Ambient-Potential Composite Ewald Method for ab Initio Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulation

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1 pubs.cs.og/jctc Ambe-Poel Compose Ewld Mehod fo b Io Quum Mechcl/Molecul Mechcl Molecul Dymcs Smulo Tmohy J. Gese d D M. Yok* Cee fo Iegve Poeomcs esech d Depme of Chemsy d Chemcl Bology, uges Uvesy, Pscwy, New Jesey , Ued Ses ABSTACT: A ew ppoch fo pefomg Pcle Mesh Ewld b o quum mechcl/molecul mechcl (/MM) smulos wh exeded omc obl bss ses s peseed. The ew ppoch, he Ambe-Poel Compose Ewld (CEw) mehod, does o pefom he /MM eco wh Mullke chges o elecosclly f chges. Ised he ucle d eleco desy ec decly wh he MM evome, bu me h vods he use of dese Foue sfom gds. By pefomg he elecoscs wh he udelyg desy, he CEw mehod vods self-cosse feld sbles h hve bee ecoueed wh smple chge mppg pocedues. Poel of me foce (PMF) pofles of he p-opheyl phosphe dssoco eco explc solve e compued fom PBE0/6- G* /MM molecul dymcs smulos wh vous elecosc poocols. The CEw pofles e show o be sble wh espec o el-spce Ewld cuoff, whees he PMFs compued fom uced d swched elecoscs poduce fcs. PBE0/6-G**, AM/d-PhoT, d DFTB /MM smulos e pefomed o geee wo-dmesol PMF pofles of he phosphoyl sesefco ecos wh ehoxde d pheoxde levg goups. The semempcl models coecly poduce coceed ehoxde mechsm, whees PBE0 coecly poduces sepwse mechsm. The b o eco bes gee moe closely o expeme h he semempcl models. The flue of Mullke-chge /MM- Ewld s lyzed.. INTODUCTION The goous eme of log-ged elecoscs s essel fo pope modelg of bologcl pocesses soluo. 5 Oe echque fo cludg log-ge elecoscs s Ewld s mehod, 6 whch eplces pmy u cell composed of Guss chges o fom fe peodc lce. The peodc Guss chge desy s esolved ple-wve bss, whece he elecosc poel s edly clculed. The ple-wve poel s he modfed wh sho-ged coecos o ccou fo he Guss chge peeo d hus ecove he elecosc poel of he po chge sysem. The compuol pefomce of Ewld s mehod ws gely mpoved wh he dve of he Pcle Mesh Ewld 7 0 (PME) mehod, whch hs become he de fco sdd fo molecul mechcl (MM) foce feld molecul dymcs (MD) smulos. Alhough he PME mehod ws oglly fomuled fo po chges, hs bee exeded houghou he yes o hdle Ces 6 d sold hmoc 7,8 mulpoles fo s pplco wh he AMOEBA polzble foce feld 9 d he modfed dvde-d-coque quum mechcl foce feld 0 (FF). Befoe he wdesped dopo of PME, elecosc foce uco, swchg, d shfg wee fequely used.,4 Elecosc cuoff mehods wee le foud o poduce fcs he popees of we 5 8 d he sucul sbly of lge bomolecules. 9 Cosequely hs bee suggesed h ew models o be pmezed usg cuoff elecoscs. 6 The eme of elecoscs wh quum mechcl/ molecul mechcl (/MM) models hs followed oe of wo geel pescpos: elecosc embeddg d mechcl embeddg.,4 Mechcl embeddg s subcve pdgm, wheeby he quum mechcl () ego s epeseed by MM-logue, he elecoscs e compued eely wh MM chges, d he ego s oduced by emovg he MM-logue self-eegy d eplcg wh he gs-phse eegy. I hs sese mechcl embeddg c be vewed s ype of ONIOM mehod. 5 7 Alhough mechcl embeddg s smple o mpleme, suffes fom he mjo dwbck h he chge desy does o decly polze o he MM evome; heoe, he elecosc embeddg mehod s sed ofe used. Elecosc embeddg decomposes he ol eegy o MM/MM, /, d /MM ddve compoes. The /MM eco explcly cludes he elecoscs bewee he chge desy d he MM po chges mog ohe ecos, cludg v de Wls foces heeby polzg he eleco desy. Combed /MM MD pplcos hve bee domed by he use of semempcl Hmlos; fo exmple, AM/ d-phot, 8 DFTB, 9,40 d eled models, 4 4 becuse he hgh cos of b o wve fuco mehods hs ofe eceved: Febuy, 06 Publshed: My, Amec Chemcl Socey 6 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

2 Joul of Chemcl Theoy d Compuo pecluded he bly o ob he mou of sscl smplg ecessy fo mkg megful compso wh expeme. 44 Neveheless b o /MM mehods 45 hve foud pplcos 46 hough he clculo of sgle po eeges, 47 NM chemcl shfs, 48 geomey opmzos, 49 dbc poel eegy sufces, 50 udge elsc bd phwys, 5 fe empeue sg mehods, 5 54 mulple me sep smulos, 55 d o coec poel of me foce (PMF) fee eegy sufces obed fom semempcl / MM clculos Applcos of semempcl /MM mehods ouely employed elecosc embeddg wh uced /MM elecosc cuoffs 60 6 ul he developme of he semempcl /MM-Ewld mehod peseed by Nm, 6 whch ws depedely epoed by ccd; 64 boh of whch wee flueced by he mehod peseed sevel yes po by Go. 65 Moe ecely hese mehods hve bee dped fo use wh PME 66 d semempcl X-Pol models. 67 The use of explc lce summos c lso be foud he leue. 68 Cosdeg h Ewld mehods hve dolly bee mplemeed fo po-chge dsbuos, he semempcl /MM-Ewld mehods hve chose o use Mullke chge epeseo of he ego o pefom he logge ecos. Whe hs ppoch ws ppled o b o /MM, ws foud h he use of Mullke o Loẅd chges cused self-cosse feld (SCF) covegece poblems whe o-mml omc obl (AO) bss ses wee used. 69,70 Ths hs moved he use of ChElPG 7 o ohe elecosc poel chge fg pocedues o poduce sble b o /MM-Ewld jecoes. 69,70,7 76 Howeve, pplcos of b o /MM ofe sll foego he use Ewld summos, peg sed o model he log-ge elecoscs wh eco feld mehod 77,78 o pefom el-spce elecosc uco, shfg, o smoohg. 5 54,79 84 ece wok hs dvoced Å el-spce swched elecosc cuoff mehod usg he mmum mge coveo. 85 egebly we hve oced h my uhos hve fled o epo he sze of he /MM obod cuoff h hey hve used, d ohe dels defg how he elecoscs wee pefomed. Ab o /MM mehods wll become fequely used he e fuue s FFs 80,86 88 d fee eegy coeco mehods mue d s hdwe echology coues o mpove. To hs ed, we queso f he effo plced o he evluo of he udelyg b o clculo s o somewh wsed by pefomg he /MM eco wh omc pl chges he h he ucle d b o eleco desy. Choosg he pl chges o model he elecosc poel cely helps o lleve hs coce o he exe h hose chges deed epoduce he poel, bu he effo equed o pefom he f could sed hve bee spe o mehod h vods chge fg logehe. Afe ll, he eso why he commuy s esog o pl chge fg s becuse cble leve fo evlug he Ewld sum b o /MM smulos hs ye o be elzed. The pcple complco ecoueed dec dpo of he Ewld o PME mehods s boe fom he eleco desy s pd chges e he ucle, whch eques uccepbly lge umbe of ple wves o esolve. Eve f oe could pefom Ewld s mehod usg o moe ple wves h wh s foud o be ccepble puely MM pplco, he lyc evluo of ech AO poduc s Foue coeffces would sll be vey cosly. The compuol effo would be fuhe mplfed by hvg o e-evlue he Ewld poel ech sep of he SCF pocedue. A wholly ew ppoch s eeded. 84,89 I hs wok, we pese ew b o /MM-Ewld mehod clled he Ambe-Poel Compose Ewld mehod, o Compose Ewld (CEw) mehod fo sho. The ew mehod does o eque moe ple wves h wh s ypclly used wh pue-mm pplcos. The lyc evluo of AO-poduc Foue coeffces e voded by umeclly egg he Ewld ecpocl-spce poel o he molecul qudue gd omlly used o compue he desy fucol heoy (DFT) exchge-coelo fucol. The log-ge ecos bewee he ego d s peodc eplcs e compued fom uced Tylo sees h s expded bou MM po-chge epeseo of he ego. Ths ppoxmo does o ffec he eco bewee he d MM egos; oly ffecs he eco of he ego wh s ow peodc mges. As cosequece of hs ppoxmo, he Ewld cobuo o he Fock mx does o chge dug he SCF pocedue, d he evluo of he ple-wve Ewld poel becomes logous o oe-me evluo of locl desy ppoxmo (LDA) exchge-coelo fucol. We compe he ew Ewld mehod o sevel ohe elecosc poocols by pefomg umbell wdow smulos o compue he PMF pofles of he p-opheyl phosphe dssoco eco. We show how vous elecosc poocols ffec he MM-we solvo oud chged egos. We exed d elboe o he lyss fs dscussed by Holde e l. 69,70 o elucde he flue of Nm s semempcl /MM-Ewld mehod 6 whe s ppled o o-mml bss se b o mehods. We compe he compuol cos of he ew mehod o elecosc embeddg s fuco of he umbe of oms. Flly, we compue wo-dmesol PMF pofles fo phosphoyl sesefco ecos volvg ehoxde d pheoxde levg goups o compe he phwys poduced by AM/d-PhoT, 8 DFTB, 9,40 d he PBE0/6-G** hybd fucol DFT mehod. 90,9. METHODS.. The /MM Eegy. I hs wok, we cosde /MM sysem h cos loclzed ego; fo exmple, smll solue molecule MM solve, o cve se wh lge bomolecule. Fuhemoe, we suppose h he clculo s pefomed ude peodc boudy codos, whose el- d ecpocl-spce lce vecos e,, d *, *, *, especvely. The ol poel eegy of he /MM sysem s E(, P) = E ( ) + E ( ) + E (, P) boded LJ elec Eboded elec( ) + Ek(, P) + Exc(, P) + Eex(, P) () whee s he se of omc coodes d P s he sglepcle desy mx (see eq ). The vous compoes of he eegy e defed below. The boded eegy, E boded, s he colleco of MM ems descbg he bods, gles, d osos bewee covlely lked MM oms d hose combos of MM d oms whch co les oe MM om: 6 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

