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5 Electronic Properties: Vertical Excitation Energies, Structure, and Frequencies in Excited Electronic States
7b ! 8b
8a ! 9a
7a,b ! 9a,b
8a ! 10a
8a ! 11a
7a,b ! 9a,b
8a ! 12a
6,5b ! 8,9b
EOM-CCSD/6-311(2 ỵ ,2 ỵ )G(d,p)34.
MRCI þ D/ANO(2 þ )ÃÃ 35.
excitation energies computed by TD-DFT with N07D basis set for the structures optimised at the respective DFT/N07D level.
for plots of Molecular Orbitals see ﬁgure 5.
Experimental results from Ref. [57–59].
TABLE 6.15 Vertical Excitation (VE) Energies, (in eV) and Oscillator Strength of the First Eight Doublet Excited Electronic States
of the Vinyl Radical
INTERPLAY OF STEREOELECTRONIC VIBRATIONAL
FIGURE 6.5 Vinyl radical, frontier molecular orbitals involved in electronic transitions into
the ﬁrst eight doublet excited electronic states.
systematically overestimates vertical excitation energies. The main discrepancy is the
underestimation of excitation energy for pÃ p transition, large enough to reverse
the ordering of second and third electronic states. Nevertheless, it should be mentioned
that accurate computation of electronic excitations is still a challenging task even for
the most elaborate (and expensive) ab initio methodologies. In summary, TD-DFT is
normally able to provide quite reliable information about the nature and properties of
highly excited electronic states at a reasonable cost.
6.5.3 Case Studies: Structures and Frequencies of Vinyl Radical in First Three
Doublet Excited Electronic States
Electronic excitation can lead to signiﬁcant changes of the geometry structure and
vibrational properties of molecular systems. Vinyl radical stands as an example where
such modiﬁcations are particularly enhanced. Computational studies of geometry
changes induced by electronic excitations require geometry optimization in excited
electronic states. This can be accomplished at TD-DFT level in a quite easy manner, in
particular without the need of reducing the number of molecular orbitals considered as
often necessary in multireference approaches (CASSCF).61 It is also possible to
compute TD-DFT harmonic frequencies from numerical differentiation of analytical
gradients. Moreover, computational studies provide information on the electron
density in both electronic states in a way that it is possible to analyze in detail its
changes upon excitation, and describe appropriately their inﬂuence on radical
properties. Such approach has been chosen to study ﬁrst three doublet electronic
states of vinyl radical with the TD-CAMB3LYP model. Changes in geometry structure
and frequencies upon excitation are reported in Tables 6.16 and 6.17, while the plots of
electron density difference between excited and ground states are shown in Fig. 6.6.
The ﬁrst excited electronic state is related to the transfer of an electron from a p
orbital into the SOMO leading to a lone pair. This induces strong repulsive interaction
and leads to the strong deviation of HC1C2 angle, and slight elongation of C–C bond.
The TD-CAMB3LYP results are in remarkable agreement with their CASSCF and
EOM-CCSD counterparts. Such transition inﬂuences mostly the frequency related to
the n7 mode, the C–H out of plane bending (see Fig. 6.7). Next excited state of 2 A 00
symmetry can be described as the pÃ n transition, and leads to changes of opposite
tendency than for 12A00 state, increase of HC1C2 angle, and smaller frequency for n7
mode. The lowest state of 2A0 symmetry is related to the pÃ p transition thus
weakening of C–C bond. Indeed the most pronounced change in the structure is a C–C
bond elongation by about 0.2 A, while all other geometry parameters remain almost
unchanged from their ground-state values. Conversely, this state is characterized by
most pronounced changes in frequency values mainly for modes n7 and n8 related to
out-of plane bending.
6.6 VIBRONIC SPECTRA
Valuable information on the physical–chemical properties of radicals can be often
obtained by photoelectron studies in which the electron is detached, so that open-shell
systems can be created. Moreover, excited electronic states of radicals can be studied
by absorption spectroscopy in the UV–vis regions. An analysis of the resulting
experimental spectra can be even more difﬁcult than for ground-state IR or Raman
ones. The additional factors can be related to the often not trivial identiﬁcation of
electronic band origin, possible overlap of several electronic transitions and nonadiabatic effects. Although such complications are challenging also for the theoretical
approaches, some5,62 examples show already their interpretative efﬁciency.
