trunk/multipole/dielectric_new.dvi

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phvb7t»Real–Uüspace“electršÂ“ostatics“f˜or“m˜ultipoles.‘ µIII.“Dielectric“Pr˜oper=ltiesŽŸ‘$"ó»ló
phvr7tºMadan–¹ Lamichhane,Ÿû¤çó2»ló     
phvr7tÝ1Ž›  :¢ºThomas“P…&arsons,Ÿû¤çÝ2Ž˜ºKathie“E.“NeÂ“wman,Ÿû¤çÝ1Ž˜ºand“J£Ü.“DanielŽ¤‘$"GezÑóelterŸû¤çÝ2Ž‘Ÿû¤ç,Ž‘
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ptmri7tÚDepartment–[´of“Physics,‘²¡UniveršáIsity“of“NotrŽVe“Dame˜,‘²¡NotrŽVe“Dame˜,Ž¡‘$"IN‘46556Ž¡‘$"Ÿû¤çÞ2)Ž‘,!ÈÚDepartment–5of›5Chemistry“and“BiocÑóhemistry‘ÿW,‘BIUniveráIsity“of˜NotršŽVe“DameáI,‘BINotr˜e“DameáI,Ž¡‘$"IN‘46556ŽŸÀ‘$"óT3{Ù
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ptmr7tëT(Dated:‘dü14–¼ÌApril“2016)ŽŽŸ€‘$"ó3{Ù
ptmr7t¹In–^ the“rst“twáIo‘^!papers“in“this“series,‘µ¨we“deš³3vÑóeloped“ne˜w‘^!shifted“potential“(SP),“gradientŽ¡‘$"shifted–·force›·(GSF),“and˜T‘ÿ
Baylor“shifted˜force“(TSF)‘¶çreal-space˜methods“for˜multipole“in-Ž¡‘$"teractions–in›condensed“phase˜simulations.‘ÊfHere,‘GŠwe“discuss˜the“dielectric˜properties“ofŽ¡‘$"
uids–\ëthat›\êemerÈ¹ge“from˜simulations“using˜these“methods.‘ÏMost“electrostatic˜methods“(in-Ž¡‘$"cluding–éõthe›éôEwáIald“sum)“require“correction˜to“the“conducting“boundary˜
uctuation“formulaŽ¡‘$"for–3the›3static“dielectric“constants,‘Ãand˜we“discuss“the˜deri•³3v“ation–3of“these˜corrections“forŽ¡‘$"the–ò›ne³3w›òœreal“space“methods.‘³ÙFÑóor˜quadrupolar“
uids,‘õIthe˜analogous“material“property˜is“theŽ¡‘$"quadrupolar–uÊsusceptibility‘ÿ8Y.‘Š>As“in“the“dipolar“case,‘‘nthe“
uctuation“formula“for“the“quadrupo-Ž¡‘$"lar–ö`susceptibility›ö_has“corrections˜that“depend˜on“the˜electrostatic“method˜being“utillized.Ž¡‘$"One–ŽÃof“the“most›ŽÄimportant“eects“measured“by“both“the˜static“dielectric“and“quadrupolarŽ¡‘$"susceptibility–‹;is“the›‹<ability“to“screen“charÈ¹ges“embedded˜in“the“
uid.‘ZW‘ÿ
Be˜use“potentials“ofŽ¡‘$"mean–¤force“between›¤€solv³3ated“ions“to“discuss˜ho³3w“geometric“fáIactors“can˜lead“to“distance-Ž¡‘$"dependent–screening“in“both“quadrupolar“and“dipolar“
uids.Ž ÎrÌŸù°‰ff$"ŽŸFÊŸü^ÀóS3{Ùff
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ptmr7tßElectronic‘€mail:‘˜gezelter@nd.edu.ŽŽŸ’á¹1ŽŽŽŒ‹* ·€ ý/€ ²€ ýp€ó,ÓßŒ˜
ptmb7t×I.Ž‘ªúINTR£ÜODUCTIONŽŸ%{)‘¹OvšÑóer–Úthe“past“se³3v˜eral–Úyears,‘á°there“has–Úbeen“increasing“interest“in“pairwise‘Úor“\real“space"“meth-Ž¤ods–Ê+for›Ê,computing“electrostatic“interactions“in˜condensed“phase“simulations.Ÿû¤çÞ1Ž‘€Ÿû¤ç{10ŽŽ‘J,¹These“techniquesŽ¡were–“initially›“
de³3vÑóeloped“by˜W‘ÿ
Bolf“Úet“al.‘qo¹in“their˜wšáIork“to³3w˜ards“an›“
óÆ›Â
zptmcm7yÂO‘õ½óÑÞ¾V
zptmcm7t¼(óˆóì?
zptmcm7m¿N‘¼)“¹Coulombic˜sum.Ÿû¤çÞ1ŽŽ‘“¹W‘ÿ
Bolf‘¨ð'‘ÿWsŽ¡method–`„of“using›`…cuto“neutralization“and“electrostatic˜damping“is“able“to˜obtain“eÑóxcellent“agree-Ž¡ment–with“Madelung“enerÈ¹gies“in“ionic“crystals.Ÿû¤çÞ1ŽŽŽ©A6‘¹Zahn–MZÚet“al.Ÿû¤çÞ2ŽŽ‘       MZ¹and“Fennell“and“Gezelter‘M[eÑóxtended“this“method“using“shifted“force“approxima-Ž¡tions–Wat›Wthe“cuto˜distance“in˜order“to“conservÑóe˜total“enerÈ¹gy˜in“molecular˜dynamics“simulations.Ÿû¤çÞ7ŽŽŽ¡¹Other–”÷recent“adv³3ances“in“real-space“methods“for‘”øsystems“of“point“charÈ¹ges“haÂ“všÑóe“included“e˜xplicitŽ¡elimination–of“the“net“multipole“moments“inside“the“cuto“sphere“around“each“charÈ¹ge“site.Ÿû¤çÞ8Ž‘€Ÿû¤ç,10ŽŽŽŸA5‘¹In–•éthe›•êpre³3vious“twáIo“papers“in˜this“series,‘»cwe˜de³3vÑóeloped“three“generalized“real˜space“methods:Ž¡shifted–bœpotential“(SP),›bgradient“shifted“force“(GSF),“and“T‘ÿ
Baylor˜shifted“force“(TSF).Ÿû¤çÞ11Ž‘  Ÿû¤ç,12ŽŽ‘"œ¹TheseŽ¡methods›L&e•³3v“aluate˜electrostatic˜interactions˜for˜charÈ¹ges‘L'and˜higher˜order˜multipoles˜using˜a˜nite-Ž¡radius–ªcuto›ªsphere.‘¶The“neutralization“and˜damping“of“local˜moments“within“the˜cuto“sphereŽ¡is–uüa›uûmultipolar“generalization˜of“W‘ÿ
Bolf‘¨ð'‘ÿWs˜sum.‘CIn˜the“GSF‘uÝand˜TSF‘uÞmethods,‘“zadditional˜terms“areŽ¡added–3ˆto“the›3‡potential“enerÈ¹gy“so“that“forces“and˜torques“also“v³3anish“smoothly“at˜the“cuto“radius.Ž¡This–ensures“that“the“total“enerÈ¹gy“is“conservÑóed“in“molecular“dynamics“simulations.Ž¦‘One–xGof“the“most“stringent“tests“of“anÑóy‘xFne³3w“electrostatic“method“is“the“delity“with“which“thatŽ¡method–Vˆcan›V‰reproduce“the“bÂ“ulk-phase“polarizability˜or“equi•³3v“alently‘ÿ8Y,‘¬*the–Vˆdielectric˜properties“ofŽ¡a–˜
uid.‘aBefore“the›—advÑóent“of“computer˜simulations,‘ðþKirkwáIood“and“Onsager˜de³3vÑóeloped“
uctu-Ž¡ation–GÐformulae›GÏfor“the˜dielectric“properties˜of“dipolar˜
uids.Ÿû¤çÞ13Ž‘        Ÿû¤ç,14ŽŽ‘Ð¹Along˜with“projections˜of“theŽ¡frequencšÑóy-dependent–Ždielectric“to“zero‘frequenc˜y‘ÿ8Y,‘òthese“
uctuation“formulae‘are“no³3w“widely“usedŽ¡to–predict“the“static“dielectric“constant“of“simulated“materials.Ž¦‘If–¿%we“consider“a›¿$system“of“dipolar“or“quadrupolar“molecules“under“the˜in
uence“of“an“eÑóxterÂ“-Ž¡nal–/eld“or“eld“gradient,‘{_the“net“polarization“of“the“system“will“larÈ¹gely“be“proportional“to“theŽ¡applied–4Xperturbation.Ÿû¤çÞ15Ž‘     Ÿû¤ç{18ŽŽ‘4W¹In“simulations,‘Anthe›4Wnet“polarization“of“the˜system“is“also˜determined“byŽ¡the–binteractions“Úbetween“¹the“molecules.‘ûwTherefore“the“macroscopic“polarizablity“obtained“fromŽ¡a–ö”simulation“depends“on›ö“the“details“of“the“electrostatic“interaction˜methods“that“were“emploáIyedŽ¡in–¿the“simulation.‘õT‘ÿ
Bo“determine“the‘¿ rele•³3v“ant–¿phð©ysical“properties“of“the“multipolar“
uid“from“theŽ¡system–zw
uctuations,‘™the›zvinteractions“between“molecules˜must“be˜incorporated“into˜the“formalismŽ¡for–the“bÂ“ulk“properties.ŽŽŸ’á2ŽŽŽŒ‹      0 ·€ ý/€ ²€ ýp€‘¹In›+Êmost–+Ësimulations,‘v¼bÂ“ulk“materials˜are˜treated“using˜periodic˜replicas“of˜small˜reÑógions,‘v½andŽ¤this–‘lle³3vÑóel›‘mof“approximation˜requires“corrections“to˜the“
uctuation“formulae˜that“were˜deri³3vÑóed“forŽ¡the–<}bÂ“ulk“
uids.‘mÆIn“1983“Neumann“proposed“a“general“formula‘<|for“e•³3v“aluating–<}dielectric“propertiesŽ¡of–U)dipolar›U*
uids“using˜both“EwáIald˜and“real-space“cuto˜methods.Ÿû¤çÞ19ŽŽ‘
Õ)¹Steinhauser˜and“NeumannŽ¡used–K”this“formula“to“e•³3v“aluate–K”the“corrected“dielectric‘K“constant“for“the“Stockmayer“
