{X}}(\ce{X}), \\
E_{\text{BSSE}}(\ce{Y}) &= E^{\ce{XY}}(\ce{Y}) - E^{\ce{Y}}(\ce{Y}),
\end{split}
\label{eq:2}
$$
which, when subtracted from $\eqref{eq:sherrill-1}$, gives $\eqref{eq:sherrill-3}$. More correctly, considering that the geometry for each step is at the final cluster geometry and not the isolated geometry, the above is [4]
$$
\begin{split}
E_{\text{BSSE}}(\ce{X}) &= E_{\ce{XY}}^{\ce{XY}}(\ce{X}) - E_{\ce{XY}}^{\ce{X}}(\ce{X}), \\
E_{\text{BSSE}}(\ce{Y}) &= E_{\ce{XY}}^{\ce{XY}}(\ce{Y}) - E_{\ce{XY}}^{\ce{Y}}(\ce{Y}).
\end{split}
\label{eq:sherrill-10} \tag{Sherrill 10}
$$
The CP-corrected total energy is the full dimer energy with BSSE removed from each monomer is then
$$
\begin{split}
E_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}} &= E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - E_{\text{BSSE}}(\ce{X}) - E_{\text{BSSE}}(\ce{Y}), \\
&= E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - \left[ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X}) - E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X}) \right] - \left[ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y}) - E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y}) \right].
\end{split}
\label{eq:sherrill-15} \tag{Sherrill 15}
$$
Note that I have modified which geometry is used for each monomer in $\eqref{eq:sherrill-15}$. All monomers are calculated at the supermolecule geometry. This is convenient for two reasons: 1. We are only interested in removing the BS
{ X}}(\ce{X }), \\ 
 E_{\text{BSSE}}(\ce{Y }) & = E^{\ce{XY}}(\ce{Y }) - E^{\ce{Y}}(\ce{Y }), 
 \end{split } 
 \label{eq:2 } 
 $ $ 
 which, when subtracted from $ \eqref{eq: sherrill-1}$, gives $ \eqref{eq: sherrill-3}$. More correctly, considering that the geometry for each step is at the concluding bunch geometry and not the detached geometry, the above is [ 4 ] 
 $ $ 
 \begin{split } 
 E_{\text{BSSE}}(\ce{X }) & = E_{\ce{XY}}^{\ce{XY}}(\ce{X }) - E_{\ce{XY}}^{\ce{X}}(\ce{X }), \\ 
 E_{\text{BSSE}}(\ce{Y }) & = E_{\ce{XY}}^{\ce{XY}}(\ce{Y }) - E_{\ce{XY}}^{\ce{Y}}(\ce{Y }). 
 \end{split } 
 \label{eq: sherrill-10 } \tag{Sherrill 10 } 
 $ $ 
 The CP - corrected total department of energy is the full dimer energy with BSSE removed from each monomer is then 
 $ $ 
 \begin{split } 
 E_{\text{tot }, \ce{\widetilde{XY}}}^{\text{CP } } & = E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY }) - E_{\text{BSSE}}(\ce{X }) - E_{\text{BSSE}}(\ce{Y }), \\ 
 & = E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY }) - \left [ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X }) - E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X }) \right ] - \left [ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y }) - E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y }) \right ]. 
 \end{split } 
 \label{eq: sherrill-15 } \tag{Sherrill 15 } 
 $ $ 
 notice that I have modify which geometry is used for each monomer in $ \eqref{eq: sherrill-15}$. All monomers are calculated at the supermolecule geometry. This is commodious for two reasons: 1. We are only interested in removing the b
{X}}(\ce{D}), \\
E_{\text{BSSE}}(\ce{Y}) &= E^{\ce{XY}}(\ce{Y}) - E^{\ce{Y}}(\ce{Y}),
\end{splnr}
\label{xq:2}
$$
which, when suctracted from $\eqref{eq:sherrilp-1}$, tives $\eqref{eq:sherrill-3}$. More zorrectly, consideeing rhat the gxkmetry njr ezgh stzp is at the finsl cluster geometry and tog che isolated geometry, the above is [4]
$$
\bqgin{splot}
F_{\text{BSSE}}(\ce{X}) &= G_{\ce{XY}}^{\be{VY}}(\ce{S}) - E_{\ce{XY}}^{\ce{X}}(\ce{X}), \\
E_{\text{BSSE}}(\ce{Y}) &= E_{\ce{XY}}^{\de{XY}}(\ce{Y}) - E_{\ce{XY}}^{\ce{Y}}(\ce{Y}).
