Quasiparticle

Four-Center Two-Electron Coulomb Integral

Directory Structure

• df_hf_int: Directory where the three-center two-electron integrals are saved.
• V_0.h5: HDF5 file containing the three-center two-electron intagrals.

Integrals Definition

Chemist’s Notation

In chemist’s notation, the integral is represented as
$$
\begin{equation}
U_{ijkl} = (ij\vert kl) = \int\int dr_1 dr_2 \phi^*_i(r_1)\phi_j(r_1)\frac{1}{\vert r_1-r_2\vert}\phi^*_k(r_2)\phi_l(r_2)
\end{equation}
$$

• Function: chem_four_center_integral computes this integral.
• Used for scGW

Physicist’s Notation

In physicist’s notation, the integral is defined as
$$
\begin{equation}
V_{ijkl} = \langle ij\vert kl\rangle = \int\int dr_1 dr_2 \phi^*_i(r_1)\phi_j^*(r_2)\frac{1}{\vert r_1-r_2\vert}\phi_k(r_2)\phi_l(r_1)
\end{equation}
$$

• Function: phys_four_center_integral computes this integral.
• Used for GF2

Simplification through Density Fitting

The expression for $U_{ijkl}=\sum_{Q} V_{ij}(Q)V_{kl}(Q)$ and $V_{ijkl}=\sum_{Q} V_{il}(Q)V_{jk}(Q)$ suggests a simplification technique known as density fitting or resolution of the identity (RI), where the complex four-center integrals are approximated using a sum over simpler terms involving fewer centers. This approach significantly reduces computational cost.
$$
\begin{equation}
V_{ij}(Q) = (ij\vert Q) = \int \int dr_1dr_2 \phi_i(r_1) \phi_j(r_1) \frac{1}{\vert r_1-r_2\vert}\chi_{Q}(r_2)
\end{equation}
$$

Usage

Read the integrals from df_hf_int

V_Qij = read_integrals(path_to_df)

Compute the coulomb integral in chemist’s notation, $U_{ijkl}$

U_ijkl = chem_four_center_integral(V_Qij)

Compute the coulomb in physicist’s notation, $V_{ijkl}$

V_ijkl = phys_four_center_integral(V_Qij)