Spin matrices four-component

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

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... 

The similarity of this formalism with that conventionally used in nonrela-tivistic quantum chemistry is obvious all we have done is to replace the usual spin-orbital basis by one comprising four-component spinors. Slater s rules for the construction of determinantal matrix elements are still applicable, so that any hamiltonian matrix element may be written in the form... 

Early four-component numerical calculations of parity-violating effects in diatomic molecules which contain only one heavy nucleus and which possess a Si/2 ground state have been performed by Kozlov in 1985 [149] within a semi-empirical framework. This approach takes advantage of the similarity between the matrix elements of the parity violating spin-dependent term e-nuci,2) equation (114)) and the matrix elements of the hyperfine interaction operator. Kozlov assumed the molecular orbital occupied by the unpaired electron to be essentially determined by the si/2, P1/2 and P3/2 spinor of the heavy nucleus and he employed the matrix elements of e-nuci,2) nSi/2 and n Pi/2 spinors, for which an analytical expres-... 

In order to understand the physical meaning of the equation, the electron mass m and the speed of light c, which are both unity in atomic units, are explicitly written in this and the next sections. The a in Eq. (6.55) is called the Pauli spin matrix (Pauli 1925). This equation is invariant for the Lorentz transformation, because the momentum p = —/ V is the first derivative in terms of space. More importantly, this equation rsqu-irts four-component wavefunctions. 

If the scalar approximation (neglecting spin-orbit-coupling terms) is activated, the computational demands turn out to be rather different. The operation count for the scalar-relativistic variants is also given in Table 14.2. Most important, only real matrix operations are required since one can employ real basis functions and the Hamiltonian operators are also real. Then, spin is a good quantum number and the spin symmetry can be exploited so that dimensions of all two- and four-component matrix operators are reduced by half. Finally, since the spin-orbit components of the relativistic potential matrix, i.e., W, Wy, and W, are neglected, the number of matrix multiplications required for the orthonormal basis transformation is decreased from ten to four. 

The relativistic basis is no longer the set of products of orbital functions with a and spin functions, but general four-component spinors grouped as Kramers pairs. Likewise, the operators are no longer necessarily spin free. If we apply the time-reversal operator to matrix elements of we can derive some relations between matrix elements... 

Both the reduced permutational symmetry and the need to account for nondiagonal matrix elements over spin functions will complicate any computational scheme. However, the difficulties introduced at this stage are certainly less serious than the integral handling introduced in the four-component calculations. 

A similar analysis can be applied to the use of the spin-free modified Dirac method, whether in its frill four-component form or in the reduced form of the matrix NESC method. In particular, the spin-orbit and spin-other-orbit integrals have the same properties, although different forms. 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

All the off-diagonal matrix elements of the spin-orbit coupling in the >, Tl> [ basis are thus reduced by the factor y, and we use the experimentally observed quenching to calculate Ej j and the corresponding geometrical distortion (14). In the Cs2NaYClg host lattice the total spread of the four spin-orbit components of T2 is 32 cm whereas crystal field theory without considering a Jahn-Teller effect predicts a total spread of approximately 107 cm-. ... 

To find the relative intensities of the four AB lines we use the results of time-dependent perturbation theory, as described in Section 2.3. For N spins, the probability Pmn of a transition between states m and n is given by a matrix element of the x component of magnetization or spin ... 

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