Density response function, invertability

This is true for an arbitrary nonconstant potential variation. We therefore see that the eigenvalues of when we regard x as an integral operator, are negative. Moreover we see that the only potential variation that yields a zero density variation is given by 8v = C where C is a constant. This implies that is invertible. We will go more closely into this matter in a later section where we will prove the same using the explicit form of the density response function. 

Now equation (98) implies that Av(r) = 0 which together with equation (99) implies that 8v(r) = C. We have therefore shown that only constant potentials yield a zero-density variation, and therefore the density response function is invertible up to a constant. One should, however, be careful with what one means with the inverse response function. The response function defines a mapping V - v4 from the set of potential variations from a nondegenerate ground state, which we call 8V and is a subset of L3/2 + L°°, to the set of first order densities variations, which we call... 

Nevertheless, this idea is sufficient to make a useful statement about the functional derivative. Let us therefore take some 8m G 8A Then, by the invertability of the density response function, there is a unique 8v modulo a constant, such that... 

We therefore find that A < 0. However, we already know that A = 0 is only possible if/(r) is constant. We have therefore obtained the result that if a nonzero density variation is proportional to the potential that generates it, i.e., 8n = A 8v, then the constant of proportionality A is negative. This is exactly what one would expect on the basis of physical considerations. In actual calculations one indeed finds that the eigenvalues of x are negative. Moreover, it is found that there is a infinite number of negative eigenvalues arbitrarily close to zero, which causes considerable numerical difficulties when one tries to obtain the potential variation that is responsible for a given density variation. We finally note the invertability proof for the static response function can be extended to the time-dependent case. For a recent review we refer to Ref. [15]. 

Equation 24.14 provides an alternative definition of the electronic responses they are derivatives of the energy s relative to the field E. Note that the response of order n, the nth derivative of the response to the perturbation, is the n + 1th derivative of the energy relative to the same perturbation. Hence, the linear response a t is a second derivative of the energy. Because the potential (E) and the density (p) are uniquely related to each other, the field can be formulated as a function of the dipole moment p. The expansion of the field in function of p can be obtained from Equation 24.12 which can be easily inverted to give... 

In Section 11.1 we viewed the overpotential as the stimulus (or perturbation), which causes the reaction to proceed at a certain rate. We could equally well have inverted the roles and looked at the current density as the stimulus and the resulting overpotential as the response. In this case the value of the overpotential is an indication of the sluggishness of the reaction. To be more precise, the ratio between the response T] and the perturbation i is a measure of the effective resistance of the system to proceed at the desired rale. In the example given earlier, applicable to small perturbations, the ratio (ri/j) is constant, but in general it is a function of potential, as implied by Fig. lA. It will be noted that r /i has the dimensions of an electrical resistance, and indeed it represents the total reaction resistance R. For the simple case discussed here we can write... 

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