Harmonic spring model

Harmonic spring models and scaling arguments29 lead to simple predictions of a power law dependence of the bending rigidity on the thickness of the membrane kc with p between 2 and 3. 

To enable an atomic interpretation of the AFM experiments, we have developed a molecular dynamics technique to simulate these experiments [49], Prom such force simulations rupture models at atomic resolution were derived and checked by comparisons of the computed rupture forces with the experimental ones. In order to facilitate such checks, the simulations have been set up to resemble the AFM experiment in as many details as possible (Fig. 4, bottom) the protein-ligand complex was simulated in atomic detail starting from the crystal structure, water solvent was included within the simulation system to account for solvation effects, the protein was held in place by keeping its center of mass fixed (so that internal motions were not hindered), the cantilever was simulated by use of a harmonic spring potential and, finally, the simulated cantilever was connected to the particular atom of the ligand, to which in the AFM experiment the linker molecule was connected. 

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

The bead model for polymer simulations. The heads may he connected by stiff rods or by harmonic springs. 

The usual structure of off-lattice chain models is reminiscent of the Larson models the water and oil particles are represented by spheres (beads), and the amphiphiles by chains of spheres which are joined together by harmonic springs... 

Bead-spring models without explicit solvent have also been used to simulate bilayers [40,145,146] and Langmuir monolayers [148-152]. The amphi-philes are then forced into sheets by tethering the head groups to two-dimensional surfaces, either via a harmonic potential or via a rigid constraint. 

Figure 1. The N-particle ding-a-ling model. Odd particles can move freely in between two collisions, while even particles are bounded by a harmonic spring.

The diode model consists of two segments of nonlinear lattices coupled together by a harmonic spring with constant strength kint (see Fig. 6). Each segment is described by the (dimensionless) Hamiltonian ... 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

Electrons in metals at the top of the energy distribution (near the Fermi level) can be excited into other energy and momentum states by photons with very small energies thus, they are essentially free electrons. The optical response of a collection of free electrons can be obtained from the Lorentz harmonic oscillator model by simply clipping the springs, that is, by setting the spring constant K in (9.3) equal to zero. Therefore, it follows from (9.7) with co0 = 0 that the dielectric function for free electrons is... 

The harmonic oscillator model does not take into account the real nature of chemical bonds, which are not perfect springs. The force constant k decreases if the atoms are pulled apart and increases significantly if they are pushed close together. The vibrational levels, instead of being represented by a parabolic function as in equation (10.3), are contained in an envelope. This envelope can be described by the Morse equation (Fig. 10.5) ... 

The most important chemical applications of the harmonic oscillator model are to the vibrations of molecules. Figure 3.7 shows how we can regard a diatomic molecule as two nuclei held together by a spring which represents the effects of the electrons forming the chemical bond. There are two difficulties we need to discuss, before the results of the previous section can be applied. 

In the treatment of two atoms connected together, a simple harmonic oscillator model can be adopted involving the two masses connected with a spring having a force constant fk. Thus, the vibrational frequency in wavenumbers 2 depends from the reduced mass p, from fk with c being the velocity of light. 

Near the equilibrium bond length qe the potential energy/bond length curve for a macroscopic balls-and-spring model or a real molecule is described fairly well by a quadratic equation, that of the simple harmonic oscillator (E = ( /2)K (q — qe)2, where k is the force constant of the spring). However, the potential energy deviates from the quadratic (q ) curve as we move away from qc (Fig. 2.2). The deviations from molecular reality represented by this anharmonicity are not important to our discussion. 

The simplest of these approaches includes Gaussian Network Models (GNM) or Elastic Network Models (ENM) which assume that the native state represents the minimum energy configuration. A structure is represented as a network of beads connected by harmonic springs.12,13 One bead represents one residue and is usually centered on the position of the Ca carbon. Single parameter harmonic interactions are assigned to bead pairs which fall within a certain cutoff distance Rc. In case of proteins, Rc is usually around 8-10 A. The representation of the molecule in the... 

Figure 2 In the shell model, a core charge z, + q, is attached by a harmonic spring with spring constant kj to a shell charge - ij,. For a neutral atom, Zi = 0. The center of mass is at or near the core charge, but the short-range interactions are centered on the shell charge. (Not drawn to scale the displacement dj between the charges is much smaller than the atomic radius.)...

The bead—spring model allows the introduction of finite extensibility of the (harmonic springlike) bonds by the introduction of a form of a harmonic-type potential... 

The definition of a theory as a set of hypotheses that has passed a test of experimental verification is uncontroversial. However, the equation of theories and models proposed by Zumdahl and Petrucci and Harwood is less straightforward, I think. It surely is true that all but the most grandiose of scientists would admit that their theories were approximations to reality, and so, to the extent that a model requires a specified list of approximations, all theories are models. However, not all models are theories. If I make the approximation of treating molecules as perfect spheres or springs as massless, I am creating a model that will make subsequent calculation easier or comprehension of the results easier, but I presumably do not believe these approximations to be true in my theory of what is occurring in reality. Chemists will talk of the harmonic-oscillator model as a mathematically convenient approximation for the interpretation of vibrational spectra, but I do not think many people would consider this to be a theory of vibrational spectroscopy. 

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