How do I find out the total number of stereoisomers of 3,6-dimethylpiperazine-2,5-dione?

As far as I can see, the compound has two chiral carbons. Would the nitrogens make a difference? They seem chiral as well (three different groups and a lone pair).

My book says there are 3 total stereoisomers, but I expected more. It has considered only two chiral carbons, and nothing has been said about the nitrogens. Why?

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    $\begingroup$ Nitrogen's lone pair is conjugated with C=O and effectively doesn't exist. $\endgroup$ – Ivan Neretin Apr 28 '18 at 15:06
  • $\begingroup$ Oh, right! Good observation! $\endgroup$ – epsilon-emperor Apr 28 '18 at 15:08
  • $\begingroup$ Is that the only reason? How can you be so sure? Do you have a source? $\endgroup$ – epsilon-emperor Apr 28 '18 at 15:13
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    $\begingroup$ Yes I think that's the only reason. I can be so sure because I've learned some chemistry in my life. Source? What source? $\endgroup$ – Ivan Neretin Apr 28 '18 at 15:30
  • $\begingroup$ The meso compound in this example is quite interesting looking... $\endgroup$ – Zhe Apr 28 '18 at 15:41

I believe this is a sophomore organic chemistry text book question, so I won't elaborate possible stereoisomers involving lone pairs on nitrogen atoms. Nonetheless, 3,6-dimethylpiperazine-2,5-dione has 2 chiral carbon centers so that it should theoretically have $2^2 = 4$ stereoisomers. If you look at the molecule, it has center of symmetry (i):

enter image description here

Therefore, two of the isomers are identical and we call it meso-isomer. Thus, total isomers of this molecule are three: ($3R,6R$)-, ($3S,6S$)-, and ($3R,6S$)- (($3S,6R$)- is essentially ($3R,6S$)-).


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