Notation

Dirac Notation

One electron integrals

$\langle p | \hat{h} | q \rangle = \int \phi^*_p(\mathbf{r}) \hat{h} \phi_q(\mathbf{r}) d\mathbf{r}$

Dirac notation for two electron integrals

$\langle pq | rs \rangle = \int \phi^*_p(\mathbf{r}_1) \phi^*_q(\mathbf{r}_2) \frac{1}{r_{12}} \phi_r(\mathbf{r}_1) \phi_s(\mathbf{r}_2) d\mathbf{r}_1 d\mathbf{r}_2$

Mulliken notation

$\left(pr|qs\right) = \int \phi^*_p(\mathbf{r}_1)\phi_r(\mathbf{r}_1) \frac{1}{r_{12}} \phi^*_q(\mathbf{r}_2) \phi_s(\mathbf{r}_2) d\mathbf{r}_1 d\mathbf{r}_2$

ERI

$\langle pq||rs \rangle = \langle pq | rs \rangle - \langle pq | sr \rangle = \left(pr|qs\right) - \left(ps|qr\right)$

CC Notation

Normal order

Normal ordered string

$\{a^\dagger_p a_q\}$

Examples:

$\begin{aligned} & \{a^\dagger_p a_q\} = a^\dagger_p a_q \\ & \{a_q a^\dagger_p\} = -a^\dagger_p a_q \\ & \{a^\dagger_p a^\dagger_q a_r a_s\} = a^\dagger_p a^\dagger_q a_r a_s \\ & \{a^\dagger_p a_sa^\dagger_q a_r\} = -a^\dagger_p a^\dagger_q a_r a_s = a^\dagger_p a^\dagger_q a_s a_r \\ \end{aligned}$

Determinant

$|\Phi_0\rangle = a^\dagger_p a_q |0\rangle$