Crystal energy, quantum mechanical effects

The reason for the formation of a lattice can be the isotropic repulsive force between the atoms in some simple models for the crystalhzation of metals, where the densely packed structure has the lowest free energy. Alternatively, directed bonds often arise in organic materials or semiconductors, allowing for more complicated lattice structures. Ultimately, quantum-mechanical effects are responsible for the arrangements of atoms in the regular arrays of a crystal. 

Classical electrostatic modeling based on the Coulomb equation demonstrated that the model system chosen could account for as much as 85% of the effect of the protein electric field on the reactants. Several preliminary computations were, moreover, required to establish the correct H-bond pattern of the catalytic water molecule (WAT in Fig. 2.6). Actually, in the crystal structure of Cdc42-GAP complex [60] the resolution of 2.10 A did not enable determination of the positions of the hydrogen atoms. Thus, in principle, the catalytic water molecule could establish several different H-bond patterns with the amino acids of the protein-active site. Both classical and quantum mechanical calculations showed that WAT, in its minimum-energy conformation,... 

Figure 13.4 Torsional potential energy surfaces about the two C-O bonds linking the anomeric centers of sucrose at the MM3 level (a), 2-tetrahydrofuranyl-2-tetrahydropyranyl ether at the MM3 level (b), the same ether at the HF/6-31G(d) level (c), and the sum of the difference between the last two with the first (d). Thus, the last surface may be viewed either as the effect of the sucrose hydroxyl groups on the energy surface, evaluated at the MM3 level, added to the framework surface calculated at the ab initio level, or as an MM3 surface that has been partially conected quantum mechanically. Solid triangles represent anomeric torsions in sucrose units found in various X-ray crystal structures. Note that the hybrid surface is the only one that clusters the large majority of these triangles within low-energy contours...

It appears from the above discussion that the most satisfactory approach to the quantum mechanical calculation of lattice energies is that developed by Yamashita, in which the parameters of the outer wave functions of the ions are adjusted by a variational method to minimize the total energy of the crystal. Orthogonalization of the simple free ion wave functions seems to produce a result rather worse than that achieved by ignoring the correction. No doubt with the availability of electronic computers Yamashita s method will be extended to crystals in addition to LiF, where it may be necessary to adjust the wave functions of both the ions by a variational method, to allow for the effect of the crystal field. This will produce an exceedingly tedious calculation. Yamashita (1S2) has also used the method described above to show that the 0 ion is stable in the MgO crystal, though not in the gas phase. 

Kelvin (the zero point motion). This latter effect is explained by quantum mechanics, and it can in turn explain absorption features of impurities in crystalline matrices. The presentation of the fundamental vibrational modes of crystals is based on the harmonic approximation, where one only considers the interactions between an atom or an ion and its nearest neighbours. Within this approximation, an harmonic crystal made of N ions can be considered as a set of 3N independent oscillators, and their contribution to the total energy of a particular normal mode with pulsation ivs (q) is ... 

---