Potential well depth

Finally, anionic well potential calculations were performed in molecule cases. The molecule selected for illustration was diatomic CO, and the well potential reference for a molecule must be zero at the region, infinitely apart from the molecule in contrast to the solid case. The molecule CO has interatomic separation of 0.11282 nm and an O " radius of 0.122 nm, respectively. Basis numerical atomic orbitals were ls-2p on both C and O atoms. The calculated well potential depth was —0.36 Eh which was shallower than the value of about —1 Eh usually used. 

FIG. 3 Schematic of the two-dimensional square-well potential u x) of depth e, width d, and period I (from Ref. 48). 

In order to see whether the results are sensitive to the exact shape of the potential field, some calculations have been made in which the field w r) was replaced by a square well. The depth of the well was taken equal to the value (Eq. 31) of w(o) for an L-J-D- field, while the radius was taken equal to the value (at— a) valid for hard spheres. In this approximation the free volume is equal to m (a —or)3, and hence in formula 38... 

However, serious drawbacks of model 3 are that (i) the proportion r of the rotators should be fitted that is, it is not determined from physical considerations and (ii) the depth of the well, in which a polar particle moves, is considered to be infinite. Both drawbacks were removed in VIG (p. 305, 326, 465) and in Ref. 3, where it was assumed that (a) The potential is zero on the bottom of the well (/(()) = 0 at [ fi < 0 < P], where an angle 0 is a deflection of a dipole from the symmetry axis of a cone, (b) Outside the well the depth of the rectangular well is assumed to be constant (and finite) U(Q) = Uq at [— ti/2 < 0 < ti/2]. Actually, two such wells with oppositely directed symmetry axes were supposed to arise in the circle, so that the resulting dipole moment of a local-order region is equal to zero (as well as the total electric moment in any sample of an isotropic medium). 

In Figure 3.16, we show the potential contours in the entrance and exit channels. In these plots, the diatomic distance is held fixed at the equilibrium bond length and the PES is plotted against Jacobi distance and angle (R, 0). A vdW well of depth 2.5 kcal/mol is observed in the entrance channel with a bent configuration. In the exit channel, a collinear vdW well of depth 2.0 kcal/mol is found. Variational calculations of the vibrational states reveal that several bound states exist in these wells [128]. More pertinent to the present discussion, the vdW wells can also lead to the existence of resonance states in both the entrance and exit channels. 

Figure 4 RPAE calculated results for the Xe 4d photoionization cross section of free Xe, o 4dee, as well as of Xe C6o calculated in the framework of both the 5-potential model, a s [37] and A-potential model, a4 A [33], Also shown, for comparison, are calculated data [33], marked ct4 5a, obtained for the 4d photoionization cross section of Xe Cgo with an artificially reduced thickness of the Cgg cage from A = 1.9 au to A = 0.5 au, deepened potential depth, UgQ = 25.9 eV, and changed inner radius Rc = 6.389 au, in order to simulate the 5-potential model but keep the binding strength of the cage potential unchanged (see the main text body).

Mass Accommodation Coefficient. For a given molecule the mass accommodation coefficient is a physical constant which depends only on the temperature and on the nature of the liquid surface. The process of the molecule entering the liquid phase might proceed as follows. Since the surface of water is non-rigid it is likely that a molecule which strikes the surface achieves thermal accommodation with near-unit probability. The molecule is bound to the surface in a potential well of depth aU, where aUs is the binding energy of the molecule to the liquid surface. 

In Fig. 8, curve (1) represents the dependence of surface tension on electrolyte concentration, based on the present model. It is assumed that OH- ions are adsorbed on the surface, with an apparent equilibrium constant K i)U = 1 X10 10 M, and that there are a cut-off 4=4 A for the cations, and a potential well of depth W1=0.5kT... 

However, doubly ionized oxygen, O2-, in Cu oxides, emits an electron in a vacuum, but is to be stabilized in an ionic crystal, and the author found that delocalization of electrons on the oxygen site causes the antiferromagnetic moment on the metal site. The analysis was performed by changing width and depth (including zero depth) of a well potential added to the potential for electrons of oxygen atom in deriving numerical trial basis functions (atomic orbitals). (The well potential was not added to copper atom.) The radial part of trial basis function was numerically calculated as described in the previous... 

