Unperturbed potential

It is important to know how accurate the transition probability from first-order time-dependent perturbation theory is. Following the method of Oppenheimer (1928), we show that our choice of unperturbed potentials minimizes the error estimation term. Therefore, it is the optimum choice. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

Pushing can be best be applied to a MC and MD system if one has in mind a reaction coordinate, r(c[) / i.e. some function of the coordinates c[ that, because it takes on a rather limited range of values, suggests that the system is trapped in a rather 1imited part of configuration space. A Monte Carlo run under the unperturbed potential U would yield a fairly narrow distribution of values of r, representable as a histogram, h(r) ... 

Figure 1.4 The phase S(x,f) of the wave packet as defined in Eq. (3). The initial state is the bound state of the unperturbed potential given in Eq. (6).

FIGURE 9.2 Potential distribution i/ (x) across two interacting plates 1 and 2 at constant surface charge density a calculated with Eqs. (9.31)-(9.33) for Kh—1 (upper) and 2 (lower). The dotted lines, which stand for unperturbed potentials, correspond to the case of infinite separation (k/i = oo). 

FIGURE 10.1 Schematic representation of the potential distribution (soUd line) across two interacting parallel dissimilar plates 1 and 2 at separation h. Dotted line is the unperturbed potential distribution ath = oo. unperturbed surface potentials of plate 1 and 2, respectively. 

For hpotential distribution near the higher charged plate (plate 1) equals the unperturbed potential distribution in the absence of plate 2 (see Fig. 10.2) so that the net force acting plate 1 becomes zero. 

FIGURE 11.2 Scaled potential distribution y x) across two likely charged plates 1 and 2 at scaled separation Kh = 2 with unperturbed surface potentials yoi = 2 and >>o2 = 1-5. yi(x) and y2ix) are, respectively, scaled unperturbed potential distributions around plates 1 and 2 in the absence of interaction at Kh = oo. 

The asymptotic expression for the unperturbed potential of plate i i= 1, 2) at a large distance h from the surface of plate i is given by... 

Here Q ta ( = 1, 2) is the hypothetical point charge that would produce the same asymptotic potential as produced by sphere i and is the asymptotic form of the unperturbed potential at a large distance R from the center of sphere i in the absence of interaction. The hypothetical point charge geffi gives rise to the potential... 

It can be shown that the solution ij/(x) to Eqs. (13.2)-(13.4) subject to Eqs. (13.6) and (13.7) is given by a linear superposition of the unperturbed potential ij/f x) produced by membrane 1 in the absence of membrane 2 and the corresponding unperturbed potential (x) for membrane 2, which are obtained by solving the hnearized Poisson-Boltzmann equations for a single membrane, namely,... 

Thus, we need only to obtain the unperturbed potential distribution produced within sphere 1 by sphere 2. By using the following relation... 

Consider the double-layer interaction between two parallel porous cylinders 1 and 2 of radii and a2, respectively, separated by a distance R between their axes in an electrolyte solution (or, at separation H = R ai—a2 between their closest distances) [5]. Let the fixed-charge densities of cylinders 1 and 2 be and Pfix2. respectively. As in the case of ion-penetrable membranes and porous spheres, the potential distribution for the system of two interacting parallel porous cylinders is given by the sum of the two unperturbed potentials... 

As the zeroth-order approximate solutions il/f x) and 1/ 2° (x), we choose the unperturbed potentials produced by plates 1 and 2 in the absence of interaction (i.e., when they are isolated), which are (see Chapter 1)... 

FIGURE 14.2 The unperturbed potentials j/f and j/f and the correction terms j/f and (k=l,2,. . Squares (dotted lines) mean that is the image potential of with respect to plate i, while is the image potential of with respect to plate j... 

This expression coincides with that obtained by the linear superposition of the unperturbed potentials and (Eq. (11.26)). Note that if spheres 1 and 2... 

In case (c), the image interaction energy carries both characters of cases (a) and (b). When the unperturbed surface potentials and the radii of the two spheres become similar, these two contributions from cases (a) and (b) tend to cancel each other so that the total image interaction for case (c) becomes small, as shown in Fig. 14.6, in which the interacting spheres are identical Ka = ku2 = 5 and i/ oi = 4 02)- In the opposite case where the difference in the two unperturbed potentials is large, the image interaction for case (c) is determined almost only by the larger unperturbed surface potential. 

A simple approximate analytic expression for P ih) can be obtained using the linear superposition approximation (LSA) (Chapter 11). In this approximation, y h/2) in Eq. (15.34) is approximated by the sum of the asymptotic values of the two scaled unperturbed potentials ys(T) that is produced by the respective plates in the absence of interaction. For two similar plates. 

This approximation is correct in the limit of large Kh. It follows from Eq. (1.37), the value of the unperturbed potential of a single plate at x = h/2 is given by... 

Br /Br has been found to be the same as in the gas phase,28 indicating unperturbed potential energy surfaces. For CHsBr the fragment translational energy decreased as the coverage increased beyond 1 ML, though the distribution remained narrow.23 This... 

This is the entire formal structure of classical statistical mechanical perturbation theory. The reader will note how much simpler it is than quantum perturbation theory. But the devil lies in the details. How does one choose the unperturbed potential, y How does one evaluate the first-order perturbation It is quite difficult to compute the quantities in Equation P5 from first principles. Most progress has been made by some clever application of the law of corresponding states. It is not the aim of this chapter to follow this road to solution theory any further. 

Here R and r are the translational and the vibrational coordinates, respectively, 7 is the Jacobi angle defined as ( R f ), UL and M are the reduced masses of the reagent diatomic and the triatom systems, respectively, and U(Rr) is the potential that governs the motion of the three interacting atoms. More details can be found in Ref. 27c, but here we would like to add a few words regarding W (Rr), the unperturbed potential which is used to calculate the unperturbed function It will be defined as an isotropic separable potential in R and r, namely ... 

Explicit Expressions up to Third Order in the Unperturbed Potential... 

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