Allene and pentatetraene cations as models for

THE JOURNAL OF CHEMICAL PHYSICS122, 144320s2005d

Allene and pentatetraene cations as models for intramolecular chargetransfer: Vibronic coupling Hamiltonian and conical intersections

Andreas MarkmannTheoretische Chemie, Physikalisch-Chemisches Institut, Ruprecht-Karls-Universität Heidelberg,Im Neuenheimer Feld 229, 69 120 Heidelberg, Germany

Graham A. WorthDepartment of Chemistry, King’s College London, Strand, London WC2R 2LS, United Kingdom

Lorenz S. CederbaumTheoretische Chemie, Physikalisch-Chemisches Institut, Ruprecht-Karls-Universität Heidelberg,Im Neuenheimer Feld 229, 69 120 Heidelberg, Germany

sReceived 12 August 2004; accepted 13 January 2005; published online 13 April 2005d

We consider the vibronic coupling effects involving cationic states with degenerate components thatcan be represented as charge localized at either end of the short cumulene molecules allene andpentatetraene. Our aim is to simulate dynamically the charge transfer process when one componentis artificially depopulated. We model the Jahn–Teller vibronic interaction within these states as wellas their pseudo-Jahn–Teller coupling with some neighboring states. For the manifold of these states,we have calculated cross sections of theab initio adiabatic potential energy surfaces alongallnuclear degrees of freedom, including points at large distances from the equilibrium to increase thephysical significance of our model.Ab initio calculations for the cationic states of allene andpentatetraene were based on the fourth-order Møller–Plesset method and the outer valence Green’sfunction method. In some cases we had to go beyond this method and use the more involvedthird-order algebraic diagrammatic construction method to include intersections withsatellite states.The parameters for a five-state, all-mode diabatic vibronic coupling model Hamiltonian wereleast-square fitted to these potentials. The coupling parameters for the diabatic model Hamiltonianare such that, in comparison to allene, an enhanced preference for indirect charge transfer ispredicted for pentatetraene. ©2005 American Institute of Physics. fDOI: 10.1063/1.1867433g

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I. INTRODUCTION

The vibrational energy level structure of various ioniand neutral molecules has been studied by valence pelectron spectroscopy.1–6 Photoelectron transitions generaoccur to more than one final electronic state. These arecoupled through the nuclear motion, an effect known abronic coupling. Whenever strong vibronic interactionspresent, the Born–Oppenheimer approximation breaks di.e., the nuclear motion cannot be considered on onetronic potential energy surfacesPESd alone. Instead, nuclemotion has to be monitored simultaneously on all surfthat participate in the coupling.

The photoionization spectrum bears the signature ovibronic coupling, more so if a conical intersection7–10 isinvolved, where the coupling between states is particuimportant.11 This field of research has received increasattention over recent years.12–17A particular class of conicaintersections is formed by the Jahn–TellersJTd systems.18,19

In these, the electronic degeneracy enforced by symmeunstable with respect to nuclear distortions. Systemwhich interactions between a degenerate and a nondegate state exist that lift the degeneracy are labeled pseJahn–TellersPJTd systems.11,20

In this paper, we will consider the vibronic couplingfects involving cationic states with degenerate compon
that can be represented as charge localized at either end
0021-9606/2005/122~14!/144320/15/$22.50 122, 1443

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the short cumulene molecules allenesC3H4d and pentateraene sC5H4d. Extending previous theoretical studiesallene,21–23we model the JT vibronic interaction within thestates as well as their PJT coupling with some neighbostates. For the manifold of these statessof which there arfived, we have calculated cross sections of theab initio adia-batic PESs. A comparison between the vibronic coupmodel Hamiltonians of these states and the dynamical bior of these two molecules shall serve to formulate heurrules for the changes in the PESs and coupling strewhen the carbon chain is extended.

The vibronic coupling in the intervening cumulene,tatrienesC4H4d, has also been studied previously.24 This mol-ecule, however, hasD2h symmetry rather than theD2d sym-metry of allene and pentatetraene. As a result the statthis molecule are not readily comparable with those conered here.

The ground and excited state of the allene cation anground andfirst twoexcited states of the pentatetraene ca

are of the2E type, i.e.,X 2E and A 2E in allene andX 2E,

A 2E, andB 2E in pentatetraene. These states are eachally degenerate at theD2d symmetry equilibrium geometrythe neutral molecule. The JT effect in the highest occu

molecular orbital sHOMOd-ionized X 2E electronic mani
offolds of allene and pentatetraene was first theoretically inves-
© 2005 American Institute of Physics20-1

http://dx.doi.org/10.1063/1.1867433

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144320-2 Markmann, Worth, and Cederbaum J. Chem. Phys. 122, 144320 ~2005!

tigated in Ref. 25, where the impact ofE3B JT couplingwas identified and dynamical calculations were performepredict photoelectron spectra for these states that comwell to experimental results.26 E3B is a relatively rare formof JT coupling in which the orbital degeneracy is lifted bnondegenerate pair of vibrations.

We label the main cationic states of the molecule

X, A, B, etc., while satellite statesswhich may be energet

cally between main statesd are given the labelsS1, S2, etc.

Here we are interested in the statesA 2E in allene andB 2E inpentatetraene. The hole charge is localized at one endmolecule in the components of these states. We are aimuse the model Hamiltonians constructed in dynamical calations in which one component of these states is artificdepopulated. These simulations will serve as models fotramolecular charge transfer. We will therefore refer to thstates as charge transfer states in the following. Ultrcharge transfer and, more specifically, electron transfercesses between molecular systems play an important rbiological and biochemical systems.27–29 They have alsbeen studied as candidates for future molecular comtional devices experimentally30–36 as well astheoretically,37–40forming an active field of current resear

In the photoelectron spectrum of allene, a sharp p

ascribed to theX 2E manifold and a broader structure

signed to the interactingA 2E/ B 2B2/ C 2A1 states exist. Ithe gap between these two structures, a feature is foun

is assigned to a satellite stateS1 by Baltzeret al.This view iscontested by Bawaganet al.41 who assert that interactio

between X 2E and S1 plays a significant role in itformation.42 Although this may be the case, we are ccerned with states that are energetically above and sepfrom these states, so that we did not include them inconsiderations explicitly in order to arrive at a modelcan be parametrized and on which dynamics can beformed with justifiable effort.

This means that we consider theX 2E/ S1 and

A 2E/ B 2B2/ C 2A1 bands as separatesas was done in prevous studies21,22,43d. Note that the effect of other states cpling to the explicitly included states is taken into accounthe extent that it influences their shapes, as this is whfitted.

The spectral band in the energy range we are interin has a number of well-resolved peaks at lower energiesan unstructured region at higher energies.41,44,45The loss ostructure is ascribed to the strong vibronic interactiontween the electronic states.45

The JT effect in theA 2E electronic manifold of allenhas been investigated by Woywod and Domcke43 using atwo-state, four-mode model. They managed to repromost of the low-energy part of the spectrum. Accordingthe coupling parameters extracted from their model, thesigned the low-energy structures to progressions alonH–C–H bending vibrational normal modesn2PA1 and n7

PB2. However, this assignment required a considerable
pirical adjustment of the linear vibronic coupling constants
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of these modes. In their experimental publication, Yanetal.44 had assigned the same structure to progressionsthe n2 andn3 normal modes.

