Hamiltonian operator singularities

The usual Hilbert-space requirement of continuous gradients is not appropriate to Coulombic point-singularities of the potential function u(r) [ 196]. This is illustrated by the cusp behavior of hydrogenic bound-state wave functions, for which the Hamiltonian operator is... 

As may be seen by comparing Eqs. [103] and [105], the no-pair spin-orbit Hamiltonian has exactly the same structure as the Breit-Pauli spin-orbit Hamiltonian. It differs from the Breit-Pauli operator only by kinematical factors that damp the 1/rfj and l/r singularities. 

The NpPolMe basis sets were developed recently (10) for the investigation of relativistic effects using the DK transformed hamiltonian (13, 18-20). This is the spin-averaged no-pair approximation which reduces the 4-component relativistic one-electron hamiltonian to a 1-component form without introducing strongly singular operators. NpPolMe basis sets indirectly incorporate some relativistic effects on the wave function. Let us note that both PolMe and NpPolMe contracted sets share the same exponents of primitive Gaussians. Contraction coefficients are, however,... 

However, these benefits come at a price. Both Vgg and Vxc and their contributions to the transformations obviously change at each self-consistent iteration so the net effect is that some very complicated operator products, involving both momentum and direct space representations, must be done at every iteration. What Rosch and co-workers noticed [44] was that the singular part of the Hamiltonian Vxe of course does not change from iteration to iteration, so they attempted an incomplete DKH transformation which retained only V g and incorporated, therefore, the bare electron-electron interactions in the transformed Hamiltonian. 

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

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]. 

The convergence of this series is better than the convergence of the standard Born series because the singularities caused by the resonances have been transferred into the resonant term Ti z). To produce results of a general validity (spectroscopy and dynamics) we derive an exact expression of the transition operators that extend the model investigated in Section 2. The unperturbed part of the Hamiltonian is chosen in the form... 

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