Application of our method to the LB94 potential produces a competitively accurate exchange GGA. In Chapter 3 we described the construction of another GGA from the same model potential, using a different approach. In that case, we started with the LB94 potential, completed it to a functional derivative, and then recovered the parent functional. Here, we first assign a functional to the LB94 potential and then use that functional to obtain the functional derivative. Since neither completion to a functional derivative nor line integration is unique for stray potentials, the Q-revLB94 functional and the functional of Chapter3 are different. The approach based on line integrals is more flexible.
Our method is also related to the procedure for constructing the “unambiguous energy density” proposed by Burke et al. [27, 28]. For exchange potentials, Burke’s
85
method amounts to defining the functional,
EX[ρ] =− 3 4π Z dr0 Z dr∇ ·[ρ(r)∇vX([ρ];r)] |r−r0| . (5.19)
This expression is essentially a gauge transformation of the Levy–Perdew virial en- ergy density carried out using the Helmholtz decomposition. Burke and co-workers in- tended Eq. (5.19) to be used with functional derivatives of standard density-functional approximations (such as LDA, PBE, and BLYP) as a way of eliminating the ambigu- ity of conventional energy densities. But, of course, their method may be also used to construct energy functionals from stray model potentials. Unfortunately, Eq. (5.19) is not very practical because it requires real-space integration overrfor each grid point
r0. Our method involves only a one-dimensional quadrature at every real-space grid point, so it is easier to implement and has a lower computational cost.
5.5
Conclusion
Assigning energy to stray model potentials always involves some degree of arbitrari- ness. Exchange energies computed via the Levy–Perdew virial relation (i.e., by line integration along a path of uniformly scaled density) are quite reasonable, provided that a model potential has a realistic shape. However, Levy–Perdew energies are not invariant with respect to molecular translations and rotations. By choosing an inte- gration path that does not involve coordinate scaling, one can ensure proper invariance of the energies obtained from a stray model potential, but often at the expense of the accuracy of the resulting energy expression.
In this work, we adopt a different view of line integration as an instrument for de- signing new density-functional approximations using stray model potentials as start- ing points. The resulting functionals are not expected to automatically yield good energies and may require fine-tuning. The modification may be as simple as adjust- ing a parameter to satisfy an exact constraint. This strategy proved quite effective in deriving the Q-revLB94 functional starting with the LB94 model potential. This new functional is properly invariant and yields better energies than the Levy–Perdew relation.
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Self-interaction correction scheme
for Kohn–Sham potentials
6.1
Introduction
In the Kohn–Sham density functional theory, electrons move in an electric field de- scribed by a multiplicative Kohn–Sham potential vs(r). The electronic part of vs is the Hartree-exchange-correlation (HXC) potential vHXC given by Eq. (1.14). The vHXC describes interaction of each electron with the field created by the remaining N −1 electrons in the system, and behaves asymptotically as (N −1)/r. The vHXC is further partitioned into a sum of the Hartree vH [Eq. (1.15)] and the exchange- correlation vXC potentials [Eq. (1.16)]. Of these two, the Hartree term includes the spurious self-interaction of each electron with itself and has the N/r asymptotic de- cay. This implies that the exact exchange-correlation potential should fall off as−1/r. Unfortunately, most approximations tovXC decay faster than −1/rand hence do not cancel out the self-interaction part of vH completely. The resulting error in vHXC, termed self-interaction error, causes the effective Kohn–Sham potential to be more repulsive than it should be at intermediate and larger [1,2]. One immediate conse- quence of this aberration is a collapse of the virtual Kohn–Sham orbital eigenvalue spectrum, which translates into poor description of response properties [3–6].
In this Chapter, we propose a self-interaction correction (SIC) motivated by a Reprinted in part with permission from A. P. Gaiduk, D. Mizzi, and V. N. Staroverov, “Self- interaction correction scheme for approximate Kohn–Sham potentials”, Phys. Rev. A 86, 052518 (2012). Copyright 2012, American Physical Society.
Reprinted in part with permission from A. P. Gaiduk, D. S. Firaha, and V. N. Staroverov, “Im- proved electronic excitation energies from shape-corrected semilocal Kohn–Sham potentials”,Phys. Rev. Lett.108, 253005 (2012). Copyright 2012, American Physical Society.
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fractional charge perspective on density functional theory [7]. We demonstrate appli- cation of our method to the calculation of vertical ionization and excitation energies. We also emphasize a unifying point of view according to which our method, the Slater transition-state technique [8–11], and the Xαapproximations [9] are all different forms of self-interaction correction for approximate exchange-correlation potentials.