\rnd{split}
\label{eq:sherrill-10} \tah{Shegrill 10}
$$
The CP-correched total ebergr is the full dimer enegyy with BSSG removed from each monomer is tfen
$$
\beyin{split}
E_{\tezt{rot}, \we{\widetilde{EY}}}^{\text{BP}} &= E_{\ce{\widetilde{XY}}}^{\ce{XF}}(\ce{XY}) - R_{\text{BSSE}}(\ce{X}) - C_{\text{USSE}}(\xe{Y}), \\
&= E_{\ce{\widetilde{XY}}}^{\ce{EY}}(\ce{XY}) - \left[ E_{\ce{\widejilde{XY}}}^{\ce{XF}}(\cz{X}) - E_{\ce{\widetilde{XY}}}^{\ce{X}}(\xe{Z}) \rigvt] - \neft[ W_{\ce{\didttikds{XY}}}^{\ce{XJ}}(\ce{B}) - E_{\ce{\widetjlde{XY}}}^{\ce{Y}}(\ce{T}) \right].
\end{split}
\labek{ez:wherrill-15} \tag{Sgerrilj 15}
$$
Gote that I have modified which geometrj is used for each monomer un $\eqref{eq:sherrill-15}$. Alp monomerf are calculated at the supermolecule geometry. Thhs is zonrckieng flr two reasons: 1. We are only interested in remjbimg the BS
{X}}(\ce{X}), \\ E_{\text{BSSE}}(\ce{Y}) &= E^{\ce{XY}}(\ce{Y}) - E^{\ce{Y}}(\ce{Y}), $$ when subtracted $\eqref{eq:sherrill-1}$, gives $\eqref{eq:sherrill-3}$. geometry each step is the final cluster and not the isolated geometry, the is [4] $$ \begin{split} E_{\text{BSSE}}(\ce{X}) &= E_{\ce{XY}}^{\ce{XY}}(\ce{X}) - E_{\ce{XY}}^{\ce{X}}(\ce{X}), \\ E_{\text{BSSE}}(\ce{Y}) &= E_{\ce{XY}}^{\ce{XY}}(\ce{Y}) E_{\ce{XY}}^{\ce{Y}}(\ce{Y}). \end{split} \label{eq:sherrill-10} \tag{Sherrill 10} $$ The CP-corrected total energy is the full energy BSSE from monomer is then $$ \begin{split} E_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}} &= E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - E_{\text{BSSE}}(\ce{X}) - E_{\text{BSSE}}(\ce{Y}), \\ &= E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) \left[ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X}) - E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X}) \right] - \left[ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y}) E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y}) \right]. \end{split} \label{eq:sherrill-15} 15} $$ Note that I modified geometry is for monomer $\eqref{eq:sherrill-15}$. All monomers calculated at the supermolecule geometry. This is convenient for two reasons: 1. We are only interested in the BS
{X}}(\ce{X}), \\
E_{\text{BSSE}}(\ce{Y}) &= E^{\ce{XY}}(\ce{Y}) - E^{\Ce{Y}}(\ce{Y}),
\end{sPlit}
\lAbeL{eq:2}
$$
WhIch, wHen sUbtracted from $\eQRef{eQ:sherrill-1}$, gives $\eqref{eq:sHerriLl-3}$. mOre cORrEctly, ConsideRInG THat ThE gEomEtRY fOr eacH stEp is at tHe final cluSteR gEometry and noT ThE isolated gEomEtry, the above Is [4]
$$
\bEgin{spLiT}
E_{\tEXt{BSSe}}(\ce{x}) &= E_{\ce{Xy}}^{\ce{XY}}(\cE{x}) - E_{\ce{XY}}^{\Ce{X}}(\ce{X}), \\
E_{\teXt{bsSE}}(\ce{Y}) &= e_{\Ce{XY}}^{\ce{Xy}}(\CE{Y}) - e_{\ce{Xy}}^{\ce{Y}}(\ce{Y}).
\end{split}
\laBEl{EQ:sherrill-10} \tag{ShErrill 10}
$$
thE cP-CORreCteD total enerGy Is the FUll dimeR EnERGY wiTH BSSE removed fRom each monoMEr iS then
$$
\bEgIn{sPLit}
E_{\teXt{tot}, \Ce{\WIdeTilde{XY}}}^{\text{cP}} &= E_{\cE{\widetildE{XY}}}^{\ce{Xy}}(\Ce{XY}) - E_{\teXT{BSSE}}(\ce{x}) - E_{\text{bSSe}}(\ce{y}), \\
&= E_{\ce{\WIdEtIldE{Xy}}}^{\Ce{Xy}}(\Ce{xY}) - \lEFt[ E_{\Ce{\widetiLdE{Xy}}}^{\ce{XY}}(\Ce{X}) - E_{\CE{\WIDetiLde{xY}}}^{\ce{x}}(\ce{X}) \rIght] - \left[ E_{\ce{\wiDetIlde{xy}}}^{\ce{xY}}(\ce{Y}) - e_{\ce{\wiDetiLdE{XY}}}^{\ce{y}}(\ce{Y}) \riGht].
