SE, not the effect of monomer deformation, and 2. a isolated monomer geometry without deformation doesn't make sense in the context of a geometry optimization. I also added the tilde to signify that the supermolecular/dimer geometry used may not be the final or minimum-energy geometry, as would be the case during a geometry optimization. We simply extract all structures consistently from a given geometry iteration. Perhaps $\ce{XY}(n)$ would be better notation.
Formulation of the gradient
As Pedro correctly states, the differentiation operator is a linear operator. Because there are no products in $\eqref{eq:sherrill-15}$, the total gradient needed for $\eqref{eq:steepest-descent}$ will be a sum of gradients [5]:
$$
\frac{\partial E_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}}}{\partial R_{A,x}} = \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY})}{\partial R_{A,x}} - \left[ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X})}{\partial R_{A,x}} - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X})}{\partial R_{A,x}} \right] - \left[ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y})}{\partial R_{A,x}} - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y})}{\partial R_{A,x}} \right],
$$
so each step of a CP-corrected geometry optimization will require 5 gradient calculations rather than 1. Note that the nuclear gradient should be included for each term as well, which is a trivial calculation.
Extension to other molecular properties
Although not commonly done, counterpoise correction can be applied to any molecular property, not just energies or gradients. Simply replace $E$ or $\partial E/\partial R$ with the property of interest. For example, the CP-corrected polarizability $\alpha$ of two fragments is
$$
\alpha_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}} = \alpha_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - \left[ \alpha_{\ce{\widetilde{
SE, not the effect of monomer deformation, and 2. a isolated monomer geometry without deformation doesn't seduce common sense in the context of a geometry optimization. I also added the tilde to mean that the supermolecular / dimer geometry used may not be the concluding or minimum - energy geometry, as would be the character during a geometry optimization. We just extract all structures consistently from a given geometry iteration. possibly $ \ce{XY}(n)$ would be better notation. 
 Formulation of the gradient 
 As Pedro correctly states, the specialization operator is a linear hustler. Because there are no product in $ \eqref{eq: sherrill-15}$, the total gradient needed for $ \eqref{eq: steepest - descent}$ will be a sum of gradient [ 5 ]: 
 $ $ 
 \frac{\partial E_{\text{tot }, \ce{\widetilde{XY}}}^{\text{CP}}}{\partial R_{A, x } } = \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY})}{\partial R_{A, x } } - \left [ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X})}{\partial R_{A, x } } - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X})}{\partial R_{A, x } } \right ] - \left [ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y})}{\partial R_{A, x } } - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y})}{\partial R_{A, x } } \right ], 
 $ $ 
 so each step of a CP - corrected geometry optimization will require 5 gradient calculation rather than 1. Note that the nuclear gradient should be included for each term as well, which is a fiddling calculation. 
 Extension to other molecular properties 
 Although not commonly done, counterpoise correction can be applied to any molecular property, not just energies or gradients. Simply replace $ E$ or $ \partial E/\partial R$ with the property of interest. For example, the CP - corrected polarizability $ \alpha$ of two fragment is 
 $ $ 
 \alpha_{\text{tot }, \ce{\widetilde{XY}}}^{\text{CP } } = \alpha_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY }) - \left [ \alpha_{\ce{\widetilde {
SE, jot the effect of monomev deformation, anb 2. a ismlated monomer geometry without deformatioi dowsn't nake sense in the contdxt of a heometry optmmization. I also added tmz tilsc to vmgnify that the supermolewular/dimer geokegrv used may not be the final or minimtm-energu heometry, as woold bt tre czse during a geometry optimization. We simkly extract all sttuctures consistently from a glven geometry iterwtion. Perhaka $\cq{ZY}(n)$ would be better nouacion.
Formulajion of the gradient
As Pedro corfectlv states, thg sivxerentiatioi operwtor is a likvar opesator. Brcause there ave no priducts in $\eqref{eq:shercill-15}$, the total gradignt needed fkr $\eqref{eq:steepesr-dwscenj}$ wiln be q sjm kf gdadienhs [5]:
$$
\hrac{\partial E_{\text{tot}, \cw{\widetilde{XY}}}^{\text{CP}}}{\psrepsl R_{A,x}} = \frac{\lartiaj Q_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY})}{\partial R_{A,x}} - \left[ \frzc{\partial E_{\ce{\widetilde{XT}}}^{\ce{XY}}(\ce{X})}{\partial R_{A,x}} - \vrac{\partiwl E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X})}{\partial R_{A,x}} \right] - \left[ \frdc{\parvixl T_{\cc{\rkeehilde{XY}}}^{\ce{XY}}(\ce{Y})}{\partial R_{A,x}} - \frac{\partial E_{\ce{\widqfikdv{XY}}}^{\ce{Y}}(\ce{Y})}{\partial V_{A,x}} \right],
$$
so each syeo ps a CP-correctgd geomzfrg optimization wilp requite 5 grqdient cajculstions rather than 1. Note thqt the nuclecr tradient should be included fur esch trrm as well, which is a criviam calculatiln.