Due to its high zero-point kinetic energy, Ps is supposed to dig a cavity, or "bubble" in liquids [56, 57]. Various levels of approximation are possible for a quantum mechanical approach to the problem the potential well (of depth U) constituting the bubble may be considered or not as infinite and/or rigid [58-60]. Some typical values of (rigid) well depth and radius are given in Table 4.2 [61] the bubble radius, Rb, remains in a rather narrow range, about 0.3—0.45 nm, independently of the solvent or temperature. 

Now if the ball is actually a molecule in a potential energy well of depth E, the occurrence of heat oscillations at the frequency ksT / h allows an estimation of the limit storage time in terms of the theory of monomolec ular reactions ... 

For the calculation of second virial coefficients in Equations 6 and 9, the actual distribution of the adsorption potential in the cavity was replaced by rectangular potential well. The depth of the well was chosen in such a manner that the following relation would be correct. 

The properties of the above system at modest particle concentrations are relatively simple to model, because the grafted octadecyl layer is thin compared to the particle radius and because the particle-particle interactions are weak enough that the properties of the dispersion are not sensitive to the detailed shape of the particle-particle interaction potential. These considerations have motivated the use of a simple square-well potential as a model of the particle-particle interactions (Woutersen and de Kruif 1991) (see Fig. 7-3). This potential consists of an infinite repulsion at particle-particle contact (where D — 0), bounded by an attractive well of width A and depth e. There are no interactions at particle-particle gaps greater than A. Near the theta point, the well depth s depends on temperature as follows (Hory and Krigbaum 1950) ... 

In this ansatz there is no Coulomb potential but an attractive well of depth e, which is switched on for infinitely short intervals after each period 2rc/co. Since the potential dies out for x oo, the asymptotic states are easily identified they are defined by the asymptotic kinetic energy E of the particle and its phase 2tct relative to the periodic oscillations of the potential in time. 

Positronium in condensed matter can exist only in the regions of a low electron density, in various kinds of free volume in defects of vacancy type, voids sometimes natural free spaces in a perfect crystal structure are sufficient to accommodate a Ps atom. The pick-off probability depends on overlapping the positronium wavefunction with wavefunctions of the surrounding electrons, thus the size of free volume in which o-Ps is trapped strongly influences its lifetime. The relation between the free volume size and o-Ps lifetime is widely used for determination of the sub-nanovoid distribution in polymers [3]. It is assumed that the Ps atom is trapped in a spherical void of a radius R the void represents a rectangular potential well. The depth of the well is related to the Ps work function, however, in the commonly used model [4] a simplified approach is applied the potential barrier is assumed infinite, but its radius is increased by AR. The value of AR is chosen to reproduce the overlap of the Ps wavefunction with the electron cloud outside R. Thus,... 

A typical model potential used to describe the scattering process 233—235 include an attractive component, such as a square well of depth D and width d and a repulsive component in the form of a stationary sinusoidal hard wall (or infinitely hard corrugated surface ). Then the potential V(r) is given by... 

In order to figure out the nature of their bonding, we performed the calculation for the sulfur fluorides, SF2, SF4, SF5 and S2F2. The molecular structures which were taken from Ref. 9 are shown in Fig 1. We took the symmetry of these molecules as C2v for SF2 and SF4, Ofi for SF5 and Cg for S2F2. The number of sample points used in the numerical integration was taken up to 500 per atoms for each calculation. Self-consistency within 0.0005 electrons was obtained for the final orbital populations. A well potential with width of 7.0 and depth of 0.5 was added. 

A quantitative comparison between the mean field prediction and the Monte Carlo results is presented in Fig. 15. The main panel plots the inverse scattering intensity vs. xN. At small incompatibility, the simulation data are compatible with a linear prediction (cf. (48)). From the slope, it is possible to estimate the relation between the Flory-Huggins parameter, x, and the depth of the square well potential, e, in the simulations of the bond fluctuation model. As one approaches the critical point of the mixture, deviations between the predictions of the mean field theory and the simulations become apparent the theory cannot capture the strong universal (3D Ising-like) composition fluctuations at the critical point [64,79,80] and it underestimates the incompatibility necessary to bring about phase separation. If we fitted the behavior of composition fluctuations at criticality to the mean field prediction, we would obtain a quite different estimate for the Flory-Huggins parameter. 

Minami proposed a diffusion path model, in which a path is defined by connectivities of sites with different potential depths. At least three distinct pairs of interactions are present in AgI-Ag2Mo04 glasses, which characterize such potential wells. The Ag" ion migrates in a minimum energy path defined by the connectivities of the wells. 

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