3 Joul of Chemcl Theoy d Compuo E ( ) = E ( ) + E ( ) + E ( ) boded bod gle oso = k ( ) b bods + k ( θ θ ) bc gles kbcd, + [ + cos( ϕ ϕ )] bcd 0, bcd bcd osos b b eq, b bc bc eq, b The k vlues e foce coss, d b, θ bc, d ϕ bcd e bod leghs, gles, d oso gles, especvely. The Led-Joes (LJ) eegy, E LJ, s explcly evlued fo ll ps of oboded oms wh cuoff cu, d log-ge coeco s ppled o ccou fo he dspeso beyod he obod cuoff: E C =, b C ( ) LJ b> b boded b < cu Nom ypes uv, b π V NN u v C 6, b 6 b () 6, uv d 6 cu () whee u,v dex om ypes d N u s he umbe of oms of ype u. The pmed summo excludes ps whee d b e boh, b s ssumed o be he mmum mge dsce bewee oms d b, d V = s he u cell volume. The peodc elecosc eegy, E elec,s Eelec(, P) = Eelec[ q] q()( q + ) = d d whee = + + s lce slo d q() s he ol chge desy (see eq 5). The MM om chge desy s colleco of sc po chges (see eq 6), d he chge desy cosss of he omc ucle d elecos (see eq 7). The oo fo he elecosc eegy show eq 4 pesumes he sdd coveo of excludg he fe Coulomb self-eegy of he po chges wheeve hose ems my ppe. Tol chge desy: q() = q () + q () MM chge desy: q MM (4) MM (5) () = q δ( ) MM (6) chge desy: q () = Zδ( ) ρ() = Zδ( ) P χ ( ) χ ( ) Tol eleco (umbe) desy: μ μ ν ν ρ () = ρ + ρ α () β () (8) (7) Sp-esolved (umbe) desy: ρ () = ϕ () ϕ () Sp-esolved molecul obl: ϕ () = C χ ( ) μ μ μ μ (9) (0) Aomc obl epeseo of he sgle-pcle desy mx: α β P = P + P () Sp-esolved desy mx: P = C C μ ν () whee Z s ucle chge, χ() s AO bss fuco, d d C e he sp-esolved occupo umbes d MO coeffces, especvely. The deses d MOs defed by eqs 7 0 hve bee we s fuco of ; howeve, we emphsze h hese ems lso deped o he omc posos hough he use of om-ceeed bss fucos, d hs depedece mus be cosdeed whe evlug he omc foces. The pmees wh E boded e chose o mplcly ccou fo he elecosc ecos bewee he boded oms, bu hose ecos e explcly cluded eq 4 fo ool coveece. Theoe, E boded elec deoes coeco h emoves he explc elecosc ecos bewee he ps of MM oms ppeg wh E boded, qq b Eboded elec( ) = b boded b b MM () The emg eegy ems eq e he eleco ecos h do o decly couple o he MM evome. No-ecg eleco kec eegy: Ek(, P) = PT (4) T = χ ( ) χ ( ) d μ μ ν ν Hee Fock exchge eegy: chf/x = Eex(, P) P P ( μ ν) α, β Eleco epulso egl (EI): ( μ ν ) = d d χ ( ) χ ( ) χ ( ) χ ( ) μ μ ν ν μ μ ν ν Desy fucol heoy exchge-coelo eegy: E (, P) = e ( ; ρ, ρ, γ, γ, γ ) d xc = xc xc α β αα αβ ββ we ( ; ρ, ρ, γ, γ, γ ) α β αα αβ ββ (5) (6) (7) (8) 6 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

4 Joul of Chemcl Theoy d Compuo whee γ = ρ () ρ (), w s molecul qudue wegh, d e xc s le combo of exchge d coelo fucol egds, e () = c e () + c e () xc DFT/x x DFT/c c (9) whee we hve smplfed he e xc oo fo bevy. The esuls show he pese wok e evlued wh PBE0/6- G* o PBE0/6-G**. PBE0 s he Pedew Buke Ezehof hybd fucol, 90,9 whch uses he geelzed gde ppoxmo fucol descbed 9. The PBE0 coeffces e c DFT/x = 0.75, c DFT/c =, d c HF/x = 0.5. The summo ppeg eq 8 s umecl ego of he DFT fucol pefomed o molecul qudue gd. The molecul qudue gd s uo of omc qudue gds, d omc gd s sees of cocec dscezed sphees. Th s, ech omc qudue po s eleme wh se of gul pos {Ω} h fom sphecl shell of dus d whch s eheed o om. The omc gd po locos d weghs e = + d, Ω, d w omc, = w d, w Ω,, especvely, whee he oo deoes he gd po wh he se of pos eheed o om. My ypes of dl qudue ules hve bee developed, cludg hose bsed o Guss Chebyshev, Guss Legede, d Eule Mclue schemes I he pese wok, we use Guss Lguee d Lebedev 97 ules fo he dl d gul qudues o fom omc gds cossg of 5580 pos pe hevy om d 496 pos pe hydoge. I pcple, ech omc gd eges ll-spce; howeve, ech omc gd oly smples he egd dequely e he especve cees. Theoe, he molecul qudue weghs w w(,) = Γ (,)w omc,, oduce spl po fuco Γ,o vod ove-coug of he egd whe wo o moe omc gds smple he sme spl e. Specfclly, we use he fuzzy Vooo pog scheme poposed by Becke, 98 whch s summzed by eqs 0 6. The elve sze of he Vooo s bsed ccodg o he om s Bgg Sle dus, BS,. P(, ) Γ (, ) = P (, ) b b (0) P(, ) = [ ppp ( ( ( νb)))] b px ( ) = x x b b b b ν (, ) = μ + ( μ ) u b b = = u b b u ( BS, / BS, b) ( / ) + μ (, ) = b () () () (4) BS, BS, b (5) b b (6) Fomlly, oe would eed o oolly ccou fo peodcy wh eqs 5 8. If he eleco desy exeds o moe h hlf he box legh, he oe c evlue hese eegy ems s we whle ssumg mmum mge coveo. Ths s ypclly mplemeed by slg he ego o he cee of he smulo box, d he wppg he MM oms oud. Fo gve se of coodes, oe mus olely mmze he eegy wh espec o he MO coeffces SCF pocedue, ude he cos h he MOs em ohooml o ech ohe, ϕ () ϕ ()d = δ j j,t C S C = I whee S s he AO ovelp mx, S (7) = χ ( ) χ ( ) d (8) μ μ ν ν d I j = δ j s he dey mx. Ude hese coss, he opml se MO coeffces c be show o obey he ooh Hll equo, F C = S C E (9) whee E e he sp-esolved obl egevlues d F s he AO epeseo of he sp-esolved Fock mx, F = E P (0) Equo 9 s geelzed egevlue poblem, whch c be educed o sdd fom by oducg sfomo mx, X = U s / () whee U d s e he egevecos d egevlues of he ovelp mx, especvely; S = U s U T. I follows h X T S X = I, d,oao,oao,oao () F C = C E whee F,OAO s he ohooml omc obl (OAO) bss epeseo of he Fock mx,,oao T () F = X F X d C,OAO e he coespodg MO coeffces, C = X C,OAO (4) fo he OAO bss defed by he sfomo OAO ν χ () = χ ( ) μ X μ μ (5) The AO Fock mx elemes e he pl devves show eq 0, ppled o eq : = μ ν + E F Tj P ( ) P exc + wχ ( χ μ μ) ( ν ν) ρ ( ) + w ( χ ( ) χ ( )) e ρ + xc exc ( ρ ) ( ) γ ( ) γ ( ) μ μ ν ν elec whee deoes he sp h s pllel o. (6) 64 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

