Fourth-order perturbative extension of the singles-doubles coupled-cluster method

Conjugation vs hyperconjugation in molecular structure of acroleinSvitlana V.Shishkina a ,⇑,Anzhelika I.Slabko b ,Oleg V.Shishkin a ,caDivision of Functional Materials Chemistry,SSI ‘Institute for Single Crystals’,National Academy of Science of Ukraine,60Lenina Ave.,Kharkiv 61001,Ukraine bDepartment of Technology of Plastic Masses,National Technical University ‘Kharkiv Polythechnic Institute’,21Frunze Str.,Kharkiv 61002,Ukraine cDepartment of Inorganic Chemistry,V.N.Karazin Kharkiv National University,4Svobody Sq.,Kharkiv 61077,Ukrainea r t i c l e i n f o Article history:Received 4August 2012In final form 16November 2012Available online 29November 2012a b s t r a c tAnalysis of geometric parameters of butadiene and acrolein reveals the contradiction between the Csp 2–Csp 2bond length in acrolein and classical concept of conjugation degree in the polarized molecules.In this Letter the reasons of this contradiction have been investigated.It is concluded that the Csp 2–Csp 2bond length in acrolein is determined by influence of the bonding for it p –p conjugation and antibonding n ?r ⁄hyperconjugation between the oxygen lone pair and the antibonding orbital of the single bond.It was shown also this bond length depends on the difference in energy of conjugative and hyperconjuga-tive interactions.Ó2012Elsevier B.V.All rights reserved.1.IntroductionButadiene and acrolein belong to the most fundamental mole-cules in the organic chemistry.They are canonical objects for the investigation of phenomena of p –p conjugation between double bonds and polarization of p -system by heteroatom [1].According to many experimental [2–16]as well as theoretical studies [13,17–23]the molecular structure of butadiene is determined by conjugation between p -orbitals of two double bonds and may be described as superposition of two resonance structures (Scheme 1).The presence of zwitterionic structure causes the shortening of the central single Csp 2–Csp 2bond as compare with similar unconjugated bond [24].Acrolein differs from butadiene by presence of the oxygen atom instead terminal methylene group.According to classical concepts of organic chemistry such replacement should causes polarization of p -system due to presence of highly polar C @O bond [25].This leads to significant increase of the contribution of the zwitterionic resonance structure (Scheme 1)reflecting strengthening of conju-gation between p -systems of double bonds.Therefore the central Csp 2–Csp 2bond must be shorter in acrolein as compared with one in butadiene.However numerous investigations of acrolein by experimental [26–28]and theoretical methods [26,27,29–36]demonstrate an opposite situation:the Csp 2–Csp 2bond length varies within the range 1.469Ä1.481Åin acrolein as compared with 1.454Ä1.467in butadiene.Based on these data one can conclude that conjugation between double bonds in acrolein is weaker than in butadiene.At that time the rotation barrier obtained from quan-tum-chemical calculations is higher in acrolein [26,27],confirming stronger conjugation between double bonds.Thus,results of experimental and theoretical investigations demonstrate the con-tradiction between the strengthening of the conjugation in acrolein as compare with butadiene and the values of the Csp 2–Csp 2bond length in these molecules.Recently such illogical situation was observed also in derivatives of cyclohexene containing conjugated endocyclic and exocyclic double bonds [37].It was assumed that elongation of the Csp 2–Csp 2bond in cycloxen-2-enone as compare with one in 3-methylene-cyclohexene is caused by the influence of n ?r ⁄hyperconjugation.In this Letter we demonstrate the results of the investigation of intramolecular interactions in butadiene and acrolein which ex-plain the experimentally observed contradiction between the length of the Csp 2–Csp 2bond and degree of conjugation in acrolein.2.Method of calculationsThe structures of all investigated molecules were optimized using second-order Møller-Plesset perturbation theory [38].The standard aug-cc-pvtz basis set [39]was applied.The character of stationary points on the potential energy surface was verified by calculations of vibrational frequencies within the harmonic approximation using analytical second derivatives at the same level of theory.All stationary points possess zero (minima)or one (saddle points)imaginary frequencies.The verification of the calculation method was performed using optimization of butadi-ene and acrolein by MP2/aug-cc-pvqz,CCSD(T)/cc-pvtz and CCSD(T)/6-311G(d,p)methods [40].The geometry of saddle points for the rotation process was lo-cated using standard optimization technique [41].The barrier of the rotation in all molecules was calculated as the difference be-tween the Gibbs free energies at 298K of the most stable s-trans0009-2614/$-see front matter Ó2012Elsevier B.V.All rights reserved./10.1016/j.cplett.2012.11.032Corresponding author.Fax:+3805723409339.E-mail address:sveta@ (S.V.Shishkina).conformer and saddle-point conformation.All calculations were performed using the G AUSSIAN 03program [42].The intramolecular interactions were investigated within the Natural Bonding Orbitals theory [43]with N BO 5.0program [44].Calculations were performed using B3LYP/aug-cc-pvtz wave func-tion obtained from single point calculations by G AUSSIAN 03program.The conjugation and hyperconjugation interactions are referred to as ‘delocalization’corrections to the zeroth-order natural Lewis structure.For each donor N BO (i )and acceptor N BO (j ),the stabiliza-tion energy E (2)associated with delocalization (‘2e-stabilization’)i ?j is estimated asE ð2Þ¼D E ij ¼q iF ði ;j Þ22j À2i;where q i is the donor orbital occupancy,e j and e i are the diagonal elements (orbital energies),and F (i,j )is the off-diagonal N BO Fock matrix element.3.Results and discussionThe equilibrium geometry of s-trans and s-cis conformers of butadiene and acrolein calculated by MP/aug-cc-pvtz method (Ta-ble 2)agrees very well with obtained earlier results [2–23,25–34]and data of higher and more computationally expensive methods (Table 1).It can be noted that the C–C bond length in acrolein(Tables 1and 2)is longer as compared with butadiene in all sta-tionary points on the potential energy surface.Such relation does not agree with the conception of the resonance theory [45–47].Analysis of intramolecular interactions in both molecules using N BO theory indicates that acrolein differs from butadiene by pres-ence of intramolecular interaction between lone pair of the oxygen atom and antibonding orbital of the C–C bond (Figure 1)as well as the polarization of one double bond containing more electronega-tive ually the interactions between lone pair and antibonding orbital of single bond are stronger than the interac-tions between the C–H bond and antibonding orbital [48]and they can influence geometrical characteristics.Such type of interactions is named by anomeric effect and it is studied very well for the case when the central bond between interacted orbitals is single [48,49].It is investigated in details [49]an influence of classical anomeric effect on conformation of the substituents about central single bond as well as on values of bond lengths.In the case of hyperconjugation interactions along double bond in acrolein orien-tation of substituents around it is determined by its double charac-ter.Therefore,n ?r ⁄hyperconjugative interaction can influence on the bond lengths of interacted ones only.It can assume the dou-ble character of the central bond must also promote some strengthening of this influence due to shorter distance between the lone pair of the oxygen atom and antibonding orbital of the C–C bond.Results of N BO analysis of intramolecular interactions in butadi-ene and acrolein demonstrate that the energy of n ?r ⁄interaction between lone pair of the oxygen atom and antibonding orbital of the C–C bond in acrolein is twice as high of the energy of r ?r ⁄interaction between the C–H bond and antibonding orbital of the C–C bond in butadiene (Table 2).At that time the energy of n ?r ⁄interaction is close enough to the energy of p –pTable 2The equilibrium geometries (bond lengths,Åand C @C–C @X (X =CH 2,O)torsion angle,deg.),transition state of the rotation process,bond length alternation (BLA)parameter,related energy (D E rel ,kcal/mol),related stability (D G 298,kcal/mol)and energy of strongest intramolecular interactions (E (2),kcal/mol)for butadiene and acrolein optimized by MP2/aug-cc-pvtz method.The wave function calculated by b3lyp/aug-cc-pvtz method was used for N BO analysis.ConformerBond lengths (Å)C @C–C @X torsion angle deg.BLA (Å)D E rel(kcal/mol)D G 298(kcal/mol)E (2)(kcal/mol)C @CC–C C @X p –pn ?r ⁄(C–C)r ?r ⁄(C–C)Butadiene s-trans 1.341 1.453 1.340180.0+0.1120030.74–8.67gauche 1.340 1.465 1.34036.8+0.125 2.83 2.8921.04–8.94TS 1.3361.4801.336101.8+0.1446.446.102.32–10.66Acrolein s-trans 1.339 1.469 1.219180.0+0.1300027.7518.06–s-cis 1.338 1.481 1.2190.0+0.143 2.26 2.2224.6319.05–TS1.334 1.492 1.21692.5+0.1588.017.46–18.78–Acrolein +BH 3s-trans 1.341 1.449 1.235180.0+0.1080033.12 2.4511.43s-cis 1.340 1.460 1.2340.0+0.120 2.45 2.3829.61 2.6011.43TS1.334 1.476 1.23193.7+0.1429.258.61–2.3911.55Table 1The Csp 2–Csp 2bond length in butadiene and acrolein optimized by different quantum-chemical methods.Method of calculationCsp 2–Csp 2bond length (Å)D (Csp 2–Csp 2)(Å)ButadieneAcrolein MP2/aug-cc-pvtz 1.453 1.4690.016MP2/aug-cc-pvqz 1.451 1.4670.016CCSD(T)/cc-pvtz1.461 1.4780.017CCSD(T)/6–311G(d,p)1.4681.4870.019S.V.Shishkina et al./Chemical Physics Letters 556(2013)18–2219conjugation between two double bonds.Therefore it can assume that the influence of p–p conjugation and n?r⁄hyperconjuga-tion on the C–C bond length should be comparable.However two strongest intramolecular interactions in the acrolein differ from each other:p–p conjugation between double bonds causes the shortening of the C–C bond in contrary to n?r⁄hyperconjugation which leads to the elongation of the C–C bond owing to the popu-lation of its antibonding orbital.Taking into account this situation it is possible to conclude that length of the Csp2–Csp2single bond in acrolein is determined by balance of two opposite factors namely p–p conjugation and n?r⁄hyperconjugation which may be considered as bonding and antibonding interactions for this bond(Figure1).In this case the length of the Csp2–Csp2bond in acrolein depends on the con-tribution of each of these factors.The changing of the delocaliza-tion of the structures due to influence of intramolecular interactions can be analyzed easier by mean of the bond length alternation(BLA)parameter(Table2).The analysis of BLA shows the presence of n?r⁄hyperconjugative interaction in acrolein what results in the increasing of alternation of double bonds as