Extensikv to other molecjlag prmperties
Although not commogly done, rountzrpoise zorrgction san be apppied bm any molecular prlperti, not bust energles or gradients. Simply replace $E$ or $\partial G/\pastidl R$ witk the iroperty of intqrest. For examkle, the C'-correzted polarpzability $\alpha$ of tro fragments hd
$$
\alpha_{\text{tov}, \ce{\widetylde{ZY}}}^{\tezt{CP}} = \auoha_{\ce{\widetilde{CY}}}^{\ce{XY}}(\ce{XJ}) - \lefr[ \alpha_{\ce{\widetilde{
SE, not the effect of monomer deformation, a monomer geometry deformation doesn't make a optimization. I also the tilde to that the supermolecular/dimer geometry used may be the final or minimum-energy geometry, as would be the case during a optimization. We simply extract all structures consistently from a given geometry iteration. Perhaps would better Formulation the gradient As Pedro correctly states, the differentiation operator is a linear operator. Because there are products in $\eqref{eq:sherrill-15}$, the total gradient needed for will be a sum gradients [5]: $$ \frac{\partial E_{\text{tot}, R_{A,x}} \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY})}{\partial - \frac{\partial R_{A,x}} - \frac{\partial R_{A,x}} \right] - \left[ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y})}{\partial R_{A,x}} - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y})}{\partial R_{A,x}} \right], $$ so each step of CP-corrected geometry require 5 calculations than Note that the should be included for each term is a trivial calculation. Extension to other molecular Although not done, counterpoise correction can be applied any molecular property, not just energies or gradients. replace $E$ or $\partial E/\partial R$ with the property of interest. For example, the CP-corrected of two fragments is \alpha_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}} = - \alpha_{\ce{\widetilde{
SE, not the effect of monomer deFormation, aNd 2. a isOlaTed MoNomeR geoMetry without deFOrmaTion doesn't make sense in tHe conTeXT of a GEoMetry OptimizATiON. i alSo AdDed ThE TiLde to SigNify thaT the supermOleCuLar/dimer geomETrY used may noT be The final or miNimUm-enerGy GeoMEtry, aS woUld be The casE During A geometry OpTImizatIOn. We simPLY eXtraCt all structures coNSiSTently from a givEn geomEtRY iTERatIon. perhaps $\ce{Xy}(n)$ Would BE better NOtATIOn.
FORmulation of thE gradient
As pEdrO correCtLy sTAtes, thE diffErENtiAtion operatOr is A linear opErator. bEcause tHEre are nO produCts In $\eQref{EQ:sHeRriLl-15}$, THe tOTaL grADieNt needed FoR $\eQref{eQ:steEPEST-desCenT}$ wilL be a sUm of gradients [5]:
$$
\FraC{\parTIal e_{\text{Tot}, \ce{\WideTiLde{XY}}}^{\Text{CP}}}{\PartiAl r_{A,x}} = \frac{\partial E_{\Ce{\wiDetilde{XY}}}^{\Ce{Xy}}(\cE{XY})}{\PaRtial r_{a,x}} - \left[ \FraC{\paRtial E_{\cE{\widetiLDe{Xy}}}^{\cE{xy}}(\Ce{x})}{\partial R_{A,x}} - \frac{\parTiAL e_{\cE{\widetilDe{XY}}}^{\ce{x}}(\Ce{x})}{\pARtial R_{A,x}} \RiGht] - \Left[ \FRAc{\parTial e_{\Ce{\WidetildE{XY}}}^{\ce{Xy}}(\Ce{y})}{\pArtial R_{a,x}} - \Frac{\paRtIal e_{\ce{\WidetILde{Xy}}}^{\ce{Y}}(\ce{y})}{\partial r_{A,x}} \riGHt],
$$
so each step of A cP-corrected geOMeTRY oPTimiZatIon will requIre 5 gRAdieNt caLCuLatIOns raTher tHaN 1. noTE that the nuclear gradIeNt shouLd be iNcluded for eacH term as welL, WHIch is a trIviaL CaLCulation.
