Hard square model

This set of exponents agrees with those of the hard square model (Huse, 1982). This model is defined as follows consider placing hard squares of linear dimension /2a on a square lattice of lattice spacing a, such... 

Fig. 32 (contd.). (b) Typical configurations of the hard square model at two values of H, 0 — 0.36 (b) and 0 = 0.375 (c), for a lattice of linear dimension L = 40 and periodic boundary conditions. Points show the centers of the hard squares. The largest cluster of the c(2x2) structure is indicated by connecting the points. From Binder and Landau (1980). 

Approximating the real potential by a square well and infinitely hard repulsive wall, as shown in figure A3.9.2 we obtain the hard cube model. For a well depth of W, conservation of energy and momentum lead [H, 12] to the very usefiil Baule fomuila for the translational energy loss, 5 , to the substrate... 

Fig. 19. Simulation results for both the soft-sphere model (squares) and the hard-sphere model (the crosses), compared with the Carnahan-Starling equation (solid-line). At the start of the simulation, the particles are arranged in a FCC configuration. Spring stiffness is K = 70,000, granular temperature is 9 = 1.0, and coefficient of normal restitution is e = 1.0. The system is driven by rescaling.

Figure 8.3 Convergence with number of binomial moments of hydration free energy predicted using several default models for a spherical solute with distance of closest approach 3.0 A for water oxygen atoms. Identifications are diamonds (dash-dot lines), hard-sphere default HS), crosses (short dash line), Lennard-Jones LJ) default squares (long dash line), Poisson default triangles (dotted line), cluster Poisson default and circles (solid line), flat default. For this circumstance, yth-order binomial moments are non-zero through j = 9, and the horizontal line is the prediction with all nine moments included. Among the predictions at j = 2, the best default model is the Lennard-Jones case. But with the hard-sphere model excepted, the differences are slight. See Hummer et al. (1996), Gomez et al. (1999) and Pratt etal. (1999) for details of the calculations.

Am/ T —> oo, (x/T finite an occupation of nearest neighbor sites becomes strictly forbidden, and a hard-square exclusion results. Thus this transition is the end-point of the phase diagram shown in fig. 28a. But at the same time, it is the end-point of a line of tricritical transitions obtained in the lattice gas model when one adds an attractive next-nearest neighbor interaction pnnn and considers the limit R = Ainn/Am - 0 (Binder and Landau, 1980, 1981 fig. 32). 

Here, Ae is the charge transferred from one reactant to the other, t and r2 are the radii of the two (spherical) reactants, r12 is, as before, the center-to-center distance, often approximated18 as the sum of + r2, and Ds and Dop are the static and optical (square of refractive index) dielectric constants of the solvent, respectively. This model for /lout treats both the reactants as hard spheres (i.e., the hard sphere model). For other shapes, more complex models are needed, which are rarely used by reaction chemists.21... 

The simplest attractive hard-sphere model is the square-well potential, for which the energy is constant (and negative) over some range extending beyond the hard repulsive core outside of this range the energy is zero, that is. 

Consider the intermolecular potential curve of a molecule represented on the one hand by the hard sphere model (with repulsive wall k, s = 2.4 A) and on the other by the square well model (with well between = 2.4 A... 

Figure 5. Dimensionless total structure versus absolute wave vector for a N = 6429 polyethylene melt just above its melting temperature. The solid circles are the x-ray scattering data, and the line is the PRISM prediction based on a hard-core model with dcH, = 3.9A. The solid square at Ac = 0 represents the experimental value based on the measured isothermal compressibility and liquid density.

D. Hard-sphere models can be extended for reactive colUsions [e.g., B. H. Mahan, J. Chem. Educ. 51,308 and 377 (1974)]. Since the hard-sphere potential is purely repulsive, the one refinement that is needed is to modify the hard-sphere potential so as to mimic a chemical bond. We do so by adding a square well, of depth —D and finite range, to the atom atom potential. It will then look like (i) below. For an A + BC colhsion the hard-sphere potential energy surface will then look like (ii). 

Fig. 3.18 a STM image of the Co(1010)-(2 x 4)-40 surface (Reprinted with permission from [110]), b the hard-sphere model for the Ru(1010)-(2 x 4) 0 strucmre [117], and c the bond configuration [111] with the dotted square framing the tetrahedron. The lack of one (Co, Ru) atom for the tetrahedron is compensated by a virtual bond between 0 and the electron cloud labeled 2, which sharpens the tip of the honeycomb-like bumps in the STM images (Reprinted with permission from [1])... 

Figure 18 Plot of light-scattering dissymmetry ratio [7(45°)//(135°) against concentration c for a polystyrene-i)/oc/c-poly(ethylene/propylene) (37000 59 700 M ) copolymer in decane at 30 °C. The experimental points are indicated by circles ( ). The full curve and squares ( ) indicates the behaviour predicted theoretically using a hard-sphere model (reproduced...

Adding only one axial ligand to a square antiprism will result in a monocapped square antiprism (MSAP). The CN is 9 and the symmetry C4V For the most stable MSAP 0A 7O.r and 0b = 125.7° in the hard sphere model. Two 0 values are necessary to describe the structure. Because of the axial ligand, the (capped) square top face is larger than the square bottom face. Examples of rare-earth systems with a MSAP coordination polyhedron are the oxyhalogenides ROX (R = La, Y, Gd and X=C1, Br, I) (Holsa and Porcher 1981, 1982a). A square antiprism and a monocapped square antiprism are shown in fig. 15. 

P.J. Gemperline and E. Cash, Advantages of soft versus hard constraints in self-modeling curve resolution problems. Alternating Least squares with penalty functions. Anal. Chem., 75, 4236 (2003). 

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