Accurate Analytic Potential Energy Function and Spectroscopic Study for G 1 Π g State of Dimer 7 Li 2

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1 Commun. Thor. Phys. (Bijing, China) 47 (2007) pp c Intrnational Acadmic Publishrs Vol. 47, No. 6, Jun 15, 2007 Accurat Analytic Potntial Enrgy Function and Spctroscopic Study for G 1 Π g Stat of Dimr 7 Li 2 SHI D-Hng, 1,2, MA Hng, 2 SUN Jin-Fng, 2 and ZHU Zun-Lu 2 1 Collg of Physics & Elctronic Enginring, Xinyang Normal Univrsity, Xinyang , China 2 Collg of Physics & Information Enginring, Hnan Normal Univrsity, Xinxiang , China (Rcivd Jun 26, 2006) Abstract Th rasonabl dissociation limit for th G 1 Π g stat of dimr 7 Li 2 is dtrmind. Th quilibrium intrnuclar distanc, dissociation nrgy, harmonic frquncy, vibrational zro nrgy, and adiabatic xcitation nrgy ar calculatd using a symmtry-adaptd-clustr configuration-intraction mthod in complt activ spac in Gaussian03 program packag at such numrous basis sts as G, G(2df,2pd), G(2df,p), cc-pvtz, G(3df,3pd), CEP-121G, G(2df,pd), G(d,p),6-311G(3df,3pd), D95(3df,3pd), G(3df,2p), G(2df), G(df,pd) D95V++, and DGDZVP. Th complt potntial nrgy curvs ar obtaind at ths sts ovr a wid intrnuclar distanc rang and hav last squars fittd to Murrll Sorbi function. Th conclusion shows that th basis st G(2df,p) is a most suitabl on for th G 1 Π g stat. At this basis st, th calculatd spctroscopic constants T, D, E 0, R, ω, ω χ, α, and B ar of V, V, cm 1, nm, cm 1, cm 1, cm 1, and cm 1, rspctivly, which ar in good agrmnt with masurmnts whnvr availabl. Th total 50 vibrational lvls and corrsponding inrtial rotation constants ar for th first tim calculatd and compard with availabl RKR data. And good agrmnt with masurmnts is obtaind. PACS numbrs: Df, Ar Ky words: vibrational frquncy, dissociation nrgy, Li 2, ab initio calculations 1 Introduction On of th most important problms in th fild of atomic and molcular physics is to obtain rliabl physical modls of bonding potntials so as to thoroughly undrstand molcular spctroscopic proprtis, sinc quantitativ molcular spctroscopic knowldg can b drivd from th molcular analytic potntial nrgy function (APEF). [1] Nowadays, with th dvlopmnt of modrn quantum chmical mthods and powrful computrs, ab initio calculations can giv mor and mor accurat potntial nrgy curv (PEC) with srial points, spcially for small molculs. And it is availabl to fit ths ab initio data using a suitabl analytic function with a root-man squar rror (RMSE) lowr than th chmical accuracy (1.0 Kcal/mol). [2] Thus, thr is a long-standing intrst in simpl yt rliabl analytical modls of chmical potntials, which should idally consist of physically idntifiabl trms and which should b basd on a limitd numbr of availabl thortical data. Li 2 is th simplst stabl homonuclar diatomics, nxt to H 2. Numrous thortical and xprimntal invstigations [3 13] hav bn publishd about its ground and low-lying xcitd singlt and triplt stats. Howvr, studis on th highly xcitd stat G 1 Π g can hardly b found in th litraturs. [14 16] Th most rcnt xprimnts hav bn mad by Brnhim t al., [16] who publishd two possibl dissociation nrgis, on is 6390 cm 1 if th rasonabl dissociation limit corrsponds to (2 2 P )+ (2 2 P ), th othr is 7507 cm 1 if th rasonabl dissociation limit corrsponds to (2 2 P ) + (3 2 S). Ths valus wr basd on th most likly ground potntial valu of 8450 cm 1 givn by Konowalow t al. [17] Th most rcnt calculations hav bn prformd by Potau t al., [15] who only calculatd th quilibrium intrnuclar sparation R, dissociation nrgy D, harmonic frquncy ω, and adiabatic xcitation T. It is a pity that thy did not