Linear regression correction to first principle theoretical calculations Improved descriptors and enlarged training set

Transcription

1 Chmical Physics Lttrs 49 (25) Linar rgrssion corrction to first principl thortical calculations Improvd dscriptors and nlargd training st Xu-Mi Duan a, Zhn-Hua Li a,b, Hai-Rong Hu a, Guo-Liang Song a, Wn-Ning Wang a,b, Guan-Hua Chn a,b,c, Kang-Nian Fan a,b, * a Shanghai Ky Laboratory of Molcular Catalysis and Innovativ Matrials, Dpartmnt of Chmistry, Cntr for Chmical Physics, Fudan Univrsity, Shanghai 2433, China b Virtual Laboratory of Computational Chmistry, Computr Ntwork Information Cntr, Chins Acadmy of Scincs, Bijing, China c Dpartmnt of Chmistry, Th Univrsity of Hong Kong, Hong Kong, China Rcivd 22 January 25; in final form 13 April 25 Availabl onlin 9 Jun 25 Abstract Th linar rgrssion corrction prviously dvlopd to rduc quantum chmical calculation rrors [X.M. Duan, G.L. Song, Z.H. Li, X.J. Wang, G.H. Chn, K.N. Fan, J. Chm. Phys. 121 (24) 786] has bn furthr improvd by using nw dscriptors obtaind from natural bond orbital analysis and an nlargd training st of 35 organic, inorganic molculs and radicals. Th nw schm is bttr suitd for corrcting raction barrirs. Upon linar rgrssion corrction, th man absolut dviation for th nw st dcrass from 284.1, 8.2, 12.4 kcal/mol to 7.3, 3.3, 2.7 kcal/mol for th /6-31G(d), /6-31G(d), and / 6-311G(2d,d,p) mthods, rspctivly, and th man absolut dviation of 12 barrir hights for six hydrogn transfr ractions is rducd from 5.3 to 2.9 kcal/mol for th /6-311G(2d,d,p) mthod. Ó 25 Elsvir B.V. All rights rsrvd. 1. Introduction * Corrsponding author. Fax: addrss: knfan@fudan.du.cn (K.-N. Fan). A grat succss of computational chmistry in th last two dcads is th succssful prdiction of th thrmochmical proprtis of small- and mdium-sizd molculs. Th calculatd proprtis ar oftn comparabl to xprimntal masurmnts, and occasionally vn bttr than th xprimntal countrparts. To achiv such accuracy lctron corrlation ffcts must b xplicitly considrd in th quantum chmical calculations [1 4]. Aftr dcads of fforts, a varity of mthods suitabl for tackling th corrlation problm, including th configuration intraction (CI) [5], th coupld-clustr (CC) procdurs [6] and th Gaussian 2 (G2) [7 9], Gaussian 3 (G3) [1,11] algorithms, hav bn dvlopd. Taking G3 mthod as an xampl, its man absolut dviation (MAD) on th hats of formation of 222 molculs is only 1.5 kcal/mol [12]. Howvr, ths procdurs ar most computational rsourc consuming and ar still inapplicabl to complx systms. Thus, a balanc has to b found btwn accuracy and fficincy. Dnsity-functional thory (DFT), spcially thos hybrid mthods, such as [13 16], surly offrs promising altrnativs. Howvr, all DFT calculations mploy approximatd xchang-corrlation (XC) functionals, and thr is no systmatic way to improv ths functionals. Morovr, th rrors of DFT calculations ar accumulatd with th siz of th molcul [17]. Rcntly, Chn t al. and w proposd two smi-mpirical procdurs, th nural-ntwork (NEURON) schm of Chn t al. [18] and our linar rgrssion corrction (LRC) approach [19] to improv th rsults of quantum chmical mthods. For th mthod, th /$ - s front mattr Ó 25 Elsvir B.V. All rights rsrvd. doi:1.116/j.cpltt

