Inverse functional derivative

We now introduce the so-called direct correlation function, which is defined in terms of the inverse functional derivative in (C.7). To do this, we view if as a functional of the density, which we write symbolically as (//(i /1 p(1>). Now, the external potential is produced by preparing a system with an arbitrary local density p 1 It is for this reason that it is necessary to work in the grand ensemble where an arbitrary density change may be envisaged see Percus (1964)f. 

Figure 3. Comparison of EXAFS envelope functions derived from the inverse transforms in Figure 2 for silica supported copper and ruthenium-copper catalysts. Reproduced with permission from Ref. 8. Copyright 1980, American Institute of Physics.

The frontier orbitals responses (or bare Fukui functions) f (r) and the Kohn-Sham Fukui functions (or screened Fukui functions)/, (r) are related by Dyson equations obtained by using the PRF and its inverse [32]. Indeed, by using Equation 24.57 and the chain rule for functional derivatives in Equation 24.36, one obtains... 

From this equation it follows that dg,A Pa is diagonal in the spin indices. We will therefore in the following put density variation 5p (r) determines the potential variation 5vs,(r) only up to a constant (see also [66] ). To find an explicit expression for the above functional derivative we must find an expression for the inverse density response function i A. In order to do this we make the following approximation to the Greens function (see Sharp and Horton [39], Krieger et al. [21]) ... 

With as complex an expression as this it is clearly a waste of time to derive an inverse function and the integrals involved can be computed numerically. In fact, much of the numerical work can be done on the desk-top computer using Simpson s rule and watching out for regions of extreme curvature. 

When the exponent refers to a root rather than a power i.e. y — xl v, we use the fact that the derivative is defined in terms of a ratio and consider the root as the inverse function of the power yq. Let us illustrate this for the square root function y — y/x, ori = y2. We calculate first ... 

We have already seen an example of how to calculate the derivative of inverse functions when dealing with fractional powers. In the case of the exponential function the corresponding inverse is the logarithmic function y = ln(x). Its derivative follows from Eq. (25) ... 

The latter coefficient addresses the question of the response of the system density to an applied external field, and is called the susceptibility. The final functional derivative relation that we will use views this susceptibility - Eq. (6.36) - as a matrix, considers the matrix inverse, and is a generalization of the partial derivative relation... 

The value of the second summand is considerably lower than 1 (unity) at /fMe0 10 p° m0o2-. Therefore, the derivative of the inverse function can be calculated using the following well-known transformation ... 

Let f be a continuously differentiable function of x, both having the same dimension n greater than zero. If the derivative f (x) is non-zero at x = xq for which yo = f(xo), then there exists a continuous inverse function f ""(y), which maps an open set Y containing yo to an open set X containing xq. 

One apparent limitation of using the R-matrix propagation is that only (inverse) log derivative information is carried along—the wave-function is only determined asymptotically with the imposition of b.c. s. This makes the accumulation of overlap integrals for photodissociation, for example, somewhat more complicated. At this Workshop, however,... 

The partial derivatives of F of course, are unknown. But the application of the inverse function theorem [26, 27] allows one to rewrite Equation 11.11 as Equation 11.12 ... 

Fig. 6.3. Grained reaction probability function/( )=, for the dissociation of N2O the discrete points represent a continuation of the calculation begun in Fig. 6.2, with all internal motions included, and the solid line is the same function derived from the inverse Laplace transform, i.e. equations (4.9) or (6.2).

In these equations, we use the fact that Op, bpp,. .. and Oq, f qq,. .. are essentially derivatives of two inverse functions (0 and P). (dQ/dP)s and (9 2/9P )s are the first and second derivative of the adsorption isotherm written in the dimensional form [56] at steady state defined by P and Q. It can be shown that the low-frequency asymptotic values of the third- and higher-order functions are proportional to the third- and higher-order derivatives of the adsorption isotherm... 

This remark demands great care and consideration. Through the signal-wavelet inversion formula, derived later on, we can represent the (physical) wavefunction as a superposition of dual basis functions and wavelet transform coefficients. We symbolically denote this, for the dyadic representation (Sec. 1.3.2), by 9(b) = i J2j,i Thus at a given point b, the... 

Care must be exercised when looking at the inverse of a functional derivative. One cannot simply invert the functional derivative, but must ensure that the inverse satisfies the identity... 

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