KS equations

The current iteration s density is used in the KS equations to detennine tlie Hamiltonian -1/2V + V(r)... 

These new ij). are used to compute a new density, which, in turn, is used to solve a new set of KS equations. This process is continued until convergence is reached (i.e. until the (ji. used to detennine the... 

This is because no four-indexed two-electron integral like expressions enter into the integrals needed to compute the energy. All such integrals involve p(r) or the product p(/)p(r) because p is itself expanded in a basis (say of M functions), even the term p(r)p(r) scales no worse than tvF. The solution of the KS equations for the KS orbitals ([). involves solving a matrix eigenvalue problem this... 

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

The KS equation (Eq. 23) when expressed in terms of a plane-wave basis set takes a very simple form ... 

By construction, KS functionals are well-defined, (but note that the XC KS term is defined with the help of two distinct paths) and give well-defined functional derivatives, so their variation proceeds as in the preceding section, leading superficially to the standard KS equations. 

Of course, this self-correction error is not limited to one electron systems, where it can be identified most easily, but applies to all systems. Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of approximate functionals in which they explicitly enforced equation (6-34) by substracting out the unphysical self-interaction terms. Without going into any detail, we just note that the resulting one-electron equations for the SIC orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS equations do not share the same potential for all orbitals. Rather, the potential is orbital dependent which introduces a lot of practical complications. As a consequence, there are hardly any implementations of the Perdew-Zunger scheme for self-interaction correction. 

Recall the central ingredient of the Kohn-Sham approach to density functional theory, i. e., the one-electron KS equations,... 

Applying the variational principle to the energy given by Eq. 1, Kohn and Sham reformulated the density functional theory by deriving a set of one-electron Hartree-like equations leading to the Kohn-Sham orbitals v().(r) involved in the calculation of p(r)15. The Kohn-Sham (KS) equations are written as follows ... 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

In principle, the KS equations would lead to the exact electron density, provided the exact analytic formula of the exchange-correlation energy functional E was known. However, in practice, approximate expressions of Exc must be used, and the search of adequate functionals for this term is probably the greatest challenge of DFT8. The simplest model has been proposed by Kohn and Sham if the system is such that its electron density varies slowly, the local density approximation (LDA) may be introduced ... 

If no correlation is introduced (ec = 0), the KS equations reduce to the well known Xa method proposed by Slater22 as a simplification of the Hartree-Fock scheme with a local exchange operator ... 

By construction, the exact TD density of the interacting system can then be calculated from a set of noninteracting, single-particle orbitals fulfilling the TD-KS Equation 8.4 and reads... 

These two variations can become equivalent if the role of the noninteracting potential is played by the functional derivative of F [n]. Thus, the GS density riQsir) and, consequently, TsCnos] can be calculated from the solutions of the so-called KS equations [compare Eq. (40)] ... 

Knowledge of all terms of the KS potential in Eq. (51) for a large distance from an atomic or molecular center is interesting both for setting the proper asymptotic behavior for the solutions of the KS equations (50), and for checking the accuracy of approximations for v (r) and v (r) in this region. 

The differential virial theorem (169) for noninteracting systems can alternatively be obtained [31], [32] by summing (with the weights fj ) similar relations obtained for separate eigenfunctions 4>ja(r) of the one-electron Schrodinger equation (40) [in particular the KS equation (50)]. Just in that way one can obtain, from the one-electron HF equations (33), the differential virial theorem for the HF (approximate) solution of the GS problem, as is shown in Appendix B, Eq. (302), in a form ... 

By applying all above considerations to the HF method posed as the DFT in Section 2.4, where the equivalent noninteracting electron problem leads to the HF-KS equations (70), we obtain from Eqs. (174) and (175)... 

Long-range asymptotic properties of KS orbitals were recalled in Sect. 4, as dictated by asymptotic properties of all potentials of KS equations. A new, simple and direct method was applied to obtain the asymptotic form of the exchange potential. For pure-state systems the known results were confirmed in Sects. 4.2 and 4.3, while for mixed-state systems a new, exact result was obtained in Sect. 5.6. 

So how accurate are DFT calculations It is extremely important to recognize that despite the apparent simplicity of this question, it is not well posed. The notion of accuracy includes multiple ideas that need to be considered separately. In particular, it is useful to distinguish between physical accuracy and numerical accuracy. When discussing physical accuracy, we aim to understand how precise the predictions of a DFT calculation for a specific physical property are relative to the true value of that property as it would be measured in a (often hypothetical) perfect experimental measurement. In contrast, numerical accuracy assesses whether a calculation provides a well-converged numerical solution to the mathematical problem defined by the Kohn-Sham (KS) equations. If you perform DFT calculations, much of your day-to-day... 

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