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The allyl radical revisited: a theoretical study of the electronic spectrum
Chemical Physics Letters 380 (2003) 689–698 www.elsevier.com/locate/cplett
The allyl radical revisited: a theoretical study of the electronic spectrum Francesco Aquilante *, Kasper P. Jensen, Bj€ orn O. Roos Department of Theoretical Chemistry, Chemical Center, Lund University, P.O. Box 124, Lund S-22100, Sweden Received 7 July 2003; in final form 16 September 2003 Published online: 9 October 2003
Abstract In this Letter, we report the electronic spectrum of the allyl radical, obtained with multiconfigurational perturbation theory (MS-CASPT2). The assignment of the spectrum is in accordance with experiment to within 0.2 eV. We have computed the complete first Rydberg series and the beginning of the second Rydberg series. A new valence-excited 2 B1 state has been found which has hitherto been hidden by Rydberg transitions. A rationalisation of the electronic spectrum is provided in terms of resonance forms in ground and excited states. This model shows that while a multiconfigurational wavefunction is necessary to qualitatively model the system, the large ionic character of the valence electronic states makes an accurate treatment of the dynamical correlation necessary for a quantitative description of the spectrum. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction The allyl radical is the simplest hydrocarbon p-radical and serves a prominent function in chemistry as a fundamental test case and starting point for more extensive work on radical chemistry [1]. In addition to having the p-electronic features of more complicated p-radicals, it is itself an important species in many polymerisation and photochemical reactions [2], including the Cope rearrangement [3] and formation of phenyl groups.
*
Corresponding author. Fax: +46-46-222-45-43. E-mail address: [email protected] (F. Aquilante).
Many of these reactions are poorly understood, and the detailed knowledge of the electronic structure of allyl is a prerequisite for understanding these dynamics. Allyl displays a higher (C2v ) symmetry than most other hydrocarbon p-radicals and is therefore easier to analyse in terms of the electronic structure. This is in particular true from a theoretical perspective where the use of the full symmetry in principle enables the direct assignment of configurations and states in the electronic spectrum, which may be difficult in an experimental spectrum of close-lying states without point group symmetry as a distinct separator of states. One of the interesting features of the allyl radical is the instability of the RHF potential energy
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surface, i.e., there are lower unsymmetric solutions with UHF. The true geometry of the radical is C2v , but UHF can only recover a Cs symmetry due to inclusion of spin polarisation. Both HF methods are highly unsuitable, however, since it is well known that the wavefunction of the radical is multiconfigurational. Recent theoretical work on the allyl radical therefore concentrates on multiconfigurational MO or VB approaches. This includes the assignment of the vibrational absorption spectrum of the ground state and the first two excited states with CASSCF [4], and a CASSCF study of the simplest cyclisation reaction imaginable, the cyclisation of allyl radical from the first excited state (2 B1 ) [5]. Generalised valence bond methods have been applied to explain the order of methylene rotation barriers in the allyl cation, anion, and radical (37.8, 23.1, and 12.6 kcal/mol, respectively) [6]. The electronic spectrum of the allyl radical remains incomplete. Until now, the most complete experimental spectrum has been derived by a twophoton resonant ionisation spectroscopy study [7]. It has the features of describing the vibrational structure of some of the low-lying Rydberg states of this molecule but misses a complete identification of the electronic states in the Rydberg region. The significant problems with assignment of the electronic spectrum of this simple molecule are due to the closeness in energy of the states [8]. In particular, in the Rydberg region there is an overlap with valence excited states, as we show in this Letter. As a consequence of these problems, theoretical studies with multiconfigurational wavefunctions have been encouraged to assign the 2 B1 state in more detail [9]. In this work, the spectrum has been assigned for the valence region and the start of the second Rydberg series has been identified. It has been concluded earlier and henceforth assumed that there is only one valence excited state in the electronic spectrum of allyl radical [7,9]. We have found a second valence excited 2 B1 state in the Rydberg region, which has not been assigned experimentally, probably because of the assignment problems in this part of the spectrum. We have also found the origin of the first Rydberg series to be the transition to the 1 2 A1 state. Finally, we
have located the starting point of the second Rydberg series as the 32 A2 state with an energy 8.36 eV above the ground state.
2. Methods and computational details Geometry optimisations were performed employing the generally contracted basis functions of the atomic natural orbital type (ANO-L) [10], which are well designed for the treatment of polarisation effects and are flexible enough to describe both ground and valence excited states [11,12]. The contraction scheme C[4s3p2d]/H[3s2p] was employed from the primitive set C[14s9p4d]/ H[8s4p]. The geometries of the ground and valence excited states were optimised at the CASPT2 level, using a numerical gradient procedure. All the structures were computed at the C2v symmetry and the active space employed for the geometry optimisations comprises all the electrons of the p-system. The molecule is placed in the zx-plane. The employed active space for the geometry optimisations can be labelled (0201/3) within the irreps (a1 b1 b2 a2 ) of the C2v point group as seen in Table 1. The obtained structure for the ground state was subsequently used to compute the vertically excited states. In order to avoid erroneous mixing between valence and Rydberg states, the ANO-L basis set used in the geometry optimisation was supplemented with a set of 1s1p1d diffuse
Table 1 CASSCF wavefunctions employed to compute the valence and Rydberg excited states of the allyl radical Active spacea
States
No. CSFs
No. rootsb
4201/3 0601/3 0421/3 0402/3 1201/5 0201/3
2
40 76 36 34 5 1
4 4 2 3 1 1
A1 ð1a2 ! 3s; 3pz ; 3dz2 ; 3dx2 y 2 Þ B1 ðvalence; 1a2 ! 3py ; 3dyz Þ 2 B2 ð1a2 ! 3px ; 3dxz Þ 2 A2 ð1a2 ! 3dxy ; 1b1 ! 3py Þ 2 A1 ð6a1 ! 1a2 Þ 4 A2 ð1b1 ! 2b1 Þ 2
a Number of active orbitals (/total number of active electrons) within the irreps a1 b1 b2 a2 , respectively. b Number of roots requested in the state-averaged CASSCF calculation.
