Second and Higher Derivatives

The derivative dy I dx of a function y = f(x) is also a function, which in turn has its own derivative. This second derivative gives the slope of the tangent curves to dy I dx. It is generally written as d2y/dx2 or f (x). It is calculated by applying the definition of a derivative (Equation 2.1) two separate times. Thus, to find the second derivative of the function y = x3, recall that we showed the first derivative is 3x2 (Equation 2.3). Equation 2.4 showed that the derivative of x2 is 2x. Equation 2.11 then implies that the derivative of 3x2 is 6x. Therefore, we have 

fourth, and higher derivatives are defined in a similar way. 

---

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.29) gives directly an equation for determining the (first-order) response, which is structurally the same as eq. (10.36). For an HF wave function, an equation of the change in the MO coefficients may also be formulated from the Hartree-Fock equation, eq. (3.50). 

Unlike the treatment described in the previous section, the structure factors are not linear functions of the unknowns Xj, as was the case for the observations described by expression (4.3). The equations of the preceding section can only be retained by the approximation that all the second and higher derivatives are zero. The price paid for this assumption is that the equations are no longer exact, so that a single calculation no longer leads to the minimum of the error function. If the deviations from linearity are not pronounced, the minimum may still be reached, but a number of iterations may be necessary to achieve convergence. 

Probably even more important to computational quantum chemistry is the development in the analytical evaluation of first, second and higher derivatives of the potential energy with respect to nuclear coordinates . These analytical derivative methods are indispensable to the location and characterization of the stationary points (minima or transition states) on the potential energy surface and have greatly advanced the scope of applicability of ab initio calculations. Ab initio calculations are in a position to predict many new types of the heavier group 15 compounds and provide valuable information for the interpretation of complex experimental data. 

Prior to calculation of a dissimilarity score, spectra can be treated mathematically, either to improve their noise characteristics or to enhance differences for very similar spectra. One mode of pretreatment, from classical spectroscopy, involves the use of derivatives, especially of the second and higher derivatives, to enhance the fine structure of absorbance bands. Smoothing of spectral data to reduce noise is another pretreatment mode that could potentially improve the selectivity of spectral matching. 

A final point in the use of iterative least squares concerns the possible interaction between two or more of the coordinates to be adjusted during the fitting. In iterative least squares the effect of second and higher derivatives of the moments with respect to the coordinates is ignored. 

If the adjustment in the coordinates is small, the effects of the second and higher derivatives are not important. However, under some circumstances the effect of the mixed second derivative may be crucial. It may prove necessary to uncouple certain sets of coordinates and alternate least squares adjustment of each set until the structure is determined as well as possible. This problem is particularly acute with hydrogen coordinates because of the low mass and, consequently, small contribution to the moments. We have found it necessary on occasion to alternate fitting the coordinates of hydrogen atoms and heavy atoms because of the effects of interaction. As a result of this kind of effect, it is necessary to iterate a nonlinear least squares problem essentially to convergence. It is sometimes possible to be misled by the results of early iterations. 

In response to a question by pliVa, schaefer indicated that the calculation of one geometry of HNC or HCN (with 6343 configurations included) required about 20 min on an IBM-360/195. Calculations were made for about 10 geometries each on HCN and HNC therefore, the total problem required about 7 h of computer time, pliva also inquired if second and higher derivatives of the energy with respect to intemuclear distance had been obtained, schaefer indicated these had been calculated for HCN and HNC since this information was needed to test the rrkm theory for the unimolecular reaction. 

Gaussian can also predict some other properties dependent on the second and higher derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included j automatically in every Hartree-Fock frequency calculation. ... 

Since partial derivatives are also functions of the independent variables, they can themselves be differentiated to give second and higher derivatives. These are written, for example, as... 

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.32) gives directly an... 

With these coefficients, accuracy of (6.134) and (6.137) is better than 2% in the whole range of current densities (Figure 6.23). Note that the second and higher derivatives (fj/dQ experience discontinuity at g = 1. 

Pulay s paper opened the way for analytic second and higher derivatives of the SCF energy. Earlier papers had suggested that this might be prohibitively expensive [7], but the development of an efficient method to solve the couple perturbed HF (CPHF) equations, made the calculation of SCF second derivatives practical [8]. As a consequence, vibrational force constants and frequencies could be calculated routinely and efficiently. Third and fourth geometric derivatives of the SCF energy followed after a few years [9-12]. The solution of the CPHF equations (in their full or reduced Z-vector form [13]) also made post-SCF first derivatives practical and cost-effective. 

Many molecular properties, such as vibrational frequencies, IR and Raman intensities, NMR shielding constants, etc., can be formulated in terms of second and higher derivatives with respect to geometry and applied fields [1-4]. Such calculations are now practical and routine using analytic derivatives at the SCF level and a few correlated methods. For some levels of theory, analytic second and higher derivatives are not yet available or are too complicated to code. In these cases, the force... 

The partial derivatives can be calculated from Eq. (4.5). It is obvious that this equation is linear with respect to concentrations because all second and higher derivatives with respect to concentration are equal to zero. It should be stressed that the dependence of a current on the potential is strongly nonlinear however, it can be linearized for very small potential perturbations. In such a case, only linear terms might be kept, and Eq. (4.15) becomes... 

A variety of expressions involving second and higher derivatives has been derived by Tschoegl. For example, as an alternative to equation 5 ... 

Functions of two or more Variables can be differentiated partially more than once with respect to either variable while holding the other constant to yield second and higher derivatives. For example. 

---