Local exact decoupling

Since most quantum chemical calculations apply a linear combination of atom-centered basis functions, we may employ these basis functions to construct atomic projectors as is done in charge and spin population analysis [760]. For those Hamiltonian matrix blocks for which relativistic corrections are important, we need to derive a relativistic expression to evaluate them. The locality is then exploited in basis-function space. While it is clear that heavy-atom diagonal blocks of an operator matrix will require a relativistic description, the treatment of heavy-atom off-diagonal blocks depends on their contribution to physical observables. 

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This approach was employed by Peralta et al. [649,661] and explored in detail by Thar and Kirchner [668] for the low-order DKH method. We refer to it as the diagonal local approximation to the Hamiltonian (DLH). It is clear that the DLH approximation can be applied to all relativistic exact-decoupling approaches. Obviously, the DLH approximation will work best at large interatomic distances. For example, it is a good approximation for heavy-atom molecules in solution for which it was conceived in Ref. [668]. Then, A represents the group of atoms that form one of the solute and/or solvent molecules. 

It turned out [670] that a local approximation to the exact-decoupling transformation is the best option. This ansatz allows us to discuss all three exact-decoupling approaches, X2C, DKH, and BSS, on the same footing. A similar local approximation was developed for the BSS transformation [671,672,761]. We may approximate the unitary transformation HJ,... 

D. Peng, M. Reiher. Local Relativistic Exact Decoupling. J. Chem. Phys., 136 (2012) 244108. 

This way of generating the EDM (-> IDM) collective modes guarantees their one-to-one correspondence to the respective IDM of reactants and, as such, may provide an alternative, convenient framework for describing the CT processes. Like the IDM, such localized EDM preserve the memory of the reactant interaction in M + and should lead to substantial hardness decoupling. This expectation is due to their resemblance to the PNM (IDM) of reactants, for which the diagonal (reactant) blocks of the relevant hardness matrix are exactly diagonal. Thus, with no external hardness interaction between the A and B subsets of EDM, it comes as no surprise that these collective, delocalized charge-displacement modes also bear some similarity to the PNM of M as a whole. 

A sometimes overlooked fact is that the Kohn-Sham equation is exact at this stage. It is much easier to solve than the coupled Schrodinger equations that would have to be solved for the original system, since it decouples into single particle equations. The only problem is that we have introduced the exchange-correlation energy, which is an unknown quantity, and which must be approximated. Fortunately, it will turn out to be relatively easy to find reasonably good local approximations for it. 

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