Angular spinor component

The structure (113) is a consequence of the block structure of the Dirac operator (98). We note that the off-diagonal kinetic operator can be written 

Inserting numerical values for the Clebsch-Gordon coefficients gives 

It follows that the angular density distribution is independent of the sign of k, since 

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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 orbital angular momentum quantum numbers, = I and corresponding, respectively, to the large and to the small components of the Dirac spinor are equal to... 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

To construct the Dirac-Fock equations, it is assumed that the wave function for an atom having N electrons may be expressed as an antisymmetrized product of four-component Dirac spinors of the form shown in Eq. (9). For cases where a single antisymmetrized product is an eigenfunction of the total angular momentum operator J2, the JV-electron atomic wave function may be written... 

For spinors the time reversal operation is not just complex conjugation. To find the effect of the time reversal operator T on a general angular momentum state j, m) we note that T anticommutes with any cartesian component of the angular momentum operator ... 

Here P and Q are the radial large and small components of the wavefunction, the angular functions are 2-component spinors, the quantum number k = 2 - j) j + 1/2), -j < rrij < j, and the phase factor i is introduced for convenience in some atomic applications because it makes the radial Dirac equation real. 

In Eq. (1), the projection operators ljm> Dirac spinors, refers to the so-called residual RECP term, where L and J are taken as one larger than the largest angular momentum quantum numbers of the core electrons. These RECPs are calculated in numerical form, but are re-expressed as expansions in Gaussian-type orbitals (GTOs) to facilitate their use in molecular electronic structure codes that employ GTO basis sets for representing the valence electrons. 

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... 

The structure of the Dirac Hamiltonian in Eq. (6.26) demands an analysis of the action of the operator cr p on the two components of the 4-spinor given in Eq. (6.51). We rewrite the product (orbital angular momentum operator in analogy to the nonrelativistic case above. According to Eq. (4.93) we understand that a way must be found to create a vector product from position and linear momentum, r x p. Recalling Eq. (4.177), we may rewrite the operator product (cr p) as... 

Since the structure of the many-electron Dirac Hamiltonian operator for atoms allows us to separate radial and angular degrees of freedom (as in the case of the one-electron atoms solved analytically in chapter 6), the coordinates in the 4-spinor s components may be separated into radial and spherical spinor parts. 

Here, P and Q are the radial large and small components of the wave function. The factor of i has been introduced to make the radial components real. The angular functions are two-component spinors, that is, a product of angular and spin functions the spin variable r has been explicitly shown. 

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. 

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