uid“using“twáIoŽ¡dierent–methods:‘¸PEwáIald-K”|ornfeld“(EK)“and“reaction“eld“(RF)“methods.Ÿû¤çÞ20ŽŽŽ¡‘¹Zahn–äJÚet›äIal.Ÿû¤çÞ2ŽŽ‘äJ¹utilized“this˜approach“and˜e•³3v“aluated–äJthe“correction˜fáIactor“for“using˜damped“shiftedŽ¡char•È¹ge-char“ge–„¨kšáIernel.‘2This“w˜as“later‘„§generalized“by“Izv•Ñóek˜o“v–„¨Úet›„§al.¹,Ÿû¤çÞ21ŽŽ‘¨¹who“noted“that˜the“eÑóxpressionŽ¡for–üthe›ýdielectric“constant“reduces˜to“widely-used˜conducting“boundary“formula˜for“real-spaceŽ¡methods–that“haÂ“všÑóe“rst“deri•³3v“ati“v˜es–that“v³3anish“at“the“cuto“radius.Ž¡‘One–%of“the›%primary“topics“of“this“paper“is˜the“deri•³3v“ation–%of“correction“fáIactors“for˜the“three“ne³3wŽ¡real–URspace“methods.‘¸FW‘ÿ
Be“nd“that“the“correction“formulae“for“dipolar“molecules“depends“not“onlyŽ¡on–0the“methodology›/being“used,‘<bÂ“ut“also“on˜whether“the“molecular“dipoles“are˜treated“using“pointŽ¡charÈ¹ges–or“point“dipoles.‘¸PW‘ÿ
Be“deri³3vÑóe“correction“fáIactors“for“both“cases.Ž¡‘In›€¤quadrupolar–€¥
uids,‘àÍthe“relationship˜between˜quadrupolar“susceptibility˜and“the˜dielectricŽ¡constant–©ñis›©ðnot“as“straightforwáIard˜as“in“the“dipolar˜case.‘¶"The“eecti³3vÑóe“dielectric˜constant“de-Ž¡pends–   ‰on“the›       ˆgeometry“of“the“eÑóxternal˜(or“internal)“eld“perturbation.Ÿû¤çÞ22ŽŽ‘
‰ˆ¹Signicant“eorts“haÂ“vÑóeŽ¡been–lxmade“to“increase›lwour“understanding“the“dielectric“properties“of˜these“
uids,Ÿû¤çÞ15Ž–        Ÿû¤ç,23Ž“,24ŽŽ‘#lx¹althoughŽ¡a–general“correction“formula“has“not“yet“been“de³3vÑóeloped.Ž¡‘In–Lbthis“paper›Lawe“deri³3vÑóe“general“formulae˜for“calculating“the“quadrupolar˜susceptibility“ofŽ¡quadrupolar–¥%
uids.‘§¿W›ÿ
Be“also“e•³3v“aluate–¥%the“correction“fáIactor‘¥&for“SP‘þ«,“GSF˜,“and“TSF‘¤¹methods“forŽ¡quadrupolar–®þ
uids›®ýinteracting“via˜point“charÈ¹ges,‘½point“dipoles˜or“directly˜through“quadrupole-Ž¡quadrupole‘interactions.Ž¡‘W‘ÿ
Be–½Õalso“calculate“the›½Ôscreening“behaÂ“vior“for“twáIo“ions“immersed“in˜multipolar“
uids“to“estimateŽ¡the–7distance“dependence›6of“charÈ¹ge“screening“in“both“dipolar˜and“quadrupolar“
uids.‘ôW‘ÿ
Be“use“threeŽ¡distinct–methods“to“compare“our“analytical“results“with“computer“simulations“(see“Fig.‘¸P1):Ž©‘1.ŽŽŽ‘responses–of“the“
uid“to“eÑóxternal“perturbations,Ž¤0‘2.ŽŽŽ‘
uctuations–of“system“multipole“moments,“andŽ¡‘3.ŽŽŽ‘potentials–of“mean“force“between“solv³3ated“ions,Ž¦‘Under–7the“in
uence“of“weak“eÑóxternal“elds,‘cDthe“bÂ“ulk“polarization“of“the“system“is“primarilyŽŸa–†Elinear“response“to“the“perturbation,‘§×where“proportionality“constant“depends“on“the“electrostaticŽŽŸ’á3ŽŽŽŒ‹8 ·€ ý/€ ²€ þÌëNŸ„á"ï<PSfile="Schematic.eps" llx=0 lly=0 urx=720 ury=362 rwi=4662 ŽŽŸÀëTFIG.–D1.‘ScDielectric“properties“of“a“
uid“measure›Cthe“response“to“eÕúxternal“electric“elds˜and“gradients“(left),Ž¤Àor–*“internal›*”elds“and˜gradients“generated˜by“the“molecules˜themselvÕúes“(center),‘For˜elds“produced˜by“em-Ž¡bedded–èions“(right).‘FOThe“dielectric“constant“(óWˆóì?
ó3
zptmcm7mëW‘è§ëT)“measures›çall“three“responses“in“dipolar“
uids˜(top).‘FPInŽ¡quadrupolar›ý’liquids–ý‘(bottom),‘
Äthe“rele•¹ëv“ant˜bÇóulk˜property–ý‘is˜the˜quadrupolar˜susceptibility“(ëWŸ¤zóXˆóì?
zptmcm7mëXQŽ‘F¦ëT),‘
Äand“the˜ge-Ž¡ometry–¼Ìof“the“eld“determines“the“eecti¹ëvÕúe“dielectric“screening.ŽŽŽŸ¿ž¹interactions–
Ÿbetween›
 the“multipoles.‘Ø/The“
uctuation“formulae˜connect“bÂ“ulk˜properties“of˜the“
uidŽ¤to–5Üequilibrium›5Ý
uctuations“in˜the“system˜multipolar“moments“during˜a“simulation.‘YæThese“
uctua-Ž¡tions–_ùalso›_údepend“on˜the“form“of˜the“electrostatic“interactions˜between“molecules.‘Ø<Therefore,‘wøtheŽ¡connections–5Vbetween›5Wthe“actual“bÂ“ulk“properties˜and“both“the“computed˜
uctuation“and“eÑóxternalŽ¡eld–responses“must“be“modied“accordingly‘ÿ8Y.ŽŸßÏ‘The–†ûpotential“of“mean“force“(PMF)‘†—alloš³3ws“calculation‘†üof“an“eecti˜vÑóe“dielectric“constant“orŽ¡screening–™ŸfáIactor›™ from“the˜potential“enerÈ¹gy“between˜ions“before“and˜after“dielectric˜material“isŽ¡introduced.‘5Computing–—¢the“PMF‘—{between›—¡embedded“point“charÈ¹ges˜is“an“additional“check˜on“theŽ¡bÂ“ulk–properties“computed“via“the“other“twáIo“methods.ŽŸ1[ƒ×II.Ž‘UôTHE–REAL-SP‘ÿ¬A‘ÿWCE“METHODSŽŸ$Â‘¹In–^the“rst›_paper“in“this“series,‘´vwe˜deri³3vÑóed“interaction“enerÈ¹gies,‘´vas“well“as˜eÑóxpressions“for“theŽ¡forces–»¥and›»¤torques“for˜point“multipoles“interacting˜via“three“ne³3w˜real-space“methods.Ÿû¤çÞ11ŽŽ‘;¤¹The“T‘ÿ
BaylorŽŽŸ’á4ŽŽŽŒ‹$^ ·€ ý/€ ²€ ýp€¹shifted-force–/$(TSF)‘.×method“modies“the‘/%electrostatic“káIernel,›zí¿f‘s³¼(¿r•F¦¼)‘S=‘S1¿=r“¹,˜so–/$that“all‘/%forces“andŽ¤torques–go“smoothly“to“zero“at“the“cuto“radius,Ž©#yš’¹_ß¿UŸûN‘jyÞTSFŽŸ=éóˆóì? 
zptmcm7mÀabŽŽ‘â¼=‘ª¨¿MŸÌÌÀaŽ–¿MŸŠÀbŽ“¿fŸÌÌÀnŽ“¼(¿rF¦¼)¿:Ž’Æ¹(1)ŽŽŽŸ#y™Here–¯tthe›¯smultipole“operator˜for“site˜¿a¹,‘ÛQ¿MŸÌÌÀaŽ‘¹,‘ÛPis“eÑóxpressed˜in“terms˜of“the“point˜charÈ¹ge,‘ÛQ¿CŸÌÌÀaŽ‘¹,‘ÛPdipole,Ž¡×DŽ‘©ùŸÌÌÀaŽ‘
©ù¹,–¨Ëand›’þquadrupole,‘¨Ìól‚ÎR6cmss12ëlQŸÌÌÀaŽ‘¹,“for˜object˜¿a¹,‘¨Ìetc.‘“úBecause˜each˜of˜the˜multipole˜operators˜includes˜a˜set˜ofŽ¡implied–‡øgradient“operators,›©öan“approximate‘‡ùelectrostatic“káIernel,˜¿fŸÌÌÀnŽ‘¼(¿rF¦¼)“¹is“T‘ÿ
BaylorÂ“-eÑóxpanded“aroundŽ¡the–-cuto“radius,‘xàso“that“¿n‘T¼+‘S1“¹deri•³3v“ati“vÑóes›-v“anish˜as–-€¿r‘˜ÎÂ!‘R(¿rŸÌÌÀcŽ‘~ö¹.‘@ÓThis“ensures˜smooth˜con…&vÑóerÈ¹genceŽ¡of›^Úthe–^ÛenerÈ¹gy‘ÿ8Y,‘vforces,‘v‘and“torques˜as˜molecules˜leaÂ“vÑóe“and˜reenter˜each“others˜cuto˜spheres.‘ÔßTheŽ¡order–ýjof“the›ýkT‘ÿ
Baylor“eÑóxpansion“is“determined“by“the˜multipolar“order“of“the“interaction.‘°That“is,Ž¡smooth–"*quadrupole-quadrupole›"+forces“require˜the“fth˜deri•³3v“ati“vÑóe‘"*to˜v“anish–"*at˜the“cuto˜radius,‘*µsoŽ¡the–appropriate“function“T‘ÿ
Baylor“eÑóxpansion“will“be“of“fth“order‘ÿW.ŽŸÓ3‘FÑóollo³3wing–S!this“procedure“results›S"in“separate“radial“functions“for“each˜of“the“distinct“orientationalŽ¡contribÂ“utions–Y¢to‘Y¡the“potential.‘Å5FšÑóor“e˜xample,›°
in“dipole-dipole“interactions,˜the“direct‘Y¡dipole“dotŽ¡product–(×DŸÌÌÀaŽ‘ª©Â‘ª©×DŸŠÀbŽ‘¹)“is“treated“dierently“than“the“dipole-distance“dot“products:Ž¦‘Y P¿UŸì«ó-ÓßŒ˜   
ptmb7tØDŸY™óˆóì?
