Jacobi rotations
By now, we have seen many examples of unitary matrices/operators that are generated by anti-Hermitian matrices/operators. For the transformation matrix for real, singlet rotations, we have \[\begin{align} \require{physics} \vb{U} &= \exp(-\boldsymbol{\kappa}) \\ &= \exp(-\sum_{p>q}^K \kappa_{pq} \vb{E}^-_{pq}) % \thinspace , \end{align}\] in which \(\vb{E}^-_{pq}\) (with \(p>q\)) form a basis for the anti-symmetric matrices: \[\begin{equation} \vb{E}^-_{pq} = \begin{pmatrix} 0 & \cdots & 0 & \cdots & 0 \\ \vdots & 0 & \cdots & -1 & \vdots \\ 0 & \vdots & \ddots & \vdots & 0 \\ \vdots & 1 & \cdots & 0 & \vdots \\ 0 & \cdots & 0 & \cdots & 0 \end{pmatrix} \thinspace , \end{equation}\] and its elements are \[\begin{equation} \qty[\vb{E}^-_{pq}]_{rs} % = \delta_{pr} \delta_{qs} % - \delta_{ps} \delta_{qr} \end{equation}\] and we furthermore have \[\begin{equation} \qty[% \vb{E}^-_{pq} % \vb{E}^-_{rs}% ]_{tu} % = \delta_{pt} \qty( % \delta_{qr} \delta_{su} % - \delta_{qs} \delta_{ru} % ) % - \delta_{qt} \qty( % \delta_{pr} \delta_{su} % - \delta_{ps} \delta_{ru} % ) % \thinspace , \end{equation}\] from which we can calculate the commutator \[\begin{equation} \qty[ % \comm{\vb{E}^-_{pq}}{\vb{E}^-_{rs}} % ]_{tu} % = \qty[ % \qty( % \delta_{rq} \vb{E}^-_{ps} % - \delta_{ps} \vb{E}^-_{rq} % ) % - \qty( % \delta_{sq} \vb{E}^-_{pr} % - \delta_{pr} \vb{E}^-_{sq} % ) % ]_{tu} % \thinspace , \end{equation}\] which, unsurprisingly, has exactly the same form as equation \(\eqref{eq:commutator_E-}\) for the operators \(\hat{E}^-_{pq}\).
These anti-symmetric matrices \(\vb{E}^-_{pq}\) form the Lie algebra \(\solie(K)\) that has \(K(K-1)/2\) parameters, and the corresponding orthogonal matrices form the Lie group \(\SOg(K)\). Illustrating this point even further, we remember that fact that a Lie algebra is the tangent vector space of the Lie group at the identity. Indeed, we can show that for any orthogonal matrix \(\vb{U}\): \[\begin{equation} \pdv{\vb{U}}{\kappa_{pq}} \eval_{\boldsymbol{\kappa}=\vb{0}} = - \vb{E}^-_{pq} \thinspace , \end{equation}\] or in element notation (Siegbahn et al. 1981): \[\begin{equation} \qty[\pdv{\vb{U}}{\kappa_{pq}} \eval_{\boldsymbol{\kappa} = \vb{0}}]_{tu} = \delta_{pu} \delta_{qt} - \delta_{pt} \delta_{qu} \thinspace , \end{equation}\] which is exactly the same result as in (Siegbahn et al. 1981), even though they define \(\vb{U} = \exp(\boldsymbol{\kappa})\). The reason for this is that in the article, the orbital transformation is written as \(\phi_p' = \qty[\vb{U} \phi]_p\), which is the transpose of the equation we use (cfr. equation \(\eqref{eq:spinor_transformation_matrix_expression}\)), which means that the extra difference in factor \((-1)\) in the exponential can be recovered since \(\boldsymbol{\kappa}^\dagger = - \boldsymbol{\kappa}\).