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
functions, contracted from a set of 8s8p8d primitives. The general procedure for obtaining this set of contracted diffuse basis functions is described in detail elsewhere [12]. The set of Rydberg functions were placed in the charge centroid of the molecular cation. The vertical excitation energies have been computed by means of the CASPT2 approach [13,14]. The CASSCF wavefunction is used as a reference to determine the first-order wavefunction and the second-order energy. In the CASSCF calculation, all the carbon 1s core electrons were kept frozen in the form of the ground-state SCF wavefunction. Furthermore, these orbitals were not correlated at the second-order level. The active spaces employed, the number of configuration state functions (CSFs), and the number of state-averaged CASSCF roots are shown in Table 1. All excitation energies were obtained using the ground state energy computed with the same active space of the corresponding excited states. The inclusion of Rydberg orbitals removes erroneus mixing between valence and Rydberg states. Active spaces for Rydberg states were obtained according to the symmetry species to which they belong. The choice of the active space for the 2 B1 states (0601/3) is not trivial and has been guided by an accurate checking of the convergence of the results enlarging the active space. The second-order perturbation treatment was carried out using the multistate extension of the CASPT2 method. In this so-called MS-CASPT2 method [15], the space spanned by the SACASSCF roots is used to construct an effective Hamiltonian, where the diagonal elements correspond to the CASPT2 energies and the off-diagonal elements introduce a coupling of second order in the dynamical correlation energy. Allowing the states to interact under the influence of the dynamic correlation, the MS-CASPT2 approach provides an adequate description of the chemical situations where the CASSCF wavefunction is not a good reference or where a strong mixing occurs between the reference state and CAS-CI states, such as in the case of strong valence-Rydberg mixing. Nonetheless, to avoid an artificially large interaction among the CASPT2 roots, the weight
691
of the reference function for the excited states must be very close to that for the ground state, since the MS-CASPT2 approach is sensitive (more than the normal CASPT2) to the good convergence of the CASSCF wavefunction and to the absence of intruder states. Finally, the CASSCF state interaction (CASSI) method [16,17] was used to obtain the transition properties. MS-CASPT2 excitation energies and PMCAS-CI transition dipole moments were used in the oscillator strength formula 2 f ¼ l2ab DEab 3
ð1Þ
calculated with respect to the states a and b. The PMCAS-CI is a wavefunction (perturbatively modified CAS-CI) obtained by linear combination of the CAS states in the MS-CASPT2 wavefunction. Even in situations where the addition of the MS-CASPT2 corrections to the CASPT2 energies is of minor importance, the second-order interaction among CAS states can significantly affect the transition properties. All calculations were performed with version 5.4 of the MO L C A S quantum chemistry software [18].
3. Results and discussion 3.1. Electronic structure and geometry The ground state of allyl is 2 A2 which is domi2 1 nated by the ð1b1 Þ ð1a2 Þ configuration (92%), corresponding to the RHF wavefunction. The geometries optimised at the CASPT2 level of theory are shown in Table 2. As shown in Table 4, a further contribution to the 2 A2 ground state CASSCF wavefunction arises from the configuration where one of the ð1b1 Þ electrons is excited to the ð2b1 Þ MO. Although this configuration gives a much smaller contribution to the total wavefunction (6%), it has important consequences for the character of the ground state of the allyl radical, as will be shown below. As is observed in Fig. 1, the three MOs which constitute the p space of the allyl radical have distinctly differing bonding characters. The ð1b1 Þ
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Table 2 Equilibrium geometries for the ground and the first excited state of the allyl radical
2
1 A2 Expt.a 12 B1 a
\C2 C1 C3 (°)
) C1 –C2 (A
) C1 –H1 (A
) C2 –H2 (A
) C2 –H02 (A
124.8 124.0 123.1
1.384 1.386 1.455
1.081 1.087 1.077
1.075 1.082 1.074
1.078 1.085 1.076
Ref. [23].
Fig. 1. Molecular orbitals of the p system of allyl radical. Upper left: 1b1 . Upper right: 2b1 . Bottom: 1a2 .
MO is a bonding orbital along the C–C bonds, ð1a2 Þ is non-bonding and ð2b1 Þ is antibonding. It is energetically favourable for the ð1a2 Þ non-bonding p orbital to distort and develop a weakly bonding character along one of the C–C bonds and a corresponding antibonding character along the other C–C bond. This is the reason behind the instability observed in the RHF wavefunction for this system. When the spin symmetry constraint has been released at the UHF level an unphysical broken symmetry solution appears. The ground state geometry of the allyl radical at the UHF level is in
fact computed to be of Cs symmetry rather than C2v (one of the C–C bonds being shorter). At the CASSCF level of theory, this singlet instability is efficiently removed by the use of a multiconfigurational wavefunction, which distributes the three p electrons in all the orbitals of the p system. The geometry of the molecule is thus correctly predicted to be of C2v symmetry by the multiconfigurational approach which is able to treat the near degeneracy problem. We also optimised the 2 B1 state which was found higher in energy by about 3 eV. The main geometrical parameters are
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
displayed in Table 2. The available experimental data and our results are in good agreement. The for the bond distances are accurate to 0.002 A carbon–carbon bonds and to 0.007 A for carbon– hydrogen bonds. The angle between the three carbon atoms \C2 C1 C3 is computed to be 124.8°, which is 0.8° larger than experiment. Recently published B3LYP results obtained with a larger basis set including diffuse functions [4] exhibit less accurate predictions, especially regarding the CCC angle. Bond lengths are in better agreement with the experimental results than the UCCSD(T) available data [19]. The main shortcomings of such methods lie in the lack of description of the multiconfigurational character of the ground state. The CASPT2 geometry optimisation on the other hand gives accurate results because it recovers the remaining (dynamical) correlation effects on top of the correct treatment of the near degeneracy problem at the CASSCF level. No experimental data are available for the geometry of the first excited state. This may imply that this state is not stable and will isomerise. The appearance of diffuse vibronic lines in the 12 B1 12 A2 band indicates that isomerisation to cyclopropyl radical may indeed take place [20]. By comparison with the ground state results, it is worth noticing the lengthening of the C–C bond due to the inclusion of the 2b1 antibonding orbital in the CSFs, which contributes to the composition of the 12 B1 state. 3.2. Atomic spin population from CASSCF The spin polarisation at the CASSCF level describes the correlation energy involved in separating electrons in the same angular and radial regions of space (Fermi correlation). We calculated the spin populations from a Mulliken analysis of the CASSCF wavefunctions to obtain a diagnostic of this correlation, which is very significant in the allyl radical. These spin populations are shown in Table 3. Due to the absence of r-MOs in the active space, r–p spin polarisation effects are not taken into account in our CASSCF/CASPT2 calculation. The ground state has the unpaired electron distributed over the atoms C2 and C3 of Fig. 2 (0.575 on each atom). The central carbon has a small ()0.179) and negative value of the spin density. The ground state
693
Table 3 CASSCF p (3,3) 2pp spin populations in the allyl radical State 2
1 A2 (GS) 12 B1 a
C1
C2 (C3 )
C1 /C2
Expt.a
)0.179 0.856
0.575 0.075
)0.311 11.413
)0.282
Ref. [19] Ratio between the measured H1 /H2 hfc constants.
Fig. 2. Allyl radical. C2v symmetry.