zptmcm7mÁaŽ‘ØDŸ¹8ÁbŽŽ‘~ö¼(¿ršF¦¼)–ª¨=“ÂŸ÷àI‘
(¼1Ž‘33Ÿ]#‰zÙÓéŸ
ýö4¿‘øÐŸäóÑÞ¾V 
zptmcm7t½0ŽŽŽŽŽ‘£¼[Ž‘âŒ(Ž‘#Šr×DŸÌÌÀaŽ‘ª©Â‘ª©×DŸŠÀbŽ‘¼)ŽŽ‘KƒÙ¿vŸä½21Ž‘ €¼(¿r˜¼)–ª©+“(Ž‘R×DŸÌÌÀaŽ‘ª©Âž¹Z‘U'¼^ŽžF¦“×rŽŽ‘þœ¼)ŽŽ‘%õð(Ž‘*Ö×DŸŠÀbŽ‘ª©Âž¹Z‘U'¼^ŽžF¦“×rŽŽ‘þœ¼)ŽŽ‘JA7¿vŸä½22Ž‘ €¼(¿r˜¼)]ŽŽŽ’Æ¹(2)ŽŽŽ¦In–9“standard›9’electrostatics,‘Gøthe“twáIo˜radial“functions,‘G÷¿vŸä½21Ž› €¼(¿rF¦¼)“¹and“¿vŸä½22Ž˜¼(¿ršF¦¼)‘9’¹are“proportional“to“¼1¿=r˜Ÿû¤ç½3Ž‘F¦¹,‘G÷bÂ“utŽ¡the•Ñóy›&«haÂ“v“e–&ªdistinct˜radial“dependence˜in“the˜TSF‘&¡method.‘,OCareful˜choice“of˜these“functions˜makáIesŽ¡the–Íforces“and“torques›Ív³3anish“smoothly“as“the“molecules“drift“beÑóyond“the˜cuto“radius“(e³3vÑóen“whenŽ¡those–molecules“are“in“dierent“orientations).ŽŸÓ3‘A‘¯©second–¯½and›¯¾some³3what“simpler˜approach“in…&vÂ“olvÑóes“shifting˜the“gradient˜of“the˜Coulomb“poten-Ž¡tial–for“each“particular“multipole“order…&,ŽŸ#y™‘PB¿UŸûN‘jyÞGSFŽŸ=éÀabŽŽ‘ ¼=‘ª¨ŸàóŽCq
zptmcm7vÅXŽŽŽ‘OÒ¼[Ž‘¢»¿U›jy¼(×r¿;–UTëlA¿;“ëlB¼)–ª©Â“¿U˜¼(¿rŸÌÌÀcŽž¹Z‘)t¼^ŽžF¦‘~ö×rŽŽ‘        Òé¿;–UTëlA¿;“ëlB¼)–ª©Â“¼(¿r‘ñOÂ“¿rŸÌÌÀcŽ‘~ö¼)ž¹Z‘ª~^ŽžF¦×rŽŽ‘þœÂ“r¿U˜¼(¿rŸÌÌÀcŽž¹Z‘)t¼^ŽžF¦‘~ö×rŽŽ‘        Òé¿;–UTëlA¿;“ëlB¼)]ŽŽŽ’Æ¹(3)ŽŽŽ¦where–áthe“sum“describes“a“separate“force-shifting“that“is“applied“to“each“orientational“contribÂ“u-Ž¡tion–Æ…to‘Æ†the“enerÈ¹gy‘ÿ8Y,‘ø'i.e.‘à¿vŸä½21Ž›
F†¹and“¿vŸä½22Ž˜¹are“shifted“separately‘ÿ8Y.‘áIn“this›Æ†eÑóxpression,ž¹Z‘¢¤¼^ŽžF¦‘ø&×rŽŽ‘
Ÿ¹is“the˜unit“vÑóectorŽ¡connecting–the›twáIo“multipoles˜(¿a“¹and˜¿b¹)“in˜space,‘$gand˜ëlA“¹and˜ëlB“¹represent˜the“orientations˜the“mul-Ž¡tipoles.‘¥žBecause–¤othis›¤pprocedure“is“equi•³3v“alent˜to–¤ousing“the“gradient˜of“an“image˜multipole“placedŽ¡at–Äåthe›Äæcuto“sphere˜for“shifting“the˜force,‘öthis˜method“is“called˜the“gradient˜shifted-force“(GSF)Ž¡approach.ŽŽŸ’á5ŽŽŽŒ‹,8 ·€ ý/€ ²€ ýp€‘¹Both–—¤the“TSF‘—}and“GSF‘—~approaches“can“be“thought“of“as“multipolar“eÑóxtensions“of“the“originalŽ¤damped–ãúshifted-force›ãû(DSF)‘ãóapproach“that˜wáIas“de³3vÑóeloped˜for“point˜charÈ¹ges.‘®øThere˜is“also˜a“multi-Ž¡polar–¿ÁeÑóxtension›¿Âof“the“W‘ÿ
Bolf˜sum“that˜is“obtained“by˜projecting“an“image˜multipole“onto˜the“surfáIaceŽ¡of–}the›|cuto“sphere,‘\and“including˜the“interactions“with˜the“central“multipole“and˜the“image.‘òÆThisŽ¡eecti³3vÑóely–shifts“only“the“total“potential“to“zero“at“the“cuto“radius,Ž¤3ÛF’“¿UŸûN‘jyÞSPŽŸ=éÀabŽŽ‘—%¼=‘ª¨ŸàÅXŽŽŽ‘OÒ¼[Ž‘¢»¿U›jy¼(×r¿;–UTëlA¿;“ëlB¼)–ª©Â“¿U˜¼(¿rŸÌÌÀcŽž¹Z‘)t¼^ŽžF¦‘~ö×rŽŽ‘   Òé¿;–UTëlA¿;“ëlB¼)]ŽŽŽ’Æ¹(4)ŽŽŽ¡where–˜the›™sum“agð©ain˜describes“separate˜potential“shifting“that˜is“done˜for“each˜orientational“contri-Ž¤bÂ“ution–\Ðto‘\Ïthe“eneršÈ¹gy‘ÿ8Y.‘Î¾The“potential“ener˜gy›\Ïbetween“a˜central“multipole˜and“other˜multipolar“sitesŽ¡goes–Óysmoothly›Ózto“zero˜as“¿r‘fÊÂ!‘ $¿rŸÌÌÀcŽ‘~ö¹,‘XbÂ“ut˜the“forces“and˜torques“obtained˜from“this˜shifted“potentialŽ¡(SP)–approach“are“discontinuous“at“¿rŸÌÌÀcŽ‘~ö¹.ŽŸI‘All–Ýythree›Ýzof“the“ne³3w“real˜space“methods“share“a˜common“structure:‘sBthe˜v³3arious“orientationalŽ¡contribÂ“utions–¨to“multipolar“interaction“enerÈ¹gies“require“separate“treatment“of“their“radial“func-Ž¡tions,‘Øðand–zYthese›zZare“tabÂ“ulated“for˜both“the˜raÑów“Coulombic˜káIernel“(¼1¿=rF¦¹)“as˜well“as˜the“dampedŽ¡káIernel›ÿ–(¼erfcŽ‘¥Ô(¿‘þ÷r•F¦¼)¿=r“¹),‘?|in˜the˜rst˜paper˜of˜this˜series.Ÿû¤çÞ11ŽŽ‘
–¹The˜second˜paper˜in˜this˜series˜e•³3v“aluatedŽ¡the–        ^delity“with“which›   ]the“three“ne³3w“methods“reproduced“EwáIald-based“results˜for“a“number“ofŽ¡model–Üsystems.Ÿû¤çÞ12ŽŽ‘
\¹One“of“the“major“ndings“wáIas“that“moderately-damped“GSF‘Ûäsimulations“pro-Ž¡duced–ÅÑnearly“identical“behašÂ“vior“with‘ÅÐEwáIald-based“simulations,‘÷Eb˜ut“the“real-space“methods“scaleŽ¡linearly–with“system“size.ŽŸSýR×III.Ž‘îDIPOLAR–FLUIDS“AND“THE“DIELECTRIC“CONST‘þëANTŽŸ)ƒ
‘¹Dielectric–{Ìproperties“of“a“
uid“arise“mainly“from‘{Ëresponses“of“the“
uid“to“either“applied“elds“orŽ¡transient–<Îelds“internal›<Íto“the“
uid.‘n¹In“response“to“an“applied˜eld,‘Lthe˜molecules“haÂ“vÑóe“electronicŽ¡polarizabilities,‘ÙSchanges›Ûto–Üinternal“bond“lengths,‘ÙRand“reorientations“to³3wáIards“the˜direction“of“theŽ¡applied–‘<eld.‘“dThere“is“an“added“complication“that“in“the“presence“of‘‘=eÑóxternal“eld,‘§cthe“perturbationŽ¡ešÑóxperienced–Õjby“an˜y›Õksingle“molecule“is“not“only“due˜to“eÑóxternal“eld“bÂ“ut“also“to˜the“elds“producedŽ¡by–the“all“other“molecules“in“the“system.ŽŽŸ’á6ŽŽŽŒ‹;w ·€ ý/€ ²€ ýp€×A.‘Response–to“ExterÑónal“PÂ“erturbationsŽŸ#9ó‘¹In–24the›25presence“of“uniform˜electric“eld“×E¹,‘~Âan“indi³3vidual˜molecule“with“a˜permanent“dipoleŽ©moment–¿pŸÌÌÀoŽ‘¹will“realign“along“the“direction“of“the“eld“with“an“aÂ“všÑóerage“polarization“gi³3v˜en“byŽ¤ªY’ÉB$Âh×pÂiŽ’Ûè›¼=‘ª¨¿Ÿä½0Ž‘¿ŸÌÌÀpŽ‘¬Ì×E¿;Ž’Æ¹(5)ŽŽŽ¡where–Œ¿ŸÌÌÀpŽ‘WÂ¼=‘ªõ¿pŸÌÌÀoŽŽ‘‘[Ÿû¤ç½2Ž‘‘[¿=¼3¿Ÿä½0Ž‘¿kŸÌÌÀBŽ‘ÿ½¿T‘m¹is“the“contribÂ“ution“to“molecular“polarizability‘‹due“solely“to“reorientationŽ¦dynamics.‘¢ Because–½pthe“applied“eld“must‘½oo•Ñóv“ercome–½pthermal“motion,‘ÊÀthe“orientational“polarizationŽ¦depends–in…&vÑóersely“on“the“temperature.Ž¦‘LikáIe³3wise,‘Ta–˜©condensed“phase“system“of“permanent“dipoles“will“also“polarize“along“the“directionŽ¦of–an“applied“eld.‘¸PThe“polarization“density“of“the“system“isŽ¡’Ìì5×PŽŽ’ÖëÚ¼=‘ª¨¿ŸÌÌÀoŽ‘¿ŸÌÌÀDŽ‘ÿ{×E¿:Ž’Æ¹(6)ŽŽŽŸªXThe–Â·constant“¿ŸÌÌÀDŽ‘
Â2¹is›Â¸the“macroscopic“polarizability‘ÿ8Y,‘óewhich“is“an“emerÈ¹gent˜property“of“the“dipolarŽ¦
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Ê+¹is˜distinct“from“the˜dipole“susceptibility‘ÿ8Y,‘ý[¿ŸÌÌÀDŽ‘ÿ{¹,‘ý\which˜is“theŽ¦quantity–most“directly“related“to“the“static“dielectric“constant,“¿‘©Ÿ¼=‘ª¨1–ª©+“¿ŸÌÌÀDŽ‘ÿ{¹.ŽŸ+Žo×B.‘Fluctuation‘F³3ormulaŽŸ#9ó‘¹FÑóor–býa›bþsystem“of“dipolar˜molecules“at“thermal“equilibrium,‘{½we“can˜dene“both“a˜system“dipoleŽ¦moment,›×M–ª¨¼=“ŸâÅPŽ‘8ôŸ–ÂÀiŽ‘ŽÅ×pŸÌÌÀiŽ‘}¹as˜well˜as˜a˜dipole˜polarization˜density‘ÿ8Y,˜×P“¼=“Âh×MÂiŽ‘£»¿=V‘·¹.Ž¦‘In–7the“presence›6of“applied“eld“×E¹,‘„the“polarization“density˜can“be“eÑóxpressed“in˜terms“of“
