CCD Intermediates

$$ t^{ab}_{ij} \epsilon^{ab}_{ij} = \langle ab \vert \vert ij\rangle + \frac{1}{2} \sum_{cd} \langle ab \vert \vert cd \rangle t^{cd}_{ij} + \frac{1}{2} \sum_{kl} \langle kl \vert \vert ij \rangle t^{ab}_{kl} + \hat{P}(ij \vert ab) \sum_{kc} \langle kb \vert \vert cj \rangle t^{ac}_{ik} + \frac{1}{4} \sum_{klcd} \langle kl \vert \vert cd \rangle t^{cd}_{ij} t^{ab}_{kl} + \hat{P}(ij) \sum_{klcd} \langle kl \vert \vert cd \rangle t^{ac}_{ik} t^{bd}_{jl} - \frac{1}{2} \hat{P}(ij) \sum_{klcd} \langle kl \vert \vert cd \rangle t^{dc}_{ik} t^{ab}_{lj} - \frac{1}{2} \hat{P}(ab)\sum_{klcd}\langle kl \vert \vert cd \rangle t^{ac}_{lk} t^{db}_{ij} $$

Define

$$I_1 = \langle kl || ij \rangle + \frac{1}{2} \sum_{cd} \langle kl || cd \rangle t^{cd}_{ij}$$$$I_2 = \langle kb || cj \rangle + \frac{1}{2} \sum_{cd} \langle kl || cd \rangle t^{db}_{lj}$$$$I_3 = \sum_{lcd} \langle kl || cd \rangle t^{cd}_{jl}$$$$I_4 = \sum_{kld} \langle kl || cd \rangle t^{bd}_{kl}$$

And substitute to find:

$$ t^{ab}_{ij} \epsilon^{ab}_{ij} = \langle ab \vert \vert ij\rangle + \frac{1}{2} \sum_{cd} \langle ab \vert \vert cd \rangle t^{cd}_{ij} + \frac{1}{2} \sum_{kl} t^{ab}_{kl} {I_1}^{kl}_{ij} + \hat{P}(ij) \hat{P}(ab) \sum_{kc} t^{ac}_{ik} {I_2}^{kb}_{cj} - \frac{1}{2} \hat{P}(ij) \sum_{k} t^{ab}_{ik} {I_3}^k_j - \frac{1}{2} \hat{P}(ab) \sum_c t^{ac}_{ij} {I_4}^b_c $$