Three-dimensional conjugated systems

Three-dimensional (3D) conjugated systems do not exist but one can conceive 3D organic semiconductors comparable to the inorganic ones (Ge, GaAs). There, besides contributions from points Eg and lines Ej, will also contain contributions from critical sur-... 

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

By definition chirality involves a preferred sense of rotation in a three-dimensional space. Therefore, it can only be affected by a modification of the nonscalar fields appearing in the rate equations. For a reaction-diffusion system [equations (1)] these fields are descriptive of a vector irreversible process, namely, the diffusion flux J of constituent k in the medium. According to irreversible thermodynamics, the driving force conjugate to diffusion is... 

In inorganic semiconductor crystals with three-dimensional system of conjugated bonds (for example, for the system of sp3 bonds in crystals with tetrahedral cells [27]) delocalization of electron/hole wave functions sharply increases and, accordingly, exchange interaction between these particles decreases. A distance between electron and hole in such bulk crystals, which... 

In order to examine the stability of the equilibrium points it is customary to separate the three-dimensional system Eqs. (6) to (11) into a fast subsystem involving V and n and a slow subsystem consisting of S. The z-shaped curve in Fig. 2.7b shows the equilibrium curve for the fast subsystem, i.e. the value of the membrane potential in the equilibrium points (dV/dt = 0, dn/dt = 0) as a function of the slow variable S, which is now to be treated as a parameter. In accordance with common practice, those parts of the curve in which the equilibrium point is stable are drawn with full lines, and parts with unstable equilibrium points are drawn as dashed curves. Starting from the top left end of the curve, the equilibrium point is a stable focus. The two eigenvalues of the fast subsystem in the equilibrium point are complex conjugated and have negative real parts, and trajectories approach the point from all sides in a spiraling manner. 

For polycyclic Jt-systems, there is not always a correlation between aromatic character and the total number of tt-electrons, as is the case for monocyclic annulenes91. In fullerenes, which are not only polycyclic but also three-dimensional, such a correlation is even less apparent. These carbon alio tropes embody completely conjugated spheroidal it-systems, so die carbon skeletons are boundary-less, and large numbers of Kekule structures can be drawn92. The aromaticity of fullerenes has been investigated theoretically and substantiated experimentally by using NMR studies93. 

The term density matrix arises by analogy to classical statistical mechanics, where the state of a system consisting of N molecules moving in a real three-dimensional space is described by the density of points in a 6N-dimensional phase space, which includes three orthogonal spatial coordinates and three conjugate momenta for each of the N particles, thus giving a complete description of the system at a particular time. In principle, the density matrix for a spin system includes all the spins, as we have seen, and all the spatial coordinates as well. However, as we discuss subsequently we limit our treatment to spins. For simplicity we deal only with application to systems of spin % nuclei, but the formalism also applies to nuclei of higher spin. 

These do not, so far, constitute industrially important monomers. Nevertheless, they do have some technical importance which is documented by the number of published studies of their polymerization. Acetylenes yield chains with a conjugated system of double bonds with semiconducting properties. It is probably just this possibility of conjugation in the generated chain that prevents formation of three-dimensional structures. Acetylene as such is a potentially tetrafunctional monomer. 

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