gav any othr information, such as th spctroscopic constants ω χ, E 0, and B, vibrational lvls G(ν), and inrtial rotation constants B ν. And no APEF can b found in th publications to dat. In this papr, w attmpt to calculat th accurat APEF and driv th main molcular spctroscopic proprtis. Compard with prvious thoris, this work shows xcllnt agrmnt with xprimnts, [16] and is mor complt ffort than prvious thoris. Thus, it is ncouraging. In this papr, th main spctroscopic constants, total 50 vibrational lvls and corrsponding inrtial rotation constants of th G 1 Π g stat for dimr 7 Li 2 ar calculatd using a symmtry-adaptd-clustr configurationintraction (SAC-CI) mthod [18,19] in full activ spac in Gaussian03 program packag. [19] In th nxt sction, w dscrib in dtail th dissociation limit and rcalculat th xprimntal dissociation nrgy for this stat. In Sc. 3, Th projct supportd by National Natural Scinc Foundation of China undr Grant No Corrsponding author, scattring@sina.com.cn

2 No. 6 Accurat Analytic Potntial Enrgy Function and Spctroscopic Study for G 1 Π g Stat of Dimr 7 Li w prsnt th computd rsults and mak som usful discussion about thm. Concluding rmarks ar mad in Sc Dissociation Limit for th G 1 Π g Stat Obviously, th APEF is diffrnt whn an lctronic stat is dissociatd in diffrnt channls. In ordr to corrctly calculat and dscrib th APEF of an lctronic stat, w must dtrmin its rasonabl dissociation limit. Th configuration of th ground Li atom is 1s 2 2s 1. Whn on lctron in 2s orbital is xcitd, th configurations 1s 2 2p 1, 1s 2 3s 1, 1s 2 3p 1, 1s 2 4s 1, 1s 2 3d 1 and so on can b formd. Whn an atom is in th configuration 1s 2 2s 1, 1s 2 3s 1 or 1s 2 4s 1, its atomic group rprsntation is 2 S g. And whn an atom is in th configuration 1s 2 2p 1 or 1s 2 3p 1, its atomic group rprsntation is 2 P u. If two Li atoms in th dissociation limit ar both in th ground stat, basd on th group thory and atomic and molcular raction statics, [20] th rprsntation 2 S g can b rsolvd into thos of D h (Li 2 ) as follows: Thir dirct product and rduction ar 2 S g 2 Σ + g. (1) 2 Σ + g 2 Σ + g 1 Σ + g 3 Σ + u. (2) It is obvious that quation (2) contains 1 Σ + g. According to th principl of rvrsibility for th microscopic procss, [20] th dissociation limit for th ground stat X 1 Σ + g may b Li 2 (X 1 Σ + g ) Li( 2 S g ) + Li( 2 S g ). (3) Whn two Li atoms in th dissociation limit ar both in th configuration 1s 2 2p 1, basd on th abov-mntiond rason, th rprsntation 2 P u is rsolvd into thos of D h (Li 2 ) as follows: 2 P u 2 Σ + u 2 Π u. (4) Thir dirct product and rduction ar 2 Σ + u 2 Π u ( 2 Σ + u 2 Π u ) 1 Σ + g (2) 1 Σ u 1 Π g 1 Π u 1 g 3 Σ + u (2) 3 Σ g 3 Π g 3 Π u 3 u. (5) Equation (5) contains 1 Π g. According to th principl of rvrsibility for th microscopic procss, [20] th dissociation limit for th G 1 Π g stat may b Li 2 (G 1 Π u ) Li( 2 P u ) + Li( 2 P u ). (6) Whn on Li atom in th dissociation limit is in th 1s 2 2s 1 and th othr in th 1s 2 2p 1, sinc 2 S g 2 Σ + g, 2 P u 2 Σ + u 2 Π u and 2 Σ + g 2 Σ + u 2 Π u ) 1 Σ + u 3 Σ + u 1 Π u 3 Π u, according to th principl of rvrsibility for th microscopic procss, [20] th dissociation limit may also b Li 2 (G 1 Π u ) Li( 2 S g ) + Li( 2 P u ). (7) Fig. 1 Potntial nrgy curvs of th ground and xcitd stats for a molcul. Fig. 2 PEC of th xcitd dimr Li 2. As shown in Fig. 1, th atomic xcitation nrgy E a for an xcitd stat of a molcul quals, E a = D + T D 0. (8) Hr, D is th dissociation nrgy for a givn xcitd stat. D 0 is th dissociation nrgy for th ground stat and quals 8517 cm 1 (about V) for dimr 7 Li 2, which has takn into considration th vibrational zro nrgy. [21] T is th adiabatic xcitation nrgy from th ground to a givn stat. And E a is th atomic xcitation nrgy sum

