Potential function square-well

A monolayer can be regarded as a special case in which the potential is a square well however, the potential well may take other forms. Of particular interest now is the case of multilayer adsorption, and a reasonable assumption is that the principal interaction between the solid and the adsorbate is of the dispersion type, so that for a plane solid surface the potential should decrease with the inverse cube of the distance (see Section VI-3A). To avoid having an infinite potential at the surface, the potential function may be written... 

Equation XVII-78 turns out to ht type II adsorption isotherms quite well—generally better than does the BET equation. Furthermore, the exact form of the potential function is not very critical if an inverse square dependence is used, the ht tends to be about as good as with the inverse-cube law, and the equation now resembles that for a condensed him in Table XVII-2. Here again, quite similar equations have resulted from deductions based on rather different models. 

We use the off-lattice MC model described in Sec. IIB 2 with a square-well attractive potential at the wall, Eq. (10), and try to clarify the dynamic properties of the chains in this regime as a function of chain length and the strength of wall-monomer interaction. 

From a structural point of view the OPLS results for liquids have also shown to be in accord with available experimental data, including vibrational spectroscopy and diffraction data on, for Instance, formamide, dimethylformamide, methanol, ethanol, 1-propanol, 2-methyl-2-propanol, methane, ethane and neopentane. The hydrogen bonding in alcohols, thiols and amides is well represented by the OPLS potential functions. The average root-mean-square deviation from the X-ray structures of the crystals for four cyclic hexapeptides and a cyclic pentapeptide optimized with the OPLS/AMBER model, was only 0.17 A for the atomic positions and 3% for the unit cell volumes. 

The solid and dashed curves, shown in Fig. 57a for L = 1.0 and 1.42 A, respectively, demonstrate that the form of the self-consistent potential well, found here, resembles, generally speaking, the form of the hat-curved well (see Fig. 20). The potential function t/(p) (433) widens near its bottom, if the H-bond length L increases (cf. solid and dashed curves calculated, respectively, for L = 1.85 and 1.54 A). Note that the cosine-squared (CS) potential well63 is substantially more concave than the self-consistent potential given by Eq. (433) (see dashed-and-dotted curve in Fig. 57a). 

This equation acknowledges that real molecules have size. They have an exclusion volume, defined as the region around the molecule from which the centre of any other molecule is excluded. This is allowed for by the constant b, which is usually taken as equal to half the molar exclusion volume. The equation also recognizes the existence of a sphere of influence around each molecule, an interaction volume within which any other molecule will experience a force of attraction. This force is usually represented by a Lennard-Jones 6-12 potential. The derivation below follows a simpler treatment (Flowers Mendoza 1970) in which the potential is taken as a square-well function as deep as the Lennard-Jones minimum (figure 2a). Its width x is chosen to give the same volume-integral, and defines an interaction volume Vx around the molecule, which will contain the centre of any molecule in the square well. This form of molecular pair potential then appears in the Van der Waals equation as the constant a, equal to half the product of the molar interaction volume and the molar interaction energy. 

The properties of electrons in a periodic potential are demonstrated with the use of a simple square well potential, the Kronig-Penney model. The potential is zero for 0 < x < a and Vq for —b < x < 0, i.e. it has a period (a + b). The Schrodinger equations for the two regions follow directly and lead on substitution of a Bloch function to ... 

Figure 4 Illustration of resonances in a one-dimensional square-well potential. The two lowest solid lines are bound-state wave functions, whereas the upper two solid lines illustrate resonance wave functions. The dashed curve represents a non-resonant scattering state . Shown is the modulus square of x the scaling is different for the different wave functions. In the numerical example in the text, Vi = 8 and V2 = 12.

Although Eq. (16.12) is based on an intemiolecular potential function that is in detail unrealistic, it nevertheless often provides an excellent fit of second-virial-coefficieiitdata. An example is provided by argon, for which reliable data for B are available over a wide temperature range, from about 85 to 1000 K. The correlation of these data by Eq. (16.12) as shown in Fig. 16.3 results from the parametervaluese/k = 95.2 K,/ = 1.69,andJ = 3.07 x 10 cm. This empirical success depends at least in part on the availability of three adjustable parameters, and is no more tlian a limited validation of the square-well potential. Use of tins potential does illustrate by a very simple calculation how the second virial coefficient (and hence tire vohnne of a gas) may be related to molecular parameters. 

This function is represented by the dark lines in Fig. 3.1. Infinite potential energy constitutes an impenetrable barrier. The particle is thus bound to a potential well, sometimes called a square well. Since the particle cannot penetrate beyond the endpoints x = 0 or x = a, we must have... 

The minimum number of nodes strictly depends on the number of unknowns in the approximating functions as well as the element type and number of known data. Should the system consist only from the continuity equations and no data imposed on electric potential, one can find the number of required collocation points as follows. Given that, the square elements are placed in the regular grid with J elements along x-axis and K elements along y-axis and 6 unknown real coefficients in every element, one finds the total number of unknowns Nunkn as ... 

The potential is the same as the square-well potential when R < dup + rsw sw is the switching distance, a and b are chosen so that Enoe is a smooth function at the point R = dup + rgw and c is the asymptote. When tiie violation is very large, the constraint energy will only increase linearly with distance, so that the potential is effective at long range. 

The external wave function with the potential (27) is proportional to the exact external solution of the square-well potential. The solution, which is continuous at r = R, is given by... 

Figure 26. Exact values of %c for R = oo and SFSS calculations of, (R=10) (indistinguishable lines) and X, =io as a function of 8 for the minimum energy state of the square-well potential.

Fig. 4.20. Ring-puckering potential function for trimethylene imine. The squared wave functions illustrate the definite left well <— right well identifications of the first four levels.

To facilitate the analytic solution, the square well potential is taken to be infinitely narrow and infinitely strong, resulting in a delta function at r = L. This problem reduces to the solution of the particle-particle viewpoint of polyatomic liquids, for a hard sphere fluid with a delta function attraction... 

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