The orbital rotation generator
More specifically, we are most often interested in unitary transformations (also: rotations) of the underlying spinor basis because unitary transformations keep orthonormal bases orthonormal. The unitary transformation matrix is written as \(\require{physics} \vb{U}\): \[\begin{equation} \vb{U} \vb{U}^\dagger = \vb{U}^\dagger \vb{U} = \vb{I} \thinspace . \end{equation}\] If the elementary second quantization operators \(\hat{c}^\dagger_P\) and \(\hat{c}_P\) are linked to the transformed spinors \(\set{\phi'_P}\) and \(\hat{a}^\dagger_P\) and \(\hat{a}_P\) are linked to the original set \(\set{\phi_P}\), we can show that the second quantization commutation relations are invariant with respect to spinor rotations: \[\begin{equation} \comm{\hat{c}^\dagger_P}{\hat{c}_Q} = \comm{\hat{a}^\dagger_P}{\hat{a}_Q} = \delta_{PQ} \thinspace . \end{equation}\]
What do the transformed creation and annihilation operators look like? The derivation in (T. Helgaker, Jørgensen, and Olsen 2000) is based on the generation of a unitary matrix by an anti-Hermitian matrix \(\boldsymbol{\kappa}\) in the following way: \[\begin{align} & \vb{U} = \exp(-\boldsymbol{\kappa}) % & \boldsymbol{\kappa}^\dagger = - \boldsymbol{\kappa} \thinspace . \end{align}\] Using this parametrization of the spinor rotation, we can show (T. Helgaker, Jørgensen, and Olsen 2000) that the rotation of the elementary second quantization operators can be written as: \[\begin{align} & \hat{c}^\dagger_P = \exp(-\hat{\kappa}) \thinspace \hat{a}^\dagger_P \thinspace \exp(\hat{\kappa}) \\ % & \hat{c}_P = \exp(-\hat{\kappa}) \thinspace \hat{a}_P \thinspace \exp(\hat{\kappa}) \thinspace , \end{align}\] in which we have introduced the all-important generator of orbital rotations (T. Helgaker, Jørgensen, and Olsen 2000): \[\begin{equation} \hat{\kappa} = % \sum_{PQ}^M \kappa_{PQ} \hat{E}_{PQ} \end{equation}\] and the parameters \(\kappa_{PQ}\) are the elements of the anti-Hermitian matrix \(\boldsymbol{\kappa}\). This all follows from the fact that the \(n\)-folded nested commutators have simple expressions: \[\begin{align} & {}_n\comm{ \hat{a}^\dagger_P }{ \hat{\kappa} } = (-1)^n \sum_Q^M \hat{a}^\dagger_Q \qty[ \boldsymbol{\kappa}^n ]_{QP} \\ % & \comm{\hat{a}^\dagger_P}{ \hat{\kappa} }_n = \sum_Q^M \hat{a}^\dagger_Q \qty[ \boldsymbol{\kappa}^n ]_{QP} \thinspace . \end{align}\] For further reference, we thus have \[\begin{align} & \exp(-\hat{\kappa}) \thinspace \hat{a}^\dagger_P \thinspace \exp(\hat{\kappa}) = \sum_Q^M \hat{a}^\dagger_Q U_{QP} \\ % & \exp(-\hat{\kappa}) \thinspace \hat{a}_P \thinspace \exp(\hat{\kappa}) = \sum_Q^M \hat{a}_Q U_{QP}^* \end{align}\] and \[\begin{align} & \exp(\hat{\kappa}) \thinspace \hat{a}^\dagger_P \thinspace \exp(-\hat{\kappa}) = \sum_Q^M \hat{a}^\dagger_Q U^\dagger_{QP} \\ % & \exp(\hat{\kappa}) \thinspace \hat{a}_P \thinspace \exp(-\hat{\kappa}) = \sum_Q^M \hat{a}_Q U_{QP}^{\dagger *} \thinspace . \end{align}\]
There are two different ways we can rewrite this orbital rotation generator.
We may write the orbital rotation generator in terms of the parameters \(\kappa_{PQ}\) (restricted to \(P \geq Q\)) and their complex conjugates in the following way: \[\begin{equation} \hat{\kappa} = \sum_{P>Q}^M \qty( \kappa_{PQ} \hat{E}_{PQ} - \kappa_{PQ}^* \hat{E}_{QP} ) + \sum_P^M \kappa_{PP} \hat{N}_P \thinspace , \end{equation}\] in which the parameters \(\kappa_{PQ}\) (restricted to \(P \geq Q\)) are now free to be varied independently.
Let us write every complex parameter in terms of its real and imaginary components: \[\begin{equation} \kappa_{PQ} = {}^{\text{R}} \kappa_{PQ} + i {}^{\text{I}} \kappa_{PQ} \thinspace , \end{equation}\] in which \[\begin{equation} {}^{\text{R}} \kappa_{PQ} = \Re \kappa_{PQ} \end{equation}\] and \[\begin{equation} {}^{\text{I}} \kappa_{PQ} = \Im \kappa_{PQ} \end{equation}\] are both real parameters without any restrictions, such that the anti-Hermiticity of the matrix \(\boldsymbol{\kappa}\) is ensured by requiring \[\begin{align} & {}^{\text{R}} \kappa_{PQ} = - {}^{\text{R}} \kappa_{QP} \\ & {}^{\text{I}} \kappa_{PQ} = {}^{\text{I}} \kappa_{QP} \thinspace , \end{align}\] which means that the real matrix \({}^{\text{R}} \boldsymbol{\kappa}\) should be antisymmetric, and the real matrix \({}^{\text{I}} \boldsymbol{\kappa}\) should be symmetric. The generator of spinor rotations can then be written as \[\begin{equation} \hat{\kappa} = \sum_{P>Q}^M {}^{\text{R}} \kappa_{PQ} \hat{E}^-_{PQ} + i \qty( \sum_{P}^M {}^{\text{I}} \kappa_{PP} \hat{N}_P % + \sum_{P>Q}^M % {}^{\text{I}} \kappa_{PQ} \hat{E}^+_{PQ} ) \thinspace , \end{equation}\] in which there are \[\begin{equation} \frac{M(M-1)}{2} + M + \frac{M(M-1)}{2} = M^2 \end{equation}\] real, free parameters \({}^{\text{R}} \kappa_{PQ}\) and \({}^{\text{I}} \kappa_{PQ}\). For the special case of rotations of real spinors, we can set \({}^{\text{I}} \kappa_{PQ} = 0\), such that we can use the following generator for real spinor rotations: \[\begin{equation} {}^{\text{R}} \hat{\kappa} = \sum_{P>Q}^M {}^{\text{R}} \kappa_{PQ} \hat{E}^-_{PQ} \thinspace , \end{equation}\] which is parametrized by the \(M(M-1)/2\) real parameters \({}^{\text{R}} \kappa_{PQ}\).
In the special case of employing a spin-orbital basis, we can rewrite and further analyse the orbital rotation generator.