This model was extended by Mahapatraet al. to a threestate, ten-mode21 and a three-state, all-mode22 model. Inthese publications, the interplay between JT and PJT e

sthe latter involving theB 2B2 state above theA 2E stated wasdemonstrated and it was ascertained that, additionallyn2

and n7, the n3PA1 has to be taken into account to explfeatures of the photoelectron spectrum. After minor adments to some parameters of the model Hamiltonian, aagreement with the experimental photoelectron spectruBaltzeret al.45 was achieved.

Experimental data at the desired resolution is presunavailable for the pentatetraene bands of interest. Wethus develop a model Hamiltonian for allene exclusivfrom ab initio data so that a fully analogous fitting proccan be performed for pentatetraene, giving models suifor a meaningful comparison.

Mahapatraet al. parametrized their vibronic couplinHamiltonian using finite-difference derivatives of calculaPESs at the equilibrium geometry of the molecule. Insteaevaluating the calculated PESs exclusively near the eqrium geometry, in a study on the photoelectron spectrubutatriene, Cattariuset al.24 used a least-square fitting produre for the parameters of the vibronic coupling Hatonian. This extended fit approach takes the overall shathe PESs better into account and led to reproducible paeters which model the surfaces to a good accuracyfrom the equilibrium geometry. In the present article weout to use this extended fit approach to construct mHamiltonians for the states of interest in allene and penraene. Theab initio PESs were calculated using outerlence Green’s function methodsOVGFd and, where necesary, the compatible third-order algebraic diagrammconstructionfADCs3dg method.

Allene and pentatetraene have 15 and 21 degrees odom, respectively. We present here model Hamiltonianstak-ing all degrees of freedom into account. The results of wavepacket propagation simulations with these Hamiltoniansbe presented in a forthcoming publication. The Hamiltonalone express many properties of the molecules andtherefore of their own interest. They will be discusseddetail in this article.

Section II discusses the basic properties of the vibrcoupling Hamiltonian and the method used to parametraccording to the data fromab initio calculations. Sections Iand IV describe the PESs calculated and model Hamiltoextracted for allene and pentatetraene, respectively. SeV contains a comparison and discussion of the results fotwo molecules.

II. THEORETICAL BACKGROUND

A. The vibronic coupling Hamiltonian

The states representable as localized at either end

molecules discussed in this work are theA 2E state of allen˜ 2
and theB E state of pentatetraene. Neighboring states at

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144320-3 Allene and pentatetraene cations as models J. Chem. Phys. 122, 144320 ~2005!

higher ionization energies are a2B2 and a2A1 state in eacmolecule. The calculated ionization energies of these sare listed in Tables I and V. These tables also featuretional states which are needed for fitting the2E surfacesThese are denotedS for satellite. Their properties are dtailed in the appropriate sections for allene and pentatetr

The coupling between the states selected and the vtional modes is modeled by the bilinear model vibronic cpling Hamiltonian

H = H01 + seijdi,j=1,…,5. s1d

Here H0=Tn+V0 is a harmonic oscillator plus quartic corection Hamiltonian modeling the neutral ground state omolecule.1 denotes the 535 unit matrix. The square matrseijdui,j=1,…,5 is the coupling matrix. The 15 vibrational modof allene have the irreducible representations

G = 3A1 + B1 + 3B2 + 4E. s2d

The corresponding 21 normal modes of pentatetraene a

G = 4A1 + B1 + 4B2 + 6E. s3d

Thus

Tn = oniPA1,B1,B2

−vi

2

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2 + oniPE

vi

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+ oniPA1,B1,B2

ji

4!Qi

4 + oniPE

ji

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4 + Qiy4 d. s5d

Qa denotes themass-frequency scaledcoordinate of the degree of freedoma. vi are the harmonic angular frequencof the normal modes andji are quartic terms accounting fanharmonicities in the ground state potential of the nemolecule due to the use of rectilinear coordinates, ascussed below.

The elementseij of the coupling matrix are expandeda Taylor series up to second order with additional fouorder diagonal elements. Using selection rules derivedsymmetry considerations,11,20 they can be parametrizedfollows:

e11 = E0s1d + o

niPA1,B2

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1

2oi,j

gi js1dQiQj +

1

4!oi

«is1dQi

4,

s6d

e22 = E0s2d + o

niPA1,B2

kis2dQi +

1

2oi,j

gi js2dQiQj +

1

4!oi

«is2dQi

4,

s7d

ekkù3 = E0skd + o

niPA1

kiskdQi +

1

2oi,j

gi jskdQiQj +

1

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«iskdQi

4,

s8d

si-

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e12 = e21 = oniPB1

lis12dQi , s9d

e13 = e31 = oniPEsA8d

lis13dQix, e23 = e32 = o

niPEsA8d

lis13dQiy ,

s10d



niPEsA8d

lis14dQiy ,

s11d



niPEsA9d

lis15dQiy ,

s12d

e34 = e43 = oniPB2

lis34dQi , s13d

wherekis2d=ki

s1d for ni PA1, kis2d=−ki

s1d for ni PB2, gi js2d=gi j

s1d

for ni ,n j of the same symmetry,gi js2d=−gi j

s1d for ni PA1, n j

PB2 and«is2d=«i

s1d for all i , j . The energies

E0skd =5

EE, k = 1,2,

EB2, k = 3,

EA1, k = 4,

ES, k = 56 s14d

are the vertical ionization energies for the states considvi are the angular frequencies of the ground state harmapproximation for the modei, ki

skd are the linear intrastacoupling constants in the statek for the modei, li

skld are thelinear JT and PJT coupling constants between the stakand l for the modei, gi j

skd are the intrastate bilinear coupliconstants, and«i

skd are the intrastate quartic coupling cstants.

Relations of the formni PG have to be understood“the ith mode is of symmetryG.” Qix and Qiy denote thexand y components of the degenerate vibrationalE modesEsA8d andEsA9d distinguish theE modes according to the symetry of the component of theE state inCs that is lowesenergetically away from the equilibrium. Allene and ptatetraene each have one mode where this componentA9,while in all otherE modes it isA8. This distinction is discussed in more detail below.

The coefficientsgiiskd specify a frequency change due

coupling in statek and modei while the«iskd specify a chang

in the corresponding quartic coefficient.As will be detailed in Sec. III below, the fitting proce

yielded zero quartic corrections«iskd, k=1, 2. The other«i

skd,k=3, 4, 5 are only small corrections, so the quartic termthe model Hamiltonian are mainly due to anharmonicpresent in the ground state potential rather than coupThis means that first-order coupling terms dominate thepling process in allene and pentatetraene.

Higher order coupling constantssfor example, bilineaoff-diagonal or general quadrilineard were not used to param
etrize the vibronic coupling Hamiltonian. We will demon-

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strate that the fit achieved with the model HamiltonianEqs. s6d–s13d is already satisfactory, supporting the notthat the vibronic coupling Hamiltonian Taylor expandedcertain order represents a good approximation to the phycircumstances.

Note that all coupling parameters involving the fouand fifth state,k=4, 5, are additional to the ones usedMahapatraet al.21–23 for allene, as there exclusively t

statesA 2E and B 2B2 were taken into account. The newintroduced states mainly serve to better fit the topolog

the A 2E andB 2B2 states. They are thus representatives fmultitude of states coupling with these.

