ReaxFF potential functions

Transcription

1 ReaxFF otental functons Suortng nformaton for the manuscrt Nelson, K.D., van Dun, A.C.T., Oxgaard, J.., Deng, W. and Goddard III, W.A. Develoment of the ReaxFF reactve force feld for descrbng transton metal catalyzed reactons, wth alcaton to the ntal stages of the catalytc formaton of carbon nanotubes. Ths document contans all the general ReaxFF-otental functons. In the current ReaxFF code all the energy contrbutons n ths document are calculated regardless of system comoston. All arameters that do not bear a drect hyscal meanng are named after the artal energy contrbuton that they aear n. For examle, val and val are arameters n the valence angle otental functon. Parameters wth a more drect hyscal 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 l over under tors con H bond vdwaals Below follows a descrton of the artal energes ntroduced n equaton (. val 6- en coa C Coulomb. Bond Order and Bond nergy A fundamental assumton of ReaxFF s that the bond order between a ar 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, -bonds and double bonds. = σ π ππ = ex bo r bo σ r o ex bo3 ex bo5 r bo π r o r bo6 ππ r o ( Based on the uncorrected bond orders, derved from quaton, an uncorrected overcoordnaton can be defned for the atoms as the dfference between the total bond (

2 order around the atom and the number of ts bondng electrons Val. = Val (3a neghbours( = ReaxFF then uses these uncorrected overcoordnaton defntons to correct the bond orders usng the scheme descrbed n quatons (a-f. To soften the correcton for atoms bearng lone electron ars a second overcoordnaton defnton boc (equaton 3b s used n equatons e and f. Ths allows atoms lke ntrogen and oxygen, whch bear lone electron ars after fllng ther valence, to break u these electron ars and nvolve them n bondng wthout obtanng a full bond order correcton. boc = Val boc (3b neghbours( = f σ π ππ = = = σ π = σ f ππ =, Val f f (, f (, f (, π (, f ππ (, f (, f (, (, f (, f (, f (, Val f (, f (, f (, Val (a Val f (, f (, (, f 3 (b ( 3 f (, = ex( boc ex( boc (c f 3 (, = [ ( ] ln boc ex ( boc ex boc (d f (, = ex( boc 3 ( boc boc boc5 (e f 5 (, = ex( boc 3 ( boc boc boc 5 (f 6-

3 A corrected overcoordnaton can be derved from the corrected bond orders usng equaton (5. = Val (5 neghbours( = quaton (6 s used to calculate the bond energes from the corrected bond orders. ( be ( bond = D e σ σ ex be σ D π e π D ππ ππ e (6 3. Lone ar energy quaton (8 s used to determne the number of lone ars around an atom. 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. e = Val e (7 neghbours( = e n l, = nt ex l e nt e (8 For oxygen wth normal coordnaton (total bond order=, e =, equaton (8 leads to lone ars. As the total bond order assocated wth a artcular O starts to exceed, 6-3

4 equaton (8 causes a lone ar to gradually break u, causng a devaton l, defned n equaton (9, from the otmal number of lone ars n l,ot (e.g. for oxygen, 0 for slcon and hydrogen. l = n l, ot nl, (9 Ths s accomaned by an energy enalty, as calculated by equaton (0. l = l l l ex 75 ( (0. Overcoordnaton For an overcoordnated atom ( >0, equatons (a-b mose an energy enalty on the system. The degree of overcoordnaton s decreased f the atom contans a broken-u lone electron ar. Ths s done by calculatng a corrected overcoordnaton (equaton b, takng the devaton from the otmal number of lone ars, as calculated n equaton (9, nto account. over = nbond = ovun D σ e lcorr Val lcorr lcorr ( ex ovun (a lcorr = l neghbours( l π ovun3 ex ovun ( ( ππ = (b 5. Undercoordnaton For an undercoordnated atom ( <0, we want to take nto account the energy contrbuton for the resonance of the π-electron between attached under-coordnated atomc 6-

5 centers. Ths s done by equatons where under s only mortant f the bonds between under-coordnated atom and ts under-coordnated neghbors artly have π-bond character. under = ovun5 ex ex( lcor ( ovun6 ovun lcor ( ovun7 ex ovun8 ( l neghbours = ( π ππ 6. Valence Angle Terms 6. Angle energy. Just as for bond terms, t s mortant that the energy contrbuton from valence angle terms goes to zero as the bond orders n the valence angle goes to zero. quatons (3a-g are used to calculate the valence angle energy contrbuton. The equlbrum angle Θ o for Θ k deends on the sum of π-bond orders (S around the central atom as descrbed n quaton (3d. Thus, the equlbrum angle changes from around 09.7 for s 3 hybrdzaton (π-bond=0 to 0 for s (π-bond= to 80 for s (π-bond= based on the geometry of the central atom and ts neghbors. In addton to ncludng the effects of π-bonds on the central atom, quaton (3d also takes nto account the effects of over- and under-coordnaton n central atom, as determned by equaton (3e, on the equlbrum valency angle, ncludng the nfluence of a lone electron ar. 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 non-metals. The functonal form of quaton (3f s desgned to avod sngulartes when S=0 and S=. The angles n quatons (3a- (3g are n radans. val [ ( Θ ( Θ ] { } = f 7 ex (3a ( f 7 ( k f8 ( val val val o k f 7 ( = ex( val 3 val (3b 6-5

