Asymptotic Behavior of Exchange-Correlation Potentials

While the behavior of the exchange-correlation potential Vxc (recall from equation (5-16) that the exchange-correlation potential Vxc is defined as the functional derivative of the 

21 Tozer and Handy empirically define far in terms of 4.7 times the Bragg-Slater radius of the corresponding 

---

Density Functional Theory (DFT) has become a powerful tool for ab-initio electronic structure calculations of atoms, molecules and solids [1, 2, 3]. The success of DFT relies on the availability of accurate approximations for the exchange-correlation (xc) energy functional Exc or, equivalently, for the xc potential vxc. Though these quantities are not known exactly, a number of properties of the exact xc potential vxc(r) are well-known and may serve as valuable criteria for the investigation of approximate xc functionals. In this contribution, we want to focus on one particular property, namely the asymptotic behavior of the xc potential For finite systems, the exact xc potential vxc(r) is known to decrease like — 1/r as r —oo, reflecting also the proper cancellation of spurious self-interaction effects induced by the Hartree potential. 

A common problem for both methods lies in the use of potentials that do not possess the correct net attractiveness. This can have the consequence that continuum feamres appear shifted in energy. In particular, there is evidence that the LB94 exchange-correlation potential currently used for the B-spline calculations, although possessing the correct asymptotic behavior for ion plus electron, is too attractive, and near threshold features can then disappear below the ionization threshold. An empirical correction can be made, offsetting the energy scale, but this can mean that dynamics within a few electronvolts of threshold get an inadequate description or are lost. There is limited scope to tune the Xa potential, principally by adjustment of the assumed a parameter, but for the B-spline method a preferable alternative for the future may well be use of the SAOP functional that also has correct asymptotic behavior, but appears to be better calibrated for such problems [79]. 

The quality of the TD-DFT results is determined by the quality of the KS molecular orbitals and the orbital energies for the occupied and virtual states. These in turn depend on the exchange-correlation potential. In particular, excitations to Rydberg and valence states are sensitive to the behavior of the exchange-correlation potential in the asymptotic region. If the exchange-correla-... 

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

Similarly, as the case with response properties discussed in the previous section, imposing the correct asymptotic behavior of vxc improves the agreement between the numerical values of —sH0M0 and the experimental ionization potentials.85,90,91,94 For these reasons, the ionization potentials and electron affinities are usually obtained as energy differences (ASCF) in calculations using common approximations to the exchange-correlation functional. The discussed hereafter numerical values were obtained in this way. 

The asymptotic behavior of the exchange-correlation potential far from the molecule has been identified as the key factor determining the accuracy of the ionization potentials of anions and electron affinities of neutral molecules.5 Recently, Wu et al.91 proposed a variational method, which enforces the correct long-range behavior of vxc. Indeed, a noticeable improvement compared to the Kohn-Sham results derived using conventional approximations (LDA-, GGA-, and hybrid functionals) was reported for atoms (H, He, Li, Be, B, C, N, O, and F) and diatomics (BeH, CH, NH, OH, CN, BO, NO, OO, FO, and FF). The still significant discrepancies between the experimental and calculated ionization potentials (or electron affinities) were attributed to errors of the exchange-correlation potential in the molecular interior. 

Gisbergen, S.J.A. van, Osinga V.P., Gritsenko O.V., Leeuwen R. van, Snijders J.G. and Baerends E. J., Improved density functional theory results for frequency-dependent polarizabilities, by the use of an exchange-correlation potential with correct asymptotic behavior. J.Chem.Phys. (1996) 105 3142-3161. 

---