Nonlocal operator

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

Importantly, this term is a derivative (nonlocal) operator on the nuclear coordinate space. 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

When is an eigenvalue of r(.B),. E is a pole. The corresponding operator, r(JS), is nonlocal and energy-dependent. In its exact limit, it incorporates all relaxation and differential correlation corrections to canonical orbital energies. A normalized DO is determined by an eigenvector of T Epou) according to... 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

Although the pseudopotential is, from its definition, a nonlocal operator, it is often represented approximately as a multiplicative potential. Parameters in some chosen functional form for this potential are chosen so that calculations of some physical properties, using this potential, give results agreeing with experiment. It is often the case that many properties can be calculated correctly with the same potential.43 One of the simplest forms for an atomic model effective potential is that of Ashcroft44 r l0(r — Rc), where the parameter is the core radius Rc and 6 is a step-function. 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

Now the Fock operator is supplemented by the self-energy operator E(E). This operator depends on an energy parameter E and is nonlocal. All... 

Resonances are nonlocal in time and space. We expect that for systems that have resonances there are new types of flucmations associated with the nondistribu-tivity of At. To see this, consider the position and momentum operators defined by... 

Equation (12.17) represents the required boundary condition. It should be emphasized that it is essentially nonlocal both in space and time. In general, the numerical implementation of the operator in the right hand side of Eq. (12.17) is a nontrivial task. 

For the GS, the HK theorems" guarantee thatEq. (10) of different exact theories all deliver the same GS density in spite of distinct mathematical structures of Oeff (r [p]) within different theoretical approaches " (i.e. local vs. nonlocal operators). The reason is simple the density is one-to-one mapped on to the GS wavefimction, regardless of how the exact wavefimction and the exact density are calculated. 

Nuclear size corrections of order (Za) may be obtained in a quite straightforward way in the framework of the quantum mechanical third order perturbation theory. In this approach one considers the difference between the electric field generated by the nonlocal charge density described by the nuclear form factor and the field of the pointlike charge as a perturbation operator [16, 17]. 

The use of clipping has a profound effect on the observed correlation function and its relation to the relaxation function. Because clipping is a nonlocal operation, the numerical factor/which relates Ck(t) to coherence area observed by the photodetector. 

Fractional dynamics is a made-to-measure approach to the description of temporally nonlocal systems, the kinetics of which is governed by a selfsimilar memory. Fractional kinetic equations are operator equations that are mathematically close to the well-studied, analogous Brownian evolution equations of the Klein-Kramers, Rayleigh, or Fokker-Planck types. Consequently, methods such as the separation of variables can be applied. More-... 

The characteristic changes brought about by fractional dynamics in comparison to the Brownian case include the temporal nonlocality of the approach manifest in the convolution character of the fractional Riemann-Liouville operator. Initial conditions relax slowly, and thus they influence the evolution of the system even for long times [62, 116] furthermore, the Mittag-Leffler behavior replaces the exponential relaxation patterns of Brownian systems. Still, the associated fractional equations are linear and thus extensive, and the limit solution equilibrates toward the classical Gibbs-B oltzmann and Maxwell distributions, and thus the processes are close to equilibrium, in contrast to the Levy flight or generalised thermostatistics models under discussion. 

The formal definitions of the nonlocal operators x ar d e can be expressed in the form of their application to a generic F(r) function ... 

In going from static to dynamic descriptions we have to introduce an explicit dependence on time in the Hamiltonian. Both terms of the Hamiltonian (1.2) may exhibit time dependence. We limit our attention here to the interaction term. Formally, time dependence may be introduced by replacing the set of response operators collected into Q(r, r ) with Q(r, r, t) and maintaining the decomposition of this operator we presented in Section 1.1.2. For simplicity we reduce Q(r, r, t) to the dielectric component under the form P(r, t). With this simplification we discard both dielectric nonlocality and nonelec-trostatic terms, which actually play a role in dynamical processes, especially dispersion and nonlocality. 

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