Shape-consistent effective potentials

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... 

Figure 1. Radial plots of the s and p terms of a Li shape consistent effective potential. (Data are from ref. 68.)...

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

Each ECP is unique in the way it is developed, and generally the method used to construct effective core potentials is either the shape-consistent method or the energy-adjusted extraction method. The former method defines the ECPs by solving an eigenvalue problem from the all-electron reference calculation, while the latter involves constructing ECPs so that they reproduce observables. The LANL2, SBKJC, and CRENBL ECPs are all deemed shape-consistent, while the SDB ECP is energy-adjusted. ... 

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... 

The problem lies in the assumption required to derive the selection rule that the u=l and u=0 surfaces are the same shape and are merely displaced vertically as we have illustrated in Fig. 2. For HF HF on the contrary, the intermolecular potential is highly anisotropic and rotational excitation of the fragments results in an effective potential which is shallow and may actually cross other surfaces. This has been demonstrated in calculations of Halberstadt et al.. The surfaces taken from their work are shown in Fig. 4. The curve crossing yields relaxation times orders of magnitude more efficient than those calculated by our selection rule. It is a challenge to the theorists to model the predissociation process, consistent with experiment, that allows both HF molecules to rotate on fragmentation. Clearly anisotropic effects will play an important role in understanding vibrational predissociation in other systems as well-for example, in the electronically excited state of OH Ar by Lester et al.. ... 

The field shape consists of two symmetric half-cycles with zero field amplitude at the pulse center. The upper panel of Fig. 5.13 shows the effective potential curves Vf R,t) obtained under the control field (to = 51 fs). The plotted curves at t = 44 fs represent the maximum shifts of the PECs in the first half-cycle of the control pulse, and those at t = 58 fs represent the maximum shifts in the later half-cycle of the control pulse. In the first half-cycle of the pulse, the control field, through the positive transition amplitude function /xn(7 ), acts to shift the Vn R) potential curve upwards, and thus the dynamical crossing position Rx t) is first shifted to the left. It reaches the leftmost position R ett at the height of the control field, and then is shifted right and restored to 7 cross by t = to- The lower panel of Fig. 5.13 shows the resulting time-dependence of Rx(t). The fact that the... 

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... 

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. 

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