Root-mean-square gradient

You should terminate a geometry optimization based upon the root-mean -square gradient, because the number of eycles needed to m in iiTi i/,e a rn oleculc varies accord in g to th c in itial forces on the... 

In setting up an optimization calculation, you can use two convergence criteria the root-mean-square gradient and the number of optimization cycles. 

On the other side, the chemical stmctures of the molecules of Figure 3.45 were computed by the Hyper Chem Software (7 Release version) (Hypercube, 2002) providing that their 3D stmctures were optimized by AMI semiempirical calculation while molecular d5mamics involved the Polak-Rebier algorithm to reach to 0.01 root mean square gradient. The results are as well displayed in Table 3.72 as LogP, POL, and for the hydrophobic, electronic and steric parameters of Eq. (11), respectively. 

For multi-dimensional potential energy surfaces a convenient measure of the gradient vector is the root-mean-square (RMS) gradient described by... 

The parameter used to design rapid mix and flocculation systems is the root mean square velocity gradient G, which is defined by equation... 

On average, the reflectivity decays in proportion to 0 or since both interfaces are sharp. However, if either surfaces is rough, then marked deviations are seen and the reflectivity is damped by a factor of exp -2k o, where a is the root-mean-square roughness. Thus, the reflectivity is very sensitive to surface roughness and to concentration gradients at interfaces. 

Figure 22 Root-mean-squared error progression plot for Fletcher nonlinear optimization and back-propagation algorithms during training.-----Fletcher routine —A—Gradient search.

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

Fig. 4.11 Instantaneous nondimensional velocity profiles in a circular duct with a purely oscillating pressure gradient. The average root-mean-square velocity, averaged over one full period, shows a region of high velocity away from the centerline. These solutions were computed in a spreadsheet with an explicit finite-volume method using 16 equally spaced radial nodes and 200 time steps per period. The plotted solution is that obtained after 10 periods of oscillation.

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