compare with butadiene.Clear estimation of influence of both interactions on geometri-cal parameters of molecule may be performed by comparison of properties of single C–C bond and BLA parameter in equilibrium s-trans conformation and in situations where one or both intramo-lecular interactions are absent.It is well known that p–p conjugation between double bonds decreases appreciably up to disappearing(in acrolein)in the tran-sition state for the rotation around single bond process(Figure2). The data of N BO analysis for butadiene and acrolein in the transition state confirm this evidence(Table2).As expected the absence of p–p conjugation results in the elongation of the Csp2–Csp2bond and increasing of BLA as compare with equilibrium geometry.At that time single C–C bond remains longer in the transition state for acrolein as compare with one for butadiene’s transition state.1.4921.4691.4491.476π−π is present n σ* is presentπ−π is absentn σ* is absent without π−πwithout n σ∗Figure2.Influence of p–p⁄conjugation and n?r⁄hyperconjugation on the C–Cbond length in acrolein.Table3The energy(E(2),kcal/mol)of the conjugative(bonding)and hyperconjugative(antibonding)intramolecular interactions influencing the Csp2–Csp2bond length in butadiene, acrolein and its complex with BH3.Molecule Bonding interactions E(2)(kcal/mol)Antibonding interactions E(2)(kcal/mol)Butadienes-trans BD(2)C1-C2–BD(2)C3-C430.74BD(1)C1-H5–BD(1)C2-C38.67 BD(1)C2-H7–BD(1)C3-H87.68BD(1)C4-H9–BD(1)C2-C38.67 gauche BD(2)C1-C2–BD(2)C3-C421.04BD(1)C1-H5–BD(1)C2-C38.94 BD(1)C2-H7–BD(1)C3-C4 5.36BD(1)C4-H9–BD(1)C2-C39.01BD(1)C3-H8–BD(1)C1-C2 5.36TS BD(1)C1-C2–BD(1)C3-C4 3.5BD(1)C1-H5–BD(1)C2-C310.66 BD(1)C1-C2–BD(2)C3-C4 3.46BD(1)C4-H9–BD(1)C2-C310.66BD(1)C3-C4–BD(2)C1-C2 3.46BD(2)C1-C2–BD(2)C3-C4 2.32BD(1)C3-H8–BD(2)C1-C29.57BD(1)C2-H7–BD(2)C3-C49.57Acroleins-trans BD(1)C1-C2–BD(1)C3-O4 2.73BD(1)C1-H5–BD(1)C2-C38.25 BD(2)C1-C2–BD(2)C3-O427.75LP(2)O4–BD(1)C2-C318.06BD(1)C2-H7–BD(1)C3-H8 5.95s-cis BD(2)C1-C2–BD(2)C3-O424.63BD(1)C1-H5–BD(1)C2-C38.76 BD(1)C2-H7–BD(1)C3-O4 4.05LP(2)O4–BD(1)C2-C319.05BD(1)C3-H8–BD(1)C1-C2 5.01TS BD(1)C1-C2–BD(2)C3-O4 2.98BD(1)C1-H5–BD(1)C2-C39.07 BD(1)C3-O4–BD(2)C1-C2 4.48LP(2)O4–BD(1)C2-C318.78BD(1)C3-H8–BD(2)C1-C2 5.85BD(1)C2-H7–BD(2)C3-O4 6.16Acrolein+BH3s-trans BD(1)C1-C2–BD(1)C3-O4 3.07BD(1)C1-H6–BD(1)C2-C38.09 BD(2)C1-C2–BD(2)C3-O433.12BD(1)C2-C3–BD(1)O4-B511.43BD(1)C2-H8–BD(1)C3-H9 5.97LP(1)O4–BD(1)C2-C3 2.45 s-cis BD(1)C1-C2–BD(1)C3-H9 4.98BD(1)C1-H6–BD(1)C2-C38.51 BD(2)C1-C2–BD(2)C3-O429.61BD(1)C2-C3–BD(1)O4-B511.43BD(1)C2-H8–BD(1)C3-O4 4.41LP(1)O4–BD(1)C2-C3 2.60 TS BD(1)C1-C2–BD(1)C3-O40.55BD(1)C1-H6–BD(1)C2-C39.05 BD(1)C1-C2–BD(2)C3-O4 3.01BD(1)C2-C3–BD(1)O4-B511.55BD(2)C1-C2–BD(1)C3-O4 4.90LP(1)O4–BD(1)C2-C3 2.39BD(1)C3-H9–BD(2)C1-C2 6.13BD(1)C2-H8–BD(2)C3-O4 6.8820S.V.Shishkina et al./Chemical Physics Letters556(2013)18–22It is additional argument about the influence of n?r⁄hypercon-jugation on the C–C bond length through the C@O double bond.In contrary to p–p conjugation n?r⁄hyperconjugation is present in all stationary points on the potential energy surface for acrolein(Table2).But this interaction can be shielded by for-mation of dative bond involving lone pair of the oxygen atom and unoccupied orbital of Lewis acid,for example,BH3.The formed O–B bond has r-character and the energy of its interaction with antibonding orbital of the central C–C bond is very close to the en-ergy of similar C–H?r⁄(C–C)interaction in butadiene(Table2). The absence of n?r⁄hyperconjugation results significant short-ening of the Csp2–Csp2bond and decreasing of BLA in all stationary points for acrolein.It is more interesting that the C–C bond in acro-lein becomes shorter and p–p conjugation becomes stronger as compare with ones in butadiene in the case of absence of n?r⁄hyperconjugative interaction(Table2)what agrees well with the resonance theory.This evidence is confirmed also by values of BLA parameter.It is very interesting the situation when both strong intramolec-ular interactions are absent namely acrolein with shielded by BH3 lone pair in the transition state for the rotation process.In absence of p–p conjugative and n?r⁄hyperconjugative interactions the C–C bond length is almost equal to mean value for length of this bond for s-trans and s-cis conformers of acrolein with both interac-tions(Table2).This fact confirms that the C–C bond length in acro-lein in the equilibrium state is determined by balance of p–p conjugation and n?r⁄hyperconjugation.Taking into account the opposite influence of two types of intra-molecular interactions on the C–C bond one may assume that its length depends on the difference in energy of bonding and antibonding interactions for this bond.In such case all bonding for C–C bond and antibonding for it interactions(Table3)must be taken into account.Specially,this is important for transition states where p–p conjugative interaction is minimal and r(c-H)–p interaction appears instead it.This interaction has bond-ing for Csp2–Csp2bond character and it is weaker as compare with p–p interaction.Analysis of relation between C–C bond length and total energy of intramolecular interaction influencing on it demon-strates good correlation between them(Figure3)with correlation coefficient aboutÀ0.93.The barrier of the rotation around ordinary C–C bond is also sensitive to intramolecular interactions.The absence of n?r⁄hyperconjugation in acrolein results the increase of conjugation in molecule what leads to the increase of the rotation barrier (Table2).4.ConclusionsResults of quantum-chemical calculations demonstrate the structure of acrolein does not correspond to conventional views about influence of the polarization of p-system by the oxygen atom.According to classic viewpoint this effect should lead to in-crease of conjugation between double bonds and shortening of central single C–C bond as compared to butadiene.However,anal-ysis of intramolecular interactions shows that the geometry of acrolein is determined by counteraction of p–p conjugation and n?r⁄hyperconjugation.The energies of these interactions are very close but ones influence on the C–C bond lengths in opposite directions.Conjugation promotes the shortening of the central sin-gle bond due to the overlapping of the p-orbitals of two double bonds.In the contrary the n?r⁄hyperconjugation causes the elongation of the C–C bond due to the population of its antibonding orbital.The absence of conjugation in the transition state for the rotation about the C–C bond process results in the elongation of the single bond in conjugated system.In turn the shielding of n?r⁄hyperconjugation by the formation of dative bond between lone pair of oxygen atom and vacant orbital of Lewis acid causes the shortening of the C–C bond in acrolein.The C–C bond length correlates well with the difference between two strong intramolec-ular interactions.The absence of both interactions does not almost change the C–C bond length.Thus,these data clearly indicate that molecular structure of conjugated systems containing heteroatoms is determined by not only p–p conjugation but also by n?r⁄hyperconjugation.References[1]F.A Carey,R.J.Sundberg,Advanced Organic Chemistry.Part A:Structure andMechanisms,Springer,Virginia,2007.[2]Yu.N.Panchenko,Yu.A.Pentin,V.I.Tyulin,V.M.Tatevskii,Opt.Spectrosc.13(1962)488.[3]A.R.H.Cole,G.M.Mohay,G.A.Osborne,Spectrochim.Acta23A(1967)909.[4]K.Kuchitsu,T.Fukuyama,Y.Morino,J.Mol.Struct.1(1967–1968)463.[5]R.L.Lipnick,E.W.Garbisch Jr.,J.Am.Chem.Soc.95(1973)6370.[6]Yu.N.Panchenko,Spectrochim.Acta31A(1975)1201.[7]Yu.N.Panchenko,A.V.Abramenkov,V.I.Mochalov,A.A.Zenkin,G.Keresztury,G.J.Jalsovszky,J.Mol.Spectrosc.99(1983)288.[8]W.Caminati,G.Grassi,A.Bauder,Chem.Phys.Lett.148(1988)13.[9]M.E.Squillacote,T.C.Semple,P.W.Mui,J.Am.Chem.Soc.107(1985)6842.[10]Y.Furukawa,H.Takeuchi,I.Harada,M.Tasumi,Bull.Chem.Soc.Jpn.56(1983)392.[11]B.R.Arnold,V.Balaji,J.W.Downing,J.G.Radziszewski,J.J.Fisher,J.Michl,J.Am.Chem.Soc.113(1991)2910.[12]J.Saltiel,J.-O.Choi,D.F.Sears Jr.,D.W.Eaker,F.B.Mallory,C.W.Mallory,J.Phys.Chem.98(1994)13162.[13]K.W.Wiberg,R.E.Rosenberg,J.Am.Chem.Soc.112(1990)1509.[14]J.Saltiel,D.F.Sears Jr,A.M.Turek,J.Phys.Chem.A105(2001)7569.[15]M.S.Deleuze,S.Knippenberg,J.Chem.Phys.125(2006)104309-1.[16]P.Boopalachandran,N.C.Craig,ane,J.Phys.Chem.A116(2012)271.[17]H.Guo,M.Karplus,J.Chem.Phys.94(1991)3679.[18]R.Hargitai,P.G.Szalay,G.Pongor,G.Fogarasi,J.Mol.Struct.(THEOCHEM)112(1994)293.[19]G.R.De Maré,Yu.N.Panchenko,J.V.Auwera,J.Phys.Chem.A101(1997)3998.[20]J.C.Sancho-García,A.J.Pérez-Jiménez,F.Moscardó,J.Phys.Chem.A105(2001)11541.[21]N.C.Craig,P.Groner,D.C.McKean,J.Phys.Chem.A110(2006)7461.[22]D.Feller,K.A.Peterson,J.Chem.Phys.126(2007)114105.[23]D.Feller,N.C.Craig,A.R.Maltin,J.Phys.Chem.A112(2008)2131.[24]D.Feller,N.C.Craig,J.Phys.Chem.A113(2009)1601.[25]H.-B.Burgi,J.D.Dunitz,Structure Correlation,vol.2,VCH,Weinheim,1994.[26]K.B.Wiberg,R.E.Rosenberg,P.R.Rablen,J.Am.Chem.Soc.113(1991)2890.[27]K.B.Wiberg,P.R.Rablen,M.Marquez,J.Am.Chem.Soc.114(1992)8654.[28]K.Kuchitsu,T.Fukuyama,Y.Morino,J.Mol.Struct.1(1967–1968)463.[29]G.Celebre,M.Concistré,G.DeLuca,M.Longeri,G.Pileio,J.W.Emsley,Chem.Eur.J.11(2005)3599.[30]R.J.Loncharich,T.R.Schwartz,K.N.Houk,J.Am.Chem.Soc.109(1987)14.[31]G.R.DeMare,Yu.N.Panchenko,A.J.Abramenkov,J.Mol.Struct.160(1987)327.S.V.Shishkina et al./Chemical Physics Letters556(2013)18–2221[32]G.R.DeMare,Can.J.Chem.63(1985)1672.[33]Y.Osamura,H.F.Schaefer III,J.Chem.Phys.74(1981)4576.[34]C.E.Bolm,A.Bauder,Chem.Phys.Lett.88(1982)55.[35]B.Mannfors,J.T.Koskinen,L.-O.Pietilä,L.Ahjopalo,J.Mol.Struct.(THEOCHEM)393(1997)39.[36]J.I.García,J.A.Mayoral,L.Salvatella,X.Assfeld,M.F.Ruiz-López,J.Mol.Struct.(THEOCHEM)362(1996)187.[37]S.V.Shishkina,O.V.Shishkin,S.M.Desenko,J.Leszczynski,J.Phys.Chem.A112(2008)7080.[38]C.Møller,M.S.Plesset,Phys.Rev.46(1934)618.[39]R.A.Kendall,T.H.Dunning Jr.,R.J.Harrison,J.Chem.Phys.96(1992)6792.[40]W.H.Hehre,L.Radom,P.V.R.Schleyer,J.A.Pople,Ab initio Molecular OrbitalTheory,Wiley,New York,1986.[41]P.Culot,G.Dive,V.H.Nguyen,J.M.Ghuysen,Theor.Chim.Acta82(1992)189.[42]M.J.Frisch et al.,G AUSSIAN,Inc.,Wallingford CT,2004.[43]F.Weinhold,in:P.V.R.Schleyer,N.L.Allinger,T.Clark,J.Gasteiger,P.A.Kollman,H.F.Schaefer III,P.R.Schreiner(Eds.),Encyclopedia of Computational Chemistry,vol.3,John Wiley&Sons,Chicheste,UK,1998.1792–1792. [44]E.D.Glendening,J.K.Badenhoop,A.E.Reed,J.E.Carpenter,J.A.Bohmann,C.M.Morales,F.Weinhold,N BO5.0Theoretical Chemistry Institute,University of Wisconsin,Madison,WI,2001.[45]E.D.Glendening,F.Weinhold,put.Chem.19(1998)593.[46]E.D.Glendening,F.Weinhold,put.Chem.19(1998)610.[47]E.D.Glendening,J.K.Badenhoop,F.Weinhold,put.Chem.19(1998)628.[48]A.J.Kirby,Stereoelectronic Effects,Oxford University Press,New York,1996.[49]I.V.Alabugin,K.M.Gilmore,P.W.Peterson,WIREs Computational MolecularScience1(2011)109.22S.V.Shishkina et al./Chemical Physics Letters556(2013)18–22。

