Potential: “effective

The form of the potential for the system under study was discussed in many publications [28,202,207,208]. Effective pair potentials are widely used in theoretical estimates and numerical calculations. When a many-particle interatomic potential is taken into account, the quantitative description of experimental data improves. For example, the consideration of three-body interactions along with two-particle interactions made it possible to quantitatively describe the stratification curve for interstitial hydrogen in palladium [209]. Let us describe the pair interaction of all the components (hydrogen and metal atoms in the a. and (j phases) by the Lennard Jones potential cpy(ry) = 4 zi [(ff )12- / )6], where Sy and ai are the parameters of the corresponding potentials. All the distances ry, are considered within c.s. of radius r (1 r R), where R is the largest radius of the radii of interaction Ry between atoms / and /). 

The interstitial sublattice comprises interstitial hydrogen atoms and free internodes (vacancies V), whereas the metal lattice is composed of palladium atoms and vacancies V. The potential energy of the system can be presented as [195] 

The distances between atoms appearing in the potentials can be expressed through the lattice constants for nearest neighbors ljh, where k — x, y, z are the components of the vectors between nearest neighbors / and h are the number of atoms at various distances. Thus, the total potential energy of the system can be expressed through the parameters 2 and minimized with respect to them  

The function hh( h) should be calculated by using actual approximations. To calculate the thermodynamic and kinetic characteristics of the system in a self-consistent manner, it is necessary to use the quasi-chemical approximation, because it takes into account nearest-order correlations (which are absent in the mean-field approximation) [80]. 

---

To rationalize the two-centre approximation , the effective potential is written as... 

Most TB approaches are not charge self-consistent. This means that they do not ensure that the charge derived from the wavefiinctions yields the effective potential assumed in their calculation. Some methods have been developed which yield charge densities consistent with the electronic potential [14, H and 16]. 

Remler D K and Madden P A 1990 Molecular dynamics without effective potentials via the Car-Parrinello approach Mol. Phys. 70 921-66... 

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

The most important molecular interactions of all are those that take place in liquid water. For many years, chemists have worked to model liquid water, using molecular dynamics and Monte Carlo simulations. Until relatively recently, however, all such work was done using effective potentials [4T], designed to reproduce the condensed-phase properties but with no serious claim to represent the tme interactions between a pair of water molecules. 

From an electrochemical viewpoint, stable pit growtli is maintained as long as tire local environment witliin tire pit keeps tire pit under active conditions. Thus, tire effective potential at tire pit base must be less anodic tlian tire passivation potential (U ) of tire metal in tire pit electrolyte. This may require tire presence of voltage-drop (IR-drop) elements. In tliis respect the most important factor appears to be tire fonnation of a salt film at tire pit base. (The salt film fonns because tire solubility limit of e.g. FeCl2 is exceeded in tire vicinity of tire dissolving surface in tlie highly Cl -concentrated electrolyte.)... 

The effective potential matrix for nuclear motion, which is a diagonal matrix for the adiabatic electronic set, is given by... 

Consider the generalized distribution Pq(r ) to be generated in the Gibbs-Boltzmann canonical ensemble (9 = 1) by an effective potential W,(r /3) which is defined... 

Given this effective potential, it is possible to define a constant temperature molecular dynamics algorithm such that the trajectory samples the distribution Pg(r ). The equation of motion then takes on a simple and suggestive form... 

The effective force derived from the effective potential W(r ) has a number of interesting properties. It is of the form F,(r /3) = =... 

With the above definitions, there is no additional overall phase factor to be included in (27). Eqs. (24)-(27) are the CSP approximation.Like TDSCF, CSP is a separable approximation, using a time-dependent mean potential for each degree of freedom. However, the effective potentials in CSP... 

In addition to wavepackets propagated along effective potentials corresponding to individual trajectories, we also propagate in CI-CSP the CSP wavefunc-tions, governed by the average potentials Vj(qj,t) of Eq. (24). The CI-CSP ansatz for the total wavepacket of the system is then as follows ... 

