Mean field models

C) All mean-field models of electronic. structure require large corrections. Essentially all ab initio quantum chemistry approaches introduce a mean field potential F that embodies the average interactions among the electrons. The difference between the mean-field potential and the true Coulombic potential is temied [20] the "fluctuationpotentiar. The solutions Ef, to the true electronic... 

The most elementary mean-field models of electronic structure introduce a potential that an electron at r would experience if it were interacting with a spatially averaged electrostatic charge density arising from the N- 1 remaining electrons ... 

A) HOW THE MEAN-FIELD MODEL LEADS TO ORBITALS AND CONFIGURATIONS... 

The one-electron additivity of the mean-field Hamiltonian gives rise to the concept of spin orbitals for any additive bi fact, there is no single mean-field potential different scientists have put forth different suggestions for over the years. Each gives rise to spin orbitals and configurations that are specific to the particular However, if the difference between any particular mean-field model and the fiill electronic... 

Hamiltonian is Hilly treated, corrections to all mean-field results should converge to the same set of exact states. In practice, one is never able to treat all corrections to any mean-field model. Thus, it is important to seek particular mean-field potentials for which the corrections are as small and straightforward to treat as possible. 

In the most connnonly employed mean-field models [25] of electronic structure theory, the configuration specified for study plays a central role in defining the mean-field potential. For example, the mean-field... 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

Before moving on to discuss metliods that go beyond the single-configuration mean-field model, it is important to examine some of the computational effort that goes into carrying out an SCE calculation. 

The Mean-Field Model, Which Forms the Basis of Chemists Pictures of Electronic Structure of Molecules, Is Not Very Accurate... 

To provide further insight why the SCF mean-field model in electronic structure theory is of limited accuracy, it can be noted that the average value of the kinetic energy plus the attraction to the Be nucleus plus the SCF interaction potential for one of the 2s orbitals of Be with the three remaining electrons in the s 2s configuration is ... 

Mean-field models are obviously approximations whose aeeuraey must be determined so seientists ean know to what degree they ean be "trusted". For eleetronie struetures of atoms and moleeules, they require quite substantial eorreetions to bring them into line with experimental faet. Eleetrons in atoms and moleeules undergo dynamieal motions in whieh their eoulomb repulsions eause them to "avoid" one another at every instant of time, not only in the average-repulsion manner that the mean-field models embody. The inelusion of instantaneous spatial eorrelations among eleetrons is neeessary to aehieve a more aeeurate deseription of atomie and moleeular eleetronie strueture. 

In summary, the dynamieal interaetions among eleetrons give rise to instantaneous spatial eorrelations that must be handled to arrive at an aeeurate pieture of atomie and moleeular strueture. The simple, single-eonfiguration pieture provided by the mean-field model is a useful starting point, but improvements are often needed. 

Corrections to the mean-field model are needed to describe the instantaneous Coulombic interactions among the electrons. This is achieved by including more than one Slater determinant in the wavefunction. 

Several methods have been employed to study chemical reactions theoretically. Mean-field modeling using ordinary differential equations (ODE) is a widely used method [8]. Further extensions of the ODE framework to include diffusional terms are very useful and, e.g., have allowed one to describe spatio-temporal patterns in diffusion-reaction systems [9]. However, these methods are essentially limited because they always consider average environments of reactants and adsorption sites, ignoring stochastic fluctuations and correlations that naturally emerge in actual systems e.g., very recently by means of in situ STM measurements it has been demon-... 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

The presented scheme offers several extensions. For example, the model gives a clear route for an additional inclusion of entanglement constraints and packing effects [15]. Again, this can be realized with the successful mean field models based on the conformational tube picture [7,9] where the chains do not have free access to the total space between the cross-links but are trapped in a cage due to the additional topological restrictions, as visualized in the cartoon. 

Figure 6.2a shows chronoamperometric transients for CO oxidation recorded on three different stepped electrodes for the same final potential. Clearly, the electrode with the higher step density is more active, as it oxidizes the CO adlayer in a shorter period of time. Figure 6.2b shows a fit of a transient obtained on a Pt(15, 15, 14) electrode (terrace 30 atoms wide) by both the mean field model [(6.5), solid line] and the N G model [(6.6), dashed line]. The mean field model gives a slightly better fit. More importantly, the mean field model gives a good fit of all transients on all electrodes. 

The temperature dependence of the magnetic hyperfine splitting in spectra of interacting nanoparticles may be described by a mean field model [75-77]. In this model it is assumed that the magnetic energy of a particle, p, with volume V and magnetic anisotropy constant K, and which interacts with its neighbor particles, q, can be written... 

Figure 2 shows that, for all but the shortest chains, the Flory-Vrij analysis predicts a slightly higher melting temperature than the present mean-field model. Both approximations are give values higher than the simulation results, but the overall agreement is reasonable. 

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

Gruen, D. W. R. and Haydon, D. A. (1981). A mean-field model of the alkane-saturated lipid bilayer above its phase transition. II. Results and comparison with... 

Figure 4 Overview of several theoretical predictions for the SE Brueckner-Hartree-Fock (continuous choice) with Reid93 potential (circles), self-consistent Green function theory with Reid93 potential (full line), variational calculation from [9] with Argonne Avl4 potential (dashed line), DBHF calculation from [16] (triangles), relativistic mean-field model from [22] (squares), effective field theory from [23] (dash-dotted fine).

Subsequently Furnstahl [20] in a more extensive study pointed out that within the framework of mean field models (both non-relativistic Skyrme as well as relativistic models) there exists an almost linear empirical correlation between theoretical predictions for both 04 and its density dependence, po, and the neutron skin, All lln — Rp, in heavy nuclei. This is illustrated for 208Pb in Fig. 5 (from ref.[20] a similar correlation is found between All and po). Note that whereas the Skyrme results cover a wide range of All values the RMF predictions in general lead to AR > 0.20 fm. 

Figure 5. Neutron skin thickness versus <24 for 208Pb for a variety of mean field models (from [20]). The circles correspond to results for the Skyrme force, the squares to RMF models with mesons, and the triangles to RMF with point couplings the shaded area indicates the range of NR values consistent with the present empirical information for 208Pb.

The above observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems. Here we want to point out that this correlation can be understood naturally in terms of the Landau-Migdal approach. To this end we consider a simple mean-field model (see, e.g., ref.[16]) with the Hamiltonian consisting of the single-particle mean field part Hq and the residual particle-hole interaction Hph-... 

Relativistic mean-field model based on 517(3) chiral symmetry [53] ( XSU(3) ), see Fig. 5... 

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