Grid-Free Techniques to Handle the Exchange-Correlation Potential

5 Grid-Free Techniques to Handle the Exchange-Correlation Potential 

This step is similar to what we have done in equation (7-7) where we obtained the matrix representation of the Kohn-Sham operator. If we insert expression (7-14) for the charge density in terms of the LCAO functions and make use of the density matrix P defined in equation (7-15), we arrive at 

While the computational work for setting up the matrix representation R of p(r) scales formally as N4, this can be cut down to N3 using again the trick introduced in section 7-3 by expanding the density in terms of an atom centered, orthonormalized auxiliary basis set cok (recall equation (7-25)). Let us review this simplification under a slightly different perspective. The starting point is again 

Since we have chosen the coK to be orthonormal, the expansion coefficients cK are related to the density matrix according to 

Consequently, the final equation for the now only approximate matrix element R becomes 

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