Two Possible Routes for the Derivation

After introducing a set of basis functions, we can rewrite the matrix elements occurring in the total energy as matrix products introduced in section 10.2. For the differentiation of the total electronic energy with respect to the molecular spinor coefficients we write the Lagrangian functional as 

we need to set the first derivative of L with respect to the molecular spinor coefficients equal to zero. 

This equation may be written in the well-known compact form as a generalized matrix eigenvalue problem. 

It is important to note that the introduction of a finite basis set produced a Fock matrix where the position dependence has been integrated out — in contrast to the atomic case of chapter 9, where all Coulomb and exchange operators feature a clear position dependence. 

Similar to nonrelativistic Hartree-Fock theory, the Dirac-Roothaan Eqs. (10.61) are solved iteratively until self-consistency is reached. However, because of the properties of the one-electron Dirac Hamiltonian entering the Fock operator, molecular spinors representing unphysical negative-energy states (recall section 5.5) show up in this procedure. As many of these negative-continuum 

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