Implementation of the Douglas-Kroll Transformation

The remaining issue to be considered is that of implementation. Performing all the integrations in momentum space is a tedious business, even given that the operators 

The evaluation of the matrix elements of the momentum-dependent operators is achieved using a theorem for functions of a matrix. If we have a matrix B whose eigenvalues b form a diagonal matrix represented by [fc] and whose eigenveetors are X, any function of the matrix can be expressed in terms of the function of the eigenvalues by 

If we choose B to be the matrix of p, it is straightforward to generate the matrix of any algebraic function of —for example, the kinematie faetors A and f . If we 

The key issue here is the accuracy of the representation of p. If the basis set is too small, there could be a serious loss of accuracy. However, for a reasonably large primitive basis, the same basis could be used for the representation of p as for the molecular calculations. Even more usefully, since the kinematic factors do not change the symmetry of the atomic basis functions, they can be used to redefine the contraction coefficients. This redefinition essentially generates a contracted basis set in the modified Dirac representation. 

Redefining the contraction coefficients does involve a further level of approximation, but for a reasonably large basis set this approximation should not be too severe. 

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