The Pauli Approximation

One of the purposes of this work is to make contact between relativistic corrections in quantum mechanics and the weakly relativistic limit of QED for this problem. In particular, we will check how performing plane-wave expectation values of the Breit hamiltonian in the Pauli approximation (only terms depending on c in atomic units) we obtain the proper semi-relativistic functional consistent in order ppl mc ), with the possibility of analyzing the separate contributions of terms with different physical meaning. Also the role of these terms compared to next order ones will be studied. 

Neglecting spin-orbit contributions (smaller than other relativistic corrections for the ground state of atoms, and zero for closed-shell ones), the Breit hamiltonian in the Pauli approximation [25] (weak relativistic systems) can be written for a many electron system as ... 

While using (4.14) and exact wave functions. This supports the conclusion of Drake [83] that for electric dipole transitions, by considering the commutator of with the atomic Hamiltonian in the Pauli approximation, we obtain Qwith relativistic corrections of order v2/c2 (see (4.18)-(4.20)). However, for many-electron atoms and ions, one has to use approximate (e.g., Hartree-Fock) wave functions, and then this term gives non-zero contribution, conditioned by the inaccuracy of the model adopted. 

The AREP has the advantage that it may be used in standard molecular calculations that are based on A-S coupling. The AREP may be interpreted as containing the relativistic effects included in the Dirac Hamiltonian, with the exception of spin-orbit coupling. This form is the same as that presented by Kahn et al. (33) which is based on the relativistic treatment of Cowan and Griffin (34). The Hamiltonian employed by Cowan and Griffin is based on the Pauli approximation to the Dirac Hamiltonian with the omission of the spin-orbit term. 

For the nucleus-electron interaction the Coulomb potential, either for a point nucleus or a finite nucleus, is used directly. The relativistic contributions to the interaction operator are obtained approximately, using the Pauli approximation. They are... 

Using the Pauli approximation of the functions g,f, one can find again the well-known coefficients (see [3], Chap. 64) but now deduced directly from an exact relativistic calculation. 

However, from a theoretical and also a practical point of view, several teachings may be deduced. In particular, with the retardation, the Pauli approximation of the Dirac theory is no longer in strict agreement with the Schrodinger theory, as it is the case with the dipole approximation, when for example two states pl/2 and p3/2 are considered as unified in a single state p. Such a feature has a nonnegligible incidence. 

Thus, this second approximation corresponds to the transformation of the system (10.1) into the system of the radial functions in the Pauli approximation for which, in (10.1a) the left hand side is replaced by (2/aa)f(r), (10.1b) remaining unchanged. 

The first approximation coincide with the Pauli approximation only for the discrete spectrum, but not for the continuum. One can expect that the first approximation has a weak incidence on the result, independently of the level of energy considered in the continuum, but the second one is directly related to the value of the number n in respect with Za and may lead to important differences for the weak values of n, that is, the high values of the energy. So in what follows, we mainly use the first approximation, the second one being devoted only to the verification of the results, by a passage to the well known nonrelativistic expressions (see [5], Sect. 71) of the matrix elements in the dipole approximation. 

We can deduce from relations established in Sect. 9.2 that a direct passage of the vectors T- -(k) of the transitions sl/2 — pl/2 and sl/2 — pl/2 to a vector T- -(k) of a transition s —p is not possible. In other words, one of the effect of the retardation is to break the possibility to find an equivalence between the Pauli approximation and the Schrodinger theory, and the reason lies on the incidence of the retardation on the spherical parts of the Dirac wave functions, related to the presence of the spin. The incidence is already sensible, in the transitions of the discrete spectrum (see (9.38), (9.39), (9.40)) and this incidence may be amplified in the contribution of the continuum, independently of the incidence of the chosen values for the radial functions. 

The historically first reduction of the Dirac equation to two-component form is the Pauli approximation, which can be obtained from Eq. (26) by trancating the series expansion for cu after the first two terms, and eliminating the energy dependence by means of a systematic expansion in c. The result is the familiar Pauli Hamiltonian... 

The above discussion leads to a natural approximation called the Pauli approximation to the Dirac equation which is tantamount to ignoring the small Q components. This is the method used in practice for most of the chemical problems. 

In the above method, one could introduce the Pauli approximation by neglecting the small Q component spinors of the Dirac equation. This leads to RECPs expressed as two-component spinors. The use of non-relativistic kinetic energy operator for the... 

From Eq. (4.171) we understand that Eq. (6.29) already features a separation into radial r and angular, cp) variables. All angular variables are contained in the operator product a 1), which is essentially the spin-orbit coupling operator known from the Pauli approximation. The remaining operators can all be expressed by the radial variable r alone. 

Historically, the first derivations of approximate relativistic operators of value in molecular science have become known as the Pauli approximation. Still, the best-known operators to capture relativistic corrections originate from those developments which provided well-known operators such as the spin-orbit or the mass-velocity or the Darwin operators. Not all of these operators are variationally stable, and therefore they can only be employed within the framework of perturbation theory. Nowadays, these difficulties have been overcome by, for instance, the Douglas-Kroll-Hess hierarchy of approximate Hamiltonians and the regular approximations to be introduced in a later section, so that operators such as the mass-velocity and Darwin terms are no... 

We have already noted in chapter 13 that the Pauli approximation produces a spin-orbit coupling operator that maybe employed in essentially one-component, i.e., nonrelativistic or scalar-relativistic, methods via perturbation theory. Of course, this is an approximation compared with fully fledged four-component methods, but it can be a very efficient one that requires less computational effort without significant loss of accuracy. 

We have already discussed in chapters 12 and 13 that low-order scalar-relativistic operators such as DKH2 or ZORA provide very efficient variational schemes, which comprise all effects for which the (non-variational) Pauli Hamiltonian could account for (as is clear from the derivations in chapters 11 and 13). It is for this reason that historically important scalar relativistic corrections which can only be considered perturbatively (such as the mass-velocity and Darwin terms in the Pauli approximation in section 13.1), are no longer needed and their significance fades away. There is also no further need to develop new pseudo-relativistic one- and two-electron operators. This is very beneficial in view of the desired comparability of computational studies. In other words, if there were very many pseudo-relativistic Hamiltonians available, computational studies with different operators of this sort on similar molecular systems would hardly be comparable. 

It is now clear that in the many-electron atoms relativity causes a different effect on the energies and wave functions of electrons with different quantum numbers. Actually, they are so complicated that they cannot be taken into account correctly by the Pauli approximation nor by any other perturbation theories. 

The relativistic correction to the potential is no more singular than the potential itself in this limit and therefore will support bound states. In the small momentum limit, when the electron is far from the nucleus, the potential goes as 1 /r and is therefore a short-range potential. It can be seen that the kinematic factors provide a cutoff to the potential that is absent in the Pauli approximation and that permits variational calculations with the free-particle Foldy-Wouthuysen transformed Hamiltonian. 

In the Pauli approximation, the Hamiltonian eorreet to second order is linear in the potential, and the separation of the zeroth-order Hamiltonian from the perturbation is... 

Here we are explicitly taking into account the fact that the metric has a relativistic correction, a feature that in the Pauli approximation was avoided with the consequence of the appearance of pathological operators. The metric in fact is a combination of projectors onto the large and small components. Using the definitions from (15.40),... 

Just as in the Pauli approximation, we will expand the normalization operator in a series. The normalized wave function is given by (17.7), and the normalization operator from (17.10) expressed in the new expansion parameter is... 

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