Can double and triple bonds, etc., extend beyond the plane in a Lewis structure, and if so, how are they represented? I.e., how would 1 and 2 in this image be drawn if they were doubles or triples?
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Yes, chemical bonds of higher order can be extended from the plane in some rare cases. There are IUPAC recommendations existing at least for double bonds suggesting to depict them simply as doubled wedge bonds if it's really necessary. From [1, p. 1930] (emphasis mine):
ST-1.5 Tetrahedral stereocenters including higher-order bonds
Tetrahedral configuration requires a tetrahedral arrangement of ligands around a central point. It does not require that all of those ligands be connected to a central atom using single bonds. It is perfectly reasonable to have one or more ligands connected to the central atom using higher-order bonds. This is most commonly seen with chiral sulfoximides, phosphates, and related compounds, each of which is commonly drawn with one or more double bonds. Stereocenters of this type should be indicated by the appropriate placement of bold and/or hashed wedged bonds connecting the central atom to the singly bonded ligands. Although wedged and hashed double bonds have appeared in the literature, they are rarely required for chemical clarity, and their use is infrequent enough to be confusing to the casual viewer (see ST-2.5).
Further from [1, pp. 1940-1942] (emphasis mine):
ST-2.5 Trigonal pyramidal Polyhedral symbol: TPY-3 Number of ligands: 3
In rare cases, a trigonal pyramidal atom with at least one double bond may be flanked by other stereocenters at the other ends of all of its single bonds. In such cases, one of the plain bonds should be changed to a wedged bond even though it joins two stereocenters. It is also acceptable to use a solid wedged or hashed wedged double bond, but that would not be preferred because wedged double bonds are so rarely seen in the literature.