Equivalent local potential

Fig. 11.3 illustrates the relative momentum profile of the 15.76 eV state in a later experiment at =1200 eV, compared with the plane-wave impulse approximation with orbitals calculated by three different methods. The sensitivity of the reaction to the structure calculations is graphically illustrated. A single Slater-type orbital (4.38) with a variationally-determined exponent provides the worst agreement with experiment. The Hartree-Fock—Slater approximation (Herman and Skillman, 1963), in which exchange is represented by an equivalent-local potential, also disagrees. The Hartree—Fock orbital agrees within experimental error. 

Due to this perturbation, the electron density will acquire a time-depen-dent change. In the same way as for the time-independent system we assume that an equivalent system of noninteracting electrons moving in a local potential (now time-dependent) can be found, the orbital densities of which build the exact total time-dependent density. We write the KS potential as the sum of the original KS potential plus the time-dependent perturbation, vs<7(ri)+dvS(7(ri,t) and the KS orbitals as (r,) exp (- ) + <50ia (r, f). 

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. 

As a consequence, any orbital functional E c[ tpjT is an (implicit) functional of p-f and pi, provided the orbitals come from a local potential. Having this in mind, we may equivalently start out with an approximation for the xc energy functional depending explicitly on the set of KS orbitals, i.e. 

Again this is an equivalent-local approximation since the momentum representation (3.40) of a local potential depends only on K. 

Fig. 8.8 shows that the coupled-channels-optical method with the equivalent-local polarisation potential (McCarthy and Shang, 1992) gives a good semiquantitative description of the experimental data of Williams (1976 ) for elastic differential cross sections below the n—2 threshold. At energies just below the n=3 threshold the resonances affect the n=2 excitations. Fig. 8.9 shows the energy dependence of the integrated cross sections for the 2s and 2p channels. Since a resonance is a property of the compound system, not the channel, the resonances observed in... 

The equivalent-local form of the coupled-channels-optical method does not give a satisfactory description of the excitation of triplet states (Brun-ger et al, 1990). Here only the exchange part of the polarisation potential contributes. The equivalent-local approximation to this is not sufficiently accurate. It is necessary to check the overall validity of the treatment of the complete target space by comparing calculated total cross sections with experiment. This is done in table 8.8. The experiments of Nickel et al. (1985) were done by a beam-transmission technique (section 2.1.3). The calculation overestimates total cross sections by about 20%, due to an overestimate of the total ionisation cross section. However, an error of this magnitude in the (second-order) polarisation potential does not invalidate the coupled-channels-optical calculation for low-lying discrete channels. 

In the absence of independent measurements of the total cross section the total ionisation cross section gives an estimate of the validity of the equivalent-local polarisation potential used for the coupled-channels-optical calculation of fig. 8.13. The calculated value at 40 eV is 5.2 nal, compared with 4.66+0.47 nal measured by Karstensen and Schneider (1975). 

It is useful to test approximations for the total ionisation cross section of helium, since it is a common target for the scattering and ionisation reactions treated in chapters 8, 10 and 11. Fig. 10.15 compares the data reported as the experimental average by de Heer and Jansen (1977) with the distorted-wave Born approximation and the coupled-channels-optical calculation using the equivalent-local polarisation potential. Cross sections... 

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. 

As in the Hartree-Fock theory, the OEL equations for a given orbital functional E are equivalent to the conditions (a Q i) = 0, / < TV < u, assuming orthonormal orbitals, where Q is the effective Hamiltonian defined by orbital functional derivatives. Equivalently, (8< /< ,) = 0, i N. If a local potential vxc(r)... 

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. 

If one works out this expression one obtains equations that are identical to equations (316) and (317). These equations were first derived by Talman and Shadwick [45]. Since in our procedure we optimized the energy of a Slater determinant wavefunction under the constraint that the orbitals in the Slater determinant come from a local potential, the method is also known as the optimized potential method (OPM). We have therefore obtained the result that the OPM and the expansion to order e2 are equivalent procedures. The OPM has many similarities to the Hartree-Fock approach. Within the Hartree-Fock approximation one minimizes the energy of a Slater determinant wavefunction under the constraint that the orbitals are orthonormal. One then obtains one-particle equations for the orbitals that contain a nonlocal potential. Within the OPM, on the other hand, one adds the additional requirement that the orbitals must satisfy single-particle equations with a local potential. Due to this constraint the OPM total energy ) will in general be higher than the Hartree-Fock energy Fhf, i.e., Ex > E. We refer to Refs. [46,47] for an application of the OPM method for molecules. 

For nonvanishing quadratic Jahn-Teller coupling, the adiabatic potentials V exhibit the threefold symmetry of the point group. The lower sheet, in particular, exhibits three equivalent local minima which are separated by three equivalent saddle points. 

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