Symmetry restrictions in the orbital basis

To allow for spin-symmetry constraints in the rotation operator (3.3.3), we express k in the orbital basis and introduce spin teisor operators. We first consider the contributions to (3.3.3) from excitation operators that are diagonal in the orbital indices 

Treating the nondiagonal orbital excitation operators in the same way, we arrive at the following expression for k  

Here we have isolated the diagonal elements but made no separation of real and imaginary parts. All parameters are complex except and which have only imaginary components. 

If we insist on using tensor operators as well as on separating real and imaginary rotations, the antisymmetry of k imposes the following constraints mi the oibital excitation parameters 

Alternatively, we may write k in terms of the Cartesian triplet operators (2.3.27). Since the Cartesian components obey the simple conjugation relation (2.3.28), we now obtain the following more symmetric form for a general anti-Heimitian operator k  

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