Vertical transition energies vs. absorption maxima: Illustration with the UV absorption spectrum of...

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy xxx (2013) xxx–xxx

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Spectrochimica Acta Part A: Molecular andBiomolecular Spectroscopy

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Vertical transition energies vs. absorption maxima: Illustration with the UVabsorption spectrum of ethylene

Benjamin Lasorne a,⇑, Joaquim Jornet-Somoza a, Hans-Dieter Meyer b, David Lauvergnat c,Michael A. Robb d, Fabien Gatti a

a Institut Charles Gerhardt Montpellier, Université Montpellier 2, CC 15001, Place Eugène Bataillon, 34095 Montpellier, Franceb Ruprecht-Karls Universität, Physikalisch Chemisches Institut, Theoretische Chemie, Im Neuenheimer Feld 229, 69120 Heidelberg, Germanyc Laboratoire de Chimie Physique, Bât. 349, Université Paris-Sud, 91405 Orsay, Franced Department of Chemistry, Imperial College, SW7 2AZ London, United Kingdom

h i g h l i g h t s

� Calibrating vertical transition energycalculations against experimentalabsorption maxima is not alwaysadequate.� The discrepancy between both

quantities can be larger than theexpected accuracy of 0.1 eV even inthe harmonic case.� The first Rydberg and valence

transitions of ethylene are twoexamples of such a discrepancy.� The valence transition is nonvertical

and too anharmonic, and the averagetransition energy is a safer estimate.� The band origin of the Rydberg

transition seems to have beenconfused for the absorptionmaximum.

1386-1425/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.saa.2013.04.078

⇑ Corresponding author. Tel.: +33 467144619.E-mail address: [email protected]

Please cite this article in press as: B. Lasorne etethylene, Spectrochimica Acta Part A: Molecula

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:UV–visible spectroscopyAbsorption maximumExcited-state quantum chemistryVertical transition energyFranck–Condon factorsEthylene

a b s t r a c t

We revisit the validity of making a direct comparison between measured absorption maxima andcomputed vertical transition energies within 0.1 eV to calibrate an excited-state level of theory. This isillustrated on the UV absorption spectrum of ethylene for which the usual experimental values of7.66 eV (V N) and 7.11 eV (R(3s) N) cannot be compared directly to the results of electronic structurecalculations for two very different reasons. After validation of our level of theory against experimentaldata, a new experimental reference of 7.28 eV is suggested for benchmarking the Rydberg state, andthe often-cited average transition energy (7.80 eV) is confirmed as a safer estimate for the valence state.

� 2013 Elsevier B.V. All rights reserved.

Introduction

When dealing with excited-state quantum chemistry methods,calculated vertical transition energies are often compared to exper-

ll rights reserved.

(B. Lasorne).

al., Vertical transition energier and Biomolecular Spectroscop

imental absorption maxima in order to validate the adequacy ofthe chosen level of theory (see, e.g., Refs. [1–3]). The rule of thumbis that the difference between both values should be no more thanabout 0.1 eV (about 800 cm�1), which is the order of magnitude ofthe accuracy expected from such methods. However, it is wellknown that these two quantities are not identical in theirdefinitions (see Appendix and, e.g., Ref. [4]). For example, a simple

s vs. absorption maxima: Illustration with the UV absorption spectrum ofy (2013), http://dx.doi.org/10.1016/j.saa.2013.04.078

http://dx.doi.org/10.1016/j.saa.2013.04.078

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2 B. Lasorne et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy xxx (2013) xxx–xxx

multidimensional harmonic model with no frequency change norDuschinsky rotation predicts the vertical transition energy as thecentroid of a convolution of Poisson distributions [5,6] that maydiffer significantly from the absorption maximum if there is a largeshift between the equilibrium positions of the two potential energysurfaces. Owing to this and, of course, to the fact that several kindsof other effects can cumulate to account for larger discrepancies,one should thus be careful when benchmarking excited-statemethods against experimental absorption maxima. The UV–visibleabsorption spectrum of ethylene is a striking example of a such adiscrepancy, as illustrated in the present work.

