Spinors potential calculations

An important advantage of ECP basis sets is their ability to incorporate approximately the physical effects of relativistic core contraction and associated changes in screening on valence orbitals, by suitable adjustments of the radius of the effective core potential. Thus, the ECP valence atomic orbitals can approximately mimic those of a fully relativistic (spinor) atomic calculation, rather than the non-relativistic all-electron orbitals they are nominally serving to replace. The partial inclusion of relativistic effects is an important physical correction for heavier atoms, particularly of the second transition series and beyond. Thus, an ECP-like treatment of heavy atoms is necessary in the non-relativistic framework of standard electronic-structure packages, even if the reduction in number of... 

Lee and JcLean have carried out all-electron Dirac four-component spinor LCAS-MS SCF calculations on AgH and AuH. The relativistic effects increase the dissociation energies by 0.08 eV and 0.42 eV in these molecules, while the bond lengths contrast by 0.08 A and 0.25 A. These values for AuH confirm the earlier effective potential calculations. 

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

The DC-CASCI (12, 4) calculation was performed to construct reference functions. The active space includes the molecular spinors, which have atomic nature of 6s1/2, 6p1/2, 6py2 of T1 and 1 s1/2 of H, and two virtual molecular spinors. The DC-CASPT2 (10, 12, 110) calculation followed and this choice of active space provided smooth potential curves for four low-lying states of T1H at the DC-CASPT2 level. 

The diatomic ThO has been detected in the vapor phase over a mixture of Th and Th02 at high temperatures and a D of -9.00 eV, an ionizational potential of 6.0 eV and a R value of 1.8403 angstrom are estimated for ThO (18, 28-29). Ab initio DF SCF calculations in which each molecular spinor (MS) is expressed as a linear combination of (4-component) atomic spinors (LCAS) ( ) as well as the corresponding non-relativistic limit (NRL) calculations were performed for the ground state of ThO at five internuclear separations viz 3.077, 3.477, 3.877, 4.277 and 4.677 au. 

The Dirac-Fock one-centre method was the first approximation used for relativistic molecular structure calculations and is now only of historical importance. In this method the electron-electron interaction is handled exactly and the one-electron wave functions are four component Dirac spinors. On the other hand both the nuclear potentials and all the one-electron orbitals are expanded about a single common centre taken to be the position of the nucleus of the heaviest atom of the system under consideration. Because of this expansion, the method is restricted to hydrides XH and even for them the expansion is only slowly convergent. Nevertheless, experience gained with non-relativistic calculations has shown surprisingly good results for equilibrium distances of X-H bonds and for force constants. 

It is clear that relativistic molecular structure calculations have the potential to generate a large amount of data. One inherits all of the problems of non-relativistic quantum chemistry amplified by the need to accommodate the spinor structure and the inevitable involvement of components beyond the... 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

Orbital (spinor) energies from AREP-HF (REP-KRHF) calculations at the optimized geometries, and ionization potentials from photoelectron spectroscopy for methyl halides and carbon tetrahalides. Units are in eV, and the degeneracy number is set in parenthesis. 

A self-consistent scalar-relativistic (SR) version of the LCGTO-DF method has also been developed recently." "The SR variant employs a unitary second-order Douglas-Kroll-Hess (DKH) "" transformation for decoupling large and small components of the full four-component spinor solutions to the Dirac-Kohn-Sham equation. The approximate DKH transformation, very appropriate and efficient for molecular calculations, has been implemented this variant utilizes nuclear potential-based projectors and leaves the electron-electron interaction untransformed. 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

Pitzer and coworkers have carried out relativistic calculations on a number of diatomics such as Xe2, Xcj, TIH, Au2, Auj, PbS, PbSe", etc. These calculations were carried out with an LCAS-MS (linear combination of atomic spinor—molecular spinor) approach with the relativistic effective potentials. Many of these calculations were at the level of single-configuration SCF. In the earlier calculations, the spin-orbit coupling was ignored at the SCF stage and introduced using a semi-empirical procedure. 

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