Spinor molecular

Lee and McLean have considered full relativistic all-electron solutions to the Dirac equation for AgH and AuH. In this method, four-component, allelectron spinors are obtained using a LCAS-MS (linear combination of atomic spinor—molecular spinor) method. These authors employ a Slater-type basis for AgH and AuH. However, such relativistic all-electron calculations do not seem to be practicable for molecules other than diatomic hydrides at present. 

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. 

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

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]. 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

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. 

Goulomb interaction integrals over molecular orbitals can be written as a sum of similar interaction integrals with G-spinor overlap densities and... 

Whilst this demonstrates that calculations using the methods of this paper may prove very useful in studies of molecules containing only low-Z atoms, a major objective has been to study systems containing heavier atoms. So far, only a limited number of molecular calculations have been carried out with BERTHA at the DHF level, mainly in connection with studies of hyperfine and PT-odd effects in heavy polar molecules such as YbF [33] and TIF [13]. The reader is referred to the literature for an assessment of these calculations and for technical details on the construction of basis sets which must not only describe molecular bonding properly but also give a good representation of spinors close to the heavy nuclei to handle the short-range electron-nuclear electroweak interactions. 

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. 

All the above restoration schemes are called nonvariational as compared to the variational one-center restoration (VOCR, see below) procedure proposed in [79, 80]. Proper behavior of the molecular orbitals (four-component spinors) in atomic cores of molecules can be restored in the scope of a variational procedure if the molecular pseudoorbitals (two-component pseudospinors) match correctly the original orbitals (large components of bispinors) in the valence region after the molecular RECP calculation. As is demonstrated in [69, 44], this condition is rather correct when the shape-consistent RECP is involved to the molecular calculation with explicitly... 

In the inner core region, the pseudospinors are smoothed, so that the electronic density with the pseudo-wave function is not correct. When operators describing properties of interest are heavily concentrated near or on nuclei, their mean values are strongly affected by the wave function in the inner region. The four-component molecular spinors must, therefore, be restored in the heavy-atom cores. 

All molecular spinors 4>p can be restored as one-center expansions in the cores using the nonvariational one-center restoration (NOCR) scheme [26, 27, 90, 92, 93, 94, 19, 97, 98] that consists of the following steps ... 

Finally, the atomic two-component pseudospinors in the molecular basis are replaced by equivalent four-component spinors and the expansion coefficients from Eq. (6.3) are preserved ... 

The molecular four-component spinors constructed this way are orthogonal to the inner core spinors of the atom, because the atomic basis functions used in Eq. (6.4) are generated with the inner core shells treated as frozen. 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

Two-step calculation of molecular properties. To evaluate one-electron core properties (hyperfine structure, P,T-odd effects etc.) employing the above restoraton schemes it is sufficient to obtain the one-particle density matrix, Dpq, after the molecular RECP calculation in the basis of pseudospinors p. At the same time, the matrix elements Wpq of a property operator W(x) should be calculated in the basis of equivalent four-component spinors p. The mean value for this operator can be then evaluated as ... 

The results obtained with the one-center expansion of the molecular spinors in the T1 core in either s p, s p d or s p d f partial waves are collected in Table 4. The first point to notice is the difference between spin-averaged SCF values and RCC-S values the latter include spin-orbit interaction effects. These effects increase X by 9% and decrease M by 21%. The RCC-S function can be written as a single determinant, and results may therefore be compared with DF values, even though the RCC-S function is not variational. The GRECP/RCC-S values of M indeed differ only by 1-3% from the corresponding DF values [89, 127] (see Table 4). 

M is calculated in Ref. [127] using two-center molecular spinors, corresponding to infinite... 

Representation theory of molecular point groups tells us how a rotation or a reflection of a molecule can be represented as an orthogonal transformation in 3D coordinate space. We can therefore easily determine the irreducible representation for the spatial part of the wave function. By contrast, a spin eigenfunction is not a function of the spatial coordinates. If we want to study the transformation properties of the spinors... 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

It has to be noted that the relation between the elements of 0(3)+ (also called SO(3), the group representing proper rotations in 3D coordinate space) and SU(2) (the special unitary group in two dimensions) is not a one-to-one correspondence. Rather, each R matches two matrices u. Molecular point groups including symmetry operations for spinors therefore exhibit two times as many elements as ordinary point groups and are dubbed double groups. 

We exemplify the procedure of determining the spinor transformation properties under molecular point group operations for the Czv double group. Other double groups can be treated analogously. The character tables of the 32 molecular double groups may be found, e.g., in Ref. 68. 

Calculated by a Configuration Interaction Method Using Determinants of Two-Component Molecular Spinors Test Calculations on Rn and T1H. 

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