Relativistic shape-consistent

Conventional shape-consistent effective potentials (67-70), whether relativistic or not, are typically formulated as expansions of local potentials, U (r), multiplied by angular projection cperators. The expansions are tnmcated after the lowest angular function not contained in the core. The last (residual) term in the expansion typically represents little more than the simple ooulombic interaction between a valence electron and the core (electrons and corresponding fraction of the nuclear charge) and is predominantly attractive. The lower A terms, on the other hand., include strongly... 

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]. 

DFT-Based Pseudopotentials. - The model potentials and shape-consistent pseudopotentials as introduced in the previous two sections can be characterized by a Hartree-Fock/Dirac-Hartree-Fock modelling of core-valence interactions and relativistic effects. Now, Hartree-Fock has never been popular in solid-state theory - the method of choice always was density-functional theory (DFT). With the advent of gradient-corrected exchange-correlation functionals, DFT has found a wide application also in molecular physics and quantum chemistry. The question seems natural, therefore Why not base pseudopotentials on DFT rather than HF theory ... 

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

Shape consistent relativistic effective core potentials... 

Ah initio relativistic effective core potentials can be derived from the relativi.stic all-electron Dirac-Fock solution of the atom the.se potentials are called the relativistic effective core potentials (RECP). and have been extensively used by several investigators to study the electronic structure of polyatomics containing very heavy atoms. The shape-consistent RECP method formulated by Christiansen, Lee, and Pitzer differs from the Phillips-Kleinman method in the representation of the nodeless pseudo-orbital in the inner region. The one-electron valence equation in an effective potential Vq of the core electrons can be expressed as... 

One further distinguishes ECPs by the kind of their adjustment, i.e., energy-consistent PPs (see Section 6.3.1) and shape-consistent PPs/MPs (see Section 6.3.2). Furthermore, ECPs are categorized by the size of their core, e.g., one differs between f-in-valence small-core [21,22] and f-in-core large-core PPs (LPP) [7-12] for the f elements. Finally, the accuracy of the underlying AE reference data determines the ECP type, e.g., for early actinides scalar-relativistic Wood-Boring (WB) [22] or relativistic multiconfiguration Dirac-Hartree-Fock (MCDHF) [23,24] small-core PPs (SPP) are available. 

Table 6.3 Shape-consistent relativistic PPs and valence basis sets for lanthanides and actinides...

There have been a number of basis sets for lanthanide and actinide elements previously reported in the literature that are based on relativistic effective core (ECP) potentials, or pseudopotentials (PP). These can be most easily categorized by the type of underlying ECP used (a) shape consistent pseudopotentials, (b) energy consistent pseudopotentials, and (c) model potentials. 

We have compared results from Fig. 17 with those of several other workers and found good agreement. Bhandari states that a vertical line moving at high speed assumes the shape of a hyperbola [21], Mathews and Lakshmanan criticize the concept of relativistic rotation and introduce the train paradox [22], When a fast-moving train is studied, should one imagine each boxcar to be rotated or the train as a whole rotated What happens to the stationary rails Finally, they conclude that the rotated appearance is not self consistent. We agree with this statement. The train is easily visualized in our Fig. 17b as one of the horizontal rows of deformed squares. From this, it is obvious that the distortion of the total train cannot be explained solely by rotation. 

Thus, at the modern level of the relativistic electronic structure theory, the problem of defining ground states of elements heavier than 122 remains. Very accurate correlated calculations of the ground states with inclusion of the quantum electrodynamic (QED) effects at the self-consistent field (SCF) level are needed in order to reliably predict the future shape of the Periodic Table. At the time of writing, an accepted version of the Table is that of Fig. 1, with the superactinides comprising elements Z = 122 through 155 as suggested in [1, 2]. 

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