First derivatives of the exchange-correlation energy

The importance of analytic derivative methods in quantum chemistry cannot be overstated. Analytic methods have been demonstrated to be more efficient than are corresponding finite difference techniques. Calculation of the first derivatives of the energy with respect to the nuclear coordinates is perhaps the most common these provide the forces on the nuclei and facilitate the location of stationary points on the potential energy hypersurface. Differentiating the electronic energy with respect to a parameter x (which may be, but is not required to be, a nuclear coordinate), leads to the well-known expression 

The use of parentheses around x in the derivative of Exc means that only the explicit dependence on x is differentiated (i.e. the MO coefficients are not differentiated). Note that the above equations are valid for HF as well, if the XC energy is defined to be the exact HF exchange energy. However, taking Exc according to Eq. (15) formally yields 

We must emphasize the word formally above, because the XC energy is computed numerically, not analytically. This brings us to the third problem associated with the treatment of grids, namely that of differentiation of the points and weights. For a numerical implementation, the derivative is properly obtained from Eq. (21), yielding 

We have previously given [37] first derivative expressions for the weight formulae which are suitable for implementation. Along with the explicit nuclear dependence of the weights and functional values, their implicit dependence on the nuclear coordinates through the grid points must also be differentiated. For example, 

In the preceding equation, the parentheses are used to contrast the explicit nuclear dependence with the implicit dependence through r, not the MO coefficients. The grid point derivative in the second term is 

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