Self consistent mean field

A recent survey analyzed the accuracy of tliree different side chain prediction methods [134]. These methods were tested by predicting side chain conformations on nearnative protein backbones with <4 A RMSD to the native structures. The tliree methods included the packing of backbone-dependent rotamers [129], the self-consistent mean-field approach to positioning rotamers based on their van der Waals interactions [145],... 

P Koehl, M Delame. A self consistent mean field approach to simultaneous gap closure and side-chain positioning m protein homology modelling. Nature Struct Biol 2 163-170, 1995. R Samudrala, J Moult. A graph-theoretic algorithm for comparative modeling of protein structure. J Mol Biol 279 287-302, 1998. 

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. 

Fig. 3 Self-consistent mean-field results for the density profile (normalized to unity) of a brush for different values of the interaction parameter f). In a the distance from the grafting surface is rescaled by the scahng prediction for the brush height, h, and in b it is rescaled by the unperturbed polymer radius Rq. As increases, the density profiles approach the parabolic profile (shown as dashed lines)...

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

In addition to the above effects, the intermolecular interaction may affect polymer dynamics through the thermodynamic force. This force makes chains align parallel with each other, and retards the chain rotational diffusion. This slowing down in the isotropic solution is referred to as the pretransition effect. The thermodynamic force also governs the unique rheological behavior of liquid-crystalline solutions as will be explained in Sect. 9. For rodlike polymer solutions, Doi [100] treated the thermodynamic force effects by adding a self-consistent mean field or a molecular field Vscf (a) to the external field potential h in Eq. (40b). Using the second virial approximation (cf. Sect. 2), he formulated Vscf(a), as follows [4] ... 

Equation (69) or (71) does not contain the self-consistent mean field potential Vscf(a), indicating that the thermodynamic force does not contribute to the steady-state stress or viscosity and thus explaining why r 0 for aqueous xanthan solutions shown in Fig. 19 is independent of Cs. However, this force may play a role in the stress in a non-steady-state flow through Vscf (a), as can be seen from Eqs, (61) and (62). 

From the theoretical viewpoint, much of the phase behaviour of blends containing block copolymers has been anticipated or accounted for. The primary approaches consist of theories based on polymer brushes (in this case block copolymer chains segregated to an interface), Flory-Huggins or random phase approximation mean field theories and the self-consistent mean field theory. The latter has an unsurpassed predictive capability but requires intensive numerical computations, and does not lead itself to intuitive relationships such as scaling laws. 

Fig. 2.44 Phase diagram for a conformationally-symmetric diblock copolymer, calculated using self-consistent mean field theory. Regions of stability of disordered, lamellar, gyroid, hexagonal, BCC and close-packed spherical (CPS), phases are indicated (Matsen and Schick 1994 ). All phase transitions are first order, except for the critical point which is marked by a dot.

Fig. 6.52 Interfarial excess in thin films of blends of a dPS-P2VP diblock (A PS = 391, jVP2Vp = 68) with PS homopolymer (NK = 6440) (Dai et al. 1992). Since the homopolymer is much longer than the diblock, the diblock forms a dry brush. The circles are the results from forward recoil spectrometry, the lines correspond to theoretical calculations. The dashed line was computed using the theory of Leibler (1988), and the solid line is from the self-consistent mean field calculation of Shull and Kramer (1990).

The theory of Jones and Richmond based on the self-consistent mean field theory predicts Eq. (B-122), which may be rewritten... 

James and Keenan [1959] used a mean-field approximation for potential (7.57). Yamamoto et al. [1977] proposed an expanded JK method (EJK), which takes into account higher-order terms in expansion (7.59). In the self-consistent mean-field approximation we have (see also Kobashi et al. [1984])... 

Fig. 1 Phase diagram of self-assembled structures in AB diblock copolymer melt, predicted by self-consistent mean field theory [31] and confirmed experimentally [33]. The MesoDyn simulations [34, 35] demonstrate morphologies that are predicted theoretically and observed experimentally in thin films of cylinder-forming block copolymers under surface fields or thickness constraints dis disordered phase with no distinct morphology, C perpendicular-oriented and Cy parallel-oriented cylinders, L lamella, PS polystyrene, PL hexagonally perforated lamella phase. Dots with related labels within the areal of the cylinder phase indicate the bulk parameters of the model AB and ABA block copolymers discussed in this work (Table 1). Reprinted from [36], with permission. Copyright 2008 American Chemical Society...

Despite the simple and universal structure of the nonrelativistic Hamiltonian for N interacting electrons, it produces a broad spectrum of physical and chemical phenomena that are difficult to conceptualize within the full -electron theory. Starting with the work of Hartree [162] in the early years of quantum mechanics, it was found to be very rewarding to develop a model of electrons that interact only indirectly with each other, through a self-consistent mean field. A deeper motivation lies in the fact that the relativistic quantum field theory of electrons is... 

Most of the theories were based on the self-consistent mean field approximation.12-32 The other ones include the scaling analysis, - molecular dynamics simulations and Monte Carlo simulations.40,41 The self-consistent mean field theories12-32 were developed along the following three lines (1) on the basis of a lattice model,12-22 (2) on the basis of a diffusion type equation,23-28 and (3) analytical approaches.29-32... 

Analytical self-consistent mean field theories were developed independently by Zhulina el al.29 30 and Milner et al.31,32 They are based on the assumption that for large stretchings of the grafted chains with respect to their Gaussian dimension, one can approximate the set of conformations of a stretched grafted chain by a set of most likely trajectories, and predict for such cases a parabolic density profile. In the calculations of the interactions between the two brushes, the interdigitation between the chains was ignored. 

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