The linear coefficientski andli can only be nonzero iat the equilibrium geometryQ=0, the cross product of threpresentations of the involved electronic statesa andb andthe modeni contains the totally symmetric representati.e.,

Ga 3 Gni3 Gb $ A1. s15d

Analogous relations hold for the bilinear coefficientsgi j .The mechanism of vibronic coupling is the same in b

molecules: The symmetric direct product of theE representation in D2d decomposes as

E 3 E = A1 + B1 + B2. s16d

While totally symmetric modes of representationA1 cannolift the degeneracy of the2E electronic manifold, theB1 andB2 modes can, i.e., they displayE^ B Jahn–Teller activity inthe charge-localized state. TheB1 modes can thus be rgarded as “coupling” modes and theB2 modes as “tuningmodes.11 Although in anE^ B Jahn–Teller system thesebels are arbitrary, they are convenient for allene and penraene, as these molecules each have only oneB1 mode.

B. Ab initio calculations

Geometry optimization and vibrational normal modeculations in the neutral ground state were performed afourth-order Møller–Plesset perturbation theorysMP4d levelemploying the 6-311G* Gaussian basis set.46 The dimensionless normal coordinatesQi are obtained from the masweighted normal coordinates by multiplication withsquare root of the angular frequencyÎvi.

47 Subsequentlypoints of the neutral ground state along the rectilinear nomode coordinates of both molecules were calculated witsame method.

To gauge their accuracy, the MP4 ground state enewere spot checked against results of coupled cluster siand doubles with triple excitation correction48 calculations, amethod which is regarded as one of the most accuraquantum chemistry today. The differences in energy toMP4 results were of the order of 0.01 eV, i.e., the Mresults are highly accurate.

In previous work,21–23,25,43all coefficients not excludeby symmetry were set to the corresponding first or sederivatives of the adiabatic potentials or differences the
at the equilibrium geometryQ=0, e.g., forni PA1
l

t-

l

ss

n

f

ki =1

2US ]VE

]QiDU

Q=0. s17d

Afterwards, some of these values were adjusted in oto better approximate the experimental photoelectron strum.

We would like to extend this model, which is accurnear the neutral ground state equilibrium, to coordinawayfrom the equilibrium geometry. A corresponding mowill also be presented for pentatetraene and its paramcompared to those determined for allene.

In order to do this, we have calculated cross sectionthe adiabatic PESs along the rectilinear normal coordinof the molecules. Generally speaking, it would be desirto use normal mode coordinates at every point away fromequilibrium rather than rectilinear coordinates. However,would complicate the kinetic energy operator consideramaking dynamical simulations involving all normal mocomputationally unviable. Since the dynamics and the chtransfer process simulated by the artificial depopulatioone component of the charge transfer state are dominatthe conical intersection, this is not so serious an approxtion as it may seem.

In order to allow a meaningful comparison betweenmolecules, we have not adjusted the parameters to fiexperimental results but instead rely fully on the paramderived from theab initio results. A comparison with thexperimental photoelectron spectrum for allene shall smerely to gauge the physical relevance of the model. Deof the fit procedure will become apparent as we describab initio data and the fitting in the following sections.

For the calculation of the cationic PESs, we have uthe ab initio OVGF methodsRefs. 49 and 50d as imple-mented in theGAUSSIAN 98 program package.46 This methodis based on the ionization of a single valence electronits Hartree–Fock orbital and determines the ionization enby a Green’s function approach. The ionization energthen added to the MP4 ground state energy at each datato construct the cationic PESs.

For some of the normal modes ofE symmetry, we had tgo beyond OVGF and use the more involved but morecise ADCs3d methodsRef. 51d to gain more appropriate rsults. This method takes two-hole–one-particle states exitly into account for the purpose of subsequent Grefunction calculations which means that the basis from wstates can be constructed by linear combination is huincreased.satellite statescan be detected with ADCs3d whichdo not constitute ionization from a single electronic orbbut consist of a linear combination of several electronicfigurations, many of which are two-hole–one-particle. Ingions where interaction with satellite states is weak, ADCs3dreproduces the results of OVGF. It is therefore readily uas natural extension to OVGF. In cases where satellite splay a minor role, the OVGF data can then be retaine
order to save numerical effort.

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III. FIVE-STATE, ALL-MODE MODEL VIBRONICCOUPLING HAMILTONIAN FOR ALLENE

When taking the topology of the potential surfaces afrom the equilibrium configuration of the neutral molecinto account, anharmonicities even in the ground state ptial become visible. These are mainly due to the use oftilinear coordinates which are not strictly normal modegeometries away from the equilibrium, particularly for nmal modes with rotational character.

Figure 1 shows data points from ourab initio MP4 cal-culations of the ground state potential along the in-planewag moden11sEd ssee Fig. 2d of allene. The harmonic potetial predicted by the normal mode frequency is shown in1sad and a fit using an additional fourth-order termfwithcoefficientj11 in Eq. s5dg is shown in Fig. 1sbd. It is obviousfrom the figure that the fourth-order term accuratelyscribes the deviation from the harmonic model. The cortion of the ground state PES viaji is performed corresponingly along all normal modes.

During the fitting along other normal modes it turnedthat the fourth-order constantji extracted from a fit of thground state usually sufficed to yield a good first-order fithe cationic states. Quartic corrections«i

skd were needed onfor some modesi and there only fork=3, 4, 5. This meanthe vibronic coupling Hamiltonian is almost linear, as tdominating fourth-order corrections appear in the mground state potentialV0 in Eq. s5d rather than in the coupling terms, Eqs.s6d–s13d.

Table I shows the OVGF and ADCs3d energies above thneutral ground state equilibrium geometrysin eVd of thestates of interest for allene. They are between 15 eV andeV. It can be seen that the results agree quite well qutively. TheE and theB2 state are slightly further apart froeach other in the ADCs3d result. The ADCs3d energies arused to fit the Hamiltonian as they derive from a higher lmethod.

For the purpose of least-square fitting the vibronic cpling model Hamiltonian to theab initio data, we have useand extended the program package VCHAM that hadused to fit a model potential for butatriene by Cattariuetal.24 This program starts from an initial guess for the diabHamiltonian of the form Eqs.s6d–s13d and least square opmizes its parameters by a standard conjugate gramethod.52 The least-square deviation from theab initio datais calculated by numerical matrix diagonalization of theabatic model Hamiltonian at every geometry where anab
initio data point is given.
--

l

-

7-

n

t

We have extended VCHAM to allow it to handlearbitrary number of electronic statessversus two statesbutatriened and fourth-order coefficients in the HamiltoniAlso, we have introduced constrained optimization in oto force certain relationships between coefficients fitted.is important to guarantee that the symmetry relations nunderneath Eq.s13d are satisfied.

Figure 2 shows a selection of the PESs of allene asome normal modes.53 We have also calculated potentialergy points where the molecule is distorted along the aof two normal modes, one ofA1 and one ofB2 symmetryThese lines were fitted using the bilinear coefficientsgi j

skd.The PESs along the totally symmetricA1 modes

n1,… ,n3 exemplified in the upper row of Fig. 2 are asymetric functions of the normal coordinate. The regions wthe discrepancy between the present harmonic plus foorder model becomes noticeable is vibrationally inaccesfor the wave packet placed initially at the ground state elibrium, so we can continue using this modelsnote energscaled. The mode shown in the second row of Fig. 2 iscoupling moden4sB1d sthe twisting moded. Here, the resuling adiabatic PESs are symmetric functions of the nocoordinate.

