ANALYTICAL FUNCTIONAL FORM AND FITTING PROCEDURE

Transcription

1 Elctronc Supplmntary Matral (ESI) for Phycal Chmtry Chmcal Phyc. Th journal th Onr Soct 6 Supplmntary Informaton Ttl: at contant calculaton of th GH + OH/O GH + H O/HO racton ung an ab nto bad full-dmnonal potntal nrgy urfac Author: J. Epnoa-Garca* C. angl and J.C. Corchado ANALYTICAL FUNCTIONAL FOM AN FITTING POCEUE a) Analytcal potntal Th n analytcal PES functon dcrbng th GH + OH GH + H O ytm bad on our urfac for th mlar CH + OH hydrogn abtracton racton hch bacally a valnc bond-molcular mchanc (B-MM) potntal nrgy urfac. W hall tart by prntng mathmatcal dtal and trm ncludd. Th potntal nrgy for a gvn gomtry gvn by th um of four trm: trtch val op atr (S) Th frt trm rprnt a trtchng potntal dfnd a th um of four London- Eyrng-Polany (LEP) trm (S) trtch GH GO hr H tand for on of th four grman hydrogn O th attacng oygn atom and th dtanc btn th to ubcrpt atom. For th a of compltn th potntal H O ( GH GO J ( ) J ( ) J ( ) J ( ) J ( ) J ( ) GH H O ) Q( H O GH ) Q( HO GO ) Q( GO H O ) GO GH (S) E ( E ( ) ) Q( J ( p p E ( ) E ( ) ) (S) E ( ) E ( ) ) (S5) p p (S6) (S7)

2 hch nclud fttng paramtr: th trplt and nglt docaton nrg and th qulbrum bond dtanc and th Mor paramtr. To ntroduc mor flblty th GH paramtr a functon of th G-H dtanc tanh c GH GH GH agh bgh (S8) bng th avrag of th four GH dtanc GH (S9) Wth th modfcaton GH chang moothly along th racton path from t valu at grman to t valu at th GH radcal. Th cond trm val rprntng th harmonc potntal dfnd a th um of harmonc potntal val j j j j j (S) hr j ar rfrnc angl and j and ar forc contant. Th j trm ar GH allod to rla from ractant to product hr paramtr j p GH a p b ( GH GH ) p b ( GHj GH ) b ( ) p b ( ) GH GH GHl Th contant forc dfnd a a functon of both th A GH GH GH GH a and b ar fttng H O (S) and dtanc A p A (S) p H O A aa (S) H O H O aa aa p aa (S) hr four n adjutabl paramtr ar ncludd namly aa. Wth rpct to th rfrnc angl thy ar alo allod to rla from grman = 9.7º to th GH radcal GH =.º

3 o ( j ( / ) S GH ) ( ung th follong tchng functon S S S ( GH GH ) S ) S ( GHj ( GHj ) GH tanha GH GH p GH B C tanha p B C GH ) (S5) (S6) (S7) GH GH GH In total th val potntal nd trm to b fttd n th calbraton proc. Th thrd trm op rprnt a quadratc-quartc potntal op f j h j j j j j (S8) ho am to corrctly dcrb th out-of-plan moton of GH radcal n th product. Th forc contant f and h ar allod to rla from ractant hr th GH fragmnt ho a pyramdal tructur to product hr t ho a quaplanar gomtry by man of th S tchng functon GH hr f GH h S GH S f f (S9) GH GH S h h (S) GH tanh (S) GH GH GH GH and dvaton from th rfrnc angl j ar fttng paramtr. Th angl maurng th aco r rj rl rj r r r r j l j r r j (S) th r r j r and r l bng th vctor gong from th G atom to th j and l hydrogn atom rpctvly. Th rfrnc angl j ar dfnd n Eq. (S5). Fnally th atr product formd n th racton dfnd by th um of to trm: ) a Mor functon dcrbng th O H j bond ( M ) p ( ) M OH OH OHj OH (S)

4 hr OH and OH ar rpctvly th OH docaton nrgy th OH OH qulbrum bond lngth (.97 Å) and th Mor paramtr to corrctly dcrb th OH radcal vbratonal frquncy; and ) a harmonc bndng trm ( harm ) dcrbng th HOH bndng mod harm K ( o ) (S) o hr th HOH qulbrum angl (.7 ) and th ummaton a tndd to th four hydrogn atom of grman to mantan th PES ymmtrc th rpct to th prmutaton of th four atom. To nclud mor flblty n th functon and provd th corrct aymptotc valu th forc contant K a allod to rla ung th tchng functon hr K o K a tanh OH ( OHj OH ) (S5) o K a and ε OH ar paramtr fttd to rproduc th valu of th atr vbratonal frqunc along th racton path. Thu th atr functon nclud paramtr n th calbraton proc. Th n PES thrfor dpnd on paramtr and rpctvly to dcrb th trtchng th valnc trm th out-of-plan potntal and th atr functon. Th paramtr gv grat flblty to th functonal form and n addton th B-MM functonal form rman phycally ntutv. Not that th urfac ymmtrc th rpct to th prmutaton of th four hydrogn n grman and t provd not only th nrgy of th ractv proc but alo th analytcal frt nrgy drvatv.. th gradnt hch rprnt a computatonal advantag n ntc and qua-clacal dynamc calculaton. b) Fttng procdur. Wth th functonal form prvouly dcrbd and ung a nput nformaton all th tatonary pont and th racton path at th ab nto lvl th paramtr r fttd. Follong our rcnt or on PES n polyatomc ytm -5 n th fttng procdur thr ar to quntal tp th frt bng manual and th cond automatd. In th frt on gvn that th paramtr n th B-MM potntal ar drctly rlatd to phycal and chmcal proprt by contructon (gomtr vbratonal frqunc othrmcty or barrr hght) attmptd manually to ft th paramtr to rproduc th proprt. Th a proc n tp laborou and dffcult bcau