3 1116 SHI D-Hng, MA Hng, SUN Jin-Fng, and ZHU Zun-Lu Vol. 47 of th sparatd atoms for a givn xcitd stat in th dissociation limit whn th atomic xcitation nrgy sum of th sparatd atoms in th dissociation limit for th ground stat is st to zro. It has bn provd that quation (1) is th Li 2 ground-stat dissociation limit, in which th two sparatd atoms ar both in th ground stat. [22] According to Tabl 1, [23] th xcitation nrgy sum of th two Li atoms in th dissociation limit quals zro. Tabl 1 Enrgy lvls of two lithium atoms in svral lctronic configurations. [23] Configuration Excitation nrgy (cm 1 ) Configuration Excitation nrgy (cm 1 ) (1s 2 2s 1 ) + (1s 2 2s 1 ) 0 (1s 2 2s 1 ) + (1s 2 3s 1 ) (1s 2 2s 1 ) + (1s 2 2p 1 ) (1s 2 2p 1 ) + (1s 2 2p 1 ) (1s 2 2s 1 ) + (1s 2 3p 1 ) (1s 2 2s 1 ) + (1s 2 3d 1 ) (1s 2 2p 1 ) + (1s 2 3p 1 ) Th bst calculatd T and D valus for th stat G 1 Π g in this invstigation ar obtaind at th basis st G(2df,p), which ar V and V, rspctivly. Thrfor, E a = V ( cm 1 ) can b obtaind. According to Tabl 1, w conclud that th two Li atoms in th dissociation limit for this stat must b both in th configuration 1s 2 2p 1. Thus th rasonabl dissociation limit for th G 1 Π g stat must b Li 2 (G 1 Π u ) Li( 2 P u ) + Li( 2 P u ). (9) In th nxt, w rcalculat th xprimntal G 1 Π g stat dissociation nrgy according to th most rcnt ground stat dissociation nrgy [21] and th G 1 Π g stat T valu (about cm 1 ) masurd in Rf. [16]. In th light of Eq. (8), D = 6457 cm 1 (about V) is attaind. 3 Rsults and Discussion Th calculations dscribd hr ar prformd in Gaussian03 program packag. [19] By gomtry optimization (OPT) calculations for th G 1 Π g stat, w hav attaind th R valus at such basis sts as D95(3df,3pd), G(2df,2pd), 6-311G(3df,3pd), G(3df,2p), DGDZVP, G(2df,p), CEP-121G, G(2df,pd), G(3df, 3pd), G(2df), G(df,pd), G(d,p), D95V++, G, and cc-pvtz. Som of th rsults ar tabulatd in Tabl 2. At th sam tim, by fin singl-point nrgy scanning (SPES) calculations nar th xprimntal quilibrium position at th sam basis st at a vry tiny intrval of 0.01 a 0, w hav obtaind th R valus, too. Som of th rsults ar also tabulatd in Tabls 2 and 3. Obviously, from Tabl 2 w can s that th two approachs, namd OPT and SPES, giv diffrnt quilibrium intrnuclar sparations at th sam basis st. It is asily undrstood, bcaus th uniqu GSUM algorithm usd in th SPES calculations is incompltly idntical with th on usd in th OPT computations. [18,19] It is th rason that th rsult obtaind by SPES calculations is quit intgratd into PEC, and all th spctroscopic proprtis including th quilibrium intrnuclar sparation can b drivd from APEF, thus th rsult obtaind by th SPES calculations should b mor rasonabl. Tabl 2 Comparison of quilibrium intrnuclar sparations (in nm) obtaind by OPT and by SPES calculations for th G 1 Π g stat of dimr 7 Li (df,pd) G(2df,2pd) G G(2df,pd) G(3df,2p) G(2df,p) OPT SPES Thn, w comput th PECs for this stat at th abov-mntiond basis sts through th quilibrium positions obtaind by SPES at 0.3 a 0 intrvals. In ordr to guarant PEC convrgnc, th calculatd intrnuclar sparation rang should b larg nough at ach basis st. Murrll Sorbi (M-S) function is a widly usd potntial nrgy function, whos form is [24] n V (ρ) = D (1 + a i ρ i) xp( a 1 ρ), (10) i=1 whr ρ = R R, R is th intrnuclar distanc of diatomics. R is rgardd as a fixd paramtr in th fitting procss, which is attaind by fin SPES calculations in this papr. Th paramtrs a i ar dtrmind by th fitting mthod using ab initio data.

