Quaternions modified Dirac

In the quaternion modified Dirac equation the spin-free equation is thereby obtained simply by deleting the quaternion imaginary parts. For further details, the reader is referred to Ref. [13]. 

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... 

L. Visscher, T. Saue. Approximate relativistic electronic structure methods based on the quaternion modified Dirac equation. /. Chem. Pkys., 113(10) (2000)3996-4002. 

The kinetic energy matrix is a full matrix of dimension rp- in both cases. As for the other two, the unmodified Dirac kinetic energy matrix is a quaternion matrix, with ArP unique quantities, whereas the spin-fl-ee modified Dirac kinetic energy matrix is a real matrix with rp unique quantities, and the resultant reduction is a factor of 4. However, if an uncontracted basis is used, the spin-free modified Dirac kinetic energy matrix is symmetric, and is the same as the pseudo-large-component overlap matrix. 

The overlap and potential energy matrices are matrices of dimension for each component, both in the modified and unmodified Dirac method, and both are Hermitian. The unmodified Dirac matrices are quaternion matrices, and there are n 2n — 1) unique quantities (real numbers) in each matrix. The modified spin-free Dirac matrices are real, with n n + l)/2 unique quantities. The reduction in the number of unique quantities is a factor of approximately 4. 

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