Mixed derivatives

Mixed derivatives refer to cross terms if the energy is expanded in more than one perturbation. There are many such mixed derivatives which translate into molecular properties, below are a few examples. 

The change in the dipole moment with respect to a geometry displacement along a normal coordinate is approximately proportional to the intensity of an IR absorption. In the so-called double harmonic approximation (terminating the expansion at first order in the electric field and geometry), the intensity is (except for some constants) 

In the double harmonic approximation, only fundamental bands can have an intensity different from zero. Including higher-order terms in the expansion allows calculation 

The intensity of a Raman band in the harmonie approximation is given by the derivative of the polarizability with respect to a normal coordinate. 

The mixed derivative of an external and an internal magnetic field (nuclear spin) is the NMR shielding eonstant, r. 

There is a further class of important derivatives that can be traced neither back to a capacity nor a susceptibility. The most well known of these is the thermal expansion coefficient. Here the differential of an extensive variable with respect to a nonenergy-conjugated variable appears, such as 

This type of differential can be converted into other expressions by using the Maxwell relations. The thermal expansion coefficient a originates naturally from the Gibbs free energy, since for the Gibbs free energy the set (T. p, n) are the natural variables. 

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Requiring that the mixed derivative is independent of the order of the differentiation yields... 

In this section we reveal some properties of difference operators approximating the Laplace operator in a rectangle and derive several estimates for difference approximations to elliptic second-order operators with variable coefficients and mixed derivatives. 

Equations with mixed derivatives. In this section we consider problem (23) involving the elliptic operator L with mixed derivatives... 

Example 2 The statement of the first boundary-value problem for the parabolic equation with mixed derivatives in the parallelepiped Go = 0 < a < L, a =1,2,..., p is... 

Mixtures of lithium with polychiorinated, polybrominated, polyiodised and mixed derivatives are extremely sensitive, to impact. 

These schemes require the calculations of the second and mixed derivatives, which normally result in poor accuracy when the computations are performed on discrete data. For noisy data, computed values of H and K depend on the finite element scheme used to calculate the first, second, and mixed derivatives. 

Producing a reasonably good accuracy for analytically defined surfaces, this scheme of calculation is very inaccurate when the field is specified by the discrete set of values (the lattice scalar field). The surface in this case is located between the lattice sites of different signs. The first, second, and mixed derivatives can be evaluated numerically by using some finite difference schemes, which normally results in poor accuracy for discrete lattices. In addition, the triangulation of the surface is necessary in order to compute the integral in Eq. (8) or calculate the total surface area S. That makes this method very inefficient on a lattice in comparison to the other methods. 

If the addition is made at the flour mill this grade of flour can be supplied directly to the bakery for use. This solution suits the large plant bakery and the domestic baker. Adding a concentrate to a mix derived from a bag of baker s flour is a solution that suits a small baker or a supermarket in-store bakery. There is then no need to stock an extra grade of flour for a product that will probably have a small sale. 

The second member of the 11 2 series is a mixed derivative (d2E/dN8v(r)),... 

It has been established that by mixing derivatives of acids having opposite rotation, the compounds had opposite, configurations. The (+) malic acid (XXIV) and (-) mercaptosuccinic acid (XXV) had opposite configurations. 

Applying Schwarz s theorem on sequential differentiation of mixed derivatives ... 

As follows from (2.3.70), it is impossible to separate variables fj unless Da = 0, and mixed derivatives prevent finding of the exact solution. In the particular case DA = 0 an explicit solution of (2.3.70) is known. 

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