Self-consistent potential

We have carried out impurity calculations for a zinc atom embedded in a copper matrix. We first perform self consistent band theory calculations on pure Cu and Zn on fee lattices with the lattice constant of pure Cu, 6.76 Bohr radii. This yields Fermi energies, self consistent potentials, scattering matrices, and wave functions for both metals. The Green s function for a system with a Zn atom embedded in a Cu matrix... 

For the conduction electrons, it is reasonable to consider that the inner-shell electrons are all localized on individual nuclei, in wave functions very much like those they occupy in the free atoms. The potential V should then include the potential due to the positively charged ions, each consisting of a nucleus plus filled inner shells of electrons, and the self-consistent potential (coulomb plus exchange) of the conduction electrons. However, the potential of an ion core must include the effect of exchange or antisymmetry with the inner-shell or core electrons, which means that the conduction-band wave functions must be orthogonal to the core-electron wave functions. This is the basis of the orthogonalized-plane-wave method, which has been successfully used to calculate band structures for many metals.41... 

The previous result is an important one. It indicates that there can be yet another fruitful route to describe lipid bilayers. The idea is to consider the conformational properties of a probe molecule, and then replace all the other molecules by an external potential field (see Figure 11). This external potential may be called the mean-field or self-consistent potential, as it represents the mean behaviour of all molecules self-consistently. There are mean-field theories in many branches of science, for example (quantum) physics, physical chemistry, etc. Very often mean-field theories simplify the system to such an extent that structural as well as thermodynamic properties can be found analytically. This means that there is no need to use a computer. However, the lipid membrane problem is so complicated that the help of the computer is still needed. The method has been refined over the years to a detailed and complex framework, whose results correspond closely with those of MD simulations. The computer time needed for these calculations is however an order of 105 times less (this estimate is certainly too small when SCF calculations are compared with massive MD simulations in which up to 1000 lipids are considered). Indeed, the calculations can be done on a desktop PC with typical... 

Figure 11. A schematic representation of the mean-field approximation, a central issue in the self-consistent-field theory. The arrows symbolically represent the lipid molecules. The head of the arrow is the hydrophilic part and the line is the hydrophobic tail. On the left a two-dimensional representation of a disordered bilayer is given. One of the lipids has been selected, as shown by the box around it. The same molecule is depicted on the right. The bilayer is depicted schematically by two horizontal lines. The centre of the bilayer is at z = 0. These lines are to guide the eye the membrane thickness is not preassumed, but is the result of the calculations. Both the potential energy felt by the head groups and that of the tail segments are indicated. We note that in the detailed models the self-consistent potential profiles are of course much more detailed than shown in this graph...

The improvement came in the form of the coherent-potential approximation (CPA) (Soven 1967, Taylor 1967, Velicky et al 1968), which remedied the lack of self-consistency exhibited by the ATA. The crux of this approach is that each lattice site has associated with it a complex self-consistent potential, called a coherent potential (CP). The CP gives rise to an effective medium with the important property that removing that part of the medium belonging to a particular site, and replacing it by the true potential, produces, on average, no further scattering. Because the CPA is used for our discussion of chemisorption on DBA s, its mathematical formulation is given below. 

Fig. 14.5 A schematic block diagram illustrating the procedure for calculating self-consistent potentials.

The dynamical history of stress-relaxation in a star-linear blend begins life in just the same way as a star-star blend,because when t r gp the linear chain relaxation is dominated by pathlength fluctuation and behaves as a two-arm star with M =Mii /2. So very early Rouse fluctuation (Eq. 25) crosses over to activated fluctuation in self-consistent potentials. These are calculated via the coordinate transformation used in the star-star case above. For example, the effective potential for the star component in this regime is... 

Stochastic aggregation does not emerge for oppositely charged particles, when electroneutrality holds due to conditions nk(t) = nB(f) = n(f), particle charge ea = — eB = e. Let us introduce, following the Debye-Hiickel method, the self-consistent potential (J> through Poisson equation... 

Therefore, the approximate treatment of the A+B — 0 reaction for charged particles inavoidably requires a combination of several approximations the Kirkwood superposition approximation for the reaction terms and the Debye-Hvickel approximation for modification of the drift terms with self-consistent potentials. Not discussing here the accuracy of the latter approximation, note... 

The solid and dashed curves, shown in Fig. 57a for L = 1.0 and 1.42 A, respectively, demonstrate that the form of the self-consistent potential well, found here, resembles, generally speaking, the form of the hat-curved well (see Fig. 20). The potential function t/(p) (433) widens near its bottom, if the H-bond length L increases (cf. solid and dashed curves calculated, respectively, for L = 1.85 and 1.54 A). Note that the cosine-squared (CS) potential well63 is substantially more concave than the self-consistent potential given by Eq. (433) (see dashed-and-dotted curve in Fig. 57a). 

