# 13.15 The Many-Body Expansion Method

The many-body expansion (MBE) for a system of $N$ monomers is given by

 $E=\sum_{I=1}^{N}\mbox{E_{I}}+\sum_{I}^{N}\sum_{J>I}^{N}\mbox{\Delta E_{IJ}% }+\\ \sum_{I}^{N}\sum_{J>I}^{N}\sum_{K>J}^{N}\mbox{\Delta E_{IJK}}+\\ \cdots,$ (13.53)

in which $E_{I}$ represents the energy of monomer $I$, $\Delta E_{IJ}$ = $E_{IJ}$ $-$ $E_{I}$ $-$ $E_{J}$ is a two-body correction for dimer $IJ$, and $\Delta E_{IJK}$ = $E_{IJK}$ $-$ $\Delta E_{IJ}$ $-$ $\Delta E_{IK}$ $-$ $\Delta E_{JK}$ $-$ $E_{I}$ $-$ $E_{J}$ $-$ $E_{k}$ is a three-body correction for trimer $IJK$, etc. In a large system and/or a large basis set, truncation of this expression at the two- or three-body level may dramatically reduce the amount of computer time that is required to compute the energy. Convergence of the MBE can be accelerated by embedding the monomer ($E_{I}$), dimer ($E_{IJ}$), trimer ($E_{IJK}$), $\ldots$ calculations in some representation of the electrostatic potential of the rest of the system. A simple means to do this is via atom-centered point charges that could be obtained when the $E_{I}$ terms are calculated; this is the so-called electrostatically-embedded many-body expansion (EE-MBE),207, 785, 786, 530 which we will denote as EE-MBE($n$) when the expansion is truncated at $n$-body terms. MBE($n$) and EE-MBE($n$) are available in Q-Chem, with analytic gradients, up to five-body terms ($n=5$).

It is well known that the interaction energies of non-covalent clusters are usually overestimated—often substantially—owing to basis-set superposition error (BSSE), which disappears only very slowly as the basis sets approach completeness. The widely used Boys-Bernardi counterpoise procedure corrects for this by computing all energies, cluster and individual monomers, using the full cluster basis set. (In clusters with more than two monomers, the obvious generalization of the Boys-Bernardi counterpoise correction is sometimes called the “site–site function counterpoise” correction or SSFC.) Note, however, that basis-set extrapolation is still necessary for high-quality binding energies. In $(\rm H_{2}O)_{6}$, for example, a counterpoise-corrected MP2/aug-cc-pVQZ calculation is still $\approx 1$ kcal/mol from the MP2 basis-set limit.783 Fortunately, the MBE allows for use of large basis sets in order to perform basis-set extrapolations in sizable clusters,783, 784 and one can employ a counterpoise correction that is consistent with an $n$-body expansion in order to obtain an $n$-body approximation to the Boys-Bernardi counterpoise-corrected supersystem energy. Two such corrections have been proposed: the many-body counterpoise correction, MBCP($n$),783, 784 and the $n$-body Valiron-Mayer function counterpoise correction, VMFC($n$).449 The two approaches are equivalent for $n=2$ but the MBCP($n$) method requires far fewer subsystem calculations starting at $n=3$ and is thus significantly cheaper, while affording very similar results as compared to VMFC($n$).783, 784

## 13.15.0.1 Job Control

A MBE($n$) calculation is requested by setting MANY_BODY_INT = TRUE in the $rem section. The level of theory used for the fragments will be whatever is specified in the$rem section. Researchers who use Q-Chem’s MBE code are asked to cite Ref. 785, 530. In addition, please cite Ref. 783 for the MBCP($n$) method.

A MBE($n$) calculation is requested by setting MANY_BODY_INT = TRUE in the $rem section. The level of theory used for the fragments will be whatever is specified in the$rem section. Researchers who use Q-Chem’s MBE code are asked to cite Ref. 785, 530. In addition, please cite Ref. 783 for the MBCP($n$) method.

MANY_BODY_INT
Perform a MBE calculation.
TYPE:
BOOLEAN
DEFAULT:
FALSE
OPTIONS:
TRUE Perform a MBE calculation. FALSE Do not perform a MBE calculation.
RECOMMENDATION:
NONE

Additional MBE-specific options, such as the order of the expansion ($n$), are specified in a $mbe input section, as described below. Order Specifies the order of the many-body expansion. INPUT SECTION:$mbe
TYPE:
INTEGER
DEFAULT:
None
OPTIONS:
$n$ Perform an MBE($n$) calculation.
RECOMMENDATION:
Orders $n\leq 5$ are available.