A two-dimensional (CC stretching and torsion) model of poten-tial energy surfaces and vibronic couplings fitted to ab initio data atthe multireference configuration interaction (MRCI) level of theorywas proposed in Ref. [7]. It was based on a 17-state multireferencecalculation coming from a nonstandard multiconfiguration self-consistent field (MCSCF) treatment presented in Refs. [8,9]. Adia-batic energies were produced with the Gaussian [10] and Molpro[11] quantum chemistry programs. This model was used to runsubsequent quantum dynamics simulations with the HeidelbergMCTDH package [12–15] and thus produce the absorption spectrafor the V N and R(3s) N transitions of 12C2

1H4 ethylene at zerotemperature. The MRCI potential energy scans along both coordi-nates from the Franck–Condon point are shown in Fig. 1.

Our MRCI vertical transition energies have values at 7.92 eV(V N) and 7.40 eV (R(3s) N), i.e., about 0.3 eV higher than thereceived absorption maxima in both cases (7.66 eV and 7.11 eV,respectively). However, most features of our computed spectra

Fig. 1. MRCI energy scans from Franck–Condon point. Black: ground state N; red:valence state V; green: Rydberg state R(3s). Upper panel: along torsional mode.Lower panel: along stretching mode. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

Please cite this article in press as: B. Lasorne et al., Vertical transition energieethylene, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscop

agree with the experimental data within a few 0.01 eV in thelow-energy range, thus validating the adequacy of our level of the-ory. In fact, the experimental values usually cited as referencesshould not be compared directly to the results of electronic struc-ture calculations. As discussed in Ref. [4], the absorption maximumof the first valence transition V N occurs at 7.66 eV while thebest calculations give vertical excitation energies in the range7.8–8.0 eV [16–19,3]. Although the valence state V of ethylene hap-pens to be surprisingly difficult to converge (see, e.g., Ref. [19] andreferences therein), this is not the explanation for such a discrep-ancy. There are other good reasons for these two quantities notto match: (i) the transition is nonvertical [20], as the V potentialenergy surface is a double well, unstable with respect to twistingin the planar region (top of the barrier) and decreasing towardperpendicular geometries much below the Franck–Condon point;(ii) it is highly anharmonic since it involves a large torsional dis-placement and is very flat with respect to stretching (see Fig. 1).This is an example where the ‘‘0.1 eV rule of thumb’’ does not hold.In their paper, Davidson and Jarzecki suggested a correctionscheme for calculating the appropriate intensity-weighted averageenergy of the transition, thus providing a value of 7.80 eV that is acorrect estimate of the vertical transition energy. We also revisitedthe R(3s) N transition. The usually cited value of 7.11 eV [16,18]corresponds to the band origin (0–0 transition) [21–24]. Followingour recent theoretical investigation [7], we suggest that the actualabsorption maximum occurs at 7.28 eV (one quantum in the CCstretching mode), which should thus be the value against whichto calibrate vertical transition energies within 0.1 eV instead of7.11 eV.

Absorption spectrum of ethylene

In this section we compare absorption maxima and verticaltransition energies for the first two valence and Rydberg electronictransitions of ethylene, V N and R(3s) N. It is shown that thereference values commonly cited in the literature for benchmark-ing electronic structure calculations of states V and R(3s) are notadequate. The valence transition is significantly nonvertical andinvolves too much anharmonicity for using the absorptionmaximum, and the average transition energy should be used in-stead. For the Rydberg transition, the reference value correspondsto the 0–0 band origin, whereas the actual absorption maximumoccurs with one quantum in the CC stretching mode.

Ground-state zero-point energy

Calculating the zero-point energy (ZPE) in the ground electronicstate is required to define the effective zero for transition energies.Using our aforementioned two-dimensional model, we obtained areduced ZPE of 0.1669 eV (1346 cm�1).

