Basis functions 2-spinor

Relativistic generalization for G-spinor basis functions of the well-known McMurchie-Davidson algorithm [3] for direct evaluation of interaction integrals. 

The basis functions of this operator are the two-component spinor variables. Guided by the two-dimensional Hermitian structure of the representations of the Poincare group, we may make the following identification between the spinor basis functions 4>a(a = 1,2) of this operator and the components ( , H )(k = 1,2, 3) of the electric and magnetic fields, in any particular Lorentz frame ... 

By incorporating these symmetries in the 4-spinor basis functions, as we have done in our BERTHA code [50-54], we can make substantial computational economies in computing interaction integrals. The angular stracture of Dirac 4-spinors described here is also exploited by the major computer package TSYM, which utilizes projection operators to construct relativistic molecular symmetry orbitals for double valued representations of point groups [77-79]. 

To solve the Kohn-Sham equation, a spinor basis that is composed of the direct product of the atomic basis functions (usually Gaussian-type basis functions) and the one-electron spin functions, aa), where a) is the atomic basis function and a) = la) and P) are the one-electron spin functions. The molecular orbitals can be expanded as linear combinations of the spinor basis functions,... 

M. S. KeUey, T. Shiozaki. Large-scale Dirac-Fock-Breit method using density fitting and 2-spinor basis functions. J. Chem. Phys., 138 (2013) 204113. 

In the Dacre and Elder method, the reduction in the integral number comes from the relations between the symmetry-equivalent atoms. Consider two symmetry-equivalent atoms, A and B, each of which has two s /2 2-spinor basis functions, Xk and xx. which we label ka and kb, and A,a and Xb. The relations between the one-electron potential energy integrals (for example) that do not follow fi om Hermitian conjugation are... 

Appendix A Cartesian Gaussian Spinors and Basis Functions... 

In terms of Cartesian Gaussian spinors, the basis functions can be defined as a linear combination of the following Gaussian spinors [57] ... 

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]. 

The application of G-spinor basis sets can be illustrated most conveniently by constructing the matrix operators needed for DCB calculations. The DCB equations can be derived from a variational principle along familiar nonrelativistic lines [7], [8, Chapter 3]. It has usually been assumed that the absence of a global lower bound to the Dirac spectrum invalidates this procedure it has now been established [16] that the upper spectrum has a lower bound when the trial functions lie in an appropriate domain. This theorem covers the variational derivation of G-spinor matrix DCB equations. Sucher s repeated assertions [17] that the DCB Hamiltonian is fatally diseased and that the operators must be surrounded with energy projection operators can be safely forgotten. 

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]. 

Equation (29) is an analog of Eq. (5) for the case of atomic basis functions represented as two-component spinors. 

The symmetry group of relativity theory tells the story. For the irreducible representations of the Poincare group (of special relativity) or the Einstein group (of general relativity) obey the algebra of quaternions. The basis functions of the quaternions, in turn, are two-component spinor variables [17]. 

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