ReaxFF Potential Functions

Transcription

1 ReaxFF Potental Functons Supportng nformaton for the manuscrpt A ReaxFF Reactve Force Feld for Molecular Dynamcs Smulatons of Hydrocarbon Oxdaton by Kmberly Chenoweth Adr CT van Dun and Wllam A Goddard III Ths document contans all the general ReaxFFpotental functons In the current ReaxFF code all the energy contrbutons n ths document are calculated regardless of system composton All parameters that do not bear a drect physcal meanng are named after the partal energy contrbuton that they appear n For example p val and p val are parameters n the valence angle potental functon Parameters wth a more drect physcal meanng lke the torsonal rotatonal barrers V V V 3 bear ther more recognzable names Overall system energy quaton descrbes the ReaxFF overall system energy system = bond over under val tors pen con coa H! bond C trple vdwaals Below follows a descrpton of the partal energes ntroduced n equaton Bond Order and Bond nergy Coulomb A fundamental assumpton of ReaxFF s that the bond order between a par of atoms can be obtaned drectly from the nteratomc dstance r as gven n quaton In calculatng the bond orders ReaxFF dstngushes between contrbutons from sgma bonds pbonds and double p bonds % = = exp p bo $ r r o p bo exp p % bo3 $ / % exp p bo5 $ r r o r r o p bo p bo6 / / Based on the uncorrected bond orders derved from quaton an uncorrected

2 overcoordnaton Δ can be defned for the atoms as the dfference between the total bond order around the atom and the number of ts bondng electrons Val neghbours = Val $ 3a = ReaxFF then uses these uncorrected overcoordnaton defntons to correct the bond orders usng the scheme descrbed n quatons af To soften the correcton for atoms bearng lone electron pars a second overcoordnaton defnton Δ boc equaton 3b s used n equatons e and f Ths allows atoms lke ntrogen and oxygen whch bear lone electron pars after fllng ther valence to break up these electron pars and nvolve them n bondng wthout obtanng a full bond order correcton boc = Val boc neghbours $ 3b = f!!! = = =! = $ f!! = $ % Val $ f $ f $ f!!! $ f $ f $ f $ f $ f $ f $ f 5 Val f f f Val 5 5 a Val f! f f 3 b 3 f = expp boc $ expp boc $ c f 3 = % $ ln p boc $ exp p $ boc [ exp p boc $ ] d f = expp boc3 $ p boc $ $ boc p boc5 e

3 f 5 = expp boc3 $ p boc $ $ boc p boc5 f A corrected overcoordnaton Δ can be derved from the corrected bond orders usng equaton 5 neghbours = Val $ 5 = quaton 6 s used to calculate the bond energes from the corrected bond orders p be bond = D e $ % $ exp p be D e $ D e $ 6 3 Lone par energy quaton 8 s used to determne the number of lone pars around an atom Δ e s determned n quaton 7 and descrbes the dfference between the total number of outer shell electrons 6 for oxygen for slcon for hydrogen and the sum of bond orders around an atomc center neghbours e = Val e $ 7 = n = nt e e % exp p e nt 3 % / 6 $ 3 $

4 For oxygen wth normal coordnaton total bond order= Δ e = equaton 8 leads to lone pars As the total bond order assocated wth a partcular O starts to exceed equaton 8 causes a lone par to gradually break up causng a devaton Δ defned n equaton 9 from the optmal number of lone pars n opt eg for oxygen for slcon and hydrogen = n opt! n 9 Ths s accompaned by an energy penalty as calculated by equaton = p exp $75 Overcoordnaton For an overcoordnated atom Δ > equatons ab mpose an energy penalty on the system The degree of overcoordnaton Δ s decreased f the atom contans a brokenup lone electron par Ths s done by calculatng a corrected overcoordnaton equaton b takng the devaton from the optmal number of lone pars as calculated n equaton 9 nto account over = nbond $ = p ovun D e % % corr Val corr exp p ovun % corr a corr = neghbours % p ovun 3 $ exp p ovun $ $ %% / = 3 b

