Vibrational analysis for the nuclear electronic orbital method

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 21 1 JUNE 2003 ARTICLES Vbratonal analyss for th nuclar lctronc orbtal mthod Tzvtln Iordanov and Sharon Hamms-Schffr Dartmnt of Chmstry, Pnnsylvana Stat Unvrsty, Unvrsty Park, Pnnsylvana Rcvd 11 Fbruary 2003; acctd 5 March 2003 Th mthodology for a vbratonal analyss wthn th nuclar lctronc orbtal NEO framwork s rsntd. In th NEO aroach, scfd nucl ar tratd quantum mchancally on th sam lvl as th lctrons, and mxd nuclar lctronc wav functons ar calculatd varatonally wth molcular orbtal mthods. Both lctronc and nuclar molcular orbtals ar xrssd as lnar combnatons of Gaussan bass functons. Th NEO otntal nrgy surfac dnds on only th classcal nucl, and ach ont on ths surfac s otmzd varatonally wth rsct to all molcular orbtals as wll as th cntrs of th nuclar bass functons. Th NEO vbratonal analyss nvolvs th calculaton, rojcton, and dagonalzaton of a numrcal Hssan to obtan th harmonc vbratonal frquncs corrsondng to th classcal nucl. Ths analyss allows th charactrzaton of statonary onts on th NEO otntal nrgy surfac. It also nabls th calculaton of zro ont nrgy corrctons and thrmodynamc rorts such as nthaly, ntroy, and fr nrgy for chmcal ractons on th NEO otntal nrgy surfac. Illustratv alcatons of ths vbratonal analyss to a srs of molculs and to a nuclohlc substtuton racton ar rsntd Amrcan Insttut of Physcs. DOI: / I. INTRODUCTION Rcntly th nuclar lctronc orbtal NEO mthod has bn dvlod for th calculaton of mxd nuclar lctronc wav functons. 1 In ths mthod, scfd nucl ar tratd quantum mchancally on th sam lvl as th lctrons. Both lctronc and nuclar molcular orbtals ar xrssd as lnar combnatons of Gaussan bass functons, and th varatonal mthod s utlzd to mnmz th nrgy wth rsct to all molcular orbtals, as wll as th cntrs of th nuclar bass functons. Sgnfcant corrlaton ffcts may b ncludd usng a multconfguratonal slfconsstnt-fld MCSCF mthod. On advantag of th NEO aroach s that nuclar quantum ffcts ar ncororatd durng th lctronc structur calculaton, rathr than subsquntly calculatd as a corrcton factor. In addton, th Born Onhmr saraton of lctrons and nucl s avodd, and xctd vbratonal lctronc stats may b calculatd. Ths mthod s comutatonally ractcal for a wd rang of chmcal ractons, and ts accuracy may b mrovd systmatcally. Th NEO aroach s artcularly usful for systms n whch only a subst of th nucl ar tratd quantum mchancally. For xaml, oftn only th hydrogn nucl ar chosn for th quantum subsystm, whl all othr nucl ar ncludd n th classcal subsystm. Th scfd quantum nucl ar tratd analogously to th lctrons, and rorts of th systm ar avragd ovr th mxd nuclar lctronc wav functon. Th coordnats of th cntrs of th nuclar bass functons as wll as th coffcnts of th molcular orbtals ar otmzd varatonally. As a rsult, th NEO otntal nrgy surfac dnds xlctly on only th coordnats of th classcal nucl and has rducd dmnsonalty rlatv to th full nuclar otntal nrgy surfac. Morovr, th locaton and charactr of th statonary onts may b substantally dffrnt for th NEO otntal nrgy surfac than for th full nuclar otntal nrgy surfac. Prvously a numbr of grous hav nvstgatd smlar aroachs. Tachkawa, Naka and co-workrs hav dvlod th nuclar orbtal molcular orbtal NOMO mthod. 2,3 Thy hav mlmntd th NOMO HF Hartr Fock Rfs. 2 and 3. NOMO CIS confguraton ntracton mthod wth sngls Rf. 4, and NOMO MBPT many-body rturbaton thory Rf. 5 aroachs. Krbch and Gross hav dvlod a nuclar lctronc mthod basd on dnsty functonal thory. 6 Shgta and coworkrs hav also xlord rlatd mthods. 7 9 To our knowldg, ths rvous grous hav not calculatd th Hssans or rformd vbratonal analyss wthn th nuclar lctronc framwork. A vbratonal analyss s rqurd for th nvstgaton of chmcal ractons on th vbratonal lctronc otntal nrgy surfac. In ths ar, w rsnt th mthodology to rform a NEO vbratonal analyss and dscuss th hyscal mlcatons of such an analyss. Th mthodology nvolvs th calculaton of a numrcal Hssan, th rojcton of ths Hssan to rmov th ovrall translatonal and rotatonal motons, and th dagonalzaton of th rojctd Hssan to dtrmn th harmonc vbratonal frquncs corrsondng to th classcal nucl. Th analyss of ths frquncs allows th charactrzaton of statonary onts on th NEO otntal nrgy surfac. Scfcally, mnma hav all ral frquncs, transton stats hav on magnary frquncy, and hghr-ordr saddl onts hav mor than on magnary frquncy. Ths analyss also nabls th calculaton of zro ont nrgy corrctons and thrmodynamc rorts such as nthaly, ntroy, and fr nrgy changs for ractons /2003/118(21)/9489/8/$ Amrcan Insttut of Physcs Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js

