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J. Phys. Chem. A 2010, 114, 8852–8857
A Coupled Cluster Benchmark Study of the Electronic Spectrum of the Allyl Radical† Taylor J. Mach,‡ Rollin A. King,§ and T. Daniel Crawford*,‡ Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, and Department of Chemistry, Bethel UniVersity, St. Paul, Minnesota 55112 ReceiVed: March 13, 2010; ReVised Manuscript ReceiVed: May 31, 2010
We have investigated 15 excited states of the allyl radical, including the lowest three valence states (two doublets and one quartet) and the n ) 3 Ry series, using coupled cluster methods that approximate the correlation effects of connected triple excitations. The quality of the excitation energies is measured on the basis of comparison to existing theoretical and experimental data, as well as on the basis of three diagnostics related to spin contamination and the overall level of excitation of a given state. Basis-set effects are significant for states exhibiting substantial Rydberg character, and the use of molecule-centered diffuse functions appears to provide an accurate description of such states, while avoiding the computational expense of basis sets in which diffuse functions are added to every atom in the molecule. In contrast to earlier observations for linear carbon-chain radicals, coupled cluster methods compare well to both theoretical predictions and experimental band origins, where discrepancies in the latter are sometimes attributable to structural relaxation in the excited state. One of the three lowest 2B1 excited states exhibits a twisting of the terminal methylene groups to yield a C2-symmetry minimum. The most challenging states for coupled cluster methods are of A2 symmetry, where both spin contamination and basis-set effects are appreciable. Introduction 1
In a 2000 report describing cavity ring-down spectra measured from the products of an electrical discharge through benzene vapor, Ball, McCarthy, and Thaddeus speculated that the carrier of a particularly strong absorption band at 4429.29 ( 0.04 Å could be associated with the lowest-energy transition of a small hydrocarbon radical. Among the possible candidates they offered were species related to the allyl radical, C3H5. Motivated by this report, we carried out our own investigation of the first 2B1 valence excited state of C2V allyl using the equation-of-motion coupled cluster singles and doubles (EOMCCSD) model. We found that the choice of reference determinant had a surprisingly large impact on the computed vertical transition energy, with spin-restricted and -unrestricted determinants differing both from one another (3.43 and 3.63 eV, respectively, with a polarized double-ζ basis set) and from the origin transition near 3 eV, most recently measured by Fischer and Chen.2 While part of the discrepancy between theory and experiment in this case could be attributed to adiabatic relaxation (specifically, a twisting of the terminal methylene groups to yield a C2-symmetric 2B excited state3), the large variation with the choice of orbitals indicated that both spin contamination4,5 and electron correlation effects could be significant. Indeed, our subsequent implementation of a coupled cluster method including triple excitations (CC3) for excited states of open-shell molecules6 demonstrated that, in agreement with earlier work by Szalay and Gauss7 on spin-restricted coupled cluster methods for excited states, both effects are important. The question remains open, however, as to the impact of these effects for higher-lying excited states, including those with significant Rydberg character. †
Part of the “Klaus Ruedenberg Festschrift”. * To whom correspondence should be addressed. E-mail:
[email protected]. ‡ Virginia Tech. § Bethel University.
