Non-local

There are complicating issues in defmmg pseudopotentials, e.g. the pseudopotential in equation Al.3.78 is state dependent, orbitally dependent and the energy and spatial separations between valence and core electrons are sometimes not transparent. These are not insunnoimtable issues. The state dependence is usually weak and can be ignored. The orbital dependence requires different potentials for different angular momentum components. This can be incorporated via non-local operators. The distinction between valence and core states can be addressed by incorporating the core level in question as part of the valence shell. For... 

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

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]. 

One current limitation of orbital-free DFT is that since only the total density is calculated, there is no way to identify contributions from electronic states of a certain angular momentum character /. This identification is exploited in non-local pseudopotentials so that electrons of different / character see different potentials, considerably improving the quality of these pseudopotentials. The orbital-free metliods thus are limited to local pseudopotentials, connecting the quality of their results to the quality of tlie available local potentials. Good local pseudopotentials are available for the alkali metals, the alkaline earth metals and aluminium [100. 101] and methods exist for obtaining them for other atoms (see section VI.2 of [97]). 

I he function/(r) is usually dependent upon other well-defined functions. A simple example 1)1 j functional would be the area under a curve, which takes a function/(r) defining the curve between two points and returns a number (the area, in this case). In the case of ni l the function depends upon the electron density, which would make Q a functional of p(r) in the simplest case/(r) would be equivalent to the density (i.e./(r) = p(r)). If the function /(r) were to depend in some way upon the gradients (or higher derivatives) of p(r) then the functional is referred to as being non-local, or gradient-corrected. By lonlrast, a local functional would only have a simple dependence upon p(r). In DFT the eiK igy functional is written as a sum of two terms ... 

To obtain a reliable value of from the isotherm it is necessary that the monolayer shall be virtually complete before the build-up of higher layers commences this requirement is met if the BET parameter c is not too low, and will be reflected in a sharp knee of the isotherm and a well defined Point B. For conversion of into A, the ideal adsorptive would be one which is composed of spherically symmetrical molecules and always forms a non-localized film, and therefore gives the same value of on all adsorbents. Non-localization demands a low value of c as c increases the adsorbate molecules move more and more closely into registry with the lattice of the adsorbent, so that becomes increasingly dependent on the lattice dimensions of the adsorbent, and decreasingly dependent on the molecular size of the adsorbate. 

A dramatic decrease in the magnitude of the magnetic susceptibility anisotropy is observed on going from thiirane to the open-chain analog, dimethyl sulfide, and has been attributed to non-local or ring-current effects (70JCP(52)5291). The decrease also is observed to a somewhat lesser extent in oxirane relative to dimethyl ether. 

F Melo, E Feytmans. Assessing protein structures with a non-local atomic interaction energy. JMol Biol 277 1141-1152, 1998. 

P. Tarazona, U. Marini Bettolo Marconi, R. Evans. Phase equilibria of fluid interfaces and confined fluids. Non-local versus local density functionals. Mol. Phys (50 573-595, 1987. 

In addition to the entropy term we assume that there is an extra local coupling between the fields via and, in addition to the coulombic coupling which is long range, we assume the existence of a short-range non-local coupling via We can choose several functional forms to... 

It is seen that the symmetry of the non-coulombic non-local interaction in the bulk phase forces the symmetry of the localized interaction with the wall. If we omitted the surface Hamiltonian and set / = 0 we would still obtain the boundary condition setting the gradient of the overall ionic density to zero. The boundary condition due to electrostatics is given by... 

Selecting the values of the parameters for the calculations we have in mind a 1 1 aqueous 1 m solution at a room temperature for which the Debye length is 0.3 nm. We assume that the non-local term has the same characteristic length, leading to b=. For the adsorption potential parameter h we select its value so that it has a similar value to the other contributions to the Hamiltonian. To illustrate, a wall potential with h = 1 corresponds to a square well 0.1 nm wide and 3.0 kT high or, conversely, a 3.0 nm wide square well of height 1.0 kT. 

Two mechanisms which contribute to GMR have been identified, a "non-local" mechanism and a "quantum" mechanism. To understand the first or non-local, mechanism it is necessary to understand that on the scale of the electron mean free path (possibly 10 to 20 nanometers at room temperature) electrical conduction is a non-local phenomenon. Electrons may be accelerated by an electric field in one region and contribute to the current in other regions. To a good approximation they may viewed as contributing to the current until they are scattered. 

The first mechanism is illustrated by Figures 1 to 4. These figures show the calculated non-local conductivity of a system consisting of 10 atomic layers of copper sandwiched between 11 atomic layers of cobalt on either side. The electric field is applied paralle to the planes which causes the current to flow parallel to the planes as well. The figures show the non-local layer depe lent conductivity, 0 1,J), which is the current of spin s electrons induced in atomic layer I by an electric field applied to layer J. 

Note that because of the different electronic structure for majority and minority Co, the nature of the non-local conductivity is different in the two spin channels. For majority Co, the electronic structure is rather similar to that in Cu, but for minority Co, most of the Fermi energy electrons have low velocities which lead to short mean free paths and hence to localized conductivities, i.e. a strong peak for I=J and a rapid decrease in the conductivity as a function of I-J. ... 

Figure 1 Non-local layer dependent conductivity for majority electrons for parallel alignment of the cobalt moments. The scattering rate is assumed to be high so that the electron lifetime is relatively short (4.8X10 sec).

Figure 3 shows the calculated conductivity for one of the channels when the cobalt moments on either side of the copper layer are aligned anti-parallel. The spin channel for which the conductivity is shown in Figure 3 is locally the majority channel in the cobalt layer to the left of the copper (spin parallel to the Co moment) and locally minority to the right of the copper (electron spin anti-parallel to the local Co moments). The non-local conductivity for the other spin channel for the case in which the cobalt moments are antiparallel is the mirror image of the conductivity shown in Figure 3. 

Figure 3 Non-local layer dependent conductivity for one spin channel for antiparallel alignment of the cobalt moments. This spin channel is locally the majority in the cobalt on the left side of the sample.

Note that majority electrons that are accelerated by the electric field in one of the cobalt layers contribute to the current, not only in that layer (I = J) but in other layers as well, including the copper layers and the cobalt layers on the other side of the copper. On the other hand, minority electrons that are accelerated by a field in one of the cobalt layers contribute very little to the conductivity in the copper or in the cobalt on the other side of the copper. For anti-parallel alignment of the moments, electrons that are accelerated by the field in one cobalt layer contribute to the current in that layer and in the cobalt, but not in the other cobalt layer. The difference in the lolal current due to both channels between parallel and anti-parallel alignment is almost entirely non-local. It comes from those electrons that are accelerated by the applied electric field in one cobalt layer and propagate across the copper to the other cobalt layer where they contribute to the current. It is clear from Figures 1-4 that this process occurs primarily for majority electrons and for the case of parallel alignment. 

The non-local nature of the contributions to the GMR can be seen in Figure 4 which shows the difference in the total non-local conductivity between the cases of parallel and anti-parallel alignment of the cobalt mom its. Thus Figure 4 shows the sum of the conductivities in Figures 1 and 2 minus the sum of the conductivities in Figure 3 and its mirror image. Note that the contributions to Ao are entirely non-local. 

---