Localization hypothesis

Further pragmatic moves are described in details in numerous books and reviews of which we cite the most concise and recent Ref. [82], Two further hypotheses are an important complement to the above cited theorems. One is the locality hypothesis, another is the Kohn-Sham representation of the single determinant reference state in terms of orbitals. The locality has been seriously questioned by Nesbet in recent papers [83,84], however, it remains the only practically implemented solution for the DFT. The single determinant form of the reference state in its turn guarantees that all the averages of the electron-electron interaction appearing in this context are in fact calculated with the two-electron density given by the determinant term in Eq. (5) with no cumulant. 

Assuming the existence of such a Frechet functional derivative [26,102] constitutes the locality hypothesis. If this hypothesis were valid, the OEL and Kohn-Sham equations would be equivalent, determining the same model or reference state. 

An orbital functional derivative in general defines a linear operator such that 7 = vplocal potential function could be computed directly from the sum rule,... 

This formula was used by Slater [385] to define an effective local exchange potential. The generally unsatisfactory results obtained in calculations with this potential indicate that the locality hypothesis fails for the density functional derivative of the exchange energy Ex [294],... 

The Lagrange multiplier p. determined by normalization, is the chemical potential [232], such that pt = dE/dN when the indicated derivative is defined. This derivation requires the locality hypothesis, that a Frechet derivative of Fs p exists as a local function (r). 

The locality hypothesis can be tested in a noninteracting model, in which the functional Fs is replaced by 7 . The kinetic energy orbital functional is T = ni 0 If 10... 

If the locality hypothesis is valid, then = vT(r). and the Thomas-Fermi equation... 

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. 

Two further hypotheses are an important complement to the theorems cited above. One is the locality hypothesis, and the other is the Kohn-Sham representation of the... 

Little is known of the actual mechanism. A mode of reaction is possible, in which the oxygen atom at the top of the ozone molecule with a formal positive charge (p. 230) reacts with an electron pair, not localized in a bond but on one carbon atom, and in which the ozone therefore reacts by an electrophilic mechanism (Wibaut, Sixma and Kampschmidt). However, in order to explain the differences between the reaction course for ozonization and for other electrophilic reactions, e.g., bromination and nitration with pyrene, these authors assume also an interaction of one of the other oxygen atoms with the adjacent carbon atom. The net result is, however, about the same as that predicted by the bond localization hypothesis. 

Figures 14 and 15 show the analogous results for a 6-31G full optimization calculation on disiloxane and the acceptor adsorption of water onto the disiloxane bridge. In agreement with our localization hypothesis, the SiO and OH bond distances in Figure 15 agree closely with the corresponding distance in Figure 126.

In Kohn-Sham (KS) density functional theory (DFT), the occupied orbital functions of a model state are derived by minimizing the ground-state energy functionals of Hohenberg and Kohn. It has been assumed for some time that effective potentials in the orbital KS equations are always equivalent to local potential functions. When tested by accurate model calculations, this locality assumption is found to fail for more than two electrons. Here this failure is explored in detail. The sources of the locality hypothesis in current DFT thinking are examined, and it is shown how the theory can be extended to an orbital functional theory (OFT) that removes the inconsistencies and paradoxes. 

These conditions could be consistent if a Frechet functional derivative vf(r) = b/- /bp(r) exists for every well-defined density functional F[p], The locality hypothesis assumes this to be true. Various tests of this hypothesis are considered here. 

It can be concluded that the locality hypothesis fails for the functional derivative of Ts. Thus there is no exact Thomas-Fermi theory for A > 2. In contrast, as will be shown below, use of the linear operator t in the KS equations is variationally correct. 

If the UHF and KS equations were equivalent, on multiplying by f/> and summing, they would imply vx(r)p(r) = Y.i Mi Xr)Vxlocal exchange potential of Slater [ 11 ]. It is well known that the Slater potential gives relatively poor results for atoms [10]. It is clearly not equivalent to the Fock exchange operator in UHF equations for more than two electrons. It can be concluded that the locality hypothesis fails for Ex in the UHF model for typical atoms. The restriction to a local exchange potential vx(r) in the KS equations is inconsistent with an exact theory. 

