Computational Chemistry

6. Use the K-S operatorh^KSand the basis functions {f} to calculate Kohn–Sham
   matrix elementshrs(cf. Fock matrix elementsFrs(Section 5.2.3.6),

hrs¼ frjh^

KS

jfs

DE

(7.27)

and assemble a Kohn–Sham matrix, the square matrix ofhrselements.

7. Orthogonalize the KS matrix, diagonalize it to get a coefficients matrixC^0 and an
   energy levels matrixe, and transformC^0 toC, the matrix of the coefficients that
   give the KS orbitals as a weighted sum of the original non-orthogonal basis
   functions (cf. Section 5.2.3.6.2). We now have the first-iteration values of the
   energy levelseiand the KS molecular orbitalsci(getting the coefficients is
   equivalent to getting the MOs, once a basis set is in hand, sincecKSi ¼

P

cfbasis).

8. Use the first-iteration values of the KS MOs to calculate an improvedr:

r 0 ¼rr¼

X^2 n

i¼ 1

jcKSi ð 1 Þj^2

(see Eq.*7.22)

9. Go back to step 4, but with the improved, first-iterationrinstead of the guess. At
   the new step 7 we will have the second-iteration values of the energy levelsei
   and the KS molecular orbitalsci(and the first-iterationr, from the first applica-
   tion of step 8). Check them for significant change. If these do not differ (within
   specified limits) from the first-iteration values, and the first-iteration r is
   unchanged from the guess we started with, stop. If they differ, go through the
   process again, to get the third-iteration values of the energy levelseiand the
   KS molecular orbitals, and the second-iterationr. Check for significant change;
   and so on.


10. When the iterations have satisfactorily converged, calculate the energy using
    Eq.7.21.


11. The geometry can be optimized with the aid of derivatives of the energy with
    respect to geometry, as outlined in Section 2.4. Any method in which the
    calculated energy varies with the geometry canin principleoptimize geometry.

7.2.3.4 The Exchange-Correlation Energy Functional: Various Levels
of Kohn–Sham DFT

We have to consider the calculation of the fourth term, the problem term, in the
KS operator of Eq.7.23, the exchange-correlation potentialvXC(r). This is defined as
the functional derivative [ 36 , 37 ] of the exchange-correlation energy functional,
EXC[r(r)], with respect to the electron density functional (Eq.7.23). The exchange-
correlation energyEXC[r(r)], a functional of the electron density functionr(r), is a
quantity which depends on thefunctionr(r) and on just what mathematical form the

7.2 The Basic Principles of Density Functional Theory 459