2 9490 J. Chm. Phys., Vol. 118, No. 21, 1 Jun 2003 T. Iordanov and S. Hamms-Schffr on th NEO otntal nrgy surfac. Furthrmor, th numrcal Hssan may b usd n th gnraton of mnmum nrgy aths, or ntrnsc racton coordnats, on th NEO otntal nrgy surfac. Ths ar rsnts llustratv alcatons of th vbratonal analyss to a srs of rrsntatv molculs, ncludng HCN, th rotonatd watr dmr, and trazn, as wll as to th racton rofl for an dntty nuclohlc substtuton (S N 2) racton. II. THEORY Th thortcal formulaton for th NEO mthod s rsntd n Rf. 1. Ths rvous ar ncluds th quatons rqurd for th mlmntaton of th NEO HF Hartr Fock, th NEO CI confguraton ntracton, and th NEO MCSCF multconfguratonal slf-consstnt-fld mthods, as wll as th quatons for th NEO HF and NEO MCSCF gradnts. Th DZSPDN nuclar bass st s also dscrbd n Rf. 1. Th NEO mthodology has bn mlmntd n th GAMESS lctronc structur rogram. 10 Th rsnt ar focuss on th NEO HF mthod, although th analyss s drctly alcabl to othr NEO mthods such as NEO CI and NEO MCSCF. Ths scton summarzs th NEO HF mthod and rsnts th thortcal formulaton for th NEO vbratonal analyss. A. Summary of th NEO HF mthod In th NEO aroach, th systm s dvdd nto thr arts: N lctrons, N quantum nucl, and N c classcal nucl. Th quatons rsntd n Rf. 1 ar drvd for frmonc quantum nucl, but th xtnson to bosons s straghtforward. At th Hartr Fock lvl th total nuclar lctronc wav functon can b aroxmatd as a roduct of sngl-confguratonal lctronc and nuclar wav functons: tot r,r 0 r 0 r, whr 0 (r ) and 0 (r ), rsctvly, ar antsymmtrzd wav functons dtrmnants of sn orbtals rrsntng th lctrons and frmonc nucl such as rotons. Hr r and r dnot th satal coordnats of th lctrons and quantum nucl, rsctvly. Th total nrgy xrsson for a rstrctd Hartr Fock RHF tratmnt of th lctrons and a hgh-sn onshll tratmnt of th quantum nucl s gvn by N /2 E2 N N /2 h j N /2 h 1 N 2 2 j j j j N j N /2 N j j2 whr th ntgrals ar dfnd as h j dr 1 *1h 1 j 1 j j, h j dr 1 *1h 1 j1, N c h A ZA r A, h 1 N c 2 ZA Z, 6 2M A r A j k l dr 1 dr 2 *1 j 1 1 r 12 k *2 l 2, and analogously for two-artcl ntgrals nvolvng th quantum nucl. Hr th satal molcular orbtals for th lctrons ar dnotd and th satal molcular orbtals for th quantum nucl ar dnotd. Th unrmd ndcs, j rfr to lctrons, th rmd ndcs,j rfr to quantum nucl, and th ndcs A, B rfr to classcal nucl. Th masss, chargs, and dstancs, rsctvly, ar dnotd by M, Z, and r wth th arorat subscrts. Th satal orbtals for th lctrons and th quantum nucl ar xandd n Gaussan bass sts whr thr ar lctronc bass functons and nuclar bass functons: 1 c 1, 1 c 1. 9 Hr th unrmd ndcs,,, rfr to lctronc bass functons, and th rmd ndcs rfr to nuclar bass functons. Th Hartr Fock quatons hav bn drvd usng th standard varaton mthod to mnmz th nrgy n Eq. 2 wth rsct to both th lctronc and nuclar molcular orbtals. Th rsultng Hartr Fock Roothaan quatons ar F c S c, 1,...,N /2, 10 F c S c, 1,...,N, 11 whr th ovrla matrx lmnts ar dfnd as S dr 1 *1 1, 12 S dr 1 *1 1, 13 and th Fock matrx lmnts may b xrssd as F h G F P, P. h G Hr th dnsty matrx lmnts ar dfnd as Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js