The potential importance of allyl both in hydrocarbon combustion8 and in the interstellar medium9,10 has been noted many times in the literature. However, allyl has also been carefully investigated both theoretically and experimentally for decades for the more fundamental reason that it serves as an archetype for the electronic structure of π-conjugated radicals. The classic molecular orbital analysis of alternant hydrocarbon radicals by Dewar and Longuet-Higgins11 in 1954 predicted that conjugated systems such as the allyl radical should exibit two valence excited states: a weak transition from the doubly occupied bonding orbital into the singly occupied nonbonding orbital, and a stronger transition from the latter into the unoccupied antibonding orbital. Self-consistent field computations a year later by Longuet-Higgins and Pople,12 using the results of previous Pariser-Parr-Pople computations13-16 confirmed this result for the allyl radical, where the corresponding bonding, nonbonding, and antibonding π-type orbitals transform, respectively, as the b1, a2, and b1 irreducible representations of the C2V point group. Thus both doublet valence states should exhibit 2B1 terms. Additional transitions are also expected to include a quartet (4A2) involving all three valence orbitals, as well as doublets corresponding to the 3s, 3p, and 3d components of the first Rydberg series. In 1966, Currie and Ramsay17 carried out the earliest experimental investigation of an excited state of the allyl radical via flash photolysis of allyl bromide, and reported that the origin of the first 2B1 state lies at ca. 3.04 eV (a result that was confirmed and refined by cavity ring-down absorption measurements by Tonokura and Koshi18 in 2000). In 1968, Callear and Lee19 reported the UV absorption spectrum of allyl in the range of 227-220 nm (5.47-5.64 eV), which they tentatively assigned as the transition to the lowest 2B2 state, now understood to be part of the 3p Rydberg band. In the first reported laser detection of allyl in 1985, Hudgens and Dulcey20 identified the 2A1 state at 4.97 eV using resonance-enhanced multiphoton ionization (REMPI) spectroscopy and characterized it as a 3s state with a
10.1021/jp102292x 2010 American Chemical Society Published on Web 06/21/2010
Electronic Spectrum of the Allyl Radical quantum defect of 0.92 (typical of s-type Rydberg states21). More recently, Fischer, Chen, and co-workers have refined most of these results through a series of multiphoton ionization spectroscopic studies,22-26 the results of which are partly summarized in a 2002 review.2 Following the original theoretical work of the Cambridge group described above,11,12,15 ab initio computational investigations of the excited states of allyl began to appear in the 1980s, starting with the configuration interaction (CI) analysis of Ha, Baumann, and Oth.27 Ten years later, Yamaguchi reported complete active space self-consistent field (CASSCF) computations3 that predicted that the first valence excited state exhibits a symmetry-breaking distortion of the methylene groups mentioned earlier, and followed this pathway to the final cyclization product, the cyclopropyl radical. A year later, Oliva and coworkers28 carried out a spin-coupled valence bond (SCVB) study of allyl excited states, with the largest differences between their results and the CI data of Ha et al. appearing for the 2B2 Rydberg states (ca. 0.5 eV). The most advanced computational analysis to date of allyl’s vertical electronic spectrum was carried out by Aquilante, Jensen, and Roos29 in 2003 using single and multistate secondorder perturbation theory based on a complete active space selfconsistent field wave function (CASPT2 and MSCASPT2, respectively). Using an atomic natural orbital basis set with an additional group of 1s1p1d charge-centered diffuse functions, they identified five 2A1, two 2A2, four 2B1, and two 2B2 states, as well as the lowest 4A2 state, and reported a careful analysis of the valence versus Rydberg character of each. Of particular note is their assignment of the 4 2B1 state at 6.90 eV as the second B1 valence state, which they described as possibly hidden by the broad bands of the upper transitions of allyl’s first Rydberg series. The purpose of this work is to test the ability of coupled cluster methods, namely the EOM-CCSD and CC3 methods noted above, to produce accurate vertical excitation energies for this paradigmatic hydrocarbon radical. The reliability of coupled cluster methods for excited states of closed-shell molecules is well established,30,31 but recent studies6,7,21,32,33 have demonstrated that open-shell species present a much greater challenge. In particular, the lack of spin adaptation in conventional formulations of coupled cluster models can lead to dramatic errors in some cases.34 The plethora of available experimental35 and theoretical data for allyl excited states, as well as its importance in combustion and interstellar chemistry, make it a useful test case for such methods. To that end, we report the extension of our earlier investigation, which focused only on the lowest valence 2B1 state, to the valence and first series of Rydberg excited states of the allyl radical approaching the ionization limit of 8.13 eV.36,37 Computational Details The 2A2 ground-state geometry of the allyl radical was optimized at the coupled cluster singles, doubles plus perturbative triples [CCSD(T)]38 level of theory using spin-unrestricted (UHF),39,40 spin-restricted (ROHF),41-44 and quasi-restricted (QRHF)44,45 open-shell reference wave functions. Dunning’s double- and triple-ζ correlation-consistent basis sets (cc-pVDZ and cc-pVTZ)46 were used in all cases. Structural optimizations of selected excited states were carried out with the EOM-CCSD approach utilizing analytic gradient methods47 and the cc-pVDZ basis set augmented with two sets of s-, p-, and d-type diffuse functions (d-aug-cc-pVDZ).48 Vertical excitation energies were calculated using the UHFCCSD(T)/cc-pVTZ optimized ground-state geometry, with
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Figure 1. Key parameters of the optimized structures (bond distances in Ångstroms, angles in degrees) of four states of the allyl radical: (a) the ground 2A2 state [UHF-CCSD(T)/cc-pVTZ] and the 2 2B1 and 3 2 B1 excited states (EOM-CCSD/d-aug-cc-pVDZ, first and second sets of parentheses, respectively), all of which are C2V; (b) the C2-symmetry 1 2B excited state (EOM-CCSD/d-aug-cc-pVDZ).
