vibronic bands in the homo-lumo excitation of linear polyyne molecules

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Vibronic bands in the HOMO-LUMO excitation oflinear polyyne moleculesTo cite this article Tomonari Wakabayashi et al 2013 J Phys Conf Ser 428 012004

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Vibronic bands in the HOMO-LUMO excitation of

linear polyyne molecules

Tomonari Wakabayashi and Yoriko WadaDepartment of Chemistry Kinki University Higashi-Osaka 577-8502 Japan

E-mail wakabachemkindaiacjp

Naoya Iwahara and Tohru SatoDepartment of Molecular Engineering Kyoto University Kyoto 615-8510 Japan

E-mail tsatosclkyoto-uacjp

Abstract Hydrogen-capped linear carbon chain molecules namely polyynes H(CequivC)nH(nge2) give rise to three excited states in the HOMO-LUMO excitation Electric dipoletransition from the ground state is fully allowed to one of the three excited states while forbiddenfor the other two low-lying excited states In addition to the strong absorption bands in the UVfor the allowed transition the molecules exhibit weak absorption and emission bands in the nearUV and visible wavelength regions The weak features are the vibronic bands in the forbiddentransition In this article symmetry considerations are presented for the optical transitions inthe centrosymmetric linear polyyne molecule The argument includes Herzberg-Teller expansionfor the state mixing induced by nuclear displacements along the normal coordinate of themolecule intensity borrowing from fully allowed transitions and inducing vibrational modesexcited in the vibronic transition The vibronic coupling considered here includes off-diagonalmatrix elements for second derivatives along the normal coordinate The vibronic selection rulefor the forbidden transition is derived and associated with the transition moment with respectto the molecular axis Experimental approaches are proposed for the assignment of the observedvibronic bands

1 IntroductionVibronic interaction creates finest fingerprints in the electronic spectra of polyatomic moleculesA symmetry forbidden transition becomes weakly allowed by the mixing of states due tothe vibronic coupling resulting in the intensity borrowing from fully allowed transitions Anoticeable example is found for the AlarrX transition in benzene for which some absorptionbands are observed around 260 nm [1]

Here we discuss a series of linear hydrocarbon molecules consisting of a conjugated sp-carbon chain terminated by a hydrogen atom at each end The centrosymmetric linear moleculesnamely polyynes H(CequivC)nH (nge2) can be produced experimentally by laser ablation of carbonparticles in organic solvents [2 3] The series of polyyne molecules exhibits a systematic trendin the UV absorption spectra shifting to longer wavelengths according to the molecular size n[4] In addition to the strong absorption bands in the UV weak absorption features have beenidentified in the near UV region [5] These features are associated with intrinsically forbiddenbut vibronically allowed transitions in the molecule [6]

XXIst International Symposium on the JahnndashTeller Effect 2012 IOP PublishingJournal of Physics Conference Series 428 (2013) 012004 doi1010881742-65964281012004

Published under licence by IOP Publishing Ltd 1

2 The HOMO-LUMO excitation in linear polyyne moleculesThe ground state electron configuration in the polyyne molecule is such that doubly degenerateπ orbitals constitute a ladder of orbital energy levels in which the gerade and ungerade orbitalsappear alternately above and below the HOMO-LUMO gap eg (πu)4(πg)4(πu)4(πg)0(πu)0 for C10H2 and (πg)4(πu)4(πg)4(πu)0(πg)0 for C12H2 Non-degenerate σg and σu orbitalslocate upper and lower sections of the π orbitals Since the neutral polyyne molecule has aclosed shell electronic structure its ground state is totally symmetric Σ+

g The HOMO-LUMOexcitation gives rise to three excited states πuotimesπg = Σ+

uoplusΣminusuoplus∆u for both series of C4m+2H2

(πu for HOMO) and C4mH2 (πg for HOMO) Electric dipole transition from the ground state isfully allowed to the Σ+

u state while forbidden to the low-lying Σminusu or ∆u state

HOMO

LUMO

ϕ ϕ ϕ ϕ ϕ x-x y-y x-y y-x0

(π) (π) (π) (π) (π) (π) (π) (π) (π) (π)x y x y x y x y x y

π

π

u

g

π

π

u

g

(a)ϕ (Σu

+) = ϕx-x + ϕy-y

ϕ (Σuminus) = ϕx-y minus ϕy-x

ϕ (∆u) = ϕx-y + ϕy-x

ϕ (∆u) = ϕx-x minus ϕy-y

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

Σ

Σ∆

Σ∆

Σ∆ΣΣ

g

u

u

g

g

u

uu

g

+

minus

minus

minus

+

minus

58(c)

allo

wed

forb

idde

nσ

π

π

π

π

π

π

σ

σ

g

u

g

u

g

u

g

u

g

π

π

π

π

π

σ

u

u

g

u

u

g

(b)

Σu+oplus Σu

minusoplus ∆u Σg+oplus Σg

minusoplus ∆g Σu+oplus Σu

minusoplus ∆u Πu

HOMO rarr LUMO HOMO rarr LUMO + 1 HOMO minus 1 rarr LUMO + 1 HOMO rarr σg

Figure 1 (a) Electron configurations for the ground state (left panel) and excited statesstemming from the HOMO-LUMO excitation (right panel) in the linear polyyne moleculeC4m+2H2 in Dinfinh point group symmetry Four singly excited electron configurations in the πundashπg transition ϕxminusx ϕyminusy ϕxminusy and ϕyminusx constitute an orthogonal set of linear combinationsfor three symmetry species πuotimesπg = Σ+

uoplusΣminusuoplus∆u (b) Configurations for single-electron

excitations providing various excited states as represented for C6H2 Among the states froma πundashπg excitation for the HOMO-LUMO excitation Σminus

u and ∆u come close in energy whileΣ+

u occupies always the upper most energy level (c) Calculated transition energies for C10H2

(TDDFT) The first allowed transition Σ+uharrΣ+

g has an oscillator strength of fz=58

Figure 1(a) illustrates the electron configuration for the ground state (left panel) and those for


2

singly excited states stemming from the HOMO-LUMO excitation Since HOMO and LUMOare doubly degenerate there are four possible electron configurations within the spin-singletmanifold In the Dinfinh point group symmetry the four configurations are classified into threesymmetry species Σ+

u Σminusu and ∆u Each of these states are represented by a linear combination

of the singly excited configurations The Σ+u state contains excitations of an electron between

π orbitals in the same direction while the Σminusu state contains excitations between π orbitals

perpendicular to each other The doubly degenerate ∆u state includes both the parallel andperpendicular excitations