3 J. Chm. Phys., Vol. 118, No. 21, 1 Jun 2003 Th nuclar lctronc orbtal mthod 9491 N /2 P 2 c c *, P N c c *, and th standard on-artcl and two-artcl arts ar dfnd as h dr 1 *1h 1 1, 18 h dr 1 *1h 1 1, 19 G P 1 2, 20 G P. 21 For fxd classcal nucl and quantum nuclar bass functon cntrs, th lctronc and nuclar Hartr Fock Roothaan quatons can b solvd tratvly to slf consstncy utlzng convrgnc acclrators rvously dvlod for lctronc structur thory. 11 In our calculatons, th nuclar Fock quatons ar fully convrgd aftr ach st n th tratv rocdur for th soluton of th lctronc Fock quatons. Othr tratv schms ar ossbl and may b mor ffcnt for dffrnt tys of systms. Th nrgy gradnt wth rsct to th nuclar coordnats for th NEO HF mthod n th atomc orbtal bass s E 1 2 h P P 1 2P P P P 4P 1 P P W S,HF W S,HF h P P P. 22 whr R n s th nuclar coordnat or coordnat of th cntr of th nuclar bass functon and th nrgy wghtd dnsty matrx s dfnd by N /2 W,HF 2 N,HF W c c, c Not that and. c ar th molcular orbtal nrgs dfnd n Eqs. 10 and 11, rsctvly. Analogous quatons may b drvd n th sam mannr for a rstrctd on shll Hartr Fock ROHF or unrstrctd Hartr Fock UHF tratmnt of th lctrons. For smlcty, th xamls n ths ar nvolv an RHF tratmnt of th lctrons and a hgh-sn on-shll tratmnt of th quantum nucl. Th gnral concts ar alcabl to othr sn stats, as wll as to multconfguratonal wav functons. B. Vbratonal analyss Th NEO otntal nrgy surfac E NEO (R class ) dnds xlctly on only th classcal nuclar coordnats R class. Each ont on th NEO otntal nrgy surfac s otmzd varatonally wth rsct to th molcular orbtal coffcnts c, c and th nuclar bass functon cntrs R bfcnt. In our mlmntaton, ach ont on th NEO otntal nrgy surfac s obtand by th otmzaton of a nuclar bass functon surfac E nucbf (R bfcnt ;R class ) wth rsct to th nuclar bass functon cntrs. Th surfac E nucbf (R bfcnt ;R class ) dnds xlctly on th nuclar bass functon cntrs and aramtrcally on th classcal nuclar ostons. Each ont corrsondng to a fxd R bfcnt and R class on ths nuclar bass functon surfac s otmzd varatonally wth rsct to th molcular orbtal coffcnts by solvng th Hartr Fock Roothaan quatons n Eqs. 10 and 11 usng tratv tchnqus. Each ont on th NEO otntal nrgy surfac E NEO (R class ) s obtand usng gomtry otmzaton tchnqus to mnmz E nucbf (R bfcnt ;R class ) wth rsct to th nuclar bass functon cntrs R bfcnt. Th gradnt wth rsct to th nuclar bass functon cntrs for th surfac E nucbf (R bfcnt ;R class )s gvn by Eq. 22. Th NEO otntal nrgy surfac may b dscrbd n trms of th locaton and charactrzaton of th statonary onts. Th statonary onts on th NEO otntal nrgy surfac ar dfnd to b gomtrs at whch th gradnts of th nrgy wth rsct to th classcal nuclar coordnats R class ar zro. Not that by dfnton th gradnts wth rsct to th nuclar bass functon cntrs ar zro at all onts on th NEO otntal nrgy surfac. Th gradnt wth rsct to th classcal nuclar coordnats for ach ont on th NEO otntal nrgy surfac s gvn by Eq. 22. Th gradnt xrsson s th sam for th nuclar bass functon cntrs and th classcal nuclar coordnats bcaus th nuclar bass functon surfac E nucbf (R bfcnt ;R class ) s otmzd wth rsct to th molcu- Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js

4 9492 J. Chm. Phys., Vol. 118, No. 21, 1 Jun 2003 T. Iordanov and S. Hamms-Schffr lar orbtal coffcnts, and th NEO surfac E NEO (R class )s otmzd wth rsct to th molcular orbtal coffcnts and th nuclar bass functon cntrs. Th statonary onts on th NEO otntal nrgy surfac ar locatd usng gomtry otmzaton tchnqus. 10 Ths statonary onts may b charactrzd by calculatng a Hssan matrx.., scond drvatv matrx for th NEO otntal nrgy surfac. Th lmnts of th Hssan matrx K can b calculatd numrcally wth th followng xrsson: 12 K j 2 E NEO dr dr j R 0 class g jg 0, 25 R 0 class 0 whr th subscrt R class dnots th coordnats of th classcal nucl corrsondng to th ont on th NEO otntal nrgy surfac at whch th scond drvatv s bng valuatd. Th gradnts ar dfnd as R class,. 