Pauli Hamiltonian validity

The problem is that the turn-over rule is valid only if the integrand vanishes at the boundaries. This is the case, e.g. for a homogeneous magnetic field, but not for the magnetic field created by a (point) nucleus. In the former case we get the same result as from the Pauli Hamiltonian (128)... 

While it is true that for singular external Potentials hke the Coulomb potential of a point charge, an expansion as in Eq. (35) is not valid in the vicinity of the nuclei, it is time to remind that the singularity of the potential is only one of the sources of the problem. Even for non-singluar potentials the Pauli Hamiltonian is unbound through the... 

In order to unify, in fhe spirit of quantum defect theory, the treatment of discrefe and confinuous spectra in the presence of discrete Rydberg and valence states and of resonances, Komninos and Nicolaides [82, 83] developed K-mafrix-based Cl formalism that includes the bound states and the Rydberg series, and where the state-specific correlated wavefunc-tions (of the multi-state o) can be obtained by the methods of the SSA. The validity and practicality of fhis unified Cl approach was first demonstrated with the He P° Rydberg series of resonances very close to the n = 2 threshold [76], and subsequently in advanced and detailed computations in the fine-structure spectrum of A1 using fhe Breit-Pauli Hamiltonian [84, 85], which were later verified by experiment (See the references in Ref. [85]). 

There are several ways in which we can develop a perturbation series for the ZORA equation. The first is simply to ignore the normalization—a perfectly valid procedure since the wave function is only defined up to a multiplicative constant. This we will do later in the present section. The second is to follow the same procedure as in the development of the Pauli Hamiltonian in chapter 17, and the third is to start from the Foldy-Wouthuysen transformation, as in the development of chapter 16. The last two of these both explicitly involve the normalization. We will commence here with the procedure used in chapter 17. 

Here erf, and erf are the Pauli matrices, that act on the proton (deuteron) wavefunctions potential wells of the proton (deuteron) potential energy profile of the th H-bond (in accordance with the available diffraction data the short H-bonds O-H- O with two off-center sites of a proton are considered). The parameters of Hamiltonian (1), i.e., the tunneling parameter Cl and the Ising parameters J,p describe the motion of a proton (deuteron) along H-bond and the effective pair interactions of these particles, respectively. When Cl is substantially smaller than Jy the static approximation (Cl = 0) becomes valid. Instead of equation (1) one has in this case... 

In the previous two sections, we have presented the Breit-Pauli perturbation Hamiltonian for one- and two-electron relativistic corrections of order 1/c to the nonrelativistic Hamiltonian. But there is a problem for many-electron systems. For the perturbation theory to be valid, the reference wave function must be an eigenfunction of the zeroth-order Hamiltonian. If we take this to be the nonrelativistic Hamiltonian and the perturbation parameter to be 1/c, we do not have the exact solutions of the zeroth-order equation. 

If we wish to incorporate some level of relativistic effects into the zeroth-order Hamiltonian, we cannot start from Pauli perturbation theory or direct perturbation theory. But can we find an alternative expansion that contains relativistic corrections and is valid for all r that is, can we derive a regular expansion that is convergent for all reasonable values of the parameters The expansion we consider in this chapter has roots in the work by Chang, Pelissier, and Durand (1986) and HeuUy et al. (1986), which was developed further by van Lenthe et al. (1993, 1994). These last authors coined the term regular approximation because of the properties of the expansion. 

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