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ýö‘    ñë¼3¿ŸÌÌÀoŽ‘¿V‘·kŸÌÌÀBŽ‘ÿ½¿TŽŽŽŽ‘GV×EŽ’Æ¹(7)ŽŽŽŸí”This–¡Ohas›¡Pstructural“similarity“with“the˜Boltzmann“aÂ“vÑóerage“for˜the“polarization“of“a˜single“molecule.Ž¦Here‘Âh×MŸû¤ç½2ŽŽ‘SóÂiŽ–Nh‘ª©h×MÂiŽŽ‘NhŸú‘ì½2Ž“¹measures–
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X-Â!‘‚…¼0¹,‘ÝSthe˜ensemble“aÂ“vÑóerage˜of“both“the˜net“dipole˜moment“Âh×MÂiŽ‘M¹and“dipolarŽ¦polarization–Ë×P›Ë¹tends“to˜v³3anish“bÂ“ut˜Âh×MŸû¤ç½2ŽŽ‘SóÂiŽ‘#9ô¹does˜not.‘      ŸThe˜macroscopic“dipole˜polarizability“canŽ¦therefore–be“written“as,ŽŸDH’º7â¿ŸÌÌÀDŽ‘        ª#¼=Ÿ÷àI‘ÝÛÂh×MŸû¤ç½2ŽŽ‘SóÂiŽ‘,C‘ª©h×MÂiŽŽ‘NhŸú‘ì½2ŽŽ‘ÝÛŸ]#‰zÙ?ï¹Ÿ
ýö‘    ñë¼3¿ŸÌÌÀoŽ‘¿V‘·kŸÌÌÀBŽ‘ÿ½¿TŽŽŽŽŽ’Æ¹(9)ŽŽŽŽŸ’á7ŽŽŽŒ‹Fj ·€ ý/€ ²€ þ¼˜PŸš¡"ï:PSfile="Tensors.eps" llx=0 lly=0 urx=731 ury=376 rwi=4662 ŽŽŸÀëTFIG.–“k2.‘èÛIn“the›“lreal-space“electrostatic˜methods,‘  the“molecular˜dipole“tensoræ,–      ógÓßŒ˜
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“Ž‘yÌëU(ëWr@wëU)ëT,“which˜can–þhaÇóvÕúe“distinct˜forms“if˜the“molecule“is˜represented“byŽ¡charÍges,–¼Ìdipoles,“or“point“quadrupoles.ŽŽŽŸ#Kp¹This–Ÿrelationship›žbetween“a“macroscopic“property˜of“dipolar“material“and˜microscopic“
uctuationsŽ¤is–true“e³3vÑóen“when“the“applied“eld“×EŽŽ‘
«¨Â!‘ª¨¼0¹.ŽŸ<´Œ×C.‘CorrÈ¹ection‘F³3actorsŽŸ%ß«‘¹In–Tthe“presence“of›Ua“uniform“eÑóxternal“eld“×EŸû¤çóÆ›Â       
zptmcm7yÃŽ‘¹,‘ªthe“total“electric“eld“at“×r˜¹depends“on“the“polarÂ“-Ž¡ization–density“at“all“other“points“in“the“system,Ÿû¤çÞ19ŽŽŽŸ(ñ(’€¢˜×E¼(×r¼)–ª¨=“×EŸûNÃŽ‘¼(×r¼)–ª©+Ÿ÷àI‘ÇÑ1Ž‘ÝÜŸ]#‰zÙÓéŸ
ýö4¿‘øÐŸÌÌÀoŽŽŽŽŽ‘:LŸñŠÅZŽ‘)VE¿d‘Ÿ·×rŸûNÃ0Ž›ù™×T¼(×r“Â“×rŸûNÃ0Ž˜¼)“Â“×P¼(×rŸûNÃ0Ž˜¼)Ž‘œþ¿:Ž’À¹(10)ŽŽŽŸ(ñ)×T–¹is“the“dipole“interaction“tensor“connecting“dipoles“at“×rŸû¤çÃ0Ž‘ù™¹with“the“point“of“interest“(×r¹).ŽŸ¥¸‘In–üYsimulations›üXof“dipolar˜
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òŸú…¢½2Ž‘+`F¿d‘Ÿ·×rŽŽŽŽŽ’À¹(46)ŽŽŽŸ!ü.inte•Ñógrated›Kwo“v“er–Kxthe˜interaction˜vÂ“olume.‘|#Note˜that“this˜geometrical˜fáIactor“is˜also˜required“to˜computeŽ¡eectiš³3vÑóe–dielectric“constants“e˜všÑóen“when“the“eld“gradient“is“homogeneous“o˜v˜er“the“entire“sample.Ž¡‘T‘ÿ
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uids˜are“not˜continuum“dielectrics,Ž¡the–eecti³3vÑóe“dielectric“constant“is“a“function“of“the“interionic“separation,Ž¤"0ì’¬hß¿‘þ÷¼(¿r•F¦¼)‘ª¨=Ÿ÷àI‘ÝÛ¿uŸÌÌ½raÝvwŽ›
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Vdirect“char•È¹ge-char“ge–
Winteraction“potential“that“is“in˜use“during“the“simulation.Ž¤¿‘þ÷¼(¿rF¦¼)–¨›¹may›¨œv³3ary“considerably˜from“the˜bÂ“ulk“estimates˜at“short˜distances,‘ºalthough“it˜should“con…&vÑóerÈ¹geŽ¡to–the“bÂ“ulk“v³3alue“as“the“separation“between“the“ions“increases.ŽŸ.M¥×VI.Ž‘TóSIMULA‘þÜ,TION‘METHODOLOGYŽŸ#Ÿ—‘¹T‘ÿ
Bo–æCtest›æBthe“formalism˜de³3vÑóeloped“in˜the“preceding˜sections,‘ëhwe“haÂ“vÑóe˜carried“out˜computer“simu-Ž¡lations–gousing›gpthree“dierent˜techniques:‘‡.i)˜simulations“in˜the“presence˜of“eÑóxternal“elds,‘Lii)“equi-Ž¡librium–ó calculations“of“box“moment“
uctuations,‘öand“iii)“potentials“of“mean“force“(PMF)‘óbetweenŽ¡embedded–¥áions.‘šEIn“all“cases,‘·çthe“
uids“were“composed“of“point“multipoles‘¥àprotected“by“a“Lennard-Ž¡Jones–potential.‘¸PThe“parameters“used“in“the“test“systems“are“gi³3vÑóen“in“table“III.Ž©e¤‘The–¥‘rst›¥’of“the“test“systems˜consists“entirely“of“
uids˜of“point“dipolar“or˜quadrupolar“moleculesŽ¡in–g'the“presence›g(of“constant“eld“or“eld“gradients.‘íÆSince“there“are“no˜isolated“charÈ¹ges“within“theŽ¡system,›{athe–b´di³3vÑóerÈ¹gence“of‘bµthe“eld“will“be“zero,˜Úi.eÑó.Ÿü
‘<•¿~ŽŸRó‘àmÂrŽŽ‘=ð‘Ï8×E‘á}¼=‘á~0¹.‘àlThis“condition‘bµcan“be“satised“byŽ¡using–the“relati³3vÑóely“simple“applied“potential“as“described“in“the“supporting“information.Ž¦‘When–úa›ûconstant“electric“eld“or˜eld“gradient“is˜applied“to“the˜system,‘Iøthe˜molecules“alignŽ¡along–¼õthe“direction“of“the“applied›¼ôeld,‘Ê^and“polarize“to“a“deÑógree˜determined“both“by“the“strength“ofŽ¡the–Ãleld“and›Ãmthe“
uid'‘ÿWs“polarizibility‘ÿ8Y.‘¤W‘ÿ
Be“haÂ“vÑóe“calculated˜ensemble“aÂ“vÑóerages“of“the˜box“dipole“andŽ¡quadrupole–Çümoments“as›Çýa“function“of“the“strength˜of“the“applied“elds.‘¥¤If“the˜elds“are“sucientlyŽ¡weak,‘RÄthe›response–is“linear˜in“the“eld˜strength,‘RÄand“one˜can“easily“compute˜the“polarizabilityŽ¡directly–from“the“simulations.Ž¦‘The–9second›9set“of˜test“systems˜consists“of˜equilibrium“simulations˜of“
uids˜of“point˜dipolar“orŽ¡quadrupolar–Ÿ›molecules›Ÿšsimulated“in˜the“absence˜of“anšÑóy“e˜xternal–Ÿšperturbation.‘— The“
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uids.‘ŸThe›´!box–´"multipolar“moments˜were“computed˜as“simple“sums˜o•Ñóv“er–´"the˜instantaneous“molec-Ž¡ular–Bmoments,‘’—and›B
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ptmr7tër2Ž‘€ëT)ŽŽ¡‰ff™€¦‘ã—Dipolar‘¼Ì
uid‘Ÿ†j„¿ÿffŽ‘WLF3.41Ž‘xó>0.2381‘     À•Ÿ†j„¿ÿffŽ’×VF-ŽŽŽ’9þ˜1.4026ŽŽŽ’ˆóŸ-Ž’œ|>-Ž’¶aÐ-‘Q˜Ÿ†j„¿ÿffŽ’ñ¢Ü39.948ŽŽŽ’=`11.613Ž’döm11.613Ž’‰Ð0.0ŽŽ¤¿ÿ‘Quadrupolar‘¼Ì
uid‘Ÿ†j„¿ÿffŽ‘Ty2.985Ž‘{°0.265‘}aŸ†j„¿ÿffŽ’×VF-ŽŽŽ’G:B-ŽŽŽ’ƒîZ0.0Ž’—vù0.0Ž’ª8-2.139‘Ÿ†j„¿ÿffŽ’îæ18.0153ŽŽŽ’:£>43.0565Ž’b9 43.0565Ž’‰Ð0.0ŽŽ¡‘E¬mqŽŽ‘K&ŸûÅþër+ŽŽŽ‘R)Ÿ†j„¿ÿffŽ‘Z       ëT1.0Ž’)¤0.1‘öúŸ†j„¿ÿffŽ’ÓU·+1ŽŽŽ’G:B-ŽŽŽ’ˆóŸ-Ž’œ|>-Ž’¶aÐ-‘Q˜Ÿ†j„¿ÿffŽ’ô_¨22.98ŽŽŽ’J›µ-Ž’r2-Ž’ŽÕG-ŽŽ¡‘D„qŽŽ‘J”˜ŸûÅþër{ŽŽŽ‘R)Ÿ†j„¿ÿffŽ‘Z ëT1.0Ž’)¤0.1‘öúŸ†j„¿ÿffŽ’Ô™z-1ŽŽŽ’G:B-ŽŽŽ’ˆóŸ-Ž’œ|>-Ž’¶aÐ-‘Q˜Ÿ†j„¿ÿffŽ’ô_¨22.98ŽŽŽ’J›µ-Ž’r2-Ž’ŽÕG-ŽŽŸìÐ‰ff™€ŽŽŽŽŽï ps: currentpoint grestore movetoŽŽŽŽŽŽŽŸ’Þ¹18ŽŽŽŒ‹ã ·€ ý/€ ²€ ýp€‘¹The–Ognal“system›Ohconsists“of“dipolar“or˜quadrupolar“
