Mixed second derivatives

The shielding constant, crA, is related to a mixed second derivative of the energy with respect to an internal magnetic nuclear moment due to spin, and external magnetic field, Bext. 

Finally, the mixed second derivatives define the nuclear Fukui function (NFF) indices ... 

The important functional relationships of thermodynamic systems also permit second derivatives to be evaluated. For example, the derivative function z- = z-(xi, x2,. , xn) of (1.6) can be differentiated with respect to a second variable Xj to give the mixed second derivative of z with respect to xL and Xj,... 

Another important extension of the theory concerns NMR chemical shift. Yamazaki et al. proposed a theory for computing the chemical shift of solvated molecules [17]. The nuclear magnetic shielding tensor o-x of a nucleus X can be represented as a mixed second derivative of the free energy A with respect to the magnetic field B and the nuclear magnetic moment mx ... 

Since the order of differentiation in mixed second derivatives is immaterial, these equations give... 

This form, however, does not seem to confer any advantage. On the contrary, it is possible to take advantage of the asymmetry if the perturbation parameters are of different nature (Pulay, 1983a) e.g. in the calculation of infrared intensities, mixed second derivatives with respect to the nuclear coordinates and the external electric field are needed. As there are only three electric field components but many nuclear coordinates, it is advantageous to choose the... 

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. 

Elements of the chemical shift tensor ct for a particular nucleus t) are given by the mixed second derivative of the energy with respect to the nuclear magnetic moments [/,/n)] and the magnetic field [B,]... 

While using an activity coefficient model will provide a quantitative relationship between the mutual solubilities, we can get a qualitative understanding of how the presence of one dissolved species affects others by examining the interrelation between mixed second derivatives. In particular, the Maxwell equations in Chapter 8 and some of the pure fluid equations in Chapter 6 were derived by examining mixed second derivatives of thermodynamic functions. Another example of this is to start with the Gibbs energy and note that at constant temperature, pressure, and all other species mole numbers,... 

However, since S is a function of T and V (dS is an exact differential) the mixed second derivatives must be equal hence we have... 

Now the same consideration applies to U the mixed second derivatives are equal. This reduces the preceding equation to... 

In Eq. (9.39) we have a rather messy expression for the pressure dependence of the entropy at constant temperature. To simplify matters, we again form the mixed second derivatives and set them equal. Differentiation of Eq. (9.38) with respect to pressure at constant temperature yields... 

T dpdT T dT dp Since the mixed second derivatives of H are also equal this equation reduces to... 

As discussed earlier, mixed second derivatives of well-behaved functions are independent of the order of differentiation. This leads to Euler s reciprocity relation... 

The ESR equivalent of the NMR shielding is called the %-tensor, and can be considered as the mixed second derivative with respect to the electron spin and an external magnetic field. It is a 3 x 3 tensor, and can be written as the diagonal component for the free electron plus a small correction due to the molecular... 

We call the hyperbolic system (2.17) and (2.18) a reaction-Cattaneo system. Eu and Al-Ghoul have derived such systems from generalized hydrodynamic theory [9, 7, 8, 6]. Reaction-Cattaneo systems can also be obtained from extended irreversible thermodynamics [223], see for example [282]. If we differentiate (2.17) with respect to t and (2.18) with respect to x and eliminate mixed second derivatives, we obtain the so-called reaction-telegraph equation. 

Differentiating (2.29) with respect to t and (2.30) with respect to x and eliminating the mixed second derivatives, we obtain the telegraph equation... 

These relations are important when deriving an equation of state by statistical methods. In addition, they can naturally lead to another set of useful identities through the derivation of the mixed second derivatives of the thermodynamic potentials (known as Maxwell relations) ... 

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