4 No. 6 Accurat Analytic Potntial Enrgy Function and Spctroscopic Study for G 1 Π g Stat of Dimr 7 Li Tabl 3 Equilibrium constants for th G 1 Π g stat of 7 Li 2. Sourc T (V) D (V) R (nm) ω (cm 1 ) E 0 (cm 1 ) Exprimnts [16] G(2df,p) G(2df,2pd) G(df,pd) G(2df,pd) G(3df,2p) G(3df,3pd) G G(2df) G(d,p) D95V G(3df,3pd) cc-pvtz CEP-121G D95(3df,3pd) DGDZVP By itrating a systm of normal quations basd on a last-squars fitting, th paramtrs a i and D in Eq. (10) ar fittd at various basis sts. In ordr to attain satisfactory rsults, w try it from n = 3 to n = 8, and find th bst rsults for n = 6. From th fitting rsults, th quadratic forc constant f 2 is calculatd, f 2 = D (a 2 1 2a 2 ). (11) According to th RKR mthod, w hav ω = f 2 8πcµR 2, (12) whr µ and c ar th rducd mass of dimr 7 Li 2 and th vlocity of light in vacuum, rspctivly. Th calculatd ω valus ar listd in Tabl 3, too. Tabl 3 tabulats T, D, R, ω, and th vibrational zro nrgy E 0 of th G 1 Π g stat at various basis sts. From ths rsults w find that th basis st G(2df,p) [25,26] is an xcllnt on, sinc th calculatd T, R, and ω valus at this basis st ar in agrmnt with th masurmnts [16] within V or 0.015%, nm or 0.003%, and 1.3 cm 1 or 0.567%, rspctivly, though D is somwhat largr than th masurmnts [16] by nm or 1.579%. Thus, furthr calculations will b prformd using th APEF obtaind at th basis st G(2df,p). Tabl 4 Paramtrs of M-S APEF for 7 Li 2(G 1 Π g) at th SAC-CI/ G(2df,p) lvl of thory. D (V) R (nm) a 1 (nm 1 ) a 2 (nm 2 ) a 3 (nm 3 ) a 4 (nm 4 ) a 5 (nm 5 ) a 6 (nm 6 ) RMSE (V) Th APEF paramtrs at G(2df,p) ar all tabulatd in Tabl 4 for intgrality. At th sam tim, in ordr to invstigat th PEC dtails of this stat, th ab initio data ar tabulatd in Tabl 5, and th fitting rsults and th curv of th ab initio calculatd points ovr th intrnuclar sparation rang from about 2.4a 0 to 37a 0 ar intuitivly illustratd in Fig. 2, too. In ordr to valuat th fitting quality of APEF at G(2df,p), w calculat th RMSE, RMSE = 1 N (V APEF V ab initio ) N 2, (13) i=1 whr V APEF and V ab initio ar nrgis attaind by th fitting and ab initio calculations, rspctivly. N is th numbr of fittd points (hr N = 116). Th prsnt RMSE for th G 1 Π g stat is only V ( 0.06 Kcal/mol). Obviously, our fitting accuracy about th APEF is gratly suprior to th chmical accuracy (1.0 Kcal/mol). [2] Thus, th APEF of th G 1 Π g stat is crdibl.