uids“with“twáIo˜oppositely“charÈ¹ged“ionsŽ¤embedded–Ó·within“the›Ó¸
uid.‘©These“ions“are“constrained˜to“be“at“xÑóed“distance˜throughout“a“simula-Ž¡tion,‘´although–¡thešÑóy“are‘¡allo³3wed“to“mo˜v˜e›¡freely“throughout“the˜
uid“while“satisfying˜that“constraint.Ž¡Separate–æsimulations“were“run‘æat“a“range“of“constraint“distances.‘¯¬A‘æ
dielectric“screening“fšáIactor“w˜asŽ¡computed–Âôusing“the“ratio“between“the“potential“between‘Âõthe“twáIo“ions“in“the“absence“of“the“
uidŽ¡medium–and“the“PMF“obtained“from“the“simulations.Ž© ‘W‘ÿ
Be–nAcarried“out“these›nBsimulations“for“all“three“of“the“ne³3w˜real-space“electrostatic“methods“(SP‘þ«,Ž¡GSF‘ÿ
B,–åÂand›åÃTSF)‘å‡that“were˜de³3vÑóeloped“in“the˜rst“paper“(Ref.‘i—11Ž‘i—)˜in“the“series.‘i—The“radius˜of“theŽ¡cuto–sphere“wšáIas“tak˜en“to“be“12“Ÿþ"„ŽŽ“A.“Each“of‘
the“real“space“methods“also“depends“on“an“adjustableŽ¡damping–jparameter“¿‘a¹(in“units“of“¼lengthŽ‘"UŸûG3Ã½1Ž‘.„¹).‘ÎW‘ÿ
Be“haÂ“vÑóe“selected“ten“dierent“v³3alues“of“dampingŽ¡parameter:‘50.0,–íï0.05,“0.1,“0.15,›íî0.175,“0.2,“0.225,“0.25,˜0.3,“and–¾Y0.35“Ÿþ"„ŽŽ“AŸû¤çÃ½1Ž‘¼‡¹in“our‘¾Xsimulations“ofŽ¡the–õ¢dipolar“liquids,›3while“four“v³3alues‘õ£were“chosen“for“the“quadrupolar“
uids:‘£•0.0,˜0.1,‘3
0.2,˜andŽ¡0.3‘Ÿþ"„ŽŽ‘AŸû¤çÃ½1Ž‘þ/¹.Ž¦‘FÑóor–Ùeach›Ùof“the˜methods“and“systems˜listed“abo•Ñóv“e,‘àÙa–Ùreference“simulation˜wáIas“carried˜out“usingŽ¡a–~multipolar›implementation“of“the˜EwšáIald“sum.Ÿû¤çÞ36Ž‘ Ÿû¤ç,37ŽŽ‘Æ~¹A‘}def˜ault“tolerance‘of“¼1‘Â‘¼10Ÿû¤çÃ½8Ž‘®¹kcal/mol“w˜asŽ¡used–Nin“all“EwšáIald“calculations,‘N"resulting“in“Ew˜ald“coecient“0.3119“Ÿþ"„ŽŽ“AŸû¤çÃ½1Ž‘      }¹for“a“cuto“radius“ofŽ¡12‘±äŸþ"„ŽŽ–±äA.›±åAll“of˜the“electrostatics˜and“constraint“methods˜were“implemented˜in“our˜group'‘ÿWs“openŽ¡source–Å+molecular“simulation“program,‘övOpenMD,Ÿû¤çÞ38Ž‘    Ÿû¤ç,39ŽŽ‘…+¹which“wáIas“used“for“all“calculations“in“thisŽ¡wáIork.Ž¦‘Dipolar–G°systems“contained“2048“Lennard-Jones-protected“point“dipolar“(Stockmayer)“moleculesŽ¡with–€Žreduced“density‘€¿‘þ÷Ÿû¤çÃŽ‘3¼=›4¦0¿:¼822¹,‘štemperature“¿T‘lƒŸû¤çÃŽ‘   ¡)¼=˜1¿:¼15¹,‘šmoment“of“inertia‘€¿I‘épŸû¤çÃŽ‘¼=˜0¿:¼025¹,‘šand“dipoleŽ¡moment–¢ó¿‘þ÷Ÿû¤çÃŽ‘Su¼=‘TŸöâÂpŽ‘SvŸöâ‰wÇRéŸ     û¼3¿:¼0ŽŽŽ‘¦_¹.‘™KThese“systems“were›¢òequilibrated“for“0.5˜ns“in“the˜canonical“(NVT)‘¢Ûensemble.Ž¡Data–(ÿcollection›(þwáIas“carried˜out“o•Ñóv“er˜a–(ÿ1˜ns“simulation˜in“the˜microcanonical“(NVE)‘(²ensemble.Ž¡Box–.Sdipole“moments‘.Twere“sampled“e³3všÑóery“fs.‘CJF˜or“simulations‘.Twith“e˜xternal“perturbations,‘yèeldŽ¡strengths–äíranging›äìfrom“¼0–‘—Â“¼10‘‘–Â“¼10Ÿû¤çÃ½4Ž‘ã¹V/Ÿþ"„ŽŽA–äíwith˜increments“of˜¼10Ÿû¤çÃ½4Ž‘ã¹V/Ÿþ"„ŽŽA“were˜carried“out˜for“eachŽ¡system.Ž¦‘Quadrupolar–úIsystems“contained›úH4000“linear“point“quadrupoles“with“a˜density“¼2¿:¼338‘g¿=¼cmŽ‘©ïŸúŠ@½3Ž‘%¤8¹at“aŽ¡temperature–¡Cof“500“K.“These›¡Bsystems“were“equilibrated“for“200“ps“in“a˜canonical“(NVT)‘¡+ensemble.Ž¡Data–Mcollection“wáIas›MŒcarried“out“o•Ñóv“er–Ma˜500“ps“simulation“in“the˜microcanonical“(NVE)‘Myensemble.Ž¡Components–öof›÷box“quadrupole“moments˜were“sampled“e³3vÑóery˜100“fs.‘—¢FÑóor˜quadrupolar“simulationsŽ¡with–™ÝeÑóxternal“eld‘™Ügradients,‘ÀTeld“strengths“ranging“from“¼0–ã¥Â“¼9‘ã¦Â“¼10Ÿû¤çÃ½2Ž‘˜¹V/Ÿþ"„ŽŽAŸû¤ç½2Ž‘™Ü¹with–™Ýincrements“ofŽ¡¼10Ÿû¤çÃ½2Ž‘þ/¹V/Ÿþ"„ŽŽAŸû¤ç½2Ž‘¹were–carried“out“for“each“system.ŽŽŸ’Þ19ŽŽŽŒ‹êÞ ·€ ý/€ ²€ ýp€‘¹T‘ÿ
Bo–Q†carry›Q‡out“the“PMF‘Q0simulations,‘¥ètwáIo“of“the“multipolar˜molecules“in“the“test˜system“wereŽ¤con…&vÑóerted–Ginto‘FqŽŽ‘      FŸûu?Þ+ŽŽŽ‘£û¹and‘FqŽŽ‘
?PŸûu?Þ{ŽŽŽ‘~¡¹ions“and›Fconstrained“to“remain˜at“a˜xÑóed“distance“for˜the“duration˜of“theŽ¡simulation.‘uÆThe–”}constrained“distance“wáIas“then›”|v³3aried“from“5{12“Ÿþ"„ŽŽ“A.“In˜the“PMF‘”Wcalculations,‘¹œallŽ¡simulations–¿ëwere“equilibrated›¿ìfor“500“ps“in“the˜NVT‘¿Úensemble˜and“run“for“5“ns˜in“the“microcanon-Ž¡ical–(NVE)“ensemble.‘¸PConstraint“forces“were“sampled“e³3vÑóery“20“fs.ŽŸ4*Ý×VII.Ž‘ ÿíRESUL‘þåfTSŽ¤$Š(A.‘Dipolar‘
uidŽ¡‘¹The–³macroscopic›³polarizability“(¿ŸÌÌÀDŽ‘ÿ{¹)˜for“the˜dipolar“
uid˜is“sho³3wn˜in“the˜upper“panels˜in“Fig.‘ž¯3.Ž¤The–»polarizability“obtained“from“the“both“perturbation“and“
uctuation“approaches“are“in“eÑóxcellentŽ¡agreement–{with›|each“other‘ÿW.‘ÈÂThe˜data“also˜sho³3w“a˜stong“dependence“on˜the“damping˜parameter“forŽ¡both–á|the›á{Shifted“Potential˜(SP)‘áBand“Gradient˜Shifted“force“(GSF)‘áAmethods,‘Ûwhile˜T‘ÿ
Baylor“shiftedŽ¡force–(TSF)“is“larÈ¹gely“independent“of“the“damping“parameter‘ÿW.Ž©P5‘The–Xðcalculated›Xñcorrection“fáIactors˜discussed“in“section˜III‘C“are“sho³3wn˜in“the˜middle“panels.Ž¡Because–ýzthe“TSF‘ý9method“has›ýy¿A–7z¼=“1–ýz¹for˜all“v³3alues“of“the“damping“parameter…&,‘<Øthe“computed“po-Ž¡larizabilities–ºtneed“no“correction“for“the“dielectric“calculation.‘ç«The“vš³3alue“of“¿A“¹v˜aries“with“theŽ¡damping–ößparameter“in‘öàboth“the“SP›ö and“GSF˜methods,‘4˜and“inclusion“of“the‘öàcorrection“yields“di-Ž¡electric–ºvestimates›ºw(sho³3wn“in˜the“lo³3wer“panel)˜that“are˜generally“too“larÈ¹ge˜until“the˜damping“reachesŽ¡Â›”í¹0.25‘”ìŸþ"„ŽŽ‘”ìAŸû¤çÃ½1Ž‘þ/¹.‘wAbo•Ñóv“e˜this–”ìv³3alue,‘ú(the˜dielectric“constants˜are˜generally“in˜good“agreement˜withŽ¡pre³3vious–simulation“results.Ÿû¤çÞ19ŽŽŽŸP4‘¹Figure–6b3“also“contains“back-calculations“of“the‘6apolarizability“using“the“reference“(EwáIald)“sim-Ž¡ulation–-\results.Ÿû¤çÞ19ŽŽ‘
\¹These“are“indicated‘-[with“dashed“lines“in“the“upper“panels.‘@cIt“is“clear“that“theŽ¡eÑóxpected–¿Ípolarizability›¿Îfor“the“SP‘¿and“GSF‘¿œmethods˜are“quite“close˜to“results“obtained˜from“theŽ¡simulations.‘±®This›ìindicates–ìthat“the“correction˜formula“for“the“dipolar˜