26 R class, j R 0 class, j Although th calculatons rsntd n ths ar wr obtand wth Eq. 25, whch rqurs a sngl dslacmnt of coordnat R class, j by an amount, th scond drvatv can also b calculatd symmtrcally wth a doubl dslacmnt. Th mass-wghtd Hssan K s dfnd as g j ENEO K j 1 K j. 27 m m j Dagonalzaton of th mass-wghtd Hssan lads to th st of gnvalus, whch ar usd to calculat th harmonc vbratonal frquncs W hav mlmntd th numrcal calculaton of th Hssan and th dtrmnaton of th harmonc vbratonal frquncs wthn th NEO framwork. Not that for ach coordnat dslacmnt rqurd to calculat th numrcal Hssan.., Eq. 25, th nrgy must b otmzd wth rsct to th nuclar bass functon cntrs to rman on th NEO otntal nrgy surfac. For th xact Hssan, th gnvalus corrsondng to ovrall translaton and rotaton of th ntr molcul ar zro bcaus th nrgy and gradnt ar nvarant to ths motons. As a rsult, for nonlnar lnar molculs, sx fv of th frquncs ar zro. For th numrcal Hssan, th gnvalus corrsondng to ovrall translatonal and rotatonal motons ar not xactly zro du to numrcal rror. Th Hssan matrx can b rojctd to mak ths gnvalus dntcally zro: K P IPKIP, 29 whr I s th dntty matrx and P s th rojcton matrx dfnd n Rf. 13. W hav mlmntd ths rojcton rocdur and th Sayvtz condtons 14 for th NEO mthod. Not that only th classcal nucl ar ncludd n th calculaton of th rojcton matrx.., th masss of th quantum nucl ar nglctd. W mhasz that th Hssan matrx for th NEO otntal nrgy surfac s fundamntally dffrnt from th Hssan matrx for th full nuclar otntal nrgy surfac bcaus ach ont on th NEO otntal nrgy surfac s otmzd wth rsct to th nuclar bass functon cntrs. Th otmzaton wth rsct to th nuclar bass functon cntrs mantans th translatonal and rotatonal nvaranc of th classcal nucl. Th hyscal sgnfcanc of ths otmzaton may b llustratd wth th datomc molcul HCl. If th H nuclus s tratd quantum mchancally n th NEO aroach, th NEO otntal nrgy surfac dnds on only th thr coordnats of th Cl atom. Ths thr coordnats corrsond to th translatonal dgrs of frdom for th molcul. Thus, th HCl molcul wthn th NEO framwork s analogous to a sngl atom n a convntonal lctronc structur calculaton for whch all nucl ar tratd classcally. As xctd, th dagonalzaton of th NEO Hssan matrx for HCl rsults n thr zro gnvalus, ndcatng that th NEO nrgy s translatonally nvarant for HCl. If th nuclar bass functon cntrs wr not otmzd for ach st of th numrcal Hssan calculaton, howvr, th gnvalus of th HCl Hssan matrx would no longr b zro and th NEO nrgy would not b translatonally nvarant. An analogous argumnt may b ald to largr molculs n trms of both translatonal and rotatonal nvaranc. An altrnatv aroach that would also mantan translatonal and rotatonal nvaranc would b to dfn th ostons of th nuclar bass functon cntrs wth rsct to th ostons of scfd classcal nucl.., to constran th dstanc btwn th nuclar bass functon cntr and th Cl atom for th HCl molcul. Ths altrnatv aroach s roblmatc for a hydrogn transfr racton, whr th oston of th hydrogn nuclar bass functon cntr rlatv to th classcal donor and acctor nucl changs durng th racton. Morovr, th otmzaton of th nuclar bass functon cntrs avods th bas ntroducd by ths altrnatv aroach. Th analyss of th Hssan can b usd to charactrz statonary onts on th NEO otntal nrgy surfac and to calculat zro ont nrgy corrctons. A NEO ont wth all ostv frquncs s a mnmum, a NEO ont wth only on magnary frquncy s a transton stat, and a NEO ont wth mor than on magnary frquncy s a hghrordr saddl ont. Th zro ont nrgy ZPE.., zro tmratur vbratonal nrgy corrcton s ZPE 1 2 h, 30 whr th summaton s ovr all ostv frquncs. Th Hssan may also b usd to calculat thrmodynamc quantts such as vbratonal nthaly, ntroy, and fr nrgy changs. Th vbratonal contrbutons to th nthaly, ntroy, and fr nrgy ar 12,15 H vb R h 2k B h k B 1 xh /k B T1, 31 Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js

5 J. Chm. Phys., Vol. 118, No. 21, 1 Jun 2003 Th nuclar lctronc orbtal mthod 9493 S vb R h k B T 1 xh /k B T1 ln1xh /k B T, 32 G vb H vb TS vb, 33 whr h s Planck s constant, k B s Boltzmann s constant, R s th dal gas constant, and th summatons ar ovr all ostv frquncs. Th translatonal and rotatonal contrbutons to th thrmodynamc quantts can b calculatd n th usual way for th NEO structurs. In gnral, th quantum nucl should b ncludd n th calculaton of th translatonal and rotatonal thrmodynamc quantts.., for th calculaton of th cntr of mass and th momnts of nrta. Thus, for th calculaton of translatonal and rotatonal thrmodynamc quantts, th molcul should b vwd as