Overlap between ONVs in different general spinor bases

Suppose we have a set of two orthonormal spinor bases, \(\{ \require{physics} \phi_{P, \thinspace \text{of}} \}\) and \(\{ \psi_{P, \thinspace \text{on}} \}\), which are used to construct ONV bases in which the ONVs \(\ket{\text{on}}\) and \(\ket{\text{of}}\) are expressed. Here, we will determine the following overlap element of these ONVs: \[\begin{equation} \braket{\text{on}}{\text{of}} \thinspace . \end{equation}\]

Since these spinor bases are both orthonormal, they must be related by a unitary transformation: \[\begin{equation} \phi_{P, \thinspace \text{of}} = \sum_Q^M \phi_{Q, \thinspace \text{on}} \thinspace U_{QP} \thinspace , \end{equation}\] with \[\begin{align} U_{PQ} &= \braket{\phi_{P, \thinspace \text{on}}}{\phi_{Q, \thinspace \text{of}}} \\ &= \vb{C}^\dagger_{\text{on}} \begin{pmatrix} \vb{S} & \vb{0} \\ \vb{0} & \vb{S} \\ \end{pmatrix} \vb{C}_{\text{of}} \thinspace , \end{align}\] where \(\vb{S}\) is the overlap matrix of the scalar basis that is used for the expansion of the \(\alpha\)- and \(\beta\)-components of both orthonormal bases.

The elementary creation operators transform in the same way: \[\begin{equation} \hat{c}^\dagger_P = \sum_Q^M \hat{a}^\dagger_{Q} U_{QP} \thinspace , \end{equation}\] which means that, if the state \(\{ \phi_{P, \thinspace \text{of}} \}\) is represented by the ONV \(\ket{\vb{m}}\), we can rewrite the operator creation string \(\hat{C}^\dagger(\vb{m})\) as: \[\begin{align} \hat{C}^\dagger(\vb{m}) &= \prod_P^M (\hat{c}^\dagger_P)^{m_P} \\ &= \hat{c}^\dagger_{P_1} \hat{c}^\dagger_{P_2} \cdots \hat{c}^\dagger_{P_N} \thinspace , \end{align}\] where \(P_i\) represents the spinor index that is occupied by electron number \(i\). Using the transformation formula, we can subsequently write: \[\begin{align} \hat{C}^\dagger(\vb{m}) &= \qty( \sum_{Q_1}^M \hat{a}^\dagger_{Q_1} U_{Q_1 P_1} ) \qty( \sum_{Q_2}^M \hat{a}^\dagger_{Q_2} U_{Q_2 P_2} ) \ldots \qty( \sum_{Q_N}^M \hat{a}^\dagger_{Q_N} U_{Q_N P_N} ) \\ &= \sum_{Q_1}^M \sum_{Q_2}^M \ldots \sum_{Q_N}^M U_{Q_1 P_1} U_{Q_2 P_2} \ldots U_{Q_N P_N} \thinspace \hat{a}^\dagger_{Q_1} \hat{a}^\dagger_{Q_2} \ldots \hat{a}^\dagger_{Q_N} \thinspace . \end{align}\] Due to the fermion anticommutation relations, the indices \(Q_1, Q_2, ldots, Q_N\) should all be different: \[\begin{equation} Q_1 \neq Q_2 \neq \ldots \neq Q_N \thinspace , \end{equation}\] so they can be seen as the elements of another occupation number vector \(\vb{k}\). Many of these are actually equal, up to a sign factor, which we will collect into a determinant of the following matrix7 \(\vb{U}(\vb{k}, \vb{m})\):

  • The rows of \(\vb{U}(\vb{k}, \vb{m})\) are those of \(\vb{U}\) with an index that refers to an occupied spin-orbital in \(\vb{k}\).
  • The columns of \(\vb{U}(\vb{k}, \vb{m})\) are those of \(\vb{U}\) with an index that refers to an occupied spin-orbital in \(\vb{m}\);

Finally, this means that we can rewrite the creation string \(\hat{C}^\dagger(\vb{m})\) as: \[\begin{equation} \hat{C}^\dagger(\vb{m}) = \sum_{\vb{k}} \det\qty( \vb{U}(\vb{k}, \vb{m}) ) \hat{A}^\dagger(\vb{k}) \thinspace , \end{equation}\] where \(\hat{A}^\dagger(\vb{k})\) creates an ONV \(\ket{\vb{k}}\) expressed in the spinors that are projected ‘on’: \(\{ \psi_{P, \thinspace \text{on}} \}\), such that we can write the expansion of \(\ket{\text{on}}\) inside an ONV basis spanned by all excitations \(\ket{\vb{k}}\) as \[\begin{equation} \ket{\vb{m}} = \sum_{\vb{k}} \det\qty( \vb{U}(\vb{k}, \vb{m}) ) \ket{\vb{k}} \thinspace , \end{equation}\] where the individual coefficients are given by: \[\begin{equation} \braket{\vb{k}}{\vb{m}} = \det\qty( \vb{U}(\vb{k}, \vb{m}) ) \thinspace . \end{equation}\]

  1. We should note that the selection of the rows and columns is consistent with the notation \(\vb{U}(\vb{k}, \vb{m})\)↩︎