1、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
2、;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
3、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 expectation
4、s.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 anthra
5、cene 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 semiconduct
6、ing 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
7、 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 exc
8、ited 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 presen
9、t 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 ex
10、citedstate 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 direc
11、tion.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 thatt
12、hose 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 i
13、n 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 s
14、pite 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
15、-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 creati
16、on,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 axi
