Four-component molecular spinors

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. 

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]. 

In principle, all four-component molecular electronic structure codes work like their nonrelativistic relatives. This is, of course, due to the formal similarity of the theories where one-electron Schrbdinger operators are replaced by four-component Dirac operators enforcing a four-component spinor basis. Obviously, the spin symmetry must be treated in a different way, i.e. it is replaced by the time-reversal symmetry being the basis of Kramers theorem. Point group symmetry is replaced by the theory of double groups, since spatial and spin coordinates cannot be treated separately. 

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. 

In our four-component molecular approach, thus, we use spin-coupled, kinetically balanced, generally contracted Gaussian-type spinors (GTSs) as basis functions. The basis expansion is... 

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, ... 

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. 

For the relativistic description of molecules this means that each molecular spinor ipi r) entering a Slater determinant is to be expanded in a set of four-component atomic spinors (r). 

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]. 

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... 

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. 

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 ... 

In the four-component case, a one-electron molecular function is not a scalar function, but a four-component vector called molecular spinor. 

On the other hand, in the pioneering DHF and post-DHF program package MOLFDIR [3] and the well-developed four-component relativistic program package DIRAC [4], the molecular four-component spinors are expanded into decoupled... 

The expansion of four-component one-electron functions into a set of global basis functions can be done in several ways independent of the particular choice of the type of the basis functions. For instance, four independent expansions may be used for the four components. However, we might also relate the expansion coefficients of the four components to each other. In contrast to these expansions, the molecular spinors can also be expressed in terms of 2-spinor expansions. This latter ansatz appears to be quite common and will be described in greater detail now. Obviously, analogous thoughts also apply for the first two possibilities (for a detailed discussion compare Dyall etal. 1991a). 

We have recently developed an efficient computational scheme for the four-component method that employs four-component contraction for molecular basis spinors and the new atomic spinor (AS) integral algorithm [130-132]. In the following sections we will briefly introduce our new relativistic scheme. 

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. 

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