The experimental fundamentals [24,25] of the torsional mode(m4) and stretching mode (m2) are 1026 cm�1 and 1625 cm�1, respec-tively. The corresponding two-dimensional ZPE is 1325 cm�1.Fundamentals are raw vibrational transition energies that partlyaccount for anharmonicity. Computing the experimental ZPEinvolves a post-treatment where the effect of anharmonicity isremoved. Received values for those [26,25] are 1044–1047 cm�1

(torsion) and 1655–1656 cm�1 (stretching). The estimated two-dimensional ZPE is 1349–1352 cm�1, which is thus in very goodagreement with our theoretical value (within about 5 cm�1).

Valence transition

Despite its apparent simplicity, ethylene is known for havingexcited states that are significantly difficult to converge. The best



0

10

20

30

40

50

60

70

80

90

6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2

Inte

nsit

y / a

rbitr

ary

units

Frequency / eV

Fig. 2. Simulated absorption spectra for the R(3s) N transition (green line) andthe V N transition (red line) [7]. The experimental absorption spectrum is shownfor comparison (black line) [24]. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

Table 1Computed stretching progression 2v

0 of torsional doublets 40;20 for the electronic

transition R(3s) N. Transition energies are given in eV with respect to the ground-state two-dimensional ZPE (experimental values [24] within brackets). The relativeintensities are the squared overlaps of the vibrational ground state in N with thevibrational eigenstates in R(3s) relative to 21

0.

Transition Energy Relative intensity

0–0 7.09 (7.11) 0.77

420

7.15 (7.17) 0.53

210

7.27 (7.28) 1.00

21042

07.33 (7.34) 0.61

220

7.45 (7.45) 0.67

22042

07.51 (7.50) 0.37

230

7.63 (7.61) 0.25

23042

07.69 (7.67) 0.19

B. Lasorne et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy xxx (2013) xxx–xxx 3

theoretical estimates to date for the vertical transition energy ofV N have been calculated within 7.8–8.0 eV [16–19,3] whereasthe absorption maximum is measured at 7.66 eV [21–24]. As dis-cussed in Refs. [20,17,4], this transition is nonvertical and tooanharmonic for directly comparing the vertical transition energyto the absorption maximum. In that case, the average energy ofthe absorption spectrum is a safer estimate of the vertical transi-tion energy than the absorption maximum as illustrated in Ref.[4]. These authors suggested a correction scheme for calculatingthe appropriate intensity-weighted average energy of the transi-tion, thus providing a value of 7.80 eV that better complies withtheoretical vertical transition energies.

With a vertical transition energy DV0 = 7.92 eV at the MRCIlevel of theory [7] we produced an absorption spectrum with amaximum, DE0v0 , occurring at 7.70 eV. This value is only 0.04 eVhigher than the experimental value, 7.66 eV. In addition, our calcu-lated average transition energy (see Appendix ), DE0 ¼ 7:86 eV, isonly 0.06 eV higher than Davidson and Jarzecki’s value [4],7.80 eV. This validates the adequacy of our ab initio calculationsand two-dimensional model for quantum dynamics simulations.In this pathological case, the vertical transition energy is 0.26 eVhigher than the absorption maximum. In contrast, the averagetransition energy is a safer estimate of the vertical transitionenergy within 0.1 eV.

The adequacy of our level of theory was also confirmed by cal-culating the band origin, DE00. The energy difference between the Vand N minima is 5.47 eV. The ground-state two-dimensional ZPE iscalculated at 0.1669 eV (1346 cm�1) and the excited-state one at0.1482 eV (1195 cm�1). Accounting for differential ZPE effect, wecomputed the band origin at 5.45 eV, which nicely matches thevalue extrapolated from experiments (not directly measured be-cause the intensity is too low) of 5.5 eV given in Ref. [23].

We also calculated the first two excited-state fundamentalswith respect to the band origin: 861 cm�1 for torsional excitationand 1319 cm�1 for stretching excitation. The low-energy range ofthe experimental spectrum shows a vibrational progression spacedby about 700–900 cm�1 [21–23]. Our torsional frequency thussupports the current interpretation that this structure correspondsto a primary progression in the torsional mode (see, e.g., Refs.[23,27,24,28]).