Second derivatives are then given by \[\begin{align} \pdv[2]{% \vb{U}% }{ \kappa_{pq} }{ \kappa_{rs} } % \eval_{% \boldsymbol{\kappa} = \vb{0}% } % &= \frac{1}{2} \qty( % \vb{E}^-_{pq} \vb{E}^-_{rs} % + \vb{E}^-_{rs} \vb{E}^-_{pq} % ) \\ &= \frac{1}{2} (1 + P_{pq,rs}) \vb{E}^-_{pq} \vb{E}^-_{rs} % \thinspace , \end{align}\] or in element notation (Siegbahn et al. 1981): \[\begin{equation} \begin{split} \qty[ % \pdv[2]{\vb{U} % }{ \kappa_{pq} }{ \kappa_{rs} } % \eval_{ % \boldsymbol{\kappa} = \vb{0} % } % ]_{tu} % = \frac{1}{2} \bigg( % & \delta_{qr} ( % \delta_{pt} \delta_{su} % + \delta_{st} \delta_{pu}% ) % - \delta_{pr} ( % \delta_{qt} \delta_{su} % + \delta_{st} \delta_{qu} % ) \\ &- \delta_{qs} ( % \delta_{pt} \delta_{ru} % + \delta_{tr} \delta_{pu} % ) % + \delta_{ps} (% \delta_{qt} \delta_{ru} % + \delta_{tr} \delta_{qu} % ) % \bigg) % \thinspace . \end{split} \end{equation}\]
If we only pick one \(\kappa_{pq}\) and set the rest to zero, we can show that this the corresponding orthogonal transformation is given by \[\begin{align} \vb{J}^{pq}(\kappa_{pq}) % &= \exp(-\kappa_{pq} \vb{E}^-_{pq}) % \label{eq:jacobi_as_exponential} \\ &= \begin{pmatrix} 1 \\ & \ddots \\ & & 1 \\ & & & \cos\kappa_{pq} & & \cdots & & \sin\kappa_{pq} \\ & & & & 1 \\ & & & \vdots & & \ddots & & \vdots \\ & & & & & & 1 \\ & & & -\sin\kappa_{pq} & & \cdots & & \cos\kappa_{pq} \\ & & & & & & & & 1 \\ & & & & & & & & & \ddots \\ & & & & & & & & & & 1 \end{pmatrix} % \label{eq:jacobi_matrix} % \thinspace , \end{align}\] which is called a Jacobi rotation (Poelmansphd?): an orthogonal transformation in a two-dimensional subspace of a \(K\)-dimensional space (Raffenetti et al. 1992). Writing out the effect of the transformation on two orbitals \(p\) and \(q\) (in which we only define Jacobi rotations for \(p>q\)): \[\begin{align} & \phi'_p = \cos(\kappa_{pq}) \phi_p - \sin(\kappa_{pq}) \phi_q \\ & \phi'_q = \sin(\kappa_{pq}) \phi_p + \cos(\kappa_{pq}) \phi_q % \thinspace , \end{align}\] i.e. the coefficients of the transformed orbitals can be found in one column, we can see that the cosines and sines appear only in the \(p\)- and \(q\)-th rows and columns: \[\begin{equation} \label{eq:jacobi_matrix_elements} \qty[ % J^{pq}(\kappa_{pq}) % ]_{rs} % = \delta_{rs} % + ( % \delta_{rp} \delta_{sp} % + \delta_{rq} \delta_{sq} % ) ( % \cos\kappa_{pq} - 1 % ) % + ( % \delta_{rp} \delta_{sq} % - \delta_{rq} \delta_{sp} % ) \sin\kappa_{pq} % \thinspace . \end{equation}\] Following (Raffenetti et al. 1992), we can also write the matrix elements of the Jacobi rotation matrix \(\vb{J}^{pq}\) as \[\begin{equation} \qty[ % \vb{J}^{pq}(\kappa_{pq}) % ]_{rs} % = \delta_{rs} + \Delta_{rs}^{pq}(\kappa_{pq}) % \thinspace , \end{equation}\] where \[\begin{equation} \label{eq:Delta_pq_rs_Jacobi} \Delta_{rs}^{pq}(\kappa_{pq}) % = ( % \delta_{rp} \delta_{sp} % + \delta_{rq} \delta_{sq}% ) ( % \cos\kappa_{pq} - 1 % ) % + ( % \delta_{rp} \delta_{sq} % - \delta_{rq} \delta_{sp} % ) \sin\kappa_{pq} % \thinspace . \end{equation}\]