Spin, spinor

Since the spinors formed from the two spin-orbit components have the same >, we may take a linear combination of them. One way of combining them results in pure-spin spinors ... 

These combinations are now either totally bonding or totally antibonding, thanks to the fact that one atomic spinor has two components of different sign while the other has the same sign for the two components. The molecular spinors are not symmetric the a part is skewed towards one atom and the ir towards the other. Linear combinations of these spinors will return pure-spin spinors as before. The numerical factors for the a and n orbitals have not been included in these spinors because they have no precise meaning here. 

Spin waves, 753 Spin wave states, 757 Spin zero particles, 498 "Spinor space, 428 Spinors, 394,428, 524 column, 524 Dirac... 

This is the most general expression obtained from a set of natural spin orbitals written in spinor form as... 

The purpose of spin-polarized DFT is again to describe the system (molecule, cluster,...) with an auxiliary noninteracting system of one-particle spinors xJ/j,. ..,v]/N. The ground state density matrix of this noninteracting system... 

Spin-dependent operators are now introduced. The external potential can be an operator Vext acting on the two-component spinors. The exchange-correlation potential is defined as in Eq. [27], although Exc is now a functional Exc = Exc[pap] of the spin-density matrix. The exchange-correlation potential is then... 

In most cases of interest, the spin density is collinear that is, the direction of the magnetization density m(7) is the same over the space occupied by the system it is customary to identify this as the -direction. The Hamiltonian is then diagonal if the external potential is diagonal, which allows one to decouple the spin f and spin J, components of the spinors and to obtain two... 

It exhibits a complicated spin-dependent structure arising from the Dirac four spinor, while it is reduced to a simple form,... 

This non-relativistic equation in terms of four-component spinors has been studied in detail by Levy-Leblond [44,45], who has shown that it results automatically from a study of the irreducible representations of the Gahlei group and that it gives a correct description of spin. It is easy to see that in the absence of an external magnetic field, equation (63) is equivalent to the Schrodinger equation in the sense that after elimination of the small component ... 

Table I also contains an analysis of the orbital character of these five energy levels. These were determined from the four-component spinors by neglecting the two lower, "small," components, and by assuming that the radial functions depend only upon , i.e. that the radial functions for pi/2 and p3/2> or for da/2 and ds/2> are the same. The orbitals may then be written in "Pauli" form as products of (complex) spherical harmonics and spin functions. Populations are equal to the squares of the absolute magnitudes of the coefficients listed in Table I. [For all but 17e3g, an additional orbital (not shown) is occupied which has the same energy but the opposite spin pattern (i.e. a and 3 are interchanged).]...

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

The spin density should follow from the density matrix (38), which includes the spin variables. As in (42), Qa(x x ) will be a sum of terms containing the various spinor components, summed over all spin-orbitals in the natural expansion. A typical term will be... 

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. 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 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). 

The theory so far is incomplete, however, because it has two SU(2) algebras that both act on the same Fermi spinor fields, and only one Higgs mechanism is used to compute the vacuum expectations for both fields. To improve the theory, consider that each SU(2) acts on separate spinor field doublets and that there are two Higgs fields that compute separate physical vacua for each SU(2) sector independently. The Higgs fields will give 2x2 vacuum diagonal expectations. If two entries in each of these matrices are equal, the resulting massive fermions in each of the two spinor doublets are identical. If the spin in one doublet... 

N. Cufaro-Petroni, P. Gueret, and J. P. Vigier, Second-order wave equation for spin /2 Fields 8-Spinors and canonical formulation, Found. Phys. 18(11) (1988). 

Spin is typically treated as a quantum phenomenon an easily accesible and readable account is given by Ohanian [107]. However, the possibility that spin may be a phenomenon with classical overtones has been a recurrent one [79,107-111,122-124], The connection between the classical polarization of light and quantum mechanics was noted long ago by Fano [125], while the connection between polarization and Clifford algebra for spinors was noted more recently [126], Finally, some philosophers have suggested that spin is a mere property of space [127],... 

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