5 th paramtr ar not ndpndnt and uually rfttng of th prvouly fttd paramtr ndd. In th frt approach ung th GH GO H O GH H O H O GH f GH GH h a b OH and ε OH paramtr attmptd to rproduc th gomtr vbratonal frqunc and clacal racton nrgy of ractant and product at th rfrnc ab nto CCS(T)/cc-pTZ lvl. In th cond approach amd to rproduc th addl pont proprt (gomtry vbratonal frqunc th pcal attnton to th magnary valu and th barrr hght hch vry mportant n ntc calculaton) at th am ab nto lvl ung th GO GH H O GO GO and c GH paramtr. In th thrd approach ft th paramtr of th tchng functon hch dcrb chang along th racton path A A (S) S S (S6S7) and S (S). Thu th paramtr aa aa aa aa A B C A B and C r obtand. Not that th tp ld to a frt t of paramtr hch r ud a th tartng pont for a n manual rft. Th proc a rpatd untl a raonabl convrgnc th th valu of th rfrnc ab nto lvl a obtand for all th analyzd proprt. Whn on condr that th fttd paramtr raonably rproduc th man topologcal charactrtc of th ractv ytm from ractant to product an automatd procdur ud. Hovr t ll non that to fnd a global mnmum for th ft vry hard and to ma mattr or th fnal rult dpndnt on th ntal paramtr; on uually obtan a numbr of dtnct t of ntal paramtr hch ar qually probabl and good. Th rprnt a rou lmtaton of th procdur. Thrfor hav dvlopd an automatd mult-bgnnng lat-quar procdur (local optmzaton mthod) hch gv th paramtr that mnmz (locally) th functon E( ) F( p) (S6) hr for a t of coordnat of th atom {}; E() and F( p) rprnt th potntal nrgy and th analytcal functon dpndng on th paramtr {p} rpctvly. To olv th lmtaton of th global fttng at lat n part adoptd th follong approach hch ta nto account th frt and cond drvatv of th nrgy gradnt g() and Han H() and a t of ght g and H

6 E( ) F( p) H H ( ) F( p) g F( p) g( ) (S7) Th algorthm bgn th th Poll' mthod for local optmzaton and thn appl a mult-bgnnng mthod to arch for a global optmum. Th choc of th ght crtcal n th mnmzaton proc. Thu for ntanc f focu on th dcrpton of th tranton tat rgon ll gv hghr ght to th magnary vbratonal frquncy and to pont nar th addl pont. A gradnt and Han ar ncludd n th fttng procdur th nformaton mut b provdd a nput data at th ab nto lvl. Thu th gradnt and Han for 5 pont along th mnmum nrgy path r computd at th CCS(T)/cc-pTZ lvl. At th lvl th vbratonal frqunc r numrcally calculatd and to comput a Cartan coordnat Han appromatly nrg ( quard and coordnat pr atom tm 7 atom) r ncary. A 5 pont r condrd along th path 5 ( tm 5) nrgy pont r calculatd hch r ud n th fttng proc. Onc both tp manual and automatd r fnhd too th prnt valu of th 5 paramtr a th bt t of th fttng algorthm. Th valu ar ltd n Tabl S and a rprntaton of th n PES plottd n Fgur S. A ha bn clarly hon n th Supplmntary Informaton th mnmzaton proc not a trval ta; t tm conumng and th fnal t of paramtr dpt rproducng th rfrnc ab nto data ud a nput nformaton not unqu. Fnally a notd n th prnt papr th n potntal nrgy urfac can b obtand upon rqut from th author pror to t publcaton n th POTLIB lbrary. 7

7 Tabl S. Fnal fttng paramtr for th PES-5 urfac. Paramtr alu GH.55 Å GH 86.5 cal mol - GH.5 cal mol - a GH.95 Å - b GH. Å - c GH. Å - HO.976 Å HO.98 cal mol - HO.867 cal mol - HO.8 Å - GO.95 Å GO.58 cal mol - GO.5589 cal mol - GO.676 Å -.9 Å Å - A.5879 Å - B.66 Å - C.8899 Å A.9787 Å - B.589 Å - C.8955 Å GH f.5 mdyn Å rad - GH h.95 mdyn Å rad - GH.6 mdyn Å rad - a.6 mdyn Å rad - b 5.7 Å - aa.776 Å - aa Å - aa Å - aa Å - K o a.7 mdyn Å rad - OH.8 Å - o.7 dgr

8 Fgur S. rprntaton of th PES-5 urfac. tanc n Å and nrg n cal mol -.