7b ! 8b
8a ! 9a
Geometry Changes Upon Excitation
7a,b ! 9a,b
TABLE 6.16 Structures, and Adiabatic
Excitation (AE) Energies (in eV) of the First Three Doublet Electronic Excited States of the Vinyl
Radical. Bond Lengths are in A and Angles in Degrees
TABLE 6.17 Vibrational frequencies (in cmÀ1) of Vinyl Radical in its Ground
and First Three Doublet Excited Electronic States
7b- > 8b
7a,b- > 12a,b
8a- > 12a
FIGURE 6.6 Vinyl radical: plots of the difference in electron density between the ground and
the ﬁrst three doublet excited electronic states. The regions that have lost electron density as a
result of the transition are shown in dark blue, and the bright yellow regions gained electron
INTERPLAY OF STEREOELECTRONIC VIBRATIONAL
Normal modes of vinyl radical.
6.6.1 Theoretical Background
A vibronic stick spectrum can be obtained by summing the intensities of the lines of
absorption or emission. For a given incident energy, these intensities are proportional to
the square of the transition dipole moment integral between the electronic states. Using
the Born–Oppenheimer approximation and the Eckart conditions,63 this integral can be
obtained through the analysis of the transitions between vibronic states. This, is
however, insufﬁcient due to the lack of an analytic solution for the electronic transition
dipole moment. An approximation derived from the Franck–Condon principle64–66
allows to expand the electronic transition dipole moment in a Taylor series whenever
the electronic transition is fast enough that the relative positions and velocities of the
nuclei are nearly unaltered by the molecular vibrations. This concept is sketched in
Fig. 6.8. Replacing the electronic transition dipole moment by its Taylor expansion
about the equilibrium geometry of the ﬁnal state, it is possible to compute the
probability of transition, and so the intensity of a line of absorption or emission,
by calculating the overlap integrals between the vibrational states of the initial and ﬁnal
electronic states. The so-called Franck–Condon (FC) approximation assumes that the
transition dipole moment is unchanged during the transition. The Herzberg–Teller
(HT) approximation takes into account a linear variation of the transition dipole
moment along normal coordinates. For most systems, the FC and HT approximations
are sufﬁcient to correctly describe both absorption and emission spectra.
FIGURE 6.8 The Franck–Condon principle is shown by the vertical dotted line. The clear
rectangle shows schematically the broader possibilities of transitions when the Herzberg–Teller
approximation is used.
In the framework of the Franck–Condon principle,64–66 time-independent ab initio
approaches to simulate vibronic spectra are based on the computation of overlap
integrals (known as FC integrals), between the vibrational wave functions of the
electronic states involved in the transition. The computation of FC integrals requires a
detailed knowledge of the multidimensional PES of both electronic states or, within
the harmonic approximation, at least computation of equilibrium geometry structures
and vibrational properties. For Herzberg–Teller calculations also the transition dipole
moment and its ﬁrst derivatives are required. Moreover, it is necessary to take into
account mixing between the normal modes of the initial and the ﬁnal states, using the
linear transformation proposed by Duschinsky67:
Q ẳ JQ0 ỵ K
where Q and Q0 represent the mass-weighted normal coordinates of the initial and ﬁnal
electronic states, respectively. The Duschinsky matrix J describes the projection of the
normal coordinate basis vectors of the initial state on those of the ﬁnal state and
represents the rotation of the normal modes upon the transition. The displacement
INTERPLAY OF STEREOELECTRONIC VIBRATIONAL
vector K represents the displacements of the normal modes between the initial-state
and the ﬁnal-state structures.
6.6.2 Computational Strategy
Till recently, computations of vibronic spectra have been limited to small systems or
approximated approaches, mainly as a consequence of the difﬁculties to obtain
accurate descriptions of excited electronic states of polyatomic molecules and to
computational cost of full dimensional vibronic treatment. Recent developments in
electronic structure theory for excited states within the time-dependent density
functional theory (TD-DFT)53,54 and resolution-of-the-identity approximation of
coupled cluster theory (RI-CC2)68 and in effective approaches to simulate electronic
spectra11,69–74 have paved the route toward the simulation of spectra for signiﬁcantly
Recently integrated approaches, capable of accurately simulating one-photon
absorbtion (OPA) or one-photon emission (OPE) vibronic spectra and at the same
time easily accessible to nonspecialists, have been introduced.11 The computational
strategy is based on an effective evaluation method72,73 able to select a priori the
relevant transitions to be computed. The details of the procedure used to compute the
spectrum can be found in Refs11,72–74. In brief, simulation of vibrationally resolved
electronic spectra starts with the computation of the equilibrium geometries, frequencies, and normal modes for both electronic states involved in the transition. The
computational tool has been set within the harmonic approximation, but a simple
correction scheme to derive excited state’s anharmonic frequencies from ground-state
data has been implemented.74 The simplest computation of Franck–Condon spectrum
requires the following data: Cartesian coordinates of the atoms, atomic masses, energy
of the ground and excited states, frequencies for the two electronic states involved in
the transition, and normal modes for the two electronic states, expressed by the atom
6.6.3 Case Studies: Electronic Absorption Spectrum of Phenyl Radical
The electronic absorption spectrum of phenyl radical is an interesting example where
computational approaches can be compared to the experimental spectrum assigned to
the above-mentioned radical on the basis of the strict correlation of intensity evolution
in simultaneously measured IR and UV–vis spectra,2 for which several independent
precursors gave consistent results.
