The spin-orbital rotation generator
When employing a spin-orbital basis rather than a general spinor basis, we can discuss the orbital rotation generator in more detail.
Unrestricted spinor rotations
In a spin-orbital basis, we write the \(M\) Pauli spinors with either an \(\alpha\) or a \(\beta\)-component, which means that the spinor rotations are now given by: \[\begin{equation} \require{physics} \begin{pmatrix} \phi'_{1 \alpha} & 0 & \cdots & \phi'_{K \alpha} & 0 \\ % 0 & \phi'_{1 \beta} & \cdots & 0 & \phi'_{K \beta} \end{pmatrix} = \begin{pmatrix} \phi_{1 \alpha} & 0 & \cdots & \phi_{K \alpha} & 0 \\ % 0 & \phi_{1 \beta} & \cdots & 0 & \phi_{K \beta} \end{pmatrix} \vb{U} \thinspace , \end{equation}\] where \(\vb{U}\) is still an \((M \times M)\)-unitary matrix. We can now arrange the \(\alpha\)- and \(\beta\)-spinors as, so that: \[\begin{equation} \begin{pmatrix} \phi'_{1 \alpha} & \cdots & \phi'_{K \alpha} & 0 & \cdots & 0 \\ % 0 & \cdots & 0 & \phi'_{1 \beta} & \cdots & \phi'_{K \beta} \end{pmatrix} = \begin{pmatrix} \phi_{1 \alpha} & \cdots & \phi_{K \alpha} & 0 & \cdots & 0 \\ % 0 & \cdots & 0 & \phi_{1 \beta} & \cdots & \phi_{K \beta} \end{pmatrix} \vb{U} % \thinspace . \end{equation}\] This small rearrangement has some very interesting and far-reaching consequences:
We can write the transformation using block matrices: \[\begin{equation} \begin{pmatrix} \boldsymbol{\phi}'_{\alpha} & \vb{0} \\ \vb{0} & \boldsymbol{\phi}'_{\beta} \end{pmatrix} = \begin{pmatrix} \boldsymbol{\phi}_{\alpha} & \vb{0} \\ \vb{0} & \boldsymbol{\phi}_{\beta} \end{pmatrix} \begin{pmatrix} \vb{U}^{\alpha \alpha} & \vb{U}^{\alpha \beta} \\ \vb{U}^{\beta \alpha} & \vb{U}^{\beta \beta} \end{pmatrix} \thinspace , \end{equation}\] where the separate blocks \(\vb{U}^{\alpha \alpha}, \vb{U}^{\alpha \beta}, \vb{U}^{\beta \alpha}\) and \(\vb{U}^{\beta \beta}\) are of dimension \(K \times K\). In principle, these blocks could have different dimensions, but for simplicity we take \(K = K_\alpha = K_\beta\)
We notice that due to the form of the spinors (no \(\beta\)-components for the first \(K\) spinors and no \(\alpha\) components for the last \(K\) spinors), we must disallow the \(\alpha\)- and \(\beta\)- spatial orbital mixing: \[\begin{align} & \vb{U}^{\alpha \beta} = \vb{0} % & \vb{U}^{\beta \alpha} = \vb{0} \thinspace . \end{align}\] This means that, effectively, we can use the following \((2K \times 2K)\)-unitary matrix to perform rotations in an unrestricted formalism: \[\begin{equation} \label{eq:unrestricted_rotation_matrix} \vb{U} = \begin{pmatrix} \vb{U}^{\alpha \alpha} & \vb{0} \\ \vb{0} & \vb{U}^{\beta \beta} \end{pmatrix} \thinspace . \end{equation}\] Since this unitary matrix is diagonal, the \(\vb{U}^{\alpha \alpha}\)- and \(\vb{U}^{\beta \beta}\)-blocks must be unitary themselves. This means that we can effectively separately (and differently) transform the \(\alpha\)- and \(\beta\)- spatial orbitals. Moreover, rotating the spatial orbitals automatically ensures that the corresponding spinors are rotated as well.
