Exchange-correlation energy and potential matrix

As we noted in the introductory section, the working equations in KS theory are of the same form as those in HF theory, merely with the HF exchange potential redefined. In deriving an expression for the DFT XC potential corresponding to Eq. (15), the calculus of variations may be used to obtain 

Upon introducing the LCAO approximation and then minimizing the total energy in Eq. (20) with respect to the MO coefficients (subject to the orthonormality of the MO s), the familiar algebraic equations for the canonical orbitals, analogous to the Pople-Nesbet equations [56] in HF theory, are obtained, with the XC matrix elements given by [57] 

Compute contributions to and G v End loop over S End loop over A 

Given a surviving quadrature sphere, the grid points which it contains are treated as a batch, and the remainder of the algorithm is structured to vectorize over these. Recall that in the SG-1 grid [49] each sphere supports a Lebedev set typically containing 194 points. This is a satisfactory vector length on most computers of interest. 

The second step in the algorithm is to construct a list of those basis functions which are close to the current sphere. A basis function is considered to be close to a sphere if and only if its overlap with the surface of the sphere is non-negligible (to within a desired accuracy e). Clearly, this definition implies that each sphere has only 0(1) close basis functions we shall denote by h the average value of this number, which is typically on the order of 30 or so for routine calculations. In the third step of the algorithm, the values of ) and are computed on the sphere for each close basis function. This requires O(hN) work, and is not computationally expensive. 

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