CCDT t3 contributions from CCAlgebra

The $(t_2t_3)$ terms

$$+P(ij)P(iw)\frac{1}{2} \sum_{cdkl} \langle kl || cd \rangle t_{ik}^{cd} t_{jwl}^{abz}$$$$+P(iw)P(jw)\frac{1}{4} \sum_{cdkl} \langle kl || cd \rangle t_{ij}^{cd} t_{wkl}^{abz}$$$$+P(ab)P(az)\frac{1}{2} \sum_{ckld} \langle kl || cd \rangle t_{kl}^{ca} t_{ijw}^{dbz}$$$$+P(ab)P(az)P(ij)P(iw)\frac{1}{1} \sum_{ckdl} \langle kl || cd \rangle t_{ik}^{ca} t_{jwl}^{dbz}$$$$+P(ab)P(az)P(iw)P(jw)\frac{1}{2} \sum_{cdkl} \langle kl || cd \rangle t_{ij}^{ca} t_{wkl}^{dbz}$$$$+P(az)P(bz)\frac{1}{4} \sum_{klcd} \langle kl || cd \rangle t_{kl}^{ab} t_{ijw}^{cdz}$$$$+P(az)P(bz)P(ij)P(iw)\frac{1}{2} \sum_{kcdl} \langle kl || cd \rangle t_{ik}^{ab} t_{jwl}^{cdz}$$$$+P(ab)P(az)P(iw)P(jw)\frac{1}{2} \sum_{cdkl} \langle kl || cd \rangle t_{ijk}^{cda} t_{wl}^{bz}$$$$+P(ab)P(az)\frac{1}{4} \sum_{cdkl} \langle kl || cd \rangle t_{ijw}^{cda} t_{kl}^{bz}$$$$+P(az)P(bz)P(ij)P(iw)\frac{1}{2} \sum_{ckld} \langle kl || cd \rangle t_{ikl}^{cab} t_{jw}^{dz}$$$$+P(az)P(bz)P(iw)P(jw)\frac{1}{1} \sum_{ckdl} \langle kl || cd \rangle t_{ijk}^{cab} t_{wl}^{dz}$$$$+P(az)P(bz)\frac{1}{2} \sum_{cdkl} \langle kl || cd \rangle t_{ijw}^{cab} t_{kl}^{dz}$$$$+P(ij)P(iw)\frac{1}{4} \sum_{klcd} \langle kl || cd \rangle t_{ikl}^{abz} t_{jw}^{cd}$$$$+P(iw)P(jw)\frac{1}{2} \sum_{kcdl} \langle kl || cd \rangle t_{ijk}^{abz} t_{wl}^{cd}$$

The linear $(t_3)$ terms

$$+P(ba)P(za)\frac{1}{1} \sum_{c} \langle a || c \rangle t_{ijw}^{cbz}=0$$$$+P(iw)P(jw)\frac{-1}{1} \sum_{k} \langle k || w \rangle t_{ijk}^{abz}=0$$$$+P(za)P(zb)\frac{1}{2} \sum_{cd} \langle ab || cd \rangle t_{ijw}^{cdz}$$$$+P(iw)P(ij)\frac{1}{2} \sum_{kl} \langle kl || jw \rangle t_{ikl}^{abz}$$$$+P(ba)P(za)P(iw)P(jw)\frac{-1}{1} \sum_{ck} \langle ak || cw \rangle t_{ijk}^{cbz}$$

The linear $(t_2)$ terms

A bug in the code misrepresented the two body interaction

$$+P(ba)P(bz)P(iw)P(jw)\frac{-1}{1} \sum_{c} \langle ax || cx \rangle t_{ij}^{cb}$$$$+P(az)P(bz)P(ij)P(iw)\frac{1}{1} \sum_{k} \langle kx || j x\rangle t_{ik}^{ab}$$

The quadratic $(t_2)$ terms

$$+P(ab)P(az)P(iw)P(jw)\frac{1}{1} \sum_{ck} \langle k || c \rangle t_{ij}^{ca} t_{wk}^{bz}$$$$+P(iw)P(jw)P(ba)P(za)\frac{1}{2} \sum_{cdk} \langle ak || cd \rangle t_{ij}^{cd} t_{wk}^{bz}$$$$+P(bz)P(ba)P(ij)P(iw)P(za)\frac{-1}{1} \sum_{ckd} \langle ak || cd \rangle t_{ik}^{cb} t_{jw}^{dz}$$$$+P(ab)P(az)P(iw)P(ij)P(wj)\frac{1}{1} \sum_{ckl} \langle kl || cj \rangle t_{ik}^{ca} t_{wl}^{bz}$$$$+P(ab)P(az)P(iw)P(jw)\frac{-1}{2} \sum_{ckl} \langle kl || cw \rangle t_{ij}^{ca} t_{kl}^{bz}$$