Space groups model building

Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ah initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). 

The picture of molecules being composed of structural units, functional groups , which behave similarly in different molecules forms the very basis of organic chemistry. The drawing of molecular structures where alphabetic letters represent atoms and lines represent bonds is used universally. Organic chemists often build ball and stick, or CPK space-filling, models of their molecules to examine their shapes Force, field methods are... 

To chemists, it makes perfect sense to think of electrons as being assigned to particular orbitals, and therefore the task of modeling the electronic ground state means building a wavefunction that describes these orbitals. The better the description of the orbitals (i.e. the better the basis set), the better the resulting calculation. This type of basis set is referred to as a localized basis set, and it can equally well be used to describe molecular crystals as well as isolated molecules, provided that the basis set is replicated by the space-group symmetry operations that describe how all molecules are located with respect to one another in the crystalline lattice. 

We have introduced an important concept here - the unit cell. In crystallography, the unit cell represents the budding block from which the infinite three-dimensional crystal lattice is built. If we are to model solid-state systems we must make use of a similar concept, from which we can build an infinite array of rephcas positioned in accordance with the crystallographic space-group symmetry operations. 

Clearly, the highlighted carbon is a chiral centre (it has four different groups attached to it). For this reason, the two protons Ha and Hb can never be in the same environment. The fact that there is free rotation around all the single bonds in the molecule is irrelevant. This can best be appreciated by building a model of the molecule. Having done so, look down the molecule from left to right as drawn and rotate the C-0 bonds so that Ha and Hb rotate. It should now be clear why these two protons can never occupy the same space and are therefore not equivalent. 

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