a stabilizing interaction. So you will want your angle to be between 180° ($\pi$) and 135° ($3\pi/4$) or so to have an interaction worth calling a hydrogen bond.

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    $\begingroup$ So why is there at least some intramolecular hydrogen bonding in ethylene glycol? ncbi.nlm.nih.gov/pubmed/16316257 $\endgroup$ Commented Apr 3, 2017 at 18:26
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    $\begingroup$ I have seen plenty of examples of molecules completing a six or five membered ring through H bond. How does it happen then? $\endgroup$
    – Sawarnik
    Commented Apr 3, 2017 at 19:20

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