Hamiltonian Douglas-Kroll, second-order

In the Douglas-Kroll-Hess spin-free relativistic Hamiltonians (second-order and third-order) [11,13], the T andF operators in Eq. (4) are... 

Operators that result from a DK transformation are directly given in the momentum representation. Hess et al. [29,31] developed a very efficient strategy to evaluate the corresponding matrix elements in a basis set representation it employs the eigenvectors of the operator as approximate momentum representation [29,31]. In practice, the two-component DK Hamiltonian is built of matrix representations of the three operators p, V, pVp + id(pV x p). This Douglas-Kroll-Hess (DKH) approach became one of the most successful two-component tools of relativistic computational chemistry [16,74]. In particular, many applications showed that the second-order operator 2 Is variationally stable [10,13,14,31,75,76,87]. 

The reduction of the relativistic many-electron hamiltonian by expansion in powers of the external field is the second-order Douglas-Kroll transformation [29], and has been used with success by Hess and co-workers [30]. The operators which result from this transformation are non-singular, but the integrals over the resulting operators are complicated and have to be approximated, even for finite basis set expansions. The reduction of the Dirac-Coulomb-Breit equation to two-component form using direct perturbation theory has been described by Kutzelnigg and coworkers [26, 27, 31], Rutkowski [32], and van Lenthe et al. [33]. 

Douglas and Kroll (1974) modified the elimination of the positronic one-particle states in a way that leads to relativistic operators suitable for the variational approaches used in quantum chemistry. Instead of expansions in powers of d the transformation is constructed to lead to expansions in powers of the external potential. The ideas of Douglas and BCroll were followed and implemented by Hess (1986), Jansen and Hess (1989a) and Samzow et al. (1992). Correct to second order in the potential the Douglas-Kj-oll-Hess (DKH) one-electron Hamiltonian is... 

In most current applications of the Douglas-Kroll transformation, the Hamiltonian is truncated at second order in the successive unitary transformation, that is, Ui, and the resulting Hamiltonian can be written in... 

The second-order one-electron Douglas-Kroll Hamiltonian has found wide application in quantum chemistry programs through approximations that are discussed in the next two sections. Although it is a considerable improvement on the first-order Hamiltonian, for some heavy elements the error is significant. Hamiltonians through fifth order have been derived by Nakajima and Hirao (2000). The third-order Hamiltonian is given by... 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

To illustrate, we use the second-order Barysz-Sadlej-Snijders transformation, which is more transparent than the Douglas-Kroll transformation. Introducing a perturbation parameter X, the Hamiltonian including the electric perturbation is... 

---