Overlap between ONVs in restricted and unrestricted spin-orbital bases

Suppose we have a set of RHF spin orbitals \(\{ \phi_{p \sigma} \}\) and a set of UHF spin orbitals \(\{ \psi_{p \sigma} \}\). Both sets of orbitals are constructed as a linear expansion of AOs \(\{ \chi_{\mu} \}\): \[\begin{equation} \require{physics} \phi_{p \sigma} = \sum_{\mu}^K \chi_{\mu} C_{\mu p} \end{equation}\] and \[\begin{equation} \psi_{p \sigma} = \sum_{\mu}^K \chi_{\mu} C^{\sigma}_{\mu p} \thinspace . \end{equation}\]

Expressing the UHF orbitals in terms of the RHF orbitals yields: \[\begin{equation} \psi_{p \sigma} = \sum_{q}^K \phi_{q \sigma} T^{\sigma}_{qp} \thinspace , \end{equation}\] where the RHF to UHF transformation matrices \(\vb{T}^{\sigma}\) are given by \[\begin{equation} \vb{T}^{\sigma} = \vb{C}^{-1} \vb{C}^\sigma \thinspace . \end{equation}\] Indeed, we can show that \[\begin{align} \vb{T}^{\sigma} \vb{T}^{\sigma \dagger} &= \vb{C}^{-1} \vb{C}^\sigma \vb{C}^{\sigma \thinspace \dagger} \vb{C}^{-1 \thinspace \dagger} \\ & = \vb{I} \thinspace , \end{align}\] using the theory of non-orthogonal spinor bases. Furthermore, we can confirm that the elements of \(T^\sigma_{pq}\) are actually the overlap between the UHF and RHF spin-orbitals: \[\begin{align} T^\sigma_{pq} &= \qty[ \vb{C}^{-1} \vb{C}^\sigma ]_{pq} \\ &= \qty[ \vb{C}^\dagger \vb{S} \vb{C}^\sigma ]_{pq} \\ &= \braket{\phi_{p \sigma}}{\psi_{p \sigma}} \thinspace , \end{align}\] and where \(\vb{S}\) represents the overlap matrix of the AOs.

The elementary creation operators expressed in the UHF spin-orbital basis can then be written as: \[\begin{equation} \hat{c}^\dagger_{p \sigma} = \sum_q^K \hat{a}^\dagger_{q \sigma} T^{\sigma}_{qp} \thinspace , \end{equation}\] so that we can write any ONV with respect to the unrestricted spin-orbitals as: \[\begin{equation} \ket{\vb{m}_\alpha \vb{m}_\beta} = \hat{C}^\dagger_{\alpha}(\vb{m}_\alpha) \hat{C}^\dagger_{\beta}(\vb{m}_\beta) \ket{\text{vac}} \thinspace , \end{equation}\] where \(\hat{C}^\dagger_{\sigma}\) creates the \(\sigma\)-part of the ‘unrestricted’ ONV: \[\begin{equation} \hat{C}^\dagger_{\sigma}(\vb{m}_\sigma) = \prod_p^K \qty( \hat{c}^\dagger_{p \sigma} )^{m_{\sigma, p}} \thinspace . \end{equation}\]

Expressing the UHF-related creation operators with respect to their RHF counterparts, we find the following intermediary result: \[\begin{equation} \hat{C}^\dagger_{\sigma}(\vb{m}_\sigma) = \sum_{q_1}^K \cdots \sum_{q_{N_\sigma}}^K T^{\sigma}_{q_1 \thinspace m_\sigma(1)} \cdots T^{\sigma}_{q_{N_\sigma} \thinspace m_\sigma(N_\sigma)} \thinspace \hat{a}^\dagger_{q_1 \sigma} \cdots \hat{a}^\dagger_{q_{N_P} \sigma} \thinspace , \end{equation}\] which represents an \(N_\sigma\)-fold summation. In this summation all summation indices \(q_i\) must be different due to the fermion creation operators \(\hat{a}^\dagger_{p \sigma}\): \[\begin{equation} q_1 \neq q_2 \neq \ldots \neq q_{N_P} \thinspace , \end{equation}\] for a non-zero result. Furthermore, many of the strings of creation operators are equal up to a sign (due to transposition of the fermion creators) that can be collected in a determinant. We will now introduce the \((N_\sigma \times N_\sigma)\)-matrix \(\vb{T}^\sigma(\vb{k}_\sigma, \vb{m}_\sigma)\), composed8 of the following:

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

Taking all this into account, the ‘unrestricted’ \(\sigma\)-creation string can be rewritten as a linear combination of ‘restricted’ creation strings \(\hat{A}^\dagger_{\sigma}(\vb{k}_\sigma)\) \[\begin{equation} \hat{C}^\dagger_{\sigma}(\vb{m}_\sigma) = \sum_{\vb{k}_\sigma} \det\qty[ \vb{T}^\sigma(\vb{k}_\sigma, \vb{m}_\sigma) ] \hat{A}^\dagger_{\sigma}(\vb{k}_\sigma) \thinspace , \end{equation}\] where \(\hat{A}^\dagger_{\sigma}(\vb{k}_\sigma)\) yields the ONV when acting on top of the vacuum: \[\begin{align} \hat{A}^\dagger_{\sigma}(\vb{k}_\sigma) \ket{\text{vac}} &= \prod_{p}^K \qty(\hat{a}^\dagger_{p \sigma})^{k_{p \sigma}} \ket{\text{vac}} \\ &= \ket{\vb{k}_\sigma} \thinspace . \end{align}\]

Using these results, the ‘unrestricted’ ONV may be expressed as follows: \[\begin{equation} \ket{\vb{m}_\alpha \vb{m}_\beta} = \sum_{\vb{k}_{\alpha} \vb{k}_{\beta}} \det\qty[ \vb{T}^\alpha(\vb{k}_\alpha, \vb{m}_\alpha) \vb{T}^\beta(\vb{k}_\beta, \vb{m}_\beta) ] \ket{\vb{k}_\alpha \vb{k}_\beta} \thinspace , \end{equation}\] where the expansion coefficients are given by: \[\begin{equation} \braket{\vb{k}_\alpha \vb{k}_\beta}{\vb{m}_\alpha \vb{m}_\beta} = \det\qty[ \vb{T}^\alpha(\vb{k}_\alpha, \vb{m}_\alpha) \vb{T}^\beta(\vb{k}_\beta, \vb{m}_\beta) ] \thinspace . \end{equation}\]

  1. We should note that the selection of the rows and columns is consistent with the notation \(\vb{T}^\sigma(\vb{k}_\sigma, \vb{m}_\sigma)\), i.e. a row refers to the first argument and a column refers to the second.↩︎