\enD{sPlit}
\label{eq:sherRill-15} \Tag{SherriLl 15}
$$
NOtE thAt i have MOdifieD whIch GeometrY is used FOr eAcH MONoMer in $\eqref{eq:sherriLl-15}$. aLL mOnomers aRe calcULaTeD At the supErMolEculE GEometRy. ThIS iS conveniEnt for TWo ReAsons: 1. We ArE only iNtEreSteD in reMOvinG the BS
{X}}(\ce{X}), \\
E_{\text{ BSSE}}(\ce {Y})&=E^{ \c e{XY }}(\ ce{Y}) - E^{\c e {Y}} (\ce{Y}),
\end{split}\labe l{ e q:2} $$ 
whic h, when su b t rac te dfro m$ \e qref{ eq: sherril l-1}$, giv es$\ eqref{eq:she r ri ll-3}$. Mo recorrectly, c ons iderin gtha t  thegeo metry  for e a ch ste p is at t he finalc lusterg e om etry  and not the isol a te d  geometry, the  above  i s  [ 4 ] 
$$ 
\b egin{split }E_{\t e xt{BSSE } }( \ c e {X} )  &= E_{\ce{XY }}^{\ce{XY} } (\c e{X})-E_{ \ ce{XY} }^{\c e{ X }}( \ce{X}), \\ 
E_{ \text{BSS E}}(\c e {Y}) &= E_{\ce{ XY}}^{ \ce {XY }}(\ c e{ Y} ) -  E _ {\c e {X Y}} ^ {\c e{Y}}(\c e{ Y} ).
\e nd{s p l i t }
\l abe l{eq :sher rill-10} \tag {Sh erri l l 1 0}
$$ 
TheCP-c or recte d tota l ene rg y is the full d imer  energy w ith  B SSE  r emove d  fromeac h m onomeris then $$\b e g i n{ split}
E_{\text{to t} , \c e{\widet ilde{X Y }} }^ { \text{CP }}  &=  E_{ \ c e{\wi deti l de {XY}}}^{ \ce{XY } }( \c e{XY})-E_{\te xt {BS SE} }(\ce { X})- E_{\ text{BSS E}}(\ c e{Y}), \\
&= E _ {\ce{\widetil d e{ X Y }} } ^{\c e{X Y}}(\ce{XY} ) -\ left [ E_ { \c e{\ w ideti lde{X Y} } }^ { \ce{XY}}(\ce{X}) -E_ {\ce{\ widet ilde{XY}}}^{\ ce{X}}(\ce { X } ) \right ] -\ le f t[ E_{\ce{\wid etild e{XY}}}^{\ c e{XY}}(\ ce{Y} ) - E_{\ ce{\widet i l de{XY}}} ^{\ ce{ Y}} (\c e { Y} ) \right].
\e n d {spl it }
\labe l{e q:sherr ill -15 } \ tag {S herrill 1 5}
$$
No te  t ha tI h ave m o dified w hi chge ome try i s  usedfor e achmo no m erin $\eq r ef { e q:sh er ri ll-1 5}$ .All m onom e rsare cal culated a t t h e su pe rm olecule  geometry. Th is  is conven ie ntfor tw o reasons:  1. We are only interes t ed in r emo vingtheBS
{X}}(\ce{X}), \\
E_{\text{BSSE}}(\ce{Y})_&= E^{\ce{XY}}(\ce{Y})_- E^{\ce{Y}}(\ce{Y}),
\end{split}
\label{eq:2}
$$
which, when subtracted_from $\eqref{eq:sherrill-1}$,_gives_$\eqref{eq:sherrill-3}$. More_correctly,_considering that the_geometry for each_step is at the_final cluster geometry_and_not the isolated geometry, the above is [4]
$$
\begin{split}
E_{\text{BSSE}}(\ce{X}) &= E_{\ce{XY}}^{\ce{XY}}(\ce{X}) - E_{\ce{XY}}^{\ce{X}}(\ce{X}), \\
E_{\text{BSSE}}(\ce{Y}) &=_E_{\ce{XY}}^{\ce{XY}}(\ce{Y})_- E_{\ce{XY}}^{\ce{Y}}(\ce{Y}).
\end{split}
\label{eq:sherrill-10}_\tag{Sherrill_10}
$$
The_CP-corrected total energy is the_full dimer energy with BSSE_removed from_each monomer is then
$$
\begin{split}
E_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}} &= E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) -_E_{\text{BSSE}}(\ce{X})_- E_{\text{BSSE}}(\ce{Y}), \\
&=_E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - \left[ E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X}) - E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X}) \right] - \left[_E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y}) - E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y}) \right].
\end{split}
\label{eq:sherrill-15} \tag{Sherrill 15}
$$
Note_that I have_modified_which_geometry is used for_each monomer in $\eqref{eq:sherrill-15}$. All monomers_are calculated at the supermolecule geometry._This is convenient for two reasons: 1._We are only interested in removing_the BS