Table 4 Composition of the PMCAS-CI wavefunctions of the valence excited states of the allyl radical State 2
1 A2 (GS) 12 B1 42 B1 32 A1 14 A2
Principal configurationa 2
2
2
1
ð6a1 Þ ð1b1 Þ ð4b2 Þ ð1a2 Þ ð1b1 Þd ! ð2b1 Þu ð1b1 Þd ! ð1a2 Þd ð1a2 Þu ! ð2b1 Þu ð1b1 Þd ! ð1a2 Þd ð1a2 Þu ! ð2b1 Þu ð6a1 Þd ! ð1a2 Þd ð1b1 Þd ! ð2b1 Þu
CASSCF (%) 92 6 (+)45 ())45 (+)45 (+)45 92 100
The label u and d refer to electrons with the spin up and down respectively. a Natural orbital occupancy with respect to the ground state principal configuration.
electronic configuration in fact has the unpaired electron in the 1a2 orbital, which has no contribution from any of the C1 AOs. This means that the contributions to the spin density of the central carbon atom arise only as an effect of the p–p spin polarisation. Such spin polarisation is a common effect of Fermi correlation present in all manyelectron systems and can be described at the unrestricted HF, KS, or CASSCF/CASPT2 level. For the first excited state 2 B1 , the main contributions to the wavefunction come from the CSFs ð1b1 Þ1 ð1a2 Þ2
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1
and ð1b1 Þ ð2b1 Þ . The spin population is now concentrated more on the central carbon (0.856), due to the single occupation of the b1 orbitals in the 12 B1 state. The last two columns in Table 3 give the calculated and experimental values of the ratio between the atomic spin populations of the two non-equivalent carbon atoms. According to the model proposed by Mc-Connell [21], the values for the hyperfine coupling (hfc) constants for the hydrogen atoms in open-shell, planar p systems are proportional to the spin populations in the 2pp AOs of the carbon atom to which they are bonded. On the other hand, hfc constants are proportional to the spin densities at the hydrogen nuclei through the Fermi-contact term. Even though these spin densities are dominated by the spin populations in the 1s AOs of the hydrogen atoms, an accurate computation of hfc constants is still a challenge for many quantum chemical methods. In light of that, the validity of the McConnell hypothesis makes possible computations of hfc from 2pp spin populations and vice versa. The CASSCF method has shown to give reasonably accurate 2pp spin densities, which match the experimental findings. Previous results carried out with several DFT and Post-HF methods [19] have not been able to give more accurate values of the 2pp spin populations for the carbon atoms of the allyl radical. 3.3. Electronic spectrum The main result of the present Letter, the analysis of the valence and Rydberg spectrum of the allyl radical calculated at different levels of theory, is reported in Table 5. The 14 A2 state, the lowest quartet, is reported for the sake of completeness. The first excited state is computed to be of 2 B1 symmetry at 3.26 eV above the ground state. The vertical transition to this state shows a low oscillator strength, in agreement with the low-lying peak of weak intensity detected experimentally at 3.07 eV. As shown in Table 4, this electronic state has a strong multiconfigurational character. It is described by the out-of-phase combination of the singly-excited configurations ð1b1 Þ ! ð1a2 Þ and
ð1a2 Þ ! ð2b1 Þ. As pointed out by previous theoretical studies [9,22] and also given in Table 4, this is a feature of the 12 B1 state in common with the 42 B1 state in the allyl radical. It is worth noticing the contribution from the dynamical correlation effects introduced by the CASPT2 correction to the CASSCF excitation energy. For the 12 B1 state the decrease in excitation energy due to dynamical correlation is approximately 0.37 eV. We have rationalised this effect in terms of the resonance forms shown in Fig. 3. These are not intended to resemble the resonance forms used in VB theories because the spin coupling of the configurations is not taken into account. The resonance forms presented here are obtained by decomposing the MO configurations in AO configurations and limiting the analysis to the p-system only. The stabilising effect of the dynamical correlation on valence excited states can be associated with the stronger ionic character compared to the ground state. Fig. 3 shows the resonance forms that can be associated with the two configurations involved in the 2 B1 CASSCF wavefunctions: CSF1 corresponds to the singlyexcited configuration ð1b1 Þ ! ð1a2 Þ and CSF2 to the ð1a2 Þ ! ð2b1 Þ configuration. By comparison, the resonance forms for the ground state are also displayed. Both the ground state and the 2 B1 state are highly ionic, with an additional contribution from a tri-radical structure in the latter case. The total number of ionic forms is the same for the two states. This indicates in turn that the stabilising contribution of the dynamical correlation to the energy of the 2 B1 state is of the same order of magnitude as in the ground state. The lower excitation energy of the 2 B1 state must be explained qualitatively by the analysis of the natural orbital occupancies for this state compared to the ground state. The occupancies of the three valence p orbitals in the ground state and in the 2 B1 state are displayed in Table 6. It is shown how the occupancy for the two NOs of b1 symmetry changes drastically from the ground to the excited state. These two orbitals correlate by means of the socalled left–right correlation effect. This contribution to the total correlation energy is thus larger for the 2 B1 state and can be related to the decrease of the excitation energy for this state when dy-
Table 5 Vertical excitation energies/eV and oscillator strengths in the electronic spectrum of the allyl radical CASSCF
CASPT2
MSCASPT2
Experimental
Oscillator strength
Experimentale
Valence excited states 12 B1 14 A2 32 A1 42 B1
3.70 ðxÞ 4.53 7.78 7.57 ðxÞ
3.33 5.89 6.22 6.59
3.32
3.07a
6.4 104
1.3 103
Rydberg states 1a2 ! R 12 A1 ð1a2 ! 3sÞ 22 B1 ð1a2 ! 3py Þ 22 A1 ð1a2 ! 3pz Þ 12 B2 ð1a2 ! 3px Þ 32 B1 ð1a2 ! 3dyz Þ 42 A1 ð1a2 ! 3dx2 y 2 Þ 52 A1 ð1a2 ! 3dz2 Þ 22 B2 ð1a2 ! 3dxz Þ 22 A2 ð1a2 ! 3dxy Þ
4.47 5.37 4.94 5.10 6.04 5.76 5.82 5.77 5.75
ðyÞ ðzÞ
5.11 5.83 5.65 5.76 6.55 6.51 6.61 6.56 6.60
5.11 5.73 5.65 5.76 6.36 6.51 6.61 6.56 6.61
Rydberg states 1b1 ! R 32 A2 ð1a2 ! 3py Þ
7.58 ðzÞ
8.34
8.36
a
ðxÞ ðyÞ ðxÞ
6.90
0.118 4.97b (5.39)5.90)c
0.113
0.23
(6.19)6.30)d
1.3 102 0.084
0.4 103 0.05
3.4 105 2.1 105 1.7 103
Ref. [24]. Ref. [25]. c Ref. [26]. d Ref. [9]. Rm e Ref. [9]; The experimental oscillator strength is computed by the formula f ¼ 4.3 109 mif edm from a spectrum recorded in an Ar matrix. b
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
State
695
696
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
Fig. 3. Main resonance structures in some of the valence states of allyl radical. Upper: 2 A2 ground state. Middle: 2 B1 (1b1 ! 1a2 ). Bottom: 2 B1 (1a2 ! 2b1 ).
