Kohn-Sham matrix elements

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),... 

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

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... 

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... 

Note that these coefficients are spin-dependent because of the exchange-correlation potential. The ADFT Kohn-Sham matrix elements are given by... 

Thus, the ADFT Kohn-Sham matrix elements are independent of the density matrix. As a result, only the approximated density and the corresponding density derivatives, in the case of gradient-corrected functionals, are numerically calculated on a grid. 

The corresponding modified Kohn-Sham matrix elements are given by... 

The perturbed Kohn-Sham matrix elements can be calculated straightforwardly... 

The only other modification one has to take care of is the construction of the total density matrix Pi = + Pi ) to build the Fock (or Kohn-Sham) matrix elements. 

In order to distinguish the and coefficients we name them Coulomb and exchange-correlation coefficients, respectively. The corresponding Kohn-Sham matrix elements are defined as ... 

Compared to the standard LCGTO kernel ixv fxc uT) the scaling of the kernel calculation is reduced by almost two orders of magnitude in the ADPT approach. With this result we now rewrite the perturbed Kohn-Sham matrix elements in molecular orbital representation as ... 

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... 

Note that each cycle of the inner loop does not require computing a new Kohn-Sham matrix, with the expensive Coulomb and exchange-correlation contributions, so that the time spent for optimizing is usually not problematic in terms of computational time, especially if an initial guess can be provided. So far, we have not specified the form of the weight function w(r) and its corresponding matrix elements that are used in the cDFT constraint We now turn to this point. 

For the optimal set of Dirac-Kohn-Sham spinors the gradient must disappear, and this can be achieved by rotating to the spinor set that diagonalizes the Dirac-Kohn-Sham matrix with elements... 

NBO analysis can be used to quantify this phenomenon. Since tire NBOs do not diagonalize the Fock operator (or tire Kohn-Sham operator, if the analysis is carried out for DFT instead of HF), when the Fock matrix is formed in the NBO basis, off-diagonal elements will in general be non-zero. Second-order perturbation tlieory indicates that these off-diagonal elements between filled and empty NBOs can be interpreted as the stabilization energies... 

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