2 316 X.-M. Duan t al. / Chmical Physics Lttrs 49 (25) root-man-squar (RMS) dviations of th calculatd hats of formation for 18 organic molculs wr dcrasd from mor than 1 kcal/mol to about 3 kcal/ mol upon DFT-NEURON and DFT-LRC corrctions. Upon -LRC corrction, th RMS dviations of th calculatd hats of formation wr dcrasd dramatically from mor than 35 kcal/mol to just 6 kcal/mol. Ths procdurs hav also bn succssfully applid to prdicting lctron affinitis (EA), ionization potntials (IP) and absorption nrgis [2]. Although vry succssful, ths procdurs ar unsuitabl for calculating potntial nrgy profils of chmical ractions, sinc th physical dscriptors mployd ar th numbrs of bonding lctrons, lon pair lctrons, cor lctrons, or atoms, which will ithr cancl or rsult in non-continuous potntial nrgy surfac. It is wll known that a chmical raction can b dscribd as a rarrangmnt of th chmical bonding pattrn btwn atoms. To charactriz a chmical raction at th lctronic lvl along a chosn raction coordinat, on must thrfor dscrib th continuous lctronic rdistribution from th ractant-lik to productlik bonding pattrn. For this purpos, w us natural bond orbital (NBO) population analysis [21 25] to obtain th physical dscriptors in th prsnt procdur. Anothr limitation of our prvious LRC mthod is that th training st contains only closd-shll organic molculs, which limits th application rang of th mthod. Thus, in th prsnt Lttr, th training st is nlargd to includ 222 hats of formation in th G3/ 99 tst st [12], and also thos usd in our prvious work (with duplicatd ons dltd). Th nw training st contains 35 hats of formation of small- and mdium-sizd organic, inorganic molculs, and radicals. 2. Th linar rgrssion corrction mthod In th LRC, th nrgy of a molcul MðA na ; B nb ; Þ is calculatd by [19] E LRC ðmþ ¼E calc þ X a i x i þ c ZPE ð1þ i and that of atom A by E LRC ðaþ ¼E calc þ X b i x i ; ð2þ i whr E calc is th calculatd lctronic nrgy by a mthod without any corrction; E LRC is th nrgy at K aftr linar rgrssion corrction; {x i } ar physical dscriptors; {a i } and {b i } ar cofficints of th dscriptors for th molcul and th atoms, rspctivly; c is th scaling factor for zro-point vibrational nrgy (ZPE). Th physical dscriptors prviously usd, th numbrs of th lctrons (all intgrs) in diffrnt bonding nvironmnt ar now rplacd with thos calculatd from th NBO population analysis, which ar th total lctron populations of diffrnt typs of NBOs: two-cntr bonds (BD), on-cntr cor pair (CR), on-cntr valnc lon pair (LP), on-cntr Rydbrg (RY*), twocntr anti-bond (BD*), and valnc non-lwis lon pair (LP*). Th cor lctrons ar furthr dividd into svral substs according to th shll in th corrsponding atoms. Thr ar thr shlls for th cor lctrons in th molculs studid hr. W dfin CR1, CR2, and CR3 as th first, th scond and th third layr blow th valnc shll. Th numbr of th unpaird lctrons (an intgr) of atom in its ground stat is also includd as a dscriptor. In an atomization nrgy schm, th hat of formation of a molcul MðA na ; B nb ;...