EOM-CCSD49 and an open-shell version6 of the approximate triples method, CC3,50 using both UHF and ROHF reference determinants. Oscillator strengths were computed using the EOM formalism of Stanton and Bartlett.49 The initial calculations were performed with the standard aug-cc-pVDZ and augcc-pVTZ basis sets as well as their doubly augmented variants.48 Further calculations utilized frozen core and virtual orbitals, as well as two additional sets of s-, p-, and d-type moleculecentered basis functions (mcf). These uncontracted functions were placed at the molecular center of mass with the following exponents: Rs ) (0.0088, 0.0018), Rp ) (0.0071, 0.0014), Rd ) (0.0200, 0.0040), where each set of exponents was obtained by dividing the most diffuse exponent of the aug-cc-pVTZ basis consecutively by a factor of 5, both for even-tempering of the diffuse functions and to span a similarly sized orbital space as that used by Christiansen et al. in computations on furan.51 The above computations were all carried out within the C2V point group using the PSI352 and CFOUR53 quantum chemical program packages. Results and Discussion The two key parameters of the UHF-CCSD(T)/cc-pVTZ optimized structure of the ground 2A2 state of allyl are depicted in Figure 1a. The C-C distance of 1.382 Å and C-C-C angle of 124.2° compare closely to the corresponding values of 1.3869 Å and 123.96° derived by Hirota et al. from r0 rotational constants extracted from the infrared diode spectrum of the ν11 CH2 wagging vibration (with all other parameters fixed at
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Mach et al.
TABLE 1: UHF-EOM-CCSD Vertical Excitation Energies (eV) of the Allyl Radical Relative to the Ground 2A2 State with Various Basis Sets state
aug-ccpVDZ
aug-ccpVTZ
d-aug-ccpVDZ
d-aug-ccpVTZ
aug-ccpVTZ + mcf
1 2B 1 2 2B 1 3 2B 1 4 2B 1 1 2A 1 2 2A 1 3 2A 1 4 2A 1 5 2A 1 1 2B 2 2 2B 2 3 2B 2 1 4A 2 2 2A 2 3 2A 2
3.672 5.856 6.806 8.364 4.994 5.562 6.476 6.573 7.387 5.656 6.618 6.804 6.573 7.580 9.015
3.696 5.851 6.677 7.944 5.114 5.673 6.519 6.599 7.155 5.768 6.678 6.795 6.610 7.381 8.965
3.671 5.646 6.205 6.808 4.965 5.515 6.316 6.448 6.554 5.605 6.416 6.795 6.559 6.601 7.735
3.697 5.741 6.291 6.914 5.103 5.652 6.451 6.553 6.585 5.746 6.561 6.784 6.608 6.743 7.780
3.696 5.768 6.318 6.916 5.102 5.654 6.470 6.560 6.598 5.747 6.589 6.787 6.608 6.786 7.411
previous ab initio values).54 In addition, the CCSD(T) values agree with the CASPT2/ANO-L results of 1.384 Å and 124.8° reported by Aquilante, Jensen, and Roos.29 No significant differences were identified between the UHF-CCSD(T), ROHFCCSD(T), or QRHF-CCSD(T) optimized geometries. Thus, only the UHF-CCSD(T)/cc-pVTZ structure was used for the subsequent series of EOM-CCSD and CC3 vertical excitation energy computations. Table 1 reports excitation energies (eV) of the allyl radical relative to the ground 2A2 state at the UHF-EOM-CCSD level of theory with basis sets ranging from aug-cc-pVDZ to d-augcc-pVTZ. For excitations with primarily valence contributions, such as the transition to the first 2B1 state, very little basis-set dependence is observed, with a variation of at most 0.026 eV between double- and triple-ζ sets. States with significant Rydberg character, on the other hand, exhibit substantially more dependence on the quality of the basis set, as expected. The largest variations occur for the 4 2B1 and the 3 2A2 states, whose excitation energies shift downward by 1.56 and 1.28 eV, respectively, between the aug-cc-pVDZ and d-aug-cc-pVDZ basis sets. On the other hand, the use of molecule-centered functions (labeled “mcf” in the table) yields results at nearly