π-共轭分子中激发态的对称性与暗态的起源.pdf

CHINESE JOURNAL OF CHEMICAL PHYSICSVOLUME 36,NUMBER 3JUNE 27,2023ARTICLEExcitation Symmetry and Origin of Dark States in-Conjugated MoleculesJun Xie,Weidong ShengState Key Laboratory of Surface Physics and Department of Physics,Fudan University,Shanghai 200433,China(Dated:Received on November 4,2021;Accepted on March 9,2022)Utilizingtheexactdiagonalizationmethod,the optical absorption spectra oftwo-conjugated molecules,anthraceneand pyrene,are calculated in variousdielectric environments.In a confinedquantumsystemwithananisotropicgeometry,it is commonly believed thatthe first excited state is localized alongthe elongated direction.In the meantime,the dipole approximation says that thetransitions to those states localized along the elongated direction shall generally havehigher intensities.In this work we report that anthracene and pyrene would respectivelyfail these intuitive expectations.It is found that the first active transition in anthraceneis always polarized along its short axis direction.For pyrene,it is revealed that thetransition of the highest intensity is the one polarized along the short axis direction of themolecule.Furthermore,the first excited state in either anthracene or pyrene is often foundto be optically inactive,which is successfully attributed to the short-range interactions byexamining the energy spectra in varying interaction environments.Key words:Excitation symmetry,Optical absorption,Dark state,Exact diagonalizationI.INTRODUCTIONOrganic semiconducting materials have attractedmuch attention during the past tens of years 16,mostly due to their potential application in microtran-sistors and optoelectronics such as photodiodes andphotoresistors.To have better insight into the electri-cal and optical properties of these materials,one needsto first study their electronic structure and excitationstates.In these organic nanostructures,the excitonicand quasiparticle effects which are induced by many-electron interactions 711 often dominate the opticalemission and absorption processes.However,some in-teresting features such as symmetry in the excited statesand dark(inactive)states in the absorption remain un-Author to whom correspondence should be addressed.E-mail:clear despite various theoretical and numerical attempts1214.Anthracene and pyrene have the simplest nontrivial-conjugated structure among all the organic molecules.Here we present a comparative study of the excitedstates in these two different-conjugated molecules 15.By using the exact diagonalization(ED)method,ourattention will first focus on the symmetry of the excitedstates.In a confined quantum system with an anisotropicgeometry,it is commonly believed that the first excitedstate is localized along the elongated direction while thesecond excited state shall be along the short axis direc-tion.However,we here reveal that anthracene wouldfail this intuitive expectation as the first active tran-sition of the molecule is found always polarized alongthe short axis direction.While pyrene does not exactlybreak the expectation,the first noticeable transition isDOI:10.1063/1674-0068/cjcp2111221307c 2023 Chinese Physical Society308Chin.J.Chem.Phys.,Vol.36,No.3Jun Xie et al.also polarized along its short axis direction.In the meantime,the dipole approximation says thatthose transitions to the states localized along the elon-gated direction generally shall have higher intensities.While the strongest transition in anthracene is indeedpolarized along its elongated direction,however we re-veal that pyrene would fail this intuitive expectation asthe second transition in pyrene is found to be polarizedalong its short axis direction while having almost thehighest intensity among all the transitions.For many-conjugated molecules and even somekinds of low-dimensional nanostructures,it is not un-common to find that the first excitation is optically in-active 1619.In spite that various affords have beenmade,the origin of these dark excited states remainselusive.In this work,we are going to explore the ori-gin of these dark states and reveal that the short-rangepart of the many-particle interactions is responsible forthe first excitation being dark and strong long-rangeelectron-electron interactions may turn the first excita-tion in anthracene or pyrene back to bright.II.MODEL AND METHODTo capture the complicate many-particle physics inmolecules,one often resort to the Hubbard model,asgiven byH=ti,j(cicj+h.c.)+Uini,ni,(1)where ci,ci,and nj=cjcjare the creation,an-nihilation and number operators respectively at site iwith spin (or).Here i,j is a pair of the near-est sites,t is the corresponding hopping parameter,andU describes the on-site interaction.To include the in-tersite electron-electron interactions 20,one needs theextended Hubbard model as follows,H=ti,j(cicj+h.c.)+Uini,ni,+i2.5,the first ex-cited state of the molecule is a dark 1B+3ustate.Thesecond excited state is an optically active 1B2ustatewhich is asymmetrical with respect to the y axis.TheFIG.2 Energy spectra of the first few low-lying excitedstates with varying.The symmetry of the first dark state,the first and second optically active states are indicated.corresponding transition is therefore polarized along theshort axis direction of the molecule.Upward there aretwo additional dark states.Beyond that,one finds thesecond bright state which has the symmetry of 1B3u,i.e.,asymmetrical with respect to the x axis.The cor-responding transition is hence polarized along the elon-gated direction and has the highest intensity.Due tothe different symmetry,the transition energies to thefirst and second active states are seen to exhibit oppo-site dependence on the effective dielectric constant.The first excited state is seen to become dark when is larger than 2.5,hence,the origin of the first darkexcitation cannot be traced back to the long-range in-teractions.To this end,we set=to see the energyspectra in the presence of only short-range interactions.FIG.3 