Þ at K aftr linar rgrssion corrction is calculatd by " # DH 298 K f ¼ X i a i x i X i þ DH calc f ðmþþ X A X A b i x i þ c ZPE þ DE calc n A DH xp f; K ða ðgþþ n A;S DH xp 298 K ða ðsþþ; ð3þ whr n A,S is th molar ratio of th lmnt A in th molcul (M) to that in its stabl stat of aggrgation at K, i.. its standard stat, with th subscript ÔSÕ rprsnting standard stat; DE calc is th atomization nrgy of M at K without ZPE corrction; DH calc f ðmþ is th calculatd nthalpy chang of M from to K; DH xp f; K ða ðgþþ is th xprimntal hat of formation of atom A in gasous stat at K; DH xp 298 K ða ðsþþ is th xprimntal nthalpy chang of lmnt A in its standard stat from to K. Eq. (3) ffctivly corrcts th unbalancd lctron corrlation nrgis in atoms and in molcul unaccountd by a thortical mthod [19]. In ordr to dtrmin th rlativ importanc of a physical dscriptor, its partial corrlation cofficint, V j is calculatd qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V j ¼ 1 q=q j ; ð4þ in which q is th squar sum of dviations and Q j is th squar sum of dviations laving out on dscriptor x j. Th closr V j is to 1, th mor rmarkabl is th influnc of x j. 3. Computational mthods Th nlargd training st contains 35 hats of formation: 222 ar from G3/99 tst st [12], and th rmaining ar from th training st mployd in our prvious works [18,19]. Gomtry optimization and vibrational frquncy analyss ar carrid out at th /6-31G(d),

3 X.-M. Duan t al. / Chmical Physics Lttrs 49 (25) /6-31G(d) and /6-311G(2d,d,p) lvls of thoris. Th mor balancd 6-311G(2d,d,p) basis st as mployd in th modifid complt basis st modl (CBS-QB3) [26] is usd in th prsnt Lttr, which includs two sts of d polarization functions on lmnts byond th first row, on st of d polarization function on th first row lmnts, and on st of p polarization function on hydrogn as wll. Th scaling factors for ZPEs ar.9135,.986 and.99, and thos for calculating DH calc 298 KðMÞ ar.895,.9989 and.99, for th /6-31G(d), /6-31G(d) and /6-311G(2d,d,p) mthods, rspctivly [27,26]. Th NBO analysis is prformd with NBO 3.1 [28] as implmntd in th GAUSSIAN 3 packag of programs [29]. Attntions must b paid in th NBO calculations for radicals and strongly dlocalizd molculs. For xampl, for radicals NO, ClO, CN, HCO, CH 3 CO and H 2 COH th SCHOOSE kyword is ndd to spcify th corrct bonding pattrn, and for bnzn and substitutd bnzn molculs th RESONANCE kyword should b usd. All th calculations hav bn don with th GAUSSIAN 3 suit of programs [29]. 4. Rsults and discussion 4.1. Linar rgrssion corrction of th hats of formation Thortical rsults compard to thir xprimntal countrparts ar illustratd in Fig. 1. Raw calculatd DH h f s by th mthod hav hug rrors and th rrors incrasd with th molcul siz [19]. Th maximal dviation of /6-31G(d) rachs to kcal/mol for C 1 H 18 O 4. Th MAD is kcal/mol for th /6-31G(d) mthod. In comparison, /6-31G(d) and /6-311+G(2d,d,p) prform much bttr than th mthod, sinc DFT mthods hav alrady involvd som lctron corrlation corrction, but mthods hav not. From Fig. 1b and c, w can s that most raw calculatd DH h f s from DFT mthods ar largr than xprimntal valus. Som rsults vn hav larg rrors. For xampl, th raw calculatd DH h f s of Cl 2 O 2 S, SO 3, PCl 5, C 2 H 6 O 2 S, and POCl 3 molculs ar about 4 kcal/mol abov xprimntal valus for th /6-31G(d) mthod. For th /6-311G (2d,d,p) mthod, th rrors of th calculatd DH h f sof SF 6 and PF 5 molculs vn com to 6.3 and 5.1 kcal/mol, rspctivly. Compard to th xprimntal masurmnts, th MAD is 8.2 kcal/mol for th /6-31G(d) mthod, and 12.4 kcal/mol for th /6-311G(2d,d,p) mthod. It is obvious that thr ar systmatic dviations btwn th calculatd and xprimntal DH