5 Joul of Chemcl Theoy d Compuo Upo echg SCF covegece, he omc gdes /, e edly obed fom elemey ch-ule dffeeo of he eegy. MM om gdes ( MM): E = E + E + E E boded LJ elec boded elec om gdes ( ): E = E = E + S T + + E E P + E + E P P c boded LJ elec P α, β xc HF/x P P α, β, μ ν P ( μ ν) μ ν E C C Gde of he exchge-coelo eegy: δ ρ ρ = + Exc w E xc x δρ ( ) X x X P, w + exc( ) P (7) (8) (9) The x / devve hs vlue of oly f he qudue po s eheed o om. Fuhemoe, he desy s le combo of AO poducs, whose gdes ssfy χ( ) = χ( ); heoe, he ls em o he fs le of eq 7 educes o ρ x x = b ρ f Xb 0 ohewse (40) Fo bevy, we e he ede o 99 fo ddol smplfcos of eq 9. The ppedx of 99 lso cos explc expessos fo he qudue wegh devve, w/. Algohms fo compug he EIs d EI gdes e foud s 00 d 0, d he oe-eleco Guss egls c be foud he seml wok by Ob d Sk. 0 Whe he /MM boudy seves covle bod, we use he lk om ppoch descbed 66. I bef, dglg bods e cpped wh hydoge om. The lk om bod legh s fxed, d s oeo s cole wh he seveed /MM bod. The omc foces of he lk om e popged o he el d MM oms v elemey ch ule devves. Fo compleeess, we oe h ohe emes of he /MM boudy c be foud he leue, cludg he Geelzed Hybd Obl mehod, 0 06 Effecve Fgme Poel mehod, d Locl Self-Cosse Feld mehod 0 o me jus few... Ambe-Poel Compose Ewld Mehod. I s wohwhle o beg wh some clfyg emks egdg eq 4, whch my cuse cofuso fo some edes becuse he lce slo,, occus oe (bu o boh) of he chge deses. I ohe wods, oe mgh hve expeced he Coulomb self-eegy of peodc desy, q(+), o be q( + ) q( + ) d d (4) becuse he Coulomb self-eegy of peodc desy, q(), s q() q( ) d d (4) Howeve, eq 4 s fomlly fe f q() s ywhee ozeo, 4 becuse ech cell he cos some mou of self-eegy d hee e fe umbe of cells he lce. Whe pefomg e poduc of wo fucos h e ech peodc (he desy d he elecosc poel e boh peodc eq 4), he desed quy s cully he e poduc s vege pe u cell. All cells he lce e decl, so he clculo of he vege eegy pe u cell meely eques oe o chge he ge of ego eq 4 fom ll spce o he volume of oe cell : Eelec[ q] = q( + ) V q( + ) d d (4) Equo 4 s ecoveed fom eq 4 by explog he peodcy of he elecosc poel. Alhough he peodc desy s o ecessly cofed wh u cell, he combed effluece of desy poduced fom he cosuco of q( +) cuses ech cell wh he lce o co oe sce of q() h ppes o hve bee wpped o he cell boudy. Equo 4 dffes fom eq 4 oly by uwppg he desy d mkg coespodg djusme o he ego lms. Aohe possble souce of cofuso my se fom he pevlece of expessos he leue h plce lce slos he deomo he h he umeo; h s, Eelec[ q] = q() q( ) d d + (44) Equos 4 d 44 e equvle. Equo 4 s ecoveed fom eq 44 by pefomg u-subsuo h eplces u + d d d u wh eq 44 d he chgg he dummy ego vble fom u o. Oe could lso we eq 44 wh deomo of, becuse he lce summo cosdes ll uque cell slos. Ou descpo of he CEw mehod mkes feque use of Dc oo, whch s ow summzed. A fuco s we s ke, f() = f. A complex cojuge s b, f*() = f. A e poduc s bke, f*() g() d = f g. If boh fucos e peodc, he he e-poduc s ego s pefomed ove he u cell volume, V d. Fom hese defos, oe c mmedely we he peodc (o pmy ) chge desy, 65 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

6 Joul of Chemcl Theoy d Compuo q() = = q q + q d he peodc chge desy, MM (45) q( + ) = + q (46) Oe c fuhe defe mbe chge desy, ( ) q + = + q = + q q 0 0 (47) whch cosss of ll sled copes of he desy eclosg he pmy mge. The c of poducg he vous foms of he desy descbed bove s ded hough he use of speclzed elecosc opeos. Le us defe he pmy elecosc opeo, j = he peodc elecosc opeo, j = + d he mbe elecosc opeo, j = + Δ 0 (48) (49) (50) These opeos c upo peodc chge desy o poduce he pmy elecosc poel (he elecosc poel of he peodc desy), jq = q( ) d (5) he peodc elecosc poel (he elecosc poel of he peodc desy), j q = q( + ) d d he mbe elecosc poel, (5) q( + ) j q = = j q j q Δ d 0 (5) The mbe elecosc poel s poduced solely fom he peodc suoudgs. Th s, s how he elecosc poel s leed upo oducg peodcy o he sysem. The elecosc eegy of he peodc sysem (eq 4) c be decomposed o /, MM/MM, d /MM ecos s follows: Eelec[ q] = q j q = + q j q q j q + q j q MM MM MM = q j q + q j q + q j q MM MM MM + q j q Δ (54) The ls em eq 54 s he mbe eegy. I s he Coulomb eco bewee he pmy mge s ego wh he egos loced he peodc suoudg. The mbe eegy s covee o evlue becuse he chce of he mbe poel chges ech SCF sep. We shll oduce ppoxmo h vods hs coveece. To beg, oe h he ego s mbe eegy c be expessed s Tylo sees expso bou eece chge desy, q (): chge desy evlued bou he eece: q () = q () + ( q () q ()) mbe eegy evlued bou he eece: (55) = + q j q q j q q q j q Δ Δ Δ + q q j q q Δ (56) whee q q q q. Obvously, f q () q (), he he ls em eq 56 s smll. Moe mpoly, he mbe elecosc opeo oly ecs he ego wh hose loced df fee peodc cells. If he u cell ws lge h he sphee whch ccumscbes he chge desy, he hs eegy could, pcple, be pefomed v mulpole mome expsos of q (). I ohe wods, he ls em eq 56 s lso eglgble whe he mulpole momes of q () esobly ppoxme hose of q (). Theoe, ppope choce of q () s oe whch ssfes lμ q () C ()d q () C ()d lμ (57) whee C lμ () s egul sold hmoc. 5 Thee e my poel choces whch could ssfy hs codo; howeve, cosdeg h foce felds hve ledy developed he pl chges o esobly model he elecoscs, he mos covee choce would be o euse he udelyg MM omc chges fo q (). Complcg he fom of q () by usg, fo exmple, omc mulpoles o dffuse uxly bss fucos, would oly cese he ccucy of he mehod sof s hose complcos could mpove he ovell descpo of he ego s mulpole momes. Alevely, oe could mpove he mulpole momes by smply djusg he udelyg MM pl chges f ws foud o be ecessy, hus edeg ddol complcos moo. Followg hs logc, he ppoch ke he pese wok s o use se of sc po chges o ppoxme he mbe chge desy, q () = q δ( ) (58) d he uce he Tylo sees o fs-ode, q q j q q 0 Δ q j q q j q q j q Δ Δ Δ (59) (60) such h he mbe eegy becomes compose eco bewee q () wh q (), d q () wh self. Afe smplfco, he eegy becomes E CEw [q] E elec [q], 66 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