Figure 1(b) depicts some single electron excitations using a schematic orbital energy diagramfor C6H2 The left column illustrates the HOMO-LUMO excitation generating three symmetryspecies Σ+

u Σminusu and ∆u The second column shows the electron excitation from HOMO to

LUMO+1 Since the two orbitals have the same parity u the resulting three excited stateshave gerade symmetry Σ+

g Σminusg and ∆g The excitation from a lower occupied level is also

possible eg from HOMOminus1 to LUMO+1 as in the third column With this higher energyexcitation another set of excited states including the same symmetry species as the HOMO-LUMO excitation is generated In the fourth column in Fig 1(b) the excitation from HOMO(πu) to an upper lying σg orbital gives rise to a Πu state

Figure 1(c) plots the transition energy for singly excited states in C10H2 (TDDFT) Thespin-singlet excited states are the consequence of the state mixing within the same symmetryspecies The three excited states consisting mostly of the HOMO-LUMO excitation are thelowest two excited states Σminus

u and ∆u and an upper lying excited state Σ+u The Σ+

u state isthe lowest energy excited state among those to which the optical transition from the groundstate is fully allowed namely the first allowed transition The Πu state to which the electricdipole transition is also allowed from the ground state (see the fourth column in Fig 2(b)) hasmuch higher transition energy (sim14 eV by the TDDFT calculation and not shown in Fig 1(c))than that for Σ+

u (sim6 eV by TDDFT) In Fig 2(c) it is noticeable that multiple states arelocating in-between the lowest two excited states Σminus

u and ∆u and the upper Σ+u state These

states stem from the electron configurations with higher transition energies (see the second andthird column in Fig 1(b)) Among the excited states stemming from the excitation betweena pair of degenerate orbitals the state for the optically allowed transition in this case the Σ+

u

state is always the highest in its transition energy [7] Therefore several states from higherenergy excitations are located below the Σ+

u stateThe electric dipoles ex ey and ez have irreducible representations πu πu and σ+

u respectively Therefore the transition moment for Σ+

uharrΣ+g is parallel to the molecular axis

z while the transition moments in (x y) for ΠuharrΣ+g are perpendicular to the molecular axis z

3 Observations and conjecturesIn a linear molecule in Dinfinh the optical transition from the ground state Σ+

g to an excited stateother than Σ+

u or Πu is forbidden by the orbital symmetry eg ∆ularrrarrΣ+g or Σminus

ularrrarrΣ+g

This selection rule strictly applies to the vibrational 0ndash0 band The origin band is not observedfor the forbidden transition The missing origin in the optical spectra poses a difficulty for theassignment of the spectral features However in some cases transitions are weakly allowedby the electric dipole mechanism when accompanied by activation or deactivation in a specificvibrational mode

Haink and Jungen analyzed UV absorption spectra for diacetylene C4H2 and triacetyleneC6H2 in the gas phase The missing origin was located at 258 cmminus1 below the prominentband in the electronic transitions ∆ularrΣ+

g and ΣminusularrΣ+

g The observed band was associatedwith an excitation of the πg vibrational mode [8] Ding et al observed resonant two photonionization spectra for the series of polyynes C2nH2 of n=3ndash7 in the gas phase The vibronicselection rule for the forbidden transitions ∆ularrΣ+


g was mentioned based on the


3

Herzberg-Teller coupling [6] During Raman spectroscopy for the polyyne molecules C2nH2 ofn=5ndash9 in solutions [9] new optical emission band systems were observed in the near UV andvisible wavelength regions The emission spectra systematically shifted to longer wavelengthswith the increasing molecular size n [10]

24000 20000 16000

Wavenumber cm-1

absorption emission

(b) H(CequivC)7H 2817 cm-1

28000 24000 20000 16000

Wavenumber cm-1

(a) H(CequivC)6H

absorption emission

2975 cm-1

Figure 2 Vibronic bands in the absorption and emission spectra for the forbidden transition∆uharrΣ+

g in the polyyne molecules (a) C12H2 and (b) C14H2 in hexane For the emission spectrathe excitation wavelength was tuned for the allowed transition Σ+

uharrΣ+g in the UV [10]

Figure 2 shows the absorption and laser induced optical emission spectra for (a) C12H2

and (b) C14H2 in hexane [10] The vibrational progression is noted in the absorption andemission spectra showing a characteristic vibrational frequency 1800ndash2100 cmminus1 for the totallysymmetric σg stretching vibrational mode of sp-carbon chains The absorption and emissionspectra constitute a mirror image The missing origin between the absorption and emissionfeatures is a promise of the forbidden transition in the molecule The lowest energy band in theabsorption and the highest energy band in the emission are largely separated by 2975 and 2817cmminus1 for these polyynes If the pair of the absorption and emission bands belong to the sameelectronic transition and if the spectral shift due to stabilization by the solvent molecules socalled a Stokes shift is not so large the missing origin should locate in the middle of the twobands In this case such a low frequency πg mode at ωe=258 cmminus1 can not explain for thelocation of the missing origin

4 Vibronic interactions in the symmetry forbidden transitionThe aim of this article is to provide symmetry considerations on the vibronic selection rulefor the forbidden transition in the linear polyyne molecule In the following sections thetheory of vibronic transitions is summarized and applied to the polyyne molecule in Dinfinh pointgroup symmetry Vibronic transition mechanisms are based on the state mixing induced bysymmetry lowering due to nuclear displacements along the normal coordinate for a vibrationalmode Intensity borrowing is expected for the forbidden transition when an excited stateconnected to the initial or the final state by an allowed transition is involved in the mixedstates Vibrational modes that activate the optical transition namely inducing modes dependon the state symmetries The number of necessary vibrational quanta for an inducing mode isassociated with the transition moment of allowed transitions in the intensity borrowing Thevibronic selection rule is deduced for a specific case of the ∆uharrΣ+

g transition


4

5 The theory of vibronic transitions51 Vibronic interactions in the HamiltonianNuclear displacements in a molecule from their equilibrium geometry at a high symmetry causeperturbation H prime in the Hamiltonian H The unperturbed Hamiltonian H0 is constructed forthe equilibrium geometry [11] The perturbation under the distortion H prime can be expressed bya series of Taylor expansion around the equilibrium geometry by the displacements along thenormal coordinates Qk and Ql for the kth and lth vibrational modes respectively

H = H0 +H prime (1)

H prime =sumk

(partH

partQk

)0

Qk +sumkl

12

(part2H

partQkpartQl

)0

QkQl +HO (2)

