Gradient model potential

The gradient model for interfacial tension described in Eqs. III-42 and III-43 is limited to interaction potentials that decay more rapidly than r. Thus it can be applied to the Lennard-Jones potential but not to a longer range interaction such as dipole-dipole interaction. Where does this limitation come from, and what does it imply for interfacial tensions of various liquids ... 

Thus, the corresponding field or gradient of potential at a distance. v from the electrode according to the diffuse-charge model of Gouy and Chapman is given by the expression ... 

Transport coefficients of molecular model systems can be calculated by two methods [8] Equilibrium Green-Kubo (GK) methods where one evaluates the GK-relation for the transport coefficient in question by performing an equilibrium molecular dynamics (EMD) simulation and Nonequilibrium molecular dynamics (NEMD) methods. In the latter case one couples the system to a fictitious mechanical field. The algebraical expression for the field is chosen in such a way that the currents driven by the field are the same as the currents driven by real Navier-Stokes forces such as temperature gradients, chemical potential gradients or velocity gradients. By applying linear response theory one can prove that the zero field limit of the ratio of the current and the field is equal to the transport coefficient in question. 

In general, the diffusive mass flux is composed of diffusion due to concentration gradients (chemical potential gradients), diffusion due to thermal effects (Soret diffusion) and diffusion due to pressure and external forces. It is possible to include the full multicomponent model for concentration gradient driven diffusion (Taylor and Krishna, 1993 Bird, 1998). In most cases, in the absence of external forces, it is... 

Finally, semiempirical MR-CI calculations based on analytic gradients are sufficiently fast for on-the-fiy determinations of excited-state trajectories, which are essential for more detailed information on the mechanisms of organic photoreactions. Because the results of such studies depend on barrier heights, semiempirical methods will not reproduce quantitative aspects correctly, but studies based on suitable model potentials as well as systematic studies of structural variations and substituent effects might be very valuable in getting deeper insight into the details of organic photoreactions. 

DFT-Based Pseudopotentials. - The model potentials and shape-consistent pseudopotentials as introduced in the previous two sections can be characterized by a Hartree-Fock/Dirac-Hartree-Fock modelling of core-valence interactions and relativistic effects. Now, Hartree-Fock has never been popular in solid-state theory - the method of choice always was density-functional theory (DFT). With the advent of gradient-corrected exchange-correlation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore Why not base pseudopotentials on DFT rather than HF theory ... 

Fig. 9. The gradient norm squared for the model potential energy surface used in Figs. 4, S and 7. The gradient norm squared is a local minimum in many places in addition to the minima and saddle point on the potential.

Shape Corrections to Exchange-Correlation Potentials by Gradient-Regulated Seamless Connection of Model Potentials for Inner and Outer Region. 

In the second case, the core orthogonality is kept but approximated. The valence pseudospinor is no longer an eigenfunction of the Fock operator or the model potential. We can write the gradient as... 

Later, in a model where each cylindrical polymer rod was confined to a concentric, cylindrical, electroneutral shell whose volume represents the mean volume available to the macromolecule, the concept was extended to the macroscopic system itself which was considered as an assembly of electroneutral shells at whose periphery the gradient of potential goes to zero and the potential itself has a constant value. Closed analytical expressions which represent exact solutions of the Poisson-Boltzmann equation can be given for the infinite cylinder model. These solutions, moreover, were seen to describe the essence of the problem. The potential field close in to the chain was found to be the determining factor and under most practical circumstances a sizable fraction of the counter-ions was trapped and held closely paired to the chain, in the Bjerrum sense, by the potential. The counter-ions thus behave as though distributed between two phases, a condensed phase near in and a free phase further out. The fraction which is free behaves as though subject to the Debye-Hiickel potential in the ordinary way, the fraction condensed as though bound . 

Numerical simulations were performed to obtain a description of the potential and current distributions on the surface of the disk electrode and in the surrounding electrolytic solution. Figure 10 gives a schematic representation of the disk electrode used in this model. In the absence of concentration gradients, the potential <(> in the solution surrounding the electrode is governed by the Laplace equation ... 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

Most gradient optimization methods rely on a quadratic model of the potential surface. The minimum condition for the... 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

The forces in a protein molecule are modeled by the gradient of the potential energy V(s, x) in dependence on a vector s encoding the amino acid sequence of the molecule and a vector x containing the Cartesian coordinates of all essential atoms of a molecule. In an equilibrium state x, the forces (s, x) vanish, so x is stationary and for stability reasons we must have a local minimizer. The most stable equilibrium state of a molecule is usually the... 

Solution—Diffusion Model. In the solution—diffusion model, it is assumed that (/) the RO membrane has a homogeneous, nonporous surface layer (2) both the solute and solvent dissolve in this layer and then each diffuses across it (J) solute and solvent diffusion is uncoupled and each is the result of the particular material s chemical potential gradient across the membrane and (4) the gradients are the result of concentration and pressure differences across the membrane (26,30). The driving force for water transport is primarily a result of the net transmembrane pressure difference and can be represented by equation 5 ... 

Analytical gradient energy expressions have been reported for many of the standard models discussed in this book. Analytical second derivatives are also widely available. The main use of analytical gradient methods is to locate stationaiy points on potential energy surfaces. So, for example, in order to find an expression for the gradient of a closed-shell HF-LCAO wavefunction we might start with the electronic energy expression from Chapter 6,... 

The entropic force fs, given as the negative of the gradient of the spring potential energy, is represented by one of the various kinds of springs used in molecular models (Fig. 3). 

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