配位化学讲义第四章（1）价键理论、晶体场理论第三章配合物的化学键理论目标：解释性质，如配位数、几何结构、磁学性质、光谱、热力学稳定性、动力学反应性等。

三种理论：①价键理论、②晶体场理论、③分子轨道理论第一节价键理论（Valencebond theory）由L.Pauling提出要点：①配体的孤对电子可以进入中心原子的空轨道；②中心原子用于成键的轨道是杂化轨道（用于说明构型）。

一、轨道杂化及对配合物构型的解释能量相差不大的原子轨道可通过线性组合构成相同数目的杂化轨道。

对构型的解释（依据电子云最大重叠原理：杂化轨道极大值应指向配体）指向实例sp3、sd3杂化四面体顶点Ni(CO)4sp2、sd2、dp2、d3杂化三角形顶点[AgCl3]2-dsp2、d2p2 杂化正方形顶点[PtCl4]2-d2sp3杂化八面体顶点[Fe(CN)6]4-sp杂化直线型[AgCl2]-二、AB n型分子的杂化轨道1、原子轨道的变换性质考虑原子轨道波函数，在AB n分子所属点群的各种对称操作下的变换性质。

类型轨道多项式sp x xp p y yp z zd xy xyd xz xzd d yz yzd x2-y2x2-y2d z22z2-x2-y2（简记为z2）*s轨道总是按全对称表示变换的。

例：[HgI3]- (D3h群)平面三角形A1′：d z2、sE′：(p x、p y )、(d x2-y2、d xy)A 2″：p zE″：(d xz、d yz)2、σ轨道杂化方案1）四面体分子AB4（Td）[CoCl4]2-以四个杂化轨道的集合作为分子点群(Td)表示的基，确定该表示的特征标：r1r4r2r3恒等操作，χ(E)=4 C3操作，χ(C3)=1对C2、S4和σd用同样方法处理，得T d E 8C3 3C2 6S46σdΓ 4 1 00 2约化：T d E 8C3 3C2 6S4 6σdA1 1 1 1 11A2 1 1 1 -1 - 1E 2 -1 2 00 (z2, x2-y2)T1 3 0 -1 1 -1T2 3 0 -1 -11 (xy,xz,yz) (x,y,z)a(A1)=1/24(1×4+8×1×1+3×1×0+6×1×0+6×1×2)=1a(A2)=1/24 [1×4+8×1×1+3×1×0+6×(-1)×0+6×(-1)×2]=0a(E)=1/24 [2×4+8×(-1)×1+3×2×0+6×0×0+6×0×2]=0a(T1)=1/24 [3×4+8×0×1+3×(-1)×0+6×1×0+6×(-1)×2]=0a(T2)=1/24 [3×4+8×0×1+3×(-1)×0+6×(-1)×0+6×1×2]=1约化结果Γ=A1+T2由特征标表：A1T2s(p x、p y、p z)(d xy、d xz、d yz)可有两种组合：sp3（s、p x、p y、p z）、sd3（s、d xy、d xz、d yz)* 以一组杂化轨道为基的表示的特征标的简化计算规则：①不变（1）②改变符号（-1）③与其他函数变换（0）2）再以[CdCI5]3-三角双锥(D3h)为例：41325D3h E 2C33C2σh2S3 3σvΓ 5 2 13 0 3约化结果：Γ= 2A1′+A2〞+E′A1′A2〞E′s p z (p x、p y)d z2(d xy、d x2-y2)两种可能的组合：（s、d z2、p z 、p x、p y）( s、d z2、p z、d xy、d x2-y2)3）[HgI3]- ( D3h)123D3h E 2C3 3C2σh2S33σvΓ 3 0 13 0 1约化得：Γ=A1′+E′A1′E′s (p x、p y)d z2(d xy、d x2-y2)可能的组合有：（s、p x、p y）、(s、d xy、d x2-y2)、(d z2、p x、p y)、(d z2、d xy、d x2-y2)4）平面AB4型分子（D4h）例：[PtCl4]2-C2′C2″D4h E 2C4(C41,C43) C2(C42) 2C2′2C2″i 2S4σh 2σv2σdΓ 4 0 0 20 0 0 4 2 0约化得：Γ=A1g+B1g+E uA1g B1g E us d x2-y2(p x、p y)d z2两种类型：dsp2（d x2-y2、s、p x、p y）、d2p2（d z2、d x2-y2、p x、p y）5）八面体AB6(O h) 例：[Fe(H2O)6]3+O h E 8C3 6C26C4 3C2i 6S4′8S6 3σh 6σdΓ 6 0 0 2 2 0 0 0 4 2约化得：Γ=A1g+E g+T1u A1g E gT1us (d z2、d x2-y2) (p x、p y、p z)只有唯一的d2sp3杂化（d z2、d x2-y2、s、p x、p y、p z)3、π成键杂化方案在AB n分子中，原子A上要有2n个π型杂化轨道和在B原子上的2n个π原子轨道成键。

模拟心得MATERIAL STUDIO 中SORPTION 第一个课题是模拟金属有机框架和共价有机框架吸附CO2以及分离CO2/CH4,使用的软件是Material studio，使用的是Sorption模块，输入的是逸度。

单组份求逸度的MA TLAB程序，只需要在主程序窗口输入function [rho,f]=PengRobinson(P1,T,N)（P1,T,N是具体的数值）就可以得到不同的压力和温度下的逸度。

function [rho,f] =PengRobinson(P1,T,N)%+++++++++++++++++++++++++++++++++++++++++++++%PengRobinson is used to calculate the density and fugacity of single%component gas at given pressure with Peng-Robinson equation.%PengRobinson v1.00 beta include the parameter of n-alkanes(1-5), CO2(6)%and CO(7).%Where P1 means input pressure(kPa), T is temperature(K), N means the serial number of gas. rho%is density, f is fugacity.%e.g. If you wanna calculate density and fugacity of methane at 300kPa, 298k,you%need input [rho,f] =PengRobinson(300,298,1).%+++++++++++++++++++++++++++++++++++++++++++++%set physical parameters%+++++++++++++++++++++++++++++++++++++++++++++P=P1./100;t_M=[16.043 30.070 44.097 58.123 72.150 44.01 28.01];t_omiga=[0.012 0.100 0.152 0.2 0.252 0.224 0.048];t_Tc=[190.6 305.3 369.8 425.1 469.7 304.2 132.9 ];t_Pc=[45.99 48.72 42.48 37.96 33.70 73.83 34.99];%+++++++++++++++++++++++++++++++++++++++++++++Tc=t_Tc(N);Pc=t_Pc(N);omiga=t_omiga(N);M=t_M(N);%+++++++++++++++++++++++++++++++++++++++++++++R=83.14;epsilon=1-2.^(0.5);sigma=1+2.^(0.5);%+++++++++++++++++++++++++++++++++++++++++++++%calculate the Z of PR equation%+++++++++++++++++++++++++++++++++++++++++++++alpha=(1+(0.37464+1.54226*omiga-0.26992*omiga.^2)*(1-(T/Tc)^(0.5))).^2;a=((0.45724*R^2*Tc^2)/Pc)*alpha;b=0.07779.*R.*Tc./Pc;beta=b.*P./(R.*T);q=a./(b.*R.*T);Z0=zeros(length(P),1);Z1=ones(length(P),1);for k=1:length(P)while abs(Z1(k)-Z0(k))>1e-6Z0(k)=Z1(k);Z1(k)=1+beta(k)-q.*beta(k).*(Z0(k)-beta(k))./((Z0(k)+epsilon.*beta(k)).*(Z0(k)+sigma.*beta(k))); endendI=(1./(sigma-epsilon)).*log((Z1+sigma.*beta)./(Z1+epsilon.*beta));%+++++++++++++++++++++++++++++++++++++++++++++%gain the density of gas%+++++++++++++++++++++++++++++++++++++++++++++rho=(P./(Z1.*R.*T)).*M.*1e6;rho=vpa(rho,6);phi=exp(Z1-1-log(Z1-beta)-q.*I);f=phi.*P1;f=vpa(f,5);双组份的求逸度的程序：求逸度的过程和单组份的一样。