uid“(Eq.‘±®16)˜is“quite“sensi-Ž¡tiš³3vÑóe–when“the“v˜alue“of“¼A“¹de˜viates“signicantly“from“unity‘ÿ8Y.Ž¦‘These–nresults“also“suggest›man“optimal“v³3alue“for“the“damping“parameter˜of“(¿‘´ùÂ‘¶¼0¿:¼2–²:Â“¼0¿:¼3‘nŸþ"„¹ŽŽ‘nAŸû¤çÃ½1Ž‘þ/¹)Ž¡when›:ðe•³3v“aluating˜dielectric˜constants˜of˜point˜dipolar˜
uids˜using˜either˜the˜perturbation˜and˜
uc-Ž¡tuation–approaches“for“the“ne³3w“real-space“methods.Ž¦‘W‘ÿ
Be–3 haÂ“vÑóe›3also“e•³3v“aluated˜the–3 distance-dependent˜screening“f•áIactor…&,‘?ç¿‘þ÷¼(¿rF¦¼)¹,‘?èbetween˜tw“o‘3 oppositelyŽ¡charÈ¹ged–Ðions“when›ÏtheÑóy“are“placed“in˜the“dipolar“
uid.‘Æ¿These“results“were˜computed“using“Eq.‘Æ¿47ŽŽŸ’Þ20ŽŽŽŒ‹ùj ·€ ý/€ ²€ŸÃæ´Ÿ„á"‘5S4ïKPSfile="dielectricFinal_Dipole.eps" llx=20 lly=40 urx=522 ury=710 rwi=3600 ŽŽŸÀëTFIG.–2³3.‘Æ¯The“polarizability“(ëWŸ¤zëXDŽ‘F¦ëT),›P,correction‘2²fãùactor“(ëWAëT),˜and“dielectric›2²constant“(ëW‘è§ëT)“for˜the“test˜dipolar“
uid.Ž¤ÀThe–¶íleft“panels“were“computed“using“eÕúxternal“elds,‘õuand“those“on“the“right“are“the“result“of“equilibriumŽ¡
uctuations.‘H.In–]Ýthe“GSF›]³and“SP˜methods,‘†!the“corrections“are›]ÜlarÍge“in“with“small˜v¹ëalues“of“ëW‘è§ëT,‘†!and˜a“opti-Ž¡mal–’damping›’coecient“is˜e¹ëvident“around“0.25˜ŸþLKŽŽ˜AŸüyëZëV1Ž‘
·FëT.‘äÕThe“dashed˜lines“in“the˜upper“panel˜indicate“back-Ž¡calculation–¼Ìof“the“polarizability“using“the“Ewãùald“estimate“(Refs.‘dü28Ž‘úand“19Ž‘
¯þ)“for“the“dielectric“constant.ŽŽŽŽŸ’Þ¹21ŽŽŽŒ‹Ø ·€ ý/€ ²€ ÿE+õŸš¡"‘YuÃï?PSfile="screen_pmf.eps" llx=21 lly=44 urx=529 ury=707 rwi=2880 ŽŽŸÀëTFIG.–‰4.‘S¿The“distance-dependent‘‰screening“fšãùactoræ,–“lëW‘è§ëU(ëWr@wëU)ëT,“between–‰tw˜o“ions›‰immersed“in˜the“dipolar˜
uid.‘S¾TheŽ¤Àneš¹ëw–Ÿmethods“are‘Ÿ       sho˜wn“in“separate“panels,‘¤ýand“dierent“v˜alues›Ÿ   of“the“damping“parameter˜(ëW‘è§ëT)“are“indicatedŽ¡with–¯'dierent“symbols.‘`oAll“of“the“methods›¯&appear“to“be“conævÕúerÍging“to˜the“bÇóulk“dielectric“constant“(ëY‘bIëU65ëT)“atŽ¡larÍge–¼Ìion“separations.ŽŽŽŸ$]¹and–are“sho³3wn“in“Fig.‘¸P4.ŽŸF®‘The–“‹screening›“ŒfáIactor“is˜similar“to“the˜dielectric“constant,‘¸obÂ“ut“measures“a˜local“property˜of“theŽ¤ions–Ö@in›Ö?the“
uid“and“depends˜on“both“ion-dipole“and˜dipole-dipole“interactions.‘ªeThese“interactionsŽ¡depend–Î!on“the“distance›Î between“ions“as“well“as“the“electrostatic“interaction˜methods“utilized“in“theŽ¡simulations.‘± The–ëðscreening“should“con…&vÑóerÈ¹ge“to“the“dielectric‘ëñconstant“when“the“eld“due“to“ions“isŽ¡small.‘MTThis›1occurs–1¬when“the“ions“are“separated˜(or“when“the“damping“parameter“is“larÈ¹ge).‘MUIn“Fig.Ž¡4–U‘we›UobservÑóe“that˜for“the“higher˜v³3alue“of˜damping“alpha˜Úi.eÑó.‘¹¿‘Ù(¼=‘Ú10¿:¼2“Ÿþ"„¹ŽŽ“AŸû¤çÃ½1Ž‘SÀ¹and˜¼0¿:¼3“Ÿþ"„¹ŽŽ“AŸû¤çÃ½1Ž‘S¿¹and“larÈ¹geŽ¡separation–ËÈbetween“ions,‘Ö9the“screening“fáIactor“does“indeed“approach‘ËÇthe“correct“dielectric“constant.ŽŽŸ’Þ22ŽŽŽŒ‹        = ·€ ý/€ ²€ ýp€‘¹It–Ä]is›Ä\also“notable“that˜the“TSF‘Ãémethod˜agð©ain“displays˜smaller“perturbations“aÑówáIay˜from“theŽ¤correct–Q0dielectric›Q1screening“behaÂ“vior‘ÿW.‘«âW‘ÿ
Be“also˜observÑóe“that˜for“TSF‘Qmethod“yields˜high“dielectricŽ¡screening–eš³3vÑóen“for“lo˜wer“v˜alues“of“¿‘þ÷¹.Ž©+‘At–short“distances,‘ šthe“presence“of“the›ions“creates“a“strong˜local“eld“that“acts˜to“align“nearbyŽ¡dipoles–hnearly“perfectly›hin“opposition“to“the˜eld“from“the“ions.‘ðUThis“has“the˜eect“of“increasingŽ¡the–âeecti³3vÑóe›áscreening“when“the“ions˜are“brought“close“to˜one“another‘ÿW.‘õThis“eect“is˜present“e³3vÑóenŽ¡in–!)the›!*full“EwáIald˜treatment,‘)sand˜indicates“that˜the“local˜ordering“behaÂ“vior“is˜being“captured˜by“allŽ¡of–the“moderately-damped“real-space“methods.ŽŸ<ä6×B.‘Quadrupolar‘
uidŽŸ%ç‘¹The–*Æpolarizability›*Å(¿Ÿë„ÀQŽ‘ÿ{¹),‘5wcorrection“fáIactor˜(¿B¹),‘5wand“susceptibility“(¿Ÿë„ÀQŽ‘ÿ{¹)˜for“the˜quadrupolar“
uidŽ¡is–Éplotted›Êagð©ainst“damping“parameter“Fig.–¬5.“In–Équadrupolar˜
uids,‘f{both“the˜polarizability“andŽ¡susceptibility–»haÂ“vÑóe›»units“of˜¼lengthŽ‘!»ŸûG3½2Ž‘&»¹.‘éœAlthough˜the“susceptibility“has˜dimensionality‘ÿ8Y,‘éàit“is˜the“rel-Ž¡e•³3v“ant–o@measure›o?of“macroscopic“quadrupolar˜properties.Ÿû¤çÞ23Ž‘        Ÿû¤ç,24ŽŽ‘/@¹The“left˜panel“in“Fig.‘5˜sho³3ws“resultsŽ¡obtained–Åúfrom›Åùthe“applied˜eld“gradient“simulations˜whereas“the“results˜from“the˜equilibrium“
uc-Ž¡tuation–formula“are“plotted“in“the“right“panels.Ž¦‘The–ìbsusceptibility›ìcfor“the˜quadrupolar“
uid˜is“obtained˜from“quadrupolarizability˜and“a˜correc-Ž¡tion–&fáIactor›&using“Eq.‘*‚(43).‘*The“susceptibilities˜are“sho³3wn“in˜the“bottom˜panels“of“Fig.‘*5.‘*‚AllŽ¡three–òämethods:‘±Â(SP‘þ«,“GSF‘ÿ
B,›òåand“TSF)‘òáproduce“similar“susceptibilities“o•Ñóv“er–òäthe“range˜of“damping“pa-Ž¡rameters.‘]This–6ísho³3ws›6îthat“susceptibility˜deri³3vÑóed“using˜the“quadrupolarizability˜and“the˜correctionŽ¡fáIactors–are“essentially“independent“of“the“electrostatic“method“utilized“in“the“simulation.Ž¦‘A‘Þmore–ädicult‘ãtest“of“the“quadrupolar“susceptibility“is“made“by“comparing“with“direct“calcu-Ž¡lation–¨gof›¨fthe“electrostatic“screening˜using“the“potential˜of“mean˜force“(PMF).“Since˜the“eecti³3vÑóeŽ¡dielectric–Åconstant›Æfor“a“quadrupolar“
uid˜depends“on“the“geometry˜of“the“eld“and˜eld“gradient,Ž¡this–is“not“a“phð©ysical“property“of“the“quadrupolar“
uid.Ž¦‘The–é¿geometrical“fáIactor“for›é¾embedded“ions“changes“with“the“ion˜separation“distance.‘°åIt“is“there-Ž¡fore–%’reasonable“to›%“treat“the“dielectric“constant“as˜a“distance-dependent“screening“fáIactor‘ÿW.‘)Since“theŽ¡quadrupolar–^zmolecules›^ycouple“with˜the“gradient“of˜the“eld,‘~Èthe“distribÂ“ution˜of“the˜quadrupoles“willŽ¡be–vðinhomogeneously›vñdistribÂ“uted“around˜the“point˜charÈ¹ges.‘!Hence˜the“distribÂ“ution˜of“quadrupolarŽ¡molecules–yshould“be›ztakáIen“into“account“when˜computing“the“geometrical“fáIactors˜in“the“presenceŽŽŸ’Þ23ŽŽŽŒ‹? ·€ ý/€ ²€ ÿF(ªŸš¡"‘YuÃïLPSfile="polarizabilityFinal_Quad.eps" llx=9 lly=40 urx=521 ury=710 rwi=2880 ŽŽŸÀëTFIG.––ò5.‘ómThe“quadrupole“polarizability‘–ñ(ëWŸ¤zëXQŽ›F¦ëT),‘