Bernier D K and P A Madden 1990. Molecular Dynamics without Effective Potentials via the Car ParrineUo Approach. Molecular Physics 70 921-966. 

One consequence of the spin-polarized nature of the effective potential in F is that the optimal Isa and IsP spin-orbitals, which are themselves solutions of F ( )i = 8i d >i, do not have identical orbital energies (i.e., 8isa lsP) and are not spatially identical to one another (i.e., (l)isa and (l)isp do not have identical LCAO-MO expansion coefficients). This resultant spin polarization of the orbitals in P gives rise to spin impurities in P. That is, the determinant Isa 1 s P 2sa is not a pure doublet spin eigenfunction although it is an eigenfunction with Ms = 1/2 it contains both S = 1/2 and S = 3/2 components. If the Isa and Is P spin-orbitals were spatially identical, then Isa Is P 2sa would be a pure spin eigenfunction with S = 1/2. 

Molecular mechanics methods have been used particularly for simulating surface-liquid interactions. Molecular mechanics calculations are called effective potential function calculations in the solid-state literature. Monte Carlo methods are useful for determining what orientation the solvent will take near a surface. Molecular dynamics can be used to model surface reactions and adsorption if the force held is parameterized correctly. 

Microwaves may be used to ionize gases when sufficient power is apphed, but only through the intermediate process of classical acceleration of plasma electrons. The electrons must have energy values exceeding the ioniza tion potential of molecules in the gas (see Plasma technology). Ionizing radiation exhibits more biological-effect potential whatever the power flux levels (2). 

Figure 5 shows the enhanced concentration of oppositely charged ions near the charged surface, and the depleted concentration of similarly charged ions near the charged surface due to electrostatic attractions and repulsions. Both factors reduce the effective potential, /, as the distance from the surface, X, increases. The distance at which / drops to 1/ (37%) of its value at the Stem plane is called the counterion atmosphere decay distance,... 

At the shear plane, fluid motion relative to the particle surface is 2ero. For particles with no adsorbed surfactant or ionic atmosphere, this plane is at the particle surface. Adsorbed surfactant or ions that are strongly attracted to the particle, with their accompanying solvent, prevent Hquid motion close to the particle, thus moving the shear plane away from the particle surface. The effective potential at the shear plane is called the 2eta potential, It is smaller than the potential at the surface, but because it is difficult to determine 01 To usual assumption is that /q is effectively equal to which can be... 

To discover the effective potential ranges for electrochemical protection, the dependence of the relevant corrosion quantities on the potential is ascertained in the laboratory. These include not only weight loss, but also the number and depth of pits, the penetration rate in selective corrosion, and service life as well as crack growth rate in mechanically stressed specimens, etc. Section 2.4 contains a summarized survey of the potential ranges for different systems and types of corrosion. Four groups can be distinguished ... 

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

The molecular and liquid properties of water have been subjects of intensive research in the field of molecular science. Most theoretical approaches, including molecular simulation and integral equation methods, have relied on the effective potential, which was determined empirically or semiempirically with the aid of ab initio MO calculations for isolated molecules. The potential parameters so determined from the ab initio MO in vacuum should have been readjusted so as to reproduce experimental observables in solutions. An obvious problem in such a way of determining molecular parameters is that it requires the reevaluation of the parameters whenever the thermodynamic conditions such as temperature and pressure are changed, because the effective potentials are state properties. 

A full-scale treatment of crystal growth, however, requires methods adapted for larger scales on top of these quantum-mechanical methods, such as effective potential methods like the embedded atom method (EAM) [11] or Stillinger-Weber potentials [10] with three-body forces necessary. The potentials are obtained from quantum mechanical calculations and then used in Monte Carlo or molecular dynamics methods, to be discussed below. 

---