assertion: “The conventional Hartree–Fock approximation can be regarded as a
density-functional approach in the HFKS scheme with correlation completely
neglected, but not in the KS scheme. Instead of the exactnonlocalexchange
potential in the HFKS equations, the KS equations use an effectivenonlocal
potential that is not known and has to be approximated. Another trade of accuracy
for simplicity!” [ 39 ].
7.2.3.3 Solving the KS Equations
First let’s review the steps in carrying out a HF calculation (shown in detail in
Section 5.2.3.6.5). We start with a guess of the basis function coefficientsc, because
the HF operatorF^(the Fock operator) itself contains the wavefunction, which is
composed of the basis functions and their coefficients. The operator is used with the
basis functions to calculate the HF Fock matrix elementsFrs¼ frjF^jfs
which
constitute the Fock matrixF. An orthogonalizing matrix calculated from the
overlap matrixSputsFinto a formF^0 that satisfiesF^0 ¼C^0 eC^0 "^1 (Section 5.2.3.6.2).
Diagonalization ofF^0 gives a coefficients matrixC^0 and an energy levels matrixe;
transformingC^0 toCgives the matrix with the coefficients corresponding to the
original basis set expansion, and these are then used as a new guess to calculate a
newF; the process continues till it converges satisfactorily on thec’s, i.e. on the
wavefunction, and the energy levels (which can be used to calculate the electronic
energy); the procedure was shown in detail in Section 5.2.3.6.5.
The standard strategy for solving the KS eigenvalue equations, like that for
solving the HF equations, which they resemble, is to expand the KS orbitals in
terms of basis functionsf(withmfunctions in the set):
cKSi ¼Xms¼ 1csifs i¼ 1 ; 2 ; 3 ;...;m (7.26)This is exactly the same as was done with the Hartree–Fock orbitals in
Section 5.2.3.6.1, and in fact the samebasisfunctionsareoftenusedasin
wavefunction theory, although as in all calculations designedto capture electron
correlation, sets smaller than split-valence (Section 5.3.3) should not be used. A
popular basis in DFT calculations is the 6-31G*. Substituting the basis set
expansion into the KS Eqs.7.23,7.25and multiplying byf 1 ,f 2 ,...,fmleads,
as in Section 5.2.3.6.1, tomsets of equations, each set withmequations, which
can all be subsumed into a single matrixequationanalogoustotheHFequation
FC¼SCe.ThekeytosolvingtheKSequationsthenbecomes,asinthestandard
HF method, the calculation of Fock matrixelementsanddiagonalizationofthe
matrix (Section 5.2.3.6.2). In a DFT calculation we start with a guess of the
density functionr(r), because this is what we need to obtain an explicit
expression for the KS Fock operatorh^KS(Eqs.7.23,7.24,7.25). This guess is
usually a noninteracting atoms guess, obtained by summing mathematically the
7.2 The Basic Principles of Density Functional Theory 457