havng N c N atoms and thrfor 3(N c N ) dgrs of frdom. In ractc, th ostons of th quantum nucl can b aroxmatd to b thr th bass functon cntrs or th xctaton valus of th quantum nuclar coordnats for localzd molcular orbtals. Thr ar thr translatonal and thr two rotatonal dgrs of frdom for a nonlnar lnar molcul. Th rmanng dgrs of frdom ar vbratonal, whr som ar th nonzro classcal vbratonal mods calculatd from th NEO Hssan and th rmandr ar quantum vbratonal mods ncludd n th NEO vbratonal lctronc nrgy. Th quantum vbratonal and lctronc contrbutons to th thrmodynamc quantts can b calculatd from th xctd NEO vbratonal lctronc stats. Oftn th contrbutons from ths xctd stats ar nglgbl bcaus th slttngs btwn th stats ar much largr than th thrmal nrgy. Ths xctd stats ar not ncludd at th NEO HF lvl but wll b ncludd for NEO MCSCF and NEO CI calculatons of thrmodynamc quantts. Although w hav rsntd ths vbratonal analyss n th framwork of th NEO HF mthod, th analogous rocdur s alcabl to NEO xtnsons such as multconfguratonal mthods.., NEO CI and NEO MCSCF, assumng th avalablty of analytcal gradnts. Rfrnc 1 rsnts th analytcal gradnts for th NEO MCSCF mthod. Ths vbratonal analyss s also vald wth dffrnt bass sts and wth multl bass functon cntrs. All of ths mthodology has bn ncludd n th NEO mlmntaton wthn th GAMESS lctronc structur rogram. 10 III. RESULTS W hav rformd gomtry otmzatons and calculatd Hssan matrcs for a srs of molculs. For ach systm, w rformd both a convntonal lctronc structur calculaton n whch all nucl ar tratd classcally and a NEO calculaton n whch on or all hydrogn nucl ar tratd quantum mchancally. Th lctronc bass st usd for HCN, th rotonatd watr dmr, and trazn was 6-31Gd, and th lctronc bass st usd for th S N 2 racton was 6-31Gd, Rf. 20 to nclud dffus functons for anons. Th nuclar bass st usd for all NEO FIG. 1. NEO HF mnmum nrgy structur for HCN, wth rlvant dstancs ndcatd. Th dstanc nvolvng th hydrogn nuclus s obtand from th xctaton valu of th hydrogn coordnat. Th dstancs obtand from a convntonal RHF lctronc structur calculaton ar gvn n arnthss. Th hydrogn nuclar orbtal s dctd as a thr-dmnsonal contour lot wth a contour valu of 0.01 bohr 3/2. Th fgur was gnratd wth MAC MOL PLOT Rf. 30. calculatons s DZSPDN wth a sngl nuclar bass functon cntr for ach hydrogn nuclus. Th DZSPDN bass st ncluds two ach of s-, -, and d-ty Gaussans, rsultng n a total of 20 nuclar bass functons r hydrogn. 1 Th lctronc bass functons corrsondng to ach quantum hydrogn ar cntrd at th sam oston as th corrsondng nuclar bass functon cntr. W comutd th gomtrs and frquncs for HCN, H 5 O 2, and N 3 H 3. Furthrmor, w calculatd th racton rofl for an dntty nuclohlc substtuton (S N 2) racton. W do not xct ths rsults to b quanttatvly accurat bcaus thy ar calculatd at th Hartr Fock lvl. Th am of ths calculatons s to llustrat th undrlyng rncls nvolvd n th NEO vbratonal analyss rathr than to rovd quanttatvly accurat rsults. Fgur 1 dcts th NEO otmzd structur for th HCN molcul. In th NEO calculatons, th hydrogn nuclus s tratd quantum mchancally. Thrfor, th NEO otntal nrgy surfac s analogous to th sxdmnsonal convntonal surfac for a datomc molcul. Rmovng th fv ovrall translatonal and rotatonal motons, th NEO HF otntal nrgy surfac has only on vbratonal frquncy, 2543 cm 1, whch corrsonds to th C N vbratonal moton. Th C N dstanc s smlar for th convntonal RHF Å and NEO HF Å calculatons. Th rotonatd watr dmr (H 5 O 2 ) has bn th focus of many thortcal studs Ths rvous studs ndcat that ths systm s hghly snstv to th lvl of thory and th bass st. At th RHF lvl, th C s symmtry structur s a mnmum and th C 2 symmtry structur s a transton stat. For hghr lvls of thory ncludng lctron corrlaton, howvr, th C 2 symmtry structur s a mnmum and th C s symmtry structur s a transton stat. 21 For all lvls of thory, th otntal nrgy surfac s xtrmly flat for ths systm. Thus, quanttatvly accurat rsults for th rotonatd watr dmr rqur larg bass sts and th ncluson of lctronc corrlaton. W mhasz that th am of th rsnt calculatons s only to llustrat th utlty of th NEO mthodology and not to rovd quanttatvly accurat rsults for ths scfc systm. In our ntal studs of th rotonatd watr dmr, w rformd a NEO HF calculaton n whch only th cntral hydrogn nuclus s tratd quantum mchancally. W found that th C 2 symmtry structur s a mnmum and th C s symmtry structur s a transton stat 0.02 kcal/mol hghr n nrgy. For comarson, at th convntonal RHF/ Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js