DFTB+Version1.0.1U S E R M A N U A LContents1Introduction2 2Input for DFTB+42.1Main input (5)2.2Geometry (5)2.3Driver (6)2.4Hamiltonian (9)2.5Options (21)2.6ParserOptions (22)A The HSD format23A.1Scalars and list of scalars (24)A.2Methods and property lists (25)A.3Modifiers (26)A.4File inclusion (26)A.5Processing (27)A.6Extended format (28)B Unit modifiers31C Description of the gen format33D Atomic spin constants34E Dispersion constants35F Publications to cite36 Index39Chapter1IntroductionThe code DFTB+is the Fortran95successor of the old DFTB code,implementing the density functional based tight binding approach[8].The code had been completely rewritten from scratch and extended with various features.The most important features are:•Non-scc and scc calculations(with expanded range of SCC accelerators)–Cluster/molecular systems–Periodic systems(arbitrary K-point sampling,band structure calc.)•l-shell resolved calculations possible•Spin polarised calculation(collinear spin)•Geometry optimisation–Steepest descent–Conjugate gradient•Geometry optimisation constraints(in xyz-coordinates)•Molecular dynamics(Anderson thermostat)•Improvedfinite temperature calculations•Dispersion correction(van der Waals interaction)•Ability to treat f-electrons•LDA+U extension•QM/MM coupling with external point charges(smoothing possible)•OpenMP parallelisation•Automatic code validation(autotest system)•New user friendly,extensible input format(HSD or XML)•Dynamic memory allocation•Additional tool for generating cubefiles for charge distribution,molecular orbitals,etc. (Waveplot)Chapter2Input for DFTB+DFTB+can read two formats,either XML or the Human-friendly Structured Data format(HSD). If you are not familiar with HSD format yet,a detailed description is given in appendix A.The inputfile for DFTB+must be named dftb_in.hsd or dftb_in.xml.The inputfile must be present in the working directory.To prevent ambiguity,the parser refuses to read any input if bothfiles are present.After processing the input,DFTB+creates afile of the parsed input,either dftb_pin.hsd or dftb_pin.xml.This contains the user input as well as any default values for unspecified options. All values are given in the default internal units.Thefile also contains the version of the current input parser.You should always keep thisfile,since if you want to exactly repeat your calculation with a later version of DFTB+,it is recommended to use thisfile instead of the original input.(You must of course rename dftb_pin.hsd into dftb_in.hsd or dftb_pin.xml into dftb_in.xml.)This guarantees that you will obtain the same results,even if the defaults for some non specified options have been changed.The code can also produce dftb_pin.xml from dftb_in.hsd or vice versa if required(see section2.6).The following sections list properties and options that can be set in DFTB+input.Thefirst column of each of the tables of options specifies the name of a property.The second column indicates the type of the expected value for that property.The letters“i”,“r”,“s”,“p”,“m”stand for integer, real,string,property list and method type,respectively.An optional prepended number specifies how often(if more than once)this type must occur.An appended“+”indicates arbitrary occurrence greater than zero,while“*”allows also for zero occurrence.Alternative types are separated by“|”. Parentheses serve only to delimit groups of settings.Sometimes a property is only interpreted if some condition(s)is met.If this is the case,the appro-priate conditions are indicated in the third column.The fourth column contains the default value for the property.If no default value is specified(“-”),the user is required to assign a value to that property.The description of the properties immediately follows the table.If there is also a more detailed description available for a given keyword somewhere else,the appropriate page number appears in the last column.Some properties are allowed to carry a modifier to alter the provided value(e.g.converting between units).The possible modifiers are listed between brackets([])in the detailed description of the property.If the modifier is a conversion factor for a physical unit,only the unit type is indicated (length,energy,force,time,etc.).A list of the allowed physical units can be found in appendix B.2.1Main inputThe inputfile for DFTB+(dftb_in.hsd/dftb_in.xml)must contain the following property defini-tions:Name Type Condition Default Page Geometry p|m-5 Hamiltonian m-9Additionally optional definitions may be present:Name Type Condition Default Page Driver m{}6 Options p{}21 ParserOptions p{}22Geometry Specifies the geometry for the system to be calculated.See p.5.Hamiltonian Configures the Hamiltonian and its options.See p.9.Driver Specifies a geometry driver for your system.See p.6.Options Various global options for the run.See p.21.ParserOptions Various options affecting the parser only.See p.22.2.2GeometryThe geometry can be specified either directly by passing the appropriate list of properties or by using the GenFormat{}method.2.2.1Explicit geometry specificationIf the geometry is being specified explicitely,the following properties can be set:Periodic l NoLatticeVectors9r Periodic=Yes-TypeNames s+-TypesAndCoordinates(1i3r)+-Periodic Specifies if the system is periodic in all3dimensions or is to be treated as a cluster.If set to Yes,property LatticeVectors{}must be also specified.LatticeVectors[length]The x,y and z components of the three lattice vectors if the system is periodic.TypeNames List of strings with the names of the elements,which appear in your geometry. TypesAndCoordinates[relative|length]For every atom the index of its type in the TypeNames list and its coordinates.If for a periodic system(Periodic=Yes)the modifier relative is specified,the coordinates are interpreted in the coordinate system of the lattice vectors.Example:Geometry of GaAs:Geometry={TypeNames={"Ga""As"}TypesAndCoordinates[Angstrom]={10.0000000.0000000.0000002 1.356773 1.356773 1.356773}Periodic=YesLatticeVectors[Angstrom]={2.713546 2.7135460.0. 2.713546 2.7135462.7135460. 2.713546}}2.2.2GenFormat{}You can use the generic format to specify the geometry(see appendix C).The geometry specifica-tion for GaAs would be the following:Geometry=GenFormat{2SGa As110.0000000.0000000.00000022 1.356773 1.356773 1.3567730.0000000.0000000.0000002.713546 2.7135460.0. 2.713546 2.7135462.7135460. 2.713546}It is also possible to include the gen-formatted geometry from afile:Geometry=GenFormat{<<<"geometry.gen"}2.3DriverThe driver is responsible for changing the geometry of the input structure during the calculation. Currently the following methods are available:{}Static calculation with the input geometry.SteepestDescent{}Geometry optimisation by moving atoms along the acting forces.See p.7. CongjugateGradient{}Geometry optimisation using the conjugate gradient algorithm.See p.8. VelocityVerlet{}Molecular dynamics with the velocity Verlet algorithm.See p.8.2.3.1SteepestDescent{}MovedAtoms i+|m Range{1-1} MaxForceComponent r1e-4MaxSteps i200StepSize r100.0OutputPre x s"geo_end" AppendGeometries l NoConstraints(1i3r)*{}MovedAtoms Index of the atoms which should be moved.If the index range is continuous,the Range{}method can be used.MovedAtoms=Range{16}#equivalent with MovedAtoms={123456}Negative indexes can be used to count backwards from thefinal atom:MovedAtoms=Range{1-1}#Move all atoms including the last MaxForceComponent[force]Optimisation is stopped,if the force component with the maximal absolute value goes below this threshold.MaxSteps Maximum number of steps after which the optimisation should stop(unless already stopped by achieving convergence).StepSize[time]Step size(δt)along the forces.The displacementδx i along the i th coordinate isgiven for each atom asδx i=f i2m δt2,where f i is the appropriate force component and m is themass of the atom.OutputPre x Prefix of the geometryfiles containing thefinal structure. AppendGeometries If set to Yes,the geometryfile in the XYZ-format will contain all the geome-tries obtained during the optimisation(instead of containing only the last geometry).Constraints Specifies geometry constraints.For every constraint the serial number of the atom is expected followed by the x,y,z components of a constraint vector.The specified atom is not allowed to move along the constraint vector.If two constraints are defined for the same atom, the atom will only by able to move normally to the the plane of the two constraining vectors.Example:Constraints={#Atom one can only move along the z-axis1 1.00.00.010.0 1.00.0}2.3.2ConjugateGradient{}MovedAtoms i+|m Range{1-1} MaxForceComponent r1e-4MaxSteps i200OutputPre x s"geo_end" AppendGeometries l NoConstraints(1i3r)*{}See previous subsection for the description of the properties.2.3.3VelocityVerlet{}Steps i-TimeStep r-Thermostat m-8 OutputPre x s"geo_end"Steps Number of MD steps to perform.TimeStep[time]Time interval between two MD steps.Thermostat Thermostating method for the MD simulation.See p.8.OutputPre x Prefix of the geometryfiles containing thefinal structure.ThermostatNone{}No thermostating during the run,only the initial velocities are set according a given temperature,hence an NVE ensemble should be achieved for a reasonable time step. InitialTemperature r-InitialTemperature[energy]Starting velocities for the MD will be created according the Max-well-Boltzmann distribution with the specified temperature.Andersen{}Andersen thermostat[2]sampling an NVT ensemble.Temperature r|m-ReselectProbability r-ReselectIndividually l-AdaptFillingTemp l NoTemperature[energy]Temperature of the thermostat.It can be either a real specifying a con-stant temperature through the entire run or the TemperaturePro le{}method specifying a changing temperature.(See p.9.)ReselectProbability Probability for reselecting velocity from the Maxwell-Boltzmann distribu-tion.ReselectIndividually If Yes,each atomic velocity is reselected individually with the specified probability.Otherwise all velocities are reselected at once with the specified probability.1 AdaptFillingTemp If Yes,the temperature of the electronfilling is always set to the current tem-perature of the thermostat.(The appropriate tag for the temperature of the electronfilling is ignored.)TemperaturePro le{}Specifies a temperature profile during molecular dynamics.It takes as argument one or more lines containing the type of the annealing(string),its duration(integer),and the target temperature(real),which should be achieved at the end of the given period.For example: Temperature[Kelvin]=TemperaturePro le{#Temperatures in Kconstant110.0#Setting T=10for the0th MD-steplinear500600.0#Linearly rising T in500steps up to T=600constant2000600.0#Constant T through2000stepsexponential50010.0#Exponential decreasing in500steps to T=10}The annealing method can be constant,linear or exponential.The duration is specified in steps of the driver containing the thermostat.The temperatures are given in atomic units,unless the Temperature keyword carries a modifier.The starting temperature for every annealing period is the target temperature of the previous one.As the last step of each annealing period a calculation at the target temperature is made.No calculation is made,however,for the starting temperature(since it is calculated as the last step of the previous period).In order to enforce the calculation of the starting temperature for thefirst annealing period,a constant profile with the duration of only one step should be specified asfirst(see the example).Otherwise the starting temperature is set to0, but no calculation for that temperature is made.The number of steps for the molecular dynamics is specified by the driver containing the thermostat. The MD is stopped after it,even if the TemperaturePro le tag contains more steps.If the sum of the annealing steps is less than the number of MD steps,the target temperature of the last annealing period is used for the remaining steps.(Note,that there is always a0th step in the MD,so the number of steps specified in the temperature profile should be1greater than the number of steps for the integrator.)2.4HamiltonianCurrently only a DFTB Hamiltonian is implemented,so you must set Hamiltonian=DFTB{}. The DFTB{}method may contain the following properties:1The original Andersen thermostat uses individual reselection,but the previous implementation in DFTB reselected all of the velocitiesSCC l NoSCCTolerance r SCC=Yes1e-5 MaxSCCIterations i SCC=Yes100 OrbitalResolvedSCC l SCC=Yes NoMixer m SCC=Yes Broyden{}11 MaxAngularMomentum p-Charge r0.0 SpinPolarisation m SCC=Yes{}14 SpinConstants p SpinPolarisation=Colinear{}-Eigensolver m DivideAndConquer{}14 Filling m Fermi{}15 IndependentKFilling l periodic system No SlaterKosterFiles p|m-15 OldSKInterpolation l No KPointsAndWeights(4r)+|m periodic system-16 OrbitalPotential m{}18 ReadInitialCharges l NoElectricField p{}19 Dispersion m{}20SCC If set to Yes,a charge self consistent(scc)calculation is made.SCCTolerance Stopping criteria for the SCC.Specifies the tolerance for the maximum difference in any charge between two SCC cycles.MaxSCCIterations Maximal number of SCC cycles to reach convergence.If convergence is not reached after the specified number of steps,the program stops. OrbitalResolvedSCC If set to Yes,all three(or four)Hubbard U values for the different angu-lar momenta are used,when calculating the SCC contributions.Otherwise,the value for the s-shell is used for all angular momenta.Please note,that the old standard DFTB code was not orbital resolved,so that only the Hubbard U for the s-shell was used.Please check the documentation of the SK-files you intend as to whether they were produced with an or-bitally resolved version of the code(many of the biologicalfiles do not use orbitally resolved charges),before you switch this option to Yes.Mixer Mixer type for mixing the charges in an SCC calculation.See p.11. MaxAngularMomentum Specifies the highest angular momentum for each atom type.Several main-block elements require d-orbitals,check the documentation of the SK-files you are using to determine if this is necessary.Possible values for the angular momenta are s,p,d,f.Example:MaxAngularMomentum={Ga="p"#You can omit the quotes around theAs="p"#orbital name,if you want.