Embed
Specifies the embedding method for EE-MBE($n$).
INPUT SECTION: $mbe TYPE: STRING DEFAULT: None OPTIONS: None Do not use embedding. Charges Use atomic point charges. Density Full Coulomb embedding using monomer densities. RECOMMENDATION: Use of atomic point charges requires a$mbe_charges section to specify the charges.

Q-Chem’s implementation of EE-MBE($n$) with atomic point charges is designed to use with a $mbe_charges input section to specify fixed embedding charges. (Use of these charges is intended to accelerate convergence of the MBE by capturing many-body polarization effects and thus making the higher-order $n$-body terms smaller, although three- and four-body terms remain non-negligible even with embedding charges.530, 582) The format of the$mbe_charges section is simply a list of charges in the same order as the atoms in the $molecule section. An example is provided below. Many-body counterpoise corrections are requested with two keywords in the$mbe input section: BSSE_Type and BSSE_Order. These have only been implemented up to $n=3$, as the $n=2$ terms make by far the most significant contribution.582

BSSE_Order
Perform a many-body counterpoise correction of the MBCP($n$) or VMFC($n$) variety.
INPUT SECTION: $mbe TYPE: INTEGER DEFAULT: 0 OPTIONS: 0 Do not perform a counterpoise correction. $n$ Perform a counterpoise correction truncated at order $n$. RECOMMENDATION: Orders $n\leq 3$ are available. Use the keyword BSSE_Type to choose between MBCP and VMFC. BSSE_Type Select the type of many-body counterpoise correction, MBCP($n)$ or VMFC($n$). INPUT SECTION:$mbe
TYPE:
STRING
DEFAULT:
MBCP
OPTIONS:
MBCP Use MBCP($n$). VMFC Use VMFC($n$).
RECOMMENDATION:
The two methods are equivalent for $n=2$ but different for $n\geq 3$. MBCP($n$) contains fewer terms but generally provides comparable results as compared to the formally more complete VMFC($n$) approach.

Example 13.31  Example showing a EE-MBE(3) calculation using TIP3P charges.

$molecule 0 1 -- 0 1 O -1.126149 -1.748387 -0.423240 H -0.234788 -1.493897 -0.661862 H -1.062789 -2.681331 -0.218819 -- 0 1 O -0.254210 1.611495 -1.293845 H -1.001520 1.163510 -1.690129 H -0.153399 2.411746 -1.809248 -- 0 1 O 1.694541 -0.226287 1.705739 H 0.785920 0.073487 1.677909 H 2.047134 0.150917 2.511706 -- 0 1 O -0.864533 0.522472 1.218817 H -0.694120 1.093542 0.469789 H -1.131418 -0.310426 0.829702$end

$rem SYM_IGNORE true METHOD B3LYP BASIS cc-pVDZ MANY_BODY_INT true THRESH 14 SCF_CONVERGENCE 7$end

$mbe order 3 embed charges$end

$mbe_charges -0.834 0.417 0.417 -0.834 0.417 0.417 -0.834 0.417 0.417 -0.834 0.417 0.417$end


Example 13.32  Example of a MBCP(3) calculation.

$molecule 0 1 -- 0 1 O -1.126149 -1.748387 -0.423240 H -0.234788 -1.493897 -0.661862 H -1.062789 -2.681331 -0.218819 -- 0 1 O -0.254210 1.611495 -1.293845 H -1.001520 1.163510 -1.690129 H -0.153399 2.411746 -1.809248 -- 0 1 O 1.694541 -0.226287 1.705739 H 0.785920 0.073487 1.677909 H 2.047134 0.150917 2.511706 -- 0 1 O -0.864533 0.522472 1.218817 H -0.694120 1.093542 0.469789 H -1.131418 -0.310426 0.829702$end

$rem MANY_BODY_INT TRUE METHOD B3LYP BASIS cc-pVDZ THRESH 12 SCF_CONVERGENCE 6$end

$mbe BSSE_Order 3 BSSE_Type MBCP ! this is the default$end