7 Joul of Chemcl Theoy d Compuo ECEw[ q] = q j q + q j q MM MM + q j q + q q j q MM q j q + q j q (6) I hs me, he peodc poels oly volve sc po chge dsbuos, whch c be compued oce befoe he SCF pocedue begs. Equo 6 s ppoxmo, bu ou fomulo ws desged o educe he eo s mgude fo ypcl /MM pplcos. I should be poed ou h he uced Tylo sees expso oly effecs he eco of he ego wh s mges; does o ppoxme he eco bewee he d MM egos o he MM ego wh self. Neveheless, f he eco bewee he ego wh s peodc mges ws such h he Tylo sees could o esobly be uced, he oe could decly evlue he mbe eegy eq 54 by evlug he mulpole momes of q () ech SCF sep d he use he pomulpole PME mehod descbed 7; howeve, he smll sze of d, heoe, ecellul dsce bewee egos should mke hs dded lye of complexy uecessy mos pplcos. The peodc elecosc poel of he po chge dsbuos ppeg eq 6 c be compued wh ehe he Ewld o PME mehods. A deled heoecl developme of hese mehods c be foud 7, whose oo we dop hecefoh. Specfclly, we eseve k = π( k * + k * + k * ) (6) o dex he gul wve umbes of ple wve bss: k = e k T Fuhemoe, we shll e o B-sple weghs, T θ( = M N * ) ( ) + d d d= (6) (64) h e cosuced fom ode- Cdl B-sple fucos, M ( u) = ( )! k= 0 k ( ) [mx( u k, 0)] k d whose dscee Foue sfom coeffces, θ k = k θ (65) (66) e evlued fom ufomly spced gd. The gd cosss of N = N N N pos h e dexed by = (,, ) d posoed : = d= d N d d (67) We wll coue by summzg he ecessy equos fo PME, we below fo geec po chge dsbuo, q p (): q () = q δ( ) p p (68) I bef, he PME mehod compues he peodc poel of model Guss desy, p β q π / β e (69) d he coecs fo he sho-ge dffeece bewee he po d Guss poels. The elecosc poel of he peodc Guss desy s pefomed lyclly upo fg o ple wve bss. Theoe, he PME poel (see eq 70) decomposes o ple-wve poel (see eq 75), sho-ge coeco h emoves he Guss poel (see eq 7), coespodg sho-ge e-feld poel h eoduces he po-chges (see eq 7), d, fo chged sysems, ufom bckgoud poel (see eq 76). Sevel of hese poels c be gouped ogehe, whe covee, o f-feld (see eq 7), el-spce (see eq 74), d ecpocl-spce (see eq 77) poels. The dffeece bewee he Guss d po-chge elecosc poels s eglgble lge dsces, so oe eed oly evlue he e-feld d Guss poels wh el-spce Ewld cuoff cu oud whle ssumg mmum mge coveo. Whe ppled o eq 6, oe mus evlue he poel ll vlues whee q () s o-eglgble. I ohe wods, he e-feld d Guss poels mus be evlued fo ll mged po chges wh cu of q (). PME poel: j q = j q j q + j q + j q p p p p p = j q j q + j q p p % p = j q + j q p > = j q + j q p p % p (70) Ne-feld poel; h s, he poel cused by eveyhg wh he el-spce Ewld cuoff cu oud : q 0 < j q = p p 0 ohewse cu (7) F-feld poel; h s, he poel cused by eveyhg ousde cu : j q = j q j q + j q > p p p p = j q j q Guss poel: j q = p % p p (7) p ef( β ) q 0 ohewse cu (7) 67 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

8 Joul of Chemcl Theoy d Compuo el-spce poel: j q = j q j q = p p p p efc( β ) 0 < cu q β = 0 π 0 > cu (74) Ple-wve poel (epoled fom egul gd): j q = θ ϕ p Ufom bckgoud poel: p (75) π j q = q, β p V p (76) ecpocl-spce poel: j q = j q + j q % p p p (77) PME poel epolo cool pos: /4β k 4π e ϕ = k e k p Vk θ k 0 S p k (78) PME sucue fco; h s, he fowd dscee Foue sfom coeffces of he B-sple epoled po chges: S p k q θ = k ( ) p (79) Hvg ow oduced he e- d f-feld poels, we show how hey e used o smplfy he secod le eq 6, MM MM > MM = q j q + q j q + q MM > MM q j q + q q j q = q j q + q j q + q j q + q q j q hough he ccello of ems, q j q q j q = 0 (80) whch e equvle becuse q () s ecessly wh cu of q (). Fo he sme eso, he ls le of eq 6 meely emoves he eece desy s e-feld eegy, ssumg h he ego s sze s less h hlf he box legh: q j q + q j q = q j q > (8) The eegy expesso fo he Ambe-Poel Compose Ewld mehod (see eq 8) s obed upo eplcg he secod d hd les of eq 6 wh he ls le of eq 80 d eq 8, especvely, ECEw[ q] = q j q + q j q MM MM + q j q + q j q + q MM > MM q j q > (8) d he expdg he f-feld poels usg he secod le of eq 7: ECEw[ q] = q j q + q j q MM MM + q j q + q j q + q MM % MM q j q + q q j q MM % + q j q (8) Explc expessos fo he eegy ems ppeg eq 8 e summzed by eqs 84 96: q j q = q j q MM MM MM MM q j q = q j q % % q j q = q j q % MM q j q + q = Z j q + q + P V Nqud % % MM (84) (85) (86) (87) % V = w χ ( ) χ ( )( j q + q ) μ μ ν ν % MM = MM q j q + q = Z j q + q + P V,, MM V = q V q V μ μ ν ν β / β V = e, π χ ( ) χ ( ) d d q j q = Z j q + P V MM MM MM χ ( χ μ μ) ( ν ν) V = q d MM (88) (89) (90) (9) (9) (9) 68 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

9 Joul of Chemcl Theoy d Compuo q j q = ZZ Z + P V + P P μ ν b b b> μ ν χ ( χ μ μ ν ν V = Z ) ( ) Z d μ μ ν ν ( ) ( μ ν ) = χ ( ) χ ( ) χ ( χ μ μ ) ( ν ν ) d d (94) (95) (96) Equo 88 s pefomed umeclly usg molecul qudue gd cossg of qudue pos d poed qudue weghs w. I ohe wods, oe eges j% q MM + q me logous o LDA exchge-coelo poel. Specfclly, he ecpoclspce poel s compued he FFT gd pos, d he B- sples e used o epole he poel oo he molecul qudue gd. If β (see eq 69) ws lge, he he umecl ego of he ecpocl-spce poel show eq 88 would be ccue uless ddol omc qudue gds wee lso plced oud he e-feld MM oms; h s, he ecpocl-spce poel hs lumps he omc posos, d hose lumps look cesgly lke q/ s β. I pcce, he β vlues used ypcl /MM smulos e suffcely smll h he use of MM qudue cees s uecessy. Fo he ske of gume, eve f hs wee ssue, oe would elme he lumps by umeclly egg eq 90 log wh he ecpocl-spce poel, he h oducg ddol qudue cees. The cobuo of E CEw o he sp-esolved Fock mx s he devve wh espec o he desy mx: E P CEw Z % = V + Pμ ν ( μ ν ) + V + V V (97) Smlly, he elecosc eegy cobuo o he omc gdes s he devve E/ P. Mos of he gde ems educe o expessos volvg sdd ucle ucle, eleco ucle, d eleco epulso egls; fo bevy, eqs 98 0 summze oly hose expessos whch volve he ple wve bss d/o umecl qudue gd. θ( ) q j q = q ϕ MM MM MM qq β X b β β b b e efc( b) + π MM b b b b < b cu = θ ( ) q j q q ϕ % b (98) (99) + = q j q q % MM θ Z j q + q ( ) % MM N qud ρ θ + w ( ) ( ) j q q % MM = Nqud wθ( ) j q + q = ρ ρ + ( ) ( ) x x X Nqud θ ρ + ( ) x w ( ) j q q % MM = x θ( ) 4π e + q e k S k Vk θ k 0 % MM whee S = k k Zθ ( ) Nqud w ρ( ) θ( ) = k /4β k (00) (0) Equo 0 eques he evluo of he eleco desy o he molecul qudue gd. Ech qudue gd po desy s he B-sple epoled oo he FFT gd. The FFT-gd epeseo of he chge desy he udegoes fowd FFT o poduce se of sucue fcos. Equo 0 does o oduce ew ppoxmo; ully ses fom sdd ch-ule dffeeo of he eegy. The umecl ego of he ecpocl-spce poel vods he explc evluo of he AO-poduc Foue coeffces... Ohe /MM Elecosc Poocols. The Ambe-Poel Compose Ewld mehod (deoed by CEw, see eq 8) wll be comped o hee ohe elecosc models:. Elecosc embeddg wh uced elecoscs fo boh he MM/MM d /MM ecos (deoed by cu, see eq 0).. The /MM Mullke chge Ewld mehod oduced by Nm 6 (deoed by Ewq, see eq 04).. A model-mm Ewld eegy wh smooh, sho-ge /MM coeco (deoed by MMEw, see eq 08). The MMEw model s mxue of mechcl d elecosc embeddg. We defe obod cuoff dus oud he ego, ousde of whch he / MM ecos occu wh mechcl embeddg (cludg he log-ge Ewld eco), d sde of whch he ecos e pefomed wh elecosc embeddg. Ths ppoch s geelzed he MMEw mehod by oducg swch h smoohly sos bewee he wo lms o yeld couous poel 69 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