In general, the accuracy of a simulated spectrum depends on the quality of the
description of both the initial and the ﬁnal electronic states of the transition. This is
obviously related to the proper choice of a well-suited computational model: a reliable
description of equilibrium structures, harmonic frequencies, normal modes, and
electronic transition energy is necessary. In the study of the A2B1 X2 A1 electronic
transition of phenyl radical the structural and vibrational properties have been
obtained with the B3LYP/TDB3LYP//N07D model, designed for computational
studies of free radicals.28,29 Unconstrained geometry optimizations lead to planar
FIGURE 6.9 Phenyl radical, atom numbering scheme and plot of the electron density
difference between the A2B1 and X2 A1 electronic states. The regions that have lost electron
density as a result of the transition are shown in bright yellow, and the darker blue regions gained
structures for both ground and ﬁrst excited electronic states. The geometry parameters
are compared in Table 6.18 with the results reported by Kim et al.76 while the atom
numbering can be found in Fig. 6.9. It can be observed that the DFT results are in good
agreement with their multireference (CASSCF) counterparts.76 Both computational
models predict the same trend in the geometry changes upon electronic excitation and
agree also on their magnitude. The main geometry changes are related to the increase
of C1–C2 and C1–C6 bond lengths and decrease of C2–C1–C6 angle, in line with the
n p transfer of electron density from the aromatic ring to the carbon orbital. Indeed,
TABLE 6.18 Geometry Structure of Phenyl Radical in
the Ground X2 A1 and First
Excited A B1 Electronic States. Bond Lengths are in A and Angles in Degrees
CASSCF Ref. 
INTERPLAY OF STEREOELECTRONIC VIBRATIONAL
the excited electronic state is characterized by an electron lone pair on the carbon atom,
as conﬁrmed by the difference density plot of Fig. 6.9.
The simulated Franck–Condon Hertzberg–Teller (FC-HT) spectra (Fig. 6.10)
computed taking into account changes in structures, normal modes, and vibrational
frequencies between both electronic states closely resemble their experimental
counterparts. The most striking difference is a relative shift of both spectra. It is
worth to recall that this part of the spectrum is very weak and the measurements have
been close to the performance limit of spectrometer even by application of a multiplepass technique, as described in detail in Ref. 2. The reason for the discrepancy can be
the weak intensity of the 0–0 transition: as a matter of fact, an analysis of the
experimental spectrum from Ref. 2 shows a weak progression preceding the ﬁrst
intense band assigned to the spectrum origin. For the weakly allowed transitions the
unequivocal assignment of the 0–0 transition may be cumbersome, and the theoretical
spectrum suggests that transition assigned as 0–0 can be already a result of the
progression to the excited vibrational state of A2B1. The comparison of theoretical
spectra with the experimental spectrum shifted by $850 cmÀ1 (as to match the
transition origin reported in Ref. 77) shows a very good agreement, also for
the band position of the most intense transitions, suggesting possible revision of
the experimental data. It is interesting to recall that earlier theoretical results have not
been able to reproduce correctly the spectrum shape, discrepancies being attributed to
nonadiabatic couplings.76,78 Nevertheless, this seems to be due to the limited
dimensionality models both studies have been performed with. It is thus strongly
advisable to exploit full-dimensional vibronic models prior to analyze the possible role
of nonadiabatic effects.
FIGURE 6.10 Theoretical, convoluted and stick FC–HT spectrum of the A2B1 X2 A1
electronic transition of phenyl. The experimental1 spectrum is shown for comparison. Spectra
have been arbitrarily shifted along energy axis to achieve best match, see text for details.
6.7 CONCLUDING REMARKS
The present paper summarizes the results of systematic computational studies devoted
to the calculation of several properties of carbon-centered radicals using DFTand TDDFT approaches and the new N07D basis set. The results for a representative set of
organic free radicals seem accurate enough to allow for quantitative studies. This
ﬁnding together with the computational efﬁciency of the approach suggests that we
dispose of a quite powerful tool for the study of free radicals, especially taking into
account that the same density functional and basis set can be used for different
properties and for second and third row atoms. Furthermore, the availability of
effective discrete/continuum solvent models and of different dynamical approaches,
together with the reduced dimensions of the N07D basis set allow to perform
comprehensive analyses aimed at evaluating the roles of stereoelectronic, vibrational,
and environmental effects in determining the overall properties of large ﬂexible
radicals of current biological and/or technological interest.
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