Functional derivative, orbital

All these functional derivatives are well defined and do not involve any actual derivative relative to the electron number. It is remarkable that the derivatives of the Kohn-Sham chemical potential /rs gives the so-called radical Fukui function [8] either in a frozen orbital approximation or by including the relaxation of the KS band structure. On the other hand, the derivative of the Kohn-Sham HOMO-FUMO gap (defined here as a positive quantity) is the so-called nonlinear Fukui function fir) [26,32,50] also called Fukui difference [51]. 

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... 

The standard ROPM replacement of functional derivatives (again using the various unique equivalences) directly yields an expression for the first contribution on the right hand side of (3.14) in terms of the orbitals and eigenvalues. 

Finally we mention some basic relations which are essential in the discussion of explicitly orbital dependent functionals. Examples of such functionals are the Kohn-Sham kinetic energy and the exchange energy which are dependent on the density due to the fact that the Kohn-Sham orbitals are uniquely determined by the density. The functional dependence of the Kohn-Sham orbitals on the density is not explicitly known. However one can still obtain the functional derivative of orbital dependent functionals as a solution to an integral equation. Suppose we have an explicit orbital dependent approximation for in terms of the Kohn-Sham orbitals then... 

From the constraint at Eq. (78) it follows that the functional derivatives of ys, must contain delta functions in order to cancel the delta functions in the second part of the above equation for ri equal to iz- As yj, depends explicitly on the orbitals we therefore have to calculate the functional derivative of the Kohn Sham orbitals with respect to the density which is given by [66]... 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

The second approach to this problem is to derive orbital-based reformulations of existing algorithms based on the spatial representation of the g-density. The resulting formulations are in the spirit of the orbital-resolved Kohn-Sham approach to density functional theory. 

The relative Michael-acceptor abilities of a variety of substituted aromatic and aliphatic nitroalkenes have been elucidated by computational methods. Several global and local reactivity indices were evaluated with the incorporation of the natural charge obtained from natural bond orbital (NBO) analysis. Natural charges at the carbon atom to the NO2 group and the condensed Fukui functions derived by this method were found to be consistent with the reactivity.187... 

Here the differentiation is shown as being with respect to p(r), but note that in Kohn-Sham theory p(r) is expressed in terms of Kohn-Sham orbitals (Eq. 7.22). Functional derivatives, which are akin to ordinary derivatives, are discussed by Parr and Yang [36] and outlined by Levine [37]. 

This says that/(r) is the functional derivative (Section 1.23.2, The KohnSham equations) of the chemical potential with respect to the external potential (i.e. the potential caused by the nuclear framework), at constant electron number and that it is also the derivative of the electron density with respect to electron number at constant external potential. The second equality shows fir) to be the sensitivity of p(r) to a change in N, at constant geometry. A change in electron density should be primarily electron withdrawal from or addition to the HOMO or LUMO, the frontier orbitals of Fukui [154] (hence the name bestowed on the function by Parr and Yang). Since p(r) varies from point to point in a molecule, so does the Fukui... 

The orbital functional derivative here defines an effective Hamiltonian... 

Since H is specified, Eq. (5.3) defines Ec as a functional of the occupied orbitals of [Pg.59]

An infinitesimal unitary transformation of the orbital basis that modifies must mix occupied and unoccupied orbital functions. For a typical orbital variation, <5 = a8c . Unitarity induces 8a = —functional derivatives 8/8

[Pg.60]

Orbital Euler-Lagrange equations are determined by functional derivatives... 

An orbital functional derivative in general defines a linear operator such that 7 = vplocality hypothesis were valid, then 4 = vF(r), and the implied local potential function could be computed directly from the sum rule,... 

This defines a Gateaux functional derivative [26, 102], whose value depends on a direction in the function space, reducing to a Frechet derivative only if all e, are equal. Defining Tt = t + v, an explicit orbital index is not needed if Eq. (5.10) is interpreted to define a linear operator acting on orbital wave functions, TL — v = t. The elementary chain rule is valid when the functional... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

Because G itself is defined by an orbital functional derivative, the increment AGs is proportional to a functional second derivative. It is convenient to define a response kernel f such that A O) induces Av = I] /l - nb)[(j f b)cbj(t) +... 

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