Shifted Lennard-Jones potential

We now present results from molecular dynamics simulations in which all the chain monomers are coupled to a heat bath. The chains interact via the repiflsive portion of a shifted Lennard-Jones potential with a Lennard-Jones diameter a, which corresponds to a good solvent situation. For the bond potential between adjacent polymer segments we take a FENE (nonhnear bond) potential which gives an average nearest-neighbor monomer-monomer separation of typically a 0.97cr. In the simulation box with a volume LxL kLz there are 50 (if not stated otherwise) chains each of which consists of N -i-1... 

The potential of our model consists of two parts a shifted Lennard-Jones potential (18), Ulj, which operates between all N beads in the system and a modified harmonic potential (19), U , which links Z beads into N chains (Np= N). Each bead represents a statistical segment of the polymer which includes many monomers... 

Fig. 12. Equation of state (a) and phase diagram (b) of a bead-spring polymer model. Monomers interact via a truncated and shifted Lennard-Jones potential as in Fig. 6 and neighboring monomers along a molecule are bonded together via a finitely extensible non-linear elastic potential of the form iJpENE(r) = — 15e(iJo/

This section discusses simulations in which the parameters are explicitly mapped to experimental systems. In particular, this mapping affects ion diameter cr, rod radius r0, Bjerrum length B, and line charge density A. In order to have a rod radius different from cr this requires the introduction of a new potential for ion-rod interactions, for which a modified truncated and shifted Lennard-Jones potential has been used ... 

The molecules interact according to a cut-and-shifted Lennard-Jones potential, i/s-Lj(Os written in terms of the intermolecular distance r between two beads on two different molecules as... 

There are also many variations of the LJ 12-6 potential. One example is the computationally inexpensive tmncated and shifted Lennard-Jones potential (TSLJ), which is commonly used for molecular simulation studies in which large molecular ensembles are regarded, e.g., for investigating condensation processes [15, 16]. Another version of the LJ potential is the Kihara potential [17], which is a non-spherical generalization of the LJ model. 

Pseudo atoms on different chains and those on the same chain interact with each other via a cut-and-shift Lennard-Jones potential. Lorenz et al. [82] modeled adhesion contact and friction for perfluorinated alkylsilane SAMs. The OPLS (optimized potential for liquid simulations) all-atom force-field parameters were used. The total potential energy for the system, is represented as a sum of nonbond interactions as well as energy contributions due to the distortion of bonds... 

Extensive molecular dynamics simulations of dilute and semidilute polyelectrolyte solutions of chains with degree of polymerization N ranging from 16 up to 300 were recently performed by Stevens and Kremer [146-148] and by Liao et al, [149], In these simulations the long-range electrostatic interactions were taken into account by the Ewald summation method, including interactions with all periodic images of the system, Stevens and Kremer [146-148] have used a spherical approximation of Adams and Dubey [150] for the Ewald sum while Liao et al, [149] have applied the PME method [110], In addition to Coulombic interactions, all particles, including monomers and counterions, interacted via a shifted Lennard-Jones potential with cutoff rcui = 2 a... 

The shift makes the potential deviate from the true potential, and so any calculated thermodynamic properties will be changed. The true values can be retrieved but it is difficult to do so, and the shifted potential is thus rarely used in real simulations. Moreover, while it is relatively straightforward to implement for a homogeneous system under the influence of a simple potential such as the Lennard-jones potential, it is not easy for inhomogeneous systems containing rnany different types of atom. 

In the simulations the maxima and minima of n y are shifted to slightly smaller porewldths compared to predictions of the theory. This trend Is consistent with the fact that the 6-12 Lennard-Jones potential Is not Infinitely repulsive at an Interparticle separation of (7, whereas the 6-oo potential Is Infinitely repulsive at a. 

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))...

As an example of application of the method we have considered the case of the acrolein molecule in aqueous solution. We have shown how ASEP/MD permits a unified treatment of the absorption, fluorescence, phosphorescence, internal conversion and intersystem crossing processes. Although, in principle, electrostatic, polarization, dispersion and exchange components of the solute-solvent interaction energy are taken into account, only the firsts two terms are included into the molecular Hamiltonian and, hence, affect the solute wavefunction. Dispersion and exchange components are represented through a Lennard-Jones potential that depends only on the nuclear coordinates. The inclusion of the effect of these components on the solute wavefunction is important in order to understand the solvent effect on the red shift of the bands of absorption spectra of non-polar molecules or the disappearance of... 

Fig. 6. Grand-canonical simulation of a Lennard-Jones monomer system. Particles interact via a shifted and truncated Lennard-Jones potential of the form Elj =...

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. 

The charge yield (shown on the plots 3 and 4) is related to the mean field that detunes the dot level from the resonance. The detuning also includes the frequency shifts due to the bonding interaction in the harmonic approximation near the potential minimum. The use of the Lennard-Jones potential could be made at longer distances, where the attraction to the surface is created by the van der Waals or Casimir-Polder potentials. This "spontaneous" interaction is relevant for the shuttle with a large amplitude of vibrations close to electrodes surfaces. 

In this expression n stands for the reduced mass, l0 for the orbital angular momentum for which the phase shift rj assumes its maximum value rj0 the absolute value of the second derivative of the phase shift with respect to / at 1 = l0 is denoted by rjo, (10) is linear in the anisotropy parameters q2.6 and 9212 °f (5) the quantities S(n, l - /, , b) are integrals which are calculated and tabulated by Franssen (1973) for a range of values of the reduced energy and the reduced collision parameter b with b0 = 21/6(/0 + 1/2)/(Rmk). The parameter n takes the values 12 and 6 for the Lennard-Jones potential actually used for / - / one has to insert the values 0 and 2, the difference of the orbital angular momentum of channels which are coupled by the interaction of (5). 

The plain Lennard-Jones potential is shifted up, so that its minimum located at 21/6ct has value 0, and set to zero beyond that point. The advantage of including the r 6 contribution instead of merely using the purely repulsive r 12 is that Eq. 4 is exactly zero beyond rcut and merges smoothly to this value at rCM. The use of a smooth hard core in molecular dynamics simulations is advantageous since the force is the derivative of the potential therefore the latter should be differentiable. In fact, the derivative must also be bounded to ensure numerical stability of the discrete integrator. 

Here, r is the radial distance from the rod axis to the center of the ion, rs is a parameter that shifts the Lennard-Jones potential towards larger r, and rcut = rs + 21,6cr. In Figure 15 the relation between rs and the rod radius is... 

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)...

The great expense in calculation time due to the inevitably large particle numbers in single-file systems calls for the application of simplified potentials. Figure 1 shows the results obtained for spherical molecules diffusing in an unstructured tube [22]. Particle-particle and particle-wall interactions have been simulated by a shifted-force Lennard-Jones potential [26] and an... 

Bjornsson and Biihl have proposed an approach to modeling moleeular crystals such that crystal effects on the equilibrium geometry of the moleeule can be captured. In this approach, the molecule of interest is embedded in the presence of point charges and Lennard Jones potentials representing its neighbors in the crystal lattice, which are updated in self-consistent fashion. The novelty here is that the medium effects no longer require a previously established set of force field parameters as these are determined at the same time and as needed in the process. These have been applied to explain the gas-to-solid bond contraction of HCN-BF3 and solid state V NMR chemical shifts in VOCI3 and a vanadium catechol complex. 

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