}Charge Total charge of the system.Negative value means electron excess. SpinPolarisation Specifies if and how the system is spin polarised.See p.14SpinConstants Specifies the atom type specific constants needed for the spin polarised calcula-tions,in units of Hartrees.For every atomic species in the calculation a property with the typename must be defined,containing the spin coupling constants for that atom.The constantsmust be specified as reals in the following order up to the highest angular momentum of thegiven atom type:w ss,w sp,w sd,...,w ps,w pp,w pd,...,w ds,w d p,w dd,...Example(GGA parameters for H2O):SpinConstants={O={#Wss Wsp Wps Wpp-0.035-0.030-0.030-0.028}H={#Wss-0.072}}Several standard values of atomic spin constants are given in appendix D.Constants calcu-lated with the same density functional as the SK-files should be used.This input block maybe moved to the SK-data definitionfiles in the future.Eigensolver Specifies which eigensolver to use for diagonalising the Hamiltonian.See p.14.Filling Method for occupying the one electron levels with electrons.See p.15. SlaterKosterFiles Name of the Slater-Kosterfiles for every atom type pair combination.See15. OldSKInterpolation If set to Yes(strongly discouraged),the lookup tables for the non-scc inter-actions are interpolated with the same algorithm as in the old DFTB code.Please note,thatthe new method uses a smoother function,is more systematic,and yields better derivativesas the old one.This option serves only compatibility purpose,and might be removed in thefuture.KPointsAndWeights[relative|absolute]Contains the special k-points to be used for the Brillouin-zone integration.See p.16.For automatically generated k-point grids the modifier shouldnot be set.OrbitalPotential Specifies which(if any)orbitally dependant contributions should be added to the DFTB energy and band structure.See p.18.ReadInitialCharges If set to Yes thefirst Hamiltonian is constructed by using the charge informa-tion read from thefile charges.bin.ElectricField Specifies an external electricfield.See p.19.Dispersion Specifies which kind of dispersion correction to apply.See p.20.2.4.1MixerDFTB+offers currently the charge mixing methods Broyden{}(Broyden-mixer),Anderson{}(Anderson-mixer),DIIS{}(DIIS-accelerated simple mixer)and Simple{}(simple mixer).Broyden{}Minimises the error functionE=ω20G(m+1)−G(m)+m∑n=1ω2nn(n+1)−n(n)|F(n+1)−F(n)|+G(m+1)F(n+1)−F(n)|F(n+1)−F(n)|2,where G(m)is the inverse Jacobian,n(m)and F(m)are the charge and charge difference vector in iteration m.The weights are given byω0andωm,respectively.The latter is calculated asωm=c√F(m)·F(m),(2.1)c being a constant coefficient.[12].The Broyden{}method can be configured using following properties:MixingParameter r0.2CachedIterations i-1InverseJacobiWeight r0.01MinimalWeight r 1.0MaximalWeight r1e5WeightFactor r1e-2MixingParameter Mixing parameter.CachedIterations Number of charge vectors of previous iterations which should be kept in the memory.Older charge vectors are written to disc.If set to-1,all charge vectors will be kept in the memory.(You should only change its value if you are really short on memory.) InverseJacobiWeight Weight for the difference of the inverse Jacobians(ω0). MinimalWeight Minimal allowed value for the weighting factorsωm.MaximalWeight Maximal allowed value forωm.WeightFactor Weighting factor c for the calculation of the weighting factorsωm in(2.1). Anderson{}Modified Anderson mixer.[7]MixingParameter r0.05Generations i4InitMixingParameter r0.01 DynMixingParameters(2r)*{}DiagonalRescaling r0.01MixingParameter Mixing parameter.Generations Number of generations to consider for the mixing.Setting it too high can lead to linearly dependent sets of equation.InitMixingParameter Simple mixing parameter used until the number of iterations is greater or equal to the number of generations.DynMixingParameters Allows to specify different mixing parameters for different levels of con-vergence during the calculation.These are specified as a list of tolerances below which a given mixing factor is used.If the loosest specified tolerance is reached,the appropriate mixing parameter supersedes that specified in MixingParameter. DiagonalRescaling Used to increase the diagonal elements in the system of equations solved by the mixer.This can help to prevent linear dependencies occuring in the mixing process.Setting it to too large a value can prevent convergence.(This factor is defined in a slightly different way from Ref.[7].See the source code for more details.)Example:Mixer=Anderson{MixingParameter=0.05Generations=4#Now the over-ride the(previously hidden)default old settingsInitMixingParameter=0.01DynMixingParameters={1.0e-20.1#use0.1as mixing if more converged that1.0e-21.0e-30.3#again,but1.0e-31.0e-40.5#and the same}DiagonalRescaling=0.01}DIIS{}Direct inversion of the iterative space is a general method to acceleration iterative sequences.The current implementation accelerates the simple mix process.MixingParameter r0.2Generations i6UseFromStart l YesMixingParameter Mixing parameter.Generations Number of generations to consider for the mixing.UseFromStart Specifies if DIIS mixing should be done right from the start,or only after the nr.of SCC-cycles is greater or equal to the number of generations.Simple{}Constructs a linear combination of the current input and output charges as(1−x)ρin+xρout. MixingParameter r0.05MixingParameter Coefficient used in the linear combination.2.4.2SpinPolarisationNo spinpolarisation({})No spin polarisation contributions to the energy or band-structure.Colinear{}Colinear spin polarisation in the z direction.The initialization of the calculation is spin restricted. UnpairedElectrons i-InitialSpin p{}UnpairedElectrons Number of unpaired electrons.(Kept constant during the calculation.) InitialSpin Initialisation for spin patterns.The default code behavior is an initial spin polarisation of0.InitialSpin The initial spin distribution can be set by specifying the spin polarisation on the atoms. For atoms without explicit specification,a spin polarisation of zero is assumed.The InitialSpin property must contain one or more AtomSpin blocks with the following properties:Atoms i+|m-SpinPerAtom r-Atoms Atoms to specify an initial spin value.The Range{}method can be used to specify an continous interval of atoms.SpinPerAtom Initial spin polarisation for each atom in this InitialSpin block.Example:SpinPolarisation=Colinear{UnpairedElectrons=0.0InitialSpin={#want to start from an anti-ferromagnetic orderingAtomSpin={Atoms={1}SpinPerAtom=-1.0}AtomSpin={Atoms={2}SpinPerAtom=+1.0}}}2.4.3EigensolverCurrently the following LAPACK3.0[1]eigensolver methods are available:•Standard{}•DivideAndConquer{}(this requires about twice the memory of the other solvers)•RelativelyRobust{}(using the subspace form but calculating all states)None of them needs any parameters or properties specified.Example:Eigensolver=DivideAndConquer{}2.4.4FillingFermi{}Fills the levels according to a Fermi distribution.When using afinite temperature,the Mermin free energy(which the code prints)should be used instead of the total energy.This is given by E−T S. Temperature r AdaptFillingTemp=No0.0Temperature[energy]Electron temperature in energy units.This property is ignored for ther-mostated MD runs,if the AdaptFillingTemp property of the thermostat had been set to Yes. Example:Filling=Fermi{Temperature[K]=300}MethfesselPaxton{}Produces a Fermi-like distribution but with much lower electron entropy[16].This is useful for systems that require high electron temperatures(for example when calculating metals) Temperature r AdaptFillingTemp=No0.0Order i2Temperature[energy]Electron temperature in energy units.This property is ignored for ther-mostated MD runs,if the AdaptFillingTemp property of the thermostat had been set to Yes. Order Order of the Methessel-Paxton scheme.The1st order scheme is equivalent to Gaussian filling.2.4.5SlaterKosterFilesThere are two different ways to specify the Slater-Kosterfiles for the atom type pairs,explicit specification and using the Type2FileNames{}method.Explicit specificationEvery possible atom type pair connected by a dash must occur as property with the name of the correspondingfile as assigned value.Example(GaAs):SlaterKosterFiles={Ga-Ga="./Ga-Ga.sk"Ga-As="./Ga-As.sk"As-Ga="./As-Ga.sk"As-As="./As-As.sk"}Type2FileNames{}You can use this method to generate the name of the Slater-Kosterfiles automatically using the element names from the geometry input.You have to specify the following propertiesPre x s""Separator s""Su x s"" LowerCaseTypeName l NoPre x Prefix before thefirst type name,usually the path.Separator Separator between the type names.Su x Suffix after the name of the second type,usually extension. LowerCaseTypeName If the name of the types should be converted to lower case.Otherwise they are used in the same way,as they were specified in the geometry input.Example(for producing the samefile names as in the previous section): SlaterKosterFiles=Type2FileNames{Pre x="./"Separator="-"Su x=".sk"LowerCaseTypeName=No}2.4.6KPointsAndWeightsThe k-points for the Brillouin-zone integration can be either specified explicitely or using the KLines{}or the SupercellFolding{}methods.If the latter is used the KPointsAndWeights key-word is not allowed to have a modifier.Explicit specificationFor every k-point four real numbers must be specified:The relative coordinates of the given k-point followed by its weight.The sum of the weights is automatically normed to one by the program.Ifthe modifier absolute is set for the KPointsAndWeights keyword,absolute k-point coordinates in atomic units are expected.KPointsAndWeights={#2x2x2MP-scheme0.250.250.25 1.00.250.25-0.25 1.00.25-0.250.25 1.00.25-0.25-0.25 1.0}SupercellFolding{}This method generates a sampling set containing all the special k-points in the Brillouin zone related to points that would occur in an enlarged supercell repeating of the current unit cell.If two k-points in the BZ are related by inversion,only one(with double weight)is used(this permitted by time reversal symmetry).The SupercellFolding{}method expects9integers and3reals as parameters:n11n12n13n21n22n23n31n32n33s1s2s3The integers n i j specify the coefficients used to build the supercell vectors A i from the original lattice vectors a j:A i=3∑j=1n i j a j.The reals s i specify the point in the Brillouin-zone of the super lattice,in which the folding should occur.The coordinates must be given in relative coordinates,in the units of the reciprocal lattice vectors of the super lattice.The original l1×l2×l3Monkhorst-Pack sampling[18]for cubic lattices corresponds to a uniform extension of the lattice:l1000l2000l3s1s2s3where s i is0.0,if l i is odd,and s i is0.5if l i is even.For the2×2×3scheme,you would write for example#2x2x3MP-scheme according original paperKPointsAndWeights=SupercellFolding{2000200030.50.50.0}To get the scheme for hexagonal lattices as proposed in the erratum of the original paper[19],you should set s1and s2to0.0independent,whether l1and l2are even or odd.s3must be set in the same way as in the original scheme.So,for a2×3×4sampling you would have to set#2x3x4MP-scheme according modi ed MP schemeKPointsAndWeights=SupercellFolding{2000300040.00.00.5}It is important to note that DFTB+does not take the symmetry of your system explicitely into account.For small high symmetric systems with a low number of k-points in the sampling this could eventually lead to unphysical results.(Components of tensor properties–e.g.forces–could be finite,even if they must vanish due to symmetry reasons.)For those cases,you should explicitely specify k-points with the correct symmetry.KLines{}This method specifies k-points lying along arbitrary lines in the Brillouin zone.This is usefull when calculating the band structure for a periodic system.(In that case,the charges should be initialised from the saved charges of a previous calculation with a proper k-sampling.Additionally for SCC calculations the number of SCC cycles should be set to1,so that only one diagonalisation is done using the initial charges.)The KLines{}method accepts for each line an integer specifying the number of points along the line segment,and3reals specifying the end point of the line segment.The line segments do not include their starting points but their end points.The starting point for thefirst line segment can be set by specifying a(zeroth)segment with only one point and with the desired starting point as end point.The unit of the k-points is determined by eventual modifier of the KPointsAndWeights property.(Default is relative coordinates.)Example:KPointsAndWeights[relative]=KLines{10.50.00.0#Setting(and calculating)starting point0.50.00.0100.00.00.0#10points from0.50.00.0to0.00.00.0100.50.50.5#10points from0.00.00.0to0.50.50.510.00.00.0#Setting(and calculating)a new starting point100.50.50.0#10points from0.50.50.0to0.50.50.0}2.4.7OrbitalPotentialCurrently only the FLL{}(fully localised limit)form of the LDA+U corrections[20]are imple-mented.This particular potential lowers the energy of states localized on the specified atomic shells while raising the energy of un-occupied localised states.This particular correction is most useful for lanthanide/actinide f states and some localised d states of transition metals(Ni3d for example). Implementation s-LConstants p-Implementation Implementation type.Currently only Eschrig’s purely onsite density matrix con-tributions to the chosen functional("on-site")is implemented[9].。