namical correlation is included at the CASPT2 level. Analogous reasoning can be applied to the 42 B1 state. The 42 B1 state has the same composition as the 12 B1 state but with the difference that the combination of the two CSFs is now in-phase. For the 42 B1 state the difference between the CASSCF and the CASPT2 energy is larger (0.98 eV). This large stabilising effect must be investigated in this case also in terms of the subsequent MS-CASPT2 results. A coupling via the dynamical correlation between the 42 B1 state and the 22 B1 and 32 B1 states takes place as showed by means of the MSCASPT2 treatment. This circumstance indicates that at the CASPT2 level the 42 B1 state has a mixed valence/Rydberg character, which results in a larger stabilisation due to dynamical correlation, as compared with the 12 B1 state which is instead of pure valence character. The large oscillator strength associated with the vertical transition to this state indicates that this state should give rise to one of the main features of Table 6 CASSCF N.O. occupanciesa for the valence p-orbitals of allyl radical State
2
1 A2 (GS) 12 B1 a
Natural orbital 1b1
1a2
2b1
1.906 1.405
1.000 1.021
0.090 0.568
Computed with the active space 0601/3.
the electronic spectrum of allyl radical. However, the region around 6.90 eV is not well established experimentally, and a comparison with the present results is not possible at the moment. The 2 A1 state is the only low-lying valence excited state of r ! p character. The transition to this state is dipole-forbidden and thus not evidenciated in the spectrum. The 2 A1 state is mainly characterised by the single excitation from the highest occupied r MO to the 1a2 SOMO. Its energy lies in the middle of the Rydberg region and it is computed to be 6.22 eV at the CASPT2 level of theory. The large stabilisation due to dynamical correlation for the 2 A1 state can be explained in terms of the presence of a pair–pair interaction in the p space among the electrons of the MOs 1b1 and 1a2 which are doubly occupied in the leading configuration for this state. In the ground state a pair–pair correlation between the electrons in the active shells exists only between the 1b1 and the r MOs and it is certainly of lower strength, owing to the spatial orthogonality of the two orbitals. The importance of correlation effects, as reflected in the ionic character of the electronic states, can be elaborated by a prediction of the energy of the 14 A2 state. This state has a clear single determinantal character, as shown in Table 4. The only resonance structure associated with this electronic configuration is the tri-radical one. For this reason, the stabilisation due to the correlation energy increases the excitation energy at
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
the CASPT2 level compared to the CASSCF results. This is a confirmation that the ground state is more stabilised by dynamical correlation, as expected because of its higher ionic character. Thus, a simple resonance hybrid approach can be used with some success for predicting the qualitative nature of the different excited states and the importance of dynamical correlation effects on their relative locations. The two 2 B1 states and the 2 A1 state described above constituted the only doublet valence excited states predicted at the MS-CASPT2 level for the allyl radical. All previous theoretical studies [9,22] have not been able to predict the existence of the second of these two states. The reason behind this failure can be analysed in terms of a poor description of the multiconfigurational character of these states and an intrinsic difficulty in describing the region of the spectrum above 6 eV due to the presence of several Rydberg states. The Rydberg series associated to the excitation from the 1a2 MO is computed to start at 5.11 eV. We also predict the nature of the lower part of the Rydberg series from the assignments. The available experimental assignments have been succesfully described in terms of members of the 1a2 ! R series. This series should converge to the first ionisation energy for the allyl radical, which is at 8.13 eV. Only for the two 2 B1 Rydberg states are the corresponding transitions calculated to have considerable intensities. These results are in agreement with experimental observations [9]. Particularly, the 32 B1 ð1a2 ! 3dyz Þ state lies in the region adjacent to the 42 B1 valence state and the vertical transitions to these states are predicted to have similar intensities. In this case, the characteristic broadness of the bands associated to Rydberg transitions compared to valence excitations could have contributed to the difficulties in the detection of the 42 B1 valence excitation predicted by theory. Another Rydberg series is predicted to start at 8.36 eV, with a weak intensity. This is the starting point of the 1b1 ! R series, which is expected to arise at higher energy than the previous one due to the bonding character of the 1b1 orbital. The computed energy of this Rydberg state is higher
697
than the first ionisation potential and this explains why the second Rydberg series has not been observed with standard spectroscopic approaches. 4. Summary and conclusions The electronic spectrum of the allyl radical has been obtained using multistate multiconfigurational perturbation theory including a proper treatment of Rydberg states, dynamical, and nondynamical correlation. The assignment of the spectrum is in accordance with experiment to within 0.2 eV. Furthermore, the assignments of the first Rydberg series and the beginning of the second Rydberg series have been achieved. We have found a second valence-excited state of the allyl radical lying in the region spanned by the first Rydberg series. The state could be identified only with the MS-CASPT2 approach, which properly handles the valence-Rydberg mixing between the CAS states. The MS-CASPT2 results lead to the identification of a new valence-excited state in the Rydberg region otherwise inaccessible by other ab initio methods. Finally, we attempted in rationalising some features of the electronic spectrum in terms of resonance forms. This model shows how the strongly ionic character of the valence excited states and of the ground state shapes the quantitative spectrum of the allyl radical. Due to the transferability of this approach to similar systems, we can conclude that the ionic character of the electronic structure of p-hydrocarbon radicals implies significant dynamical correlation effects on their ground and valence excited states.
Acknowledgements This work has been supported by the Swedish Science Research Council.