plots the energy levels of the first few low-lyingexcited states as a function of the on-site interactionparameter U.In the single-particle picture where U=0and=,we find that the first excited state is brightwhile the second is a quadruple-degenerate state,alsobright.After turning on the short-range interactions,itis seen that such quadruple degeneracy breaks down andthe second excited state degenerates into four states,one bright and three dark.As the on-site interactionU becomes stronger,the transition energy to the firstexcited state increases while the energy to the lowestdark state degenerating from the quadruple-degeneratestate decreases.As 2.2,these two states are seen tocross over each other and the first excited state becomesdark.Hereby we reveal that anthracene fails the intuitiveexpectation that the first transition shall be polarizedalong the elongated direction of the system.It is be-DOI:10.1063/1674-0068/cjcp2111221c 2023 Chinese Physical Society310Chin.J.Chem.Phys.,Vol.36,No.3Jun Xie et al.FIG.3 Energy spectra of the first few low-lying excitedstates with varying on-site interaction parameter U/t in theabsence of long-range interactions(=).lieved that the abnormal polarization is due to zigzagedges of anthracene.In the meantime,we also pointout that the short-range interactions alone shall accountfor the first excited state being dark.Next we turn topyrene and see how the molecule fails the other intuitiveexpectation by the dipole approximation.FIG.4 plots the polarized absorption spectra ofpyrene in two different dielectric environments,=1.0for the case where the screening effect is totally sup-pressed,and=10.0 for medium screening strength.The most noticeable difference in comparison to an-thracene is that the first active transition in pyreneis now polarized along the elongated direction of themolecule.In the meantime,the transition inten-sity is seen to be very small compared with the sec-ond active transition that is polarized along the shortaxis direction,e.g.,I(E2)/I(E1)=9.85 at=10.0 andI(E2)/I(E1)100 at=1.0.However,the transitionsto those states localized along the elongated directionshall generally have high intensities according to thedipole approximation.Apparently,pyrene,which hasan aspect ratio of 1.44,fails this intuitive expectationby showing exactly the opposite.The energy spectra of the first few low-lying excitedstates in pyrene look very similar to those of anthraceneexcept for symmetry of the active excitations.When1.3,the first excited state is optically dark due toits symmetry of 1B+3u.As is shown in FIG.4,the firstactive excitation is polarized along the elongated direc-tion as it has a symmetry of 1B3u.The transition tothe second bright excited state has a very high intensity,and the state has a symmetry of 1B2u.FIG.4 Polarized absorption spectra of pyrene in two differ-ent dielectric environments,calculated by the exact diago-nalization method.Black dots indicate dark excitations ofenergies lower than the second transition peak.The openred dot indicates a bright excitation with very small inten-sity.Inset:schematic exhibition of the molecule.IV.CONCLUSIONIn summary,we have reported a study of optical prop-erties of two-conjugated molecules,anthracene andpyrene.We have calculated the optical absorption spec-tra in various dielectric environments by using the exactdiagonalization method based on the PPP model.Wefind that the first active transition in anthracene is al-ways polarized along its short axis direction.We pointout that as a highly elongated molecule,anthracene vi-olates a commonsense expectation that the first excitedstate in a confined quantum system with an anisotropicgeometry shall always be localized along its elongateddirection.Furthermore,we reveal that the first excitedstate in anthracene being dark is due to the short-rangeinteractions.For pyrene,we show that the second ac-tive transition polarized along the short axis direction ofthe molecule has a very high intensity,almost one orderof magnitude higher than the first one along the elon-gated direction.We point out that pyrene,an elongatedmolecule with an aspect ratio of 1.44,fails another in-tuitive expectation that the transitions to those stateslocalized along the elongated direction shall generallyhave high intensities.V.ACKNOWLEDGMENTSThis work is supported by the National Natural Sci-ence Foundation of China(No.12074078).DOI:10.1063/1674-0068/cjcp2111221c 2023 Chinese Physical SocietyChin.J.Chem.Phys.,Vol.36,No.3Dark States in-Conjugated Molecules311APPENDIX A:symmetry of statesBoth anthracene and pyrene belong to the pointgroup D2h.In the one-particle picture,for a symmetryoperation g,if one has g:|i|j,then the matrix rep-resentation of g can be expressed as j|(g)|i=1.Tocharacterize the symmetry of excited states in a many-electron system,one needs first to obtain the matrixrepresentation of each symmetry operation g on the ba-sis states of ED.Any basis state|m in ED can beexpressed as|m=ci1ci2ciucj1cj2cjd|0=ijcicj|0(A1)After the symmetry operation g,or a permutation g,|m becomes g(|m)=c g(i1)c g(i2)c g(iu)c g(j1)c g(j2)c g(jd)|0=g(i)g(j)c g(i)c g(j)|0(A2)where g(i)and g(j)is the permutation of i andj under g,respectively.For another basis state|n,|n=klckcl|0(A3)onemayobtainthematrixelementn|(g)|m=(1)w+wbycountingthenumberof exchanges of adjacent indices,wand w,in thesorting of g(i)and g(j),respectively.Let us define(g)=Trace(g),then for each sub-space with irreducible representation k,its dimensionis given bydk=1h g(g)k(g)(A4)where k(g)is the character of g in k and h is the orderof group.In this work,we find that only the subspaceswith Ag,B1g,B2uand B3usymmetry have non-zerodimension.Next one needs the projection operatorsfor irreducible representation k to block diagonalize therepresentation matrix,Pk=1h gk(g)(g)(A5)With projection operator Pk,one can define a new sym-metric basis,|k=Pk|i(A6)for subspace with irreducible 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