The third row of Fig. 2 exemplifies the PESs alongtuning modesn5,… ,n7sB2d which are symmetric with respect to the equilibrium geometry. Due to

B2 3 B2 3 A1 $ A1 s18d

linear coupling between theB2 state and theA1 state has tbe parametrized as well as the intrastate linear couplinrameterski

s1d of the E state manifold.The last row of Fig. 2 exemplifies a PJT activeE mode

In the calculation of the cationic states using the OVmethod, complications arose that are discussed in the foing paragraphs. Alternative data sets were therefore usfit these potential energy curves as described below.

The inconsistencies become apparent in Fig. 3, wthe OVGF results of the adiabatic potential energy linetheE andB2 states along the moden11sEd are presented. Thdistortion of the molecule along anE mode breaks almost amolecular symmetries, leaving only the mirror symmetrythis case in the plane of the tail wag, i.e., the molecule haCs

symmetry when moving along anE normal mode. In Fig. 3can be seen that alongn11sEd the upper branch of the eletronic E state manifold assumesA9 symmetry. TheB2 statehasA8 symmetry inCs. The main problem is that the cro

FIG. 1. Anharmonicities in the grounstate of neutral allene:sad The har-monic potential at the frequency of tn11 normal modeswhere the effect oanharmonicity is strongestd plottedagainst theab initio MP4 ground statenergiesspointsd for the softest modin allene. sbd A model potential withan admixture of fourth-order termplotted against the same data poiAbscissa in dimensionless coordina

over of the state symmetries does not coincide with the

ross

ag.atn

d by

sts

ed

rom vibrpling

qui-een

more


crossover of OVGF energies as it should do. Near the cing, the OVGF energy values are also visibly distorted.

An additional complication is found along then10sEdmode which is an out-of-plane H–C–H bending tail wHere, the electronicE state manifold splits in such a way ththe component of symmetryA9 in Cs has lower energy thathe A8 component. This anomalous behavior is denotethe superscriptA9, n10PEsA9d in Eqs. s6d–s13d. The otherEmodes are denotedEsA8d. Due to symmetry, this suggecoupling with one or several higher lying statessd of the samesymmetry, i.e.,A9. However, our OVGF calculations yieldonly states ofA8 symmetry above theE electronic state.

FIG. 2. Some normal modes of allene. Mode labels with symmetry, discoupling model Hamiltonianslinesd with ab initio data pointssfilled circlemode, a tuning mode, and a pseudo-Jahn–Teller active mode.

placement diagrams illustrating the mode and adiabatic energies derived fonicsd. From top to bottom are exemplified a totally symmetric mode, the cou

These problems indicate that for modes ofE symmetry

-TABLE I. OVGF and ADCs3d energies above the neutral ground state elibrium geometrysin eVd of the states of interest for allene. It can be sthat theE and theB2 state are further apart from each other in the ADCs3dresult. The ADCs3d energies are used to fit the Hamiltonian as they areaccurate.

State label k fin Eqs.s1d–s13dg OVGF energy ADCs3d energy

X 2E ¯ 9.82 9.92

S1E ¯ ¯ 13.96

A 2E 1,2 15.04 15.08

S2E ¯ ¯ 16.81

B 2B23 15.46 15.42

A 2A14 17.56 17.47

S3E 5 ¯ 17.69

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fof th

tar theractatelli


the OVGF method does not capture all states coupling tcharge transfer states of interest. It is well known that OVbecomes unsuitable as a computational tool once sastates are close byand interact withthe main states.50 Hencewe decided to repeat the potential surface calculations athe low-frequencyE modesswhere distortions were appaentd using the ADCs3d method.51 This method takes twohole–one-particle states into account explicitly. Satestates are now detected which are not represented by aization from a single electronic orbital but by a linear cobination of several electronic excitations, many of whichtwo-hole–one-particle. One of these satellite states is selas representative and used to fit the topology of the chtransfer states. This is the state denotedS, k=5 in Eqss6d–s13d. It has the required symmetryA9 in Cs that is needeto model coupling with the charge transferE state manifoldalong then10sEd mode. As the main cationic states are

ready labeledX, A, B, C, and so forth, we will label th

satellite statesS1, S2, S3, and so on. In this notation, th

satellite state selected isS=S3sEd and as such degeneraHowever, as it is used as a representative for the purpofitting only, this degeneracy is ignored.

In order to demonstrate that the use of ADCs3d resolvesthe problems described above, Fig. 4 shows the resultsADCs3d calculations for points along some low-frequencEmodes of allene.

Along the n9 mode, it can be seen that the PESs fOVGF fFig. 4sadg approximate the ADCs3d PESs presentealongside itfin Fig. 4sbdg very well. The solid lines drawn iFig. 4sbd are the fit lines according to the OVGF data, shito the ADCs3d energies at the equilibrium coordinate. Itremarkable that the fit appears better compared to the haccuracy ADCs3d results, particularly at the minima of theA8

branch of theA 2E state, although this fit was made usingOVGF data. This supports the view that the linear vibro

FIG. 3. Inconsistencies in the main state PESs along the tail-wagn11sEd in allene, as calculated with OVGF. The filled dots mark energiestates ofA8 symmetry inCs; the open dots mark energies of states oA9symmetry. It can be seen that the crossing points of the energies and

symmetries of the upper branch of theA 2E state and theB 2B2 state do nocoincide as they should. The energy values are visibly distorted necrossing. The problem is resolved by noting that a satellite state intwith the main states in the region where inconsistencies arise. Such sstates can be accounted for using the ADCs3d methodssee text, Sec. IIId.

coupling model Hamiltonian is a good approximation to the

e

e

g

n-

de

f

e

r

physically relevant situation. Note that theA1 state is useonly to fit the charge transfer states, so its crossing wsatellite is ignored.

The ADCs3d results shown in Fig. 4scd salong then10

moded demonstrate a more complex situation with satestates ofA8 symmetrysblack symbolsd crossing through thB2 state and the upper branch of theE state in the regiooutside 3.8 normal coordinate units fromQ=0 and aroun16.5 eV. As before, the crossing points lie at a considevibrational energy relative to the energy at equilibriumometry sabout 1.5 eV above theE state energy at the eqlibriumd, so for the purpose of wave-packet dynamic silations, this additional state can be ignored.

The data points seen below theA 2E state and crossinwith it belong to the satellite ofE symmetry that cannointeract along then10 mode by symmetry, as inCs swhere themode itself is alwaysA8d,

A9 3 A8 3 A8 ¢ A8. s19d

Other modessto be thought of as running into the paplaned may, however, lift the degeneracy and couplestates. Whether this happens is difficult to ascertain frompresent data.

Coupling effects, if present, appear to be weak, howesince the shape of the PESs can be fitted so well withpresent model Hamiltonian along the cuts through the elibrium coordinate along the other modes.sIf strong couplingwas present, it should be seen along the other modes asrun parallel to lines imagined into the paper plane at a cing point.d

The ADCs3d results shown in Fig. 4sdd salong then11

moded demonstrate that satellite states ofA8 symmetry crosthrough theB2 state and the upper branch of theE state in theregion at 4.2 normal coordinate units fromQ=0 and aroun15.5 eV. However, the crossing point lies at about 0.5above the energy at the equilibrium of the charge transEstate of interest. For the purpose of wave-packet dynsimulations that this work aims for, this additional statebe ignored. A complete model should of course take itaccount. This crossing explains the aberrant behavior oOVGF result as seen in Fig. 3. The ADCs3d results show nsuch inconsistencies, so fitting can proceed on the fivemodel using the ADCs3d data.