the same level of completeness as the d-aug-cc-pVTZ basis set. For most of the states reported in Table 1, the differences between the two basis sets is negligible, though the largest difference (0.37 eV) occurs for the 3 2A2 state, the highestenergy transition considered in this work. The principal virtue of the molecule-centered functions in describing the Rydberg character of the excited states is their computational efficiency for higher levels of theory (such as CC3): the aug-cc-pVTZ+mcf basis set contains only 271 contracted functions for allyl, versus 346 for d-aug-cc-pVTZ. Relying on the aug-cc-pVTZ+mcf basis set, Table 2 reports the corresponding excitation energies at the EOM-CCSD and CC3 levels of theory, as well as EOM-CC49 oscillator strengths (for the dipole allowed transitions) and the largest molecular orbital contributions to each excited state from the ROHF-EOMCCSD/aug-cc-pVTZ level of theory. For the latter, we have included all single-excitation coefficients larger than 0.3 and double-excitations larger than 0.2. The ground-state wave function is dominated by the Hartree-Fock configuration of ...4b226a211b211a2. The key virtual orbitals for the states reported in the table include the π antibonding 2b1 orbital mentioned earlier, as well as the 3s (7a1), 3p (8a1, 3b1, 5b2), and 3d (9a1, 10a1, 2a2, 4b1, 6b2) Rydberg orbitals, though it should be
noted that no attempt has been made to artificially separate the contributions of the valence and diffuse functions in our characterizations of the states. Indeed, we have observed that for basis sets with large numbers of diffuse functions, the Hartree-Fock orbitals themselves exhibit substantial mixing of valence and Rydberg character, and thus simple descriptions of many of the electronic states are often not possible. The quality of the CC results in Table 2 can be judged on the basis of comparison with the MSCASPT2 data of Aquilante, Jensen, and Roos29 and the available experimental transition energies, as well as three critical diagnostics: (1) the difference between UHF- and ROHF-based data (referred to as the “U-R” difference); (2) the magnitude of the approximate excitation level (AELs); (3) the expectation value of the spin-squared operator,4 〈Sˆ2〉 (reported for all 15 states at the EOM-CCSD level of theory in Table 3). The U-R difference is expected to decrease as the level of electron correlation is improved, falling to zero at the full-CI level. The AEL, which is based on projection of the excited-state wave function onto singly and doubly excited determinants, measures the overall excitation level relative to the reference determinant. A value significantly larger than 1.0 suggests appreciable double-excitation character in the excitedstate wave function, for which EOM-CCSD is generally expected to yield poor results. Finally, the spin-orbital implementation55 used in conventional open-shell coupled cluster programs, including those employed in this work, does not yield eigenfunctions of the Sˆ2 operator. While this inadequacy has been shown to be typically unimportant for ground-state wave functions,4 it can have a dramatic impact for excited states.5,7,34 This is particularly true for states involving combinations of multiple low-spin determinants, such as those arising in the allyl case from excitations out of the doubly occupied orbitals into unoccupied orbitals (as opposed to excitations involving the singly occupied orbital). Thus, values of 〈Sˆ2〉 that deviate significantly from the pure-state eigenvalues of 0.75 (for doublets) or 3.75 (for quartets) warrant skepticism in the corresponding excitation energies. The