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. Malmqvist, B.O. Roos, L. Serrano-Andres, [15] J. Finley, P.-A Chem. Phys. Lett. 288 (1997) 299. . Malmqvist, Int. J. Quantum Chem. 30 (1986) 479. [16] P.-A . Malmqvist, B.O. Roos, Chem. Phys. Lett. 155 (1989) [17] P.-A 189. [18] K. Andersson, M. Barysz, A. Bernhardsson, M.R.A. Blomberg, Y. Carissan, D.L. Cooper, T. Fleig, M.P. F€ ulscher, L. Gagliardi, C. de Graaf, B.A. Hess, G. . Malmqvist, P. Neogrady, J. Karlstr€ om, R. Lindh, P.-A Olsen, B.O. Roos, B. Schimmelpfennig, M. Sch€ utz, L. Seijo, L. Serrano-Andres, P.E.M. Siegbahn, J. St alring, T. Thorsteinsson, V. Veryazov, M. Wierzbowska, P.-O. Widmark, MO L C A S Version 5.4, Department of Theoretical Chemistry, Chem. Center, University of Lund, P.O.B. 124, S-221 00 Lund, Sweden, 2002. [19] T. Bally, D.A. Hrovat, W.T. Borden, Phys. Chem. Chem. Phys. 2 (2000) 3363. [20] K. Holtzhauer, C. Cometta-Morini, J.F.M. Oth, J. Phys. Org. Chem. 3 (1990) 219. [21] H.M. McConnell, J. Chem. Phys. 24 (1956) 632. [22] G. Levin, W.A.G. III, J. Am. Chem. Soc. 97 (1975) 1649. [23] E. Hirota, C. Yamada, M. Okunishi, J. Chem. Phys. 97 (1992) 2963. [24] L. Currie, D.A. Ramsay, J. Chem. Phys. 45 (1966) 488. [25] J.W. Hudgens, C.S. Dulcey, J. Phys. Chem. 89 (1985) 1505. [26] A.B. Callear, H.K. Lee, Trans. Faraday Soc. 64 (1968) 308.
The allyl radical revisited: a theoretical study of the electronic spectrum Francesco Aquilante *, Kasper P. Jensen, Bj€ orn O. Roos Department of Theoretical Chemistry, Chemical Center, Lund University, P.O. Box 124, Lund S-22100, Sweden Received 7 July 2003; in final form 16 September 2003 Published online: 9 October 2003
Abstract In this Letter, we report the electronic spectrum of the allyl radical, obtained with multiconfigurational perturbation theory (MS-CASPT2). The assignment of the spectrum is in accordance with experiment to within 0.2 eV. We have computed the complete first Rydberg series and the beginning of the second Rydberg series. A new valence-excited 2 B1 state has been found which has hitherto been hidden by Rydberg transitions. A rationalisation of the electronic spectrum is provided in terms of resonance forms in ground and excited states. This model shows that while a multiconfigurational wavefunction is necessary to qualitatively model the system, the large ionic character of the valence electronic states makes an accurate treatment of the dynamical correlation necessary for a quantitative description of the spectrum. Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction The allyl radical is the simplest hydrocarbon p-radical and serves a prominent function in chemistry as a fundamental test case and starting point for more extensive work on radical chemistry [1]. In addition to having the p-electronic features of more complicated p-radicals, it is itself an important species in many polymerisation and photochemical reactions [2], including the Cope rearrangement [3] and formation of phenyl groups.
*
Corresponding author. Fax: +46-46-222-45-43. E-mail address: [email protected] (F. Aquilante).
Many of these reactions are poorly understood, and the detailed knowledge of the electronic structure of allyl is a prerequisite for understanding these dynamics. Allyl displays a higher (C2v ) symmetry than most other hydrocarbon p-radicals and is therefore easier to analyse in terms of the electronic structure. This is in particular true from a theoretical perspective where the use of the full symmetry in principle enables the direct assignment of configurations and states in the electronic spectrum, which may be difficult in an experimental spectrum of close-lying states without point group symmetry as a distinct separator of states. One of the interesting features of the allyl radical is the instability of the RHF potential energy
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.09.079
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surface, i.e., there are lower unsymmetric solutions with UHF. The true geometry of the radical is C2v , but UHF can only recover a Cs symmetry due to inclusion of spin polarisation. Both HF methods are highly unsuitable, however, since it is well known that the wavefunction of the radical is multiconfigurational. Recent theoretical work on the allyl radical therefore concentrates on multiconfigurational MO or VB approaches. This includes the assignment of the vibrational absorption spectrum of the ground state and the first two excited states with CASSCF [4], and a CASSCF study of the simplest cyclisation reaction imaginable, the cyclisation of allyl radical from the first excited state (2 B1 ) [5]. Generalised valence bond methods have been applied to explain the order of methylene rotation barriers in the allyl cation, anion, and radical (37.8, 23.1, and 12.6 kcal/mol, respectively) [6]. The electronic spectrum of the allyl radical remains incomplete. Until now, the most complete experimental spectrum has been derived by a twophoton resonant ionisation spectroscopy study [7]. It has the features of describing the vibrational structure of some of the low-lying Rydberg states of this molecule but misses a complete identification of the electronic states in the Rydberg region. The significant problems with assignment of the electronic spectrum of this simple molecule are due to the closeness in energy of the states [8]. In particular, in the Rydberg region there is an overlap with valence excited states, as we show in this Letter. As a consequence of these problems, theoretical studies with multiconfigurational wavefunctions have been encouraged to assign the 2 B1 state in more detail [9]. In this work, the spectrum has been assigned for the valence region and the start of the second Rydberg series has been identified. It has been concluded earlier and henceforth assumed that there is only one valence excited state in the electronic spectrum of allyl radical [7,9]. We have found a second valence excited 2 B1 state in the Rydberg region, which has not been assigned experimentally, probably because of the assignment problems in this part of the spectrum. We have also found the origin of the first Rydberg series to be the transition to the 1 2 A1 state. Finally, we
have located the starting point of the second Rydberg series as the 32 A2 state with an energy 8.36 eV above the ground state.
2. Methods and computational details Geometry optimisations were performed employing the generally contracted basis functions of the atomic natural orbital type (ANO-L) [10], which are well designed for the treatment of polarisation effects and are flexible enough to describe both ground and valence excited states [11,12]. The contraction scheme C[4s3p2d]/H[3s2p] was employed from the primitive set C[14s9p4d]/ H[8s4p]. The geometries of the ground and valence excited states were optimised at the CASPT2 level, using a numerical gradient procedure. All the structures were computed at the C2v symmetry and the active space employed for the geometry optimisations comprises all the electrons of the p-system. The molecule is placed in the zx-plane. The employed active space for the geometry optimisations can be labelled (0201/3) within the irreps (a1 b1 b2 a2 ) of the C2v point group as seen in Table 1. The obtained structure for the ground state was subsequently used to compute the vertically excited states. In order to avoid erroneous mixing between valence and Rydberg states, the ANO-L basis set used in the geometry optimisation was supplemented with a set of 1s1p1d diffuse
Table 1 CASSCF wavefunctions employed to compute the valence and Rydberg excited states of the allyl radical Active spacea
States
No. CSFs
No. rootsb
4201/3 0601/3 0421/3 0402/3 1201/5 0201/3
2
40 76 36 34 5 1
4 4 2 3 1 1
A1 ð1a2 ! 3s; 3pz ; 3dz2 ; 3dx2 y 2 Þ B1 ðvalence; 1a2 ! 3py ; 3dyz Þ 2 B2 ð1a2 ! 3px ; 3dxz Þ 2 A2 ð1a2 ! 3dxy ; 1b1 ! 3py Þ 2 A1 ð6a1 ! 1a2 Þ 4 A2 ð1b1 ! 2b1 Þ 2
a Number of active orbitals (/total number of active electrons) within the irreps a1 b1 b2 a2 , respectively. b Number of roots requested in the state-averaged CASSCF calculation.
