Hamiltonian Dirac-Coulomb-Gaunt

Figure 5. Relativistic effects on bond lengths and binding energies of group 4 tctrahydrides XH. The bond length contraction (in A) and bond destabilization (in eV) were obtained as the difference between relativistic Dirac-Hartree-Fock calculations based on the Dirac-Coulomb-Gaunt Hamiltonian and corresponding nonrelativistic Hartree-Fock calculations [28,29].

Figure 6. Influence of relativistic corrections to the electron-electron interaction on the bond length contraction and bond destabilization of the group 4 tetrahydrides XH. The percentage of results obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian wrt. those obtained with the Dirac-Coulomb (DC) Hamiltonian has been derived from Dirac-Hartree-Fock calculations [28,29].

Reference values for the various 2-component relativistic Hamiltonians are provided by the 4-component Dirac-Coulomb Hamiltonian, but we have also included orbital energies obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian. As already mentioned, the Gaunt term brings in spin-other-orbit (SOO) interaction. Since spin-orbit interaction induced by other electrons will oppose the one induced by nuclei we see from Table 3.3 that the spin-orbit splitting of orbital levels is overall reduced. However, one should note that the Gaunt term also modifies /2 levels. 

Most 4-component relativistic molecular calculations are based on the Dirac-Coulomb Hamiltonian corresponding to the choice g = Coulomb The Gaunt term of (173) has been written in a somewhat unusual manner. The speed of light has been inserted in the numerator which clearly displays that the Gaunt term has the form of a current-current interaction, contrary to the... 

For further details the reader is referred to, e.g., a review article by Kutzel-nigg [67]. The Gaunt- and Breit-interaction is often not treated variationally but rather by first-order perturbation theory after a variational treatment of the Dirac-Coulomb-Hamiltonian. The contribution of higher-order corrections such as the vaccuum polarization or self-energy of the electron can be derived from quantum electrodynamics (QED), but are usually neglected due to their negligible impact on chemical properties. 

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)... 

Here, the current density j r) is still required, but it is now uniquely determined by the electron density, j = j[p]. If magnetic interactions between the electrons are not included — that is, when the Dirac-Coulomb Hamiltonian is employed and the Breit or Gaunt terms are omitted — then also the DFT analog of the Coulomb interaction will reduce to a functional J[p] of the density only [396]. Moreover, it now becomes possible to set up theories in which not the full 4-current is used as the fundamental variable, but only those parts of it that are related to the total electron density and the spin density. 

The two-electron operator is given in the nuclear frame and not in the reference of either electron. The spin-orbit coupling due to the relative motion of elecrons therefore splits into two parts The total interaction is the coupling of the spin of a selected reference electron with the magnetic field induced by a second electron. The spin-same orbit (SSO) and spin-other orbit (SOO) contributions arise from the motion of the reference electron and the other electron, respectively, relative to the nuclear frame and are carried by the Coulomb and Gaunt terms, respectively. For most molecular application it suffices to include the Coulomb term only, thus defining the Dirac-Coulomb Hamiltonian, but for the accurate calculation of molecular spectra the Gaunt term should be included as well. 

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 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],... 

We will evaluate the numbers of integrals required for a calculation with the unmodified Dirac Hamiltonian, and compare them with the number of integrals required for a calculation with the spin-free modified Dirac Hamiltonian, and with the number required for a nonrelativistic calculation. The spin-free Hamiltonian is formed by summing all the spin-free terms defined above, but we will consider the Coulomb term and the Gaunt and Breit terms separately. For the purpose of this evaluation, we make the following assumptions and definitions ... 

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