Rydberg transition

The best theoretical estimates to date for the vertical transitionenergy of R(3s) N have been calculated within 7.09–7.16 eV[16,18] (MRCI levels of theory with different reference spacesand excitation schemes). They compare nicely with the experimen-tal band origin, 7.11 eV. In contrast, the MRCI level of theory weused [7] (alternative reference space) gave us 7.40 eV, worryinglyoff by about 0.3 eV! However, subsequent quantum dynamics cal-culations gave us a band origin at 7.09 eV, now in almost perfectagreement with the experimental value 7.11 eV. Finding the reasonbehind such a discrepancy is thus required in order to validate ourcalculations.

Our theoretical spectrum matches the experiments quite well inthe spectral region 6–8 eV (see Fig. 2). We computed the most sig-nificant vibrational eigenstates with a partial Lanczos diagonalisa-tion and the same Hamiltonian model. An approximate assignmentis given in Table 1. The agreement with experiments falls within0.01–0.02 eV. From the relative intensities, it appears that the ac-tual experimental absorption maximum of the Rydberg transitionalone is not 7.11 eV (origin of the band, 0–0) but rather 7.28 eV.In this spectral region, both Rydberg and valence transitions over-lap. A possible confusion may thus have come from a crude sepa-ration of the two spectra, where it was assumed that the 0–0transition was the most intense when removing the contribution


due to the valence transition V N. We thus suggest that anycalculation of the vertical transition energy for R(3s) N shouldbe compared to 7.28 eV 21

0

� �instead of 7.11 eV. The ‘‘0.1 eV rule

of thumb’’ actually applies to this case, as the vertical transitionenergy is larger than the absorption maximum by only 0.12 eV.

Let us now analyse this spectrum using the model presented inAppendix to determine the origin of the remaining discrepancy.The stretching progression involves a quantum of about 0.17 eV,and each torsional doublet is separated by a double quantum ofabout 0.08 eV (both experimentally and theoretically) [24]. Wecalculated the distortion parameters, c, as the square-roots of theratios of the fundamentals. For the torsional mode, we obtainedan effective c = 0.49. This value is only indicative, as the torsionalmode is quite anharmonic (see Fig. 1). For a 40

0 transition, thevertical transition energy is larger than the absorption maximum(differential ZPE effect; see Eq. (A.15)) by about 0.05 eV. For thestretching mode, c = 0.92. Using an ad hoc value bd = 1.6 leads to

relative Franck–Condon factors jhvRð3sÞ j0Nij2

jh1Rð3sÞ j0Nij2¼ 0:8; 1:0; 0:6; 0:2 for

v = 0, 1, 2, 3. These approximately fit the relative intensities inTable 1. Note that the second components of the doublets relativeto 21

0420 follow a similar trend (0.9,1.0,0.6,0.3). For such values, the

vertical transition energy is larger than the absorption maximumby about 0.03 eV. In total, a simple two-dimensional harmonicmodel accounts for an error of about 0.08 eV out of 0.12 eV whencomparing the vertical transition energy to the absorption maxi-mum. The remaining 0.04 eV can most certainly be attributed tothe effect of anharmonicity (Morse-shaped potential energy forthe stretching mode and double-well potential energy curve forthe torsional mode; see Fig. 1). Note that the ten modes not



Fig. 3. Calculated and observed transition energies for some selected transitions in the absorption spectrum. Vertical and nonvertical transition energies computed at theMRCI level of theory are displayed for comparison.


accounted for in our study must have negligible or self-compensat-ing effects since the peaks calculated with our two-dimensionalmodel match the experimental ones almost perfectly with noartificial shift of the relative energy.

Finally, this analysis can also provide an estimate of the shift ofthe equilibrium position in R(3s) with respect to N. Using an effec-tive mass m = m(12C)/2 (see Ref. [7]) gives b = 17.0 Å�1. For aneffective bd = 1.6, we get d = 0.09 Å. From the experimental value1.34 Å in N, this leads to an effective 1.43 Å in R(3s), which exactlymatches the minimum of our ab initio potential energy curve (seeFig. 1). Note that the inferred experimental value [24] is 1.41 Å.

