Dirac spin matrices

Separability can be exploited even with admission of relativistic effects, by using the standard density matrix formalism with a simple extension to admit 4-component Dirac spin-orbitals this opens up the possibility of performing ab initio calculations, with extensive d, on systems containing heavy atoms. 

For the Dirac bispinor, the irreducible representation matrix Dab for each helicity component is a Pauli spin matrix a multiplied by ti/2. Then... 

The first term which is independent of nuclear spin dominates by several orders of magnitude in heavy elements over the second, nuclear spin dependent, term. In the above expression a and T are the standard Dirac matrices and o is the Pauli spin matrix (see e.g. Sakurait78]) stands for any nucleon either proton or neutron and the coupling constants C take on the following form in lowest order in... 

Show that the Fock-Dirac density matrix (5.3.7), for one determinant of orthonormal spin-orbitals, has the fundamental property... 

Particles spin Vz, 517 Dirac equation, 517 spin 1, mass 0,547 spin zero, 498 Partition function, 471 grand, 476 Parzen, E., 119,168 Pauli spin matrices, 730 PavM, W., 520,539,562,664 Payoff, 308 function, 309 discontinuous, 310 matrix, 309... 

We then turn to the question of how to eliminate the spin-orbit interaction in four-component relativistic calculations. This allows the assessment of spin-orbit effects on molecular properties within the framework of a single theory. In a previous publication [13], we have shown how the spin-orbit interaction can be eliminated in four-component relativistic calculations of spectroscopic properties by deleting the quaternion imaginary parts of matrix representations of the quaternion modified Dirac equation. We show in this chapter how the application of the same procedure to second-order electric properties takes out spin-forbidden transitions in the spectrum of the mercury atom. Second-order magnetic properties require more care since the straightforward application of the above procedure will extinguish all spin interactions. After careful analysis on how to proceed we... 

Bold quantities are operators, vectors, matrices or tensors. Plain symbols are scalars. a Polarizability a, P Spin functions a, p Dirac 4x4 spin matrices ap-jS Summation indices for basis functions F Fock operator or Fock matrix Fy, Eajd Fock matrix element in MO and AO basis Y Second hyperpolarizability yk Density matrix of order k gc Electronic g-factor... 

Using the spinor form (2.2) we can go from the spin orbital form (2.5) of the Fock-Dirac matrix to the orbital form ... 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

The symmetry properties of the effective Hamiltonian (2.25) or (2.27) are determined by those of the Fock-Dirac matrix. If all the occupied spin orbitals are spin rotated, the orbital part of the Fock-Dirac matrix is transformed as follows ... 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

The operator a0 is identity. Because of the anticommutation relations, the Dirac operators cannot be multiplicative operators (numbers). They are not differential operators either because of the independence of px,Py,pz,Po,x,y,z,t. But what variable (degree of freedom) do the Dirac operators act upon In the chapter dealing with the electron spin we saw that there are the Pauli matrix operators (which obey the idempotency and anticommutation) acting on a two-component wave function (the two-component spinor)... 

---