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
functions, contracted from a set of 8s8p8d primitives. The general procedure for obtaining this set of contracted diffuse basis functions is described in detail elsewhere [12]. The set of Rydberg functions were placed in the charge centroid of the molecular cation. The vertical excitation energies have been computed by means of the CASPT2 approach [13,14]. The CASSCF wavefunction is used as a reference to determine the first-order wavefunction and the second-order energy. In the CASSCF calculation, all the carbon 1s core electrons were kept frozen in the form of the ground-state SCF wavefunction. Furthermore, these orbitals were not correlated at the second-order level. The active spaces employed, the number of configuration state functions (CSFs), and the number of state-averaged CASSCF roots are shown in Table 1. All excitation energies were obtained using the ground state energy computed with the same active space of the corresponding excited states. The inclusion of Rydberg orbitals removes erroneus mixing between valence and Rydberg states. Active spaces for Rydberg states were obtained according to the symmetry species to which they belong. The choice of the active space for the 2 B1 states (0601/3) is not trivial and has been guided by an accurate checking of the convergence of the results enlarging the active space. The second-order perturbation treatment was carried out using the multistate extension of the CASPT2 method. In this so-called MS-CASPT2 method [15], the space spanned by the SACASSCF roots is used to construct an effective Hamiltonian, where the diagonal elements correspond to the CASPT2 energies and the off-diagonal elements introduce a coupling of second order in the dynamical correlation energy. Allowing the states to interact under the influence of the dynamic correlation, the MS-CASPT2 approach provides an adequate description of the chemical situations where the CASSCF wavefunction is not a good reference or where a strong mixing occurs between the reference state and CAS-CI states, such as in the case of strong valence-Rydberg mixing. Nonetheless, to avoid an artificially large interaction among the CASPT2 roots, the weight
691
of the reference function for the excited states must be very close to that for the ground state, since the MS-CASPT2 approach is sensitive (more than the normal CASPT2) to the good convergence of the CASSCF wavefunction and to the absence of intruder states. Finally, the CASSCF state interaction (CASSI) method [16,17] was used to obtain the transition properties. MS-CASPT2 excitation energies and PMCAS-CI transition dipole moments were used in the oscillator strength formula 2 f ¼ l2ab DEab 3
ð1Þ
calculated with respect to the states a and b. The PMCAS-CI is a wavefunction (perturbatively modified CAS-CI) obtained by linear combination of the CAS states in the MS-CASPT2 wavefunction. Even in situations where the addition of the MS-CASPT2 corrections to the CASPT2 energies is of minor importance, the second-order interaction among CAS states can significantly affect the transition properties. All calculations were performed with version 5.4 of the MO L C A S quantum chemistry software [18].
3. Results and discussion 3.1. Electronic structure and geometry The ground state of allyl is 2 A2 which is domi2 1 nated by the ð1b1 Þ ð1a2 Þ configuration (92%), corresponding to the RHF wavefunction. The geometries optimised at the CASPT2 level of theory are shown in Table 2. As shown in Table 4, a further contribution to the 2 A2 ground state CASSCF wavefunction arises from the configuration where one of the ð1b1 Þ electrons is excited to the ð2b1 Þ MO. Although this configuration gives a much smaller contribution to the total wavefunction (6%), it has important consequences for the character of the ground state of the allyl radical, as will be shown below. As is observed in Fig. 1, the three MOs which constitute the p space of the allyl radical have distinctly differing bonding characters. The ð1b1 Þ
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Table 2 Equilibrium geometries for the ground and the first excited state of the allyl radical
2
1 A2 Expt.a 12 B1 a
\C2 C1 C3 (°)
) C1 –C2 (A
) C1 –H1 (A
) C2 –H2 (A
) C2 –H02 (A
124.8 124.0 123.1
1.384 1.386 1.455
1.081 1.087 1.077
1.075 1.082 1.074
1.078 1.085 1.076
Ref. [23].
Fig. 1. Molecular orbitals of the p system of allyl radical. Upper left: 1b1 . Upper right: 2b1 . Bottom: 1a2 .
MO is a bonding orbital along the C–C bonds, ð1a2 Þ is non-bonding and ð2b1 Þ is antibonding. It is energetically favourable for the ð1a2 Þ non-bonding p orbital to distort and develop a weakly bonding character along one of the C–C bonds and a corresponding antibonding character along the other C–C bond. This is the reason behind the instability observed in the RHF wavefunction for this system. When the spin symmetry constraint has been released at the UHF level an unphysical broken symmetry solution appears. The ground state geometry of the allyl radical at the UHF level is in
fact computed to be of Cs symmetry rather than C2v (one of the C–C bonds being shorter). At the CASSCF level of theory, this singlet instability is efficiently removed by the use of a multiconfigurational wavefunction, which distributes the three p electrons in all the orbitals of the p system. The geometry of the molecule is thus correctly predicted to be of C2v symmetry by the multiconfigurational approach which is able to treat the near degeneracy problem. We also optimised the 2 B1 state which was found higher in energy by about 3 eV. The main geometrical parameters are
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
displayed in Table 2. The available experimental data and our results are in good agreement. The for the bond distances are accurate to 0.002 A carbon–carbon bonds and to 0.007 A for carbon– hydrogen bonds. The angle between the three carbon atoms \C2 C1 C3 is computed to be 124.8°, which is 0.8° larger than experiment. Recently published B3LYP results obtained with a larger basis set including diffuse functions [4] exhibit less accurate predictions, especially regarding the CCC angle. Bond lengths are in better agreement with the experimental results than the UCCSD(T) available data [19]. The main shortcomings of such methods lie in the lack of description of the multiconfigurational character of the ground state. The CASPT2 geometry optimisation on the other hand gives accurate results because it recovers the remaining (dynamical) correlation effects on top of the correct treatment of the near degeneracy problem at the CASSCF level. No experimental data are available for the geometry of the first excited state. This may imply that this state is not stable and will isomerise. The appearance of diffuse vibronic lines in the 12 B1 12 A2 band indicates that isomerisation to cyclopropyl radical may indeed take place [20]. By comparison with the ground state results, it is worth noticing the lengthening of the C–C bond due to the inclusion of the 2b1 antibonding orbital in the CSFs, which contributes to the composition of the 12 B1 state. 3.2. Atomic spin population from CASSCF The spin polarisation at the CASSCF level describes the correlation energy involved in separating electrons in the same angular and radial regions of space (Fermi correlation). We calculated the spin populations from a Mulliken analysis of the CASSCF wavefunctions to obtain a diagnostic of this correlation, which is very significant in the allyl radical. These spin populations are shown in Table 3. Due to the absence of r-MOs in the active space, r–p spin polarisation effects are not taken into account in our CASSCF/CASPT2 calculation. The ground state has the unpaired electron distributed over the atoms C2 and C3 of Fig. 2 (0.575 on each atom). The central carbon has a small ()0.179) and negative value of the spin density. The ground state
693
Table 3 CASSCF p (3,3) 2pp spin populations in the allyl radical State 2
1 A2 (GS) 12 B1 a
C1
C2 (C3 )
C1 /C2
Expt.a
)0.179 0.856
0.575 0.075
)0.311 11.413
)0.282
Ref. [19] Ratio between the measured H1 /H2 hfc constants.
