Higher-Order Derivatives and State Changes

We first observe that higher-order derivatives of U or S are implicitly related to changes of state from the initial state to a nearby equilibrium state along a reversible path. Suppose that we parameterize the thermodynamic state Ms 

To be specific, let us first consider the metric matrix elements 

In metric geometrical terms, this corresponds to the vector relationship 

Of course, we might also consider the changes dMy with respect to changes in the intensive variables dRk. The resulting geometrical relationship can be inferred most directly by substituting the usual expression for extensive jX ) in terms of intensive R/) ket vectors, 

Alternatively, we might have started by considering the conjugate metric elements 

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In any chemical reaction, the approaching molecular systems experiences both electron transfer (in some cases, spin polarization) and external potentials changes while the interacting system evolves towards the final state. Behind the perturbative approximation we are here concerned, and within the context of the [A( , Np, v (r), v (r)] representation of spin polarized DFT, the nonlocal descriptors are defined as first (and higher) order derivatives of the electron density of a given spin p (r) with respect to the spin external potentials Vo-(r). In particular, the symmetric linear response (or polarizability) kernels, defining the spin density... 

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