Two-Step Transformation BSS

In the BSS approach, the free-particle Foldy-Wouthuysen transformation in addition to the orthonormal transformation K is applied to obtain the four-component Hamiltonian matrix to be diagonalized. The free-particle Foldy-Wouthuysen transformation Uq is composed of four diagonal block matrices. 

It is then applied to yield a transformed four-component Hamiltonian matrix 

Since Uq is a unitary matrix, it preserves the orthonormality condition. Ho is then diagonalized by a standard hermitean eigenvalue solver. The eigenvalue equation has the same structure as Eq. (14.35) with the primed labels (L) and (S) replaced by doubly primed ones, ( ) and (S) , to indicate the change of basis by the free-particle Foldy-Wouthuysen transformation. The X matrix in this basis representation is obtained by Eq. (14.36) with the same label replacement. Analogously, the renormalization matrix reads R = I + The final decoupling transformation 

After the exact-decoupling BSS transformation has been carried out, the Hamiltonian matrix is back-transformed to the original non-orthogonal basis representation 

Compared with the X2C approach, the BSS approach requires a few more matrix multiplications. The off-diagonal terms of the X2C Hamiltonian matrix H are diagonal matrices, whereas those in the BSS Hamiltonian matrix Hq are not. Furthermore, the BSS decoupling transformation matrices are composed of more terms than the X2C expression in Eq. (14.39). 

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