Self-consistent equations for

In the noninteracting limit, the band gap is directly proportional to the dimerization gap. In fact, this prediction is violated in traris-polyacteylene, indicating the importance of electron-electron interactions - as we describe in Chapter 7. 

The ratio of the band width, W = 4t, to the band gap introduces an important concept, namely the coherence length, 

A more general scheme to derive the equilibrium bond distortions, A , without resorting to a guess about these distortions, is to require that the force per bond, fn, is zero. 

Although 1 ) is a function of A , this expression is conveniently evaluated if we use the Hellmann-Feynman theorem, which states that 

setting / = 0 gives the following self-consistent equation for A , 

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Equation (7.19) is a self-consistent equation for AEn, in the form of a sum of a pair of continued fractions (CFs). Although numerical solutions to (7.19) are feasible, we are only concerned with its qualitative features. In particular, we note that an exact WSL occurs when AEn = 0, which happens only if both CFs contain the same number of terms (apart from the trivial case 0 = 0). For the infinite chain, this situation is the case for every allowed energy, so an exact WSL is indeed found. But, for a finite chain, AEn = 0 only for the center state, which thus possesses the exact WSL energy. Therefore, the set of energies for a finite chain form only an approximate WSL. 

Taking into account the lowest diagram, we can then write down the self-consistent equations for the mean-fields, V and A ... 

The self consistent equation, for the ground state energy gap An (ky) is ... 

Theorem III (coupled self-consistent equations for the fields) The effective Dirac, Maxwell, and Newton equations, given the associated initial quantities in... 

Problem definition. Deriving the self-consistent equation for electromagnetic scalar potential in MWCNT... 

Solving (5) in the linear approximation and substituting its solutions into Eqs. (2) and (1), we come to the self-consistent equation for scalar potential in the... 

Equation (6) can be transformed into self-consistent equation for electromagnetic field produced in [1] for the transition in one zone if to fulfill expansion = (dn/ /dk)q, where q = k2 k is the photon wave... 

From these observations it becomes evident that one needs to introduce a self consistent equation for a one body force (or energy dissipation). We shall now consider various methods so far developed. 

The minimization of Eq. (8) with respect to PJ - (dself-consistent equations for the system at equilibrium ... 

By minimizing the chain energy /fgi+ iat one obtains the self-consistency equation for A(x) ... 

These equations are exact for any electronic system, if the exact functional Exc [n, n t] is used. Unfortunately, it is impossible to know this functional exactly for most systems, and approximations need to be made to apply these equations to real systems. However, once such an approximation is made, the resulting equations are straightforward to solve, being a set of self-consistent equations for the orbitals 4 a,a(r). The remainder of this article is devoted to how such approximations are constructed, and how well they perform. 

Equation 2.71 is the self-consistency equation for S. S depends on the combination ksT/a only, shown in Figure 2.15, and thus is a function of temperature and the coupling constant a. As ksT/a = 0.22019, a nematic to isotropic transition occurs. The temperature Tc is the N-I transition temperature at which S jumps from 0.4289 to zero. 

Since is a convolution expression, the self-consistent equation for d is not local in Fourier space, and therefore one can no longer seek a single effective frequency solution as in Eq. (2.27). Therefore, this diagrammatic analysis demonstrates that the optimized LHO reference system is the best possible quadratic potential with which to approximate an anharmonic potential, a fact reached independently from the GB variational perspective. Further corrections to the centroid density are thus beyond an effective potential description [3]. 

Stoner and Wohlfarth were able to derive a self-consistent equation for the spontaneous magnetization Mo in the molecular field approximation (MFA) using the model described above. Mo is then written in terms of the average occupation numbers (n ) for electrons of spin cr ... 

The minimum free energy resulting from corresponds to a constant and homogeneous field a = rge " whose overall phase is set to zero. The minimum conditions lead to self-consistent equations for the amplitude and the Lagrange parameter A in terms of the mean-field expectation values. They are evaluated by introducing quasiparticle states created by ( y = ) which diagonalize to... 

As described in Chapter 4, by using the Hellmann-Feynman theorem we can derive a self-consistent equation for A for any state,... 

A self-consistent equation for J may be obtained as a function of temperature. It is given by... 

This is the generalization of (3,14) and (3,17), and it is a self-consistent equation for the effective force constants, as the wavefunction T depends on G (or O), This equation has been used to obtain the force constants by iterations (Koehler, 1966), and once the force constants are known,... 

The NSCFT formalism comprises as a set of self-consistent equations for the equilibrium density distribution and effective field for each component. For a given system, there can be multiple solutions. The original, and still most-used, procedure is to find solutions for each possible microphase and, for each structure, a range of lattice parameters (domain sizes). The equilibrium morphology is then the phase and lattice parameter that have the lowest free energy. Metastable phases can also be identified. 

Note that Q and a are the quantities yet to be determined, while B may be chosen arbitrarily since it only fixes the initial phase of the traveling waves Qt). Since Q and a were supposed to be time independent, one may integrate (5.4.14) to obtain y/a for each a. The solution set obtained in this way determines n y/,t), which gives a via (5.4.13), and this a must coincide with the same quantity in (5.4.14). In this way, we are led to a self-consistent equation for the order parameter. 

The Maier-Saupe theory is extremely useful in understanding the spontaneous long-range orientational order and the related properties of the nematic phase. The single-molecule potential Vi(cos0) is given by Eq. (3.19) with e being volume dependent and independent of pressure and temperature. The self-consistency equation for (P2) is... 

The self-consistency equations for the determination of the temperature dependences of order parameters are given by [3.18]... 

We now see in Eq. [5] a self-consistent equation for the determination of the temperature dependence of < P2 >. The order parameter < P2 > appears on both the left and right hand sides of the equation. For every temperature T (or / ) we can use a computer to obtain the value (or values) of < P2 > that satisfies the self-consistency equation. This process has been accomplished and the results are depicted in Fig. 2. < P2 > = 0 is a solution at all temperatures this is the disordered phase, the normal isotropic liquid. For temperatures T below 0.22284 v/k, two other solutions to Eq. [5] appear. The upper branch tends... 

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