The spectral function (467) and (468) is determined by the same set of parameters of the model, as was involved in Section IX.B, combined now in the form of a single parameter p defined by the formula (469). This parameter characterizes an inhomogeneity of the self-consistent potential (p), since at... 

Fig. 3.30 An adsorbed polymer near a surface. The length a defines the range in which the surface potential can be felt. Beyond that range, there are only repulsive forces between polymers and this repulsion creates a self-consistent potential. (From ref. [142])...

For small values of e = 4 - d, a polymer chain in solution is nearly Brownian and a mean field method might reasonable results in this limit. Thus it is possible consider that the chains feel a potential V (x) which is the sum of the (attractive and repulsive) surface potential and of a self - consistent potential produced by the other chains. 

Fig. 28). Clearly the profile becomes flatter with decreasing solvent quality. Note, however, that the self-consistent potential and the end-segment distributions still satisfy Eqs. (109) and (110). 

Near the nucleus the self-consistent potential energy condition... 

In fact, because of its importance in solid-state science, a large variety of band-structure approaches have been used to calculate the electronic structure of sphalerite. These have included self-eonsistent and semiem-pirical orthogonalized-plane-wave (OPW) (Stukel et al., 1969), empirical-pseudopotential (Cohen and Bergstresser, 1966), tight-binding (Pantelides and Harrison, 1975), APW (Rossler and Lietz, 1966), and modified OPW (Farberovich et al., 1980), as well as KKR (Eckelt, 1967) methods. In a recent and extremely detailed study using a density-functional approach (specifically a method termed the self-consistent potential variation... 

As an example of the study of vacancies and self-interstitial impurities by the continued fraction expansion of Eq. (S.2S), we mention the work of Kauffer et al. These authors consider impurities in silicon and set up a model tight-binding Hamiltonian with s p hybridization, which satisfactorily describes the valence and conduction bands of the perfect crystal. A cluster of 2545 atoms is generated, and vacancies (or self-interstitial impurities) are introduced at the center of the cluster. One then takes as a seed state an appropriate orbital or symmetrized combination of orbitals, and the recursion method is started. Though self-consistent potential modifications are neglected in this paper, the model leads to qualitatively satisfactory results within a simple physical picture. 

Ho is the Hamiltonian chosen to be best suited for modelling the target states. If a single determinant is a sufficiently-accurate model for j) then the definition (7.22) is self-consistent if Hq is the Hartree—Fock Hamiltonian. However, the self-consistent potential is not the same for all target states j). The one-electron potential is discussed in chapter 5. 

Thus, we examine a single spherical grain of a radius a imbedded in a weakly ionized high pressure gas. In this case, it is natural to use the drift-diffusion (DD) approach, because collisions of plasma particles with neutrals play here a dominant role. Assuming two types of plasma particles (ions and electrons) only, we write the general time-dependent equations for the unknown ion/electron densities ri-Le and self-consistent potential < > in the form,... 

The prime at AEbanj indicates that in this approximation the self-consistent potential is calculated only once and that the band structures E(k, B2) and E(k, B32) are thus different due to the differences in the structure factors. In this model AE is determined in a procedure similar to the Huckel calculations for organic molecules. 

Under not too unusual circumstances, calculations on molecules with a high density of levels near the Fermi level converge badly or not at all with the Fermi occupation rule. The most simple conceivable system with that behavior contains just two levels, say a and b and one electron. If a is occupied, the self-consistent potential lowers b below a and vice versa. If a and b belong to the same symmetry representation, which is trivially the case for completely nonsymmetric nuclear conformations (Ci), a hybrid orbital can be found which minimizes the total energy self-consistently. It is illuminating to study the behavior of the calculations in this case with a model. One finds that the electrostatics leads to a shift in the levels depending on the population n of the level e . Defining eab = ea — eb ... 

The idea behind the replacement of the excluded volume interactions by a self-consistent potential was put forward by Edwards6 and subsequently developed by various other physicists.5 7 Concerning the critical exponent v, this method gives the same result as Flory s method (v = 3/([Pg.298]

Of course, this dissymmetry can be remedied by replacing the semi-infinite chain by a finite chain and by introducing a self-consistent field without spherical symmetry however, the remedy would be only partial. In fact, in all cases, when N is large, in any self-consistent potential V(r), the chain has a tendency to follow the lines of force. Thus, in the asymptotic limit, the chain always has the shape of a needle, which may be curved. This means that the action of an external potential cannot produce a swelling of the chain in all directions. For this, it is necessary to introduce true correlations between chain links. 

Chapter 9 contains a manual for a series of computer codes based upon the theory presented in the first 8 chapters of the book. With the programmes and the examples given there the user should be able to perform full-scale self-consistent calculations of his own. Finally, the book contains a table of self-consistent potential parameters which together with the LMTO programme will allow the user to reproduce the self-consistent energy bands of 61 metals at normal volume. 

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