Symmetry-restricted unitary transformations

The unitary transformations discussed in Section 3.2 are completely general, allowing us to express any unitarily transformed operator and state in terms of a set of independent parameters. However, in many situations less general transformations are required owing to the presence of special symmetries in the electronic system. A more general discussion of symmetry restrictions is given in Chapter 4. Here we anticipate this development by considering symmetry-restricted or symmetry-constrained unitary operators exp(— ). 

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The special class of transformation, known as symmetry (or unitary) transformation, preserves the shape of geometrical objects, and in particular the norm (length) of individual vectors. For this class of transformation the symmetry operation becomes equivalent to a transformation of the coordinate system. Rotation, translation, reflection and inversion are obvious examples of such transformations. If the discussion is restricted to real vector space the transformations are called orthogonal. 

The operator k in the form of equation (111) is needed for a general unitary transformation. Often, however, we are interested in restricted unitary operators with special properties. Thus, the following operator, where the Kp are real-valued parameters, generates real unitary (i.e., orthogonal) transformations that conserve the spin symmetry of the transformed electronic state... 

The exponential parametrization of a unitary operator is independent in the sense that there are no restrictions on the allowed values of the numerical parameters in the operator - any choice of numerical parameters gives rise to a bona fide unitary operator. In many situations, however, we would like to carry out restricted spin-orbital and orbital rotations in order to preserve, for example, the spin symmetries of the electronic state. Such constrained transformations are also considered in this chapter, which contains an analysis of the symmetry properties of unitary orbital-rotation operators in second quantization. We begin, however, our exposition of spin-orbital and orbital rotations in second quantization with a discussion of unitary matrices and matrix exponentials. 

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