Matrix-Based Quadratically Convergent SCF D-QCSCF

Density Matrix-Based Quadratically Convergent SCF (D-QCSCF) 

To minimize the energy functional (Eq. [115]) with respect to density changes 

For small changes Pa in the density P, terms higher than linear in the expansion can be discarded. We want to minimize the energy gradient of Eq. [132] as 

Neglecting all terms higher than linear and differentiating Eq. [132], we immediately arrive at the Newton-Raphson equation, which has to be solved iteratively to obtain the density update P  

The term on the right-hand side of Eq. [134] is already known from the simple energy gradient (Eq.[117]) and the left-hand side can be calculated as 

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