Breit interaction operator

Gaunt and Breit interaction operators represent potential energies due to magnetic interactions, which one would also assume to play a role in the non-relativistic many-particle theory given by the Pauli equation, Eq. (5.140). One... 

Note that the subscript on the a matrices refers to the particle, and a here includes all of the tlx, tty and components in eq. (8.4). The first correction term in the square brackets is called the Gaunt interaction, and the whole term in the square brackets is the Breit interaction. The Dirac matiices appear since they represent the velocity operators in a relativistic description. The Gaunt term is a magnetic interaction (spin) while the other term represents a retardation effect. Eq. (8.27) is more often written in the form... 

In a rigorous treatment, one replaces the one-electron operator h by the four-component Dirac-operator hjj and perhaps supplement the two-electron operator by the Breit interaction term [15]. Great progress has been made in such four-component ab initio and DPT methods over the past decade. However, they are not yet used (or are not yet usable) in a routine way for larger molecules. 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

It is known that the Breit interaction can give contributions in excess of one thousand wave numbers even to energies of transitions between lowest-lying states of very heavy elements (see, e.g., tables 7 and 8). It is also clear that the point nuclear model becomes less appropriate when the nuclear charge is increased. Therefore, the RECPs designed for accurate calculations of actinide and SHE compounds should allow one to take into account the Breit interaction and the finite size of nuclei. The most economic way is to incorporate the corresponding contributions into the RECP operator. 

The inner core electrons occupy closed shells. The only exchange part of the two-electron Breit interaction between the valence, outer core and inner core electrons, Bf and P/c, gives non-zero contribution. The contributions from Bfy and P/c, are quite essential for calculation at the level of chemical accuracy (about 1 kcal/mol or 350 cm for transition energies). This accuracy level is, in general, determined by the possibilities of modern correlation methods and computers already for compounds of light elements. Note, that the contribution from the exchange interaction is not smaller than that from the Coulomb part [29]. The inner core electrons can be considered as frozen in most physical-chemical processes of interest. Therefore, the effective operators for P/ and P/c acting on the valence and... 

Zgp O.7, Zg 0.4, Zjg 0.3. Thus, Bcc>, Bcv, and B > contributions are negligible for the chemical accuracy of calculation. Therefore, the above made estimates provide us a good background for approximating the Breit interaction by a one-electron GRECP operator that should work well both for actinides and for superheavy elements. The numerical tests of the GRECPs accounting for the Breit effects are discussed in the next section. 

By inserting the equations defining the kinetic energy operators and the pairwise interaction operators into Eq. (8) we obtain the Dirac-Coulomb-Breit Hamiltonian, which is in chemistry usually considered the fully relativistic reference Hamiltonian. 

Note that the Breit-type operators are often neglected in quantum chemistry because they yield small energy contributions in comparison to the instantaneous Coulomb interaction. However, the effects may not be negligible in highly accurate quantum chemical calculations or for spin- or magnetic-field-dependent properties such as those measured by magnetic resonance spectroscopies. 

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

In contrast to the one-electron terms, the reduction of the 4x4 Dirac-Breit Hamiltonian to the 2x2 Breit-Pauli Hamiltonian is very tedious for the two-electron terms as each interaction term has to be transformed according to the Foldy-Wouthuysen protocol. As the derivation can be found for example in Refs. (56-58) and in detail in Ref. (21), we only present here the transformed terms and discuss their dimension. The two-electron Breit-Pauli operator gBP (i, j) reads... 

Breit-Pauli Operators Using General Cartesian Gaussian Functions. II. Two-Electron Interactions. 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

The electronic spin-orbit interaction operator, referred to as the Breit-Pauli spin-orbit Hamiltonian, is given by... 

It is seen that the Breit-Pauli operator has the structure of (2) for each atomic center, but depends explicitly on the distances of the unpaired electron from the atomic centers, defined in (6). While the magnetic interaction energy is and thus of shorter range than the electrostatic interaction, it can nevertheless result in a non-negligible dependence of the SO operator on the nuclear coordinates. This effect is neglected when the empirical SO operators (2) or (3) are employed. 

The electron-electron interaction is usually supposed to be well described by the instantaneous Coulomb interaction operator l/rn. Also, all interactions with the nuclei whose internal structure is not resolved, like electron-nucleus attraction and nucleus-nucleus repulsion, are supposed to be of this type. Of course, corrections to these approximations become important in certain cases where a high accuracy is sought, especially in computing the term values and transition probabilities of atomic spectroscopy. For example, the Breit correction to the electron-electron Coulomb interaction should not be neglected in fine-structure calculations and in the case of highly charged ions. However, in general, and particularly for standard chemical purposes, these corrections become less important. 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

For molecules the evaluation of the Breit correction to the Coulomb-type electron-electron interaction operator becomes computationally highly demanding and cannot be routinely evaluated, not even on the Dirac-Fock level. To test the significance of the Breit interaction, the Gaunt term is evaluated as a first-order perturbation. It turned out that it can be neglected in most cases as can be seen from the DF 4- Bmag calculations cited in Table 2.1. 

If a multiparticle system is considered and the election interaction is introduced, we may use the Dirac-Coulomb-Breit (DCB) Hamiltonian which is given by a sum of one-particle Dirac operators coupled by the Coulomb interaction 1 /r,7 and the Breit interaction Bij. Applying the Douglas-Kroll transformation to the DCB Hamiltonian, we arrive at the following operator (Hess 1997 Samzow and Hess 1991 Samzow et al. 1992), where an obvious shorthand notation for the indices pi has been used ... 

The interaction between the electrons is described by the Coulomb-Breit [5] operator. This operator is usually considered in the zero-frequency limit where it becomes... 

This singularity certainly deserves more studies. It does survive if one takes care of the Gaunt or the Breit interaction [84]. One possibile explanation for its appearance may be that the electron interaction operator, that we have used, is only correct to 0(c ), and that adding the term of 0(c ) may cancel the singularity. Unfortunately this missing term is unknown, and there is no obvious prescription how one should evaluate it vide infra). 

This expression can be rearranged [83] to (495). To this result, valid for the Dirac-Coulomb operator, the contribution of the Gaunt or Breit interaction has to be added ... 

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