5 1118 SHI D-Hng, MA Hng, SUN Jin-Fng, and ZHU Zun-Lu Vol. 47 Tabl 5 Potntial nrgis E(R) at diffrnt intrnuclar sparations R for th G 1 Π g stat at SAC-CI/ G(2df,p) lvl of thory. R (nm) E(R) (Hartr) R (nm) E(R) (Hartr) R (nm) E(R) (Hartr) Basd on th following Eqs. (14) (18), th forc constants f 3, f 4 and thn th spctroscopic constants B, α, and ω χ ar calculatd. Th calculatd spctroscopic rsults ar tabulatd in Tabl 6. For convnint comparison, w also tabulat prsnt T, R, D, and ω valus in Tabl 6 togthr with th masurmnts [16] and othr thoris. [14,15] ( ) f 3 = 6D a 3 a 1 a 2 + a3 1, (14) 3 f 4 = D (3a a 2 1a a 1 a 3 ), (15) h B = 8πcµR 2, (16) ( α = 6B2 f3 R ) + 1, (17) ω 3f 2 ω χ = B [ f 4R 2 ( ω ) α 2 ] 8 f 2 6B 2. (18) Hraftr, w calculat th vibrational zro nrgy E 0 at various basis sts (th calculatd E 0 rsults ar tabulatd in Tabl 3), E 0 = 1 2 ω 1 4 ω χ. (19) Tabl 6 Comparison with masurmnts and othr thoris about T, R, D, ω, ω χ, B, and α for dimr 7 Li 2 (G 1 Π g) at SAC-CI/ G(2df,p) lvl of thory. Sourc T (V) R (nm) D (V) ω (cm 1 ) ω χ (cm 1 ) B (cm 1 ) α (cm 1 ) This work Exp. [16] Thory [14] Thory [15] From Tabl 6, w find that th bst R, D, and ω valus of prvious thoris ar prsntd by Potau t al. [15] Thir discrpancis dviatd from th masurmnts [16] ar % for R, 0.914% for D, and 0.07% for ω, rspctivly. Ths discrpancis ar quivalnt to ours as a whol. Sinc w prsnt th complt APEF for th first tim and furthr calculat othr main spctroscopic constants ω χ, B, α, E 0, total 50 vibrational lvls and th corrsponding inrtial rotation constants, thus w say that our calculations ar mor complt than prvious thoris and rprsnt an improvmnt. Howvr, w cannot carry out any comparison for α and E 0, bcaus no masurmnts and thoris can b found in th litraturs to th bst of our knowldg. Bsids, w considr that th ω χ valu givn in Rf. [14] is possibly unrliabl, sinc thir R, D, and ω valus ar gratly dviatd from th masurmnts. [16] Now w comput th vibrational lvl G(ν) and th corrsponding inrtial rotation constant B ν by solving th following radial Schrödingr quation of nuclar motion in th adiabatic approximation, [ h2 2µ d 2 dr 2 + h2 2µr 2 J(J + 1) + V (r) ] Ψ ν.j (r) = E ν,j Ψ ν.j (r). (20)

6 No. 6 Accurat Analytic Potntial Enrgy Function and Spctroscopic Study for G 1 Π g Stat of Dimr 7 Li Hr V (r) is th adiabatic rotationlss potntial nrgy function tabulatd in Tabl 4. ν and J ar th vibrational and rotational quantum numbrs, rspctivly. Th rotational sublvls of a givn vibrational lvl ar rprsntd by th following powr sris, [27] E ν,j = G(ν) + B ν [J(J + 1)] D ν [J(J + 1)] 2 + H ν [J(J + 1)] 3 + L ν [J(J + 1)] 4 + (21) Tabl 7 Th first 21 vibrational lvls, inrtial rotation constants and th corrsponding comparison with availabl RKR data for th 7 Li 2 (G 1 Π g) (J = 0) stat at SAC-CI/ G(2df,p) lvl of thory. ν SAC-CI/ G(2df,p) RKR data [16] G(ν) cm 1 B ν cm 1 G(ν) cm 1 B ν cm Tabl 8 Vibrational lvls and inrtial rotation constants from ν = 21 to ν = 49 for th 7 Li 2(G 1 Π g) (J = 0) stat at SAC-CI/ G(2df,p) lvl of thory. ν G(ν) (cm 1 ) B ν (cm 1 ) ν G(ν) (cm 1 ) B ν (cm 1 ) W hav obtaind a total of 50 vibrational lvls for this stat whn J = 0. For ach vibrational lvl G(ν), on inrtial rotation constant B v and six cntrifugal distortion constants D ν, H ν, L ν, M ν, N ν, and O ν ar attaind. Hr only 50 vibrational lvls and th corrsponding inrtial rotation constants togthr with availabl RKR data [16] ar tabulatd in Tabls 7 and 8 du to th lngth of th papr. From Tabl 7, on can asily find that th calculatd rsults

