Distance dependence

Use Equation VIII-1 to determine the effective mass of the cantilever if the cantilever has a spring constant C = 20 N/m, the minimum detectable force gradient is hF/dz = 4 X 10 N/m, and the frequency shift is 200 kHz. How does the frequency shift depend on distance from the surface if the force has a 1/z distance dependence ... 

R), i.e. there is no effect due to caging of the encounter complex in the common solvation shell. There exist numerous modifications and extensions of this basic theory that not only involve different initial and boundary conditions, but also the inclusion of microscopic structural aspects [31]. Among these are hydrodynamic repulsion at short distances that may be modelled, for example, by a distance-dependent diffiision coefficient... 

Deniz A A, Dahan M, Grunwell J R, Ha T, Faulhaber A E, Chemla D S, Weiss S and Schultz P G 1999 Single-pair fluorescence resonance energy transfer on freely diffusing molecules observation of Forster distance dependence and subpopulations Proc. Natl Acad. Sc/. USA 96 3670-5... 

By using an effective, distance-dependent dielectric constant, the ability of bulk water to reduce electrostatic interactions can be mimicked without the presence of explicit solvent molecules. One disadvantage of aU vacuum simulations, corrected for shielding effects or not, is the fact that they cannot account for the ability of water molecules to form hydrogen bonds with charged and polar surface residues of a protein. As a result, adjacent polar side chains interact with each other and not with the solvent, thus introducing additional errors. 

In this model of electrostatic in teraction s, two atoms (i and j) have poin t charges tq and qj. The magnitude of the electrostatic energy (V[. , [ ) varies inversely with the distance between the atoms, Rjj. fh e effective dielectric constant is . For in vacuo simulations or simulation s with explicit water rn olecules, the den om in a tor equals uRjj, In some force fields, a distance-dependent dielectric, where the denominator is uRjj Rjj, represen is solvent implicitly. 

ZINDO/1 IS based on a modified version of the in termediate neglect of differen tial overlap (IXDO), which was developed by Michael Zerner of the Quantum Theory Project at the University of Florida. Zerner s original INDO/1 used the Slater orbital exponents with a distance dependence for the first row transition metals only. Ilow ever. in HyperChein constant orbital expon en ts are used for all the available elein en ts, as recommended by Anderson. Friwards, and Zerner. Inorg. Chem. 2H, 2728-2732.iyH6. 

The above poten tial describes the m onopole-m on opole interactions of atomic charges c. and q. a distance R.. apart. Normally th CSC eh arge in teractiori s arc corn pti ted on ly for non bon ded atom s and once again the I 4 in tcraetioti s m igti t be treated differen ily from the more normal nonborided interactions (I 5 relationship or rn ore). Th c dielectric con Stan t t used in th c calculation is sometimes scaled or made distance-dependent, as described in the next section. ... 

The presence of the q B term with its implied distance dependency means that the charges depend upon the molecular geometry. Thus, should the conformation of a molecule change the atomic charges will also change. Just three parameters are required for each atom in the system (the electronegativity, the idempotential and the covalent radius). 

Li) he so-called distance-dependent dielectric models. The simplest implementation of a dis-i.iiice-dependent dielectric is to make the relative permittivity proportional to the distance. Tine interaction energy between two charges qi and qj then becomes ... 

In a second attempt to obtain more insight into the binding location of the dienophile and now also the diene, we have made use of the influence of paramagnetic ions on the spin-lattice relaxation rates of species in their proximity. Qose to these ions the spin-lattice relaxation rate is dramatically enhanced. This effect is highly distance-dependent as is expressed by Equation 5.7, describing the spin-lattice... 

Use a constant dielectric of 1.0 with TIP3P water molecules in a periodic box. Because of the parameterization of TIP3P molecules, using a distance-dependent dielectric or a value other than 1.0 gives unnatural results. 

A distance-dependent dielectric constant is commonly used to mimic the effect of solvent in molecular mechanics calculations, in the absence of explicit water molecules. 

You can use two types of dielectric functions a constant and a distance-dependent dielectric. Use constant dielectric for in vacuo systems and for molecular systems with explicit solvent molecules. 

Also use constant dielectric for MM+ and OPLS calculations. Use the distance-dependent dielectric for AMBER and BlO-t to mimic the screening effects of solvation when no explicit solvent molecules are present. The scale factor for the dielectric permittivity, 8, can vary from 1 to 80. HyperChem sets 8 to 1.5 for MM-t. Use 1.0 for AMBER and OPLS, and 1.0-2.5 for BlO-t. 

Pattemators may comprise an array of tubes or concentric circular vessels to coUect Hquid droplets at specified axial and radial distances. Depending on the pattemator, various uniformity indexes can be defined using the accumulated relative values between the normalized flow rate over a certain sector or circular region and a reference value that represents a perfectly uniform distribution. For example, using an eight-sector pie-shaped coUector, the reference value for a perfectly uniform spray would be 12.5%. The uniformity index (28) could then be expressed as foUows, where is the normalized volume or mass flow rate percentage in each 45-degree sector. 

Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fuUy developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr, 71, 472 76 [1993])... 

In an isolated two-spin system, the NOE (or, more accurately, the slope of its buildup) depends simply on where d is the distance between two protons. The difficulties in the interpretation of the NOE originate in deviations from this simple distance dependence of the NOE buildup (due to spin diffusion caused by other nearby protons, and internal dynamics) and from possible ambiguities in its assignment to a specific proton pair. Mofec-ufar modeling methods to deaf with these difficulties are discussed further below. 

---