Starting from a set of spin-orbitals (with no further restrictions), we can in principle generate the following rotations: \[\begin{equation} \hat{\kappa} = \sum_{pq}^K \sum_{\sigma \tau} \kappa_{p \sigma, q \tau} \hat{a}^\dagger_{p \sigma} \hat{a}_{q \tau} \thinspace , \end{equation}\] in which there are some generating elements, like \(\kappa_{p\beta, q\alpha}\), that do not again yield an unrestricted set of spinors. Introducing the spin tensor operators and the parameter transformations \[\begin{align} & \kappa^{0,0}_{pq} = \frac{1}{2} ( \kappa_{p \alpha, q \alpha} + \kappa_{p \beta, q \beta} ) \\ % & \kappa^{1,-1}_{pq} = \kappa_{p \beta, q \alpha} \\ % & \kappa^{1,0}_{pq} = \frac{1}{\sqrt{2}} ( \kappa_{p \alpha, q \alpha} - \kappa_{p \beta, q \beta} ) \\ % & \kappa^{1,1}_{pq} = - \kappa_{p \alpha, q \beta} \thinspace , \end{align}\] we can equivalently write the spin-orbital rotation generator as: \[\begin{equation} \hat{\kappa} = \sum_{pq}^K \kappa^{0,0}_{pq} \hat{E}_{pq} + \sum_{pq}^K \sum_{M = -1}^1 \kappa^{1,M}_{pq} \hat{T}^{1,M}_{pq} \thinspace . \end{equation}\]
Since we can decompose the complex parameters into their real and imaginary parts: \[\begin{equation} \kappa^{S,M}_{pq} = {}^{\text{R}} \kappa^{S,M}_{pq} + i {}^{\text{I}} \kappa^{S,M}_{pq} \thinspace , \end{equation}\] requiring that \(\hat{\kappa}\) is anti-Hermitian (i.e. \(\hat{\kappa}^\dagger = - \hat{\kappa}\)) leads to the following restrictions on the following real parameters: \[\begin{align} & {}^{\text{R}} \kappa^{0,0}_{pq} = - {}^{\text{R}} \kappa^{0,0}_{qp} & {}^{\text{I}} \kappa^{0,0}_{pq} = {}^{\text{I}} \kappa^{0,0}_{qp} \\ % & {}^{\text{R}} \kappa^{1,-1}_{pq} = {}^{\text{R}} \kappa^{1,1}_{qp} & {}^{\text{I}} \kappa^{1,-1}_{pq} = - {}^{\text{I}} \kappa^{1,1}_{qp} \\ % & {}^{\text{R}} \kappa^{1,0}_{pq} = - {}^{\text{R}} \kappa^{1,0}_{qp} & {}^{\text{I}} \kappa^{1,0}_{pq} = {}^{\text{I}} \kappa^{1,0}_{qp} \thinspace . \end{align}\] Using these relations, we can write the spin-orbital rotation generator as: \[\begin{equation} \begin{split} \hat{\kappa} = & \sum_{p>q}^K {}^{\text{R}} \kappa^{0,0}_{pq} ( \hat{E}_{pq} - \hat{E}_{qp} ) + i \qty( \sum_p^K {}^{\text{I}} \kappa^{0,0}_{pp} \hat{N}_p + \sum_{p}^K {}^{\text{I}} \kappa^{1,0}_{pp} \hat{S}^z_p ) \\ &+ \sum_{p>q}^K {}^{\text{R}} \kappa^{1,0}_{pq} ( \hat{T}^{1,0}_{pq} - \hat{T}^{1,0}_{qp} ) + \sum_{pq}^K {}^{\text{R}} \kappa^{1,1}_{pq} ( \hat{T}^{1,1}_{pq} + \hat{T}^{1,-1}_{qp} ) \\ & + i \qty[ \sum_{p>q}^K {}^{\text{I}} \kappa^{0,0}_{pq} ( \hat{E}_{pq} + \hat{E}_{qp} ) + \sum_{p>q}^K {}^{\text{I}} \kappa^{1,0}_{pq} ( \hat{T}^{1,0}_{pq} + \hat{T}^{1,0}_{qp} ) + \sum_{pq}^K {}^{\text{I}} \kappa^{1,1}_{pq} ( \hat{T}^{1,1}_{pq} - \hat{T}^{1,-1}_{qp} ) ] \thinspace . \end{split} \end{equation}\]
