Derivatives slope

Order 2 minimization algorithms, which use the second derivative (curvamre) as well as the first derivative (slope) of the potential function, exhibit in many cases improved rate of convergence. For a molecule of N atoms these methods require calculating the 3N X 3N Hessian matrix of second derivatives (for the coordinate set at step k)... 

Figure 7.2 Phase coexistence conditions (circles), showing phase of lowest chemical potential as a function of (a) T, (b) P. The pgsis (heavy dots), p qilid (dashes), and pso id (solid line) curves are plotted with slopes (7.21) [or (7.23)] consistent with the expected order (7.22) [or (7.24)] for the derivative slope (dp/dT)P [or (dp/dP)T].

The general requirement for matching chemical potentials thereby determines how the derivatives (slopes) of Giiq(x) and Gsol(x) must be related in order for these phases to coexist, giving rise to a hatched coexistence region. 

Figure 6.5 Method for determining / and its derivatives. Slopes were calculated using linear regression over 5 points in a data set of 500 points. Double precision was required in the computer program in order to avoid noise in the second derivative. The fraction transformed versus temperature trace was numerically generated assuming a second order reaction.

Figure 8 Cooperative and noncooperative binding of O2. (a) Binding curves of myoglobin and hemoglobin, (b) Hill plot of binding curves. The Hill coefficient munber is determined from the first derivative (slope) of the HiU plots...

Fig. 3. ( ) Overall and (o) derivative slopes of ( ) retention diagram for n-h cadecane on pc slytene...

Here t0 is the column hold-up time, F is the phase ratio and dq/dc is the derivative (slope) of the isotherm function that is evaluated at the plateau concentration Ct. The retention time of a small injection of a compound in a column equilibrated with pure mobile phase without the compound (such as for analytical chromatography) gives the retention time under linear conditions, through the equation ... 

The minimum value of G in Equation 10.9 can be found by finding the value of Xb at which the derivative (slope) of G(Xb) is equal to zero. Taking the derivative of Equation 10.9 gives... 

If we can assume that the end point coincides with the equivalence point at the steepest part of the pH curve to the right of Figure 7-4a, then the first derivative (slope) should be a maximum at the end point. Furthermore, the pH curve inflects, so that its second derivative (rate of change of slope) goes through zero at this point. (The equivalence point does not rigorously coincide with this inflection when the pH is not at 7, but the error is smaller than those we are considering in these titrations.)... 

The inflection point of the van der Waals equation at the critical point is very helpful in a mathematical sense since not only is the first-derivative zero at that point but the first-derivative (slope) changes sign on either side of the critical point so the second derivative (curvature) must also go through zero. We now embark on a series of mathematical manipulations to determine the (a, b) parameters of a given gas from the experimental critical point parameters (F, Vc, FJ. 

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