Quasiclassical trajectory calculations of collisional energy transfer in propane systemsApichart Linhananta and Kieran F.Lim*¤Centre for Chiral and Molecular T echnologies,School of Biological and Chemical Sciences,Deakin University,Geelong,V ictoria3217,Australia.E-mail:lim=.auRecei v ed6th December1999,Accepted27th January2000Published on the Web9th March2000Quasiclassical trajectory calculations of collisional energy transfer(CET)and rotational energy transfer from highly vibrationally excited propane to rare bath gases are reported.The calculations employed atomÈatom pairwise-additive Lennard-Jones,Buckingham exponential and hard-sphere intermolecular potentials to examine the dependence of CET on the intermolecular potential and to establish a protocol for future work on larger alkane systems.The role of the torsional(internal)and molecular(external)rotors in the energy-transfer mechanism were parison of the results with our earlier work on ethane]neon systems(A. Linhananta and K.F.Lim,Phys.Chem.Chem.Phys.,1999,1,3467)suggests that the internal and external rotors play a signiÐcant role in the deactivation mechanism for highly vibrationally excited alkanes.I.IntroductionGas-phase chemical reaction rates are strongly dependent on intermolecular collisional energy transfer(CET).CET is a vital component in any combustion-model and atmospheric-model systems.The only experimental CET quantities for hydrocarbon fuel molecules have been inferred““indirectlyÏÏfrom measurements of pressure-dependent reaction rates.1h3 Despite this,there have been no systematic theoretical dynamics studies of CET of hydrocarbon and halogenated hydrocarbons.In fact,most theoretical studies have been on small molecules.3h14The exceptions are the quasiclassical tra-jectory(QCT)calculations of azulene,toluene,benzene and hexaÑuorobenzene systems.15h23We have recently reported QCT calculations for highly vibrationally excited ethane in neon bath gas.24This and the recent work by Svedung et al. are,to our knowledge,theÐrst theoretical CET studies of an alkane with internal rotors.24,25Comparisons of theoretical and experimental studies of CET show that many of the dominant energy transfer mecha-nisms in small molecules are also present in large mol-ecules.3h6However,there are several di†erences between large-substrate and small-substrate behaviours.A notable example is that in the CET from a““smallÏÏsubstrate to a rare gas collider the trend He[Ne[Ar is observed,26h28 whereas the opposite trend of He\Ne\Ar is observed for ““largeÏÏsubstrates.29h39Theoretical studies of CET on small molecules employing various techniquesÈquantum,semi-classical and classical dynamicsÈall have correctly predicted the small-substrate behaviour.40,41This is not the case for large-substrate systems where QCT simulations incorrectly found the same smallsubstrate trend.15h18The discrepancy may be due to the lack of reliable data on the intermolecular potential surface involving large molecules and is most likely to be manifested in systems with the small collider helium bath gas.42,43¤Lim Pak Kwan.The aforementioned QCT calculations of large-substrate molecules have been on aromatic hydrocarbons because they have been most amenable to experimental studies using spec-troscopic probes.There have been fewer studies28,44h47on alkanes and branched-alkanes,which are the main com-ponents of common combustion fuels,and their halogenated analogues,which are important in ozone and““greenhouseÏÏchemistry.TheÐrst most obvious di†erences between alkanes and aromatics are their shapes,which are expected to a†ect the rotational energy transfer(RET).Since rotation to trans-lation(R]T)energy transfer and vibration to rotation (V]R)energy transfer are often more efficient than vibration to translation(V]T)energy transfer,this can have a strong inÑuence on the overall CET.Another crucial aspect is theÑexibility of alkanes.QCT simulations of alkanes would require the development of an efficient algorithm for sampling conformer space.Related to theÑexibility,as well as to RET,is the role of internal rotors in the CET mechanism.QCT calculations of highly vibra-tionally excited ethane in neon bath gas show that there is an interrelationship between the internal methyl rotors and the external rotation giving rise to V]torsion,R energyÑow in theÐrst collision,resulting in an““enhancedÏÏCET in sub-sequent collisions.24The same e†ect is also observed in experiments on the deactivation of highly vibrationally excited benzene and toluene,where toluene has much larger CET values than benzene.37This e†ect suggests that the torsional rotors in alkanes are important.Since the use of intramolecular torsional potential terms (nine such terms for each additional methylene unit)24plus a sampling of the conformational space may prove to be cost-prohibitive for large(r)alkane systems,there is a need to establish an e†ective protocol for QCT alkane simulations. Use of a hard-sphere potential will reduce a substrateÈcollider collision into a sequence of““sudden-impactÏÏatomÈatom encounters.Furthermore,there is no need to calculate molec-ular interactions at medium-to-large atomic separations.This paper““benchmarksÏÏCET using a hard-sphere potential against the more-commonly used Lennard-Jones andDOI:10.1039/a909614k Phys.Chem.Chem.Phys.,2000,2,1385È13921385This journal is The Owner Societies2000(Buckingham-type exponential-6models,by performing QCT calculations on the propane ]monatomic collider systems.The role of the torsional (internal)and molecular (external)rotors in the energy-transfer mechanism are reported.II.Quasiclassical trajectory calculationsA.Intermolecular potentialThe lack of knowledge of the detailed form of intermolecular potentials has always been a hindrance to quasiclassical mod-elling of CET.This is especially true for large-substrate systems,where there is a paucity of reliable theoretical and experimental data.Previous trajectory calculations of large molecules usually modelled the intermolecular potential by pairwise-additive atom Èatom potentials:7h 24,48h 50the inter-action parameters were usually obtained by semiempirical methods.Collins and coworkers have ““builtÏÏintermolecular potentials by interpolation of ab initio data:51h 53thus far they have only applied their method to relatively small polyato-mics whereas we wish to use a protocol that can be consistent-ly and easily ““scaled upÏÏfor larger alkane systems.Hence in this work three pairwise-additive atom Èatom intermolecular potentials were employed.The Ðrst intermolecular potential was the pairwise-additive Lennard-Jones (LJ)potential with atom Èatom terms given by V ij \4e ijCA p ij r ij B 12[A p ij r ijB 6D,(1)(i \C,H;j \rare gas),where is the atom Èatom centre-of-r ijmass separation,and and are the Lennard-Jones radiusp ij e ijand well depth,respectively.The LJ parameters were chosen by the method of Lim to match empirical values.16,29,54The second intermolecular potential was the pairwise-additive Buckingham exponential (exp-6)potential with atom Èatom terms given byV ij \A ij exp([c ij r ij )[C ij r ij~6,(2)where the parameter determines the repulsive steepness ofc ijthe potential.55The parameters and were chosen toA ij C ijmatch empirical values.16,29,54The last intermolecular potential was a pairwise-additive hard-sphere (HS)potentialV ij \GO ,0,r ij O r ij vdW ,r ij [r ijvdW ,(3)where is the van der Waals radius 56between atoms i andr ijvdW j .This potential is in the spirit of the e†ective mass theory.57The HS potential is tested here to determine if it can be used to derive useful qualitative information:if so then it would be a useful model for simulations of larger alkanes.The intermolecular parameters for propane ]Rg (Rg \rare gases He,Ne and Ar)potentials are given in Table 1.B.Intramolecular potentialA simple harmonic valence force Ðeld,consisting of harmonic stretches,bends and torsions,was used to describe the propane substrate:V intra\;i V stretch,i ];i V bend,i ];iV torsion,i .(4)The Ðrst two terms have been deÐned previously.15,58,59The harmonic stretching and bending force constants were obtained by the empirical prescription of Lindner:60k str,CC\4.705]102J m ~2,J m ~2,k str,CH \4.702]102k bend,CCH\6.67]10~17J rad ~2,and J rad ~2.k bend,HCH\5.61]10~17The Ðnal term in eqn.(4)is a 3-fold methyl torsional potential,which was assumed to be:V torsion,i \V 0n ;j /1n cos 2A 3qij 2B.(5)The torsional angles are the nine H ÈC ÈC ÈH or H ÈC ÈC ÈCq ijdihedral angles for each of the i th C ÈC bonds.Each carbon centre was assumed to have perfect tetrahedral geometry with C ÈC and C ÈH bond lengths of 0.1543nm and 0.1093nm,respectively.To study the e†ect of the torsion,the torsional barrier parameter was taken to be 0(free rotors)and 13.8V 0kJ mol ~1(experimental barriers).61The direction of the bond vectors was deÐned so that the staggered conformer has the lowest-energy geometry.The free-rotor model has apparent harmonic torsional ““vibrationalÏÏfrequencies of 9.2and 9.3cm ~1while the hindered-rotor model has apparent harmonic torsional ““vibrationalÏÏfrequencies of 167.4and 186.3cm ~1.These fre-quencies arise from the numerical normal mode analysis and are used in the selection of initial conditions.58,59,62The other 25vibrational frequencies compare favourably with experi-mental group frequencies of putational detailsTrajectory calculations were performed using program MARINER 58which is a customised version of VENUS96.59The LJ and exp-6potential models,selection of initial condi-tions,and general methodology are standard options in program MARINER/VENUS96.58,59,62The initial impact energy was chosen from a 300K thermal distribution.InE transthe majority of cases,the initial rotational angular momentum of propane was chosen from a thermal distribution at 300K.The rotational temperature was varied from 100to 1500K to investigate the RET of propane ]argon by the HS model.The initial vibrational phases and displacements were chosen from microcanonical ensembles at E @\41000,30000or 15000cm ~1,where E @is the rovibrational energy above the zero-point energy.These initial conditions are appropriate for comparison with the Ðrst few collisions in time-resolved infra-red Ñuorescence and ultraviolet absorption experi-ments.3h 6,29h 38,64Note that experiments measure the CET values of a cascade of collisions.The rovibrational energy dis-tribution of subsequent collisions will not be microcanonical,Table 1Intermolecular potential parametersLJ model exp-6model HS model p (e /k B )Aij Cij c ij r vdW /nm/K /kJ mol ~1/10~6kJ mol ~1nm 6/nm 1/nm H ÉÉÉHe 0.28258.0882294712479.945.50.325C ÉÉÉHe 0.291517.6931859254179745.60.345H ÉÉÉNe 0.293817.00103476168.5345.70.305C ÉÉÉNe 0.302034.156********.6945.90.325H ÉÉÉAr 0.306628.87140033519.240.80.335C ÉÉÉAr0.321658.025809650187641.00.3551386Phys .Chem .Chem .Phys .,2000,2,1385È1392but the CET behaviour of these subsequent collisions can be inferred 18,65,66from the microcanonical values.For the models employing the LJ and exp-6intermolecular potentials,trajectories were initialised with a centre-of-mass separation of 1.2nm and the classical equations of motion were integrated by the Adams ÈMoulton algorithm 58,59,62until the distance between the monatomic collider and the closest hydrogen exceeded a critical value of 1.0nm,at which point the trajectory was terminated.The initial impact param-eter b was chosen with importance sampling 16,17,58,59,62for values between 0nm and nm (He and Ne)or 0.9nmb m\0.8(Ar).These initial and Ðnal conditions were chosen by per-forming preliminary runs which showed that an insigniÐcant amount of energy was transferred at larger distances.For the HS interaction model,there is no intermolecular interaction until the point of impact,when the propane sub-strate is still described by a (near)microcanonical putationally,this is achieved by initialising trajectories as above,but translating the colliders to the point of initial contact without altering the rovibrational phases and orienta-tion.The translation was performed using an algorithm devel-oped by Alder and Wainwright 67,68to model hard-sphere Ñuid systems.After this initial point of contact,the trajectory was propagated normally.At each time step,the interatomic distances between the rare gas collider and every propane atom were checked for overlap.If an atom Èatom encounter occurred,the trajectory was projected back to the point of impact and the impulsive momentum transfer was calcu-lated.68The process was repeated until another encounter occurred or until the distance between the monatomic collider and the closest hydrogen exceeded a critical value,at which point the trajectory was terminated.Program MARINER 58was altered to implement the HS potential and trajectory-propagation algorithms.The short-ranged HS interaction per-mitted critical values as low as 0.4nm.Since the equations of motion are integrated for a comparatively short period,the HS model required much less computing time than the LJ and exp-6models.For E @\15000and 30000cm ~1,the integration time step was chosen to be 0.085fs,which is sufficient to conserve total energy to within 0.5cm ~1.This is approximately four times larger than the time step used in our previous ethane trajec-tory calculations.24Propane has less excitation per vibra-tional mode and hence energy can be conserved by larger time steps.For E @\41000cm ~1,it was necessary to employ a time step of 0.075fs to conserve energy.The numerical insta-bilities associated with the inversion of the methyl group(s)previously observed in simulations of ethane 24and toluene 16,17were not observed here.The calculations were performed on a DEC Alpha 3000/300LX workstation and an SGI Power Challenge Super-computer.In calculations that employed the LJ or exp-6intermolecular potentials,batches of 3000trajectories required approximately 60CPU hours for He collider and 100CPU hours for Ar on the workstation.The HS model decreased the required CPU time by a factor of 10:this reduction will be very signiÐcant in the study of larger alkanes.CPU time was reduced by a factor of about 4on the supercomputer.D.Rotation energy and torsional angular momentum It is well documented that rotational energy transfer is an effi-cient pathway for CET.3,24,65,66,69However,while angular momenta are well-deÐned,rovibrational coupling gives rise to an ambiguity in the deÐnition of rotational energy.Previous quasiclassical simulations employed several di†erent methods to decouple the rotational and vibrational energies.One method 11deÐnes the rotational energy asE rot \1(JI ~1J ),(6)where I and J are,respectively,the instantaneous moment of inertia and angular momentum.In a second method isE rotapproximated by the instantaneous angular momentum,but the moment of inertia is taken to be the equilibrium geometry value.11Both deÐnitions give rotational energies that oscillate with time.There is an alternative deÐnition that is valid for symmetrical top rotors:65E rot \1B effJ 2,(7)where J is the magnitude of the rotational angular momentum and is an e†ective rotational constant.This deÐnitionB effdecoupled the rovibrational energy so that the rotational energy includes only the ““adiabatic partÏÏ,whereas the ““activeÏÏpart is included with the vibrational energyE V \E [E rot,(8)where and E are,respectively,the vibrational and totalE Vinternal energies.Eqn.