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

$$E=\sum _{I=1}^{N}{E}_{I}+\sum _{I}^{N}\sum _{J>I}^{N}\mathrm{\Delta}{E}_{IJ}+\sum _{I}^{N}\sum _{J>I}^{N}\sum _{K>J}^{N}\mathrm{\Delta}{E}_{IJK}+\mathrm{\cdots},$$ | (12.63) |

in which ${E}_{I}$ represents the energy of monomer $I$, $\mathrm{\Delta}{E}_{IJ}$ =
${E}_{IJ}$ $-$ ${E}_{I}$ $-$ ${E}_{J}$ is a two-body correction for dimer $IJ$, and
$\mathrm{\Delta}{E}_{IJK}$ = ${E}_{IJK}$ $-$ $\mathrm{\Delta}{E}_{IJ}$ $-$ $\mathrm{\Delta}{E}_{IK}$ $-$
$\mathrm{\Delta}{E}_{JK}$ $-$ ${E}_{I}$ $-$ ${E}_{J}$ $-$ ${E}_{k}$ 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 (${E}_{IJK}$), $\mathrm{\dots}$ 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),^{Dahlke:2007, Richard:2014a, Richard:2014b, Lao:2016a} 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 ${({\mathrm{H}}_{2}\mathrm{O})}_{6}$, for example, a counterpoise-corrected MP2/aug-cc-pVQZ calculation is still $\approx 1$ kcal/mol from the MP2 basis-set
limit.^{Richard:2013a} Fortunately, the MBE allows for use of large basis
sets in order to perform basis-set extrapolations in sizable
clusters,^{Richard:2013a, Richard:2013b} 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($n$),^{Richard:2013a, Richard:2013b} and the $n$-body
Valiron-Mayer function counterpoise correction, VMFC($n$).^{Kamiya:2008}
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($n$).^{Richard:2013a, Richard:2013b}

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. Richard:2014a, Lao:2016a. In addition, please cite
Ref. Richard:2013a for the MBCP($n$) method.

*$rem* section. The level of theory used for the fragments will be whatever is specified in the *$rem* section.
Researchers who use Q-Chem’s MBE code are asked to cite Ref. Richard:2014a, Lao:2016a.
In addition, please cite Ref. Richard:2013a 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 $n\le 5$ 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.^{Lao:2016a, Liu:2017b}) 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.^{Liu:2017b}

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 $n\le 3$ 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 $n\ge 3$.
MBCP($n$) contains fewer terms but generally provides comparable results as compared
to the formally more complete VMFC($n$) approach.

$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

$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