6 9494 J. Chm. Phys., Vol. 118, No. 21, 1 Jun 2003 T. Iordanov and S. Hamms-Schffr FIG. 3. NEO HF mnmum-nrgy structur for trazn, wth rlvant dstancs and angls ndcatd. Th dstancs and angls obtand from a convntonal RHF lctronc structur calculaton ar gvn n arnthss. Th hydrogn nuclar orbtals ar dctd as thr-dmnsonal contour lots wth a contour valu of 0.01 bohr 3/2. Th fgur was gnratd wth MAC MOL PLOT Rf. 30. FIG. 2. NEO HF mnmum-nrgy structurs for H 5 O 2, wth rlvant dstancs and angls ndcatd for a C s statonary ont and b C 2 statonary ont. Th dstancs nvolvng th quantum hydrogn nuclus ar obtand from th xctaton valu of th hydrogn coordnat. Th dstancs and angls obtand from a convntonal RHF lctronc structur calculaton ar gvn n arnthss. Th hydrogn nuclar orbtal s dctd as a thr-dmnsonal contour lot wth a contour valu of 0.01 bohr 3/2.Th fgur was gnratd wth MAC MOL PLOT Rf Gd, lvl, th C s symmtry structur s a mnmum and th C 2 symmtry structur s a transton stat 0.15 kcal/ mol hghr n nrgy. Intrstngly, th NEO aroach altrs th natur of th Hartr Fock statonary onts n a smlar mannr as th ncluson of lctronc corrlaton. Th magnary frquncy for th convntonal RHF C 2 transton stat s 452 cm 1, and th mod s domnatd by th moton of th cntral hydrogn nuclus. In th NEO calculaton, howvr, ths cntral hydrogn s tratd quantum mchancally. Th magnary frquncy for th NEO HF C s transton stat s 149 cm 1, and th mod s domnatd by th twstng moton of th four xtror hydrogn nucl. Not that th hyscal manng of th magnary mod s fundamntally dffrnt for th convntonal RHF and NEO HF calculatons for ths systm. Fgur 2 rovds a comarson btwn th convntonal RHF and NEO HF gomtrs wth C 2 and C s symmtrs. Th most sgnfcant dffrnc s th O O dstanc for th C 2 symmtry statonary ont, whr th convntonal RHF dstanc s Å and th NEO HF dstanc s Å. W also rformd a NEO calculaton n whch all fv hydrogn nucl ar tratd quantum mchancally. As for th HCN molcul, th rsultng NEO otntal nrgy surfac s sx dmnsonal. Rmovng th fv ovrall translatonal and rotatonal motons, th sngl vbratonal mod of frquncy 673 cm 1 corrsonds to th O O vbratonal moton. Th NEO HF O O dstanc s Å, whch s vrtually dntcal to th convntonal RHF dstanc of Å. In addton, w rformd NEO calculatons for trazn (N 3 H 3 ). In ths NEO calculatons, all thr hydrogn nucl wr tratd quantum mchancally, so th NEO otntal nrgy surfac s nn dmnsonal. Rmovng th ovrall translatonal and rotatonal motons, thr ar thr molcular vbratons wth frquncs 787, 1133, and 1832 cm 1 corrsondng to th vbratonal motons of th ntrogn atoms. As shown n Fg. 3, th NEO HF gomtry s smlar to th convntonal RHF gomtry for trazn. To llustrat th calculaton of statonary onts for a chmcal racton on th NEO otntal nrgy surfac, w xamnd th followng S N 2 racton: Cl CH 3 Cl ClCH 3 Cl. In th NEO calculatons, th thr hydrogn nucl ar tratd quantum mchancally. As for trazn, th NEO otntal nrgy surfac s nn dmnsonal wth thr vbratonal mods aftr rmoval of th sx ovrall translatonal and rotatonal motons. Th NEO transton stat structur s shown n Fg. 4. Th C Cl dstanc at th transton stat s smlar for th NEO HF Å and convntonal RHF Å mthods. Th magnary frquncy s also smlar for th NEO HF 421 cm 1 ) and convntonal RHF 426 cm 1 ) mthods, and th mod s domnatd by th moton of th carbon and chlorn atoms. Not that ths magnary frquncs ar smlar bcaus th nucl tratd quantum mchancally n th NEO calculatons do not contrbut sgnfcantly to ths magnary mod. Th nrgy rofl for ths racton s llustratd schmatcally n Fg Th racton nvolvs th ntal formaton of a ractant on molcul comlx, whch has a comlxaton nrgy E com rlatv to saratd ractants. Th ractant comlx surmounts th cntral actvaton barrr FIG. 4. NEO HF transton stat for an S N 2 racton, wth th C Cl dstanc ndcatd. Th dstanc obtand from a convntonal RHF lctronc structur calculaton s gvn n arnthss. Th hydrogn nuclar orbtals ar dctd as thr-dmnsonal contour lots wth a contour valu of 0.01 bohr 3/2. Th fgur was gnratd wth MAC MOL PLOT Rf. 30. Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js