5 5 Undercoordnaton For an undercoordnated atom Δ < we want to take nto account the energy contrbuton for the resonance of the πelectron between attached undercoordnated atomc centers Ths s done by equatons where under s only mportant f the bonds between undercoordnated atom and ts undercoordnated neghbors partly have πbond character under = / p ovun5 / exp exp / p cor p ovun6 ovun cor povun7 exp$ p $ % ovun8 / neghbours =!! 6 Valence Angle Terms 6 Angle energy Just as for bond terms t s mportant that the energy contrbuton from valence angle terms goes to zero as the bond orders n the valence angle goes to zero quatons 3ag are used to calculate the valence angle energy contrbuton The equlbrum angle Θ o for Θ k depends on the sum of πbond orders S around the central atom as descrbed n quaton 3d Thus the equlbrum angle changes from around 97 for sp 3 hybrdzaton πbond= to for sp πbond= to 8 for sp πbond= based on the geometry of the central atom and ts neghbors In addton to ncludng the effects of πbonds on the central atom quaton 3d also takes nto account the effects of over and undercoordnaton n central atom as determned by equaton 3e on the equlbrum valency angle ncludng the nfluence of a lone electron par Val angle s the valency of the atom used n the valency and torson angle evaluaton Val angle s the same as Val boc used n equaton 3c for nonmetals The functonal form of quaton 3f s desgned to avod sngulartes when S= and S= The angles n quatons 3a3g are n radans val [ p!! ] { p p } = f 7 f 7 k f8 $ val val exp 3a val o k 5

6 p f 7 = exp p val 3 val 3b $ f 8 = p val 5 p val 5 angle exp p val 6 $ angle angle exp p val 7 $ exp p val 6 $ 3c S = neghbours 8 $ % exp $ n $ angle $ p val 8 n n= neghbors n n n= angle = Val angle S = f S neghbours n n= S = S p val 9 f < S < S = S p val 9 S = f S > 3d $ 3e f < S < = $ % $ exp $p val % $ S 3f { [ ]} 3g 6 Penalty energy To reproduce the stablty of systems wth two double bonds sharng an atom n a valency angle lke allene an addtonal energy penalty as descrbed n quatons a and b s mposed for such systems quaton 9b deals wth the effects of over/undercoordnaton n central atom on the penalty energy pen = p pen f 9 exp $p pen $ [ ] exp [ $p $ pen k ] a exp p pen $ exp p pen 3 $ f 9 = exp p pen 3 $ b 63 Threebody conugaton term The hydrocarbon ReaxFF potental contaned only a fourbody conugaton term see secton 7 whch was suffcent to descrbe most conugated hydrocarbon systems However ths term faled to descrbe the stablty obtaned from conugaton by the NO group To descrbe the stablty of such groups a threebody conugaton term s ncluded equaton 5 6

7 exp val exp pcoa /! 5 exp [! p ] [! p! 5 ] coa neghbours neghbours = exp! 3 exp! 3! coa pcoa pcoa n pcoa! k kn n= n= 7 Torson angle terms coa k 7 Torson rotaton barrers Just as wth angle terms we need to ensure that dependence of the energy of torson angle ω kl accounts properly for and for greater than Ths s done by quatons 6a6c tors = f $ V % k sn kl k sn kl 5 % $ cos kl V exp{ ptor k f k } cos kl V3 cos3 kl! f k kl = exp p tor f k = 6a [ ] [ exp p tor k ] exp p tor kl 6b exp p tor3 $ angle angle [ k ] [ ] exp p tor $ angle angle [ k ] exp p tor3 $ angle angle k [ ] 6c % $ 7 Four body conugaton term quatons 7ab descrbe the contrbuton of conugaton effects to the molecular energy A maxmum contrbuton of conugaton energy s obtaned when successve bonds have bond order values of 5 as n benzene and other aromatcs con = f k kl p cot [ cos kl $ sn% k sn% kl ] 7a $ f k kl = exp p cot % / exp p $ cot % k / exp p $ cot % kl / 7b 8 Hydrogen bond nteractons quaton 8 descrbed the bondorder dependent hydrogen bond term for a XH Z system as ncorporated n ReaxFF r [ ] exp p hb hb 3 Hbond = p hb exp p hb XH 7 $ o r $ HZ o / sn 8 XHZ % r HZ r hb % 8

