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Is the lone electron pair of the nitrogen atom part of the conjugated π system in the α,β-unsaturated amide pictured below?

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Is there a general rule for choosing when to include a nitrogen atom in a Hückel matrix?

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  • $\begingroup$ I have edited your post for clarity. At the risk of incurring ill-will from our German folks, I have replaced your "ue" (yes, it's correct; no, most people don't know that) in the title and added back the umlaut in the text. $\endgroup$
    – Todd Minehardt
    Commented Jul 15, 2021 at 23:14
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    $\begingroup$ Pro tip: get to know a compose key and never fear Unicode German or Greek letters again with simple and intuitive keystrokes like RAlt " u for letter ü or RAlt * p for letter π (cc @ToddMinehardt). $\endgroup$
    – andselisk ♦
    Commented Jul 15, 2021 at 23:22
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    $\begingroup$ @ToddMinehardt So to get it right - is or is not ue acceptable EN transcription for DE ü ? ( I guess such transcriptions would get hard time for CZ ones ) // An alternative is to have a short list of frequent UC characters and codes, performing either copy/paste either invoking unicode insertion by the code, like CTRL+SHIFT+U<unicode>ENTER on Linux. $\endgroup$
    – Poutnik
    Commented Jul 16, 2021 at 8:54
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    $\begingroup$ Actually I am German, so I have the ü right on my keyboard. I have just always read the "ue" spelling in English literature, so I thought I'd not confuse you with umlauts :D $\endgroup$
    – Charles Taylor
    Commented Jul 16, 2021 at 9:12
  • $\begingroup$ I guess you can include the lone pair in the Huckel matrix, but it will probably become very difficult to solve by hand. If you have a computer, that will solve it. If you ignore the lone pair, your system is only a linear chain of p-orbitals, it will give that diagonal looking matrix (not a diagonal matrix!!) that you could solve with a pen and paper. $\endgroup$
    – S R Maiti
    Commented Jul 16, 2021 at 10:41

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Setting aside the arguments over umlauts, yes you can add the nitrogen atom to the Hückel matrix; in fact, theoretically you should do so and allow the matrix eigenfunctions and eigenvalues to decide whether the nitrogen atom in this case is really conjugated. The carbonyl oxygen can also be coupled in.

But there is a catch. Since the nitrogen or oxygen atom has different atomic orbital energy levels from the carbon atoms, and the orbital overlap with other atoms will be different, you cannot just put in the same numbers that you would with a carbon atom there. Various estimates are available for these heteroatom parameters, for instance this one given by Texas A&M University which includes common heteroatoms for the diagonal elements and their bonds with carbon for the off-diagonal elements.

Once you have decided upon the matrix elements you can solve the matrix for its eigenfunctions (molecular orbitals) and eigenvalues (energy levels) the same way you would with an all-carbon system, but the equations will probably be more complex and require numerical methods to solve efficiently.

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