Potentials asymptotic structure

It follows from the definition cited that the size of the zeta potential depends on the structure of the diffuse part of the ionic EDL. At the outer limit of the Helmholtz layer (at X = X2) the potential is j/2, in the notation adopted in Chapter 10. Beyond this point the potential asymptotically approaches zero with increasing distance from the surface. The slip plane in all likelihood is somewhat farther away from the electrode than the outer Helmholtz layer. Hence, the valne of agrees in sign with the value of /2 but is somewhat lower in absolute value. 

For the self-consistent orbitals, we have determined1"4 the exact analytical asymptotic structure in the classically forbidden vacuum region of (i) the Slater potential VJ (r), (u) the functional derivative (exchange potential)... 

The asymptotic structure of the exchange potential vx(r) was derived via the relationship between density functional theory and many-body perturbation theory as established by Sham26. The integral equation relating vxc(r) to the nonlocal exchange-correlation component Exc(r, rf ) of the self-energy (r, r7 >) is... 

Now if it assumed that the asymptotic structure of the exchange-correlation potential vxc(r) is the image potential, the analytical asymptotic structure of the KS correlation potential is... 

Finally, the asymptotic structure of the Slater potential is derived1,2 as... 

The KS exchange potential coefficient aKS x(0) is essentially the image-potential value of 1/4, ranging from 0.195 to 0.274 over the metallic range of densities. Its value is precisely 0.250 for 0 = yjl, which corresponds to a Wigner-Seitz radius of rg — 4.1. The jellium model is stable for approximately this value of rs. With the assumption that the asymptotic structure of vxc (r) is the image potential, we see that the correlation contribution to this structure is an... 

For finite systems, such as atoms and molecules, the asymptotic structure of the KS exchange potential vx(r) in the classically forbidden region is due entirely to Pauli correlations5 as described by Wx (r). Thus,... 

In Fig. 5 the correlation-kinetic potential component Wt (z) is plotted. For these densities, the potential is entirely positive, possesses the correct asymptotic structure of Eq. (45) in the vacuum, and exhibits the Bardeen-Friedel oscillations. Once again, thepotential w[ z) is an order of magnitude smaller than the Pauli component Wx (z). For higher density metals (rs < 2), the correlation-kinetic contribution to vx(z) will be less significant. It will vanish entirely for the very slowly varying density case for which33,34 vx(z) = Wx (z). 

It is easy to see that since z Umoopn(z) = 1, where z = kFx, then for any value of the parameter p, the potential V ,app(r) reproduces the correct asymptotic structure of the exact Slater potential V (r) in the metal bulk ... 

Fig. 6 The approximate exchance potential v pp(z) of Eq. (64) at the surface of a metal of Wigner-Seitz radius rs = 3.24. The potential in the local density approximation (LDA) is also plotted, as is the exact asymptotic structure - ks(X0J)/z of the KS exchange potential.

The physical interpretation of the functional derivative vx(r) shows that it is comprised of a term Wx (r) representative of Pauli correlations, and a term wj (r) that constitutes part of the total correlation-kinetic contribution Wt (r). cThe exact asymptotic structure of these components in the vacuum has been determined and shown to also be image-potential-like. Although the structure of vx(r) about the surface and asymptotically in the vacuum and metal-bulk regions is comprised primarily of its Pauli component, the correlation-kinetic contribution is not insignificant for medium and low density metals. It is only for high density systems (rs < 2) that vx(r) is represented essentially by its Pauli component Wx (r). Thus, we see that the uniform electron gas result of -kF/ir for the functional derivative vx(r), which is the asymptotic metal-bulk value, is not a consequence of Pauli correlations alone as is thought to be the case. There is also a small correlation-kinetic contribution. The Pauli and correlation-kinetic contributions have now been quantified. 

For the nonuniform electron gas at a metal surface, the Slater potential has an erroneous asymptotic behavior both in the classically forbidden region as well as in the metal bulk. In the vacuum region, the Slater potential has the analytical [10] asymptotic structure [35,51] V r) = — Xs(p)/x, with the coefficient otsiP) defined by Eq. (103). In the metal bulk this potential approaches [35] a value of ( — 1) in units of (3kp/27r) instead of the correct Kohn-Sham value of ( — 2/3). Further, in contrast to finite systems, the Slater potential V (r) and the work W,(r) are not equivalent [31, 35, 51] asymptotically in the classically forbidden region. This is because, for asymptotic positions of the electron in the vacuum, the Fermi hole continues to spread within the crystal and thus remains a dynamic charge distribution [34]. 

There is, of course, much that remains to be understood with regard to the physical interpretation. For example, the correlation-kinetic-energy field Z, (r) and potential W, (r) need to be investigated further. However, since accurate wavefunctions and the Kohn-Sham theory orbitals derived from the resulting density now exist for light atoms [40] and molecules [54], it is possible to determine, as for the Helium atom, the structure of the fields P(r), < P(r), and Zt (r), and the potentials WjP(r), W (r), W (r), and W (r) derived from them, respectively. A study of these results should lead to insights into the correlation and correlation-kinetic-energy components, and to the numerical determination of the asymptotic power-law structure of these fields and potentials. The analytical determination of the asymptotic structure of either [Z, (r), W, (r)] or [if (r), WP(r)] would then lead to the structure of the other. 

Asymptotic Structure of the Kohn-Sham Exchange-Correlation Potential Vx (r)... 

In this section we discuss the asymptotic structure of the Kohn-Sham exchange-correlation potential v (i ) and its components from the work perspective (see Eq. (80)) for finite and extended systems, and then present results of application to atoms based on this understanding. 

The precise analytical asymptotic structure of the potentials W (r) and W, (r) are at present unknown and under investigation. Numerical studies [28], however, show them to decay more rapidly than the exchange potential. 

For the nonuniform electron density system at a jellium-metal surface, it is generally accepted [5-7,9,31-33] that the asymptotic structure of the Kohn-Sham exchange-correlation potential is the image potential ... 

This describes a semilogarithmic dependence between the overpotential for the opening of the polymeric structure (tjN) and the cathodic overpotential (ffc) at which it was closed. The experimental results (Fig. 56) fit Eq. (53). This equation also contains an asymptotic approach to the opening potential (rjN) when the cathodic potential of prepolarization increases. 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

Hydroxyl radical (OH) is a key reactive intermediate in combustion and atmospheric chemistry, and it also serves as a prototypic open-shell diatomic system for investigating photodissociation involving multiple potential energy curves and nonadiabatic interactions. Previous theoretical and experimental studies have focused on electronic structures and spectroscopy of OH, especially the A2T,+-X2n band system and the predissociation of rovibrational levels of the M2S+ state,84-93 while there was no experimental work on the photodissociation dynamics to characterize the atomic products. The M2S+ state [asymptotically correlating with the excited-state products 0(1 D) + H(2S)] crosses with three repulsive states [4>J, 2E-, and 4n, correlating with the ground-state fragments 0(3Pj) + H(2S)[ in... 

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