Kohn-Sham matrices

The HF and KS operators in the reciprocal space are represented by the Fock matrices F f k) and Kohn-Sham matrices F k), which are related to the matrices in the coordinate space by the relations... 

These conditions are automatically fulfilled upon diagonalization of the Fock or the Kohn-Sham matrices and the formation of the density matrix (Eq. [8]). 

Up to this point the derivation has exactly paralleled the Hartree-Fock case, which only differs in using the corresponding Fock matrix, F rather than the Kohn-Sham counterpart, Fks. By expanding fKS into its components, the individual elements of the Kohn-Sham matrix become... 

We now need to discuss how these contributions that are required to construct the Kohn-Sham matrix are determined. The fust two terms in the parenthesis of equation (7-12) describe the electronic kinetic energy and the electron-nuclear interaction, both of which depend on the coordinate of only one electron. They are often combined into a single integral, i. e ... 

What we have not discussed so far is how the contribution of the final components of the Kohn-Sham matrix in equation (7-12), i. e., the exchange-correlation part, can be computed. What we need to solve are terms such as... 

We can obtain a more direct comparison of the ab initio and Hiickel quantities in terms of the valence pi block of the NAO Fock matrix (or Kohn-Sham matrix) F(NA0), which provides the direct ab initio counterpart of (3.155) ... 

The reaction field effects are easily incorporated as an additional term in the Kohn-Sham matrix, given by ... 

Use the K-S operator hKS and the basis functions (f>) to calculate Kohn-Sham matrix elements hrs (cf. Fock matrix elements Frs (Section 5.23.6),... 

In order to complete our description of the LSA method all that remains is to specify Xl and, then, to evaluate the interaction energy. An optimum choice for Xl is determined by the variation condition which yields [10] the local space analogue of a familiar result, namely (RFU + UFR)L = 0. Here F is either the Fock or the Kohn-Sham matrix. Since R, F and U all depend upon Xl, this is a nonlinear relation that must be solved iteratively. The simplest, but least efficient, method of solution is steepest descents which corresponds to the choice... 

This diagonalization can be performed by explicit construction of the matrix Haf ) which is then diagonalized by standard methods when the basis set is not too large. For the case of large systems and/or large basis sets, we will prefer iterative techniques, like the Lanczos method [74,143-145], which avoid the explicit construction of the Kohn-Sham matrix it is sufficient in these methods to have a procedure to apply (successively) the Kohn-Sham matrix on vectors cf. Only vectors cf need then to be stored. 

Since the potential depends on the density p, this contribution to the Kohn-Sham matrix element depends on the perturbation,... 

We have here introduced the Eourier transform of the perturbed Kohn-Sham matrix... 

Numerical evaluations of Kohn-Sham matrix elements and exchange-correlation (xc) contributions to response vectors follow the same scheme. In contrast to the Coulomb and exact Hartree-Fock exchange contributions which are usually evaluated by summing analytically computed integrals between basis functions... 

N (Kohn-Sham) orbitals can be expanded into atomic orbitals according to (4). Furthermore, the expansion coefficients C, can be determined by requiring that they optimize the total (Kohn-Sham) energy. This results in the (Kohn-Sham) matrix equation similar to (5)... 

Compute the Fock (or Kohn Sham) matrix with all the occupied molecular orbitals including the frozen ones. 

However, a more significant elfect is expected on the first excited A-state complex, in which one of the iIq electrons is promoted into the ti system. About half the H-bonding attraction ( 3 kcal/mol — 1000 cm ) is lost by removal of the Uq oh interaction in the 0 spin set, but this is partially compensated by the reverse oh delocalization that is thereby opened by emptying of the tIq orbital. From the relative pre-NBO overlap integrals for the two interactions (S /S o — 0.6) and the expected MuUiken-type proportionality to energetic (Fock/Kohn-Sham) matrix elements, we might expect that the residual -40% elfect on the vertical A<— X promotion energy corresponds to a blue-shift on the order of -400 cmThe calculated vertical X A blue shifts AVn-bond at the CIS/6-311-H-G " (450 cm ) or CASNBO(7,5)/6-311-i-i-G " (250 cm ) levels are consistent with this simple estimate. 

Performing Kohn-Sham Density-Functional Calculations. How does one do a molecular density-functional calculation with (or some other functional) One starts with an initial guess for p, which is usually foimd by superposing calculated electron densities of the individual atoms at the chosen molecular geometry. From the initial guess for p(r), an initial estimate of u c( ) found from (15.127) and (15.131) and this initial v d ) is used in the Kohn-Sham equations (15.121), which are solved for the initial estimate of the KS orbitals. In solving 15.121), the flP s are usually expanded in terms of a set of basis functions Xr ( P = 2r=i to yield equations that resemble the Hartree-Fock-Roothaan equations (13.157) and (13.179), except that the Fock matrix elements = xr F x are replaced by the Kohn-Sham matrix elements = (Xr Xs), where is in (15.122) and(15.123).Thus, instead of (13.157), in KS DFT with a basis-set expansion of the orbitals, one solves the equations... 

Alternative versions where the Coulomb part of the Kohn-Sham matrix is assembled using plane waves as the auxiliary basis have also been proposed and, properly implemented, these achieve linear scaling even for small systems and for large basis sets. °... 

In Equation (11), S represents the overlap matrix, c the molecular orbital coefficient matrix, and e the Kohn-Sham orbital energies. The expansion coefficients of the approximate density, necessary for the construction of the Kohn-Sham matrix, are calculated by the minimization of... 

As can be seen from Figure 1 the construction of the Kohn Sham matrix scales already for 1,500 basis functions subquadratically. Above 2,500 basis functions the scaling of the construction of the Kohn-Sham matrix becomes linear. The CPU times quoted in the figure refer to a 1.2 GHz AMD Athlon processor. 

---