Relativistic interaction

Mn is the mass of the nucleon, jis Planck s constant divided by 2ti, m. is the mass of the electron. This expression omits some temis such as those involving relativistic interactions, but captures the essential features for most condensed matter phases. 

The Born- Oppenheimer approximation. With spin-orbit and other relativistic interactions omitted, the Hamiltonian of a polyatomic molecule is... 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

S.V. Bulanov, F. Califano, G.I. Dudnikova et al. Relativistic interaction of laser pulses with plasmas. In Reviews of Plasma Physics, edited by V.D. Shafranov (Kluwer Academic/Consultants Bureau, New York 2001), vol. 

Moszynski R, Lach G, Jaszunski M, Bussery-Honvault B (2003) Longrange relativistic interactions in the Cowan-Griffin approximation and their QED retardation Application to helium, calcium, and cadmium dimers. Phys Rev A 68 052706... 

As discussed in section 2.8, relativistic effects on the valence electronic structure of atoms are dominated by spin-orbit splitting of (nl) states into (nlj) subshells, and stabilization of s- and p-states relative to d- and f-states. Here we examine consequences of relativistic interactions for cubo-octahedral metal-cluster complexes of the type [M6X8Xfi] where M=Mo, Nb, W and X=halogen, which have a well defined solution chemistry, and are building blocks for many interesting crystal structures. 

Higher order terms in the Born series, relevant to relativistic interactions, are seldom, but occasionally (cf. e.g., Refs. [2,17]) discussed. 

The relativistic interaction Hamiltonian of the interaction of a bound electron with the nucleus is given by... 

One important step is the choice of the gauge in which to express the two-body relativistic interaction. For the exact solution we know that the result should be gauge independent but this will not hold for approximate solutions (remember the length and velocity forms for dipole radiative transition probabilities). The final result for g(l,2) of Eq. (5) reads ... 

This is the origin of the effective relativistic interaction between electrons first derived heuristically by Breit [2]. The approximation of the total interaction energy derived from the Coulomb (74) and Breit (76) interactions by retaining terms up to 0 l/c ) in an expansion in powers of 1/c is the well-known... 

Breit-Pauli effective Hamiltonian [63, 39], [64, Appendix 4] which is regularly used to describe relativistic corrections to the familiar nonrelativistic theory of atoms and molecules. It is important to realize that this widely used perturbation expansion contains less physics than the simple relativistic interactions used here. 

To summarize, we have the somewhat puzzling situation that external magnetic fields, the origin of which we don t know, give rise to a non-relativistic interaction with moving electrons and with electron spin, while magnetic fields that are obviously created by moving electrons, enter only as relativistic effects, which are comparable in size to effects of relativistic kinematics, and should not be separated from the latter. 

The quasi-relativistic model obtained by using the ZORA ansatz in combination with a fully variational derivation is the infinite-order regular approximation (lORA) previously derived by Sadlej and Snijders [46] and by Dyall and Lenthe [47]. The lORA method has recently been implemented by Klop-per et al. [48]. The ZORA model can be obtained from the lORA equation by omitting the relativistic correction term to the metric. However, the indirect renormalization contribution is as significant as the relativistic interaction operator in the Hamiltonian. This is the reason why ZORA overestimates the... 

The 12 RP fragments cap alternately the Cu4 faces of the Cu24 polyhedron, resulting in fivefold-coordinated phosphorus atoms. This structure resembles that of the recently described [Cu24(NPh)i4] anionic cluster (40). The Cu-P and Si-P distances are unremarkable. The construction principle of parallel Cu layers to form a metal-like package has also been observed for other Cu clusters (41). The main reason for the different structures of CU2PR and Li2PR clusters is the covalent character of the Cu-P bond, with the additional involvement of favorable Cu-Cu interactions. The latter are probably due to relativistic interactions (dispersion-type of interaction) (42, 43). 

Section 3 is a theoretical discussion that covers the free-ion Hamiltonian, crystal-field theory, and intensities. We discuss the effect of interelectronic Coulomb interactions and the spin-orbit interaction in determining the structure of the free-ion levels. Several additional interactions that are weaker than the above shift the free-ion levels by a few hundred wavenumbers and are necessary to give good agreement between calculated free-ion levels and experiment. These include spin-independent configuration interaction by the Coulomb interaction, spin-dependent configuration interaction by the spin-orbit interaction, and relativistic interactions that couple orbital and spin angular momenta of different electrons. 

We now begin the study of molecular quantum mechanics. If we assume the nuclei and electrons to be point masses and neglect spin-orbit and other relativistic interactions (Sections 11.6 and 11.7), then the molecular Hamiltonian is... 

The use of the Coulomb (4.18) Breit (4.19) or Gaunt (4.21) interaction operators in combination with Dirac Hamiltonians causes that the approximate relativistic Hamiltonians does not have any bound states, and thus, becomes useless in the calculations of relativistic interactions energies. This feature is known under the name of the Brown-Ravenhall disease and is a consequence of the spectrum of the Dirac Hamiltonian [38],... 

B. Swirles, Proc. R. Soc. London A, 157, 680 (1936). The Relativistic Interaction of Two Electrons in the Self-Consistent Field Method. 

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

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