Conclusions

As illustrated in this paper, one should be careful when compar-ing computed vertical transition energies to observed absorptionmaxima when calibrating excited-state quantum chemistry calcu-lations. Discrepancies between vertical transition energies andabsorption maxima larger than the usually received value of0.1 eV may occur for good or for bad reasons.

First, this rule of thumb does not apply to the valence transitionin ethylene because it is nonvertical and too anharmonic. Davidsonand Jarzecki [4] thus proposed to use the average energy of thetransition, estimated at 7.80 eV, instead of the absorption maxi-mum, observed at 7.66 eV. As summarized in Fig. 3, our calcula-tions produced a vertical transition energy at 7.92 eV, an averagetransition energy at 7.86 eV (0.06 eV higher than Davidson and Jar-zecki’s estimate), and an absorption maximum at 7.70 eV (0.04 eVhigher than the experimental value). We may probably infer that aperfect calculation of the vertical transition energy should thus fallin the range 7.86–7.88 eV. If so, this value is less than 0.1 eV offwith respect to Davidson and Jarzecki’s value, thus confirming itas a safer estimate than the absorption maximum.

Second, the agreement may seem to fail when it actually worksif the absorption maximum of the transition is not assigned cor-rectly. This is the case for the Rydberg transition in ethylene. Aftervalidation of our level of theory against experimental data within afew 0.01 eV, we have suggested here a new experimental referenceof 7.28 eV (21

0 transition) instead of 7.11 eV (0–0 transition) forbenchmarking the Rydberg state energy against the absorptionmaximum (see Fig. 3). Proposing 7.40 eV as a value for the verticaltransition energy awaits confirmation from a full-dimensionaltreatment including a possible differential ZPE effect due to theremaining coordinates.

As a definitive test, quantum dynamics simulations should berun when affordable on accurate ab initio potential energy sur-faces. They will produce a theoretical absorption spectrum thatcan be compared to the experimental one and allowing for a de-


tailed assignment of the vibronic structure. Anharmonicity andother effects such as Herzberg–Teller coordinate-dependence ofthe transition dipole or vibronic couplings among electronic statescan thus be accounted for correctly.

Acknowledgements

This work was supported by Projet blanc ANR-09-BLAN-0417 ofthe Agence Nationale de la Recherche (ANR) of the French CentreNational de la Recherche Scientifique (CNRS) and the DeutscheForschungsgemeinschaft (DFG). JJS gratefully acknowledges theANR and the Comissionat per a Universitats i Recerca del Departa-ment d’Innovació, Universitats i Empresa, de la Generalitat deCatalunya for financial support, and the Marie Curie Cofund Action(FP7) of the European Union for the Beatriu de Pinós (2010 BP-A-00248) postdoctoral fellowship.

Two-level harmonic model

In this appendix we quantify the extent of the discrepancy be-tween computed vertical transition energies and observed absorp-tion maxima in the best-case scenario, where a harmonic,adiabatic, Franck–Condon approximation is valid and withoutany significant Duschinsky rotation. We illustrate this on a one-dimensional system described with a two-level harmonic model.A quantitative estimate of the error for a single vibrational modeis provided for various shifts and distortions between both poten-tial energy curves. It is shown that the ‘‘0.1 eV rule of thumb’’ is va-lid under most circumstances but may also badly fail.