Now comes the beauty of the previously introduced spin tensor operators. In order to remain in an unrestricted framework, we have to disallow generators of the mixing form \(\kappa_{p\beta, q\alpha}\), which correspond to the generators \(\kappa_{pq}^{1,\pm 1}\). Disallowing these generators means setting them to zero, whose consequence is that the resulting spin-orbital rotation generator \[\begin{align} \hat{\kappa}^{M=0} &= \sum_{pq}^K \kappa^{0,0}_{pq} \hat{E}_{pq} + \sum_{pq}^K \kappa^{1,0}_{pq} \hat{T}^{1,0}_{pq} \\ &= \sum_{p>q}^K {}^{\text{R}} \kappa^{0,0}_{pq} ( \hat{E}_{pq} - \hat{E}_{qp} ) + i \qty( \sum_p^K {}^{\text{I}} \kappa^{0,0}_{pp} \hat{N}_p + \sum_{p}^K {}^{\text{I}} \kappa^{1,0}_{pp} \hat{S}^z_p % ) \notag \\ & \hspace{12pt} + \sum_{p>q}^K {}^{\text{R}} \kappa^{1,0}_{pq} ( \hat{T}^{1,0}_{pq} - \hat{T}^{1,0}_{qp} ) + i \qty[ \sum_{p>q}^K {}^{\text{I}} \kappa^{0,0}_{pq} ( \hat{E}_{pq} + \hat{E}_{qp} ) + \sum_{p>q}^K {}^{\text{I}} \kappa^{1,0}_{pq} ( \hat{T}^{1,0}_{pq} + \hat{T}^{1,0}_{qp} ) ] \end{align}\] commutes with \(\hat{S}^z\): \[\begin{equation} \comm{\hat{S}^z}{\hat{\kappa}} = 0 \thinspace . \end{equation}\]
We must be cautious to ‘remain’ in a spin-orbital basis: one triplet \(M=\pm 1\) rotation would be enough to end up with a general set of spinors, after which we should resort to the general spinor generator since we do not have an unrestricted set of spinors anymore. However, if we stay within the \(M=0\) component, we are effectively using a block diagonal generator \[\begin{equation} \boldsymbol{\kappa} = \begin{pmatrix} \boldsymbol{\kappa}^{\alpha \alpha} & \vb{0} \\ \vb{0} & \boldsymbol{\kappa}^{\beta \beta} \end{pmatrix} \thinspace , \end{equation}\] whose exponential negative yields the blocked form of the unitary transformation matrix \(\vb{U}\). Essentially, in order to remain in the unrestricted spinor framework, we can only transform the elementary operators as \[\begin{align} & \hat{c}^\dagger_{p \sigma} = \sum_q^K \hat{a}^\dagger_{q \sigma} U^{\sigma \sigma}_{qp} = \exp( -\hat{\kappa}^{\sigma \sigma} ) \thinspace \hat{a}^\dagger_{p \sigma} \thinspace \exp\*( \hat{\kappa}^{\sigma \sigma} ) \\ % & \hat{c}_{p \sigma} = \sum_q^K \hat{a}_{q \sigma} U^{\sigma \sigma *}_{qp} = \exp( -\hat{\kappa}^{\sigma \sigma} ) \thinspace \hat{a}_{p \sigma} \thinspace \exp( \hat{\kappa}^{\sigma \sigma} ) \thinspace , \end{align}\] with the inverse transformations being: \[\begin{align} & \hat{c}^\dagger_{p \sigma} = \sum_q^K \hat{a}^\dagger_{q \sigma} \qty(U^{\sigma \sigma})^{-1}_{qp} \\ & \hat{c}_{p \sigma} = \sum_q^K \hat{a}_{q \sigma} \qty(U^{\sigma \sigma})^{-1 *}_{qp} \thinspace . \end{align}\]
