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1

The answer to this is very deep, and is rooted in mathematics, not physics. I don't think any chemistry textbook seriously explains what an irreducible representation "really" is, precisely because it's almost impossible to teach it to the average chemist. To properly understand this, I'd suggest reading about group theory, and then representation ...

3

When you have atoms bonded all in one plane, there will be $p$ orbitals oriented perpendicular to the plane which may not interact significantly with adjacent atoms. Such orbitals would then be called nonbonding. We may compare water with carbon dioxide. Introductory textbooks often describe the oxygen as having a distorted $sp^3$ hybridization, but in ...

3

Whether to use a centered cell or the smaller primitive cell is a question of convention. The conventions are guided by making life easy. You could take a primitive triclinic cell and redefine the cell axes and lengths to make it body-centered but that would be kind of silly. There is no gain in simplifying symmetry operations, and you just made your cell ...

4

Essentially, certain combinations of the possible point-group symmetries (cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic, triclinic) and possible translational symmetries (simple, base-centered, face-centered, body-centered) end up having identical overall lattice symmetries and thus you don't get $7×4$ unique lattices. For example, suppose ...

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