Here we take both the linear and quadratic terms in order to consider all possible allowedtransitions in the intensity borrowing within the first order perturbation

52 The Herzberg-Teller expansionUsing the linear and quadratic terms in Eqn (2) the wavefunction for an electronic state isrepresented within the framework of the first order perturbation theory [12]

φ(1)n (qQ) = φ(0)

n (q 0) +summ6=n

[sumk

langφ

(0)m

∣∣∣( partHpartQk

)0Qk

∣∣∣φ(0)n

rangEn minus Em

+sumkl

langφ

(0)m

∣∣∣12( part2HpartQkpartQl

)0QkQl

∣∣∣φ(0)n

rangEn minus Em

]φ(0)

m (q 0) (3)

The electronic wavefunction φ(1)n (qQ) as a function of the electron coordinates q under the

nuclear displacements Q is represented by a linear combination of unperturbed wavefunctionsφ

(0)m (q0) at the equilibrium geometry Q=0 The wavefunction φ(0)

m is intrinsically a complexnumber thus contains a phase factor For our purpose of symmetry considerations only the realpart is presented in the following discussion

53 The Born-Oppenheimer approximationThe vibronic wavefunction ψn(qQ) in the nth electronic state is represented by a product ofthe electronic wavefunction φ(1)

n (qQ) in Eqn (3) and the vibrational wavefunction χv(Q)with vibrational quantum numbers v [13 14 15]

ψn(qQ) = φ(1)n (qQ)χv(Q) (4)

For an excited state e with vibrational quantum numbers vprime the vibronic wavefunction is

ψe = φ(0)e χvprime +

summ6=e

[sumk

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

Qk +sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

QkQl

]φ(0)

m χvprime (5)

and for the ground state g with vibrational quantum numbers vprimeprime it is

ψg = φ(0)g χvprimeprime +

summ6=g

[sumk

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

Qk+sumkl

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

QkQl

]φ(0)

m χvprimeprime (6)

where the nuclear displacements Qk and Ql are eligible for escaping the integral over theelectron coordinates


5

54 Transition momentsFor the electric dipole microρ = eρ (ρ = x y z) micro = ex+ ey + ez the transition moment betweenthe ground state and an excited state is represented as

〈microρ〉 = 〈ψg|eρ|ψe〉〈micro〉 = 〈microx〉+ 〈microy〉+ 〈microz〉 (7)

The integral is taken over all the electron coordinates q and the nuclear coordinates Q Theoscillator strength f is proportional to the square of the transition moment and the transitionenergy f prop ∆E|〈micro〉|2

55 Allowed and forbidden transitionsUsing the wavefunctions for the ground and excited states in Eqn (5) and (6) the matrixelement for the transition moment micro is calculated as

〈micro〉 = 〈ψg|micro|ψe〉 (8)

= 〈φ(0)g |micro|φ(0)

e 〉〈χvprimeprime |χvprime〉

+summ 6=e

〈φ(0)g |micro|φ(0)

m 〉[sum

k

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

〈χvprimeprime |Qk|χvprime〉

+sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minusEm

〈χvprimeprime |QkQl|χvprime〉]

+summ 6=g

〈φ(0)m |micro|φ(0)

e 〉[sum

k

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minusEg


+sumkl

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minus Eg

〈χvprimeprime |QkQl|χvprime〉] (9)

When the 0th order integral 〈φ(0)g |micro|φ(0)

e 〉 is not zero the first term predominates the transitionmoment 〈micro〉 The relevant transition is designated as allowed or a dipole allowed transitionOn the other hand when the integral 〈φ(0)

g |micro|φ(0)e 〉 vanishes the relevant electronic transition is

designated as forbidden or a symmetry forbidden transition In this case however having non-vanishing matrix elements in the latter four terms in Eqn (9) transitions between particularpairs of vibronic states can have non-zero contributions to the transition moment The selectionrule for the vibronic bands in the forbidden transition can be obtained by looking for suchcontributions

56 Franck-Condon integralsFor the allowed transition in the first term in Eqn (9) the integral for vibrational wavefunctions〈χvprimeprime |χvprime〉 is represented explicitly as

〈χvprimeprime |χvprime〉 =3Nminus5prod

j

〈vprimeprimej |vprimej〉 (10)

The magnitude of this integral is approximated using a harmonic oscillator for each vibrationalmode in the lower and upper electronic states For the jth mode the harmonic oscillator in thelower state |vprimeprimej 〉 is located at the equilibrium geometry Q=0 while that in the upper state


6

|vprimej〉 at the potential minimum in the excited state The Franck-Condon integral 〈vprimeprimej |vprimej〉 canbe calculated separately for each vibrational mode

In the allowed transition vibronic patterns are totally governed by the Franck-Condon factors|〈vprimeprimej |vprimej〉|2 Among all the possible combinations of vibrational energy levels some contributesignificantly to the vibronic pattern For an extreme case in which the potential surfaces in theupper and lower electronic states are similar and both harmonic the Franck-Condon factors|〈vprimeprimej |vprimej〉|2 are unity only for the case of vprimeprimej = vprimej providing contributions only from 0ndash0 1ndash1 transitions in all the vibrational modes

Concerning the vibronic bands in the forbidden transition four vibronic terms in Eqn (9)are relevant in which the nuclear displacements Qk and Ql are involved in the integral overthe nuclear coordinates Q as

〈χvprimeprime |Qk|χvprime〉 = 〈vprimeprimek |Qk|vprimek〉prodj 6=k


〈χvprimeprime |QkQl|χvprime〉 = 〈vprimeprimek |Qk|vprimek〉〈vprimeprimel |Ql|vprimel〉prod

j 6=kl


where the subscripts k l and j specify the normal modes Note that the quadratic term inEqn (12) includes two cases the diagonal term k = l for the overtone and the off-diagonalterm k 6= l for the combination

〈χvprimeprime |Q2k|χvprime〉 = 〈vprimeprimek |Q2

k|vprimek〉︸︷︷︸overtone

prodj 6=k

〈vprimeprimej |vprimej〉

〈χvprimeprime |QkQl|χvprime〉 = 〈vprimeprimek |Qk|vprimek〉〈vprimeprimel |Ql|vprimel〉︸︷︷︸combination

prodj 6=klk 6=l


The number of vibrational modes in the centrosymmetric linear polyyne moleculeH(CequivC)nH is 3N minus 5 = 6n+ 1 They are classified into four symmetry species σg σu πg andπu For σg (σu) only σ+

g (σ+u ) modes exist and there is no σminusg (σminusu ) mode The σg (σu) modes

correspond to the symmetric (anti-symmetric) stretching vibrations with nuclear displacementsalong the molecular axis z The πg (πu) modes correspond to the trans-bending (cis-bending)vibrations with nuclear displacements perpendicular to the molecular axis (x y) The numbersof vibrational modes are n+1 n n and n for the σg σu πg and πu modes respectively Notethat each of the πg and πu modes is doubly degenerate according to the degeneracy in thecoordinates perpendicular to the molecular axis (x y)

