Local Approximations to the Exact-Decoupling Transformation

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, 

If we now substitute the approximate unitary transformation of Eq. (14.71) into the block-diagonalization of the four-component Hamiltonian, 

They are the same as in the DLH approach, while the off-diagonal blocks are 

The cost for the assembly of the unitary transformation within the DLU approach is then of order M, where M measures the system size in terms of the number of atomic nuclei. If no local approximation was applied for the unitary decoupling transformation, according to Eq. (14.75) the calculation of the matrix will require 2M matrix multiplications. Since the number of matrices to be calculated is of order O(M ), the total cost for the relativistic transformation without local approximation is therefore of order O(M ). If the DLU approximation is applied, no summation will be needed and only two matrix multiplications will be required for each heavy-heavy block H,4b-The total cost is then of order O(M ). 

If the distance between two atoms A and B is sufficiently large, the relativistic description of can be neglected. Thus, neighboring atomic pairs may be defined according to their distances. Then, only the Hamiltonian matrix of neighboring pairs requires the transformation whereas all other pairs are simply taken in their nonrelativistic form. The total cost is then reduced to order 0(M) as the number of neighboring pairs is usually a linear function of system size M. 

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