Fig. 2. Allyl radical. C2v symmetry.
Table 4 Composition of the PMCAS-CI wavefunctions of the valence excited states of the allyl radical State 2
1 A2 (GS) 12 B1 42 B1 32 A1 14 A2
Principal configurationa 2
2
2
1
ð6a1 Þ ð1b1 Þ ð4b2 Þ ð1a2 Þ ð1b1 Þd ! ð2b1 Þu ð1b1 Þd ! ð1a2 Þd ð1a2 Þu ! ð2b1 Þu ð1b1 Þd ! ð1a2 Þd ð1a2 Þu ! ð2b1 Þu ð6a1 Þd ! ð1a2 Þd ð1b1 Þd ! ð2b1 Þu
CASSCF (%) 92 6 (+)45 ())45 (+)45 (+)45 92 100
The label u and d refer to electrons with the spin up and down respectively. a Natural orbital occupancy with respect to the ground state principal configuration.
electronic configuration in fact has the unpaired electron in the 1a2 orbital, which has no contribution from any of the C1 AOs. This means that the contributions to the spin density of the central carbon atom arise only as an effect of the p–p spin polarisation. Such spin polarisation is a common effect of Fermi correlation present in all manyelectron systems and can be described at the unrestricted HF, KS, or CASSCF/CASPT2 level. For the first excited state 2 B1 , the main contributions to the wavefunction come from the CSFs ð1b1 Þ1 ð1a2 Þ2
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1
and ð1b1 Þ ð2b1 Þ . The spin population is now concentrated more on the central carbon (0.856), due to the single occupation of the b1 orbitals in the 12 B1 state. The last two columns in Table 3 give the calculated and experimental values of the ratio between the atomic spin populations of the two non-equivalent carbon atoms. According to the model proposed by Mc-Connell [21], the values for the hyperfine coupling (hfc) constants for the hydrogen atoms in open-shell, planar p systems are proportional to the spin populations in the 2pp AOs of the carbon atom to which they are bonded. On the other hand, hfc constants are proportional to the spin densities at the hydrogen nuclei through the Fermi-contact term. Even though these spin densities are dominated by the spin populations in the 1s AOs of the hydrogen atoms, an accurate computation of hfc constants is still a challenge for many quantum chemical methods. In light of that, the validity of the McConnell hypothesis makes possible computations of hfc from 2pp spin populations and vice versa. The CASSCF method has shown to give reasonably accurate 2pp spin densities, which match the experimental findings. Previous results carried out with several DFT and Post-HF methods [19] have not been able to give more accurate values of the 2pp spin populations for the carbon atoms of the allyl radical. 3.3. Electronic spectrum The main result of the present Letter, the analysis of the valence and Rydberg spectrum of the allyl radical calculated at different levels of theory, is reported in Table 5. The 14 A2 state, the lowest quartet, is reported for the sake of completeness. The first excited state is computed to be of 2 B1 symmetry at 3.26 eV above the ground state. The vertical transition to this state shows a low oscillator strength, in agreement with the low-lying peak of weak intensity detected experimentally at 3.07 eV. As shown in Table 4, this electronic state has a strong multiconfigurational character. It is described by the out-of-phase combination of the singly-excited configurations ð1b1 Þ ! ð1a2 Þ and
ð1a2 Þ ! ð2b1 Þ. As pointed out by previous theoretical studies [9,22] and also given in Table 4, this is a feature of the 12 B1 state in common with the 42 B1 state in the allyl radical. It is worth noticing the contribution from the dynamical correlation effects introduced by the CASPT2 correction to the CASSCF excitation energy. For the 12 B1 state the decrease in excitation energy due to dynamical correlation is approximately 0.37 eV. We have rationalised this effect in terms of the resonance forms shown in Fig. 3. These are not intended to resemble the resonance forms used in VB theories because the spin coupling of the configurations is not taken into account. The resonance forms presented here are obtained by decomposing the MO configurations in AO configurations and limiting the analysis to the p-system only. The stabilising effect of the dynamical correlation on valence excited states can be associated with the stronger ionic character compared to the ground state. Fig. 3 shows the resonance forms that can be associated with the two configurations involved in the 2 B1 CASSCF wavefunctions: CSF1 corresponds to the singlyexcited configuration ð1b1 Þ ! ð1a2 Þ and CSF2 to the ð1a2 Þ ! ð2b1 Þ configuration. By comparison, the resonance forms for the ground state are also displayed. Both the ground state and the 2 B1 state are highly ionic, with an additional contribution from a tri-radical structure in the latter case. The total number of ionic forms is the same for the two states. This indicates in turn that the stabilising contribution of the dynamical correlation to the energy of the 2 B1 state is of the same order of magnitude as in the ground state. The lower excitation energy of the 2 B1 state must be explained qualitatively by the analysis of the natural orbital occupancies for this state compared to the ground state. The occupancies of the three valence p orbitals in the ground state and in the 2 B1 state are displayed in Table 6. It is shown how the occupancy for the two NOs of b1 symmetry changes drastically from the ground to the excited state. These two orbitals correlate by means of the socalled left–right correlation effect. This contribution to the total correlation energy is thus larger for the 2 B1 state and can be related to the decrease of the excitation energy for this state when dy-
Table 5 Vertical excitation energies/eV and oscillator strengths in the electronic spectrum of the allyl radical CASSCF
CASPT2
MSCASPT2
Experimental
Oscillator strength
Experimentale
Valence excited states 12 B1 14 A2 32 A1 42 B1
3.70 ðxÞ 4.53 7.78 7.57 ðxÞ
3.33 5.89 6.22 6.59
3.32
3.07a
6.4 104
1.3 103
Rydberg states 1a2 ! R 12 A1 ð1a2 ! 3sÞ 22 B1 ð1a2 ! 3py Þ 22 A1 ð1a2 ! 3pz Þ 12 B2 ð1a2 ! 3px Þ 32 B1 ð1a2 ! 3dyz Þ 42 A1 ð1a2 ! 3dx2 y 2 Þ 52 A1 ð1a2 ! 3dz2 Þ 22 B2 ð1a2 ! 3dxz Þ 22 A2 ð1a2 ! 3dxy Þ
4.47 5.37 4.94 5.10 6.04 5.76 5.82 5.77 5.75
ðyÞ ðzÞ
5.11 5.83 5.65 5.76 6.55 6.51 6.61 6.56 6.60
5.11 5.73 5.65 5.76 6.36 6.51 6.61 6.56 6.61
Rydberg states 1b1 ! R 32 A2 ð1a2 ! 3py Þ
7.58 ðzÞ
8.34
8.36
a
ðxÞ ðyÞ ðxÞ
6.90
0.118 4.97b (5.39)5.90)c
0.113
0.23
(6.19)6.30)d
1.3 102 0.084
0.4 103 0.05
3.4 105 2.1 105 1.7 103
Ref. [24]. Ref. [25]. c Ref. [26]. d Ref. [9]. Rm e Ref. [9]; The experimental oscillator strength is computed by the formula f ¼ 4.3 109 mif edm from a spectrum recorded in an Ar matrix. b
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
State
695
696
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
Fig. 3. Main resonance structures in some of the valence states of allyl radical. Upper: 2 A2 ground state. Middle: 2 B1 (1b1 ! 1a2 ). Bottom: 2 B1 (1a2 ! 2b1 ).
