Phase polygon

The complex phases of the LCAO coefficients can be visualized as a phase polygon. This is a regular polygon of unit radius with its center at the origin in the complex plane, oriented in such a way that its vertex number zero coincides with +1 on the real axis. As one proceeds from one AO of the real polygon-shaped perimeter to the next, the complex phase of the coefficient changes, in a counterclockwise fashion for 0, and in a clockwise fashion for... 

Figure 2.12. Nodal properties of the transition densities of the first four transitions in benzene, a) Representation of the complex LCAO coefficients of HOMOs 01 and 0, as well as LUMOs nd by means of a phase polygon. Each coefficient has the absolute magnitude n and the complex phase shown by a dot in the complex plane of which the real and imaginary axes are abscissa and ordinate, b) Representation of the overlap densities evaluated from the complex coefficients, and c) values of the overlap densities at the vertices of the perimeter and the resulting nodal properties.

The rate at which the complex phase changes from one AO Xn to the next AO is proportional to k. It jumps from one vertex of the phase polygon to the -th, counterclockwise for t and clockwise for 0 i. This procedure is illustrated for the HOMOs and LUMOs of benzene in Figure 2.12a. The products of AO coefficients needed for evaluating the overlap densities are easily obtained from these diagrams by simply adding the complex phases. Thus,... 

Figure C2.2.9. Polygonal domains of focal conics in a smectic A phase confined between parallel plates.

The phase-11 islands have several interesting properties. First, they have a positive surface potential relative to the surrounding unperturbed water film. The potential is highest immediately after formation and decays with time to zero. Another interesting property is the shape of the islands. Their boundaries are often polygonal, bending in angles of 120°,... 

Figure 9. Density of states of a water sample, referring to three-, four-, five- and six-member polygons and to the total of the sample (from top to bottom), as resulting from MD simulation, T=305 K. Dotted lines indicate vibrational frequencies for a single water molecule in gas phase.

WoWj/2 the body-centred cubic structure of W (1 atom in 0, 0, 0 and 1 atom in A, A, /) corresponds to a sequence of type 1 and type 4 square nets at the heights 0 and A, respectively. Note, however, that for a fall description of the structure, either in the hexagonal or the tetragonal case, the inter-layer distance must be taken into account not only in terms of the fractional coordinates (that is, the c/a axial ratio must be considered). For more complex polygonal nets, their symbolic representation and use in the description, for instance, of the Frank-Kasper phases, see Frank and Kasper (1958) and Pearson (1972). 

Figure 5.8. (a) Typical spiral pattern (phase contrast photomicrograph of (0001) face of Sic grown from the vapor phase), and spiral growth hillocks which appear as (b) polygonal and (c) conical pyramids due to narrow step separation. Part (b) is a differential interference photomicrograph, (1010), and part (c) is a reflection photomicrograph, (1011), of hydrothermally synthesized quartz. 

Fig. 52 Sequence of LC phases formed by self-assembly of T-shaped bolaamphiphiles (e.g. compounds 182), depending on the size of the semiperfluorinated lateral chain (a,b) smectic phases, (c-h) polygonal LC cylinder phases, and (i-1) Lam phases [8]...

Attaching only one lateral chain to the rod-like core (T-shaped amphiphiles 181-183) gave rise to LC honeycomb phases where the cross-section of the honeycomb walls contains two rods arranged side-by-side, i.e., the honeycombs have double walls (Fig. 61a) [8]. However, attaching two lateral chains to opposite sides of the aromatic core (X-shaped polyphiles like compound 187) generates polygonal honeycombs with walls that are only one molecule thick (Fig. 61b) [319]. As a consequence of the thinner walls, effectively more space is left available for the lateral chains inside the cells. Hence, honeycombs with smaller cells could be achieved by using two short lateral chains instead of only one chain with the same total volume [330]. 

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