Gaunt operator

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... 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

For the computational investigation of molecular systems containing heavy atoms, such as transition metals, lanthanides, and actinides, we could neglect neither relativity nor electron correlation. Relativistic effects, both spin-free and spin-orbit, increase with the nuclear charge of atoms. Therefore, instead of the nonrelativistic Schrodinger equation, we must start with the Dirac equation, which has four-component solutions. For many-electron systems, the four-component Hamiltonian is constructed from the one-electron Dirac operator with an approximated relativistic two-electron operator, such as the Coulomb, Breit, or Gaunt operator, within the nopair approximation. The four-component method is relativistically rigorous, which includes both spin-free and spin-orbit effects in a balanced way. However it requires much computational time since it contains more variational parameters than the approximated, one or two-component method. 

The simplest example of the Wigner—Eckart theorem is given by the Gaunt integral over three spherical harmonics, which is the matrix element for the transition between eigenstates m) and fm ) of a single orbital angular momentum observable due to a tensor operator Tj. We prefer to use the renormalised tensor operator C, which simplifies the expression. 

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. 

To avoid the difficult integration over the third term this operator is often further approximated to the Gaunt [6,7]... 

The transformation of AE2 to Breit-Pauli form has been known for a long time [81, 82]. (There is a small difference in the operators published by Bethe-Salpeter [81] and Itoh [82], but this does not matter, since it has no effect on the expectation value with respect to a nonrelativistic reference function). A detailed presentation of this reformulation, which is unexpectedly tedious, has been given recently [83]. The result for the Gaunt interaction is ... 

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 ... 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

Figure 4. Total SCF density difference with and without the Gaunt contribution at the T1 (left) and hydrogen (right) center for the equilibrium geometry of 186.8 pm. Positive values indicate a depletion of density after inclusion of the Gaunt operator. Units are given in a.u.

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

In the context of Breit operator (4.19) one should also mention its approximate form known as the Gaunt interaction V ... 

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],... 

The indices i,J denote electrons, whereas X,fi stand for nuclei with charges and Z j. Relativistic, quasirelativistic or nonrelativistic expressions may be inserted into this Hamiltonian for the one- and two-electron operators h and g, respectively. In some cases, e.g. the relativistic all-electron Dirac-CouIomb-(Breit/Gaunt) Hamiltonian, it is necessary to (formally) bracket the Hamiltonian by projection operators onto many-electron (positive energy) states in order to avoid problems connected with unwanted many-electron-positron (negative energy) states. 

As in the case of the unretarded Gaunt interaction, we choose the velocity operator expression r ca (instead of the momentum operator) to obtain consistent results as before and obtain the interaction energy of Eq. (3.252) promoted to the operator calculus of quantum theory. 

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 the imprinted form of the Breit interaction plus an independent exponential retardation factor. Grant and co-workers [213] refer in this context to earlier work by Brown [214] who essentially embeds the Rosenfeld arguments into the QED language with the final result being the Coulomb plus Gaunt operators times the retardation factor as derived in the previous section. 

As a final remark we may comment on the fact that we need to study the two-electron problem in the attractive external potential of an atomic nucleus, hence, as a bound-state problem. It is immediately seen that this affects, for instance, the expansion in terms of zeroth-order state functions in Eq. (8.22), where bound states of the one-electron problem rather than free-particle states become the basis for the construction of the wave function (and wave-function operators in second quantization). The situation is, however, more delicate than one might think and reference is usually made to the discussion of this issue provided by Furry [216] (Furry picture). Of course, it is of fundamental importance to the QED basis of quantum chemistry. However, as a truly second-quantized QED approach, we abandon it in our semi-classical picture and refer to Schweber for more details [165, p. 566]. Instead, we may adopt from this section only the possibility to include either the Gaunt or the Breit operators in a first-quantized many-particle Hamiltonian. 

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