Fock operator invariance under unitary transformation

Question

I know how to show that the Coulomb operator of the Fock operator is invariant under a unitary transformation of the orbitals, as on page 121 of Szabo and Ostlund, but the indices in my proof for the exchange operator are just not working. What might I be missing? I have

\begin{align} \sum_i\hat{K}_i'(1) &=\sum_i\int dr_2\chi_i'^*(2)\chi_j'(2)/r_{12} \\ &=\sum_i\int dr_2\sum_kU_{ki}^*\chi_k^*(2)\sum_lU_{lj}\chi_l(2)/r_{12}\\ &=\sum_{ikl}\int dr_2U_{ki}^*U_{lj}\chi_k^*(2)\chi_l(2)/r_{12}\\ &=\sum_{i}\int dr_2\chi_i^*(2)\chi_i(2)/r_{12}\\ &\neq\sum_{i}\int dr_2\chi_i^*(2)\chi_j(2)/r_{12}. \end{align}

Answer 1

The exchange operator is not a operator by itself, it is only defined with the orbital it is working on: $% \newcommand{\ll}{\left\langle}\newcommand{\rr}{\right\rangle} \newcommand{\lb}{\left|}\newcommand{\rb}{\right|} \newcommand{\op}[1]{\mathbf{#1}}$ $$\begin{align} && \op{F}_i &= \op{H}^\mathrm{c} + \sum_j (\op{J}_j - \op{K}_j),\\ \text{with}&& \op{J}_j\lb \phi_i\rr &= \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} \lb \phi_j(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr,\\ \text{and}&& \op{K}_j\lb \phi_i\rr &= \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} \lb \phi_i(\op{x}_1) \rr \lb \phi_j(\op{x}_2) \rr. \end{align}$$

While in the Coulomb operator case the $\lb \phi_i\rr$ doesn't do anything, so you can take it through the proof as a constant:^* \begin{align} \sum_j\op{J}_j\lb \phi_i\rr &= \sum_j \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} \lb \phi_j(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\ &= \sum_k \sum_l \sum_j U_{kj}^*U_{lj} \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} \lb \phi_l(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\ &= \sum_k \sum_l \delta_{kl} \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} \lb \phi_l(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\ &= \sum_k \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} \lb \phi_k(\op{x}_1) \rr \lb \phi_i(\op{x}_2) \rr\\ &= \sum_k \op{J}_k \lb \phi_i\rr \end{align}

That is not the case in the Exchange operator, as it switches the orbitals: \begin{align} \sum_j\op{K}_j\lb \phi_i\rr &= \sum_j \ll \phi_j(\op{x}_1) \rb r_{12}^{-1} \lb \phi_i(\op{x}_1) \rr \lb \phi_j(\op{x}_2) \rr\\ &= \sum_k \sum_l \sum_j U_{kj}^*U_{lj} \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} \lb \phi_i(\op{x}_1) \rr \lb \phi_l(\op{x}_2) \rr\\ &= \sum_k \sum_l \delta_{kl} \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} \lb \phi_i(\op{x}_1) \rr \lb \phi_l(\op{x}_2) \rr\\ &= \sum_k \ll \phi_k(\op{x}_1) \rb r_{12}^{-1} \lb \phi_i(\op{x}_1) \rr \lb \phi_k(\op{x}_2) \rr\\ &= \sum_k \op{K}_k \lb \phi_i\rr \end{align}

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^* Let $$ \mathbb{U}=\left(\begin{matrix} U_{11} & U_{12} & \dots \\ U_{21} & U_{22} & \dots \\ \vdots & \vdots & \ddots\\ \end{matrix}\right); \mathbb{U}^\dagger\mathbb{U}=\mathbb{E} \implies U_{ij}^* U_{kl}= \delta_{ij} = \begin{cases} 1; & i = j = k = l \\ 0; & i \neq j \dots\\ \end{cases} .$$