Third Unitary Transformation

The next unitary transformation U2 is applied in order to eliminate the odd term of second order of/a given by Eq. (12.12), 

W2 is conveniently and uniquely chosen to eliminate the second-order odd term. 

Since even and odd operators obey the same multiplication rules as natural numbers, i.e., even times odd is odd, etc., this is obviously an odd and antiher-mitean operator of second order in the scalar potential, which is independent of the chosen parametrizations of the unitary transformations. 

W2 is therefore a second-order integral operator in momentum space, whose action on a 4-spinor ip is defined by 

Recall that the momentum-space formulation is advantageous for the evaluation of the kinematic factors that require the evaluation of a square root of an expression involving the square of the momentum eigenvalue in Ep. 

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The Douglas-Kroll (DK) approach [153] can decouple the large and small components of the Dirac spinors in the presence of an external potential by repeating several unitary transformations. The DK transformation is a variant of the FW transformation [141] and adopts the external potential Vg t an expansion parameter instead of the speed of light, c, in the FW transformation. The DK transformation correct to second order in the external potential (DK2) has been extensively studied by Hess and co-workers [154], and has become one of the most familiar quasi-relativistic approaches. Recently, we have proposed the higher order DK method and applied the third-order DK (DK3) method to several systems containing heavy elements. 

The third-order term X 3 seems to depend on the parametrization of the unitary transformations since it explicitly contains the coefficient flg. However, the operator W3 also depends on the chosen parametrization... 

Now remove the first row and first column and repeat with the submatrix of order n — 1. After continuing the process until the reduced matrix is of third order, restore the rows and columns that had been dropped, and border the transforming matrices with ones on the diagonal and zeros elsewhere. There results, then, a unitary matrix W, the product of the Wt, such that... 

This section demonstrates how the first three unitary matrices are explicitly constructed and applied to the one-electron operator / (or to some of its parts such as + V uc)- The first transformation has necessarily to be the free-particle Foldy-Wouthuysen transformation Uo, which is followed by the transformation Ui. The third transformation U2 turns out to produce even operators that depend on the parametrization chosen for Uz- Afterwards the infinite-order, coefficient-dependence-free scheme is discussed. 

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