Frozen-core orbitals

In active-space calculations, the total orbital space is usually partitioned into external core orbitals (c), active orbitals (a), and unoccupied virtual (external) orbitals (v). (There can additionally be some frozen core orbitals that remain doubly occupied throughout the calculation.)... 

We use Slater type basis sets that are of triple- quality in the valence region. These basis sets are augmented by two (all elements in ZORA calculations elements H - Kr in QR calculations ADF standard basis V) or one (all other elements in QR calculations ADF standard basis IV) sets of polarization functions. For Pauli (QR) calculations, we use the frozen core approach (27). The (frozen) core orbitals are... 

In electron correlation treatments, it is a common procedure to divide the orbital space into various subspaces orbitals with large binding energy (core), occupied orbitals with low-binding energy (valence), and unoccupied orbitals (virtual). One of the reasons for this subdivision is the possibility to freeze the core (i.e., to restrict excitations to the valence and virtual spaces). Consequently, all determinants in a configuration interaction (Cl) expansion share a set of frozen-core orbitals. For this approximation to be valid, one has to assume that excitation energies are not affected by correlation contributions of the inner shells. It is then sufficient to describe the interaction between core and valence electrons by some kind of mean-field expression. 

Frozen-core orbitals are doubly occupied and a spin integration has been performed for the core electrons. The summation index k therefore runs over spatial orbitals only. Employing the frozen-core approximation considerably shortens the summation procedure in single excitation cases. Double excitation cases are left unaltered at this level of approximation. The computational effort can be substantially reduced further if one manages to get rid of all explicit two-electron terms. 

Not only does the frozen core approximation reduce the number of configurations, but it also reduces the computational effort required to evaluate matrix elements between the configurations which remain. Assuming that all frozen core orbitals are doubly occupied and orthogonal to all other molecular orbitals, it can be shown126 that... 

Following the suggested Gaussian theory, all systems were reoptimized at the MP2 level without any frozen core orbitals. 

In fact, for the special case of one electron outside a set of occupied inner orbitals, we can use frozen core orbitals in a formulation which is identical in form to the above theory. We assume that the HF equation has been solved- for the inner orbitals ... 

Now we freeze these orbitals and introduce the outer orbital and, since there is no self-consistency requirement for the introduced single electron, the equation for the outer electron is just the same as the HF equation for the frozen core orbitals. 

If we begin to think about the Coulomb/exchange potential generated by a nucleus and a determinant of frozen core orbitals it is clear that we should, initially at least, give separate consideration to two quantities ... 

Fomili A, Loos P-F, Sironi M, Assfeld X (2006) Frozen core orbitals as an alternative to specific frontier bond potential in hybrid quantum mechanics/molecular mechanics methods. Chem Phys Lett 427 236V240... 

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