lowest lying excited state is the valence 1 2B1 state, with primary contributions from single excitations into and out of the singly occupied 1a2 orbital, as predicted by Longuet-Higgins and Pople in 1955.12 The 1 2B1 vertical excitation energy was found to exhibit a U-R difference of 0.2 eV at the EOM-CCSD level, reduced to less than 0.05 eV at the CC3 level, in agreement with our previous work.6 Furthermore, the difference between EOM-CCSD and CC3 is negligible for the ROHF determinant, suggesting that spin contamination of the UHF reference wave function (〈Sˆ2〉 ) 0.95) is partly to blame for the discrepancy at the CCSD level. However, the spin contamination in the EOM-CCSD wave function is relatively small at 0.77 and 0.76 with UHF and ROHF references, respectively. Furthermore, the AEL for this state is close to 1: 1.07 for ROHF, but somewhat higher for UHF at 1.17. With all of these diagnostics in mind, the ROHF-CC3 vertical excitation energy is expected to be quantitatively accurate, possibly within the usual bounds of 0.2 eV expected from CC methods for closedshell systems. The MSCASPT2/ANO-L value of 3.32 eV reported by Aquilante et al.29 is 0.1 eV lower than the ROHFCC3 value obtained here, and both are somewhat higher than the well-established experimental value of 3.07.2 The remaining discrepancy between theory and experiment arises primarily from structural relaxation in the excited state identified originally by Yamaguchi.3 The EOM-CCSD/d-aug-cc-pVDZ optimized geometry of the 1 2B1 state is depicted in Figure 1b. In agreement with earlier computations, we observe a twisting of
Electronic Spectrum of the Allyl Radical
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TABLE 2: Vertical Excitation Energies (in eV, Using the aug-cc-pVTZ+mcf Basis Set), Oscillator Strengths (Unitless, in Parentheses), State Assignments, Wave Function Character, and Approximate Excitation Levels (AELs) of the Allyl Radical Relative to the Ground 2A2 State EOM-CCSD state 1 2 B1 2 2 B1 3 2 B1
4 2 B1 1 2 A1 2 2 A1 3 2 A1 4 2 A1 5 2 A1 1 2 B2 2 2 B2 3 2 B2 1 4 A2
2 2 A2 3 2 A2
excitation character
a
1b1 f 1a2 (0.72β) 1a2 f 2b1 (0.51R) 1a2 f 3b1 (0.77R) 1b1 f 1a2 (0.47β) 1a2 f 2b1 (0.32R) 1a2 f 4b1 (0.74R) 1b1 f 1a2 (0.34β) 1a2 f 2b1 (0.33R) 1a2 f 3b1 (0.33R) 1a2 f 2b1 (0.58R) 1a2 f 4b1 (0.54R) 1a2 f 3b1 (0.37R) 1a2 f 7a1 (0.84R) 1a2 f 12a1 (0.39R) 1a2 f 8a1 (0.84R) 1a2 f 13a1 (0.33R) 6a1 f 1a2 (0.83β) 1a2 f 9a1 (0.36R) 1a2 f 9a1 (0.71β) 6a1 f 1a2 (0.44β) 1a2 f 10a1 (0.30R) 1a2 f 11a1 (0.71R) 1a2 f 10a1 (0.37R) 1a2 f 5b2 (0.83R) 1a2 f 7b2 (0.41R) 1a2 f 6b2 (0.80R) 4b2 f 1a2 (0.37β) 1a2 f 9b2 (0.32R) 4b2 f 1a2 (0.86β) 1a2 f 6b2 (0.35R) 1b1 f 2b1 (0.53β) 1b1 f 2b1 (0.51R) 1b1 f 3b1 (0.31β) 1a2 f 2b1 1b1 f 1a2 (0.26Rβ) 1a2 f 2a2 (0.94R) 1b1 f 3b1 (0.82β)
CC3
UHF
ROHF
UHF
ROHF
AELb
MSCASPT2c
exptd
3.696 (0.0013)e
3.482 (0.0010)
3.482
3.436
1.07
3.32
3.07
5.768 (0.2619)
5.689 (0.2440)
5.657
5.648
1.07
5.73
5.00
6.318 (0.0555)
6.225 (0.0638)
6.204
6.196
1.07
6.36
6.916 (0.1054)
6.826 (0.1118)
6.798
6.790
1.07
6.90
5.102
4.998
4.957
4.941
1.07
5.11
5.654
5.547
5.504
5.489
1.07
5.65
6.470
6.358
6.296
6.265
1.07
6.22f
6.560
6.432
6.371
6.352
1.08
6.51
6.598
6.493
6.461
6.448
1.07
6.61
5.747 (0.0252)
5.646 (0.0242)
5.604
5.587
1.07
5.76
6.589 (0.0000)
6.487 (0.0000)
6.453
6.438
1.07
6.56
6.787 (0.0011)
6.655 (0.0011)
6.574
6.540
1.08
6.608
6.517
6.206
6.150
1.15
5.89f
6.786 (0.0002) 7.411 (0.0239)
6.687 (0.0001) 7.312 (0.0220)
6.659 7.288
6.648 7.279
1.07 1.07
6.61 8.36
4.97
5.15
6.33