{correction“fãùactor“(ëWBëT),‘
zand“susceptibility“(ëWŸ¤zëXQŽ˜ëT)‘–ñfor“the“testŽ¤Àquadrupolar–ùß
uid.‘3The“left›ùÞpanels“were˜computed“using˜eÕúxternal“eld˜gradients,‘I#and˜those“on˜the“rightŽ¡are–¹Äthe›¹Åresult“of“equilibrium“
uctuations.‘[åThe“GSF‘¹„and“SP‘¹ƒmethods˜allo¹ëw“nearly“unmodied“use˜of“theŽ¡\conducting–¼Ìboundary"“or“polarizability“results“in“place“of“the“bÇóulk“susceptibility‘ÿIÑ.ŽŽŽŸ¬P¹of–this“perturbation,ŽŸ â‘|
«¿GŽ’‡_L¼=ŽŽŸ÷¹9’”Ÿ÷ŠÅRŽ’™¦–Ÿq©ÀVŽ’¡
¿g¼(×r¼)‘UTžó¶ÂjŽ‘¨=r×EŸû¤çÃŽ‘žó¶ÂjŽŽ‘ŠrŸú…¢½2Ž‘#ßÆ¿d‘Ÿ·×rŽ’”Ÿ„3‰zÙQ”)ŸðF‘C°Ÿ÷ŠÅRŽ‘Z6Ÿq©ÀVŽ‘`ªžó¶ÂjŽ‘³“×EŸüˆ9ÃŽ‘žó¶ÂjŽŽ‘0|Ÿú…¢½2Ž‘6\Ð¿d‘Ÿ·×rŽŽŽŽŽŽŽ¤¡’‡_L¼=ŽŽŸ÷!&’”2¿‘N$Ÿ÷ŠÅRŽŸúzá‘      {0½1ŽŸöÈ‘dªÃ½1ŽŽ‘¸-Ÿ÷ŠÅRŽŸúzá‘å9ÀRŽŸöÈ‘Î³½0ŽŽ‘#:J¿rF¦Ÿû¤ç½2Ž‘F¦¿g¼(¿r‘þñ¨;‘UT¼cosŽ‘©•¿›a@¼)‘UTžó¶ÂjŽ‘¨=r×EŸû¤çÃŽ‘žó¶ÂjŽŽ‘ŠrŸú…¢½2Ž‘#ßÆ¿d–Ÿ·rÔüd“¼(cosŽ‘TA¿˜¼)Ž’”ŸF‰zÙÂ2ŸðF‘G’¥Ÿ÷ŠÅRŽ‘L©+Ÿq©ÀVŽ‘T¯Ÿžó¶ÂjŽ‘Xˆ×EŸüˆ9ÃŽ‘žó¶ÂjŽŽ‘hVqŸú…¢½2Ž‘n«Å¿d‘Ÿ·×rŽŽŽŽŽŽ’À¹(49)ŽŽŽŽŸ!Hûwhere–„$¿g¼(¿r‘þñ¨;‘UT¼cosŽ‘©•¿‘a@¼)“¹is›„%a“distribÂ“ution“function“for“the˜quadrupoles“with“respect“to“an˜origin“at“midpointŽ¤of–a“line“joining“the“twáIo“probe“charÈ¹ges.Ž¡‘The–“5eecti³3vÑóe›“4screening“fáIactor˜is“plotted˜agð©ainst“ion˜separation“distance“in˜Fig.–”6.“The‘“5screeningŽŽŸ’Þ24ŽŽŽŒ‹¬ ·€ ý/€ ²€ ÿ8ÍŸ‡Éï?PSfile="screen_all.eps" llx=39 lly=38 urx=730 ury=548 rwi=4662 ŽŽŸÀëTFIG.–f6.‘‰ÈThe“distance-dependent›escreening“f•ãùactoræ,‘vËëW‘è§ëU(ëWr@wëU)ëT,‘vÌbetween˜tw“o–fions˜immersed“in˜the“quadrupolarŽ¤À
uid.‘YWResults–™Ýfrom›™Þthe“perturbation“and“
uctuation˜methods“are“sho¹ëwn˜in“right“and“central˜panels.‘YWHere“theŽ¡susceptibility–tíis›tîcalculated“from“the“simulation˜and“the“geometrical˜fãùactor“is“e•¹ëv“aluated˜analytically‘ÿIÑ,‘¢õusingŽ¡the–R?eld“and›R>eld-gradient“produced“by“ions.‘AwThe“right“hand˜panel“sho¹ëws“the“screening˜fãùactor“obtained“fromŽ¡the–¼ÌPMF“calculations.ŽŽŽŸ,Ìš¹e•³3v“aluated–A;from›A:the“perturbation“and˜
uctuation“methods“are“sho³3wn˜in“right“and˜central“panels.Ž¤Here–èithe›èhsusceptibility“is˜calculated“from“the˜simulation“and“the˜geometrical“fáIactor˜is“e•³3v“aluatedŽ¡analytically‘ÿ8Y,‘¥»using–„–the“eld›„•and“eld-gradient“produced˜by“ions.‘FThe“right“hand“panel˜sho³3ws“theŽ¡screening–fáIactor“obtained“from“the“PMF“calculations.ŽŸfM‘W‘ÿ
Be–e²note“that“the›e³screening“fáIactor“obtained“from“both“the˜perturbation“and“
uctuation“formulaŽ¡sho³3w–ì¡good“agreement“with“the“screening“fáIactor“calculated“using“PMF‘ìdmethod.‘~3As“there“are“noŽ¡larÈ¹ge–4ùdierences“in“quadrupole-quadruople“interactions“for“v³3arious“real-space“methods,Ÿû¤çÞ11Ž‘      Ÿû¤ç,12ŽŽ‘ôù¹weŽ¡generally–good“agreement“for“the“screening“fáIactors“using“anÑóy“of“the“real“space“methods.ŽŽŸ’Þ25ŽŽŽŒ‹"5 ·€ ý/€ ²€ ýp€×VIII.Ž‘%ªçCONCLUSIONSŽŸ%õ‘¹W‘ÿ
Be–vShaÂ“vÑóe“used›vRboth“perturbation“and“
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uids.‘#The“static“dielectric˜constant“is˜the“rele•³3v“ant˜bÂ“ulk˜property‘sGforŽ¡dipolar–§Ð
uids,‘ÑÃwhile“the›§Ïquadrupolar“susceptibility˜plays“a“similar˜role“for˜quadrupoles.‘¯¿Correc-Ž¡tions–Áto›Á both“the˜static“dielectric˜constant“and“the˜quadrupolar“susceptibility˜were“deri³3vÑóed˜for“threeŽ¡ne³3w–Õreal“space›Öelectrostatic“methods,‘!Šand“these˜corrections“were“tested“agð©ainst“a˜third“measure“ofŽ¡dielectric–screening,“the“potential“of“mean“force“between“twáIo“ions“immersed“in“the“
uids.Ž©á‘FÑóor–2¾the“dipolar›2½
uids,‘?nwe“nd˜that“the“polarizability“e•³3v“aluated–2¾using“the˜perturbation“and“
uc-Ž¡tuation–…amethods“sho³3w“eÑóxcellent“agreement,‘¦¹indicating“that‘…`equilibrium“calculations“of“the“dipoleŽ¡
uctuations–~are›~good“measures˜of“bÂ“ulk˜polarizability‘ÿ8Y.‘2ZOne˜of“the˜ndings“of˜the“second˜paper“inŽ¡this–lÛseries“is›lÜthat“the“moderately“damped“GSF‘lÀand“SP‘l¿methods“were“most˜suitable“for“molecularŽ¡dynamics–and“Monte“Carlo“simulations,“respecti³3vÑóely‘ÿ8Y.Ÿû¤çÞ12ŽŽŽŸá‘¹The–aœdielectric›aconstant“e•³3v“aluated–aœusing“the˜computed“polarizability“and˜correction“fáIactorsŽ¡agrees–0©well›0ªwith“the“pre³3vious˜EwáIald-based“simulation˜resultsŸû¤çÞ19Ž‘     Ÿû¤ç,28ŽŽ‘ð©¹for“moderate˜damping“parame-Ž¡ters–in“the“range“0.25{0.3“Ÿþ"„ŽŽ“A“Ÿû¤çÃ½1Ž‘þ/¹.Ž¦‘Although–_the›`TSF‘Xmethod“alters“manÑóy˜dynamic“and“structural˜features“in˜multipolar“liquids,Ÿû¤çÞ12ŽŽŽ¡¹it–Nïis“surprisingly“good›Nðat“computing“bÂ“ulk“dielectric“properties“at“nearly“all˜ranges“of“the“dampingŽ¡parameter‘ÿW.‘j¹In–ÎfšáIact,‘µthe“correction“f˜actor…&,‘µ¿A–û¼=“1¹,‘µfor›Íthe–ÎTSF‘©method“so˜the“conducting“boundaryŽ¡formula–is“essentially“correct“when“using“this“method“for“point“dipolar“
uids.Ž¦‘The–Ð dielectric“correction“formula“(equation‘ @16)‘Ð!is“ešÑóxtremely“sensiti³3v˜e“when“the“correction“fáIac-Ž¡tor–(¿A¹)“de³3viates“from›1,‘Zand“this“behaÂ“vior˜is“also“seen“in“the“eecti³3vÑóe˜screening“of“ions“embeddedŽ¡in–the“
uid.Ž¦‘As–¡8in›¡9the“dipolar˜case,‘É†the˜quadpole“polarizability˜e•³3v“aluated–¡8from˜both“perturbation˜and“
uc-Ž¡tuation–›ûsimulations“sho³3w››úeÑóxcellent“agreement,‘Âúagð©ain“conrming“that˜equilibrium“
uctuation“cal-Ž¡culations–ATare›AUsucient“to˜reproduce“bÂ“ulk˜dielectric“properties“in˜these“
uids.‘|NThe“quadrupolarŽ¡susceptibility–Ôcalculated›Óvia“our˜deri³3vÑóed“correction˜fáIactors“tends“to˜produce“the˜same“result˜for“allŽ¡three–M?real›M@space“methods.‘ Similarly‘ÿ8Y,‘`the˜screening“fáIactor“calculated˜using“the“susceptibility˜and“aŽ¡weighted–»;geometric“fáIactor“proÑóvides“good“agreement“with“results“obtained“directly“via“potentialsŽ¡of›Sžmean–SŸforce.‘³+FÑóor“quadrupolar˜