10 Joul of Chemcl Theoy d Compuo eegy. I he lm h he swchg wdh becomes zeo, he so occus dscouously. Cuoff-bsed elecoscs (E cu [q] E elec [q]): qq b Ecu[ q] = q j q + + b MM < * b cu P q b E cu cobuo o he Fock mx: E P cu b * Z MM b b MM b < MM b>< * b b cu = V + P ( μ ν ) Zq b b χ ( χ μ μ) ( ν ν) d b cu (0) χ ( χ μ μ) ( ν ν) q d b MM b b < cu (0) Mullke chge /MM Ewld mehod (E Ewq [q] E elec [q]): Mul Ewq MM E [ q] = q + q j q + q q j q q MM Mul Mul Mul + q j q (04) Mul () = q δ( ) Mullke chge: Mul (05) Mul q = Z PS PS μ μ, ν ν μ, ν (06) E Ewq cobuo o he Fock mx: E P Ewq = V + P ( μ ν ) = Mul EEwq EEwq q + P Mul q P Mul, q Z S E q Ewq Mul μ + E q Ewq Mul b ν (07) Model-MM Ewld eegy wh sho-ge /MM coeco (E MMEw [q] E elec [q]): EMMEw[ q] = q + q j q + q MM MM + q j q q j q + q j q MM MM q j q (08) The smooh, sho-ge copoo of he /MM ecos: q j q = S ( ;, w ) Zq MM * b cu sw b MM b MM S ( w P * b; cu, sw) q χ ( χ μ μ) ( ν ν) d b b b The swched, sho-ge model-mm ecos: q j q = S w q q ( ;, ) MM * b cu sw b MM The swch off fuco: b b b b (09) (0) 0, cu S = ( ;, w ), w cu sw cu sw 4 5 0x 5x + 6 x, ohewse cu x = wsw () The dsce bewee MM om b d he ees om: * b = m{ b,, N b } () E MMEw cobuo o he Fock mx: E P MMEw Z b MM = V + P ( μ ν ) S ( ;, w ) q * b cu sw χ ( χ μ μ) ( ν ν) d b b () The expesso fo he Led-Joes eegy (eq ) d he E CEw (eq 8), E cu (eq 0), E Ewq (eq 04), d E MMEw (eq 08) elecosc mehods ll employ he symbol cu. I ll cses, hs dsce s he obod cuoff dus used by he MM pogm o geee eghbo lss. We poceed by summzg how he obod cuoff s used wh dffee mehods. I dog so, we use he phses el-spce Ewld cuoff, elecosc cuoff, d swched cuoff o d he ede s bly o ecll how he vous mehods behve he obod cuoff. If om s sde he obod cuoff, he he pwse Led-Joes eegy s compued; whees, f s ousde he obod cuoff, he s eco s modeled hough log-ge l coeco. The elecosc mehods use sgle eghbo ls fo he ego, s whole, whch s cosuced fom he uo of he dvdul om eghbo lss. Theoe, MM om s wh he obod cuoff f s dsce o y om s less h cu. The cu ppeg he E CEw d E Ewq mehods e el-spce Ewld cuoffs, whch meely deoes he dsce whee he po chge d PME Guss poels e suffcely sml h ddol el-spce coecos would hve eglgble effec. The cu ppeg he E cu mehod s elecosc cuoff h mks he dsce ousde of whch he elecosc ecos e scly (d dscouously) goed. The cu ppeg he E MMEw mehod s he oue edge of swched 60 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

11 Joul of Chemcl Theoy d Compuo cuoff h exeds fom cu w sw o cu. Isde he swchg ego ( cu w sw ), he /MM elecosc eco explcly volve he omc ucle d AO poducs. Ousde of he swched cuoff ( cu ), he /MM elecosc eco (cludg he log-ge Ewld-compoe of he eegy) s pefomed usg sc, MM po chge epeseo of he oms. The E MMEw swchg ego ( cu w sw > > cu ) smoohly sos bewee hese wo lms. Theoe, he E MMEw om foces e flueced by he Ewld poel, bu he eleco desy explcly polzes oly o he eby MM oms wh he swched cuoff. I hs pevously bee oed 69,70 h he use of Mullke chges wh E Ewq c esul SCF covegece sbly whe ppled o sdd, ll-eleco AO bss ses. I ode o mpove he SCF sbly, ou mplemeo of E Ewq holds he Mullke chges wh he cue MD sep fxed o he SCF coveged Mullke chges fom he pevous MD sep. I ohe wods, he E Ewq / q Mul chge devves ppeg eq 07 e compued oce he begg of he SCF pocedue..4. Compuol Dels. The b o code d ll elecosc mehods descbed he pevous secos wee mplemeed fom scch wh developme veso of AmbeTools 5 d efced o he Sde MD pogm, whch ws used o pefom ll smulos descbed below. 6 The oo E cu, E MMEw wd, E CEw, E Ewq s used o dsgush he elecosc poocols descbed he pevous secos d he ssoced obod cuoffs. Fo exmple, E cu 4 deoes he use of eq 0 wh cu = 4 Å, d E MMEw 4w4 s eq 08 wh cu = 4 Å d w sw = 4 Å. Smlly, E CEw 4 d E Ewq 4 e o eq 8 d eq 04, especvely, wh cu =4Å. All Ewld ecpocl-spce clculos e pefomed usg Å gd desy, d he Ewld coeffce ws chose fom he vlue of cu o epoduce dec sum olece of 0 6 u. Fgues 6 dsply smulo esuls fo sysem composed of p-opheyl phosphe (pnpp) ego uced ochedo flled wh 456 TIP4P/Ew we molecules. Sl os wee o dded o he sysem. The ol chge of he sysem s. The Ewld mehods use eulzg ufom bckgoud poel (eq 76) o ccou fo he e chge. The pnpp Led-Joes d q pmees wee obed fom he Aechmbe pogm cluded AmbeTools 5, whch chose he LJ pmees d chges fom he GAFF foce feld d AM-BCC poocol, especvely. 6 9 The sysem volume ws equlbed usg he DFTB semempcl Hmlo d he /MM Ewld mehod descbed 6 he soheml sobc esemble (NPT) fo 00 ps ( fs/sep) 98 K d m. The Beedse bos ws used o cool he pessue wh elxo me of ps, d he Lgev hemos coolled he empeue wh collso fequecy of 5 ps. The smulo cell s equlbed el-spce lce vecos e 56.4 Å. The equlbed volume d coodes wee he used fo he mcococl (NVE) d cocl (NVT) esemble smulos descbed below. Fgue shows he PBE0/6-G* /MM smulo ol eegy, elve o he fs sep, fo sees of NVE smulos h dffe oly by he choce of elecosc poocol. Ech smulo ws u fo 0 ps (0 000 seps), d hey sed fom he sme l codos. Fgue. NVE smulo eegy cosevo s fuco of me sep (0.00 ps/sep) usg vous elecosc poocols. The sysem s pnpp solue compued wh PBE0/6-G* uced ochedo of TIP4P/Ew wes. See he ex Compuol Dels fo descpo of he oo used o dsgush he vous elecosc poocols. Fgue dsplys PBE0/6-G* /MM poel of me foce pofles of he pnpp dssoco eco: [NO C 6 H 6 O- PO ] NO C 6 H 6 O +PO, whee he eco coode s he P O dsce. Ech PMF ws geeed fom 96 umbell wdow NVT smulos h sped fom PO =.4 Å o 6. Å seps of 0. Å. The umbell hmoc poels used foce cos of 50 kcl mol Å. The PMFs wee geeed fom he dsbuo of PO vlues usg he vol fee eegy pofle (vfep) mehod. 0 Ech NVT smulo ws pefomed 98 K fo ps (Fgue ), 4 ps (Fgue b d Fgue c), o 6 ps (Fgue d). The ol mou of smplg fo ech cuve ws chose o cheve esheclly plesg fgues; h s, he PMFs show Fgue d e ll suffcely sml o oe ohe h ddol smplg ws dded o emove smll umecl ose he ego whee he PMF s elvely fl, so h he cuves could be moe esly dsgushed. Alevely, he PMFs ppeg Fgue e defbly dffee such h smplg beyod ps ws o cosdeed o be pude use of esouces. We oe h he E MMEw 4w PMF ppeg Fgue b s emed PO 4 Å becuse my of he E MMEw 4w smulo wdows beyod PO > 4 fl o complee due o lge velocy wgs. Ths s expeced becuse he swchg wdh hs cse s so smll h y wes wh he swchg lye expeece lge foce, becuse he poel eegy ppoches dscouy s he wdh es zeo. The dshed, hozol les Fgue mk he expemel be soluo (9.6 kcl/mol). Fgue compes dl dsbuo fucos (DFs) bewee he phosphous d we oxyges obseved wh PBE0/6-G* /MM pnpp smulos pefomed usg sevel elecosc poocols. Fgue,b shows DFs obed fom smulos whose umbell poel s ceeed bou PO =.7 Å (he ppoxme mmum of he PMF), d Fgue c,d shows DFs obed usg umbell wdow ceeed bou PO = 4.0 Å (he ppoxme so se of he PMF). Fgue b,d show zoomed- es of pels d c, especvely, s mked by he blck boxes. The DFs wee geeed fom 500 ps of NVT 6 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