Concerning the trans-bending mode of πg symmetry in Dinfinh the degeneracy is lifted by thenuclear displacements along the πg normal coordinate to lower the symmetry into ag and bg inC2h For the ag and bg modes the planes of nuclear motions are perpendicular to each otherThis applies also to the cis-bending πu mode For a doubly degenerate πg mode the quadraticvibronic coupling in Eqn (12) is associated with an overtone π2

g for the case of a2g or b 2

g ora combination πgπ

primeg for the case of agbg This is important for considerations on the vibronic

coupling in the ΣminusuharrΣ+

g and ∆uharrΣ+g forbidden transitions

6 Symmetry considerations on the transition moment61 The intensity borrowingFor large contributions to the transition moments in Eqn (9) 〈φ(0)

g |micro|φ(0)m 〉 and 〈φ(0)

m |micro|φ(0)e 〉

must be large The former is associated with the dipole allowed transition from the groundstate such as Σ+

uharrΣ+g and ΠuharrΣ+

g The latter for the transition between excited states alsocontributes significantly The vibronic transition becomes allowed with the aid of the transitionmoment in the allowed transitions thus the mechanism is called intensity borrowing


7

In the following discussion we restrict our attention to a specific case for the ∆uharrΣ+g

transition in the linear polyyne molecule The allowed transitions to be considered are thoseconnecting either the initial or final state to another Possible intermediate states are Πu and Σ+

u

for the allowed transitions connected with the ground state Σ+g while Πg and ∆g for the allowed

transitions connected with the excited state ∆u The latter two are rationalized by the directproducts Πgotimes∆u=ΠuoplusΦu and ∆gotimes∆u=Σ+

uoplusΣminusuoplusΓu Each of the direct products includes at

least one representation corresponding to the electric dipoles Πu for (x y) and Σ+u for z These

four transitions are classified by their transition moments (x y) and z and associated with thelinear and quadratic terms in the vibronic coupling respectively The important vibronic termsin the ∆uharrΣ+

g transition are represented explicitly as

〈micro〉 = 〈Σ+g |〈χvprimeprime |micro|χvprime〉|∆u〉 (13)

=sumk

langΣ+

g

∣∣∣ex+ ey∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u

rangE(∆u)minus E(Πu)

〈vprimeprimek |Qk|vprimek〉prodj 6=k


+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg

rangE(Πg)minus E(Σ+

g )〈vprimeprimek |Qk|vprimek〉

prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u

∣∣∣12( partHpartQkpartQl

)0

∣∣∣∆u

rangE(∆u)minus E(Σ+

u )〈vprimeprimek |Qk|vprimek〉〈vprimeprimel |Ql|vprimel〉

prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g

rangE(∆g)minus E(Σ+

g )〈vprimeprimek |Qk|vprimek〉〈vprimeprimel |Ql|vprimel〉

prodj 6=kl


62 Inducing modesOnce the intermediate state symmetry is determined from considerations on the intensityborrowing the next step is to find vibrational modes Qk and Ql which provide non-zerocontributions to the vibronic coupling 〈Πu|(partHpartQk)0|∆u〉 and 〈Σ+

u |12(partHpartQkpartQl)0|∆u〉Since the unperturbed Hamiltonian H0 is totally symmetric its derivatives with respect toQk and QkQl have the same symmetry as the relevant vibrational mode symmetry Γ(Qk) andΓ(Qk)otimesΓ(Ql) For non-vanishing contributions the direct product of the vibrational speciesand the two electronic species must contain a totally symmetric species In other words thesymmetry species of the derivative must be the same as one of the symmetry species in thedirect product of the two electronic states

The linear molecule inDinfinh has four fundamental species in the vibrational modes σ+g σ+

u πgand πu For the vibronic coupling 〈Πu|(partHpartQk)0|∆u〉 the symmetry species for the electronicpart is Πuotimes∆u=ΠgoplusΦg Then the vibrational mode of interest should be πg Concerning thequadratic coupling 〈Σ+

u |12(partHpartQkpartQl)0|∆u〉 the symmetry species for the electronic part isΣ+

uotimes∆u=∆g The vibrational δg species is produced by an overtone or a combination of πg

modes πgotimesπg=σ+g (oplusσminusg )oplusδg Also by an overtone or a combination of πu modes πuotimesπu the

vibrational δg species can be produced

63 The vibronic selection rule in the ∆uharrΣ+g transition

Suppose that a single πg mode among the n πg modes is the inducing mode which is responsiblefor the main spectral features in the vibronic transition In the summation over all the vibrationalmodes Qk and Ql in Eqn (14) most terms other than those for the inducing mode Qk=Ql=πg


8

are eliminated for their null contribution

〈micro〉 =langΣ+

g


ranglangΠu

∣∣∣( partHpartQπg

)0

∣∣∣∆u


〈vprimeprimeπg|Qπg |vprimeπg

〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )〈vprimeprimeπg|Qπg |vprimeπg

〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u

∣∣∣12( part2HpartQ 2

πg

)0

∣∣∣∆u


u )〈vprimeprimeπg|Q 2

πg|vprimeπg〉prod

j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )〈vprimeprimeπg|Q 2


j 6=πg


We further focus on a specific case of the emission from the vibrationally ground state vprimej = 0for all the 3N minus 5 modes in the upper electronic state ∆u to multiple vibrational levels vprimeprimej in the lower electronic ground state Σ+

g Among the considerable number of Franck-Condonintegrals most are zero except for transitions to the vprimeprimej = 0 level An exceptional case for theHOMO-LUMO excitation in the polyyne molecule is a series of transitions for the CC stretchingσg mode providing the vibrational progression in vprimeprimeσg

Harmonic oscillator considerations on the inducing πg mode provide the vibronic selection

rule ∆vπg = plusmn1 for the linear coupling associated with the transition moment in (x y) and∆vπg = plusmn2 for the quadratic coupling associated with the transition moment in z In the caseof the emission from the vibrational vprime = 0 level in the excited ∆u state the formula in Eqn(15) is simplified as follows including the vibrational progression for vprimeprimeσg

= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )

]〈1|Qπg |0〉〈vprimeprimeσg

|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2

πg|0〉〈vprimeprimeσg

|0〉

(17)

7 Determining factors for the transition intensityFigure 3 summarizes the mechanisms for the vibronic bands in the optical (a) absorption and(b) emission in the forbidden transition ∆uharrΣ+

g in the centrosymmetric polyyne moleculeEach of the four intermediate states is connected both to the initial and final states by a dipoleallowed transition in one side and by a vibronic coupling in the other In the vibronic couplingone (two) vibrational quantum (quanta) is (are) excited in the inducing mode The appearanceof spectral features is a matter of relative amplitudes between the terms in Eqn (15) Thefactors which determine the magnitude of the vibronic term in Eqn (15) are examined in thefollowing

71 Strengths in the vibronic couplingThe primarily important factor is the vibronic coupling in the form of 〈φ(0)

m |(partHpartQk)0|φ(0)n 〉 or

〈φ(0)m |12(partHpartQkpartQl)0|φ

(0)n 〉 The former is an off-diagonal matrix element for the first derivative

Similar coupling appears in considerations on the pseudo Jahn-Teller effects [16] The latter


9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)

Figure 3 Schematic drawing for the mechanism of vibronic bands in the optical (a) absorptionand (b) emission spectra for the ∆uharrΣ+

g transition in the centrosymmetric polyyne moleculeC10H2 For the forbidden transition the intensity borrowing is necessary from the fully allowedtransition ΠuharrΣ+

g Σ+uharrΣ+

g Πgharr∆u or ∆gharr∆u as depicted by a thick line with allowed (ρ)The symbol ρ = x y or z in parenthesis represents the direction in the transition moment withrespect to the molecular axis z The vibronic coupling is depicted by a thin line with vibronic(∆vπg) The number ∆vπg = +1 or +2 in parenthesis represents the number of vibrationalquanta necessary for activating the transition by the excitation of the inducing vibrational modeπg Note that the 0ndash0 band in the ∆uharrΣ+

g transition is strictly forbidden by the mechanismthus the signal is missing both in the absorption and emission spectra

is an off-diagonal matrix element for the second derivative This is beyond the conventionalmatrix elements in considerations on symmetry breaking systems Calculations of these vibroniccoupling matrix elements are the subject in theoretical developments in the future It is ourbelief that the magnitude of the quadratic coupling 〈φ(0)

m |12(part2HpartQ2k)0|φ

(0)n 〉〈vprimeprimek |Q2

k|vprimek〉 can becomparable to the linear coupling 〈φ(0)

m |(partHpartQk)0|φ(0)n 〉〈vprimeprimek |Qk|vprimek〉

72 Transition moments in the allowed transitionFor the intensity borrowing transition moments in the allowed transition are important Inthe polyyne molecule oscillator strengths for Σ+

uharrΣ+g transitions are concentrated to the first

allowed transition in the UV sim5 eV This is regarded as the plasmon excitation with fz = 58(TDDFT for C10H2) and with an absorption coefficient ε sim 2 times 105 L molminus1 cmminus1 [4] TheΠuharrΣ+

g transition has been suggested to locate in the vacuum UV region sim8 eV close to the


10

ionization threshold [5] with fx+fy = 23 (TDDFT for C10H2) Although any allowed transitioninvolving the ∆u state has not been observed their oscillator strengths should be comparableto the other allowed transitions Note that the relation between the transition moment and theoscillator strength is |micromn|2 prop fmn∆Emn where ∆Emn denotes the transition energy

73 Energy denominatorsThe perturbation coefficient includes a denominator by the energy difference between electronicstates connected by the vibronic coupling As is seen in Figure 3 the magnitude of thedenominator for the quadratic term |E(∆u)minusE(Σ+

u )| is small Thus substantial contributionsare expected by the intensity borrowing from the Σ+

uharrΣ+g transition Terms with denominators

of |E(∆g)minusE(Σ+g )| and |E(∆u)minusE(Πu)| may also contribute to some extent The magnitude

of the denominator |E(Πg)minusE(Σ+g )| is the largest thus probably having a less contribution

Excited states for which the transition energies have been experimentally determined are the Σ+u

and ∆u states Therefore the comparison between the four denominators remains an estimate

74 Vibrational off-diagonal elements associated with the inducing modeEquation (15) contains three types of integrals for the vibrational wavefunctions in the formsof 〈vprimeprimek |Qk|vprimek〉〈vprimeprimek |Q 2

k |vprimek〉 and 〈vprimeprimej |vprimej〉 where k represents the inducing mode and j the othermodes The former two are associated with the linear and quadratic vibronic couplings Whenthe scaling factors in Qk and Q2

k are normalized or cancelled by the factors in the vibroniccoupling derivatives all the vibrational integrals are counted as order unity With a simpleestimation based on the harmonic oscillator approximation the integrals with unitary amplitudesare those for ∆vk=plusmn 1 and plusmn2 for the linear and quadratic integrals respectively For polyynemolecules the inducing mode is a trans-bending πg mode as discussed in Section 62 and 63[5 6 8 10 17] The importance of the quadratic term has been emphasized first with theobservation of the pair of absorption and emission spectra in the forbidden transition [10]

75 Franck-Condon factors ndash the vibrational progressionAmong the three types of vibrational integrals the latter one 〈vprimeprimej |vprimej〉 is the Franck-Condonintegral In the harmonic oscillator approximation the integrals for ∆vj = 0 have an unitaryamplitude However it is the electronic transition thus the difference in the molecular geometrybetween the upper and lower electronic states causes the vibrational progression in somevibrational modes Actually for the Σ+

ularrΣ+g allowed transition the vibrational progression

is conspicuous for the symmetric stretching σg mode with an increment of sim2000 cmminus1 [4 9]Also for the vibronic spectra in the ∆uharrΣ+

g forbidden transition in Fig 2(a) and (b) [6 8 10]the vibrational progression in the same σg mode is discernible

76 Possible interference effects between the termsFinally it should be emphasized again that the transition intensity is proportional to the squareof the transition moment and that the phase factor has not been considered in the presentarticle Interference effects may occur when the squared sum of the elementary transitionmoments are calculated explicitly Some vibronic features can be pronounced and some canbe diminished Here we just mention about the possibility for the interference effects Thismay force some modifications to our description on the intensity for the vibronic bands in theforbidden transition

8 The selection rule and the transition momentOur considerations on the vibronic transition in the centrosymmetric linear polyyne moleculecomplete when both the linear and quadratic vibronic couplings are included The linear term is