a Excitation character computed at the ROHF-EOM-CCSD/aug-cc-pVTZ level of theory. b AEL values computed at the ROHF-EOM-CCSD/ aug-cc-pVTZ+mcf level of theory. c Taken from Aquilante, Jensen, and Roos.29 d Band origins taken from Fischer and Chen.2 e Oscillator strengths computed with the aug-cc-pVTZ basis. f CASPT2 value taken from Aquilante et al.29
TABLE 3: 〈Sˇ2〉 Values of Excited States of the Allyl Radical UHF-EOM-CCSD
ROHF-EOM-CCSD
state aug-cc-pVDZ d-aug-cc-pVDZ aug-cc-pVDZ d-aug-cc-pVDZ 1 2 B1 2 2 B1 3 2 B1 4 2 B1 1 2 A1 2 2 A1 3 2 A1 4 2 A1 5 2 A1 1 2 B2 2 2 B2 3 2 B2 1 4 A2 2 2 A2 3 2 A2
0.773 0.769 0.777 0.784 0.784 0.782 0.789 0.797 0.783 0.793 0.791 0.798 3.498 0.808 1.303
0.773 0.775 0.773 0.776 0.784 0.781 0.785 0.781 0.798 0.792 0.786 0.798 2.770 1.509 0.796
0.760 0.752 0.753 0.757 0.755 0.753 0.758 0.759 0.754 0.760 0.759 0.759 3.549 0.772 1.276
0.760 0.752 0.752 0.752 0.755 0.753 0.755 0.753 0.760 0.759 0.756 0.760 2.613 1.690 0.763
the terminal methylene groups by roughly 23° out of plane, yielding a C2 minimum-energy structure. When this deformation is considered, the resulting adiabatic excitation energy (computed as the difference in ROHF-CC3/d-aug-cc-pVDZ singlepoint energies for the 2A2 state at its UHF-CCSD(T)/cc-pVTZ
optimized geometry and for the 1 2B1 state at its UHF-EOMCCSD/d-aug-cc-pVDZ optimized geometry) is 3.00 eV, in excellent agreement with experiment. The 2 2B1 state has been described by Fischer and Chen2 as exhibiting a potential vibronic coupling with the lowest 2A1 state. The largest orbital contribution to this state arises from excitation out of the singly occupied 1a2 into a Rydberg 3p-like orbital (3b1), but non-negligible contributions arise from the valence orbitals. The vertical excitation energy reported in Table 2 for the 2 2B1 state of 5.65 eV is significantly higher than the assignment of Fischer and Chen at 5.00 eV. Furthermore, neither the U-R difference of less than 0.1 eV, the AEL of 1.07, the 〈Sˆ2〉 of 0.75, nor the small shift between CCSD and CC3 suggests any reason for skepticism in the CC values. The relatively good agreement between MSCASPT2 and CC3 lends further confidence in the theoretical vertical excitation energy of this state. Optimization of the 2 2B1 state at the EOM-CCSD/ d-aug-cc-pVDZ level of theory yields a planar, C2V minimum, as depicted in Figure 1a. (The key structural parameters are given in parentheses in the figure.) The C-C bond length increases by 0.02 Å, and the C-C-C angle narrows by a few degrees, but otherwise the structure is very similar to that of the ground state. As a result, the ROHF-CC3 adiabatic excitation
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energy obtained for this geometry is 5.46 eV, only slightly lower than the vertical transition energy and still well above the experimental value. For comparison, we have also optimized the structure of the 3 2B1 state at the EOM-CCSD level and find that it also remains planar, undergoing similar structural changes as the 2 2B1 state, also remaining very similar to the ground state (see Figure 1a). This yields a ROHF-CC3 adiabatic excitation energy of 5.97 eV, which is only slightly lower than the vertical excitation energy reported in Table 2 of 6.20 eV. The primary orbital contributions for the 3 2B1 state involve the 3d Rydberg level, as well as significant valence and 3p Rydberg components. We also note that the computed oscillator strengths for the transitions to the 1 2B1 and 2 2B1 states match the predictions of Dewar and Longuet-Higgins more than 50 years ago11 in that the latter is much more intense than the former (f ) 0.001 and 0.244, respectively). EOM-CCSD computations indicate that this pattern also holds for the 3 2B1 and 4 2B1 states. The comparison between theory and