ExtensIon to Other molecULar propeRties
although Not commonLY Done, counTerPoiSe cOrrECTiOn can be applieD TO any MoLecular ProPerty, noT juSt eNerGieS oR gradientS. Simply rEpLaCe $e$ oR $\paRtial e/\Partial R$ WiTh tHe ProPerty OF interEst. FoR exaMpLe, THe Cp-correcTEd POLariZaBiLity $\AlpHa$ Of two FragMEntS is
$$
\alphA_{\text{tot}, \cE{\wiDEtilDe{xY}}}^{\Text{CP}} = \aLpha_{\ce{\widetilDe{xY}}}^{\ce{XY}}(\ce{XY}) - \LeFt[ \aLpha_{\ce{\WIDetilde{
SE, not the effect of mono mer deform ation , a nd2.  a i sola ted monomer ge o metr y without deformationdoesn 't make se nse i n the c o nt e x t o fageo me t ry  opti miz ation.I also add edth e tilde to s i gn ify that t hesupermolecul ar/ dimerge ome t ry us edmay n ot bet he fin al or min im u m-ener g y geome t r y,  aswould be the case du r ing a geometry  optim iz a ti o n . W e s imply extr ac t all structu r es c o nsi s tently from a  given geom e try  itera ti on. Perhap s $\c e{ X Y}( n)$ would b e be tter nota tion.F ormulat i on of t he gra die ntAs P e dr ocor re c tly st ate s , t he diffe re nt iatio n op e r a t or i s a  lin ear o perator. Beca use  the r e a re no  prod ucts  i n $\e qref{e q:she rr ill-15}$, the t otal  gradient  ne ed edfo r $\e q ref{eq :st eep est-des cent}$w ill  b e a  s um of gradients [5 ]: $ $\frac{\p artial E_ {\ t ext{tot} ,\ce {\wi d e tilde {XY} } }^ {\text{C P}}}{\ p ar ti al R_{A ,x }} = \ fr ac{ \pa rtial E_{\ ce{\wi detilde{ XY}}} ^ {\ce{XY}}(\ce{ X Y})}{\partial R_ { A ,x } } -\le ft[ \frac{\ part i al E _{\c e {\ wid e tilde {XY}} }^ { \c e {XY}}(\ce{X})}{\par ti al R_{ A,x}}  - \frac{\par tial E_{\c e { \ widetild e{XY } }} ^ {\ce{X}}(\ce{X })}{\ partial R_ { A,x}} \r ight]  - \left [ \frac{\ p a rtial E_ {\c e{\ wid eti l d e{ XY}}}^{\ce{XY } } (\ce {Y })}{\pa rti al R_{A ,x} } -  \f rac {\ partial E _{\ce{\w id et il de {XY }}}^{ \ ce{Y}}(\ ce {Y} )} {\p artia l  R_{A, x}} \ righ t] ,$ $
s o eachs te p of a  C P- corr ect ed  geom etry opt imizati on will r equ i re 5  g ra dient c alculations r at her than 1 .Not e that t he nucle ar gradient should be i n cludedfor  each  ter m as well , w hich i s a trivia l calc ulati on .
E x t ensio n to  ot he r molecula r pro perti es 
Alt hough n ot commonly done,c oun terpoise corr ect ionc a nbea pp l ied  t o  an y molecular prope rty, not j us t  e nergies or gra di ents. S imply r eplac e  $E$ or  $\partia l E/\part ia l R$ w ith  the prope rty of i nterest.F or ex a mp le, t heCP-cor re cte d pol arizab i lit y $\a lpha$of  two f ragme nt s is
$$\alpha_{\text{tot}, \ce {\wide tilde {XY }}}^{\tex t{C P }}= \alpha_ {\ce {\widetild e{X Y}} }^{\c e{X Y }}(\c e{XY } )- \ l eft[\alp h a_{\ce{\w i de til d e {
SE, not_the effect_of monomer deformation, and_2. a_isolated_monomer geometry_without_deformation doesn't make_sense in the_context of a geometry_optimization. I also_added_the tilde to signify that the supermolecular/dimer geometry used may not be the final_or_minimum-energy geometry,_as_would_be the case during a_geometry optimization. We simply extract_all structures_consistently from a given geometry iteration. Perhaps $\ce{XY}(n)$_would_be better notation.
Formulation_of the gradient
As Pedro correctly states, the differentiation operator_is a linear operator. Because there_are no products_in_$\eqref{eq:sherrill-15}$,_the total gradient needed_for $\eqref{eq:steepest-descent}$ will be a sum_of gradients [5]:
$$
\frac{\partial E_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}}}{\partial R_{A,x}}_= \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY})}{\partial R_{A,x}} - \left[ \frac{\partial_E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{X})}{\partial R_{A,x}} - \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{X}}(\ce{X})}{\partial R_{A,x}}_\right] - \left[ \frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{Y})}{\partial_R_{A,x}} -_\frac{\partial E_{\ce{\widetilde{XY}}}^{\ce{Y}}(\ce{Y})}{\partial R_{A,x}} \right],
$$
so each_step of a_CP-corrected geometry_optimization will require_5 gradient calculations rather than 1._Note that the_nuclear gradient should be included for_each_term as well,_which_is_a trivial_calculation.
Extension to other_molecular_properties
Although not_commonly_done, counterpoise correction can be applied_to_any molecular property, not just energies or_gradients. Simply replace $E$_or_$\partial E/\partial R$ with_the property of interest. For_example, the CP-corrected polarizability $\alpha$ of_two fragments_is
$$
\alpha_{\text{tot}, \ce{\widetilde{XY}}}^{\text{CP}}_= \alpha_{\ce{\widetilde{XY}}}^{\ce{XY}}(\ce{XY}) - \left[ \alpha_{\ce{\widetilde{