Main elements of F12 theory

The most simple and instructive way to study the main elements of F12 theory is to start with second-order Moller-Plesset perturbation theory, MP2. We will show in the following how the conventional, orbital-expansion based theory is supplemented by additional geminal functions and how the working equations change. We will then discuss how expensive many-electron integrals are avoided by reducing them to products of two-electron integrals, and finally, how F12 theory is transferred to second quantization, which in particular is required for the formulation of coupled-cluster theory with F12 terms. 

The preceding step to both MP2 and coupled-cluster calculations is to solve the Hartree-Fock equations. The standard approach is, of course, to solve the equations in a basis set expansion (Roothaan-Hall method), using atom-centered basis functions. This set of basis functions is used to expand the molecular orbitals and we will call it orbital basis set (OBS). It spans the computational (finite) orbital space. Occupied spin orbitals will be denoted (pi and virtual (unoccupied) spin orbitals pa- In order to address the terms that miss in a finite OBS expansion, the set of virtual spin orbitals in a formally complete space is introduced, pa- If we exclude from this space all those orbitals which can be represented by the OBS, we obtain the complementary space, with orbitals denoted cp i. The subdivision of the orbital space and the index conventions are summarized in the left part of Fig. 2. 

The second-order correlation energy can be written as a sum over pair 

the factor one half is introduced to account for double counting, the terms with 2 = b are automatically excluded in the spin-orbital formalism as = 0. 

In F12 theory, the pair function is augmented by a geminal function with explicit dependence on the interelectronic distance ri2 

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