Gaussian spinors

Appendix A Cartesian Gaussian Spinors and Basis Functions... 

The procedure for evaluating the matrix elements of the P,T-odd interaction operator H,/ with Cartesian Gaussian spinors is a bit complicated and different... 

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

Canonical ensemble, multiparticle collision dynamics, single-particle friction and diffusion, 116-118 Cartesian Gaussian spinors ... 

The basis consisted of 21sl7plld7/ Gaussian spinors [52], and the 4spd/5spd6s electrons were correlated. Table 1 shows the nonrelativistic, Dirac-Coulomb,... 

The basis consisted of 21sl7plld7/ Gaussian spinors [67], and correlated shells included 4 spdf5spd6s. Table 11 shows the nonrelativistic, Dirac-Coulomb, and Dirac-Coulomb-Breit total energies of the two ions. As expected, relativistic effects are very large, over 1100 haxtree. The nonadditivity of relativistic and correlation corrections to the energy, apparent in Table 11, has been noted above. 

The use of Gaussian spinors in relativistic electronic structure calculations the effect of the boundary of the finite nucleus of uniform proton charge distribution. Chem. Phys., 225 (1997) 239-246. 

Abstract Hilbert space, 426 Accuracy of computed root, 78 Acharga, R., 498,539,560 Additive Gaussian noise channel, 242 Adjoint spinor transformation under Lorentz transformation, 533 Admissible wave function, 552 Aitkin s method, 79 Akhiezer, A., 723 Algebra, Clifford, 520 Algebraic problem, 52 linear, 53... 

Ishikawa and coworkers [15,24] have shown that G-spinors, with orbitals spanned in Gaussian-type functions (GIF) chosen according to (14), satisfy kinetic balance for finite c values if the nucleus is modeled as a uniformly-charged sphere. 

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. 

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]. 

The l/REP(r), U ARKP(r), and terms At/f EP(r) in 11s0 of Eqs. (23), (31), and (34) or Eq. (6), respectively, are derived in the form of numerical functions consistent with the large components of Dirac spinors as calculated using the Dirac-Fock program of Desclaux (27). These operators have been used in their numerical form in applications to diatomic systems where basis sets of Slater-type functions are employed (39,42,43). It is often more convenient to represent the operators as expansions in exponential or Gaussian functions (32). The general form of an expansion involving M terms is... 

In this work the Kohn-Sham orbitals are expressed in terms of a set of gaussian-orbital functions (fj(r)) and spinors Recording to ... 

Figure 1 shows four snapshots from the (numerically calculated) time-evolution -0(t) of the initial function -tpoix) = iVexp(—x /2)(l,l). This is a spinor with Gaussian initial functions in both components. More precisely, the pictures show (according to our interpretation) the position probability density 4> x)f + 1 2 x) p. We see that the shape of the wave packet at later times shows strange distortions, similar to distortions caused by interference phenomena. Moreover, consider the expectation value of the position x in the state which is (ac-... 

The subspace of positive energies does not contain strictly localized spinors. There is no wave function that vanishes everywhere in an open region of space. All positive-energy wave packets are essentially spread over all of space. Still, there are wave packets which are approximately localized in the same sense as a Gaussian wave packet, i.e., they vanish faster than any inverse power of x, as x goes to infinity. Examples of such Gaussian-type wave packets are in Figures 3 and 4. 

In molecular calculations it is convenient to express G-spinors in terms of Spherical Gaussian-type Functions (SGTF) [107]... 

The electrostatic interactions in a molecule are determined by the structure of the density matrix D. In constructing D from an atomic orbital or atomic spinor basis, we incorporate a lot of redundant information in the Gaussian overlap charge distributions, since most of the electron density is concentrated near the nuclei. One should therefore try to transform D into a block diagonal form in which each dense block corresponds to a one-centre density so that the... 

Generally contracted Gaussian-type spinors and kinetic balance... 

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... 

We now discuss how at the DKee level of theory the transformed Hartree potential can be conveniently evaluated also for a density constructed from four-component Dirac spinors see Eqs. (6) and (26). The fitting-function (FF) technique, e.g. with an auxiliary Gaussian basis / [20,32],... 

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