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.
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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 -body GMBE, denoted as GMBE(), the energy of the supersystem is expressed as
(12.74) |
where 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.
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The supersystem’s energy is approximated using GMBE(1) as:
(12.75) |
The intersection-corrected energy for fragment is
(12.76) |
where , , , and denote energies respectively of the fragment , the intersection of fragments and , the intersection of fragments , , and , and the intersection of fragments , , , and .
Similarly, the GMBE(1) approximation for the supersystem’s density matrix is given by:
(12.77) |
where the intersection-corrected density matrix sub-blocks for subset I are defined as:
(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.