6 f 8 ( = val5 ( val 5 ( angle ( angle ex val6 ( ex val 7 angle ex val6 (3c S = neghbours( 8 ex( n angle val 8 n l, neghbors( ( π n ππ n n= Θ 0 angle = Val angle n= S = 0 f S 0 neghbours( n n= 6-6 ( (3d (3e S = S val 9 f 0 < S < S = ( S val 9 f < S < S = f S > ( = π Θ 0,0 ex val0 S (3f { [ ( ]} (3g 6. Penalty energy. To reroduce the stablty of systems wth two double bonds sharng an atom n a valency angle, lke allene, an addtonal energy enalty, as descrbed n quatons (a and (b, s mosed for such systems. quaton (9b deals wth the effects of over/undercoordnaton n central atom on the enalty energy. en = en f 9 ( ex en ( [ ] ex [ en ( k ] (a ( ( ex en3 f 9 ( = ex( en3 ex en (b 6.3 Three-body conugaton term. The hydrocarbon ReaxFF otental contaned only a four-body 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 -grou. To descrbe the stablty of such grous a three-body conugaton term s ncluded (equaton 5. neghbours( neghbours( = ex 3 ex 3 coa coa coa n coa k kn n= n= ex val ex( coa (.5 ex [ ] [ (.5 ] coa 7. Torson angle terms coa k (5

7 7. Torson rotaton barrers. Just as wth angle terms we need to ensure that deendence of the energy of torson angle ω kl accounts roerly for 0 and for greater than. Ths s done by quatons (6a-(6c. tors = f 0 (, k, kl sn Θ k sn Θ kl V ex { π tor ( k f (, k } ( cosω kl V cos3ω 3 kl f 0 (, k, kl = ex( tor f (, k = ( (6a [ ] [ ex( tor k ] ex tor kl (6b ex tor3 angle angle k ex tor3 angle angle [ ( k ] ex tor [ ( ] [ ( ] angle angle [ ( k ] (6c 7. Four body conugaton term. quatons (7a-b 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 cot [ ( cos ω kl snθ k snθ kl ] (7a f (, k, kl = ex cot ex cot k ex cot kl (7b 8. Hydrogen bond nteractons quaton (8 descrbed the bond-order deendent hydrogen bond term for a X-H Z system as ncororated n ReaxFF. o r [ ] ex hb hb3 r HZ o sn 8 r HZ r hb Hbond = hb ex( hb XH Θ XHZ (8 9. Correcton for C ReaxFF erroneously redcts that two carbons n the C -molecule form a very strong (trle bond, whle n fact the trle bond would get de-stablzed by termnal radcal electrons, and for that reason the carbon-carbon bond s not any stronger than a double bond. 6-7

8 To cature the stablty of C we ntroduced a new artal energy contrbuton ( C. quaton (9 shows the otental functon used to de-stablze the C molecule: C C = k = 0 c ( f 0.0 > 3 (9 f where 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 arameter assocated wth ths artal energy contrbuton. 0. Nonbonded nteractons In addton to valence nteractons whch deend on overla, there are reulsve nteractons at short nteratomc dstances due to Paul rncle orthogonalzaton and attracton energes at long dstances due to dserson. These nteractons, comrsed of van der Waals and Coulomb forces, are ncluded for all atom ars, thus avodng awkward alteratons n the energy descrton durng bond dssocaton. 0. Taer correcton. To avod energy dscontnutes when charged seces move n and out of the non-bonded cutoff radus ReaxFF emloys a Taer correcton, as develoed by de Vos Burchart (995. ach nonbonded energy and dervatve s multled by a Taerterm, whch s taken from a dstance-deendent 7 th order olynomal (equaton 0. Ta = Ta 7 r 7 Ta 6 r 6 Ta 5 r 5 Ta r Ta 3 r 3 Ta r Ta r Ta 0 (0 The terms n ths olynomal are chosen to ensure that all st, nd and 3 rd dervatves of the non-bonded nteractons to the dstance are contnuous and go to zero at the cutoff boundary. To that end, the terms Ta 0 to Ta 7 n equaton (0 are calculated by the scheme n equaton (, where R cut s the non-bonded cutoff radus. 7 Ta 7 = 0/R cut 6 Ta 6 = 70/R cut 5 Ta 5 = 8 /R cut Ta = 35/R cut Ta 3 = 0 Ta = 0 Ta = 0 Ta 0 = ( 6-8

9 0. van der Waals nteractons. To account for the van der Waals nteractons we use a dstance-corrected Morse-otental (quatons. a-b. By ncludng a shelded nteracton (quaton b excessvely hgh reulsons between bonded atoms (- nteractons and atoms sharng a valence angle (-3 nteractons are avoded. vdwaals = Ta D ex α f (r 3 ex r vdw α f (r 3 r (a vdw f 3 (r = r vdw γ w vdw vdw (b 0.3 Coulomb Interactons As wth the van der Waals-nteractons, Coulomb nteractons are taken nto account between all atom ars. To adust for orbtal overla between atoms at close dstances a shelded Coulomb-otental s used (quaton 3. coulomb = Ta C q q [ r 3 ( /γ 3 ] /3 (3 Atomc charges are calculated usng the lectron qulbraton Method (M-aroach. The M charge dervaton method s smlar to the Qq-scheme; the only dfferences, aart from arameter 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 aroach to account for charge overla. 6-9