Reintroducing the excitation operators \(\hat{E}^\alpha_{pq}\) and \(\hat{E}^\beta_{pq}\), we then find: \[\begin{align} \hat{\kappa}^{M=0} &= \sum_{pq}^K \kappa^{\alpha \alpha}_{pq} \hat{E}^\alpha_{pq} + \sum_{pq}^K \kappa^{\beta \beta}_{pq} \hat{E}^\beta_{pq} \\ &= \hat{\kappa}^{\alpha \alpha} + \hat{\kappa}^{\beta \beta} \\ &= \sum_{pq}^K \sum_\sigma \kappa^{\sigma \sigma}_{pq} \hat{E}^\sigma_{pq} \thinspace , \end{align}\] in which we have done the following parameter transformations: \[\begin{align} & \kappa^{\alpha \alpha}_{pq} = \kappa^{0,0}_{pq} + \frac{1}{2} \kappa^{1,0}_{pq} = \kappa_{p\alpha, q\alpha} \\ % & \kappa^{\beta \beta}_{pq} = \kappa^{0,0}_{pq} - \frac{1}{2} \kappa^{1,0}_{pq} = \kappa_{p\beta, q\beta} \thinspace . \end{align}\] Requiring an anti-Hermitian operator \(\hat{\kappa}^{M=0}\), leads to the following constraints on the real and imaginary parts of the parameters \[\begin{align} & {}^{\text{R}} \kappa_{pq}^{\sigma \sigma} = - {}^{\text{R}} \kappa_{qp}^{\sigma \sigma} & {}^{\text{I}} \kappa_{pq}^{\sigma \sigma} = {}^{\text{I}} \kappa_{pq}^{\sigma \sigma} \thinspace , \end{align}\] leading to the following decomposition of \(\hat{\kappa}^{M=0}\) in its real and imaginary parts: \[\begin{align} \hat{\kappa}^{M=0} = \sum_{p>q}^K \sum_{\sigma} {}^{\text{R}} \kappa_{pq}^{\sigma \sigma} \qty( \hat{E}^\sigma_{pq} - \hat{E}^\sigma_{qp} ) + i \qty[ \sum_p^K {}^{\text{I}} \kappa_{pp}^{\sigma \sigma} \hat{N}_p + \sum_{p>q}^K \sum_{\sigma} {}^{\text{I}} \kappa_{pq}^{\sigma \sigma} \qty( \hat{E}^\sigma_{pq} + \hat{E}^\sigma_{qp} ) ] \thinspace . \end{align}\]
If a real description is sufficient (for example in the case of a vanishing magnetic field), we can use the real part of the \(M=0\) spin-orbital rotation generator: \[\begin{align} {}^{\text{R}} \hat{\kappa}^{M=0} & = \sum_{p>q}^K {}^{\text{R}} \kappa^{0,0}_{pq} ( \hat{E}_{pq} - \hat{E}_{qp} ) + \sum_{p>q}^K {}^{\text{R}} \kappa^{1,0}_{pq} ( \hat{T}^{1,0}_{pq} - \hat{T}^{1,0}_{qp} ) \\ &= \sum_{p>q}^K \sum_{\sigma} {}^{\text{R}} \kappa_{pq}^{\sigma \sigma} \qty( \hat{E}^\sigma_{pq} - \hat{E}^\sigma_{qp} ) \\ &= \sum_{p>q}^K {}^{\text{R}} \kappa_{pq}^{\alpha \alpha} \qty( \hat{E}^\alpha_{pq} - \hat{E}^\alpha_{qp} ) + \sum_{p>q}^K {}^{\text{R}} \kappa_{pq}^{\beta \beta} \qty( \hat{E}^\beta_{pq} - \hat{E}^\beta_{qp} ) \thinspace , \end{align}\] which is the result we find for the UHF orbital rotation generator in section 10.10.1 in Helgaker (T. Helgaker, Jørgensen, and Olsen 2000).