7 J. Chm. Phys., Vol. 118, No. 21, 1 Jun 2003 Th nuclar lctronc orbtal mthod 9495 IV. CONCLUSIONS FIG. 5. Schmatc nrgy rofl for th symmtrc S N 2 racton Cl CH 3 Cl ClCH 3 Cl. E cnt to th transton stat structur, whch braks down nto th on molcul comlx and fnally nto saratd roducts. Th ovrall actvaton barrr rlatv to saratd ractants s dnotd E ovr. Th calculatons wr don wth C 3v symmtry for th ractant and ractant comlx and D 3h symmtry for th transton stat. Tabl I rsnts th NEO HF and convntonal RHF nrgs wth and wthout zro ont nrgy corrctons, as wll as th nthals and fr nrgs at 298 K corrsondng to th thr tys of barrrs dfnd n Fg. 5. Not that th am of ths calculatons s to llustrat th alcaton of th NEO aroach to chmcal ractons rathr than to obtan quanttatvly accurat rsults. Prvous studs hav shown that lctronc corrlaton s rqurd to obtan quanttatvly accurat rsults for ths S N 2 racton. 24 Th NEO HF and convntonal RHF rsults ar vry smlar for th S N 2 racton. Ths smlarty arss bcaus th hydrogn atoms ar not actvly artcatng n th chmcal racton.., bonds nvolvng hydrogn atoms ar not brokn or formd. Th NEO aroach s xctd to dffr sgnfcantly from convntonal lctronc structur aroachs for ractons nvolvng th transfr of a hydrogn nuclus. As mntond n Rf. 1, howvr, hydrogn transfr ractons rqur th us of multl bass functons and multconfguratonal wav functons. Ths ar rsnts th mthodology and llustratv alcatons for a vbratonal analyss wthn th NEO framwork. In th NEO aroach, scfd nucl ar tratd quantum mchancally on th sam lvl as th lctrons. Each ont on th NEO otntal nrgy surfac s otmzd varatonally wth rsct to th lctronc and nuclar molcular orbtal coffcnts and th nuclar bass functon cntrs. Th statonary onts on th NEO otntal nrgy surfac ar dfnd to b gomtrs at whch th gradnts wth rsct to th classcal nuclar coordnats ar zro. Ths statonary onts may b locatd usng gomtry otmzaton tchnqus. Th NEO vbratonal analyss mthodology nvolvs th calculaton, rojcton, and dagonalzaton of a numrcal Hssan to obtan th harmonc vbratonal frquncs corrsondng to th classcal nucl. Ths analyss allows th charactrzaton of statonary onts on th NEO otntal nrgy surfac, th calculaton of zro ont nrgy corrctons, and th calculaton of thrmodynamc rorts such as nthaly, ntroy, and fr nrgy for chmcal ractons on th NEO otntal nrgy surfac. Th NEO vbratonal analyss wll b artcularly usful for alcatons to hydrogn transfr ractons, whch xhbt substantal nuclar quantum ffcts. For ths tys of ractons, th transfrrng hydrogn nuclus s tratd quantum mchancally and th mnmum nrgy ath ncluds th sgnfcant nuclar quantum ffcts such as zro ont nrgy and hydrogn tunnlng. Ths aroach mantans th conctual ctur of a racton ath, whl lmnatng th dffcults assocatd wth larg curvatur of th racton ath for th transfr of lght nucl. 25 For many hydrogn transfr ractons, th nuclar wav functon corrsondng to th transfrrng hydrogn s blobal at th transton stat. 1 Th alcaton of NEO to ths tys of hydrogn transfr ractons rqurs th us of multl nuclar bass functon cntrs and multconfguratonal wav functons. Th NEO vbratonal analyss mthodology s drctly alcabl to ths tys of ractons and wll allow th locaton and charactrzaton of statonary onts, as wll as th calculaton of nrgtcs and thrmodynamc rorts, for hydrogn transfr ractons. Th alcaton of th NEO aroach to hydrogn transfr ractons s analogous to convntonal calculatons for lctron transfr ractons. In th NEO aroach, th racton coordnat of th mnmum nrgy ath for a hydrogn transfr racton dnds xlctly on only th classcal nucl, TABLE I. NEO HF barrrs for th S N 2 racton dctd n Fg. 5. Th convntonal RHF barrrs ar gvn n arnthss. Th quantts rovdd ar th NEO vbratonal lctronc nrgy E, th NEO vbratonal lctronc nrgy wth zro ont nrgy corrctons E(ZPE), th nthaly H at 298 K, and th fr nrgy G at 298 K. Barrr ty E EZPE H G Comlxaton Ovrall Cntral Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js

