KS-LCAO

All that is needed (in principle) in order to solve the KS-LCAO equations is a simple modification to standard HF-LCAO computer codes. The exchange contribution has to be replaced by the KS contribution. 

As mentioned above, a KS-LCAO calculation adds one additional step to each iteration of a standard HF-LCAO calculation a quadrature to calculate the exchange and correlation functionals. The accuracy of such calculations therefore depends on the number of grid points used, and this has a memory resource implication. The Kohn-Sham equations are very similar to the HF-LCAO ones and most cases converge readily. 

Here is a KS-LCAO calculation on water at the experimental geometry of 95.7pm and 104.5°. I chose the BLYP functional this comprises Becke s 1988... 

The calculated density functional KS-LCAO energy is denoted E(RB-LYP). 

The KS-LCAO orbitals may be visualized by all the popular methods, or one may just focus on the Mulliken population analysis indices (Figure 13.5). 

I drew attention in Chapter 12 to the fact that the Xa orbitals did not satisfy the nice properties of standard HF-LCAO ones the Koopmans theorem is not valid, and so on. The same is true of all density functional KS-LCAO calculations. In practice, it usually turns out that the KS-LCAO orbitals are very similar to ordinary HF-LCAO ones, which must mirror the fact that exchange-correlation effects are only a minor part of the total electronic energy. So the orbitals are often analysed as if they were ordinary HF orbitals (Figure 13.4). 

As an example of a mature topic, consider Density Functional Theory (DFT). DFT is far from new and can be traced back to the work of John Slater and other solid state physicists in the 1950 s, but it was ignored by chemists despite the famous papers by Hohenberg/ Kohn (1964) and Kohn/ Sham (KS) (1965). The HF-LCAO model dominated molecular structure theory from the 1960 s until the early 1990s and I guess the turning point was the release of the rather primitive KS-LCAO version of GAUSSIAN. DFT never looked back after that point, and it quickly became the standard for molecular structure calculations. So this Volume of the SPR doesn t have a self contained Chapter on DFT because the field is mature. 

The HF LCAO method for periodic systems was considered in Sections 4.1.5 and 4.1.6. We discuss here the KS LCAO method for crystals in comparison with the HF LCAO approach. The electronic energy of the crystttl (per primitive cell) as calculated within the HF approximation (Ehf) and DFT E pt) can be expressed in terms of the one-electron density matrix (DM) of the crystal dehned as P k) in terms of Bloch sums of AOs, (4.125) or as p R,R ) in coordinate space, (4.126). These expressions are ... 

For rutile T1O2 the LCAO results for MuUiken charges and the Ti-0 overlap population can be compared with those found in DFT-PW calculations [601] after projecting the eigenstates onto a localized basis set of atomic pseudo-orbitals. In KS-LCAO calculations, Qn = 1.88 and overlap populations of Ti-0 bonds are 0.11 and 0.10 for nearest and next-nearest oxygens, respectively. In PW calculations [20], Qti = 1.46, and the overlap populations are 0.35 and 0.43, respectively. This comparison shows that the calculated local properties numerical values depend on the choice of basis set used in calculations (LCAO or PW sets). Moreover, in PW calculations the procedure of projection is used to receive atomic orbitals (see Sect. 9.1.5), which is not necessary in LCAO calculations. 

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