12.13 The Generalized Many-Body Expansion Method 12.13 The Generalized Many-Body Expansion Method 12.13.2 Job Control

12.13.1 Introduction

(September 23, 2025)

The generalized many-body expansion (GMBE) method approximates the energy of a supersystem using the energies of its fragments or subsystems, determined by a distance-based threshold. ¹¹²³ Richard R. M., Herbert J. M.
J. Chem. Phys.
(2012), 137, pp. 064113. Link ^, ⁸⁹⁴ Mayhall N. J., Raghavachari K.
J. Chem. Theory Comput.
(2012), 8, pp. 2669. Link ^, ¹¹²⁴ Richard R. M., Herbert J. M.
J. Chem. Theory Comput.
(2013), 9, pp. 1408. Link This threshold can be defined by the minimum atomic distance between “unit fragments” of the supersystem. Alternatively, one may use a criterion based on the heavy-atom distance, which excludes hydrogen atoms. The resulting subsystems, identified based on this threshold, are referred to as primitive fragments (monomers). These primitive fragments serve as the foundation for generating a set of fragment calculations used to approximate the supersystem’s energy. To prevent redundancy, the principle of inclusion and exclusion (PIE) is enforced.

In the $n$ -body GMBE, denoted as GMBE( $n$ ), the energy of the supersystem is expressed as

E=\sum_{I}^{{N}\choose{n}}{\mathcal{E}}_{I}^{(n)},

(12.74)

where $N$ is the number of primitive fragments.

The GMBE implementation in Q-Chem is currently limited to first-order, GMBE(1), and employs a novel binning algorithm for efficiently generating the set of primitive fragments. ⁶⁹ Ballesteros F., Tan J. A., Lao K. U.
J. Chem. Phys.
(2023), 159, pp. 074107. Link The supersystem’s energy is approximated using GMBE(1) as:

\displaystyle E

\displaystyle\approx\sum_{I}^{n}\mathcal{E}_{I}^{(1)}.

(12.75)

The intersection-corrected energy for fragment $I$ is

\mathcal{E}_{I}^{(1)}=E_{I}^{(1)}-\sum_{{\begin{subarray}{c}J\\ (J>I)\end{subarray}}}E_{I\cap J}^{(1)}\\ +\sum_{{\begin{subarray}{c}J,K\\ (K>J>I)\end{subarray}}}E_{I\cap J\cap K}^{(1)}-\sum_{{\begin{subarray}{c}J,K,L% \\ (L>K>J>I)\end{subarray}}}E_{I\cap J\cap K\cap L}^{(1)}+\cdots,

(12.76)

where $E_{I}^{(1)}$ , $E_{I\cap J}^{(1)}$ , $E_{I\cap J\cap K}^{(1)}$ , and $E_{I\cap J\cap K\cap L}^{(1)}$ denote energies respectively of the fragment $I$ , the intersection of fragments $I$ and $J$ , the intersection of fragments $I$ , $J$ , and $K$ , and the intersection of fragments $I$ , $J$ , $K$ , and $L$ .

Similarly, the GMBE(1) approximation for the supersystem’s density matrix is given by:

\displaystyle P

\displaystyle\approx\sum_{I}^{n}\mathcal{P}_{I}^{(1)}

(12.77)

where the intersection-corrected density matrix sub-blocks for subset I are defined as:

\mathcal{P}_{I}^{(1)}=P_{I}^{(1)}-\sum_{{\begin{subarray}{c}J\\ (J>I)\end{subarray}}}P_{I\cap J}^{(1)}\\ +\sum_{{\begin{subarray}{c}J,K\\ (K>J>I)\end{subarray}}}P_{I\cap J\cap K}^{(1)}-\sum_{{\begin{subarray}{c}J,K,L% \\ (L>K>J>I)\end{subarray}}}P_{I\cap J\cap K\cap L}^{(1)}+\cdots

(12.78)

The GMBE(1) density matrix can be used to predict the supersystem’s energy through a single Fock build. Alternatively, it can serve as an initial guess in a SCF calculation for the supersystem.

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12.13.1 Introduction