Spinor components, molecular systems

The inclusion of relativistic effects is essential in quantum chemical studies of molecules containing heavy elements. A full relativistic calculation, i.e. based upon Quantum Electro Dynamics, is only feasible for the smallest systems. In the SCF approximation it involves the solution of the Dirac Fock equation. Due to the four component complex wave functions and the large number of basis functions needed to describe the small component Dirac spinors, these computations are much more demanding than the corresponding non-relativistic ones. This limits Dirac Fock calculations, which can be performed using e.g. the MOLFDIR package [1], to small molecular systems, UFe being a typical example, see e.g. [2]. 

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

The fully relativistic (four-component) LCAO calculations of molecular systems use contracted Gaussian-type spinors as the basis two scalar wavefunctions within a two-component basis spinor are multiplied by a common expansion coefficient, for dimensions n of both the large and small components the total number of variational parameters (the scalar expansion coefficients) is equal to 2n [496]. In the relativistic correlated calculations the atomic basis sets should be optimized in the atomic correlated calculations. As Almlof and Taylor showed [538], atomic basis sets optimized to describe correlations in atoms also describe correlation effects in molecules very well. The two main types of basis sets are used in correlation calculations of molecules basis of atomic natural orbitals (ANO) suggested by Ahnlof and Taylor [538] correlation-consistent (CC) basis set suggested by Dunning [462]. 

Abstract. BERTHA is a 4-component relativistic molecular structure program based on relativistic Gaussian (G-spinor) basis sets which is intended to make affordable studies of atomic and molecular electronic structure, particularly of systems containing high-Z elements. This paper reviews some of the novel technical features embodied in the code, and assesses its current status, its potential and its prospects. 

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 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. 

Aucar et al demonstrated, by means of a four-component relativistic calculation, that the origin of the diamagnetic contribution to any magnetic molecular property is due to contributions from positronic spinors in calculating the response of the system. Several approximations for the calculation of the DSO term were also investigated. As example, the DSO term for the chalcogen hydrides, XH2 (X = O, S, Se and Te), were calculated. 

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