Franck–Condon factors

Here, we recall the well-known analytical expressions ofFranck–Condon factors for a one-dimensional harmonic oscillator(see, e.g., Ref. [29] and references therein). The two electronicstates, A and B, are assumed to be uncoupled (Born-Oppenheimerapproximation) except through light-matter interaction. Thepotential energy corresponding to state B is shifted (by a term d)and distorted (by a factor c) with respect to state A. Thecorresponding Born-Oppenheimer Hamiltonian operators (normalcoordinate x, mass m) read

bHA ¼ � �h2

2m@2

@x2 þk2

x2; ðA:1Þ

bHB ¼ � �h2

2m@2

@x2 þ c4 k2ðx� dÞ2 þ C: ðA:2Þ



Table A.2Variation of v0 and v⁄ � v0 for various values of the dimensionless shift, bd, anddistortion, c.

v0 bd

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.5 0 0 0 0 1 1 1 2 2 3 40.6 0 0 0 0 1 1 2 2 3 4 50.7 0 0 0 0 1 2 2 3 4 5 60.8 0 0 0 1 1 2 3 4 5 6 80.9 0 0 0 1 1 2 3 5 6 8 10

c 1.0 0 0 0 1 1 3 4 6 8 10 121.2 0 0 0 1 2 4 6 8 11 14 171.4 0 0 0 1 3 5 8 11 14 19 231.6 0 0 0 0 3 6 10 14 19 24 301.8 0 0 0 0 4 8 13 18 24 31 392.0 0 0 0 0 6 10 16 22 30 38 48

v⁄ � v0

0.5 2 2 2 2 1 1 2 1 2 1 10.6 1 1 1 1 1 1 1 1 1 1 00.7 1 1 1 1 1 0 1 1 0 0 10.8 0 0 1 0 1 0 0 0 0 1 00.9 0 0 1 0 1 1 1 0 1 0 0

c 1.0 0 0 1 0 1 0 0 0 0 0 01.2 0 0 1 0 1 0 0 1 0 0 11.4 0 0 1 1 1 1 1 1 1 1 11.6 0 0 1 3 2 2 1 1 1 2 21.8 0 0 1 3 2 2 1 1 2 1 12.0 0 0 2 4 2 2 2 2 2 2 2


Using the harmonic frequency of state A;x ¼ffiffiffiffiffiffiffiffiffiffik=m

p, and the corre-

sponding harmonic parameter, b ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffimx=�h

p, both Hamiltonian

operators can be rescaled in terms of dimensionless coordinates,such that

bHA ¼ �hx2� @2

@ðbxÞ2þ ðbxÞ2

" #; ðA:3Þ

bHB ¼ c2 �hx2� @2

@½cbðx� dÞ�2þ ½cbðx� dÞ�2

( )þ C; ðA:4Þ

where it is clear that c2x is the harmonic angular frequency of stateB and cb its corresponding harmonic parameter. In a practical case,the distortion parameter is estimated as c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixB=xA

pif xB and xA

denote the harmonic frequencies of states B and A, respectively.Assuming zero temperature, resonant light absorption can in-

duce a transition from the initial vibrational ground eigenstate inA:

uA0ðxÞ ¼

ffiffiffiffiffiffiffiffibffiffiffiffipp

se�

b2x2

2 ; ðA:5Þ

to any final vibrational eigenstate in B:

uBvðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficb

2vv !ffiffiffiffipp

sHv ½cbðx� dÞ�e�

c2b2 ðx�dÞ22 ; ðA:6Þ

where Hv is the Hermite polynomial of degree v.The Franck–Condon factors that give the transition probabilities

and govern the relative intensities of the absorption peaks underthe Franck–Condon picture read (see, e.g., Ref. [29] and referencestherein)

jhvBj0Aij2 ¼ 12v�1v !

c1þ c2 e

�c2b2d2

1þc2

�Xbv=2c

j¼0

v2j

� �ð2j� 1Þ!! 4c2

1þ c2

� �j

Hv�2j �cbd

1þ c2

� �" #2

;

ðA:7Þ

where b c denotes the floor function (integer value), and(2j � 1)!! = 1 � 3 � � � � � (2j � 1) is a double factorial (note that(�1)!! = 0!! = 1 by definition).

Using, the explicit expression of the Hermite polynomialcontribution:

Hv�2j �cbd

1þ c2

� �¼ ðv � 2jÞ!