7 1120 SHI D-Hng, MA Hng, SUN Jin-Fng, and ZHU Zun-Lu Vol. 47 ar in good agrmnt with availabl RKR data as a whol. Thrfor, th vibrational lvls and th corrsponding inrtial rotation constants tabulatd in Tabl 8 should b rliabl. 4 Conclusions W hav attaind th rasonabl dissociation limit for dimr 7 Li 2 (G 1 Π g ), calculatd th intraction potntials using SAC-CI mthod at numrous basis sts and found that th bst potntial can b obtaind at G(2df,p). Employing prsnt potntial obtaind at G(2df,p), w hav computd th main spctroscopic constants D, E 0, R, ω, ω χ, α, and B, and for th first tim calculatd th vibrational lvls G(ν) and th inrtial rotation constants B ν. Favorabl agrmnt has bn found in comparing with availabl RKR data as a whol. Th rsults obtaind hr ar mor complt than prvious thortical invstigations, thus rprsnt an improvmnt. Rfrncs [1] K.T. Tang, J.P. Tonnis, and W. Myr, J. Chm. Phys. 95 (1991) [2] A. Aguado and M. Paniagua, J. Chm. Phys. 96 (1992) [3] M.D. Halls, H.B. Schlgl, M.J. DWitt, and G.W.F. Drak, Chm. Phys. Ltt. 339 (2001) 427. [4] C. Linton, F. Martin, A.J. Ross, t al., J. Mol. Spctrosc. 196 (1999) 20. [5] A.M. Maniro and P.H. Acioli, Int. J. Quant. Chm. 103 (2005) 711. [6] W.T. Zmk and W.C. Stwally, J. Chm. Phys. 111 (1999) [7] L. Li, G. Lazarov, and A.M. Lyyra, J. Mol. Spctrosc. 191 (1998)387. [8] A.A. Zavitsas, J. Mol. Spctrosc. 221 (2003) 67. [9] Y. Huang and R.J. LRoy, J. Chm. Phys. 119 (2003) [10] K. Urbanski, S. Antonova, A. Yiannopoulou, t al., J. Chm. Phys. 104 (1996) [11] D. Danovich, W. Wu, and S. Shaik, J. Am. Chm. Soc. 121 (1999) [12] N. Bouloufa, P. Cacciani, R. Vttr, and A. Yiannopoulou, J. Chm. Phys. 114 (2001) [13] X. Xi and R.W. Fild, J. Chm. Phys. 83 (1985) [14] D.D. Konowalow and J.L. Fish, Chm. Phys. 84 (1984) 463. [15] R. Potau and F. Spiglmann, J. Mol. Spctrosc. 171 (1995) 299. [16] R.A. Brnhim, L.P. Gold, P.B. Klly, T. Tipton, and D.K. Virs, J. Chm. Phys. 74 (1981) 749. [17] D.D. Konowalow and M.L. Olson, J. Chm. Phys. 71 (1979) 450. [18] H. Nakatsuji, M. Hada, M. Ehara, t al., SAC/SAC- CI Program Combind with Gaussian for Calculating Ground, Excitd, Ionizd, and Elctron-Attachd Stats and Singlt, Doublt, Triplt, Quartt, Quintt, Sxtt, and Sptt Spin Stats and Thir Analytical Enrgy Gradints, Kyoto Univrsity Prss, Kyoto (2002). [19] M.J. Frisch, G.W. Trucks, H.B. Schlgl, t al., Gaussian 03 Rvision A1, Gaussian Inc., Pittsburgh, PA (2003). [20] Z.H. Zhu, Atomic and Molcular Raction Statics, Scinc Prss, Bijing (1996). [21] B. Barakat, R. Bacis, F. Carrot, S. Churassy, P. Crozt, and F. Martin, Chm. Phys. 102 (1986) 215. [22] D.H. Shi, J.F. Sun, X.D. Yang, Z.L. Zhu, and Y.F. Liu, Chin. Phys. 14 (2005) [23] C.E. Moor, Atomic Enrgy Lvls, US Govrnmnts Printing Offic, Washington (1971). [24] J.N. Murrll, S. Cartr, S.C. Farantos, P. Huxly, and J.C. Varandas, Molcular Potntial Enrgy Functions, John Wily & Sons, Chichstr (1984). [25] R. Krishnan, J.S. Binkly, R. Sgr, and J.A. Popl, J. Chm. Phys. 72 (1980) 650. [26] M.J. Frisch, J.A. Popl, and J.S. Binkly, J. Chm. Phys. 80 (1984) [27] G. Hrzbrg, Molcular Spctra and Molcular Structur, Vol. 1, Van Nostrand Rinhold, Nw York (1951).