Dirac-Coulomb- Hamiltonian

Fig. 1. BLYP/uncDZ mean dipole polarizability of the mercury atom as a function of frequency. All values in atomic units. SR+SO refers to calculations based on the Dirac-Coulomb Hamiltonians, whereas SR refers to calculations in which all spin-orbit interaction has been eliminated.

A natural generalization of Eq. (6) would be to choose the parameters in all one-electron small components of the two-electron wavefunction (30) to maximize E and then to choose the parameters in all one-electron large components to minimize E. However, in order to solve variationally the eigenvalue problem of the Dirac-Coulomb Hamiltonian, Kolakowska et al [12] advocated, on the basis of rather intuitive arguments, the following mle ... 

If we refrain from such a restriction and consider a spin-operator-dependent Hamiltonian, such as the 4-component KS Hamiltonian or the Dirac-Coulomb Hamiltonian, the Hamiltonian does not commute with the square of the spin operator. The square of the spin operator and the Hamiltonian then do not share the same set of eigenfunctions, and hence spin is no longer a good quantum number. In this noncollinear framework we must therefore find a different solution and may define a spin density equal to the magnetization vector (32). 

DIRAC-COULOMB HAMILTONIAN AND TWO-COMPONENT BASIS SPINORS... 

Within the Born-Oppenheimer approximation, the total electronic Dirac-Coulomb Hamiltonian is written as... 

The use of non-relativistic basis functions in (a) requires that the SO interaction can be considered as a relatively weak perturbation of the non-relativistic Hamiltonian, which typically is the case for second- and third-row atoms and transition metals. For systems with heavier atoms, two-component relativistic electronic basis functions should be employed or the analysis should be based on the four-component Dirac-Coulomb Hamiltonian. 

Thble3.6 Bond length Rc (A), vibrational constant coe (cm-1) and binding energy De (eV) of Eka-Au hydride (111)H without (with) counterpoise correction of the basis-set superposition error. All-electron (AE) values based on die Dirac-Coulomb-Hamiltonian (Seth and Schw-erdtfeger 2000) are compared with valence-only results obtained with energy-consistent (EC) (Dolg etal. 2001) and shape-consistent (SC) (Han and Hirao 2000) pseudopotentials (PP). The numbers 19 and 34 in parentheses denote the number of valence electrons for the Eka-Au PP. 

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... 

The simplest approximation is to combine the Dirac theory with the nrl of electrodynamics, which automatically leads to the Dirac Coulomb Hamiltonian... 

We now investigate the nrl and DPT for the Dirac Coulomb Hamiltonian. For the sake of simplicity we consider a two-electron system with the Dirac-Coulomb equation ... 

This is the leading relativistic correction of 0[c ) to the energy, based on the Dirac-Coulomb-Hamiltonian. We shall later see that there is another term of 0(c ) due to the Breit interaction. 

Forming the scalar product of (462) from the left by we get E4 for the Dirac-Coulomb Hamiltonian... 

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. 

The Douglas-Kroll transformation [40] of the Dirac-Coulomb Hamiltonian in its implementation by HeB [41-45] leads to one of the currently most successful and popular forms of a relativistic no-pair Hamiltonian. The one-electron terms of the Douglas-Kroll-HeB (DKH) Hamiltonian have the form... 

Shape-consistent pseudopotentials including spin-orbit operators based on Dirac-Hartree-Fock AE calculations using the Dirac-Coulomb Hamiltonian have been generated by Christiansen, Ermler and coworkers [161-170]. The potentials and corresponding valence basis sets are also available on the internet under http //www.clarkson.edu/ pac/reps.html. A similar, quite popular set for main group and transition elements based on scalar-relativistic Cowan-Griffin AE calculations was published by Hay and Wadt [171-175]. 

Figure 21. Valence spinors of 53I from average level multi-configuration calculations using the AE Dirac-Coulomb-Hamiltonian (solid lines) and a PP valence-only model Hamiltonian (dashed lines) [193],...

Visser O, Visscher L, Aerts P J C and Nieuwpoort W C 1992 Molecular open shell configuration interaction calculations using the Dirac-Coulomb Hamiltonian the f -manifold of an embedded cluster J. Chem. Phys. 96 2910-19... 

Relativistic treatments of A-electron systems usually start from the so-called Dirac-Coulomb hamiltonian, Hoc, where... 

To explain these differences the quantum electrodynamic corrections have to be implemented. The velocity of light is finite and this means retardation of the interparticle interactions. This means that the Dirac-Coulomb Hamiltonian has to be corrected by further expressions. 

One of the most important physical corrections to the Dirac-Coulomb Hamiltonian is the replacement of the nonrelativistic Coulomb repulsion, — in with a... 

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