Xbv=2c�j

l¼0

ð�1Þl

l!ðv � 2j� 2lÞ! �2cbd

1þ c2

� �v�2j�2l

;

ðA:8Þ

and

ð2j� 1Þ!! ¼ ð2jÞ!2jj!

; ðA:9Þ

Eq. (A.7) can be recast as

jhvBj0Aij2 ¼ v !

2v�1

c1þ c2 e

�c2b2d2

1þc2Xbv=2c

j¼0

1j!

2c2

1þ c2

� �j"

�Xbv=2c�j

l¼0

ð�1Þl

l!ðv � 2j� 2lÞ! �2cbd

1þ c2

� �v�2j�2l#2

: ðA:10Þ

Absorption maximum

Here, we use the previous expressions to compare the verticaltransition energy, which can be calculated directly using a quan-


tum chemistry method, to the absorption maximum, which is ob-tained from a UV–visible spectroscopy measurement. The verticaltransition energy reads

DV0 ¼c4ðbdÞ2

2�hxþ C: ðA:11Þ

On the other hand, the absorption maximum reads

DE0v0 ¼c2ð2v0 þ 1Þ � 1

2�hxþ C; ðA:12Þ

where v0 denotes the vibrational quantum number correspondingto the largest Franck–Condon factor, jhvB

0j0Aij2.

An effective quantum number, v⁄, can be introduced such thatthe vertical transition energy is rounded to the nearest transitionenergy:

DV0 � DE0v� ; ðA:13Þ

with

v� ¼ c2ðbdÞ2 þ 1=c2 � 12

" #; ðA:14Þ

where [ ] denotes here the nearest-integer function.Under normal conditions (moderate shift and distortion), both

quantum numbers, v0 and v⁄, differ by no more than one unit,which validates the usual assumption that the vertical transitionenergy is a good estimate of the absorption maximum within afraction of ⁄x. This is assessed quantitatively in Tables A.2 andA.3 for pedagogical purposes. The worst cases shown here(bd = 1.5 and c P 1.6) lead to v⁄P 3 whereas v0 = 0. Note thatthings get worse rapidly when c becomes larger because the erroron the excited-state energy increases as c2(v⁄ � v0).

The vertical transition energy is always larger than the absorp-tion maximum, except for cases with no shift (bd = 0) and stiffen-ing distortion (c > 1), as shown in Table A.3. The no-shift caseapplies to non-totally symmetric modes when there is molecularsymmetry. If the oscillator is less stiff in the excited state, thereis a downshift of the 0–0 transition (the most intense):

DE00 ¼c2 � 1

2�hxþ C; ðA:15Þ



Table A.3Variation of the difference DV0 � DE0v0 in units of ⁄x for various values of the dimensionless shift, bd, and distortion, c.

DV0 � DE0v0 bd

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.5 0.4 0.4 0.4 0.4 0.3 0.3 0.4 0.3 0.4 0.3 0.20.6 0.3 0.3 0.4 0.5 0.2 0.4 0.2 0.4 0.3 0.2 0.10.7 0.3 0.3 0.4 0.5 0.2 0.0 0.4 0.3 0.2 0.2 0.30.8 0.2 0.2 0.4 0.0 0.4 0.2 0.1 0.1 0.3 0.5 0.20.9 0.1 0.2 0.4 0.0 0.6 0.5 0.6 0.1 0.5 0.3 0.2

c 1.0 0.0 0.1 0.5 0.1 1.0 0.1 0.5 0.1 0.0 0.1 0.51.2 �0.2 0.0 0.8 0.7 1.0 0.5 0.5 1.0 0.5 0.6 1.21.4 �0.5 0.0 1.4 1.9 1.3 1.7 1.1 1.5 2.8 1.2 2.51.6 �0.8 0.0 2.5 6.6 4.6 4.3 3.1 3.5 3.0 4.1 4.31.8 �1.1 0.2 4.1 10.7 6.9 5.8 4.0 4.9 5.1 4.7 3.72.0 �1.5 0.5 6.5 16.5 6.5 8.5 6.5 8.5 6.5 8.5 6.5

Table A.4Estimate of the difference DV0 � DE0v0 in the case of a typical ground-state vibrationof 1500 cm�1 for various values of the dimensionless shift, bd, and distortion, c. Thedifferences DV0 � DE0 are shown in the last but one column for comparison. The lastcolumn gives the excited-state frequency in cm�1.