8 9496 J. Chm. Phys., Vol. 118, No. 21, 1 Jun 2003 T. Iordanov and S. Hamms-Schffr analogous to th collctv racton coordnat n Marcus thory for lctron transfr. 26 Th NEO racton coordnat for hydrogn transfr dos not dnd on th transfrrng hydrogn coordnat for th sam rason that th convntonal racton coordnat for lctron transfr dos not dnd on th lctron coordnats. Th NEO transton stat corrsonds to th classcal nuclar confguraton for whch th transfrrng hydrogn nuclar wav functon s dlocalzd btwn th donor and acctor classcal nucl. Analogously, th convntonal transton stat for lctron transfr s th nuclar confguraton for whch th adabatc lctronc wav functon s dlocalzd btwn th donor and acctor, also dscrbd as th confguraton for whch th ractant and roduct dabatc lctronc stats ar dgnrat. In both th NEO aroach for hydrogn transfr and th convntonal aroach for lctron transfr, th magnary mod at th transton stat corrsonds to th havy-atom moton that drvs th charg transfr racton va th rorganzaton of th nvronmnt. In contrast, th magnary mod at th transton stat for a convntonal hydrogn transfr mnmumnrgy ath s domnatd by th transfrrng hydrogn coordnat tslf. Th dscrton of hydrogn transfr ractons n trms of a collctv, havy-atom racton coordnat togthr wth a quantum mchancal rrsntaton of th transfrrng hydrogn has bn shown to rovd a hyscally manngful framwork for studyng ths ractons Th alcaton of th NEO aroach to hydrogn transfr ractons wll rovd mchanstc nsght by nablng th dntfcaton of th domnant havy-atom motons contrbutng to th collctv racton coordnat. Ths motons ar sgnfcant bcaus thy corrsond to th rorganzaton of th nvronmnt ncssary for th charg transfr rocss. In addton, th NEO aroach wll allow accurat calculatons of rats and kntc soto ffcts for hydrogn transfr ractons. ACKNOWLEDGMENTS Th authors ar gratful for fnancal assstanc from AFOSR Grant No. F Thy would also lk to thank Smon Wbb and Mk Pak for many hlful dscussons. 1 S. P. Wbb, T. Iordanov, and S. Hamms-Schffr, J. Chm. Phys. 117, M. Tachkawa, K. Mor, H. Naka, and K. Iguch, Chm. Phys. Ltt. 290, H. Naka, Int. J. Quantum Chm. 86, H. Naka, K. Sodyama, and M. Hoshno, Chm. Phys. Ltt. 345, H. Naka and K. Sodyama, J. Chm. Phys. 118, T. Krbch and E. K. U. Gross, Phys. Rv. Ltt. 86, Y. Shgta, H. Nagao, K. Nshkawa, and K. Yamaguch, J. Chm. Phys. 111, Y. Shgta, Y. Ozak, K. Kodama, H. Nagao, H. Kawab, and K. Nshkawa, Int. J. Quantum Chm. 69, Y. Shgta, H. Takahash, S. Yamanaka, M. Mtan, H. Nagao, and K. Yamaguch, Int. J. Quantum Chm. 70, M. W. Schmdt, K. K. Baldrdg, J. A. Boatz t al., J. Comut. Chm. 14, P. Pulay, J. Comut. Chm. 3, H. L and J. H. Jnsn, Thor. Chm. Acc. 107, W. H. Mllr, N. C. Handy, and J. E. Adams, J. Chm. Phys. 72, A. Sayvtz, J. Chm. Phys. 7, N. Davdson, Statstcal Mchancs McGraw-Hll, Nw York, R. Dtchfld, W. J. Hhr, and J. A. Pol, J. Chm. Phys. 54, P. C. Harharan and J. A. Pol, Thor. Chm. Acta 28, M. M. Francl, W. J. Ptro, W. J. Hhr, J. S. Bnkly, M. S. Gordon, D. J. DFrs, and J. A. Pol, J. Chm. Phys. 77, W. J. Hhr, R. Dtchfld, and J. A. Pol, J. Chm. Phys. 56, T. Clark, J. Chandraskhar, G. W. Stznagl, and P. v. R. Schlyr, J. Comut. Chm. 4, Y. X, R. B. Rmngton, and H. F. Schafr III, J. Chm. Phys. 101, A. A. Aur, T. Hlgakr, and W. Klor, Phys. Chm. Chm. Phys. 2, E. F. Valv and H. F. Schafr III, J. Chm. Phys. 108, M. N. Glukhovtsv, A. Pross, and L. Radom, J. Am. Chm. Soc. 117, J. Carrngton and W. H. Mllr, J. Chm. Phys. 84, R. A. Marcus, Annu. Rv. Phys. Chm. 15, P. M. Kfr and J. T. Hyns, J. Phys. Chm. A 106, P. M. Kfr and J. T. Hyns, J. Phys. Chm. A 106, S. R. Blltr, S. P. Wbb, T. Iordanov, P. K. Agarwal, and S. Hamms- Schffr, J. Chm. Phys. 114, B. M. Bod and M. S. Gordon, J. Mol. Grahcs 16, Downloadd 17 May 2003 to Rdstrbuton subjct to AIP lcns or coyrght, s htt://ojs.a.org/jco/jccr.js