11

associated with the transition of the single quantum excitation in the inducing vibrational modewith an intensity borrowing from the allowed transitions with a transition moment perpendicularto the molecular axis (x y) On the other hand the quadratic term is associated with thetransition of the double quanta excitation in the inducing vibrational mode with an intensityborrowing from the allowed transitions with a transition moment in parallel with the molecularaxis z From the symmetry considerations on the inducing mode in Section 6 the selection rulefor the vibronic bands in the ∆ularrΣ+

g transition is establishedThe remaining questions are (1) which πg mode among the n πg modes is really the inducing

mode (2) where the missing origin is located and (3) which vibronic term is dominating in theobserved vibronic bands For answering these questions the vibronic considerations presentedin this article constitute a milestone casting a light for the direction in the future First thedirect comparison in the absorption and emission spectra provides information on the vibrationalfrequency for the inducing mode as well as for the location of the missing origin [10] Second thedirection of the transition moment is a key feature in the vibronic transition Since the transitionmoment is related to the vibronic selection rule the correlation between the transition intensityand the polarization of a linearly polarized light should be an interesting issue for understandingthe mechanisms in the vibronic transition

9 The experimental approachExperimental approaches based on the idea presented above are under way in our groupMolecular crystals containing a trace amount of highly oriented polyyne molecules are the subjectfor absorption spectroscopy using a linearly polarized light The allowed transition Σ+

ularrΣ+g is

known to have the transition moment in parallel with the molecular axis thus the absorptionshould be strong when polarization of the incident light gets along the line of the molecularaxis in the crystal If the absorption intensity for the vibronic bands in the forbidden transitionfollows the intensity for those in the allowed transition upon rotation of the polarization oflight the vibronic bands must have the transition moment in the same direction as the allowedtransition In this case the vibronic bands should be induced by the excitation of an overtonein the πg mode Therefore the missing origin should be separated from the observed vibronicband by twice the frequency of the inducing πg mode If the absorption intensities in the allowedand forbidden transitions show anti-correlation upon rotation of the polarization of light thevibronic bands in the forbidden transition must be associated with the single quantum excitationin the inducing πg mode with the transition moment perpendicular to the molecular axis

One of the critical issues for such experiments is to prepare an appropriate single crystalin which the linear polyyne molecules are highly oriented Transparency down to shorterwavelengths sim220 nm is also a necessary condition We are trying to prepare such a crystallineform of molecular solids in which the size-selected polyyne molecules are embedded Preparationof the size-selected polyynes as well as cyanopolyynes is in progress in our laboratory [18] Veryrecently we found that polyyne molecules form a relatively stable molecular complex withiodine molecules in non-polar solvents For the polyyne-iodine complex the vibronic bands inthe forbidden transition were dramatically intensified in the absorption spectra [19 20] Withthe experiments in solutions and solids as well as in the gas phase information is availablefor further understanding of the mechanisms for the vibronic transition in the linear polyynemolecules

10 ConclusionsBased on the Herzberg-Teller coupling for the intensity borrowing the optical selection rulewas derived for the vibronic bands in the symmetry forbidden transition ∆uharrΣ+

g in the linearpolyyne molecules C2nH2 in Dinfinh point group symmetry In addition to the linear vibroniccoupling by the first derivative the quadratic vibronic coupling by the second derivative was


12

taken into account The quadratic coupling is associated with the vibronic transition havingthe transition moment in parallel with the molecular axis in the linear polyyne molecule Itis induced by the excitation of an overtone in the vibrational πg mode for which the intensityborrowing from the fully allowed transition Σ+

uharrΣ+g is expected

References

[1] Steinfeld J I 1985 Molecules and Radiation An Introduction to Modern Molecular Spectroscopy 2nd Edition(The MIT Press) chapter 9 pp 272ndash280

[2] Tsuji M Tsuji T Kuboyama S Yoon S-H Korai Y Tsujimoto T Kubo K Mori A and Mochida I 2002 ChemPhys Lett 355 101

[3] Tabata H Fujii M Hayashi S Doi T and Wakabayashi T 2006 Carbon 44 3168[4] Eastmond R Johnson T R and Walton D R M 1972 Tetrahedron 28 4601[5] Kloster-Jensen E Haink H-J and Christen H 1974 Helv Chim Acta 57 1731[6] Ding H Schmidt T W Pino T Guthe F and Maier J P 2003 Phys Chem Chem Phys 5 4772[7] Nakai H 2009 Rules for excited states of degenerate systems Interpretation by frozen orbital analysis (Advances

in The Theory of Atomic and Molecular Systems Conceptual and Computational Quantum ChemistryProgress in Theoretical Chemistry and Physics) eds Piecuch P Maruani J Delgado-Barrio G and WilsonS (Springer Dordrecht-Heidelberg-London-New York) pp 363ndash395

[8] Haink H-J and Jungen M 1979 Chem Phys Lett 61 319[9] Wakabayashi T Tabata H Doi T Nagayama H Okuda K Umeda R Hisaki I Sonoda M Tobe Y Minematsu

T Hashimoto K and Hayashi S 2007 Chem Phys Lett 433 296[10] Wakabayashi T Nagayama H Daigoku K Kiyooka Y and Hashimoto K 2007 Chem Phys Lett 446 65[11] Sato T Tokunaga K and Tanaka K 2006 J Chem Phys 124 024314[12] Eyring H Walter J and Kimball G E 1944 Quantum Chemistry (John Wiley amp Sons Inc) chapter 7 pp

92ndash99[13] Orlandi G and Siebrand W 1973 J Chem Phys 58 4513[14] Lin S H and Eyring H 1974 Proc Natl Acad Sci USA 71 3415[15] Lin S H and Eyring H 1974 Proc Natl Acad Sci USA 71 3802[16] Bersuker I B 2006 The Jahn-Teller Effect (Cambridge University Press) chapter 4 pp 110ndash161[17] Turowski M Crepin C Gronowski M Guillemin J-C Coupeaud A Couturier-Tamburelli I Pietri N and

Kolos R 2010 J Chem Phys 133 074310[18] Wakabayashi T Saikawa M Wada Y and Minematsu T 2012 Carbon 50 47[19] Wada Y Wakabayashi T and Kato T 2011 J Phys Chem B 115 8439[20] Wada Y Morisawa Y and Wakabayashi T 2012 Chem Phys Lett 541 54


13

Vibronic bands in the HOMO-LUMO excitation of

linear polyyne molecules

Tomonari Wakabayashi and Yoriko WadaDepartment of Chemistry Kinki University Higashi-Osaka 577-8502 Japan

