Local Decomposition of the X-Operator

In order to derive consistent expressions for relativistically transformed off-diagonal Hamiltonian blocks, we need to consider atomic approximations to the ingredients of the transformation matrices. As a first option, a local construction scheme for the matrix representation of the X-operator can be considered. 

As discussed by Dyall [720] for the NESC method, the so-called Breit-Pauli approximation to the matrix representation of the X-operator, which is used to construct the relativistic transformation matrix according to Eq. (14.29), is 

The same approach was proposed in Ref. [669] in the sense of an atomic approximation to the projection on electronic states. If the electronic spinors are then transformed to a two-component picture by a renormalization matrix, a local X2C method emerges. 

The local approximation to the X matrix operator, Eq. (14.69), works well for spectroscopic constants of diatomic molecules [728,733], but there are some drawbacks of the local-X matrix approximation to the X2C method. First, 

---