ðDV0 � DE0v0 Þ/eV ðDV0 � DE0Þ/eV c2x/cm�1

bd 0.0 1.0 2.0 4.0

c0.5 0.07 0.08 0.05 0.07 0.04 3750.7 0.05 0.07 0.05 0.04 0.04 7350.9 0.02 0.08 0.11 0.09 0.02 12151.0 0.00 0.09 0.19 0.00 0.00 15001.2 �0.04 0.15 0.19 0.10 �0.05 21601.4 �0.09 0.27 0.25 0.52 �0.13 2940


with respect to the vertical transition energy:

DV0 ¼ C; ðA:16Þ

where c2 � 1 < 0.From Table A.3, it is clear that significant discrepancies are only

observed for stiffening distortions (c > 1), which occur less com-monly than softening ones, since the overall antibonding characterof the electronic structure often increases when dealing with ex-cited electronic states. Noticeable counterexamples are vibrationsexhibiting excited-state exalted frequencies due to second-orderJahn–Teller effect (see, e.g., Refs. [30,31]). In the latter case, the ex-cited state is stiffer than the ground state due to a strongly avoidedcrossing (strong enough for the ground state to still exhibit a single– although flat – minimum). However, this usually happens alongnon-totally symmetric vibrations, hence without any shift, and theerror stays moderate as already mentioned.

In most cases, the vertical transition energy is thus a good esti-mate of the absorption maximum within about 0.1 eV (about800 cm�1), as usually assumed. This analysis was made for a singlevibrational mode. Even though the error will be small in mostcases, it may cumulate for each vibration in a polyatomic moleculeand thus become significant in a large system. The main reasons forany further discrepancy are most likely (i) the anharmonicity of thepotential energy surfaces and (ii) the breakdown of the Franck–Condon approximation (in particular the dependence of thetransition dipole on the nuclear coordinates, known as theHerzberg–Teller effect; see, e.g., Ref. [32]). Nonadiabatic effects(vibronic couplings) may also play a role.

Average transition energy

As pointed out in Ref. [4] the average energy of the transition isa safer estimate of the vertical transition energy than the absorp-tion maximum. The average kinetic energy of the initial state(vibrational ground state in the electronic ground state, uA

0ðxÞ, pro-jected onto the electronic excited state) is identical on both elec-tronic states, and the average potential energy will not lead tolarge differential effects.

More precisely, the average transition energy is defined as

DE0 ¼ h0AjbHBj0Ai � h0AjbHAj0Ai; ðA:17Þ

¼ h0AjVB � VAj0Ai; ðA:18Þ

which, in the harmonic case, reads

DE0 ¼ DV0 þc4 � 1

4�hx: ðA:19Þ

For illustration purposes, Table A.4 gives the difference ofDV0 � DE0v0 and DV0 � DE0 for a typical ground-state vibration of1500 cm�1. The average transition energy, DE0, does not depend


on the shift. The difference with the vertical transition energy,DV0, is small in all cases where the potential energy is less stiff inthe excited state (c < 1). For a shifted case (bd – 0), it seems alwayssafer to use DE0 instead of DE0v0 as an estimate of DV0 in the har-monic case. Note that anharmonicity is not expected to lead to ma-jor discrepancies when calculating the average value of thepotential energy over the spatial extent of juA

0ðxÞj2. For unshifted

cases (bd = 0 and v0 = 0), DE0 is still better than DE00 when c < 1(most usual case) and slightly worse only when c > 1. In practice,estimating DE0 from experimental data is not straightforward, asthis requires a post-treatment of the absorption spectrum to pro-duce an intensity-weighted average energy of the transition [4].

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Vertical transition energies vs. absorption maxima: Illustration with the UV absorption spectrum of...

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