12 Joul of Chemcl Theoy d Compuo Fgue. Effec of vous elecosc poocols o he poel of me foce pofles of he pnpp dssocve eco. Fgue. Pwse dl dsbuo fuco bewee phosphous d he we oxyges. The mephosphe PO sepo pes () d (c) e.70 d 4.00 Å, especvely. Pels (b) d (d) e zoomed- es of () d (c), s dced by he dshed boxes. smulos 98 K, whose jecoy ws we evey 50 seps. Fgue 4 shows he poel, kec, d ol eeges fom PBE0/6-G* /MM NVE smulos of pnpp pefomed usg he E Ewq 4 (Fgue 4) d E CEw 4 (Fgue 4b) elecosc poocols s fuco of me sep ( fs/sep). Boh jecoes s fom he sme coodes d l veloces, d umbell poel PO =.7 Å s ppled. The E Ewq 4 jecoy fls o SCF covege fe sep 08. Theoe, he E Ewq 4 jecoy s esed sep 08 fom 6 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

13 Joul of Chemcl Theoy d Compuo Fgue 4. /MM NVE smulos of he pnpp molecule (e he mmum of he PMF; h s, he PO =.7 Å wdow) usg he E Ewq 4 (lef) d E CEw 4 (gh) mehods. The plos (op) decompose he ol smulo eegy o kec (ed) d poel (gee) eegy cobuos, elve o he l codo. Boh smulos s fom he sme l codo. The molecules beeh ech plo e spshos of he solue log he especve jecoy. Ech me sep s 0.00 ps. he coodes d veloces poduced by he E CEw 4 jecoy sep 08. The E Ewq 4 jecoy g fls o SCF covege fe sep 97. The E Ewq 4 SCF flues e mked Fgue 4 by Imploso # d Imploso #. Sevel spshos of he pnpp solue fom he E Ewq 4 d E CEw 4 jecoes e show mmedely below he especve eegy pofles. Fgues 5 d 6 e-lyze he PBE0/6-G* Mullke chges d Wbeg bod odes of he fs 08 seps fom ech jecoy show Fgue 4. Specfclly, Fgue 5 d shows he SCF coveged Mullke chges poduced by he E Ewq 4 mehod usg he omc coodes fom he fs 08 seps of Fgue 4. Fgue 5e h shows he SCF-coveged Mullke chges poduced by he E CEw 4 mehod usg he omc coodes fom he fs 08 seps of Fgue 4. Smlly, Fgue 5 l d m p e he E Ewq 4 d E CEw 4 Mullke chges, especvely, evlued fom he fs 08 seps of Fgue 4b. The fom of Fgue 6 s logous; dffes fom Fgue 5 oly by dsplyg he Wbeg bod odes he h Mullke chges. Fo cly, he Mullke chges d Wbeg bod odes e evlued usg he coodes of he ee /MM sysem. Tble compes he /MM smulo es cheved usg E CEw 4 d E MMEw 4w4 whe ppled o he solved 7- om le ch show Fgue 7. The le ch s eclosed wh uced ochedo cog 508 TIP4P/Ew wes. The mgs e lsed s he vege smulo e (ps/dy) epoed by Sde fe ps (000 seps) of PBE0/6-G* NVT smulo 98 K. The smulo es e epoed s fuco of cel pocessg u (CPU) coe cou d sysem sze, cludg lk oms. The ego cosss of ll oms o he lef of he cus show Fgue 7 d he hydoge lk-oms cossg he boudes. The mgs wee pefomed o he smpede supecompug cluse he Texs Advced Compug Cee. Ech ode cos wo Iel Xeo E5-680 pocessos (8 coes/pocesso), d he odes e ecoeced wh Mellox FD IfBd echology. Alhough Iel Ph copocessos d/o Nvd gphcs cds wee vlble o he compue odes, ou code cuely oly us o he CPU. Fgue 8 s schemc of he sesefco of hydoxylkyl phosphe eses. Of pcul ees hs wok e he ecos volvg ehoxde (EO) o pheoxde (PhO), whch e epeseve of poo d ehced levg goups, especvely. The d bod lbels ppeg Fgue 8 e eco coodes used o pefom he wo-dmesol (D) umbell wdow smulos show Fgue 9. Fgue 9 c shows D PMFs of he EO phosphoyl sesefco, d Fgue 9d f shows D PMFs of he PhO phosphoyl sesefco. Fgue 9,d wee pefomed wh AM/d-PhoT (bbeved s AM/d); Fgue 9b,e wee pefomed wh DFTB; d Fgue 9b,e wee 6 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

14 Joul of Chemcl Theoy d Compuo Fgue 5. Mullke chge lyss of he E Ewq 4 d E CEw 4 SCF coveged eleco deses s fuco of me sep. Pels h d p e evlued usg he omc coodes of he E Ewq 4 d E CEw 4 jecoes show Fgue 4, especvely. pefomed wh PBE0/6-G**. The EO smulos cossed of 4 wdows h fom gd fom (, )= (.4 Å,.4 Å) o (5.6 Å,. Å) d fom (.4 Å,. Å) o (.4 Å,.0 Å) usg 0. Å spcg ehe deco. The PhO smulos cossed of 09 wdows fom (, ) = (.4 Å,.4 Å) o (5.6 Å,. Å). The umbell wdows hd foce cos of 85 kcl mol Å, d he solue ws eclosed wh uced ochedo cog 404 (EO) o 58 (PhO) TIP4P/Ew wes. The desy of he sysem ws equlbed wh DFTB NPT smulo me logous o he pnpp equlbo poocol descbed bove, d poduco ws pefomed he NVT esemble 98 K. The AM/d-PhoT d DFTB semempl models wee smpled d lyzed fo 00 ps/wdow, whees he PBE0/ 6-G** smulos wee u fo 0 ps/wdow. Alyss of he fs 0 ps chges he PBE0/6-G** bes by less h 0. kcl/mol; heoe, smplg ws suspeded 0 ps. The vfep mehod ws used o geee he D PMFs fom he dsbuos of d vlues obseved he smulos. 0 The ccle d X mks ppeg Fgue 9 e mm d so ses of he D sufces, d he coloed les coecg he soy pos e udge elsc bd mmum fee eegy ph., The D PMFs ppeg Fgue 9 e he fee eegy pofles log he mmum fee eegy ph. Ou /MM smulos ssume h he -O hs ledy bee depooed. Th s, we explcly model he eco be fom * o TS (see Fgue 8), whees he expemel be coespods o he eco fom o TS. Theoe, ou clculed fee eeges e coeced o ccou fo he fee eegy dffeece bewee d *. The e of he eco show Fgue 8 ws expemelly deemed o be k = d L mol s fo he EO d PhO levg goups, especvely, 80 C ph 8 buffe soluo. 4 The so se fee eeges sdd se coceo e compued fom he Eyg equo, ΔG = T l(kh/k B T), whee h d k B e Plck s cos d Bolm s cos, especvely. Thus, he expemel bes fo EO d PhO e 8.55 d.6 kcl/mol, especvely. To coec ou fee eeges, we mus cosde he pk of -OH ph 8. Expemel mesuemes of he - OH pk smll-molecule models of NA hve ged bewee.5 d.9, depedg o he sucue d expemel echque. 5 8 Of he vlble expemel pk d, he model mos closely esemblg hose show Fgue 8 s he deose -ehyl phosphe used 7, 64 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

15 Joul of Chemcl Theoy d Compuo Fgue 6. Wbeg bod ode dces of he E Ewq 4 d E CEw 4 SCF coveged eleco deses s fuco of me sep. Pels h d p e evlued usg he omc coodes of he E Ewq 4 d E CEw 4 jecoes show Fgue 4, especvely. Fgue 7. /MM smulo es e pefomed usg 7-om le ch solved by 508 TIP4P/Ew wes uced ochedo smulo cell. The mgs wee pefomed wh dffee egos, whch coespod o he vous cus he covle bods show he fgue. The seveed bods e cpped wh hydoge lk oms, whch e cluded he cou fo he umbe of oms. whose -OH pk ws foud o be.68 ± Assumg pk of.68, he fee eegy of he depooed -O ph 8 d 98 K s esmed o be ΔG * =.0T(pH pk ) = 6.8 kcl/mol. Theoe, 6.8 kcl/mol hs bee dded o ll compued fee eeges show Fgue 9.. ESULTS.. Compso of Elecosc Poocols. Eegy Cosevo. The df ol eegy obseved wh NVE smulos of pnpp e show Fgue. The E cu 4 d E MMEw 4w0 mehods poduce sog eegy dfs becuse he poel eegy sufces co dscoues. The E cu 4 eegy df s lge h E MMEw 4w0 becuse he E cu 4 mehod excludes he log-ge ecos eely. The E MMEw 4w4 d E CEw 4 mehods coseve he ol eegy. Boh models hve couous poel eeges d model he log-ge ecos, lbe dffee mes. The E Ewq elecosc mehod fs developed by Nm e l. 6 s o show Fgue becuse he solue mplodes fe 00 seps. The sbly of E Ewq jecoes wll be dscussed del below. I bef, he E Ewq smulos do coseve he ol eegy fo he legh of me h hey c be u. The cosevo of ol eegy s mpo fo mg sble smulo empeue. 9 Alhough he empeue c be coeced hough velocy esclg, he lck of ol 65 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

