Lennard-Jones 6-12 potential equilibrium distance

Figure 2.10. Part of the better description of the Morse and Exp.-6 potentials may be due to the fact that they have three parameters, while the Lennard-Jones potential only employs two. Since the equilibrium distance and the well depth fix two constants, there is no additional flexibility in the Lennard-Jones function to fit the form of the repulsive interaction.

Work with a neighbor. Consider the Lennard-Jones potential, as given by Eq. (1.12), for which m = 6 and w = 12. Yon wish to determine the separation distance, r, at which the maximum force occurs, in terms of the equilibrium bond distance, ro. 

Person 1 Use Eq. (1.16) with the values of m and n of the Lennard-Jones potential to solve for the constant a in terms of b and the equilibrium bond distance, ro. Now perform the determination of F ax as given by Eq. (1.15) substitute this value of a back into Eq. (1.12), differentiate it twice with respect to r (remember that ro is a constant), and set this equal to zero (determine, then maximize the force function). Solve this equation for r in terms of tq. The other constant should drop out. 

Neon, and the elements directly below it in the periodic table or the Solid State Table, form the simplest closed-shell systems. The electronic structure of the inert-gas solid, which is face-ccntercd cubic, is essentially that of the isolated atoms, and the interactions between atoms are well described by an overlap interaction that includes a correlation energy contribution (frequently described as a Van der Waals interaction). The total interaction, which can be conveniently fitted by a two-parameter Lennard-Jones potential, describes the behavior of both the gas and the solid. Electronic excitations to higher atomic states become excitons in the solid, and the atomic ionization energy becomes the band gap. Surprisingly, as noted by Pantelides, the gap varies with equilibrium nearest-neighbor distance, d, as d... 

In order to make comparison with the experiment [Park 2000], we chose the frequency = 5 meV, which corresponds to a (, m quantum dot interacting with gold electrodes via the Lennard-Jones potentials. For the electron temperatures ksT = 0.4 meV, the vibrational amplitude is xo 0.03A, and hence only the zero-point mode is active, provided O > kj j. The equilibrium distance D/2 6.2Aseparates the center-of-mass of the quantum dot from both leads. When eV < Q, the frequency shifts un are negligible, since the system "leads + dot" is cold ksT < Q, and therefore vn = F xn = xn/D < nVL. 

Lennard-Jones potential As two atoms approach one another there is the attraction due to London dispersion forces and eventually a van der Waals repulsion as the interatomic distance r gets smaller than the equilibrium distance. A well-known potential energy function to describe this behavior is the Lennard-Jones (6-12) potential (LJ). The LJ (6-12) potential represents the attractive part as r-6-dependent whereas the repulsive part is represented by an r n term. Another often used nonbonded interaction potential is the Buckingham potential which uses a similar distance dependence for the attractive part as the LJ (6-12) potential but where the repulsive part is represented by an exponential function. 

Contraction of atom-atom distances as found on the surfaces of metals is a quantum-mechanical phenomenon. If the computed equilibrium distance between atoms in an infinite chain of atoms interacting with a Lennard-Jones potential is compared with that between atoms in a diatomic molecule which interact with the same potential, one findst l ... 

Fig. 3. Reduced plot of the Lennard-Jones (12, 6) potentiaj and Morse potentials for two different values of p versus the scaled distance R/Re. Note that the scaled variables V(R)/De have the same meaning De is the dissociation energy of the pair of particles bound by the potential and the function is plotted such that the potential goes to zero at a large distance. However the scaled distances oiR and R/tr do not have quite the same meaning the equilibrium distance, (the potential minimum), for the Morse potential occurs at R = Re, while that of the Lennard-Jones potential occurs at R i =...

The dispersion and repulsion interactions form the Lennard-Jones potential (Barrer, 1978 Masel, 1996 Razmus and Hall, 1991 Gregg and Sing, 1982 Steele, 1974 Adamson and Cast, 1997 Rigby, et al., 1986), with an equilibrium distance (rf) at which point Pd +Pr= 0. This distance is taken as the mean of the van der Waals radii of the interacting pair. Once the attractive, dispersion constant. A, is known, B is readily obtained by setting dd> dr = 0 at tq. Hence, B = Afq jl. Interestingly, at ro, 4>d = — r- The most commonly used expression for calculating A is the Kirkwood-MiiUer formula ... 

The harmonic oscillator is particularly important, because any mechanical system in the vicinity of stable equilibrium can be approximated by a harmonic oscillator. If the deviations from equilibrium are small, one can set the origin at the equilibrium point and expand the potential energy in powers of the displacement. To take a simple example, for a pair of identical atoms interacting via the Lennard-Jones potential [Eq. (6)], if the distance between the atoms r is close to the equilibrium value ro = we can expand the potential in powers of m = r — ro... 

The average radius of the first solvation shell is usually less than the equilibrium distance R in the Lennard-Jones potential. Each additional intermolecular bond further stabilizes the reference molecule, so an increased density can improve the stability of the system by increasing the average coordination number. 

As a second example we take a Lennard-Jones potential with parameters fitted to the equilibrium distance a and the bond energy [11, 23], which gives... 

Interaction contributions to a typical force held. Bond stretch vibrations are described by a harmonic potential 1/ , the minimum of which is at the equilibrium distance bo between the two atoms connected by chemical bond / (the indices i,j etc. are not shown in the Figure). Bond angles and out-of-plane (Improper) angles are also described by harmonic potential terms, and V, where 00 and (q denote the respective equilibrium angles. Dihedral twists (torsional angles) are subjected to a periodic potential 1/° the respective force constants are denoted by ft s with appropriate indices. Non-bonded forces are described by Coulomb interactions, 1/ -, and Lennard-Jones potentials, + /Vdw where the latter includes the Pauli repulsion, V and the van der... 

F. The Lennard-Jones (12,6) functional form, Eq. (2.24), is often used to approximate interatomic potentials, (a) What is the equilibrium distance for the potential in units of a (b) Show that near its minimum the potential can be expanded in a harmonic form F(P) = — + k R — Rm) - (c) Express the force constant A in terms of the parameters of the potential and show that in reduced units it has the value 72. (d) Compute the vibrational frequency of the harmonic motion at the bottom of the well as a fimction of the mass and express it in reduced units. 

Figure 1.20. The Lennard-Jones potential (r) and acting force F(r) between two gas molecules. Note that the force F(r) is zero and that the potential energy F is at a minimum in the equilibrium distance r = ro.

The equilibrium distance between two molecules is achieved when the intermolecular force F(r) is 0 from (1.13) it is seen that the constant ro denotes the equilibrium distance between two molecules. Further it is seen that the Lennard-Jones potential (1.12) has minimum (r) = —for r = ro, i.e. when the molecules are at their mutual equilibrium distance. For r —> oo, (P r) and F r) — 0. 

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