Some Basics of SCF Theory

The simplest approximation used to solve the time-independent Schro-dinger equation 

With this expansion for the wave function, the expectation value using the electronic Hamiltonian (Eq. [2]) can be calculated using the Slater-Condon rules. The result is (in Dirac notation)  

Minimizing the HE expectation value (Eq. [4]) with respect to orbital rotations while imposing orthonormality constraints leads to the well-known HE equation  

We use the Mulliken notation for two-electron integrals over (real-valued) Gaussian atomic basis functions in the following  

Because the Fock matrix depends on the one-particle density matrix P constructed conventionally using the MO coefficient matrix C as the solution of the pseudo-eigenvalue problem (Eq. [7]), the SCF equation needs to be solved iteratively. The same holds for Kohn-Sham density functional theory (KS-DFT) where the exchange part in the Fock matrix (Eq. [9]) is at least partly replaced by a so-called exchange-correlation functional term. For both HF and DFT, Eq. [7] needs to be solved self-consistently, and accordingly, these methods are denoted as SCE methods. 

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After a brief introduction to some basics of SCF theories, we describe in the following four sections how Fock-type matrices can be built in a linear-scaling fashion, which is one of the key issues in SCF theories. 

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