8 9 Correcton for C ReaxFF erroneously predcts that two carbons n the C molecule form a very strong trple bond whle n fact the trple bond would get destablzed by termnal radcal electrons and for that reason the carboncarbon bond s not any stronger than a double bond To capture the stablty of C we ntroduced a new partal energy contrbuton C quaton 9 shows the potental functon used to destablze the C molecule: C C = k = c $ $ $ 3 f $ $ > 3 9 f $ $! 3 where Δ s the level of under/overcoordnaton on atom as obtaned from subtractng the valency of the atom for carbon from the sum of the bond orders around that atom and k c the force feld parameter assocated wth ths partal energy contrbuton Trple bond energy correcton To descrbe the trple bond n carbon monoxde a trple bond stablzaton energy s used makng CO both stable and nert Ths energy term only affects CO bonded pars quaton shows the energy functon used to descrbe the trple bond stablzaton energy trp = p trp exp [/ p / 5 ] trp exp / p trp neghbours k= % k / exp / p $ 5 exp [ p!! ] trp3 trp neghbours k= k / % $ Nonbonded nteractons In addton to valence nteractons whch depend on overlap there are repulsve nteractons at short nteratomc dstances due to Paul prncple orthogonalzaton and attracton energes at long dstances due to dsperson These nteractons comprsed of van der Waals and Coulomb forces are ncluded for all atom pars thus avodng awkward alteratons n the energy descrpton durng bond dssocaton Taper correcton To avod energy dscontnutes when charged speces move n and out of the nonbonded cutoff radus ReaxFF employs a Taper correcton as developed by de Vos 8

9 Burchart 995 ach nonbonded energy and dervatve s multpled by a Taperterm whch s taken from a dstancedependent 7 th order polynomal equaton Tap = Tap 7 r 7 Tap 6 r 6 Tap 5 r 5 Tap r Tap 3 r 3 Tap r Tap r Tap The terms n ths polynomal are chosen to ensure that all st nd and 3 rd dervatves of the nonbonded nteractons to the dstance are contnuous and go to zero at the cutoff boundary To that end the terms Tap to Tap 7 n equaton are calculated by the scheme n equaton where R cut s the nonbonded cutoff radus 7 Tap 7 = /R cut 6 Tap 6 = 7 /R cut 5 Tap 5 = 8 /R cut Tap = 35/R cut Tap 3 = Tap = Tap = Tap = van der Waals nteractons To account for the van der Waals nteractons we use a dstancecorrected Morsepotental quatons 3ab By ncludng a shelded nteracton quaton 3b excessvely hgh repulsons between bonded atoms nteractons and atoms sharng a valence angle 3 nteractons are avoded 3 % vdwaals = Tap D exp $ f 3r $ exp r vdw / % $ f 3r r vdw / 73 3a p f 3 r = r vdw % $ w p vdw p vdw 3b 3 Coulomb Interactons As wth the van der Waalsnteractons Coulomb nteractons are taken nto account between all atom pars To adust for orbtal overlap between atoms at close dstances a shelded Coulombpotental s used quaton 9

10 coulomb = Tap C q q [ r 3 / 3 ] / 3 Atomc charges are calculated usng the lectron qulbraton Method Mapproach The M charge dervaton method s smlar to the Qqscheme; the only dfferences apart from parameter defntons are that M does not use an teratve scheme for hydrogen charges as n Qq and that Qq uses a more rgorous Slater orbtal approach to account for charge overlap