a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we find that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark effective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coefficients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suffices to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the effective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures thefinite mass difference between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)defined by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus effectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the effective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have refined the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such effects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only briefly sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”off-shell momenta of the heavy quarks.Due to the phase redefinition of the effective heavy quark fields in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coefficient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which define the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the effective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coefficient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one first writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coefficient function.possible subtraction terms.Because of theflavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The effective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suffices to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonfield,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usfirst present the result for the nonperturbative power corrections.WefindK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its effect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we find that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our final result,there is one more effect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the effective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the effective theory[16,18].Onefinds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,wefind stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourfinal result forχ3(y)is not very different from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisfigure one can also deduce how changes in the values of the vacuum condensates would affect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Wefind that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonfield and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesfinancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientific,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisfigure and are both represented by the narrow dots.11。
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。
Painleve方程是一种非线性微分方程，其研究在非线性科学中具有重要的地位。
Painleve分析方法被广泛用于判定非线性微分方程的可积性和求解。
对于高阶的Levi方程，Painleve分析主要涉及调谐因子的计算和Painleve测试，以确定方程是否具有Painleve性质。
此外，相容性分析也被用于将方程组的解的奇异流形展式截断为有限项的形式。
对于(1+1)维Levi方程，可以利用WTC方法进行Painleve分析。
通过仿照Painleve求解常微分方程（组）调谐因子的方法，可以得到(1+1)Levi方程的调谐因子，并通过Painleve测试证明方程具有Painleve 性质。
此外，还可以通过相容性分析得出相容方程，从而将方程组的解的奇异流形展式截断为有限项的形式。
利用Schwarz导数及其性质可以得到方程的一类精确解，同时得到方程的一个自Backlund变换。
对于(2+1)维Levi方程，同样可以使用WTC方法进行Painleve分析。
特别地，对于特殊情况下的耦合(2+1)维Levi方程，可以得到其单孤子解、双孤子解及多孤子解。
对于求得的Levi方程的精确解，可以选择适当的参数，利用Matlab作出了解的图形，并根据图形分析解的性质。
在对耦合(2+1)Levi方程双孤子解图像的分析中，还可以讨论孤立波的裂变现象。
对于(1+1)维Levi方程主导项系数的情况，也可以进行Painleve测试，得到这些情况下的相容方程组。
以上内容仅供参考，如需更多信息，建议查阅相关文献或咨询数学领域的专家。
