Non-orthogonal spinor bases
Working in an orthonormal spinor basis has its advantages because only in an orthonormal spinor basis \(\require{physics} \set{\phi_P(\vb{r})}\), the spinor overlap matrix is the identity matrix: \[\begin{equation} \braket{\phi_P}{\phi_Q} = \delta_{PQ} \thinspace , \end{equation}\] and thus the two following relations are simultaneously fulfilled (Péter. R. Surján 1989):
On the one hand, we have the adjoint relation: \[\begin{equation} \hat{a}_P = \qty(\hat{a}^\dagger_P)^\dagger \thinspace , \end{equation}\] which means that the annihilator is the adjoint of the creator.
On the other hand, we have the simple elementary anticommutator \[\begin{equation} \comm{ \hat{a}_P }{ \hat{a}^\dagger_Q }_+ = \delta_{PQ} \thinspace , \end{equation}\] which simplifies the calculation of matrix elements because \(\hat{a}_P\) is considered to be the true annihilator for \(\hat{a}^\dagger_P\).
In a general spinor basis \(\set{\chi_P(\vb{r})}\) (to which the creation and annihilation operators \(\hat{b}^\dagger_P\) and \(\hat{b}_P\) are linked), the spinor overlap matrix is not necessarily the identity matrix: \[\begin{equation} \label{eq:non_orthogonal_overlap_matrix} \braket{\chi_P}{\chi_Q} = \int \dd{\vb{r}} \chi^\dagger_P(\vb{r}) \chi_Q(\vb{r}) = S_{PQ} \thinspace , \end{equation}\] so only one of both previous conditions can hold at the same time. In this equation, we call \(\vb{S}\) the spinor overlap matrix, which is Hermitian: \[\begin{equation} \vb{S}^\dagger = \vb{S} \thinspace . \end{equation}\]
How could we transform the orbitals \(\set{\chi_P(\vb{r})}\) such that the resulting set \(\set{\phi_P(\vb{r})}\) would be orthonormal? We use \[\begin{equation} \vb{C} = \begin{pmatrix} \vb{C}^\alpha \\ \vb{C}^\beta \end{pmatrix} \end{equation}\] as the coefficient matrix (transformation matrix) for this basis transformation: \[\begin{equation} \phi_P(\vb{r}) = % \begin{pmatrix} \phi_{P \alpha}(\vb{r}) \\ \phi_{P \beta}(\vb{r}) \end{pmatrix} = % \begin{pmatrix} \displaystyle \sum_\mu^{K_\alpha} \chi^\alpha_\mu(\vb{r}) C^{\alpha}_{\mu P} \\ \displaystyle \sum_\mu^{K_\beta} \chi^\beta_\mu(\vb{r}) C^{\beta}_{\mu P} \end{pmatrix} \thinspace . \end{equation}\] In general, we can derive that the transformation matrix \(\vb{C}\) that orthonormalizes the basis \(\set{\chi_P(\vb{r})}\) must satisfy \[\begin{equation} \label{eq:transformation_orthonormalization_relation} \qty(\vb{C})^\dagger \vb{S} \vb{C} = \vb{I} \qquad \qquad \vb{C} \vb{C}^\dagger = \vb{S}^{-1} \end{equation}\] and \[\begin{equation} (\vb{C}^{-1})^\dagger \vb{C}^{-1} = \vb{S} \thinspace , \end{equation}\] where \(\vb{S}\) is the spinor overlap matrix in the non-orthogonal spinor basis \(\set{\chi_P(\vb{r})}\).
A general way to build such a transformation matrix \(\vb{C}\) would be to split it up into a unitary part and an overlap part (Van Raemdonck 2017): \[\begin{align} & \vb{C} = \vb{S}^{-1/2} \vb{U}^\dagger \\ & \vb{C}^{-1} = \vb{U} \vb{S}^{1/2} \thinspace , \end{align}\] such that we obtain the so-called Löwdin-orthonormalization matrix by setting the unitary part to the identity matrix: \[\begin{align} & \vb{C} = \vb{S}^{-1/2} \\ & \vb{C}^{-1} = \vb{S}^{1/2} \thinspace . \end{align}\] If we link the orthornomal set \(\set{\phi_P}\) with the elementary second-quantization operators \(\set{\hat{a}^\dagger_P, \hat{a}_P}\), the transformation formulas for the operators become: \[\begin{align} & \hat{a}^\dagger_P = \sum_Q^M \hat{b}^\dagger_Q C_{QP} \label{eq:non-orthogonal:a^dagger_b} \\ & \hat{a}_P = \sum_Q^M \hat{b}_Q C^*_{QP} \label{eq:non-orthogonal:a_b} \end{align}\] and we obtain for the reverse transformations: \[\begin{align} & \hat{b}^\dagger_P = \sum_Q^M \hat{a}^\dagger_Q C^{-1}_{QP} \label{eq:non-orthogonal:b^dagger_a} \\ & \hat{b}_P = \sum_Q^M \hat{a}_Q C^{-1 *}_{QP} \label{eq:non-orthogonal:b_a} \thinspace , \end{align}\] according to the discussion in section \(\ref{sec:general_spinor_transformations}\). The anticommutator between the adjoint operators related to the non-orthogonal set become T. Helgaker, Jørgensen, and Olsen (2000): \[\begin{equation} \comm{ \hat{b}_P }{ \hat{b}^\dagger_Q }_+ = S_{PQ} \neq \delta_{PQ} \thinspace , \end{equation}\] which means that the adjoint operators \(\hat{b}_P\) are no longer the true annihilation operators for the creators \(\hat{b}^\dagger_P\).