Spherical harmonics spinor

Teichteil et al developed the angular part (spherical harmonic spinors) of the Pauli spinors as functions of the spatial spherical harmonics [43] ... 

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).]...

At the restoration stage, a one-center expansion in the spherical harmonics with numerical radial parts is most appropriate both for orbitals (spinors) and for the description of external interactions with respect to the core regions of a considered molecule. In the scope of the discussed two-step methods for the electronic structure calculation of a molecule, finite nucleus models and quantum electrodynamic terms including, in particular, two-electron Breit interaction may be taken into account without problems [67]. 

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]. 

Pfi denotes a projection operator on spinor spherical harmonics centred at the core k... 

In the case of valence band photoemission, the atomic model cannot directly be applied to the numerical estimation of the effect, but rather for a qualitative consideration only. For valence bands, the initial state is no longer described by a single spinor spherical harmonic as it was done in [32] but it can be expanded for a certain k value in a series of spherical harmonics [44] due to their completeness. This procedure will influence the values of the state multipoles and the dipole matrix elements in Eq. 5.6, but the general Eqs. 5.5 and 5.7 will remain unchanged. In particular, they should correcdy describe the dependence of MDAD on the angle of photon incidence. 

P K r) and 2 v( ) denote the radial parts for the upper and lower components, respectively. The corresponding angular parts consist of two-component spinor spherical harmonics... 

ELECTRONIC STRUCTURE CALCULATIONS with the projection operator onto spinor spherical harmonics... 

The advantage of the above form is that it can be fit into conventional non-rela-tivistic codes since the two-component spinor projections have been eliminated and we obtain simple Im > projections involving ordinary spherical harmonics. However, the averaging method eliminates the spin-orbit operator. Fortunately, the spin-orbit operator itself can be expressed in terms of RECPs as shown by Hafner and Schwarz (1978,1979) and Ermler et al. (1981). This form is shown below. 

The orthonormality of the spherical harmonics and of the spin eigenvectors psnts (with s = 1/2 and OTs = 1/2) ensures orthonormality of the Pauli spinors,... 

The orthonormality restriction for the spinors, Eq. (8.108), may be split into two parts in the case of atoms. The product ansatz for the spinor automatically yields orthonormal angular parts (coupled spherical harmonics cf. chapter 9). But these do not contain information about the principal quantum numbers in the composite indices i and j. For this reason, the restriction to orthonormal spinors results in the orthonormality restriction for radial functions... 

As discussed in the chapter on symmetry (chapter 6), neither orbital nor spin angular momentum provide good quantum numbers for the Dirac equation in a central field, and we must instead turn to eigenfunctions of the operators and with eigenvalues j j -1-1) and nij. For a one-electron wave function the angular momentum part can be expressed in a basis of coupled products of a spherical harmonic and a Pauli spinor Ti(mj)... 

As well as being eigenfunctions of the operators j, and K, the two-spinor angular functions are eigenfunctions of the inversion operator I with eigenvalue (—1). This follows directly from the inversion properties of the spherical harmonics. Because the I value of the spherical harmonics in the angular function for the small component differs from that of the large component by 1, the small component has the opposite parity under inversion. This fact was demonstrated in chapter 6. 

We can insert the expressions for the Clebsch-Gordan coefficients into the angular 2-spinors (and suppress the angular variables in the spherical harmonics) to get... 

Another feature that emerges from these plots is the loss of nodal structure. Because the spin-up and spin-down components of each spinor have nodes in different places, the directional properties of the angular functions are smeared out compared with the properties of the nonrelativistic angular functions. Only for the highest m value does the spinor retain the nodal structure of the nonrelativistic angular function, and that is because it is a simple product of a spin function and a spherical harmonic. The admixture of me and me + I character approaches equality as I increases and as me approaches zero, resulting in a loss of spatial directionality. The implications of this loss of directionality for molecular structure could be significant, particularly where the structure is not determined simply from the molecular symmetry or from electrostatics. 

The decision of whether to work with 2-spinors or a scalar spin-orbital basis must be made at an early stage of computer program construction because it affects all stages of the SCF process evaluation of the integrals, construction of the Fock matrix, and solution of the SCF equations. However, at each stage, the scalar spin-orbital basis can be transformed to the 2-spinor basis. Transformation of the integrals to a 2-spinor basis is not particularly difficult it is similar in principle to the transformation from Cartesians to spherical harmonics. Some efforts have been made to develop new algorithms in which these transformations are incorporated, and RKB is implemented from the start in the 2-spinor basis (Quiney et al. 1999, 2002, Yanai et al. 2002). 

In the DIRAC program, molecular spinors are expressed as a sum of regular spherical harmonics (Rif). [33] As an example the p functions are expressed as follows ... 

Expressions (4.3) and (4.4) conform to the symmetry of equation (4.1). In the following sections we give the Mulliken gross atomic populations (GAOP) [34] of the respective molecular spinors in terms of the spherical harmonics. 

Here, the projection operator P[j is set up with spinor spherical harmonics ljm, I)... 

Here, v and oc are the principle quantum numbers for valence and outer core orbitals, respectively. The maximum L and J quantum numbers are related by / = L -f 1/2. Aj is the projector on the spinor spherical harmonics ljm)... 

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