Concluding this analysis, it has become clearADCs3d data needed to be used to fit the low-frequencEmodes. The action of the linear coupling parametersli

s1kd isbest seen in the shape of theB2 state alongn9sEsA8dd sthisstate is bent upwards due to coupling with theE manifolddand in the PES of the satellite state alongn10sEsA9dd swherethe S state at the top is bent upwardsd.

TheS state,k=5, was fitted only alongn10 andn11, as itwas represented by the harmonic model along all omodes. Alongn11sEsA8dd, the PES ofS is bent downwardsmost likely due to coupling with states lying higher upenergy. Fitting this state according to the vibronic coupparadigm would necessitate involving these higher stateprobably even higher ones coupling to them and so on.

e

e

este

is neither feasible nor necessary, as satellite states generally

tralstatThelingindy

urth

theondsur-e o

altoouldll incon

s an

GF

on of

i-

del

es ofin thecou-

lingonalinentsferus-

ts. A satell

tionallyat

and isof

The

tellite state,ored.


have diminished transition matrix elements with the neuground state and therefore can be ignored as an initialfor the purpose of wave-packet dynamic simulations.satellite state modeled only plays a role due to its coupwith the E state. It is therefore sufficient to fit its shapeorder to reproduce the right recurrence behavior in thenamics. This was done by means of the quadratic and foorder coefficientsgs11ds11d

s5d and«s11ds11ds5d .

In regions where satellite states do not interact withmain states, OVGF results approximate well the corresping ADCs3d results by construction. Hence, the potentialfaces calculated with OVGF were used for the purposfitting the vibronic coupling Hamiltonian along these normmodes. For the high-frequencyE modes, the distortion duesatellite states was also negligible so that OVGF data cbe used for those as well, saving computational effort. Aall, this means that we have relied on a mixed data setsisting of OVGF results along the nondegenerate modeADCs3d results along the pseudo-Jahn–Teller activeEmodes. ADCs3d naturally extends the corresponding OVresults, so this mixed approach is consistent.

FIG. 4. Ab initio results for some low-frequencyE modes of allene.sad OVGfor the construction of the model Hamiltonian.sbd ADCs3d data pointssfilledsin Csd. The size of the symbols illustrate the pole strength, i.e., large sy

state crosses theC 2A1 state at around 4 U removed from the equilibriuminaccessible. The fit fromsad is drawn over the ADCs3d data. It can be se

the minima of theA8 branch of theA 2E state. This supports the viewtestament to its robustness against deviations in theab initio adiabatic dainterest are drawn manually over the data to guide the eye. A crossing

data points seen below theA 2E state and crossing with it belong to the sdata along then11sEd mode. Lines indicating the states of interest arepredicted from the analysis of the OVGF result, shown in Fig. 3, can beso the satellite state can be neglected. Note that theA1 state is used only

Table II shows an overview of the on-diagonal linear and

e

--

-

f

-d

quadratic parameters for allene, alongside a comparisthe normal mode frequencies from this worksvtheord andfrom experimental worksvexpt, from Ref. 54d. Table IIIshows the linear off-diagonalsli

sabdd and the on-diagonal blinear sbut not quadraticd coupling coefficientssgi j ,iÞ j

sad d, withthe corresponding values from the previous model21–23

shown for comparison.Table IV shows the quartic coefficients of the mo

Hamiltonian. Note that«is1d=«i

s2d=0, and the other«iskd are

only small corrections. This means that the quartic shapcation PESs are mainly due to anharmonicities presentground state and first-order coupling terms dominate thepling process in this model Hamiltonian.

The most important coefficients mediating the coupbetween the diabatic electronic states are the off-diagcoupling coefficientsli

sk,d. A comparison of the valuesTable III shows that the newly fitted coupling coefficil4

s12d mediating the direct coupling within the charge tranmanifold of theE state differs by about 11% to the previostudies,21–23 while the coefficientsli

s13d mediating the cou

ta pointssfilled circlesd along then9sEd mode with fit resultssolid linesd usedolsd alongn9sEd—gray squares forA8 states and black triangles forA9 states

ls mark the main states that are also approximated by the OVGF resulite

is crossing is 1 eV above the minimum of that state and hence vibraat it approximates the ADCs3d databetter than the OVGF data, particularly

the linear vibronic coupling model Hamiltonian is physically relevantints.scd ADCs3d data along then10sEd mode. Lines indicating the states

en above 4 Usaround 16.5 eVd. This energy range is inaccessible vibrationally.

ite ofE symmetry that cannot interact by symmetryfsee Eq.s19dg. sdd ADCs3dn manually over the data to guide the eye. The crossing with a sa

n above 4 Usaround 15.7 eVd. This is 0.5 eV higher than theE state at equilibriumd the fit of the charge transfer states, so crossings involving it are ign

F dasymbmbo

. Then th

thatta pois se

atelldrawsee

to ai

pling to theB2 stateswhere present in the previous studiesd

re-as rha-

dels

tn as

sagnr

eram

esfferelec

arge

ndingthat

ff

evant

f

free-rest

e

thee

d to

454659

492

6

es byplinge also that


differ by about 11%–14%. This qualitative agreement isassuring, as the experimental photoelectron spectrum wproduced very well by the model Hamiltonian due to Mapatraet al.21–23

The most important differences between the two mocome about by the inclusion of theA1 statek=4 and theA9statek=5. Coupling alongn10 between theE manifold andtheB2 state through a coupling coefficientl10

s13d is not presenin the new model, as it turns out to be symmetry forbiddediscussed above. Instead, a coupling coefficientl10

s15d mediat-ing coupling between theE manifold and theA9 state waparametrized. Note that this parameter has a larger mtude compared to the otherli

sk,d. This is due to the largeenergy difference between theE state and theA9 state at thequilibrium geometry, necessitating a larger coupling paeter to effect a comparable change in the adiabaticE state.

The newly introduced coupling parameterslis34d mediat-

ing coupling between theB2 state and theA1 state are of thsame order of magnitude as the parametersli

s13d. This meanthat they are necessary to take into account and may onew pathway for charge transfer between the localizedtronic states.

TABLE II. Linear and quadratic on-diagonal coefficients of model Hamthose for state 1 by the relations listed below Eq.s13d and parameters no

i Sym kis1d ki

s3d kis4d gi

s

1 A1 −0.4471 −0.2762 −0.4491 02 A1 −0.2527 0.4702 0.3564 −03 A1 −0.1874 −0.0237 0.0988 04 B1 −0.05 B2 0.3364 0.6 B2 0.0865 0.7 B2 0.3163 −0.8 E 0.09 E −0.010 E 0.011 E −0.0

aReference 52.