h f valus. To rduc such dviations, Eq. (3) mploying th nin dscriptors as prviously dscribd is usd to corrct th hats of formation. Fifty molculs out of 35 molculs ar st asid randomly as tsting st, and th rst ar training st. Th 3 training molculs ar randomly dividd into six substs of qual siz. Fiv of thm ar usd to train th linar rgrssion approach, and th sixth to validat its prdictions. This procdur is rpatd six tims in rotation. Th MAD and th RMS dviations ar listd in Tabl 1. Th linar rgrssion cofficints obtaind for th dscriptors ar listd in Tabl 2. Upon linar rgrssion corrction th dviations ar all substantially dcrasd. For th /6-31G(d) mthod, th MAD is rducd from to 7.3 kcal/mol, by mor than 3 tims. For th /6-31G(d) mthod, th MAD is rducd from 8.2 to 3.3 kcal/mol, and for th /6-311G(2d,d,p) mthod, it is rducd from 12.4 to 2.7 kcal/mol. For th DFT mthods, th dviation using th smallr 6-31G(d) basis st is comparabl to th largr 6-311G(2d,d,p) basis st upon linar rgrssion corrction. This implis that th truncation rror in th basis st can b gratly corrctd by th LRC approach. Thus, th smallr 6-31G(d) basis st can b usd in our schm to sav mounts of computr tim for larg molculs. A comparison btwn th linar rgrssion corrctd DH h f s and th xprimntal valus ar illustratd in Fig. 1d f. Th figurs clarly show that th corrctd rsults ar much closr to thir xprimntal countrparts for both th training and tsting sts. Th surprising ffcts of th corrction indicat that quantum chmical calculation rsults can b gratly improvd by linar rgrssion corrction approach. Morovr, whil uncorrctd calculations yild wors rsults for larg molculs compard to small ons, our LRC approach dos not discriminat against larg molculs [19]. Th dviations of larg molculs ar of th sam magnitud as thos of small molculs. Th histograms for th dviations (from th xprimnts) as shown in Fig. 2 furthr dmonstrat that our LRC approach gratly dcrass th larg systmatic calculation rrors of th DFT mthods. Tabl 3 lists th partial corrlation cofficints of all dscriptors. Examination of th data indicats that bonding lctrons ar vry important for lctron corrlation corrction for th thr mthods. Th larg partial corrlation valus of th innr-layr lctrons indicat that th changs of lctron corrlation from atoms to molculs ar larg and non-ngligibl. Gnrally, th closr th lctrons to th nuclus, th lss important ar thir contributions in chmical ractions. Th larg partial corrlation valus of RY* imply that th contribution of Rydbrg orbitals ar also important. In th NBO analysis, th NBOs ar partitiond into highand low-occupancy orbital typs. Th high-occupancy orbitals (BD, CR, LP) play th primary rol in dtrmining th proprtis of molculs. Th low-occupancy orbitals (BD*, LP* and RY*) ar consisting of th rmaining (formally unoccupid) orbitals. In th molc-

4 318 X.-M. Duan t al. / Chmical Physics Lttrs 49 (25) (a) 1 (d) /6-31(d) -3-4 /6-31G(d) training st validation st tsting st Calculatd Hf Θ (298K) (kcal/mol) (b) /6-31G(d) () /6-31G(d) training st validation st tsting st 1 (c) 1 (f) /6-311G(2d,d,p) Exprimntal H f Θ (298K) (kcal/mol) /6-311G(2d,d,p) training st validation st tsting st Fig. 1. Exprimntal vs. calculatd DH h f for all 35 molculs: (a) (c) raw calculatd rsults; (d) (f) linar rgrssion corrctd rsults. Tabl 1 Th MAD and RMS dviations (kcal/mol) bfor and aftr linar rgrssion corrction B-MAD a A-MAD b B-RMS a A-RMS b a Bfor corrction. b Aftr corrction. ular nvironmnt, thir occupancis ar not ncssarily zro. Ths xtra-valnc shll orbitals play th scondary rol in dscribing th lctron dnsity associatd Tabl 2 Optimizd cofficints (kcal/mol) of th dscriptors mployd in Eq. (3) BD(a 1 ) LP(a 2 ) CR1(a 3 ) CR2(a 4 ) CR3(a 5 ) RY*(a 6 ) BD*(a 7 ) LP*(a 8 ) Unpaird(b 1 )

