Pauli form factor

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation... 

The interaction potential above generated by the Pauli form factor may be written in terms of the spin-orbit interaction... 

Calculation of the Pauli form factor contribution follows closely the one which was performed in order a Za), the only difference being that we have to employ the second order contribution to the Pauli form factor (see Fig. 3.3) calculated a long time ago in [26, 27, 28] (the result of the first calculation [26] turned out to be in error)... 

For calculation of the Pauli form factor contribution to the Lamb shift the third order contribution to the Pauli form factor (Fig. 3.5), calculated numerically in [33], and analytically in [34] is used... 

The interaction of the virtual photon with the electron on its mass shell can be parameterized by the so-called Dirac and Pauli form factors ... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

The photon-nucleus interaction vertex is described by the Dirac (Fi) and Pauli (T2) form factors... 

This is the two-electron spin-orbit interaction operator, and reduces to the Breit-Pauli form when the limit p 0 is applied in the kinematic factors. 

The wavefunction (1), antisymmetric under exchange of space-spin variables of the two electrons, was conveniently factorized into a product of space and spin factors anti-parallel coupling of the spins (antisymmetric spin factor) then implied symmetry of the spatial factor and a consequent enhancement of electron density in the bond region. Such a factorization is unique to a 2-electron system for an 7V-electron system, a Pauli-compatible function must have the form... 

It has been realized in recent years that the lanthanide contraction is only part of the explanation for the behavior of the heavier elements. An equally important factor is relativity. On a fundamental level, relativity actually plays an integral role in quantum theory, beginning with the space-time and momentum-energy symmetric which suggested the form of the time-dependent Schrixlinger equation [cf. Sei tion 2.3]. Electron spin and the Pauli exclusion principle are, in fact, implication ... 

The orbital <[>res is a resonant orbital, in which a hot electron is attached to form a temporary anion. We assume that desorption is triggered by single orbital resonance to ([>res through an extension to multi-resonant levels is straightforward. The term n) is the occupation of the electron in the ground state, and is required to keep the Pauli principle. The coefficient r eN is the reactive eN coupling factor and assumed to be... 

As before, xi (1) is used to indicate a fxmction that depends on the space and spin coordinates of the electron labelled 1. The factor l/ /M ensures that the wavefunction is normalised we shall see later why the normalisation factor has this particular value. This functional form of the wavefunction is called a Slater determinant and is the simplest form of an orbital wave-fxmction that satisfies the antisymmetry principle. The Slater determinant is a particularly convenient and concise way to represent the wavefunction due to the special properties of determinants. Exchanging any two rows of a determinant, a process which corresponds to exchanging two electrons, changes the sign of the determinant and therefore directly leads to the antisymmetry property. If any two rows of a determinant are identical, which would correspond to two electrons being assigned to the same spin orbital, then the determinant vanishes. This can be considered a manifestation of the Pauli principle, which states that no two electrons can have the same set of quantum numbers. The Pauli principle also leads to the notion that each spatial orbital can accommodate two electrons of opposite spins. 

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