For some molecules, the mirror image molecules can’t be rotated in any way to match the originals. They are different molecules, but because the atomic distances and angles are all the same, the energy levels must all be same (in a symmetric environment).
Is it possible for this relationship to extend beyond pairs of molecules? Could there be, for instance, four molecules that share the same atomic composition, distances, and angles, but still cannot be rotated to match up with any of the other three? Is there a mathematical principle that prevents this?
Molecules can have more than one chiral centre, yes. (Where a chiral centre is an asymmetric carbon atom, usually.)
A molecule with n chiral centres will have 2[sup]n[/sup] stereoisomers.
You will end up with 2[sup]n[/sup]/2 pairs of mirror-image enantiomers, all of which are distinct stereosiomers. (The diagram here makes that clearer than my explanation, I hope.)
There is a mathematical principle that prevents this, as Lumpy essentially notes.
If two molecules located in space share the same atomic composition, distances, angles, etc., then the way in which they match up corresponds to an “automorphism” of 3d space (a function from points in space to points in space which preserves all the relevant properties of size, angles, etc.) which takes the positions of the one to the corresponding positions of the other. Ignoring the translational aspect, this amounts to an automorphism of 3d vectors. Not every automorphism of 3d vectors is a rotation, but every automorphism of 3d vectors is either a rotation or a rotation combined with a reflection (the former being the automorphisms with determinant 1, and the latter being the automorphisms with determinant -1; because of preservation of size, etc., the determinant must have absolute value 1, so these are the only possibilities). So the kinds of molecules to which anyone molecule is isomorphic are all either rotations of that molecule, or rotations of a reflection of that molecule, with no 3rd possibility available.
I believe this answers my question. I was trying to imagine if there was a transformation (besides reflection) that would convert a molecule into a distinguishably different molecule, but with all the same bonds – in particular, a transformation that would allow there to be more than one other such partner. But if automorphisms (a new term for me) are restricted as above, then there can be at most two automorphic versions of the same molecule.
As the esteemed **indistinguishable ***almost *said, it gets messier in 4- or 5-space. But as far as we know today, that’s not the environment we’re living in.
I guess the easiest way to think about his is that a molecule can have only one mirror image, and it doesn’t matter where the mirror is placed. Whenever you mirror twice you will be back at the original molecule. That limits the group to two.
As to a specific, say you have two chiral centers, an R and an R. The mirror will be S and S and have the same potential energy.
However, the R S combo will not have the same potential energy as the RR because the two centers interact with each other. You can think of it as the molecule itself being its own asymmetric environment. (of course, the S R mirror will have the same potential energy as the R S.) It works similarly when the chirality is more complicated than just specific centers.
I guess the messiest thing is pinning down the definition of rotation in this context (is “rotation” always rotation by some angle around some axis? Or is any linear orientation-preserving isometry a rotation [i.e., the rotations are precisely the elements of SO(n)]? Or some other choice of how to use the term?).
If “rotations” are taken to be precisely the elements of SO(n), it seems to me there are still only the two possibilities relating isomorphic configurations, in any positive number of dimensions: either they are related by a rotation, or they are related by a rotation and reflection.
(True story: As I went to bed last night, I was suddenly struck with the thought “Oh, shit, why’d I word that SDMB post that way? What about symmetric objects? I should clarify that there are always just the two possibilities, which sometimes merge into the same one possibility according to symmetry. And once I no longer mean “two” in the strictly distinct sense, that obviates the reason I said ‘positive number of dimensions’ rather than ‘arbitrary number of dimensions’, so I should note that was pointless. Ah, I’ll fix it in the morning…”. Well, here I am.)
There are often multiple chiral centers in a molecule. They can all be R or S sometimes, as in a peptide or other chain. The pairs that are mirror images are ensntiomers to each other. The ones that are for example R, R, S and R, S, S are called diastereomers. They have different physical properties.
I was interpreting the requirement for all distances and angles to remain the same as genuinely meaning all distances (between any two atoms in the molecule) and all angles (between any three atoms in the molecule). If I understand correctly, diastereomers will have the same distances and angles between “adjacent” atoms, but will not have all the same distances and angles at a global level.