Restricted spinor rotations
If we further more require the orbitals to be restricted (i.e. the same spatial form is used for both the \(\alpha\)- and \(\beta\)-orbitals), the spinor rotation is written in short as: \[\begin{equation} \begin{pmatrix} \boldsymbol{\phi}' & \vb{0} \\ \vb{0} & \boldsymbol{\phi}' \end{pmatrix} = \begin{pmatrix} \boldsymbol{\phi} & \vb{0} \\ \vb{0} & \boldsymbol{\phi} \end{pmatrix} \begin{pmatrix} \vb{U} & \vb{0} \\ \vb{0} & \vb{U} \end{pmatrix} \thinspace , \end{equation}\] where the total transformation matrix is of dimension \((2K \times 2K)\) and consists of diagonal and equal unitary \((K \times K)\)-blocks, which means that the transformation formulas should be given by \[\begin{align} & \hat{c}^\dagger_{p \sigma} = \sum_q^K \hat{a}^\dagger_{q \sigma} U_{qp} = \exp\*(-\hat{\kappa}^{0,0}) \thinspace \hat{a}^\dagger_{p \sigma} \thinspace \exp\*(\hat{\kappa}^{0,0}) \\ % & \hat{c}_{p \sigma} = \sum_q^K \hat{a}_{q \sigma} U_{qp}^* = \exp\*(-\hat{\kappa}^{0,0}) \thinspace \hat{a}_{p \sigma} \thinspace \exp\*(\hat{\kappa}^{0,0}) \thinspace , \end{align}\] and the inverse transformation are: \[\begin{align} & \hat{c}^\dagger_{p \sigma} = \sum_q^K \hat{a}^\dagger_{q \sigma} U^{-1}_{qp} \\ % & \hat{c}_{p \sigma} = \sum_q^K \hat{a}_{q \sigma} U^{-1 *}_{qp} \thinspace . \end{align}\] Note that, since we are working in the restricted framework, the same rotation matrix \(\vb{U}\) is used for both \(\alpha\)- and \(\beta\)-components.
In order to remain in a ‘restricted’ formulation, we require \(\hat{S}^2\) to commutate with the spinor rotation generator: \[\begin{equation} \comm{\hat{S}^2}{\hat{\kappa}} = 0 \thinspace , \end{equation}\] which, means that \(\hat{\kappa}\) may only contain singlet operators, i.e. with \(S=0\) and \(M=0\). Therefore, we find that: \[\begin{equation} \hat{\kappa}^{0,0} = \sum_{pq}^K \kappa^{0,0}_{pq} \hat{E}_{pq} \thinspace . \end{equation}\] Furthermore, decomposing the \(S=0\) and \(M=0\) orbital rotation generator into its real and imaginary parts, we find: \[\begin{equation} \hat{\kappa}^{0,0} = \sum_{p>q}^K {}^{\text{R}} \kappa^{0,0}_{pq} \hat{E}^-_{pq} + i \qty( \sum_p^K {}^{\text{I}} \kappa^{0,0}_{pp} \hat{N}_p + \sum_{p>q}^K {}^{\text{I}} \kappa^{0,0}_{pq} \hat{E}^+_{pq} ) \thinspace , \end{equation}\] using the constraints on the parameters \(\kappa^{0,0}_{pq}\). For real singlet rotations, we ultimately only need \({}^{\text{R}} \hat{\kappa}^{0,0}\): \[\begin{equation} {}^{\text{R}} \hat{\kappa}^{0,0} = \sum_{p>q}^K {}^{\text{R}} \kappa^{0,0}_{pq} \hat{E}^-_{pq} \thinspace . \end{equation}\] The parameters \({}^{\text{R}} \kappa^{0,0}_{pq}\) form a real antisymmetric matrix \(\boldsymbol{\kappa}\): \[\begin{align} & \boldsymbol{\kappa}^T = -\boldsymbol{\kappa} % & \kappa_{pq} = -\kappa_{qp} % \thinspace . \end{align}\]