experiment is significantly better for the 1 2A1 state, which itself has been identified by both theory29 and experiment20 as a 3s Rydberg state. Our analysis agrees with this assessment, as reported in Table 2. The ROHF-CC3 vertical excitation energy for the 1 2A1 state of 4.94 eV falls very close to the value of 4.97 reported by Hudgens and Dulcey20 and by Fischer and Chen.2 Again, the diagnostics considered here (the U-R difference, the AEL, and 〈Sˆ2〉) all suggest high confidence in the CC results, although adiabatic effects have not been explicitly considered for this state. Indeed, we note that no major discrepancies are observed between CC3 and MSCASPT2 for any of the 2B1 or 2A1 states; none of the differences between these two levels of theory exceeds 0.2 eV. The remaining two experimental data points correspond to the 1 2B2 and 1 4A2 states. The 1 2B2 excited state is predicted to lie ca. 5.59 eV above the ground state with a moderate oscillator strength and a primary orbital contribution involving excitation from the singly occupied orbital into the 3p Rydberg domain. The U-R difference in this case is small (0.1 eV at the CCSD level and negligible at the CC3 level), and the AEL of 1.07 and 〈Sˆ2〉 of 0.76 are close to their ideal values. However, while MSCASPT2 and CC3 differ by only 0.17 eV, both models substantially overshoot the experimental value of 5.15 eV, most likely due to adiabatic effects, though these have not been directly probed in this work. The 1 4A2 state presents additional challenges. While the agreement between theory and experiment (6.15 eV with ROHFCC3 versus 6.33 eV reported by Fischer and Chen2) is certainly better for this state than for the 1 2B2 state, both the CC3 and CASPT2 values fall significantly below their experimental counterpart. Thus, inclusion of structural relaxation in the models would serve only to worsen the discrepancy and most likely shift it beyond the preferred zone of confidence of 0.2 eV. Furthermore, for the CC methods, this state exhibits substantial spin-contamination effects, as reported in Table 3. While the smaller aug-cc-pVDZ basis set produces values of 〈Sˆ2〉 of 3.5, reasonably close to the quartet ideal of 3.75, improvement of the basis set by inclusion of a second set of diffuse functions worsens the spin contamination, yielding 〈Sˆ2〉 values of 2.8 (UHF) and 2.6 (ROHF). The AEL for this state (1.15) is also the largest ROHF-based value of those considered here and arises from the appearance of significant double-excitation contributions to the excited-state wave function involving the 1b1, 1a2, and 2b1 orbitals. While such double-excitations are necessary (in conjunction with R and β single-excitations) to
Mach et al. produce the three Ms ) 1/2 components of the quartet state,34 the spin-orbital CC formulation55 fails to maintain the proper balance of these components, with two of the three arising as single excitations and the third as a “spin-flip” double excitation.5,7 In addition, the large shift between the CCSD and CC3 levels of ca. 0.4 eV gives us little confidence in the accuracy of the CC methods for this state. A similar basis-set dependence is observed for the 2 2A2 and 3 2A2 states, while the former exhibits greater spin contamination upon addition of diffuse functions (〈Sˆ2〉 increases from 0.8 to 1.7), the latter produces the opposite (〈Sˆ2〉 decreases from 1.3 to 0.8). Conclusions The primary purpose of this work is to examine the ability of state-of-the-art coupled cluster methods to provide reliable excited-state wave functions and energies for open-shell species. The allyl radical provides an ideal test case because