5 X.-M. Duan t al. / Chmical Physics Lttrs 49 (25) (a) 6-31G(d) (b) 6-311G(2d,d,p) Frquncy 2 (c) 6-31G(d) (d) 6-311G(2d,d,p) Dviations (kcal/mol) Fig. 2. Histograms for th dviations of th mthods bfor corrction (a, b) and aftr corrction (c, d). Tabl 3 Partial corrlation cofficints of th dscriptors BD(a 1 ) LP(a 2 ) CR1(a 3 ) CR2(a 4 ) CR3(a 5 ) RY*(a 6 ) BD*(a 7 ) LP*(a 8 ) Unpaird(b 1 ) Tabl 4 Th MADs of linar rgrssion corrction for diffrnt dscriptors (kcal/mol) I II III IV V I: BD, LP, RY*, BD*, LP*, and unpaird lctrons of atom as dscriptors. II: BD, LP, CR1, CR2, CR3, and unpaird lctrons of atom as dscriptors. III: BD, LP, CR1, CR2, CR3, RY*, BD*, and LP* as dscriptors. IV: BD, LP, th sum of innr layr, RY*, BD*, LP*, and unpaird lctrons of atom as dscriptors. V: BD, LP, CR1, CR2, CR3, RY*, BD*, LP*, and unpaird lctrons of atom as dscriptors. with th atom. Th unpaird lctrons of atom hav a larg ffct on th /6-31G(d) and /6-311G(2d,d,p) calculation rsults. For both mthods th partial corrlation cofficint rachs to.99. W hav also tstd th rlativ importanc of a dscriptor by laving it out and r-training th rst. Th rsults ar listd in Tabl 4. It is th common sns that lctron corrlation nrgy of th innr-layr lctrons may not chang too much whn forming chmical bonds. On th contrary, our rsults show that th corrlation nrgy changs of th innr-layr lctrons from atoms to molcul also hav significant contribution to th ovrall corrlation nrgy changs. Without innrlayr-lctron dscriptors, th MADs incras from 7.3 to 12. kcal/mol and from 3.3 to 4.4 kcal/mol for th /6-31G(d) and /6-31G(d) mthods, rspctivly. Whn taking th sum of thr innr-layr lctrons as on dscriptor, th MADs rach to 8.8 and 4. kcal/mol, rspctivly. Ths indicat that lctron corrlation nrgy changs of diffrnt innr-layr lctrons ar not th sam and non-ngligibl whn forming a bond. On th contrary, for th /6-311G(2d,d,p) mthod, th MADs chang littl with th absnc of th thr innr-layr-lctron dscriptors, or including thir sum. Unpaird lctrons of th atoms also hav rmarkabl contributions to th corrction of corrlation nrgy at th lvl. Without unpaird-lctron dscriptor, th MAD of th mthod incrass by 6.1 kcal/mol. This rflcts a larg discrpancy in th corrlation nrgy of th unpaird lctrons bfor and aftr forming bonds at th lvl. On th contrary, for th two DFT mthods, th ffct of unpaird lctrons is not so significant. Howvr, without lowoccupancy-orbital dscriptors (RY*, BD*, and LP*), th MADs of th thr mthods all incras gratly,