namical correlation is included at the CASPT2 level. Analogous reasoning can be applied to the 42 B1 state. The 42 B1 state has the same composition as the 12 B1 state but with the difference that the combination of the two CSFs is now in-phase. For the 42 B1 state the difference between the CASSCF and the CASPT2 energy is larger (0.98 eV). This large stabilising effect must be investigated in this case also in terms of the subsequent MS-CASPT2 results. A coupling via the dynamical correlation between the 42 B1 state and the 22 B1 and 32 B1 states takes place as showed by means of the MSCASPT2 treatment. This circumstance indicates that at the CASPT2 level the 42 B1 state has a mixed valence/Rydberg character, which results in a larger stabilisation due to dynamical correlation, as compared with the 12 B1 state which is instead of pure valence character. The large oscillator strength associated with the vertical transition to this state indicates that this state should give rise to one of the main features of Table 6 CASSCF N.O. occupanciesa for the valence p-orbitals of allyl radical State
2
1 A2 (GS) 12 B1 a
Natural orbital 1b1
1a2
2b1
1.906 1.405
1.000 1.021
0.090 0.568
Computed with the active space 0601/3.
the electronic spectrum of allyl radical. However, the region around 6.90 eV is not well established experimentally, and a comparison with the present results is not possible at the moment. The 2 A1 state is the only low-lying valence excited state of r ! p character. The transition to this state is dipole-forbidden and thus not evidenciated in the spectrum. The 2 A1 state is mainly characterised by the single excitation from the highest occupied r MO to the 1a2 SOMO. Its energy lies in the middle of the Rydberg region and it is computed to be 6.22 eV at the CASPT2 level of theory. The large stabilisation due to dynamical correlation for the 2 A1 state can be explained in terms of the presence of a pair–pair interaction in the p space among the electrons of the MOs 1b1 and 1a2 which are doubly occupied in the leading configuration for this state. In the ground state a pair–pair correlation between the electrons in the active shells exists only between the 1b1 and the r MOs and it is certainly of lower strength, owing to the spatial orthogonality of the two orbitals. The importance of correlation effects, as reflected in the ionic character of the electronic states, can be elaborated by a prediction of the energy of the 14 A2 state. This state has a clear single determinantal character, as shown in Table 4. The only resonance structure associated with this electronic configuration is the tri-radical one. For this reason, the stabilisation due to the correlation energy increases the excitation energy at
F. Aquilante et al. / Chemical Physics Letters 380 (2003) 689–698
the CASPT2 level compared to the CASSCF results. This is a confirmation that the ground state is more stabilised by dynamical correlation, as expected because of its higher ionic character. Thus, a simple resonance hybrid approach can be used with some success for predicting the qualitative nature of the different excited states and the importance of dynamical correlation effects on their relative locations. The two 2 B1 states and the 2 A1 state described above constituted the only doublet valence excited states predicted at the MS-CASPT2 level for the allyl radical. All previous theoretical studies [9,22] have not been able to predict the existence of the second of these two states. The reason behind this failure can be analysed in terms of a poor description of the multiconfigurational character of these states and an intrinsic difficulty in describing the region of the spectrum above 6 eV due to the presence of several Rydberg states. The Rydberg series associated to the excitation from the 1a2 MO is computed to start at 5.11 eV. We also predict the nature of the lower part of the Rydberg series from the assignments. The available experimental assignments have been succesfully described in terms of members of the 1a2 ! R series. This series should converge to the first ionisation energy for the allyl radical, which is at 8.13 eV. Only for the two 2 B1 Rydberg states are the corresponding transitions calculated to have considerable intensities. These results are in agreement with experimental observations [9]. Particularly, the 32 B1 ð1a2 ! 3dyz Þ state lies in the region adjacent to the 42 B1 valence state and the vertical transitions to these states are predicted to have similar intensities. In this case, the characteristic broadness of the bands associated to Rydberg transitions compared to valence excitations could have contributed to the difficulties in the detection of the 42 B1 valence excitation predicted by theory. Another Rydberg series is predicted to start at 8.36 eV, with a weak intensity. This is the starting point of the 1b1 ! R series, which is expected to arise at higher energy than the previous one due to the bonding character of the 1b1 orbital. The computed energy of this Rydberg state is higher
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than the first ionisation potential and this explains why the second Rydberg series has not been observed with standard spectroscopic approaches. 4. Summary and conclusions The electronic spectrum of the allyl radical has been obtained using multistate multiconfigurational perturbation theory including a proper treatment of Rydberg states, dynamical, and nondynamical correlation. The assignment of the spectrum is in accordance with experiment to within 0.2 eV. Furthermore, the assignments of the first Rydberg series and the beginning of the second Rydberg series have been achieved. We have found a second valence-excited state of the allyl radical lying in the region spanned by the first Rydberg series. The state could be identified only with the MS-CASPT2 approach, which properly handles the valence-Rydberg mixing between the CAS states. The MS-CASPT2 results lead to the identification of a new valence-excited state in the Rydberg region otherwise inaccessible by other ab initio methods. Finally, we attempted in rationalising some features of the electronic spectrum in terms of resonance forms. This model shows how the strongly ionic character of the valence excited states and of the ground state shapes the quantitative spectrum of the allyl radical. Due to the transferability of this approach to similar systems, we can conclude that the ionic character of the electronic structure of p-hydrocarbon radicals implies significant dynamical correlation effects on their ground and valence excited states.
Acknowledgements This work has been supported by the Swedish Science Research Council.
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