16 Joul of Chemcl Theoy d Compuo Tble. Smulo es (ps/dy; fs/sep) Usg E CEw 4 d E MMEw 4w4 s Fuco of CPU Coe-Cou (N coe ) d ego Sze (N, he Numbe of Aoms) fo he Solved Ale Ch Show Fgue 7 N, E CEw 4 N, E MMEw 4w4 N coe Fgue 8. Schemc of he phosphoyl sfe (model NA sesefco) ecos wh ehoxde d pheoxde levg goups. eegy cosevo c, ove me, mfes self by dmpeg hgh-fequecy moo d sfomg o low-fequecy moos. 0 Fuhemoe, he eme of logge foces hs lso bee show o dmclly ffec he sucul sbly of poes 0 eve whe swchg fucos e oduced. The smulos pefomed he emde of hs muscp e o pefomed o me scle h s log eough o decly obseve hese pevously obseved fcs. pnpp Fee Eegy Pofles. Smulos of ecos volvg chged speces e ubquous bology. Amog he mos mpo e phosphoyl sfe ecos, whch e vl fo my cellul pocesses, fo exmple, gee egulo, cell sglg, d eegy coveso. 4 Phosphoyl sfe ecos hve bee suded exesvely by compuol mehods, 5 d mechsc sghs hve bee ged by sudyg model sysems h povde epeo of le fee-eegy elos 6 d kec soope effecs. 7 I hs seco, we exme how vous elecosc poocols ffec he phosphoyl sfe eco be of pnpp (see Fgue ). I he vcy of OP.7 Å, he mephosphe s covlely boud o opheyl such h s sgle molecule wh chge. A sepo of OP 4 Å, he solue s bee descbed s wo polyomc mooos. Oe c g sgh o he ole of solvo hese wo suos by eg o Bo o model whch, fo we, ppoxmely sles o ΔG solv Q /, whee s he dus of he o. I ohe wods, hee s gee degee of solve sblzo e he PMF mmum h hee s e he so se. Alhough vous elecosc mehods ppe o cese he be, he dffeeces e eled o he solve sblzo e he PMF mmum, whch hs bee chose o defe he zeo of fee eegy ll cses. Hd he sysem cossed of p of coueos, such s he NH 4+ Cl o NH 4+ PO sysems exmed 6, he he solve sbly of he ego would be ehced s he os wee seped, becuse he sepo of os poduces cesg dpole mome. The sees of E cu pofles show Fgue dffe skgly fom he ohe elecosc poocols. The PMFs coully cese s he mephosphe dssoces. The uced elecosc foces cuse he wes o become sucued he /MM elecosc cuoff (see Fgue ), whch cs o cee fcl solvo shell. As oe deceses he elecosc cuoff fom 8 o 0 Å, he fcl solvo shell foms close o he solue d hus ceses he sbly of he PMF mmum. The pofles do o co so se below OP < 6 Å fo wo esos:. The legh of he elecosc cuoff peves he solvo shells of ech fgeme fom dssocg fom ech ohe hs ge.. The we oud phosphous coues o dsply fclly sucued DF eve whe he mephosphe s seped (see Fgue c,d), so he dus of he effecve Bo o ceses s he mephosphe dssoces. The E MMEw mehod s eme of he log-ge /MM elecoscs s fudmelly dffee fom E cu, d hs dffeece cuses he E MMEw PMFs o exhb so ses (see Fgue b). As he wdh of he swchg ego s educed, he dssoco be ceses. Ths obsevo 66 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

17 Joul of Chemcl Theoy d Compuo Fgue 9. Fee eegy pofles of he sesefco ecos show Fgue 8. ( c) Ehoxde levg goup. (d f) Pheoxde levg goup. Ise: oe-dmesol plos of he fee eeges log he udge elsc bd ph coecg he mmum o he e-lmg so se. The hozol le he se plos mk he expemel be. ppes o be eled o how he swchg wdh ffecs he solve sucue he vcy of he swchg ego. Fgue, pels,b d c,d lluse he solve sucues of E MMEw 4w, E MMEw 4w, d E MMEw 4w4 OP =.70 Å d OP = 4.00 Å, especvely. As he swchg wdh becomes smlle, he so bewee he /MM d MM-logue/MM ecos occus moe buply, d he wes expeece gee foce he swchg ego. The degee of we sucue he swchg ego s gee OP =.70 Å becuse he solue moe closely esembles do. Ulke he E cu mehod, he fcl solvo shell dsspes s he mephosphe dssoces. The sees of E MMEw pofles show Fgue c use swchg wdh of zeo. Thus, he wdh s oo smll fo he wes o expeece foce ssoced wh he so, d he dffeece bewee MM-logue/MM d /MM ecos s oo suble o cuse dmc dffeece he so se be. Icesg he elspce Ewld cuoff fom 0 o 8 Å lowes he be fom.4 (E MMEw 0w0 ) o 0. (E MMEw 8w0 ) kcl/mol. elve o Fgue,b, he E MMEw w0 pofles show Fgue c e elvely sble d gee wh expeme much moe closely; howeve, he ede should ke oe h he elecosc poocols show Fgue c do o coseve he ol eegy NVE smulos. The sees of E CEw pofles show Fgue d exhb he gees degee of smly s fuco of obod cuoff. The bes e. kcl/mol (E CEw 0 ),.8 kcl/mol (E CEw 4 ),. kcl/mol (E CEw 8 ). Ulke he ohe elecosc poocols ppeg Fgue c, he E CEw mehod llows he Hmlo o explcly polze o boh he 67 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

18 Joul of Chemcl Theoy d Compuo sho- d log-ge elecoscs. As cosequece, E CEw 4 does o duce fclly sucued solvo shell, s c be see Fgue. Elecoc Polzo. Fgues 4 6 e used o dscuss he sympoms whch se whe empg o use he semempcl-syle /MM-Ewld mehod 6 wh b o Hmlo. To lyze he behvo of E Ewq 4 d E CEw 4,we cosuc NVE jecoes of ech (Fgue 4 d 4b, especvely), sg fom he sme l codos. The E Ewq 4 ol eegy s coseved ul sep 06, whch po he om veloces e oo lge o dequely popge wh fs me sep. Fuhemoe, we e uble o SCF covege he E Ewq 4 ego fe sep 08. The solue exhbs uusul chges he hydoge covle gles fe he fs 75 seps, dmc defomo of he pheyl g fe he fs 90 seps, d wldly ophyscl bod leghs he ls few seps whch po he solue c oly be descbed s hvg udegoe mploso. The E CEw 4 jecoy (Fgue 4b) does o ecoue y of hese sympoms. To demose h he E Ewq 4 mploso s o fluke, we esed he E Ewq 4 smulo fom he coodes ke fom sep 08 of he E CEw 4 jecoy. The E Ewq 4 jecoy g mplodes fe ddol 80 seps. The sbly of E Ewq jecoes hs pevously bee descbed by Holde e l., 69,70 whch moved hem o cococ po-chge epeseo of he chge desy usg ChElPG chge-fg pocedue. 7 The E CEw mehod descbed he pese wok ecs he ego wh s suoudgs whou esog o uxly epeseo. Neveheless, we gee wh Holde e l. h he udelyg eso fo he sbly of E Ewq jecoes ses fom he me whch he Mullke poels cobue o he Fock mx. I ohe wods, he E Ewq 4 jecoy Fgue 4 becomes ophyscl becuse he elecosc poocol spuously polzes he desy mx o he evome. Fgues 5 d 6 compe how he E Ewq 4 d E CEw 4 poocols ffec he polzo of he eleco desy by moog he Mullke chges d Wbeg bod odes ech sep of he jecoes. Fgue 5 shows he E Ewq 4 cbo Mullke chges evlued usg he fs 08 seps of he E Ewq 4 jecoy. The cbo chges dvege s he jecoy eches he fs mploso. A fs glce, ede mgh be emped o flpply dsmss Fgue 5 becuse Mullke chges e kow o exhb sesvy o bss se. 8,9 Howeve, he E CEw 4 cbo Mullke chges show Fgue 5e e evlued he sme coodes s hose used Fgue 5, d ye he E CEw 4 chges e emkbly sble eve s he solue mplodes. The cbo chges e ecompued log he E CEw 4 jecoy Fgue 5m,. Ag, he E Ewq 4 chges (Fgue 5) e moe sesve, bu hey do o dvege becuse he E CEw 4 jecoy does o mplode. The E Ewq 4 Mullke chges of he ohe oms (O,N,P,H) e elvely sble compso o cbo. We coclude he followg fom Fgue 5:. The E Ewq 4 Mullke chges of cbo e pcully sesve o geomey, d. he E Ewq 4 elecoscs mus be polzg he udelyg desy mx suspcous me, becuse he E CEw 4 Mullke chges e sble whe evlued wh he sme coodes. The C C, C O, C N, d C H Wbeg bod dces show Fgue 6 e secod mes fo lyzg he eleco polzo. The C C bod odes Fgue 6,e,,m should be bou.5 becuse of he cojugo wh he pheyl g. Howeve, some of he C C bods Fgue 6 show sog bodg behvo. The spuous E Ewq 4 C C bodg pe sgfcly les cbo s bodg o he ohe oms s oe ppoches he mploso. The E Ewq mehod s polzo s sesve becuse. hee s lge mou of C C AO ovelp, d. he Mullke poels poduce mulple, cosse vews of he exel evome s elecoc chemcl poel. Thee s lge mou of C C AO ovelp becuse he Cs fuco h s, he pmve s-fuco 6-G* hs smll Guss expoe ( ) d hee e 6 cbos close poxmy o ech ohe. To be moe pecse, hee s lge mou of Cs Cs AO ovelp. The sog Cs AO ovelp poduces e-le depedece d coespodgly smll AO ovelp mx egevlue. Cosequely, he fs colum (ssumg he egevlues e soed) of X (eq ) s sees of umbes h e lge mgude, osclle sg, d e domed by Cs chce. Alevely sed, he Cs AOs e sfomed he OAO bss (eq 5) o become osclloy d deloclzed coss he pheyl g. The e-le depedece h we hve jus descbed s expeced d oue. Mos pplcos of b o mehods wh smll molecules ecoue f smlle AO ovelp egevlues h wh s foud hee. The degee of e-le depedece he pese exmple oly becomes poblem oce he chemcl poel s cossely vewed. The elecoc chemcl poel poduced by he exel evome (he MM oms d log-ge elecoscs) s δ μ = E ex () ex δρ() (4) d polzes he eleco desy hough s cobuo o he Fock mx, Δ F = μ () χ () χ ()d ex μ ν (5) By pefomg he eco hough Mullke chges, he E Ewq mehod poduces dffee chemcl poels fo ech AO poduc: μ μ = + E Ewq EEwq () () ex Ewq, q q μ b ν (6) Ech of hese chemcl poels s cos houghou spce, bu evey AO p expeeces dffee cos. Whe eq 5 s sfomed o he OAO bss, he spl cosseces of μ Ewq, () d he mulcee deloclzo of eq 5 mlfom he chemcl poel elve o he ohe OAO bss fucos o poduce egules he eleco polzo. The E CEw mehod peseed hs wok does o poduce e polzos becuse uses sgle, cosse vew of he exel evome s chemcl poel. Compuol Effcecy. The E CEw mehod s moe expesve h E MMEw becuse E CEw polzes he ego o boh he sho- d log-ge ecos. Specfclly, E CEw eges he Ewld Guss poel v molecul qudue, d he E CEw gdes (eq 00) eque fowd 68 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6