E-mail wakabachemkindaiacjp

Naoya Iwahara and Tohru SatoDepartment of Molecular Engineering Kyoto University Kyoto 615-8510 Japan

E-mail tsatosclkyoto-uacjp

Abstract Hydrogen-capped linear carbon chain molecules namely polyynes H(CequivC)nH(nge2) give rise to three excited states in the HOMO-LUMO excitation Electric dipoletransition from the ground state is fully allowed to one of the three excited states while forbiddenfor the other two low-lying excited states In addition to the strong absorption bands in the UVfor the allowed transition the molecules exhibit weak absorption and emission bands in the nearUV and visible wavelength regions The weak features are the vibronic bands in the forbiddentransition In this article symmetry considerations are presented for the optical transitions inthe centrosymmetric linear polyyne molecule The argument includes Herzberg-Teller expansionfor the state mixing induced by nuclear displacements along the normal coordinate of themolecule intensity borrowing from fully allowed transitions and inducing vibrational modesexcited in the vibronic transition The vibronic coupling considered here includes off-diagonalmatrix elements for second derivatives along the normal coordinate The vibronic selection rulefor the forbidden transition is derived and associated with the transition moment with respectto the molecular axis Experimental approaches are proposed for the assignment of the observedvibronic bands

1 IntroductionVibronic interaction creates finest fingerprints in the electronic spectra of polyatomic moleculesA symmetry forbidden transition becomes weakly allowed by the mixing of states due tothe vibronic coupling resulting in the intensity borrowing from fully allowed transitions Anoticeable example is found for the AlarrX transition in benzene for which some absorptionbands are observed around 260 nm [1]

Here we discuss a series of linear hydrocarbon molecules consisting of a conjugated sp-carbon chain terminated by a hydrogen atom at each end The centrosymmetric linear moleculesnamely polyynes H(CequivC)nH (nge2) can be produced experimentally by laser ablation of carbonparticles in organic solvents [2 3] The series of polyyne molecules exhibits a systematic trendin the UV absorption spectra shifting to longer wavelengths according to the molecular size n[4] In addition to the strong absorption bands in the UV weak absorption features have beenidentified in the near UV region [5] These features are associated with intrinsically forbiddenbut vibronically allowed transitions in the molecule [6]


Published under licence by IOP Publishing Ltd 1






HOMO

LUMO



π

π

u

g

π

π

u

g

(a)ϕ (Σu

+) = ϕx-x + ϕy-y




6

4

2

0

Tra

nsiti

on E

nerg

y e

V

Σ

Σ∆

Σ∆

Σ∆ΣΣ

g

u

u

g

g

u

uu

g

+

minus

minus

minus

+

minus

58(c)

allo

wed

forb

idde

nσ

π

π

π

π

π

π

σ

σ

g

u

g

u

g

u

g

u

g

π

π

π

π

π

σ

u

u

g

u

u

g

(b)

Σu+oplus Σu



minusoplus ∆u Πu











2

















u









ularrrarrΣ+g








3


24000 20000 16000

Wavenumber cm-1

absorption emission


28000 24000 20000 16000

Wavenumber cm-1

(a) H(CequivC)6H

absorption emission

2975 cm-1







g transition


4


H = H0 +H prime (1)

H prime =sumk

(partH

partQk

)0

Qk +sumkl

12

(part2H

partQkpartQl

)0

QkQl +HO (2)



φ(1)n (qQ) = φ(0)

n (q 0) +summ6=n

[sumk

langφ

(0)m


)0Qk

∣∣∣φ(0)n

rangEn minus Em

+sumkl

langφ

(0)m


)0QkQl

∣∣∣φ(0)n

rangEn minus Em

]φ(0)

m (q 0) (3)







ψn(qQ) = φ(1)n (qQ)χv(Q) (4)



summ6=e

[sumk

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

Qk +sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

QkQl

]φ(0)

m χvprime (5)



summ6=g

[sumk

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

Qk+sumkl

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

QkQl

]φ(0)

m χvprimeprime (6)



5








+summ 6=e


m 〉[sum

k

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em


+sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minusEm


+summ 6=g


e 〉[sum

k

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minusEg


+sumkl

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minus Eg








j




6







j 6=kl





prodj 6=k



prodj 6=klk 6=l













m |micro|φ(0)e 〉





7



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13






HOMO

LUMO



π

π

u

g

π

π

u

g

(a)ϕ (Σu

+) = ϕx-x + ϕy-y




6

4

2

0

Tra

nsiti

on E

nerg

y e

V

Σ

Σ∆

Σ∆

Σ∆ΣΣ

g

u

u

g

g

u

uu

g

+

minus

minus

minus

+

minus

58(c)

allo

wed

forb

idde

nσ

π

π

π

π

π

π

σ

σ

g

u

g

u

g

u

g

u

g

π

π

π

π

π

σ

u

u

g

u

u

g

(b)

Σu+oplus Σu



minusoplus ∆u Πu











2

















u









ularrrarrΣ+g








3


24000 20000 16000

Wavenumber cm-1

absorption emission


28000 24000 20000 16000

Wavenumber cm-1

(a) H(CequivC)6H

absorption emission

2975 cm-1







g transition


4


H = H0 +H prime (1)

H prime =sumk

(partH

partQk

)0

Qk +sumkl

12

(part2H

partQkpartQl

)0

QkQl +HO (2)



φ(1)n (qQ) = φ(0)

n (q 0) +summ6=n

[sumk

langφ

(0)m


)0Qk

∣∣∣φ(0)n

rangEn minus Em

+sumkl

langφ

(0)m


)0QkQl

∣∣∣φ(0)n

rangEn minus Em

]φ(0)

m (q 0) (3)







ψn(qQ) = φ(1)n (qQ)χv(Q) (4)



summ6=e

[sumk

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

Qk +sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

QkQl

]φ(0)

m χvprime (5)



summ6=g

[sumk

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

Qk+sumkl

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

QkQl

]φ(0)

m χvprimeprime (6)



5








+summ 6=e


m 〉[sum

k

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em


+sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minusEm


+summ 6=g


e 〉[sum

k

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minusEg


+sumkl

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minus Eg








j




6







j 6=kl





prodj 6=k



prodj 6=klk 6=l













m |micro|φ(0)e 〉





7



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13

















u









ularrrarrΣ+g








3


24000 20000 16000

Wavenumber cm-1

absorption emission


28000 24000 20000 16000

Wavenumber cm-1

(a) H(CequivC)6H

absorption emission

2975 cm-1







g transition


4


H = H0 +H prime (1)

