Radial derivative

To overcome the primary weakness of GTO fimetions (i.e. their radial derivatives vanish at the nucleus whereas the derivatives of STOs are non-zero), it is coimnon to combine two, tliree, or more GTOs, with combination coefficients which are fixed and not treated as LCAO-MO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs are multiplied by contraction coefficients and suimned to produce a CGTO, which approximates the proper cusp at the nuclear centre. 

The approximations for the radial derivatives are substituted into the governing PDE, Equation (8.12), to give... 

Regarding accuracy, the finite difference approximations for the radial derivatives converge O(Ar ). The approximation for the axial derivative converges 0(Az), but the stability criterion forces Az to decrease at least as fast as Ar. Thus, the entire computation should converge O(Ar ). The proof of convergence requires that the computations be repeated for a series of successively smaller grid sizes. 

The natural van der Waals radii of Table 1.1 are generally found to be in good correspondence to empirical values. However, one can establish that the van der Waals surface of a bonded atom is generally somewhat ellipsoidal (rather than spherical), with major and minor axes respectively transverse and longitudinal to the bond direction. One can also evaluate the hardness (radial derivative of steric energy), charge dependence, and temperature dependence of van der Waals radii, thereby obtaining many quantitative refinements of empirical steric concepts. 

Fig. 9. The Raman spectrum of the A i (radial)-derived v(CO) modes in crystalline Mn -Rej (CO)5Br...

The action of the derivatives appearing in Eq.(17) is performed independently on each monodimensional basis function used to build the rovibrational states in the excited state, Radial derivatives are performed on numerical functions using Fourier Transforms[43], The angular derivatives have been done analytically using... 

The radial expansions on the radial faces must recognize that the face area itself depends on r (i,t.,dAr = rdrdO). Thus the r must be retained inside the radial derivatives. 

By substituting the relationships above into the Navier-Stokes equations, eliminating the terms that involve radial derivatives of density or U, and simplifying what remains, the following ordinary differential equations emerge ... 

Since dp/dz is known to be a function of z alone, its radial derivative must vanish. Therefore it must be the case that (1 /r)(dp/dr) is a constant, which we will call Ar,... 

We can use the equation of state to write dp/dz and dp/dz in terms of dT/dz, which in turn can be replaced by substitution from the energy equation. Making all these substitutions are a very tedious task they do not need to be carried out explicitly. However, were each to be done, the integrals in Eq. 7.70 would be functions of radial derivatives of the specified... 

The radial concentration scans obtained from the UV spectrophotometer of the analytical ultracentrifuge can be either converted to a radial derivative of the concentrations at a given instant of time (dc/dr)t or to the time derivative of the concentrations at fixed radial position (dc/dt)r (Stafford, 1992). The dcf dt method, as the name implies, uses the temporal derivative which results in elimination of time independent (random) sources of noise in the data, thereby greatly increasing the precision of sedimentation boundary analysis (Stafford, 1992). Numerically, this process is implemented by subtracting pairs of radial concentration scans obtained at uniformly and closely spaced time intervals c2 — G)/( 2 — h)]. The values are then plotted as a function of radius to obtain (dc/dt) f versus r curves (Stafford, 1994). It can be shown that the apparent sedimentation coefficient s ... 

A practical difficulty, encountered in this study was performing the AUC runs at temperatures higher than 40 °C at such high temperatures, the oil vapors from the diffusion pump interfere with the UV absorption optics. They circumvented this problem by using a different type of optical scanning known as the Schlieren optics, which generates the data as profiles of radial derivative of concentration distributions as a function of radius (as opposed to the concentration versus radius scans obtained from UV optics). 

The flow is practically parallel to the axis of a cylindrical reactor and the whole system is symmetrical about the axis of the cylinder. Then the fluxes E grad T and E grad y have only radial components, and the product (grad Cp) (E grad T) is just the product of the radial derivative of Cp with the magnitude of E grad T. It takes the form... 

There are two practical approaches in formulating the working difference equations for the packed tubular reactor. The simple one is to use a forward-difference equation for the axial derivative and central-difference formulas for the radial derivatives. The leading terms in the truncation error are then proportional to k2 and to kh2, where h and k are written for the radial and axial steps. This means that, in order to take advantage of the accuracy of the approximations for the radial derivatives, k must have the same order of magnitude as h2, so that k2 and kh2 will be comparable. This is a serious limitation on the length of the axial step that can be used. 

Two considerations regarding truncation error that enter into the derivation of the partial difference equations should be pointed out. In some published formulations of these equations, the first radial derivative has been approximated by a forward-difference expression (Kl, S5, Wl). This unsymmetrical formula has no advantage over a symmetrical or central-difference expression, but has a greater—lower order—truncation error. The central-difference approximation... 

A convenient choice of weight is for the new profile, giving a weight of zero on the nth profile. In the general case, the approximation to the second radial derivative at the point m,n is... 

The surface of an atomic basin has the radial derivative of the electron density equal to zero. In general there will also be points on such surfaces where the total derivative is zero, i.e, the tangential components are also zero, such a point is marked with a dot in Figure 9.2. The basin attractor is also such a stationary point on the electron density... 

Figure 18. Electron pair density, h(u), its radial derivative, h (u ), and their difference, for H, calculated using a 204 term Hylleraas-type wavefunction, as described in Ref. [71],...

Numerical solutions to PDEs must be tested for convergence as Ar and Az both approach zero. The flnite-difference approximations for radial derivatives converge O(Ar ) and those for the axial derivative used in Euler s method converge 0(Az). In principle, just keep decreasing Ar and Az until results with the desired accuracy are achieved, but it turns out that Ar and Az cannot be chosen independently when using the method of lines. Instead, values for Ar and Az are linked through a stability requirement that the overall coefficient on the central dependent variable cannot be negative ... 

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