TABLE III. Off-diagonal linear slisabdd and on-diagonal bilinearsgi j

sadd coeMahapatraet al. sRef. 22d. All values are in eV. Note that coupling alongbetween states 1 and 5. Where corresponding parameters were fittedparameters with exchanged indices are related to those shown by the

i a b lisabda li

sabdb

4 1 2 0.0917 0.1035 3 4 0.19466 3 4 0.24537 3 4 0.23148 1,2 3 0.2726 0.2459 1,2 3 0.2057 0.23710 1,2 5 0.560010 1,2 3 0.13311 1,2 3 0.0500 0.04411 1,2 4 0.2400

aThis work.b
Reference 22.
e-

i-

-

a-

IV. FIVE-STATE, ALL-MODE MODEL VIBRONICCOUPLING HAMILTONIAN FOR PENTATETRAENE

The overall qualitative features of the PESs of the ch

transfer statesB 2E and associated statesC 2B2 andD 2A1 ofthe pentatetraene cation are very similar to the correspoones in allene. As in allene, exactly one mode existsexhibits anomalous down bending of theA9 component othe E state manifold. It isn12sEsA9dd. All other E modes, owhich there are six in pentatetraenesfour in allened, areEsA8d.As before, we solved the problem of theEsA9d mode andspurious crossings of the PESs by recalculating the relPESs along the low-frequencyE modes with ADCs3d. Thesatellite state selected to fit via coupling theA9 component o

theE manifold alongn12sEsA9dd is at the energy of theS13sEdstatesabove which many satellite states existd. As before, thisstate is used merely to provide an additional degree ofdom for the fitting procedure rather than being of inteitself.

Table V shows the OVGF and ADCs3d energies abovthe neutral ground state equilibrium geometrysin eVd of thestates of interest for pentatetraene. It can be seen thatEand theB2 state are this timecloser to each other in th

ian for allenesall values in eVd. Note that parameters for state 2 are relatewn are zero.

giis3d gii

s4d giis5d vtheor vexpt.

a

1 −0.0029 0.0166 0.3937 0.392 −0.0311 −0.0265 0.1861 0.180 0.0000 −0.0117 0.1357 0.13

−0.1656 −0.0788 0.1117 0.11100.0036 −0.0020 0.3937 0.3940.0037 −0.0206 0.2532 0.252

−0.0688 0.0020 0.1800 0.178−0.0813 0.1674 0.4049 0.4055−0.0434 −0.0152 0.1281 0.1260−0.1656 −0.0112 0.1057 0.1069−0.0600 −0.0828 −0.132 0.0446 0.044

nts of model Hamiltonian for allene compared to corresponding valuoden10 between states 1 and 3 in Ref. 22 is replaced in this work by cou

ef. 22, they agree qualitatively with those from the present work. Notgtions listed below Eq.s13d and that parameters not shown are zero.

i j a gi jsada gi j

sadb

1 2 1,2 −0.0575 0.0431 3 1,2 0.00422 3 1,2 −0.0222 7 1,2 0.00085 6 1,2 0.0045 7 1,2 0.0436 7 1,2 0.0216 0.01851 2 3 −0.09892 7 3 0.01096 7 3 0.0641

iltont sho

i1d

.025

.049

.000418044000730530052164216200

fficiethe min Rrela

s inhetere

malultsde

eneally

amilt the

ramiousbra-n othe

Ss

theeing

Pa-that

e itentes

uss. As

esesulte

mil-e

moti-

, al-

e

rus-Fig.

ten-in-for

tionre.the

.

en-sym-

this

tate. It

more


ADCs3d result, unlike for allene, where the correction wathe other direction. The ADCs3d energies are used to fit tHamiltonian as they are more accurate. The states of inare between 15 and 18.4 eV.

As for allene, we present a selection of the normodes with the fitted vibronic coupling Hamiltonian resin Fig. 5. The first row exemplifies a totally symmetric moand the coupling mode which are mostly fitted as for allThe anharmonicity of the PES functions along the totsymmetric modes is most pronounced along then1 mode.The regions of large discrepancy to the present model Htonian are inaccessible by a vibrational wave packet bufit can be improved by adding cubic terms13! h1

skdQ13 to the

diagonal Hamiltonian matrix elementsekk in Eqs. s6d–s8d.This approach necessitates changing all other model paeters for this mode accordingly and introduces spurmaxima in the PESs which are, however, kept at vitionally inaccessible energies due to the balancing actiofourth-order terms. The fit using cubic terms is shown intop right panel of Fig. 5.

A quantitatively more appropriate model for the PE

TABLE IV. Quartic on-diagonal coefficients of mparameters not shown are zero, particularly«i

s1d=«is2d

i Sym ji

1 A1 0.035 132 A1

3 A1

4 B1 0.018 005 B2

6 B2

7 B2 0.002 648 E 0.050 009 E 0.006 5010 E 0.024 0011 E 0.015 50

TABLE V. OVGF and ADCs3d energies above the neutral ground sequilibrium geometrysin eVd of the states of interest for pentatetraenecan be seen that theE and theB2 state are closer to each other in the ADCs3dresult. The ADCs3d energies are used to fit the Hamiltonian as they areaccurate.

State label k fin Eqs.s1d–s13dg OVGF energy ADCs3d energy

X 2E ¯ 8.59 8.65

S1E ¯ ¯ 11.22

A 2E ¯ 11.80 11.56

S2E ¯ ¯ 12.81

]

S6E ¯ ¯ 14.93

B 2E 1,2 15.11 15.08

S7E ¯ ¯ 15.24

C 2B23 15.91 15.82

D 2A14 16.33 16.27

S8E ¯ ¯ 16.30

]

S13E 5 ¯ 18.36

st

.

-

-

f

along theA1 modes would use Morse-type potentials butpolynomial fit is satisfactory and has the advantage of bwithin the Taylor expansion paradigm.

The third row of Fig. 5 exemplifies a tuning mode.rametrization proceeds as with allene. It is remarkablealong n8, the coupling between theB2 and theA1 statessk=3, 4d becomes clearly visible to the naked eye. Herbecomes obvious that theA1 state should be included whfitting a vibronic coupling model Hamiltonian for the staof interest in the pentatetraene cation.

The last two rows of Fig. 5 exemplify degenerateEmodes. As for allene, anEsA9d mode exists and spurioOVGF results are seen along the low-frequency modebefore, this is analyzed by comparing to ADCs3d results.

OVGF and ADCs3d results for two low-frequency modof pentatetraene are compared in Fig. 6. The OVGF rshown along then14sEd mode in Fig. 6sad approximates thmain states from the ADCs3d result fFig. 6sbdg. The OVGFresults are used for the purpose of fitting the model Hatonian. The successful fit along then9sEd mode in allenbased on OVGF data and confirmed by the ADCs3d resultsthat was presented in the preceding section serves asvation for this approach.

The overall qualitative shapes agree in both results

though the upper branch of theB 2E state is predicted to b

steeper in OVGFsand actually crosses theC 2B2 state furtheaway fromQ=0d. The crossing in OVGF has no repercsion on the fitted coupling constants. It can be seen in6sbd that satellite states ofE symmetrysmarked by “sat E”dcross into theB 2E state from below. These crossings potially complicate the interaction in pentatetraene. In theterest of obtaining a computationally viable model andcomparability with allene, we decided to omit this interacin our model and leave its detailed analysis for the futu

It can be seen in Fig. 6sdd that a satellite state crosses

C 2B2 state at around 4 U removed from the equilibrium

However, theC 2B2 state is included primarily as represtative for a larger number of states of the correspondingmetry to aid the fit of the charge transfer states and so

crossing is ignored. The data points seen around theB 2E

l Hamiltonian for allenesall values in eVd. Note that

is3d «i

s4d «is5d

31 87

−0.018 00

10 36 0.010 36−0.050 00 −0.050 00−0.006 50 −0.006 50−0.010 00 −0.002 00

0.006 80

ode=0.