6 32 X.-M. Duan t al. / Chmical Physics Lttrs 49 (25) Tabl 5 Classical barrir hights (kcal/mol) at th /6-311G(2d,d,p) lvl a Ractions Bst-stimat Raw calculation -LRC V 6¼ f V 6¼ r V 6¼ f V 6¼ r V 6¼ f V 6¼ r OH + CH 4! CH 3 +H 2 O CH 3 +NH 2! CH 4 + NH C 2 H 5 +NH 2! C 2 H 6 + NH C 2 H 6 +NH 2! C 2 H 5 +NH CH 4 +NH 2! CH 3 +NH s-trans, cis-c 5 H 8! s-trans, cis-c 5 H a V 6¼ f for th forward dirction and V 6¼ r for th rvrs dirction. which implying that th ffcts of low-occupancy orbitals cannot b ignord. Although th ovrall rsults of th LRC approach ar quit satisfactory for all thr mthods, th dviations for non-hydrogn systms ar still far byond th man absolut dviation. Thr ar 53 non-hydrogn molculs in th total 35 molculs. Th MADs for this typ of molculs ar 12.7, 5.8, and 5.3 kcal/mol for th / 6-31G(d), /6-31G(d), and /6-311G(2d, d,p) mthods, rspctivly. For th two DFT mthods, th dviations of PF 5 and SF 6 ar abov 2 kcal/mol. For th mthod, th dviations for molculs containing fluorin lmnt ar spcially larg, such as for ClF 3, F 2 O, SF 6 and C 2 F 6. Th larg dviations may b arousd by th inaccurat gomtris optimizd at ths lvls of thoris and othr factors unconsidrd in th prsnt Lttr, such as spin orbital coupling and rlativistic ffcts [1,12]. Th currnt LRC approachs using NBO dscriptors furthr dcras th calculation rror of a quantum mchanical mthod compard with th original approachs using lctron pairs. Aftr rtraining using th sam dscriptors as in our prvious work [19], th MADs for th 35 molculs ar 7.4, 3.8 and 3.2 kcal/ mol for th /6-31G(d), /6-31G(d) and /6-311G(2d,d,p) mthods, rspctivly. Compard to th rsults of NBO dscriptors (7.3, 3.3 and 2.7 kcal/mol), th dviations ar all still largr Tsting of th raction barrir hights Raction barrir hights can now b corrctd using th currnt LRC approach. This is achivd by substituting th fittd cofficints back to Eqs. (1) and (2) to calculat nrgis of molculs and atoms and thn us thm to calculat raction barrirs. It should b noticd that our LRC approach is dsignd to obtain rlativly accurat nrgy of a molcul rlativ to its composing atoms and th slctd dscriptors charactriz molculs bst, whil only on dscriptor (unpaird lctrons of atom) is usd for atoms. Consquntly, th corrction cofficints obtaind from this approach ar most suitabl for molcular systms. Thr ar ight chmical ractions involving no atoms in th BH42/3 raction databas [3,31]. Howvr, two ractions of thm hav ngativ raction barrirs du to th wll-known slfintraction problm of th DFT mthods. In th nd w just choos th rmaining six ractions as our tsting sts. Th rsults prsntd in Tabl 5 ar obtaind at th /6-311G(2d,d,p) lvl. Th raw calculatd barrir hights ar all lowr than th givn bst stimats [3,31], which is typical for most DFT mthods. Aftr linar rgrssion corrction, thr is an obvious incras for vry barrir hight. Th corrctd barrir hights ar all closr to th bst-stimatd valus. Th MAD for th 12 barrir hights is rducd from 5.3 to 2.9 kcal/mol, almost to th sam accuracy as that for th hats of formation. It should b clarifid that barrir hights of ractions involving atoms can also b improvd, but th improvmnt is not as rmarkabl as for thos involving only molculs. 