Pauli Approximation / Equation

Section III is devoted to illustrating the first theoretical tool under discussion in this review, the GME derived from the Liouville equation, classical or quantum, through the contraction over the irrelevant degrees of freedom. In Section III.A we illustrate Zwanzig s projection method. Then, in Section III.B, we show how to use this method to derive a GME from Anderson s tight binding Hamiltonian The second-order approximation yields the Pauli master equation. This proves that the adoption of GME derived from a Hamiltonian picture requires, in principle, an infinite-order treatment. The case of a vanishing diffusion coefficient must be considered as a case of anomalous diffusion, and the second-order treatment is compatible only with the condition of ordinary... 

In conclusion, in this section we have proved that the Markov approximation requires some caution. The Markov approximation may be incompatible with the quantum mechanical nature of the system under study. It leads to the Pauli master equation, and thus it is compatible with the classical picture of a particle randomly jumping from one site to another, a property conflicting, however, with the rigorous quantum mechanical treatment, which yields Anderson localization. 

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... 

Table 3.1 Effective Nuclear Charge (Scaling Parameter) Z ff for Approximate Spin-Orbit Interaction Calculations Using the One-Electron Term in the Breit-Pauli Hamiltonian (Equation 3.8, developed by Koseki et...

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... 

The Pauli approximation may be used in conjunction with this method by neglecting the small component spinors Q) of the Dirac equation, leading to RECPs expressed in terms of two-component spinors. The use of a nonrelativistic kinetic energy operator for the valence region, and two-component spinors leads to Hartree-Fock-like expressions for the pseudoorbitals. Note that the V s (effective potentials) in this expression are not the same for pseudo-orbitals of different symmetry. Thus the RECPs are expressed as products of angular projectors and radial functions. In the Dirac-Fock approximation, the orbitals with different total j quantum numbers, but which have the same / values are not degenerate, and thus the potentials derived from the Dirac-Fock calculations would be y-dependent. Consequently, the RECPs can be expressed in terms of the /y-dependent radial potentials by equa-... 

The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

Because single-electron wave functions are approximate solutions to the Schroe-dinger equation, one would expect that a linear combination of them would be an approximate solution also. For more than a few basis functions, the number of possible lineal combinations can be very large. Fortunately, spin and the Pauli exclusion principle reduce this complexity. 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

Importantly, the value of the results gained in the present section is not limited to the application to actual systems. Eq. (4.2.11) for the GF in the Markov approximation and the development of the perturbation theory for the Pauli equation which describes many physical systems satisfactorily have a rather general character. An effective use of the approaches proposed could be exemplified by tackling the problem on the rates of transitions of a particle between locally bound subsystems. The description of the spectrum of the latter considered in Ref. 135 by means of quantum-mechanical GF can easily be reformulated in terms of the GF of the Pauli equation. 

The account of two-particle correlations in nuclear matter can be performed considering the two-particle Green function in ladder approximation. The solution of the corresponding Bethe-Salpeter equation taking into account mean-field and Pauli blocking terms is equivalent to the solution of the wave equation... 

The modification of the three and four-particle system due to the medium can be considered in cluster-mean field approximation. Describing the medium in quasi-particle approximation, a medium-modified Faddeev equation can be derived which was already solved for the case of three-particle bound states in [9], as well as for the case of four-particle bound states in [10]. Similar to the two-particle case, due to the Pauli blocking the bound state disappears at a given temperature and total momentum at the corresponding Mott density. 

However, it is more appropriate to provide theoretical justifications for such use. In this respect, first, we introduce the third category of decoupling of positive and negative states commonly known as the direct perturbation theory . This approach does not suffer from the singularity problems described previously. However, the four-component form of the Dirac equation remains intact. The new Hamiltonian requires identical computational effort as for the Dirac equation itself, hence it is not an attractive alternative to the Dirac equation. However, it is useful to assess the accuracy of approximate two-component forms derived from the Dirac equation such as Pauli Hamiltonian. Consider the transformation... 

Equation (1) is obtained by using an expansion in E/ 2c - Vc) on the Dirac Fock equation. This expansion is valid even for a singular Coulombic potential near the nucleus, hence the name regular approximation. This is in contrast with the Pauli method, which uses an expansion in (E — V)I2(. Everything is written in terms of the two component ZORA orbitals, instead of using the large and small component Dirac spinors. This is an extra approximation with respect to the original formalism. 

Here we have used the natural expansion (33), with spin-orbitals written in the form (29). The second term in (41), absent in a Pauli-type approximation, contains the correction arising from the use of a 4roomponent formulation it is of order (2tmoc) and is usually negligible except at singularities in the potential. As expected, for AT = 1, (41) reproduces the density obtained from a standard treatment of the Dirac equation but now there is no restriction on the particle number. 

The spin-orbit mean field (SOMF) operator (56-58) is used to approximate the Breit—Pauli two-electron SOC operator as an effective one-electron operator. Using second-order perturbation theory (59), one can end up with the working equations ... 

Equation (39) corresponds to the Boltzmann approximation of statistical physics (or the so-called Pauli equation). We shall discuss it in more detail in Section VII. 

This equation is, of course, well known and often called the Pauli equation. We recognize on the right-hand side the familiar gain and loss terms. The transition probabilities which appear in the Pauli equation correspond to the Born approximation for one-photon processes. For further reference let us summarize the main properties of this weakly coupled approximation. 

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