uids,‘¨†the“distance˜dependence“of˜the“electrostatic˜interactionŽ¡is–Bñsignicantly“reduced›Bðand“the“correction“fáIactors“are“all˜small.‘#These“points“suggest˜that“ho³3w“anŽ¡electrostatic–|method›{treats“the“cuto˜radius“become“less˜consequential“for“higher˜order“multipoles.ŽŽŸ’Þ26ŽŽŽŒ‹(_ ·€ ý/€ ²€ ýp€‘¹FÑóor›Athis–Breason,‘Qwe“rene³3w˜our“recommendation˜that˜the“moderately-damped˜GSF‘@method“is˜anŽ¤eÑóxcellent–[þchoice“for“molecular“dynamics“simulation‘[ýwhere“point-multipole“interactions“are“beingŽ¡utilized–to“compute“bÂ“ulk“properties“of“
uids.ŽŸ-¦µ×A³3ppendix–A:“Contraction“of“the“quadrupolar“tensor“with“the“traceless“quadrupole“momentŽŸ#…‚‘¹FÑóor–dquadrupolar›eliquids“modeled˜using“point˜quadrupoles,‘F¾the“interaction˜tensor“is˜sho³3wn“inŽ¡Eq.–¸P(31).“The–FÑóourier“transformation“of“this“tensor“for“×k–ª¨¼=“0‘¹is,Ž¤!â¬Ÿýœ±’¬«%¼~ŽŸcO’©÷÷¿TŸ’‘À–¿9‘c
“ŽŽŽ’Æ¶™¼(0)–ª¨=“ŸñŠÅZŽ‘§”Ÿ      q©ÀVŽ‘®¿TŸ’‘À–¿9‘c
“Ž‘ù¼(×r¼)¿d‘Ÿ·×rŽ’½X¹(A1)ŽŽŽ¡On–íUthe“basis›íVof“symmetry‘ÿ8Y,‘(ªthe˜81“elements“can“be“placed˜in“four“dierent“groups:Ÿýœ±‘F(¼~ŽŸcO‘’ú¿TŸÌÌÀaaaaŽŽŽ‘Ø—¹,Ÿýœ±‘ÛÙ¼~ŽŸcO‘(«¿TŸŠÀaaabŽŽŽ‘nH¹,Ž¤Ÿýœ±‘³.¼~ŽŸcO¿TŸŠÀaabbŽŽŽ‘E¹,–«andŸýœ±‘  /¼~ŽŸcO‘V¿TŸŠÀaabcŽŽŽ‘”¹,‘«‚where›V¿a¹,“¿b¹,“and˜¿c¹,‘«‚and˜can˜takáIe˜on˜distinct˜v³3alues˜from˜the˜set˜žó¶ÂfŽ‘
V¿x   /;–UTy‘ÿW;“zžó¶ÂgŽŽ‘(Yš¹.‘ºSTheŽ¡elements–8belonging›8to“each“of˜these“groups“can˜be“obtained˜using“permutations“of˜the“indices.Ž¡InteÑógration–1of“all“of“the›0elements“sho³3ws“that“only“the“groups“with“indices˜¿aaaaŽ‘.a¹and“¿aabbŽ‘.b¹are“non-Ž¡zero.ŽŸKŽ‘W‘ÿ
Be–can“deriš³3vÑóe“v˜alues“of“the“components“ofŸýœ±‘³.¼~ŽŸcO“¿TŸÌÌÀaaaaŽŽŽ›E¹andŸýœ±‘³.¼~ŽŸcO“¿TŸŠÀaabbŽŽŽ˜¹as“follo³3ws;Ž¤ yÉŸýœ±‘S»¼~ŽŸcO‘PR¿TŸÌÌÀx•ãx“x“xŽŽŽ‘f¯Ž¼(0)ŽŽ‘xª=ŽŽ’„§“ŸñŠÅZŽ’Š¤Ÿ      q©ÞVŽ’’ùNŸöF"ÅŽ’—÷:¼3¿vŸä½41Ž›  €¼(¿R¼)–ª©+“6¿x     /ŸûN½2Ž‘      /¿vŸä½42Ž˜¼(¿r•F¦¼)¿=r“ŸûN½2Ž‘ñO¼+‘ª©¿x        /ŸûN½4Ž‘^ƒ¿vŸä½43Ž˜¼(¿r“¼)¿=r“ŸûN½4Ž‘F¦ŸöF"ÅŽŽ’S1B¿d‘Ÿ·×rŽŽŽ©ðû‘xª¼=ŽŽ’„§“12¿‘N$ŸñŠÅZŽŸôzá‘
ÉÀrŸY™ÁcŽŽŸöÈ‘K½0ŽŽ‘ûŸïïÅŽ‘Xä¿vŸä½41Ž– €¼(¿ršF¦¼)‘ª©+Ÿ÷àI‘ÝÜ2Ž‘ÝÜŸ]#‰zÙŸ
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¿vŸä½42Ž“¼(¿r˜¼)‘ª©+Ÿ÷àI‘ÝÜ1Ž‘ÝÜŸ]#‰zÙŸ
ýö15ŽŽŽŽ‘¿vŸä½43Ž“¼(¿r˜¼)ŸïïÅŽŽ’«^€¿rF¦ŸûN½2Ž‘›ú¿d‘Ÿ·r‘ñN¼=‘ª¨12¿‘øÐ¼BŽŽŽ’½X¹(A2)ŽŽŽŽ¡andŽ¡Ÿýœ±‘Vë¼~ŽŸcO‘SM½¿TŸÌÌÀx•ãx“yyŽŽŽ‘iœø¼(0)ŽŽ‘{—l=ŽŽ’‡”ýŸñŠÅZŽ’‘éŸ     q©ÞVŽ’•æ¸ŸöF"ÅŽ’šä¤¿vŸä½41Ž›    €¼(¿R¼)–ª©+“(¿x     /ŸûN½2Ž‘³Ø¼+“¿yŸûN½2Ž‘¼)¿vŸä½42Ž˜¼(¿r•F¦¼)¿=r“ŸûN½2Ž‘ñO¼+‘ª©¿x  /ŸûN½2Ž‘      /¿yŸûN½2Ž‘UT¿vŸä½43Ž˜¼(¿r“¼)¿=r“ŸûN½4Ž‘F¦ŸöF"ÅŽŽ’t¾™¿d‘Ÿ·×rŽŽŽ¦‘{—l¼=ŽŽ’‡”ý4¿‘N$ŸñŠÅZŽŸôzá‘
ÉÀrŸY™ÁcŽŽŸöÈ‘K½0ŽŽ‘ûŸïïÅŽ‘Xä¿vŸä½41Ž– €¼(¿ršF¦¼)‘ª©+Ÿ÷àI‘ÝÜ2Ž‘ÝÜŸ]#‰zÙŸ
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¿vŸä½42Ž“¼(¿r˜¼)‘ª©+Ÿ÷àI‘ÝÜ1Ž‘ÝÜŸ]#‰zÙŸ
ýö15ŽŽŽŽ‘¿vŸä½43Ž“¼(¿r˜¼)ŸïïÅŽŽ’«^€¿rF¦ŸûN½2Ž‘›ú¿d‘Ÿ·r‘ñN¼=‘ª¨4¿‘øÐ¼BŽ‘:þ¿:ŽŽ’½X¹(A3)ŽŽŽŽ¡These–ò“inteÑógrals“yield“the“same“v³3alues“for“all“permutations“of“the“indices“in“both“tensor“elementŽ¤groups.‘TIn›Wequation–V36,‘mfor“a˜particular˜v³3alue“of˜the˜quadrupolar“polarizationŸý4B‘‹¼~ŽŸË¾˜ŸÌÌÀaaŽŽŽ‘•¹we“can˜contractŽ¡Ÿýœ±‘³.¼~ŽŸcO¿TŸ’‘Àaa
‘¿9ŽŽŽ‘ES¼(0)–¹withŸý4B‘r®¼~ŽŸË¾“Ÿ’‘À
‘¿9ŽŽŽ‘d¹,“using“the“traceless“properties“of“the“quadrupolar“moment,ŽŸ yÉŸýœ±‘K›¼~ŽŸcO‘HOm¿TŸ’‘Àx•ãx“
‘¿9ŽŽŽ‘` r¼(0)Ÿý4B‘r®~ŽŸË¾Ÿ’‘À
‘¿9ŽŽŽ‘d¼(0)ŽŽ’•Nº=ŽŽŸýœ±’£ÿy~ŽŸcO’¡LK¿TŸÌÌÀx•ãx“x“xŽŽŽ’·©L¼(0)Ÿý4B‘r®~ŽŸË¾ŸÌÌÀxãxŽŽŽ‘p¼(0)–ª©+Ÿýœ±‘]×~ŽŸcO“¿TŸÌÌÀx•ãx“yyŽŽŽ‘ùä¼(0)Ÿý4B‘r®~ŽŸË¾ŸÌÌÀyyŽŽŽ‘b>¼(0)“+Ÿýœ±‘]×~ŽŸcO“¿TŸÌÌÀx•ãx“zzŽŽŽ‘ün¼(0)Ÿý4B‘r®~ŽŸË¾ŸÌÌÀzzŽŽŽ‘dÈ¼(0)ŽŽŽ¤’•Nº=ŽŽ’¡LK12¿›øÐ¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀxãxŽŽŽ‘p¼(0)–ª©+“4¿˜¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀyyŽŽŽ‘b>¼(0)“+“4¿˜¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀzzŽŽŽ‘dÈ¼(0)ŽŽŽ¡’•Nº=ŽŽ’¡LK8¿›øÐ¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀxãxŽŽŽ‘p¼(0)–ª©+“4¿˜¼B‘UTŸöF"ÅŽŸý4B‘  Fù¼~ŽŸË¾‘ÔKŸÌÌÀxãxŽŽŽ‘DO¼(0)“+Ÿý4B‘W~ŽŸË¾“ŸÌÌÀyyŽŽŽ‘ç¼(0)“+Ÿý4B‘W~ŽŸË¾“ŸÌÌÀzzŽŽŽ‘q¼(0)ŸöF"ÅŽŽŽŽŽ¡’•Nº¼=ŽŽ’¡LK8¿‘øÐ¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀxãxŽŽŽ‘p¼(0)ŽŽ’½X¹(A4)ŽŽŽŽŽŸ’Þ27ŽŽŽŒ‹5 ·€ ý/€ ²€ ýpÜž¹Similarly–Åfor›Åa“quadrupolar“polarizationŸý4B‘8=¼~ŽŸË¾“ŸÌÌÀxãyŽŽŽ‘ô@¹in“equation“36,‘öôwe“can“contractŸýœ±‘x½¼~ŽŸcO“¿TŸ’‘Àxãy
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‘¿9ŽŽŽ’‰#¼(0)Ÿý4B‘r®~ŽŸË¾Ÿ’‘À
‘¿9ŽŽŽ‘d¼(0)ŽŽ’½Ñ[=ŽŽŸýœ±’Ì‚~ŽŸcO’ÉÎì¿TŸÌÌÀx•ãyx“yŽŽŽ’à'¼(0)Ÿý4B‘r®~ŽŸË¾ŸÌÌÀxãyŽŽŽ›i!¼(0)–ª©+Ÿýœ±‘]×~ŽŸcO“¿TŸÌÌÀxãyyxŽŽŽ‘ùä¼(0)Ÿý4B‘r®~ŽŸË¾ŸÌÌÀyxŽŽŽ˜¼(0)ŽŽŽ¤’½Ñ[=ŽŽ’ÉÎì4¿›øÐ¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀxãyŽŽŽ‘i!¼(0)–ª©+“4¿˜¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀyxŽŽŽ‘i!¼(0)ŽŽŽ¡’½Ñ[=ŽŽ’ÉÎì8¿‘øÐ¼BŸý4B‘r®~ŽŸË¾ŸÌÌÀxãyŽŽŽ‘i!¼(0)ŽŽ’½X¹(A5)ŽŽŽŽŸœ_Here,‘5/we–Ä%haÂ“vÑóe“used“the›Ä&symmetry“of“the“quadrupole˜tensor“to“combine“the˜symmetric“terms.Ž¦Therefore–we“can“write“matrix“contraction“forŸýœ±‘³.¼~ŽŸcO“¿TŸ’‘À–¿9‘c
“ŽŽŽ‘¾¢¼(0)“¹andŸý4B‘r®¼~ŽŸË¾“Ÿ’‘À
‘¿9ŽŽŽ‘d¼(0)“¹in“a“general“form,Ž¤ÃŸýœ±’žõ›¼~ŽŸcO’œBm¿TŸ’‘À–¿9‘c
“ŽŽŽ’¹¼(0)Ÿý4B‘r®~ŽŸË¾Ÿ’‘À
‘¿9ŽŽŽ‘d¼(0)–ª¨=“8¿‘øÐ¼BŸý4B‘r®~ŽŸË¾Ÿ’‘À‘¿9ŽŽŽ‘Ý¡¼(0)¿;Ž’½X¹(A6)ŽŽŽ¡which–is“the“same“as“Eq.‘¸P(37).Ž¦‘When–¿Athe“molecular›¿Bquadrupoles“are“represented“by˜point“charÈ¹ges,‘othe˜symmetry“of“theŽ¦quadrupolar–çtensor›ç‚is“same˜as“for˜point“quadrupoles“(see˜Eqs.“28˜and“31).‘°&Ho•³3we“vÑóer…&,‘ìhfor‘çmolecularŽ¦quadrupoles–|represented›|by“point“dipoles,‘–xthe˜symmetry“of“the˜quadrupolar“tensor“must˜be“handledŽ¦separately–¤º(compare›¤»equations“30˜and“31).‘¦€Although“there˜is“a˜dierence“in“symmetry‘ÿ8Y,‘Íêthe“nalŽ¦result–(Eq.“A6)“also“holds“true“for“dipolar“representations.ŽŸ+&ú×Aš³3ppendix–B:“Quadrupolar“corrÈ¹ection“factor“f˜or“the“Ewald-K˜orÑónfeld“(EK)“methodŽŸ#9ó‘¹The–G°interaction“tensor“between“twšáIo“point“quadrupoles“in“the“Ew˜ald“method“may“be“eÑóxpressed,Ÿû¤çÞ20Ž‘    Ÿû¤ç,37ŽŽŽ¡‘rà¿TŸ’‘À–¿9‘c
“Ž‘ù¼(×r¼)‘ª¨=ŽŽŽŸ÷àI’«sÒ4¿Ž’«sÒŸ]#‰zÙ
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