Spinor transformations
Let’s say we have a set of general spinors \(\require{physics} \set{\phi_P ; P=1 \cdots M}\), (to which the elementary creation operators \(\set{\hat{b}^\dagger_P}\) are related) and we would like to do a transformation of them. Let’s call the transformed set \(\set{\phi'_P ; P=1 \cdots M}\), such that we can write the transformation as \[\begin{equation} \label{eq:spinor_transformation_matrix_expression} \begin{pmatrix} \phi'_1 & \phi'_2 & \cdots & \phi'_M \end{pmatrix} = \begin{pmatrix} \phi_1 & \phi_2 & \cdots & \phi_M \end{pmatrix} \vb{T} \thinspace , \end{equation}\] in which \(\vb{T}\) is the \(M \times M\)-dimensional transformation matrix that is supposed to be non-singular, i.e. we should be able to back-transform to the original basis. Written for each of the spinors specifically, we have \[\begin{equation} \label{eq:spinor_transformation} \phi'_P = \sum_Q^M \phi_Q T_{QP} \thinspace , \end{equation}\] which means that every transformed spinor’s coefficient in the old basis can be found as the the respective column of \(\vb{T}\), i.e. \(\phi'_P\)’s coefficients can be found in the \(P\)-th column of \(\vb{T}\). If \(\set{\phi'_P}\) is again related to another spinor basis \(\set{\phi_P''}\) as: \[\begin{equation} \phi''_P = \sum_Q^M \phi'_Q R_{QP} \thinspace , \end{equation}\] the total transformation matrix between \(\set{\phi_P}\) and \(\set{\phi_P''}\) is then given by: \[\begin{equation} \vb{T}_{\text{total}} = \vb{T} \thinspace \vb{R} \thinspace . \end{equation}\]
Following Helgaker (T. Helgaker, Jørgensen, and Olsen 2000), the elementary second quantization operators \(\hat{d}^\dagger_P\) and \(\hat{d}_P\) related to the transformed set \(\set{\phi'_P}\) necessarily transform as: \[\begin{align} & \hat{d}^\dagger_P = \sum_Q^M \hat{b}^\dagger_Q T_{QP} \\ & \hat{d}_P = \sum_Q^M \hat{b}_Q T_{QP}^* \thinspace , \end{align}\] For the back-transformation, we have \[\begin{align} & \hat{b}^\dagger_P = \sum_Q^M \hat{d}^\dagger_Q T^{-1}_{QP} \\ & \hat{b}_P = \sum_Q^M \hat{d}_Q T^{-1 *}_{QP} \thinspace . \end{align}\] The elementary anti-commutation relations transform to \[\begin{align} & \comm{ \hat{d}_P }{ \hat{d}^\dagger_Q }_+ = \sum_{RS}^M \comm{ \hat{b}_R }{ \hat{b}^\dagger_S }_+ T^*_{RP} T_{SQ} \label{eq:transformed_anticommutator} \\ % & \comm{\hat{d}^\dagger_P}{\hat{d}^\dagger_Q}_+ = \comm{\hat{d}_P}{\hat{d}_Q}_+ = \comm{\hat{b}^\dagger_P}{\hat{b}^\dagger_Q}_+ = \comm{\hat{b}_P}{\hat{b}_Q}_+ = 0 \thinspace . \end{align}\]
Since the one- and two-electron integrals depend on the underlying spinor basis, a transformation of the spinors results in a transformation of the integrals. For the one-electron integrals, we find \[\begin{align} h'_{PQ} &= \sum_{RS}^M T^*_{RP} \thinspace h_{RS} \thinspace T_{SQ} \\ &= \sum_{RS}^M T^\dagger_{PR} \thinspace h_{RS} \thinspace T_{SQ} \thinspace , \end{align}\] which is equivalently written as a matrix product as to \[\begin{equation} \vb{h}' = \vb{T}^\dagger \thinspace \vb{h} \thinspace \vb{T} \thinspace . \end{equation}\] The two-electron integrals over the spinors transform as: \[\begin{equation} g'_{PQRS} = \sum_{TUVW}^M T^*_{TP} T_{UQ} \thinspace g_{TUVW} \thinspace T^*_{VR} T_{WS} \thinspace , \end{equation}\] but we wouldn’t do this transformation in practice, i.e. in a computer environment. Instead, we will break it up, as in (Miller et al. 1977) by using the intermediate contractions \[\begin{align} & a_{TURW} = \sum_V^M g_{TUVW} T^*_{VR} \\ & b_{TURS} = \sum_W^M a_{TURW} T_{WS} \\ & c_{TQRS} = \sum_U^M T_{UQ} b_{TURS} \thinspace , \end{align}\] ultimately leading to \[\begin{equation} g'_{PQRS} = \sum_T^M T^*_{TP} c_{TQRS} \thinspace . \end{equation}\]
The expectation value of a one- or two-electron operator should be independent of the spinor basis that is used. Since we have previously seen that the one- and two-electron integrals transform under a basis transformation, so must the 1- and 2-DMs. The expectation value of a one-electron operator is written as \[\begin{equation} \ev*{\hat{f}} = \sum_{PQ} f'_{PQ} D'_{PQ} \thinspace , \end{equation}\] so when we express the one-electron integrals in a different basis, we may write: \[\begin{equation} \ev*{\hat{f}} = \sum_{\mu \nu} f_{\mu \nu} D_{\mu \nu} \thinspace , \end{equation}\] if the elements of the transformed density matrix \(D'_{\mu \nu}\) are given by: \[\begin{equation} D_{\mu \nu} = \sum_{PQ} T^*_{\mu P} T_{\nu Q} D'_{PQ} \thinspace , \end{equation}\] which can be inverted to yield: \[\begin{equation} D'_{PQ} = \sum_{\mu \nu} T^{-1 *}_{P \mu} T^{-1}_{S \nu} D_{\mu \nu} \thinspace . \end{equation}\]
Similarly, we can write down the back-and-forth transformations of the 2-DM as: \[\begin{equation} d_{\mu \nu \rho \lambda} = \sum_{PQRS} T^*_{\mu P} T_{\nu Q} T^*_{\rho R} T_{\lambda S} d'_{PQRS} \end{equation}\] and \[\begin{equation} d'_{PQRS} = \sum_{\mu \nu \rho \lambda} T^{-1 *}_{T \mu} T^{-1}_{Q \nu} T^{-1 *}_{R \rho} T^{-1}_{S \lambda} d_{\mu \nu \rho \lambda} \thinspace . \end{equation}\]