(7)is a valid approximation for sym-metrical top molecules.70The main advantage of this deÐni-tion is that,classically,it is a conserved quantity.The equilibrium Cartesian principal moments of inertia of propane are kg m 2,kgI xx \1.11]10~45I yy\9.7]10~46m 2and kg m 2.Hence,propane is a goodI zz\2.97]10~46approximation of a symmetrical top and it is possible to deÐne the rotational energy by eqn.(7),with the approx-imationB eff \12hc (I xx I yy I zz)~1@3.(9)It was shown in our previous work on ethane 24that the coupling between external and internal rotors enhances the overall CET.Hence the torsional angular momentum of propane was also monitored in this work.Whereas ethane has only one torsional rotor which lies along its molecular axis,propane has two distinct and unparallel torsional rotors.The deÐnition of the torsional angular momentum introduced for ethane is generalised by calculating the rotational angular momentum of the methyl group and the associated ethyl groupJ methyl \;i /H,H,Hr i ]p iJ ethyl\;i /C,H,Hr i ]p i,(10)where is the angular momentum of the methyl groupJ methyland is the angular momentum of the associated ethylJ ethylrotor.Note that for consistency with eqn.(5),only the six atoms directly bonded to each torsional C ÈC bond have been included in the summation in eqn.(10).The torsional angular momentum is then deÐned asJ tor \o (J methyl [J ethyl)Éa o ,(11)where is a unit vector parallel to the CC torsional axis.The a CET to/from the torsional rotors was monitored by calcu-lating the average torsional angular momentum change*J tor \J tor (Ðnal)[J tor(initial).(12)E.Data analysisTrajectory data were analysed by a bootstrap algorithm 71,72in program PEERAN.16,73Some 3000È5000trajectories were performed for each potential model.This was sufficient to obtain average energy-transfer quantities with statistical uncertainties of about 10%.However,the uncertainties for the average rotational energy transfer were about 20%,due to the initial rotational-energy Boltzmann distribution (rather than an initial microcanonical distribution).Trajectory averagesPhys .Chem .Chem .Phys .,2000,2,1385È13921387deÐned by (for both overall CET and RET)S*E n TtrajS*E n T traj \1N ;i /1N bi bm(*E i )n(13)are related to experimentally obtained quantities S*E n T by ratio of collision cross-sectionsS*E n T \p b m 2p p LJ2X (2,2)RS*E n T traj (14)where is the LJ collision cross-section and is thep LJ 2X (2,2)R b mmaximum impact parameter in the trajectory simulation.This normalisation removes the ambiguity related to the elastic scattering at high impact parameter.74The input LJ param-eters were obtained from ref.29.At 300K,the LJ collision cross-section values of nm 2,0.4834nm 2p LJ2X (2,2)R \0.3976and 0.6945nm 2for propane ]He,propane ]Ne and propane ]Ar,respectively,were obtained using the program COLRATE.75This corresponds to the LJ collision frequencies of m 3s ~1,328.58]10~18m 3s ~1Z LJ,coll\523.29]10~18and 382.37]10~18m 3s ~1,respectively.In this paper,we have reported both the Ðrst and second moments of the trajectory data since the Ðrst moment is usually more useful for comparison with experiment,but the QCT second moment is statistically more reliable.74Some experiments can determine both the Ðrst and second moments of the CET probability.3,5III.Results and discussionA.The e†ect of the torsional barrierFigs.1and 2show the CET values,[S*E T and S*E 2T 1@2,and the RET values,as functions of energy E @aboveS*E RT ,zero-point energy for propane ]neon.One set of results areFig.1Dependence of energy-transfer quantities on torsional barrier for deactivation of vibrationally excited propane by neon bath gas:)Hindered-rotor (LJ);Free-rotor (LJ);Hindered-rotor (exp-6);L +…Free-rotor (exp-6).Fig.2Dependence of rotational energy transfer on torsional barrier for deactivation of vibrationally excited propane by neon bath gas:)Hindered-rotor (LJ);Free-rotor (LJ);Hindered-rotor (exp-6);L +…Free-rotor (exp-6).for the free-rotor model the other for the hindered-(V 0\0),rotor model kJ mol ~1).These results are for the LJ(V 0\13.8and exp-6intermolecular potentials.The overall deactivation,[S*E T and S*E 2T 1@2,is larger for the hindered-rotor model,similar to results for ethane ]neon.24The torsional angular momentum transfer is shownS*J torT in Fig.3.Note that for the hindered-rotor modelsS*J torT with both LJ and exp-6intermolecular potentials are virtually identical:the reason for this is unclear.Overall,S*J torTdecreases,but remains positive,with the presence of a barrier In contrast,for ethane ]neon changes from posi-V 0.S*J torT tive to negative over a similar range of values.24This di†er-V 0ence is probably due to the higher torsional moment of inertia for propane torsion compared to ethane(CH 3ÈCCH 2)This means that propane torsion has higher(CH 3ÈCH 3).density of states and can more readily gain torsional excita-tion than ethane torsion,explaining why is positiveS*J torT for propane,but negative for ethane.In ethane,the torsion acts like a vibration providing an efficient torsion ]T pathway.24The increase in [S*E T and S*E 2T 1@2(Fig.1)for the hindered-rotor model suggests that propane torsions play the same role in the CET mechanism.The RET is smaller for the propane free-rotor modelS*E RT than the hindered-rotor model (Fig.2),contrary to the ethane results.24For ethane,the torsion is aligned along the molecu-lar axis,hence any increase in methyl-rotor angular momen-tum contributes to both (internal)torsional excitation S*J torTand (external)rotational excitation The propane free-S*E RT .rotor model has Ðve (three external and two internal)indepen-Fig.3Dependence of torsional angular momentum change on tor-sional barrier for deactivation of vibrationally excited propane by neon bath gas:Hindered-rotor (LJ);Free-rotor (LJ);)L +Hindered-rotor (exp-6);Free-rotor (exp-6).Note that the two sets …of hindered-rotor results are almost identical.1388Phys .Chem .Chem .Phys .,2000,2,1385È1392dent rotors,none of which have coincident axes.The extra rotors mean that there is less energy available to the external rotors in any V ]torsion,R energy redistribution.Noteworthy is the fact that the di†erences between the free-rotor and hindered-rotor models persist up to E @\41000cm ~1.For ethane ]neon,there is an onset of near-free-rotor behaviour at E @\30000cm ~1:at E @\41000cm ~1there is no signiÐcant di†erence between the free-and hindered-rotor models.However,the larger number of vibrational modes in propane,which decreases the excitation per torsional mode,ensures that the di†erences remain even at very high excita-tion.Hence correct theoretical treatments of internal rotors become even more essential for larger molecules.B.Trajectory results for LJ and exp-6modelsThe CET results for the deactivation of highly excited propane by helium,neon and argon are shown in Fig.4,where the intermolecular interactions have been modelled by the LJ and exp-6potentials.Three important features are:(1)Energy transfer increases with increasing E @and is in accord with theoretical and experimental studies on the deac-tivation of highly vibrationally excited molecules.(2)The LJ potential results in larger CET values than the exp-6model,since the LJ potential has a much harder repul-sive part than the exp-6potential.There are numerous works which concluded that CET depends mainly on the repulsive part of the intermolecular potential and that,in general,a harder repulsive part results in larger energy transfers.9,16,17(3)The deactivator efficiency shows the trend He [Ne [Ar which,unfortunately,is in discord with experi-mental trends for Ñuorinated alkane systems.28To our knowledge,there has been no experimental study of CET in propane ]rare gas systems.““IndirectÏÏstudies of related systems include 2-bromopropane ]Ne ([S*E T \130cm ~1for E @\17000È21000cm ~1)76andFig.4Energy-transfer quantities for deactivation of vibrationally excited propane by rare gases:Helium (LJ);Neon (LJ);)K |Argon (LJ);Helium (exp-6);Neon (exp-6);Argon (exp-6).+=>isotopically-substituted cyclopropane ]He (S*E 2T 1@2\200È400cm ~1for E @D 22000cm ~1).2These CET quantities were not directly measured,but were inferred from pressure-dependent thermal reaction rates at elevated temperatures.Some more recent studies using time-resolved optoacoustic spectroscopy include ([S*E T \114cm ~1atC 3F 8]Ar E @\15000cm ~1and [S*E T \300cm ~1at E @\40000cm ~1).46These studies reveal no information about RET nor the role of torsional modes.These experimental CET quan-tities correlate well with our present calculations (Fig.4)but also indicate a need for fresh experimental studies.The decreasing trend with collider He [Ne [Ar has been observed in many other QCT studies.9,15,18,77Although the lack of qualitative agreement with experiment is disappoint-ing,these studies and the present work have used very crude intermolecular potential models.Given the lack of detailed information about polyatomic intermolecular potential sur-faces,the intention in the present and other studies has been to use a set of consistent and transferable potentials,16much in the spirit of molecular mechanics force Ðelds.Experience with simulations on other systems would suggest that the exp-6model potentials predict ““betterÏÏCET values than the LJ potentials.17Fig.5plots the RET of propane ]rare gas systems.For Ne and Ar,monotonically increases with E @,whereas forS*E RT He,it initially increases but decreases at higher excitation energy.In all cases,RET is larger for the LJ model which is in accord with the CET behaviour.Clary and Kroes 78and others 16,17,40have observed that RET is larger for heavier colliders because the collision duration is closer to the rota-tional period of the molecular substrate.Fig.6plots the torsional angular momentum transfer as a function of E @.is largest for He and smal-S*J tor T S*J torT lest for Ar,which is the same trend as for CET.This implies that,in addition to the external rotor gateway,the torsional rotor is a gateway for facile CET via V,torsion ]torsion,T.24An interesting feature of Fig.6is that seems to beS*J torT Fig.5Rotational energy transfer for deactivation of vibrationally excited propane by rare gases:Helium (LJ);Neon (LJ);)K |Argon (LJ);Helium (exp-6);Neon (exp-6);Argon (exp-6).+=>Phys .Chem .Chem .Phys .,2000,2,1385È13921389Fig.6Torsional angular momentum change for deactivation ofvibrationally excited propane by rare gases:Helium(LJ);Neon)K(LJ);Argon(LJ);Helium(exp-6);Neon(exp-6);Argon|+=>(exp-6).insensitive to the intermolecular potential.However,the factthat it depends on the type of bath gas indicates a dependenceon the mass of the deactivator.This suggests that isS*JtorTinsensitive to theÐne details of the intermolecular potentialand can be modelled by either LJ or exp-6potentials.C.Trajectory results for hard-sphere modelLJ and exp-6potentials have long-range attractive terms andare computationally expensive.Since HS is a short-rangepotential,it is computationally cheaper in terms of computertime than other potential models by an order of magnitude.Inthis section we compare the results of the short-range HS withthe longer-range potentials.Fig.7shows S*E T and S*E2T1@2for the HS model.Fig.8shows the RET for the HS model.The qualitative behavioursare the same as for the LJ and exp-6models but the energy-transfer values are several times larger than for the LJ andexp-6model.This is not surprising in view of the““hardnessÏÏof the HS potential.9,16,17Another important feature is thatS*E T and S*E2T1@2for He are several times larger than forNe and Ar.This is also true for the LJ model(Fig.3)whichindicates that the HS and LJ models tend to give CET valuesthat are much too high for helium colliders.Table2lists the average number of encounters per collision,for He,Ne and Ar colliders.This average includes onlyNC,trajectories in which collisions have occurred.As expected NCTable2Average number of atomÈatom encounters NcE@/cm~1150003000041000Propane]He 1.967 1.884 1.847Propane]Ne 3.145 2.952 2.852Propane]Ar 3.753 3.501 3.400Fig.7Energy-transfer quantities for deactivation of vibrationallyexcited propane by rare gases for the HS model:Helium;Neon;+=Argon.>is largest for Ar and smallest for He due to their reducedmasses.also decreases with increasing E@which suggestsNCthat a more highly excited substrate imparts more energy perencounter to the deactivator,reducing the collision duration.Fig.9shows S*E T,and for propane]argonS*EVT S*ERTsystems at rotational temperatures300,1000andTROT\100,1500K.In these simulations,initial excitation wasÐxed atE@\15000cm~1and the initial translational temperaturewas K.It can also be seen that RETTtrans\300S*ERTdecreases with increasing the magnitude of the vibra-TROT;tional energy transfer also decreases with increasingS*EVTThis implies that rotationally cold systems exhibitTROT.V]R,T energy transfer,whereas rotationally hot systemsexhibit V,R]R,T.It can be seen that the overall[S*E T islarger for larger which agrees with the hypothesis thatTROTthe external rotation is a facile CET path.This behaviour hasFig.8Rotational energy transfer for deactivation of vibrationallyexcited propane by rare gases for the HS model.Helium;Neon;+=Argon.>1390Phys.Chem.Chem.Phys.,2000,2,1385È1392。