H prime =sumk

(partH

partQk

)0

Qk +sumkl

12

(part2H

partQkpartQl

)0

QkQl +HO (2)



φ(1)n (qQ) = φ(0)

n (q 0) +summ6=n

[sumk

langφ

(0)m


)0Qk

∣∣∣φ(0)n

rangEn minus Em

+sumkl

langφ

(0)m


)0QkQl

∣∣∣φ(0)n

rangEn minus Em

]φ(0)

m (q 0) (3)







ψn(qQ) = φ(1)n (qQ)χv(Q) (4)



summ6=e

[sumk

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

Qk +sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

QkQl

]φ(0)

m χvprime (5)



summ6=g

[sumk

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

Qk+sumkl

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

QkQl

]φ(0)

m χvprimeprime (6)



5








+summ 6=e


m 〉[sum

k

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em


+sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minusEm


+summ 6=g


e 〉[sum

k

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minusEg


+sumkl

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minus Eg








j




6







j 6=kl





prodj 6=k



prodj 6=klk 6=l













m |micro|φ(0)e 〉





7



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13


24000 20000 16000

Wavenumber cm-1

absorption emission


28000 24000 20000 16000

Wavenumber cm-1

(a) H(CequivC)6H

absorption emission

2975 cm-1







g transition


4


H = H0 +H prime (1)

H prime =sumk

(partH

partQk

)0

Qk +sumkl

12

(part2H

partQkpartQl

)0

QkQl +HO (2)



φ(1)n (qQ) = φ(0)

n (q 0) +summ6=n

[sumk

langφ

(0)m


)0Qk

∣∣∣φ(0)n

rangEn minus Em

+sumkl

langφ

(0)m


)0QkQl

∣∣∣φ(0)n

rangEn minus Em

]φ(0)

m (q 0) (3)







ψn(qQ) = φ(1)n (qQ)χv(Q) (4)



summ6=e

[sumk

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

Qk +sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

QkQl

]φ(0)

m χvprime (5)



summ6=g

[sumk

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

Qk+sumkl

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

QkQl

]φ(0)

m χvprimeprime (6)



5








+summ 6=e


m 〉[sum

k

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em


+sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minusEm


+summ 6=g


e 〉[sum

k

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minusEg


+sumkl

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minus Eg








j




6







j 6=kl





prodj 6=k



prodj 6=klk 6=l













m |micro|φ(0)e 〉





7



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13


H = H0 +H prime (1)

H prime =sumk

(partH

partQk

)0

Qk +sumkl

12

(part2H

partQkpartQl

)0

QkQl +HO (2)



φ(1)n (qQ) = φ(0)

n (q 0) +summ6=n

[sumk

langφ

(0)m


)0Qk

∣∣∣φ(0)n

rangEn minus Em

+sumkl

langφ

(0)m


)0QkQl

∣∣∣φ(0)n

rangEn minus Em

]φ(0)

m (q 0) (3)







ψn(qQ) = φ(1)n (qQ)χv(Q) (4)



summ6=e

[sumk

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

Qk +sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em

QkQl

]φ(0)

m χvprime (5)



summ6=g

[sumk

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

Qk+sumkl

langφ

(0)m


)0

∣∣∣φ(0)g

rangEg minus Em

QkQl

]φ(0)

m χvprimeprime (6)



5








+summ 6=e


m 〉[sum

k

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em


+sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minusEm


+summ 6=g


e 〉[sum

k

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minusEg


+sumkl

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minus Eg








j




6







j 6=kl





prodj 6=k



prodj 6=klk 6=l













m |micro|φ(0)e 〉





7



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13








+summ 6=e


m 〉[sum

k

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minus Em


+sumkl

langφ

(0)m


)0

∣∣∣φ(0)e

rangEe minusEm


+summ 6=g


e 〉[sum

k

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minusEg


+sumkl

langφ

(0)g


)0

∣∣∣φ(0)m

rangEm minus Eg








j




6







j 6=kl





prodj 6=k



prodj 6=klk 6=l













m |micro|φ(0)e 〉





7



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13







j 6=kl





prodj 6=k



prodj 6=klk 6=l













m |micro|φ(0)e 〉





7



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13



u








=sumk

langΣ+

g


ranglangΠu


)0

∣∣∣∆u




+sumk

langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



prodj 6=k


+sumkl

langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


)0

∣∣∣∆u



prodj 6=kl


+sumkl

lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


)0

∣∣∣∆g



prodj 6=kl













8



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13



g


ranglangΠu


)0

∣∣∣∆u



〉prod

j 6=πg


+langΠg

∣∣∣ex+ ey∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg



〉prod

j 6=πg


+langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u




j 6=πg


+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g




j 6=πg






= 0minus 4

microxy =

[langΣ+

g

∣∣∣exey

∣∣∣Πu

ranglangΠu


)0

∣∣∣∆u


+langΠg

∣∣∣exey

∣∣∣∆u

ranglangΣ+g


)0

∣∣∣Πg


g )


|0〉

(16)

microz =

[langΣ+

g

∣∣∣ez∣∣∣Σ+u

ranglangΣ+u


πg

)0

∣∣∣∆u


u )+lang∆g

∣∣∣ez∣∣∣∆u

ranglangΣ+g


πg

)0

∣∣∣∆g


g )

]〈2|Q 2


|0〉

(17)









9

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13

8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V

forb

idde

n

πg2

πg

0

0

0

0

πg2

πg

Σg+

Σu+

Πu

Πg

∆u

∆g

allo

wed

(z)

allo

wed

(z)

allo

wed

(xy

)

allowed (xy)

vibr

onic

(+1)

vibronic (+1)

vibr

onic

(+2)

vibronic (+2)

∆u larr Σg+

absorption

(a)8

6

4

2

0

Tra

nsiti

on E

nerg

y e

V0

0

0

πg2

πg2

πg

0

πg

Σg+

Σu+

Πu

Πg

∆g

∆u

allo

wed

(z)

forb

idde

n

vibr

onic

(+2)

vibr

onic

(+1)

allo

wed

(xy

)

allo

wed

(z)

vibronic (+2)

vibronic (+1)

allowed (xy)

∆u rarr Σg+

emission

(b)



g Σ+uharrΣ+













10



















11





ularrΣ+g is






12



References










13



















11





ularrΣ+g is






12



References










13





ularrΣ+g is






12



References










13



References










13

vibronic bands in the homo-lumo excitation of linear polyyne molecules

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