«

0.0

0.0

state belong to satellite states ofE symmetry. Although they

t al-,t is

niceterrermsg onheir

rder

he

t toweraregionto aid

tates

derived, the

odes


do visibly cross with the state of interest, they are nolowed to interact with it by symmetryfEq. s19dg. As beforecoupling into the paper plane cannot be excluded bulikely to be weak.

Tables VI and VII show the parameters for the vibrocoupling model Hamiltonian for pentatetraene. Paramfrom a fit along n1 involving cubic fitting parameters apresented in Table VIII. On-diagonal bilinear coupling tethat are not quadratic, i.e., mediate coupling dependintwo different normal mode coordinates were not fit, as t

FIG. 5. Some normal modes of pentatetraene. Mode labels with symmvibronic coupling model Hamiltonianslinesd with ab initio data pointsfillecoupling mode, a tuning mode, and two pseudo-Jahn–Teller active m

influence on the potential surfaces in pentatetraene is ma

s

ginal. sThese coefficients were already small—of the oof 0.01 eV—in allene, as witnessed by Table III.d

It can be seen that theS adiabatic state along t

n12sEsA9dd mode seen in Fig. 5 is upward bent with respecthe harmonic model due to fitting the down bent of lobranch of theE state. No corresponding data pointsshown for this state, however, as it is placed at a repopulated by many satellite states and selected merelythe fitting of the charge transferE state.

Due to the large number of states crossing into the s

displacement diagrams illustrating the modes and adiabatic energiesfromrclesd. From top to bottom are exemplified a totally symmetric mode.

etry,d ci

r-

orecripthis

statethe

-e b

be

llenngth

en--

areene.enen inplingstatesin al-

dentonic

o theng

llin-cted

eslarge

n

ity ando

so

o


considered in this energy region, it is very likely that mstates should be taken into account for a complete destion of the system. However, this is outside the scope of

work. Since we are interested in a good model for theB 2E

and C 2B2 states relevant for charge transfer and theS statewas introduced only to aid this fit, this is acceptable.

V. COMPARISON OF THE TWO MODELSAND DISCUSSION

As the energy spacing between the charge transferand the states coupling most strongly to them is similar inallene and pentatetraene cationssTables I and Vd, a comparison of the expected coupling strengths can be madmeans of the off-diagonal linear coupling constantssTablesIII and VII d.

It can immediately be seen that theE manifold JT intr-astate coupling constant which mediates direct couplingtween the charge transfer states is, atl5

s12d=0.0251 eV inpentatetraene, only 37% of the corresponding value in as0.0917 eVd. This was expected due to the increased le

FIG. 6. Ab initio results along some low-frequency modes of pentatetr

used for the construction of the model Hamiltonian. The upper branchof the states agree with those found with ADCs3d. sbd ADCs3d datasfilled symsymbols mark the main states that are predicted by OVGF. It can be

mode of allene shown in Fig. 4 but the qualitative descriptions agree.

be seen that satellite states cross into theB 2E electronic manifold whicomparability with the allene model Hamiltonian. Modeling this interac

OVGF data alongn15sEd. The model qualitatively corresponds to the ADs3state at around 4 U removed from the equilibrium. However, a wave pa

that it is not likely to reach this crossing quickly enough to make it relsatellite states ofE symmetry. Although they do visibly cross with the s

of the pentatetraene molecule compared to allene.

-

s

y

-

e

The other off-diagonal PJT coupling constants of ptatetraene involving theE state,li

s1,d, ,=3,…, 5, which mediate indirect coupling between the charge transfer statesof a very similar order to the corresponding values in allOn top of this, there are sixE modes present in pentatetraalong which such activity is present, i.e., two more thaallene. This means that the preference for indirect coubetween the charge transfer states via other electronicshould be at least twice as strong in pentatetraene aslene.

A factor that should greatly influence the time-depenpopulations of the diabatic electronic states is the vibrcoupling between theB2 and A1 states alongB2 normalmodes which has previously not been considered due tabsence of theA1 state from other models. This coupliprocess transfers the wave function from theB2 to the A1

state, where coupling to theE state manifold is overaslightly weaker. This process should therefore slightly dimish the preference for the indirect charge transfer expefrom the coupling constants of the formli

s1,d, ,=3,…,5alone.

compared.sad OVGF datasfilled circlesd alongn14sEd with fit result ssolid linesdemanifold is softer than predicted by ADCs3d but the qualitative tendencid alongn14sEd. The size of the symbols indicates the pole strength, i.e.,that the agreement to the OVGF result insad is not quite as good as along then9

are drawn over the states of interestsB 2E, C 2B2, D 2A1d to guide the eye. It ca

owever, have to be neglected for the sake of computational viabilill be one of the first tasks to complete in the construction of a future mdel. scdults.sdd ADCs3d data alongn15sEd mode. A satellite state crosses theC 2B2

propagating on the flatC 2B2 potential is unlikely to gain a large momentum,

t for dynamics simulations. The data points seen around theB 2E state belong tof interest, they are not allowed to interact with it by symmetryfEq. s19dg.

aene

of thB 2Ebols

seen

Lines

ch, htion w

Cd res

cket

evantate

The number ofB2 modes in pentatetraene is 4, one more

.2,s0.25tantsrgee

linglargelinguellenethe

thanse

ss iFig

hes a

the.st oftureg for

ongtaten.

eor theaene.ctsVGFt-

are

4

t


than in allene. Thelis34d coupling constants are roughly 0

0.1, 0.36, and 0.08 eVssee Table VII for accurate numberd.In allene, the corresponding numbers are roughly 0.2,and 0.23 eV. This means that two of the coupling consare much smaller in pentatetraene while one is much laOverall, the coupling between theB2 andA1 states should bslightly stronger in pentatetraene.

This slightly stronger adverse effect on the PJT coupin pentatetraene should not, however, fully quench theincrease in preference for PJT coupling over JT coupThe photoelectron spectrum of pentatetraene should, dthe larger PJT coupling, be less structured than that of a

Overall, the number of satellite states found inADCs3d calculations for pentatetraene is much largerthat in allene. This means that the errors introduced bylecting only five states to model the charge transfer procemore severe for pentatetraene. This is demonstrated in6sbd, where a number of satellite states crosses into tEstate manifold from below. The density of satellite state

TABLE VI. Linear and quadratic on-diagonal coefficients of model Hrelated to those for state 1 by the relations listed below Eq.s13d and param

i Sym kis1d ki

s3d kis4d

1 A1 −0.3780 −0.2440 −0.24402 A1 −0.1411 −0.2266 −0.26563 A1 −0.2907 0.36584 A1 −0.0989 −0.0739 −0.02685 B1

6 B2 0.33847 B2 0.07508 B2 0.1887 0.33709 B2 0.280610 E

11 E

12 E

13 E

14 E

15 E

TABLE VII. Off-diagonal linear slisabdd and on-diagonal quarticsji ,«i

sadd cparameters for state 2 are related to those for state 1 by the relation=0.

i a b lisabd i sym

5 1 2 0.0251 1 A1

10 1 3 0.2921 2 A1

11 1 3 0.2311 3 A1

15 1 3 0.0681 4 A1

10 1 4 0.2261 5 B1

11 1 4 0.0326 6 B2

13 1 4 0.0808 7 B2

14 1 4 0.2224 8 B2

12 1 5 0.7000 9 B2

6 3 4 0.1823 10 E

7 3 4 0.1033 11 E

8 3 4 0.3620 12 E

9 3 4 0.0761 13 E

14 E

15 E

,

r.

