Potential, external

The parameter J.j is a measure of the energy of interaction between sites and j while h is an external potential or field common to the whole system. The tenn ll, 4s a generalized work temi (i.e. -pV, p N, VB M, etc), so is a kind of generalized enthalpy. If the interactions J are zero for all but nearest-neighbour sites, there is a single nonzero value for J, and then... 

In DFT, the electronic density rather than the wavefiinction is tire basic variable. Flohenberg and Kohn showed [24] that all the observable ground-state properties of a system of interacting electrons moving in an external potential are uniquely dependent on the charge density p(r) that minimizes the system s total... 

Maurits, N.M., Fraaije, J.G.E.M. Mesoscopic dynamics of copolymer melts from density dynamics to external potential dynamics using nonlocal kinetic coupling. J. Chem. Phys. 107 (1997) 5879-5889. 

Equation (3.40) is the DFT equivalent of the Schrbdinger equation. The subscript Vext indicates that this is under conditions of constant external potential (i.e. fixed nuclear po.-,ilions). It is interesting to note that the Lagrange multiplier, p, can be identified with (lu chemical potential of an electron cloud for its nuclei, which in turn is related to the... 

In Ecjuation (3.47) we have written the external potential in the form appropriate to the interaction with M nuclei. , are the orbital energies and Vxc is known as the exchange-correlation functional, related to the exchange-correlation energy by ... 

Extended-zone and reduced-zone representations of band diagram for ID lattice with no external potential. 

Fig. 3.19 The effect of a weak external potential is to lift degeneracy and create band gaps as illustrated for a 2D hexagonal lattice (compare with Figure 3.18).

A diatomie moleeule with fixed bond length R rotating in the absenee of any external potential is deseribed by the following Sehrodinger equation ... 

Anodic Protection This electrochemical method relies on an external potential control system (potentiostat) to maintain the metal or alloy in a noncorroding (passive) condition. Practical applications include acid coolers in sulfuric acid plants and storage tanks for sulfuric acid. 

Some useful exaet relations ean be obtained for speeifie fluid-wall interaetions. With this aim, we now eonsider a one-dimensional external potential, Va z). First, let us introduee the abbreviation zx,Z2,k), for the Fourier-Bessel transform of a funetion 0(zi,22, 12) where i i2 +... 

We now switch on an external potential U(r) that is slowly varying over the dimensions of the metallic box. This makes the conduction electron density... 

The paper deals with a gas of interacting metallic electrons, and examines the ground state of such a gas in the presence of an external potential U(r). 

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

Suppose there are two external potentials ui(r) and V2 r) arising from the same electron density F(r). There will be two Hamiltonians H and H2 with the same... 

The Vext operator is equal to Vne for A = 1, for intermediate A values, however, it is assumed that the external potential Vext(A) is adjusted so that the same density is obtained for both A = 1 (the real system) and A = 0 (a hypothetical system with noninteracting electrons). For the A = 0 case the exact solution to the Schrddinger equation is given as a Slater determinant composed of (molecular) orbitals, for which the... 

Assume now that two different external potentials (which may be from nuclei), Vext and Vgjjj, result in the same electron density, p. Two different potentials imply that the two Hamilton operators are different, H and H, and the corresponding lowest energy wave functions are different, and Taking as an approximate wave function for H and using the variational principle yields... 

The relation between the external potential and the external current, which is the source of the external electromagnetic field, is... 

By virtue of Eq. (10-6) clearly the external potential is renormalized in the same way... 

The propagator K can be interpreted as the Green s function for computing amplitudes for one-particle processes relative to the amplitude that the vacuum remain a vacuum under the influence of the external potential. To lowest order the amplitude that the vacuum remain a vacuum is given by... 

Table 3.2. Electrocatalytic reactions investigated in doped Zr02 solid electrolyte fuel cells with external potential application...

Table 10.1. Dependence of the ratio of hydrogenation rates obtained by method B (presence of H and external potential control) and by method C (electrocatalytic reduction).5 ...

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

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