●竞赛时间4小时。

迟到超过半小时者不能进考场。

开始考试后1小时内不得离场。

时间到，把试卷(背面朝上)放在桌面上，立即起立撤离考场。

●试卷装订成册，不得拆散。

所有解答必须写在指定的方框内，不得用铅笔填写。

草稿纸在最后一页。

不得持有任何其他纸张。

●姓名、报名号和所属学校必须写在首页左侧指定位置，写在其他地方者按废卷论处。

●允许使用非编程计算器以及直尺等文具。

第1题大自然中的金属A只存在于化合物中，主要是通式为M x(SiO4)y硅酸盐和它的氧化物。

氧化物存在多种形态，并且易形成配位数CN metal = 7的单斜结构。

温度高于1100°C，它的晶格转变为四方结构。

温度超过2000°C它转变成立方结构。

后者的晶格结构是氟化钙型，晶格常数为a0 = 507 pm.。

这个结构在室温下可以被CaO稳定。

纯的氧化物（立方结构）密度为6.27 g·cm-3。

1.画出立方结构的晶胞。

2. 这个氧化物的实验式是什么？3. 这个氧化物中金属的配位数为多少？4. 金属在氧化物中的氧化数和在硅酸盐中的一样。

给出硅酸盐的实验式。

5.这个金属是什么？给出过程。

6.写出这个金属的电子构型。

7. 该氧化物中阳离子和阴离子的配位数为多少？8.运用如下热力学数据，计算氧的电子亲和能∆EA H°(O(g) + 2 e-→ O2-(g)):∆sub H°(M) = 609 kJ·mol-1; ∆IE H°(M/M n+) = 7482 kJ·mol-1; ∆diss H°(O2) = 498 kJ·mol-1;∆lattice H°(M-oxide) = -10945 kJ·mol-1; ∆f H°(M-oxide) = -1100 kJ·mol-1;得到这个金属的过程是一个两步合成。

第一步是硅酸盐的碳氯化，即在高温下与碳和气体氯反应。