It can be shown that the response kernel is a functional second derivative. Assuming that the functional first derivative 8Fx/8p is a local function (the locality hypothesis for vx), Petersilka et al. [15] have derived the approximate formula... 

This result cannot be reconciled with that of Dirac [13] and with the structure of the second-quantized Hamiltonian [12,14]. The implication is that the locality hypothesis fails for Ex and probably for Ec, which can be incorporated in a formally exact extension of Dirac s derivation. [17] Thus restriction to local exchange and correlation potentials is inconsistent with exact linear-response theory. 

For any orbital-functional model, an optimal effective (local) potential (OEP) can be constructed following a well-defined variational formalism [24,25]. If a Frechet derivative existed for the exchange-correlation energy E,lc for ground states, it would be obtained in an OEP calculation, while the minimum energy and corresponding reference state would coincide with OFT results. Thus numerically accurate OEP calculations test the locality hypothesis. 

OEP imposes one variational constraint on OFT (a) vxc must be a local function. Hence in general /iOKP > 0FT. Because of the Green s function weighting, OEP does not imply ( lvXL. — vxcl/) = 0. i < /V < a. and the computed densities may differ [26,27]. In the UHF model, accurate calculations find that E0EP > Euhf [27-29] for more than two electrons. Table 1 shows computed values of several criteria that test the locality hypothesis by comparing the densities and wave functions ... 

When Exc p is specified, the relevant ground-state density for Hohenberg-Kohn theory is p0, computed using the equivalent orbital functional Exc in the OEL equahons, (Q — e,-)local potential w(r) in the corresponding KS equahons is determined by the KSC by minimizing T for p = p0. Assuming the locality hypothesis, that w — v is the Frechet derivative of the model ground-state functional h p — Ts[p, this implies that w = vh + vxc + v is a sum of local potentials. If i>xc in the OEL equahons was equivalent to a local potential vxc(r), the KS and OEL equations would produce the same model wave function. 

The KSC imposes two variational constraints on OFT (a) vxc must be a local function and (b) p = p0. These nested constraints imply EKSC — qep — OFr. [20] In the UHF model, a particular case of OFT, for typical atoms [29,20,10], KSC — oep > uhf for more than two electrons, and the KSC local exchange potential does not reproduce the Hartree-Fock ground state. These results confirm the failure of the locality hypothesis for vv. and demonstrate that noninteracting v-represent-ability does not imply locality. 

The failure of the DFT linear-response theory to reduce to the exact formalism of Dirac [13] in the exchange-only limit [12] is symptomatic of the inadequacy of the locality hypothesis. It will be shown here that on dropping this hypothesis a linear-response theory can be derived that is formally exact for both exchange and correlation. As will be discussed in more detail in the following Section, an exact but implicit orbital functional exists for the correlation energy Ec [31]. This produces a formally exact correlation term in the OEL equations, defined by the orbital functional derivative... 

For fixed normalization the Lagrange multiplier terms in 8Ts vanish. If these constants are undetermined, it might appear that they could be replaced by a single global constant pt. If so, this would result in the formula [22] 8Ts = J d3r p, — v(r) 8p(r). Then the density functional derivative would be a local function vr(v) such that STj/Sp = Vj-(r) = ix — v(r). This is the Thomas-Fermi equation, so that the locality hypothesis for vT implies an exact Thomas-Fermi theory for noninteracting electrons. 

At this point we want to make a brief comment upon the approach of Nesbet, reviewed in another article of this volume [9]. Nesbet s main conclusion is that one of the main fundaments of DFT, the so-called locality hypothesis—the assumption that the derivative of the density functionals can be expressed in the form of multiplicative local function—is not generally valid. As we have pointed out in a separate Comment to the Physical Review [10], we believe that the arguments of Nesbet are incorrect and that the mistake is connected to the above-mentioned question of extending the functionals into the domain of unnormalized densities. We summarize our main arguments here and refer to our Comment for further details. 

As we have pointed out in our Comment [10], the expression (15) is in combination with the density expression (16) not a density functional outside the normalization domain. Therefore, the identity (18) is not generally valid and the chain rule cannot be used in the way Nesbet does. If the variations are restricted to the normalization domain, then the constant term (orbital eigenvalue) disappears, and the locality hypothesis is restored. 

---