19 Joul of Chemcl Theoy d Compuo Foue sfom of he chge desy fom he molecul qudue gd. As show Tble, hs ddol lye of compuo slows he E CEw smulos by oly smll mou becuse he dom effo coues o volve he clculo of sdd Guss egls equed by he udelyg b o Hmlo. Fo 5-om ego, he dffeece mgs bewee E CEw d E MMEw s 4%, depedg o how my CPU coes e used. The choce of el-spce Ewld cuoff ffecs he elve cos bewee he el- d ecpocl-spce evluos, d hus could be ued o opmze he ovell cos. Specfclly, educg he sze of he el-spce Ewld cuoff ecesses he use of lge Ewld coeffces whch u eques lge umbe of ple wves o esolve. The opml choce s obed fom empcl obsevo. The segy used Ambe, whch we hve doped hs wok, s o use po/ Å FFT gd desy, d he choose he Ewld coeffce such h ef(β cu )=ε cu, whee ε s dec sum olece. Oe could gue h lge vlues of cu my be ecessy whe usg b o mehods becuse he eleco desy exeds sevel gsoms beyod he ucleus. By employg 4 Å el-spce Ewld cuoff, we e empg o peempvely dffuse h gume. Hvg sd h, we hve o ye ecoueed sceeo whee 0 Å el-spce Ewld cuoff poduces quesoble esul. I pcce, he pnpp smulos employg 4 Å el-spce Ewld cuoff slowed he clculo by 7 8% elve o he 0 Å el-spce Ewld cuoff... Compso bewee b Io d Semempcl Fee Eegy Pofles. Semempcl heoy hs see wdesped use fo geeg PMF fee eegy pofles fom /MM smulos. Cosdeg h semempcl models e ppoxmely 00 mes fse h my b o mehods, oe my vely queso whehe he ccucy of b o ppoches s woh he dded effo. Fgue 9 compes he AM/d-PhoT d DFTB semempcl model PMFs wh hose poduced by PBE0/6-G** fo he sesefco eco show Fgue 8. The mos skg dffeece occus wh he EO levg goup (Fgue 9 c). The semempcl models pedc sgle, le so se, whees PBE0 poduces wo so ses (he le so se s e coollg) d emede. I ohe wods, he AM/d- PhoT d DFTB semempcl models pedc coceed mechsm, d PBE0 pedcs sepwse mechsm. The expemel es of hs eco do o povde sgh o whehe o o emede exss; howeve, we hve used he Guss pogm 40 o pefom geomey opmzos d so se seches usg PBE0/6-G*, PBE0/6- G**, BLYP/6-G*, d BLYP/6-G** wh polzed couum model (PCM) mplc solve. 4 All of he PCM clculos h we pefomed pedc he exsece of emede d wo so ses, geeme wh ou PBE0 explc solve /MM PMF. Pevous sudes of sesefco ecos wh vous levg goups hve foud h poo (EO) d ehced (PhO) levg goups geelly follow sepwse d coceed mechsms, especvely. 6 Ideed, he sesefco eco PMFs volvg he PhO levg goup (Fgue 9d f) e pedced o udego coceed mechsm fo ll mehods. The PBE0/6-G** eco bes gee moe closely o expeme h ehe AM/d-PhoT o DFTB. The semempcl mehods pedc EO bes h e 0 kcl/mol lge h expeme, whees PBE0 s oly.6 kcl/mol lge. Smlly, he semempcl models poduce PhO bes h e 8 kcl/mol lge h expeme, d PBE0 dffes fom expeme by oly 0.8 kcl/mol. 4. CONCLUSION I hs wok, we peseed ew, compose Ewld mehod (CEw) fo /MM smulos. Ulke he semempcl- /MM Ewld mehod oduced by Nm, 6 he ew mehod poduces sble MD jecoes whe evlued wh o-mml Guss AO bss ses. The CEw mehod dffes fom he ecely descbed /MM-LEC ppoch 85 by polzg he ego o he log-ge elecoscs usg Ewld summo. I fuhe dffes fom he ppoch descbed by Holde e l. 69,70 by ecg he ego wh he exel evome decly hough he chge desy, s opposed o esog o po-chge uxly epeseo. We comped how vous elecosc poocols ffec he PMF of he p-opheyl phosphe dssoco eco. I ws show h he CEw PMFs e less sesve o he dvso of sho- d log-ge ecos h he ohe mehods h we comped. Fo exmple, he dffeece fee eegy bes bewee usg el-spce Ewld cuoff of 0 d 8 Å ws foud o be kcl/mol whe usg he CEw mehod. We comped CEw o subcve Ewld scheme (MMEw) d foud h CEw voded he sucul fcs ssoced wh swchg he /MM eco bewee elecosc d mechcl embeddg. We exmed he compuol cos of he ew mehod d foud h he copoo of Ewld sums o he clculo oly slowed he smulo e by 4% fo 5- om ego. We cofmed Holde s explo 69,70 fo he sbly ecoueed wh Mullke chge-bsed /MM Ewld mehods. 6 The use of Mullke chges poduces cosse epeseos of he elecoc chemcl poel whch, he ohogolzed omc obl epeseo of he Fock mx, become defomed d hus ely polzes he eleco desy. The ew CEw mehod polzes he desy o globlly cosse epeseo of he chemcl poel. We pefomed wo-dmesol PMFs of phosphoyl sesefco ecos wh ehoxde d pheoxde levg goups o compe PBE0/6-G** wh he AM/d- PhoT d DFTB semempcl models. We foud h he semempcl models poduced coceed eco mechsms fo he ehoxde levg goup, whees PBE0 smulos poduced sepwse mechsm, geeme wh mplc solvo clculos d pevous wok. 6,7 Fuhemoe, he PBE0/6-G** eco bes wee foud o moe closely mch expeme h he semempcl mehods by 6 7 kcl/mol fo boh levg goups. AUTHO INFOMATION Coespodg Auho *E-ml: D.Yok@uges.edu. Fudg The uhos e geful fo fcl suppo povded by he Nol Isues of Helh (No. GM07485). Compuol esouces wee povded by he Nol Isues of Helh ude G No. S0OD046 d by he Exeme Scece d Egeeg Dscovey Evome (XSEDE), whch s 69 DOI: 0.0/cs.jcc.6b0098 J. Chem. Theoy Compu. 06,, 6 6