5. Conclusions Th training st of 18 hats of formation usd in our prvious work is xpandd to 35 hats of formation including 222 hats of formation in th G3/99 tst st (with duplicatd molculs dltd). Th nw st includs 35 hats of formation of small- and mdium-sizd organic, inorganic molculs, and radicals. At th sam tim, nw dscriptors obtaind by th NBO analysis ar mployd in this Lttr. Th dscriptors ar th lctron populations of diffrnt typs of NBOs: two-cntr bonds (BD), on-cntr cor pair (CR), on-cntr valnc lon pair (LP), on-cntr Rydbrg (RY*), two-cntr anti-bond (BD*), and non-lwis valnc lon pair (LP*). Th numbr of th unpaird lctrons of atom in ground stat is also includd as th dscriptor. Aftr linar rgrssion corrction, th MADs of th calculatd DH h f s for th 35 molculs ar gratly rducd from to 7.3 kcal/mol for th /6-31G(d) mthod. In th man tim, th calculatd DH h f of th DFT mthods hav also bn improvd. Th MADs ar rducd from 8.2 to 3.3 kcal/mol and from 12.4 to 2.7 kcal/mol for th /6-31G(d) and /6-311G(2d,d,p)

7 X.-M. Duan t al. / Chmical Physics Lttrs 49 (25) mthods, rspctivly. Th currnt approach is not only an improvmnt ovr th prvious approach on calculating hats of formation, but also is suitabl for th calculation of raction barrirs. Th MAD for 12 barrir hights is rducd from 5.3 to 2.9 kcal/mol. Acknowldgmnts This work was supportd by th National Natural Scinc Foundation of China Grant Nos. ( and 24333) and th Natural Scinc Foundation of Shanghai Scinc and Tchnology Committ Grant No. (2DJ1423). Rfrncs [1] A.C. Hurly, Elctron Corrlation in Small Molculs, Acadmic Prss, Nw York, [2] S. Wilson, Elctron Corrlation in Molculs, Clarndon Prss, Oxford, [3] K. Raghavachari, Annu. Rv. Phys. Chm. 42 (1991) 615. [4] D.R. Yarkony (Ed.), Modrn Elctronic Structur Thory, World Scintific, Singapor, [5] I. Shavitt, in: H.F. Schafr (Ed.), Modrn Thortical Chmistry, vol. 3, Plnum Prss, Nw York, 1977, p [6] R.J. Bartltt, J. Phys. Chm. 93 (1989) [7] L.A. Curtiss, K. Raghavachari, G.W. Trucks, J.A. Popl, J. Chm. Phys. 94 (1991) [8] L.A. Curtiss, J.E. Carpntr, K. Raghavachari, J.A. Popl, J. Chm. Phys. 96 (1992) 93. [9] L.A. Curtiss, K. Raghavachari, J.A. popl, J. Chm. Phys. 16 (1995) [1] L.A. Curtiss, K. Raghavachari, P.C. Rdfrn, V. Rassolov, J.A. Popl, J. Chm. Phys. 19 (1998) [11] A.G. Baboul, L.A. Curtiss, P.C. Rdfrn, K. Raghavachari, J. Chm. Phys. 11 (1999) 765. [12] L.A. Curtiss, K. Raghavachari, P.C. Rdfrn, J.A. Popl, J. Chm. Phys. 112 (2) [13] A.D. Bck, J. Chm. Phys. 98 (1993) [14] C. L, W. Yang, R.G. Parr, Phys. Rv. B 37 (1988) 785. [15] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (198) 12. [16] P.J. Stphns, F.J. Dvlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chm. 98 (1994) [17] P.C. Rdfrn, P. Zapol, L.A. Curtiss, K. Raghavachari, J. Phys. Chm. A 14 (2) 585. [18] L.H. Hu, X.J. Wang, L.H. Wong, G.H. Chn, J. Chm. Phys. 119 (23) [19] X.M. Duan, G.L. Song, Z.H. Li, X.J. Wang, G.H. Chn, K.N. Fan, J. Chm. Phys. 121 (24) 786. [2] X.J. Wang, L.H. Wong, L.H. Hu, C.Y. Chan, Z.M. Su, G.H. Chn, J. Phys. Chm. A 18 (24) [21] J.P. Fostr, F. Winhold, J. Am. Chm. Soc. 12 (198) [22] A.E. Rd, F. Winhold, J. Chm. Phys. 78 (1983) 466. [23] A.E. Rd, R.B. Winstock, F. Winhold, J. Chm. Phys. 83 (1985) 735. [24] A.E. Rd, F. Winhold, J. Chm. Phys. 83 (1985) [25] A.E. Rd, L.A. Curtiss, F. Winhold, Chm. Rv. 88 (1988) 899. [26] J.A. Montgomry Jr., M.J. Frisch, J.W. Ochtrski, G.A. Ptrsson, J. Chm. Phys. 11 (1999) [27] A.P. Scott, L. Radom, J. Phys. Chm. 1 (1996) [28] E.D. Glndning, A.E. Rd, J.E. Carpntr, F. Winhold, NBO, Vrsion 3.1. [29] M.J. Frisch t al., GAUSSIAN 3, Rvision B.3, Gaussian, Inc., Pittsburgh, PA, 23. [3] Y. Zhao, B.J. Lynch, D.G. Truhlar, J. Phys. Chm. A 18 (24) [31] Y. Zhao, B.J. Lynch, D.G. Truhlar, J. Phys. Chm. A 18 (24) 4786.