Potential energy Lennard-Jones

Fig. 3. Curves calculated using (8) for a series of increasing a values. The curves were calculated using tr = 0.6 nm and e = 0.4 kj/mol. Note that for a = 0.0 the normal 6-12 Lennard Jones potential energy function is recovered.

The interaction of hydrogen (deuterium) molecules with a transition metal surface c an be conveniently described in terms of a Lennard--Jones potential energy diagram (Pig. 1). It cxxislsts of a shallcw molecular precursor well followed by a deep atomic chemisorption potential. Depending on their relative depths and positions the wells m or may not be separated by an activation energy barrier E as schematically Indicated by the dotted cur e in Fig. 1. 

The classical approach for discussing adsorption states was through Lennard-Jones potential energy diagrams and for their desorption through the application of transition state theory. The essential assumption of this is that the reactants follow a potential energy surface where the products are separated from the reactants by a transition state. The concentration of the activated complex associated with the transition state is assumed to be in equilibrium... 

Lennard-Jones gas dimensionless thermal diffusion ratio, 25 307 Lennard-Jones molecules, 25 302 Lennard-Jones potential, 7 620 23 94 Lennard-Jones potential energy function, 25 302... 

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

Lennard-Jones potential energy diagram, JABLONSKI DIAGRAM Leucine,... 

In PLPscore, both the steric and hydrogen bonding terms are calculated from a piecewise linear potential function (see Fig. 1), instead of a smooth 6-12 Lennard-Jones potential energy function. The difference between the two terms is simply in the parameter values chosen for each term. 

The Lennard-Jones potential energy function systematically describes the attractive and repulsive forces at all distances in terms of s the well depth and a the location of the potential minimum. 

AB > AB = constants in the Lennard-Jones potential-energy function for the molecular pair AB Oab is in A. 

Since the Lennard-Jones potential-energy function is used, the equation is strictly valid only for nonpolar gases. The Lennard-Jones constants for the unlike molecular pair AB can be estimated from the constants for like pairs A A and BB ... 

Figure 4. Phase coexistence of orientation-disordered and orientation-ordered phase in TeFe clusters of various sizes (a) 137 molecules (g = 300, r=93.3K, Z o = 92.5K), (b) 89 molecules (Lennard-Jones potential energy, Q = 300, r= 88 K, 5-ns run), (c) 59 molecules (Q = 200, r=76K, ro = 76K).

Figure 25.3 Lennard-jones potential energy diagram of a Hj molecule interacting with an active (full line) and an inactive metal surface (dotted line) as a schematic one-dimensional description of the activated (non-activated) hydrogen adsorption. The dashed line indicates the potential energy U(z) for a pre-dis-sociated Hj molecule (shifted by the dissociation energy E, , with respect to energy zero)...

Figure 3.2. Lennard-Jones potential energy diagram for the interaction of hydrogen with the surface of a metal of Groups 8-10 (see text for description). The lower part of the diagram shows possible configuration at three points in the chemisorption process.

The potential energy E in eq. (2.5-1) is contributed by two interactions. One is the vertical interaction adsorbate-adsorbent interaction, and the other is the horizontal interaction adsorbate-adsorbate interaction. The vertical interaction energy is the negative of the well depth of the Lennard-Jones potential energy between a molecule and all the atoms on the surface. The horizontal interaction between two adsorbed molecules is (eq. 2.4-13)... 

Figure 4.33 The Lennard-Jones potential energy curve (red) between two nonpolar molecules separated by a distance r. The upper and lower curves (blue) show the repulsive 1/r and attractive 1/r contributions, respectively.

Fig. 2.10. Comparison of Dymond-Alder (—) and Lennard-Jones (--) potential-energy functions for argon. The insert compares them in the domain where repulsive forces dominate.

Here Qi and Qj represent the two point charges, while Ry equals the distances between these two points. In some force fields, Coulombic interactions are modified by changing the dependence of the dielectric constant, e. In general, van der Waals interactions are modeled using a 6-12 Lennard-Jones potential energy term. This expression, shown in Eq. (28), consists of a repulsion and attraction term. 

The second simulation technique is molecular dynamics. In this technique, which was pioneered by Alder, initial positions of theparticles of a system of several hundred particles are assigned in some way. Displacements of the particles are determined by numerically simulating the classical equations of motion. Periodic boundary conditions are applied as in the Monte Carlo method. The first molecular dynamics calculations were done on systems of hard spheres, but the method has been applied to monatomic systems having intermolecular forces represented by the square-well and Lennard-Jones potential energy functions, as well as on model systems representing molecular substances. Commercial software is now available to carry out molecular dynamics simulations on desktop computers. ... 

The energy barrier is shown in the Lennard-Jones potential energy graphical representation (Eig. 5.7), indicating how the energy varies with distance between molecules or atoms, as, for example, for the reaction A + B R. 

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