One-electron excitation operators

Since \(\hat{a}_Q\) annihilates an electron and \(\hat{a}^\dagger_P\) creates an electron, it is often useful to observe their combined effect. \(\hat{E}_{PQ}\) is a one-electron excitation operator: \[\begin{equation} \require{physics} \hat{E}_{PQ} = \hat{a}^\dagger_P \hat{a}_Q \thinspace . \end{equation}\] The one-electron excitation operators are said to generate the unitary group, because their commutators are: \[\begin{equation} \comm{ \hat{E}_{PQ} }{ \hat{E}_{RS} } = \delta_{RQ} \hat{E}_{PS} - \delta_{PS} \hat{E}_{RQ} \thinspace . \end{equation}\]

The Hermitian adjoint of the one-electron excitation operators are: \[\begin{equation} \hat{E}^\dagger_{PQ} = \hat{E}_{QP} \thinspace . \end{equation}\]

Some commutators with the elementary operators are: \[\begin{align} & \comm{ \hat{a}_P^\dagger }{ \hat{E}_{QR} } = - \delta_{PR} \hat{a}^\dagger_Q \\ % & \comm{ \hat{a}_P }{ \hat{E}_{QR} } = \delta_{PQ} \hat{a}_R \thinspace . \end{align}\]

Some special linear combinations of excitations also deserve some attention:

• \(\hat{E}^-_{PQ}\) is an anti-Hermitian operator: \[\begin{equation} \hat{E}^-_{PQ} = \hat{E}_{PQ} - \hat{E}_{QP} \end{equation}\]

• \(\hat{E}^+_{PQ}\) is a Hermitian operator: \[\begin{equation} \hat{E}^+_{PQ} = \hat{E}_{PQ} + \hat{E}_{QP} \thinspace . \end{equation}\]

We can find the action of \(\hat{E}_{PQ}\) on an ON vector \(\ket{\vb{k}}\) to be: \[\begin{equation} \hat{E}_{PQ} \ket{\vb{k}} = \varepsilon_{PQ} \Gamma^{\vb{k}}_P \Gamma^{\vb{k}}_Q \thinspace (1 - k_P + \delta_{PQ}) k_Q \thinspace \ket{\ldots, 1_P, \ldots, \delta_{PQ_P}, \ldots} \thinspace , \end{equation}\] where \[\begin{equation} \varepsilon_{PQ} = \begin{cases} +1 \qquad P \leq Q \\ -1 \qquad P > Q \end{cases} \thinspace . \end{equation}\]