Improved xc potential

The first improved xc potential, which found an application in TDDFT, was the potential of van Leeuwen and Baerends (LB94) [38]. The proper downward shift and Coulombic asymptotics are produced in LB94 with the Becke-type [44] gradient correction on top of the LDA potential... 

A variety of modern (meta-)GGA (generalized gradient approximation) exchange-correlation (xc) energy functionals, such as the van Voorhis Scuseria functional, are evaluated. For reliable property calculations, improved xc potentials with correct asymptotic behavior, such as SAOP" and GRAC," have been developed. 

Table 1 shows a gradual improvement in the calculated a along the series vx A < vxc < vxc°P < vxc- The LDA leads, for these relatively hard systems, to a polarizability that is systematically too high, which is related to the excess energies of the occupied orbitals, see below. Note, that the GGA-BP gradient correction of the xc potential produces only a relatively small reduction in the LDA/ALDA average absolute error from 8.8% to 5.6%. The improved SAOP potential reduces the error substantially to 2.9%. Still, further significant improvement is achieved with the accurate xc potential the error from the combination (accurate vxc)/ALDA is only 1.0% (See Table 1). Therefore, the crucial improvement of the TDDFRT results for the molecules considered is achieved with just an alteration of the xc potential, while keeping ALDA for the xc kernel.

Therefore, the main route to better performance of TDDFRT is through improved approximation of the ground-state xc potential vxc, while the simple ALDA for the xc kernel can be used for many chemical applications. Improved approximations for vxc will be considered in the next section. 

Another well-known drawback of the LDA and GGA potentials is their asymptotic behavior. They decay faster than the Coulombic asymptotic behavior vxc(ri)—>1/1 1, Ir —>oo required for the accurate xc potential. In the bulk region the LDA and GGA potentials lack the pronounced atomic shell structure of the accurate potential. The improved potentials should possess these features as well as the proper depth in the bulk region, so they should be shifted downward compared to the LDA/GGA potentials. 

Further, there are asymptotically corrected XC kernels available, and other variants (for instance kernels based on current-density functionals, or for range-separated hybrid functionals) with varying degrees of improvements over adiabatic LDA, GGA, or commonly used hybrid DFT XC kernels [45]. The approximations in the XC response kernel, in the XC potential used to determine the unperturbed MOs, and the size of the one-particle basis set, are the main factors that determine the quality of the solutions obtained from (13), and thus the accuracy of the calculated molecular response properties. Beyond these factors, the quality of the... 

The linear photoresponse of metal clusters was successfully calculated for spherical [158-160, 163] as well as for spheroidal clusters [164] within the jellium model [188] using the LDA. The results are improved considerably by the use of self-interaction corrected functionals. In the context of response calculations, self-interaction effects occur at three different levels First of all, the static KS orbitals, which enter the response function, have a self-interaction error if calculated within LDA. This is because the LDA xc potential of finite systems shows an exponential rather than the correct — 1/r behaviour in the asymptotic region. As a consequence, the valence electrons of finite systems are too weakly bound and the effective (ground-state) potential does not support high-lying unoccupied states. Apart from the response function Xs, the xc kernel /xc[ o] no matter which approximation is used for it, also has a self-interaction error. This is because /ic[no] is evaluated at the unperturbed ground-state density no(r), and this density exhibits self-interaction errors if the KS orbitals were calculated in LDA. Finally the ALDA form of /,c itself carries another self-interaction error. 

Although smaller than the Hartree contribution, the remaining xc part of the electron-electron interaction should also be subjected to the DK transformation to obtain further improved two-component wave functions for calculating g values work in this direction is in progress in our group. We have good reason to assume that the difference of our g values from those calculated with the two-component KS method ZORA [112] can be rationalized by the fact that the xc potential remains untransformed in our present g tensor approach. 

Note that the KLI approximation can also be derived from Eq. (99) setting in the third term i/ a(r) = 0, and thus the KLI approximation is a mean-field approximation to the OEP equation in fact the OEP and the KLI expression differ only by a term whose density-averaged integral vanishes. A further improvement of the KLI approximation was presented in Ref. 122. Eq. (130) is an explicit expression for the XC potential in terms of KS orbitals. However it must be solved iteratively because the second term of Eq. (130), called the response term, depends on the matrix elements of the potential itself. 

The band-structure of silicon obtained in this calculation is shown in Fig. 6.10. It was calculated at the LDA equilibrium lattice constant, even in the GGA case. These band-structures exhibit the well-known band-gap problem of DFT the predicted band-gap is too small roughly by a factor of two. This is true for the LDA and the GGA. In fact, the GGA does not show a great improvement, even when the band-structure is calculated at its predicted equilibrium lattice constant (Table 6.4). The failure of these two DFT schemes in predicting the band-gap of silicon is not a surprise. Even if the true xc potential was known, the difference between the conduction and valence bands in a KS calculation would differ from the true band-gap (Eg). The true band-gap may be defined as the ground-state energy difference between the N and N l systems... 

Simple self-interaction corrected approximations have been shown to provide viable alternative to accurate polarizability calculations of long-chain polymers within DFT." The schemes have been applied to (H2) chains with n = 2-6 in comparison with HF, conventional DFT, and high-level electron correlation schemes. SIC functionals have been shown to exhibit a field counteracting term in the response part of the XC potential as a result of which the calculated polarizabilities are much improved in comparison to normal LDA and GGA functionals. In a related investigation, it has been demonstrated that a self-interaction correction implemented rigorously within Kohn-Sham theory via the optimized effective potential... 

There is also reason to believe that DFT calculations will become faster and more accurate as the field advances. This will come about not only due to improved algorithms and XC functionals but also through improvements in computational speed. As computation speed doubles every 18 months or so (according to Moore s law [%]), larger calculations that were previously thought to be intractable start to become possible with more computational power. Such an increase allows for more accurate modeling of these systems with less need for potentially inaccurate assumptions. 

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