.to.

-s.

t

higher energies is immense but their interaction withcharge transfer state is modeled by only the oneS stateCurrently, these effects have to be ignored in the interecomputational viability. This means that the need for fuimprovements of the presented model is more pressinpentatetraene than for allene.

The results of the MCTDH wave-packet dynamics alwith a detailed analysis of the evolution of the diabatic spopulations will be published in a forthcoming publicatio

VI. CONCLUSION

We have performedab initio OVGF calculations of thcharge transfer states and states interacting with them flinear carbohydrate molecules allene and pentatetrAlong the low-frequencyE modes a satellite state interawith the main states and hence we had to go beyond Oand use the more involved ADCs3d method which takes saellite states into account.

onian for pentatetraenesall values in eVd. Note that parameters for state 2rs not shown are zero.

giis1d gii

s3d giis4d gii

s5d v

0.39210.2387

0.0464 −0.0199 −0.0225 0.1801−0.0080 −0.0128 −0.0106 0.091.0276 −0.0429 −0.0665 0.0876.0031 0.0031 −0.0269 0.3920

0.0234 0.0513 −0.0248 0.27310.0476 −0.0290 −0.0319 0.1873

0.0191 −0.0103 −0.0417 0.1634−0.1024 0.0542 0.4054 0.4033

.0086 −0.1303 0.0279 −0.0053 0.1253

.0100 0.1016

.0107 −0.0178 0.0670 0.0543−0.0386 −0.0662 0.0875 0.0367

.0350 −0.0339 −0.0192 0.1300 0.0188

ients of model Hamiltonian for pentatetraenesall values in eVd. Note thated below Eq.s13d and that parameters not shown are zero, particularly«i

s1d=«is2d

ji «is3d «i

s4d «is5d

0.001 76 −0.000 94 −0.000 96 −0.001 760.018 28 0.005 64 −0.018 280.034 83 0.011 56 −0.011 36 −0.034 830.011 45 0.016 61 −0.026 10 −0.011 45

0.085 00 0.030 87 −0.012 13 −0.086 040.006 66 0.024 30 −0.020 23 −0.005 890.038 00 −0.036 000.007 51 −0.005 25 −0.001 99 −0.007 260.027 00 −0.016 43 −0.009 36 −0.027 670.018 00 −0.003 29 −0.002 49 −0.017 32

amiltete

−

−00

−−

−

−000

−0

oeffics lis

onic

ly toculetetoeen

es

estatneellite

arly

theclu-s of

delopaonlytionfeattingou-lex

n ofxten

tesamillcuas

uppoor-con

ol-u-

ascouer o

dling

raenerefortatet-trum

nt anave-raene

us-ling,withiusrkg in

ecu-

aler

m-

d

.


We have least-square fitted the data to a linear vibrcoupling Hamiltonian, Eq.s13d, using the fitting programVCHAM.24 Compared to previous work,21–23 the modeHamiltonian was extended by fourth-order terms mainlbetter describe anharmonicities in the neutral moleground state. Taking theA1 single electron ionization staimmediately above theB2 state into account allowed usextend the model by off-diagonal linear coupling betwthe E state manifold and theA1 state along theEsA8d modesand between theB2 and theA1 state alongB2 modes. Adeeper analysis of theCs configurations of the moleculperturbed by theE modes revealed that there is oneE modein each molecule, labeledEsA9d, where coupling of theE statemanifold to a higher energyS state ofA9 symmetry must bpresent. Such states were identified among the satellitepredicted by the ADCs3d method and consequently omodel state chosen to represent the multitude of satstates coupling with theE state.

In the present context of charge transfer, particulthose satellite states which cross the charge transferE statesfsee Fig. 4sdd for allene and Fig. 6scd for pentatetraenegshould be kept in mind. We find that their coupling tocharge transferE states is weak but nevertheless their insion could be of importance when studying the dynamicthe system.

An explicit inclusion of the satellite states in the mois currently beyond our computational means but the prgation time in quantum dynamics calculations scaleslinearly with the number of states so that with a continuaof the growth of computer capacity, this should becomesible in the future. Besides the actual propagation, the fiprocess performed for the construction of the vibronic cpling model Hamiltonian becomes increasingly compwith the inclusion of additional states. Further automatiothe fitting process is therefore desirable to make the esion of the model a viable option.

After the extension of the model to five electronic staas discussed above, parametrization of the model Htonian yielded a satisfactory fit to the adiabatic PESs calated ab initio. This success for such complex situationspresent in allene and pentatetraene lends resounding sto the notion of modeling the coupling Hamiltonian by fmally Taylor expanding the diabatic representation andsidering the lowest order terms.

There is only one simple JT coupling mode in each mecule and it is ofB1 symmetry. The corresponding JT copling constant fitted for pentatetraene is only about 37%large as the corresponding coefficient in allene. The PJTpling constants, on the other hand, are of the same ord

TABLE VIII. Fitting parameters along then1 modetermssall values in eVd. This fit is shown in the topto the first lines in Tables VI and VII are markedterms to avoid potential maxima in the region of

kis1d ki

s3d kis4d hi

s1d his

−0.315 −0.114 −0.189 −0.1276 −

magnitude in both molecules and their number is larger in

es

-

-

-

--

rt

-

-f

pentatetraene. The coupling between theB2 and theA1 stateswhich is by symmetry most relevant alongB2 modes shoulbe equally strong in both molecules, although the coupalong some of these modes is preferred in pentatetwhile the coupling activity of theB2 modes in allene is moor less uniform. Overall, the PJT coupling pathwaycharge transfer should be preferred more strongly in penraene than in allene. The resulting photoelectron specshould have less pronounced features. We will preseevaluation of these predictions against the results of wpacket dynamics simulations of the allene and pentatetcations in a forthcoming publication.

ACKNOWLEDGMENTS

The authors would like to thank Horst Köppel and Santa Mahapatra for helpful discussions on vibronic coupJochen Schirmer and Jörg Breidbach for their supportusing an ADCs3d implementation, and Christoph Cattarfor his introduction to the fit program VCHAM. This wohas been supported financially by the Volkswagen-Stiftunthe Schwerpunktprogramm Intramolecular and Intermollar Charge Transfer.

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entatetraene from a fit involving third-order couplingt panel of Fig. 5. The parameters changed with respecta tilde. Note that the cubic terms are offset with quarticest. Note also thatk1

s1d= k1s2d, h1

s1d=h1s2d, and «1

s1d= «1s2d.

his4d «i

s1d «is3d «i

s4d

2 −0.1345 0.030 95 0.029 95 0.021 64

for prigh

withinter

3d

0.152

10452s1999d.

Köp-

hem.

lv.

Thin

hem.

m-

, J.

. J.

em.

r

. A.

A.

hem.

. Rep.

, in-

of allfor the

ink tosec-

y


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53See EPAPS Document No. E-JCPSA6-122-304513 for an overviewmodes of allene and pentatetraene which may serve as a companioninterested reader to illustrate statements made in the text. A direct lthis document may be found in the online article’s HTML referencetion. The document may also be reached via the EPAPS homepageshttp://www.aip.org/pubservs/epaps.htmld or from ftp.aip.org in the director/epaps/. See the EPAPS homepage for more information.

54
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Documents

Allene and pentatetraene cations as models for