Several molecules per unit cell

The role of mixing of molecular states can be particularly important for excitonic bands with a small Davydov splitting in crystals with unit cells containing several molecules. 

Since both the splitting energies and intensities of excitonic transitions can be established experimentally with a high degree of accuracy, theoretical determination of those quantities is important. 

As we have shown above, for the calculation of the a l quantities E (k) from eqn (3.97) we must solve a secular equation of order at. It is clear that this procedure is quite complicated even for l = 2 and a 2. 

We wish to discuss the problem more exactly. Now the operator H3 which appears in eqn (3.88) will be taken into account, since its contribution for the crystal Hamiltonian is relevant, and will not complicate the calculations. 

The index ji( f ) in eqns (3.105) and (3.106) denotes one of the excitonic states which corresponds to the /th molecular term without mixing effects, so that denotes the summation over all excitonic states which correspond to the molecular term /. It is evident that in the new coordinates the operator AII f, being equal to AH (see eqn 3.103) only when the mixing contributions are neglected (which means the contributions proportional to the matrix elements g), is also diagonal. Taking into account these matrix ele- 

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Crystals with several molecules per unit cell... 

With the aim of obtaining general expressions for corrections to the Heitler-London approximation in crystals with several molecules per unit cell, we put eqns (3.62) into the form... 

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

L material that contains several water molecules per unit cell, see [22]. If the water molecules are completely removed from the main channel, the spectroscopic properties may change. 

Experiments at high pressure have shown that the P-T phase diagram of butadiene is comparatively simple. The crystal phase I is separated from the liquid phase by an orientationally disordered phase II stable in a narrow range of pressure and temperature. The strucmre of phase I is not known, but the analyses of the infrared and Raman spectra have suggested a monoclinic structure with two molecules per unit cell as the most likely [428]. At room temperature, butadiene is stable in the liquid phase at pressures up to 0.7 GPa. At this pressure a reaction starts as revealed by the growth of new infrared bands (see the upper panel of Fig. 25). After several days a product is recovered, and the infrared spectrum identifies it as 4-vinylcyclohexene. No traces of the other dimers can be detected, and only traces of a polymer are present. If we increase the pressure to 1 GPa, the dimerization rate increases but the amount of polymer... 

Synthetic Zeolites. Many new cry stalliue zeolites have been. synthesized and several fulfill important functions in the chemical and petroleum industries and in consumer products such as detergents. The structural formula of a zeolite is based on the crystal unit cell, the smallest unit of structure, represented by At, ((Alt), >, (SiO i t/ H 0. where n is the valence of cation M. ir is the number of water molecules per unit cell, v and r are. respectively, the number of AlOa and SiOj tetrahedra per unit cell, and y/v usually has values of 1-5. Examples of important synthetic zeolites are shown in Table I. 

X 77.8 X 51.4 A, a space group of P2i2i2j, and a Z value of 4 (molecules per unit cell) (46). Several heavy atom derivatives, whose crystalline structure are isomorphous with that of the native enzyme, have been prepared for the X-ray diffraction analysis of crystalline cytochrome c peroxidase. 

Several types of crystals contain two or more types of molecules per unit cell, the hydrates providing a very simple example, in which the solvent molecules take part in the crystal structure and help satisfy requirements for hydrogen bonding and metal coordination. Sometimes solid solutions of two compounds with similar structures may form. This situation can cause problems in refining a crystal structure, especially if it is not real-... 

The procedure common to both types of interpretation is as follows. First, the parameters of the unit cell are deduced, and the reflections are indexed second, the space group is determined (assuming that this is possible, from the fiber diagram) on the basis of reflections systematically absent. At this point, a noncrystallographic procedure is used in order to determine the density of the specimen, from which the number of molecules per unit cell is found. The intensity of the reflections is now determined by any one of several methods, but the manner in which the intensity data are used depends on the type of the pattern. This is the point of divergence previously mentioned. 

The whole crystal can be constructed by elementary translations (and eventually by appropriate space-group operations) of a basic unit (unit cell or asymmetric unit). We shall call this basic unit the motif . It is choosen as a closed shell molecule or molecular complex or even a neutral assembly of ions, containing n electrons and M nuclei. Let N be the number of unit cells within the Born-Karman boundaries and denote by Q the elementary translation vectors pointing from the origin to the L-th cell. (For the sake of notational simplicity we shall assume that each unit cell contains only one motif. Generalization to several motifs per unit cell is straightforward.)... 

For crystals with several molecules (labeled by s) per unit cell, which have symmetry operations exchanging molecules with different s values, as holds for anthracene crystals, a similar procedure yields... 

Table IV shows the cation distributions estimated for several forms of X dehydrated under various conditions. For trivalent cations (.—30 per unit cell), sites I and II are most suitable electrostatically, but the sample of La-X calcined at unspecified temperature (probably 350°C) showed all the La atoms on I, while site II was occupied by electron density explainable by 32 water molecules. This remarkable distribution yields a very stable chemical complex with each La bonded to 3 03 and 3 HoO at 2.5A and with each H20 bonded to either 2 or 3 La atoms. The H20 also is bonded weakly to 3 02 atoms of a free 6-ring.

The first problem originates from the structural complexity of zeohte frameworks which normally contain several hundred atoms per unit cell. This makes studies of the vibrational behavior of the lattice and the search for modes characteristic of special structural units even more difficult. In this case, a usual approximation is to cut out an isolated model cluster from the framework and treat it hke a molecule. In comparison with quantum mechanics, in NCA it is not necessary to saturate the dangfing bonds of the cluster by terminal pseudoatoms (vide infra). In a first attempt, based on such an assumed decoupHng of modes from the surrounding framework, Blackwell [7] predicted vibrational frequencies... 

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