The single-eonfiguration mean-field theories of eleetronie strueture negleet eorrelations among the eleetrons. That is, in expressing the interaetion of an eleetron at r [Pg.230]

Levelt Sengers J M H 1999 Mean-field theories, their weaknesses and strength Fluid Phase Equilibria 158-160 3-17 [Pg.662]

As a prelude to discussing mean-field theory, we review the solution for non-interacting magnets by setting J = 0 in the Ising Flamiltonian. The PF [Pg.529]

The neglect of fluctuations in mean-field theory implies that [Pg.534]

Table A2.3.5 Critical temperatures predicted by mean-field theory (MFT) and the quasi-chemical (QC) approximation compared with the exact results. |

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. [Pg.117]

P Koehl, M Delarue. Application of a self-consistent mean field theory to predict protein side-chains conformation and estimate their conformational entropy. J Mol Biol 239 249-275, 1994. [Pg.308]

J. Mai, W. von Niessen. The CO -(- O2 reaction on metal surfaces. Simulation and mean-field theory The influence of diffusion. J Chem Phys 95 3685-3692, 1990. [Pg.434]

Although the exact equations of state are known only in special cases, there are several usefid approximations collectively described as mean-field theories. The most widely known is van der Waals equation [2] [Pg.443]

Exponent values derived from experiments on fluids, binary alloys, and certain magnets differ substantially from all those derived from analytic (mean-field) theories. Flowever it is surprising that the experimental values appear to be the same from all these experiments, not only for different fluids and fluid mixtures, but indeed the same for the magnets and alloys as well (see section A2.5.5). [Pg.639]

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science. |

Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory. |

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. [Pg.445]

Weeks J D, Katsov K and Vollmayr K 1998 Roles of repulsive and attractive forces in determining the structure of non uniform liquids generalized mean field theory Phys. Rev. Lett. 81 4400 [Pg.556]

The assumption that the free energy is analytic at the critical point leads to classical exponents. Deviations from this require tiiat this assumption be abandoned. In mean-field theory. [Pg.538]

FIG. 9 Changes of the monolayer film critical temperature with the concentration of impurities obtained from the Monte Carlo simulations (open circles) and resulting from the mean field theory (solid line). (Reprinted from A. Patrykiejew. Monte Carlo studies of adsorption. II Localized monolayers on randomly heterogeneous surfaces. Thin Solid Films, 205 189-196, with permision from Elsevier Science.) [Pg.274]

This implies that the critical exponent y = 1, whether the critical temperature is approached from above or below, but the amplitudes are different by a factor of 2, as seen in our earlier discussion of mean-field theory. The critical exponents are the classical values a = 0, p = 1/2, 5 = 3 and y = 1. [Pg.538]

Plotting r versus 1/n gives kTJqJ as the intercept and (kTJqJ)( -y) as the slope from which and y can be determined. Figure A2.3.29 illustrates the method for lattices in one, two and tliree dimensions and compares it with mean-field theory which is independent of the dimensionality. [Pg.543]

The parameters a and b are characteristic of the substance, and represent corrections to the ideal gas law dne to the attractive (dispersion) interactions between the atoms and the volnme they occupy dne to their repulsive cores. We will discnss van der Waals equation in some detail as a typical example of a mean-field theory. [Pg.444]

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