Reference unit cell

An infinite system which is periodic in p directions (p = 1, 2, 3 for polymers, surfaces, and crystals, respectively) is described by a reference unit cell and p basis vectors, ai,. ap. These vectors point to the directions of the periodicity... 

Let Xa(r) be atomic orbitals in the reference unit cell. Then according to the periodicty of the system, the atomic orbitals in the ith unit cell are iXla(r) = X (r R/) - With the atomic orbitals, one can construct symmetrized orbitals by... 

Consequently, to study the electronic stmcture of the infinite periodic chain we must first build the N sets of Bloch orbitals associated with each of the M atomic orbitals in the reference unit cell and then combine these Bloch orbitals to generate crystal orbitals. As Bloch orbitals associated with different k wave vectors belong to different irreducible representations of the translational group, by symmetry, only those Bloch orbitals associated with the same value of k can combine to give M different crystal orbitals for each k wave vector. 

The electronic orbital energy becomes a function of the FBZ points and we obtain what is known as band structure of the energy levels. This band structure decides the electronic properties of the system (insulator, semiconductor, metal). We will also show how to carry out the mean field (Hartree-Fock) computations on infinite periodic systems. The calculations require infinite summations (interaction of the reference unit cell with the infinite crystal) to be made. This creates some mathematical problems, which will be also described in the present chapter. 

The last sum in (2.16) is actually independent of / for the following reason. Here ij lk, I k ) depends on (/ — t), or on the distance between the unit cells considered, rather than on the actual position of any one of the unit cells in the lattice. The exponential argument depends on this difference only. In other words, the value of the sum over / will be the same, no matter what the value of / is set to be. We can therefore set / = 0 on some reference unit cell. Then we write... 

In the periodic systems the basis sets are chosen in such a way that they satisfy the Bloch theorem. Let a finite number of contracted GTFs be attributed to the atom A with coordinate in the reference unit cell. The same GTFs are then formally associated with all translationally equivalent atoms in the crystal occupying positions rA + (In (In is the direct lattice translation vector). For the crystal main region of N primitive unit cells there are N tha Gaussian-type Bloch functions (GTBF)... 

Consider a molecular crystal, like, for example, the one whose layers are depicted in Fig. 5.5. Each molecule is a recognizable entity composed of a number of atoms, Aat, each one in turn described by a number, Aorb.i, of atomic orbitals expressed in gaussian expansions as in equations 3.40-3.42. There are Z entire molecules in the unit cell. A reference unit cell (called the Ref-cell) is chosen, which contains a basis set of Nbs atomic orbitals, xj, with Nbs = Z J Aorb.i- The corresponding real and reciprocal space can then be defined (Section 5.4) ... 

Compositional range of C0 ri. O2 Phases and crystal system Space group References Unit-cell parameters... 

All atoms-in-molecules (AIM) methods partition the total electron density p r) into a set of components pa( a) normally associated with the system s atoms. The system s atoms A are located at positions. Ra, in a reference unit cell, U. For a non-periodic system (e.g. a molecule), U is any parallelpiped enclosing the entire electron distribution. The reference unit cell has k = k2 = = 0... 

It is thus tempting to define the first saturated layer as being one monolayer, and this often done, causing some confiision. One therefore also often uses tenns like saturated monolayer to indicate such a single adsorbate layer that has reached its maximal two-dimensional density. Sometimes, however, the word saturated is omitted from this definition, resulting m a different notion of monolayer and coverage. One way to reduce possible confiision is to use, for contrast with the saturated monolayer, the tenn fractional monolayer for the tenn that refers to the substrate unit cell rather than the adsorbate size as the criterion for the monolayer density. 

These simple examples serve to show that instinctive ideas about symmetry are not going to get us very far. We must put symmetry classification on a much firmer footing if it is to be useful. In order to do this we need to define only five types of elements of symmetry - and one of these is almost trivial. In discussing these we refer only to the free molecule, realized in the gas phase at low pressure, and not, for example, to crystals which have additional elements of symmetry relating the positions of different molecules within the unit cell. We shall use, therefore, the Schdnflies notation rather than the Hermann-Mauguin notation favoured in crystallography. 

The many commercially attractive properties of acetal resins are due in large part to the inherent high crystallinity of the base polymers. Values reported for percentage crystallinity (x ray, density) range from 60 to 77%. The lower values are typical of copolymer. Poly oxymethylene most commonly crystallizes in a hexagonal unit cell (9) with the polymer chains in a 9/5 helix (10,11). An orthorhombic unit cell has also been reported (9). The oxyethylene units in copolymers of trioxane and ethylene oxide can be incorporated in the crystal lattice (12). The nominal value of the melting point of homopolymer is 175°C, that of the copolymer is 165°C. Other thermal properties, which depend substantially on the crystallization or melting of the polymer, are Hsted in Table 1. See also reference 13. 

Currently, there are about 197,500 entries in the National Institute of Standards and Technology (NIST) Crystal Data File. An exhaustive search takes about one minute. Unit cell parameters are very definitive. Usually only one or a few hits are found and the appropriate Hterature reference(s) are Hsted. If no hits are found, the stmcture has not been previously reported. 

Concerning the VDW parameters, the ability to directly apply previously optimized values makes convergence criteria unnecessary. If VDW parameter optimization is performed based on pure solvent or crystal simulations, then the heats of vaporization or sublimation should be within 2% of experimental values, and the calculated molecular or unit cell volumes should be also. If rare gas-model compound data are used, the references cited above should be referred to for a discussion of the convergence criteria. 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

We refer to the planes which close the box parallel to the boundary as the terminating planes. Between the terminating planes, the box includes one or more translational unit cells of the system parallel to the boundary. We therefore have to assume that the system is periodic parallel to the boundary, but this assumption is anyway necessary for practical schemes of calculation. The terminating planes should be far enough away from the interface to be locally in the undistorted bulk material. 

The unit cell of the hydrates crystallizing in Structure II is rather complicated, and for a detailed description the reader is referred to the original publications.6 48 Its composition is characterized by ... 

The unit-cell edge length of lithium fluoride is 401.8 pm. What is the smallest angle at which the x-ray beam generated from a molybdenum source (X = 71.07 pm) must strike the planes making up the faces of the unit cell for the beam to be diffracted from those planes Refer to Major Technique 3 on x-ray diffraction, which follows this set of exercises. 

The slightly galactosylated mannans are essentially linear polymers. As a result of their cellulose-like (1 4)-/3-D-mannan backbone, they tend towards self-association, insolubility, and crystallinity. Crystallographic study of C. spectabilis seed GaM [180] with a Man Gal ratio 2.65 1 suggested an orthorhombic unit cell with lattice constants of a = 9.12, b = 25.63, and c = 10.28 the dimension b was shown to be sensitive to the degree of galactose substitution and the hydration conditions [180 and references therein, [191]]. 

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