it is both paradigmatic for small unsaturated hydrocarbons and challenging for theoretical models. While coupled cluster theory is widely regarded for its high level of accuracy for molecular properties of closed-shell molecules (including structure, thermodynamic constants, and spectra) open-shell molecules present many outstanding challenges. Electronically excited states are particularly difficult to describe, in part because conventional coupled cluster models typically utilize “spin-orbital” formulations (meaning that R and β contributions are factorized and treated separately55) that lead to a lack of spin adaptation and often a lack of balanced treatment of low-spin determinantal contributions to the excited-state wave function. In this work, we have investigated 15 excited states of the allyl radical with high-level coupled cluster methods including the effects of connected triple excitations, using diagnostics related to spin-contamination and the overall excitation level to measure the reliability of the excitation energies. Our results compare well to both multistate CASPT2 data of Roos and coworkers29 and experimental band origins.2 For the latter, the largest discrepancies appear to arise from structural relaxation. We have investigated this effect in the three lowest 2B1 states, one of which exhibits twisting of the terminal methylene groups to yield a C2 minimum, while the others remain C2V. However, even the inclusion of such adiabatic effects does not account for the largest discrepancy between theory and experiment of 0.46 eV for the 2 2B1 state. The valence 1 4A2 state provides a clear example of the difficulties that can be encountered by conventional coupled cluster methods when dealing with excited states of open-shell molecules. Starting from a doublet radical ground state, a zerothorder description of a quartet excited state involves three Ms ) 1 /2 determinants, each with three unpaired electrons. Two of these arise from comparable single excitations out of the highest doubly occupied orbital (1b1 in the case of allyl radical), but the third arises from a double excitation: simultaneous R-spin excitation out of the singly occupied orbital (1a2) and β-spin excitation from the doubly occupied orbital back into the 1a2 (a “spin-flip” double).5,7 In a spin-adapted formulation, all three of these determinants would be treated on an equal footing, because they are all part of the same electronic configuration. However, in a spin-orbital formulation, such as that used in most open-shell coupled cluster implementations, the “spin-flip” double is not described in a balanced manner as compared to the singles, leading to errors of both electron correlation effects and spin-contamination. Furthermore, while triple excitations serve to diminish this problem somewhat, they cannot completely eliminate its effects. For the 1 4A2 state, in particular,
Electronic Spectrum of the Allyl Radical while the comparison between the ROHF-CC3 excitation energy (6.15 eV) and experiment (6.33 eV) is reasonable, structural relaxation will likely increase the discrepancy between the two. Acknowledgment. This work was supported by a grant from the U.S. National Science Foundation (CHE-0715185). T.J.M. was supported by a fellowship from the Institute for Critical Technology and Applied Science (ICTAS) at Virginia Tech. We are grateful to Prof. A. I. Krylov (University of Southern California) for helpful discussions and to Dr. Andrew Simmonett (University of Georgia) for access to his CheMVP program, which was used to produce the molecule images in Figure 1. References and Notes (1) Ball, C. D.; McCarthy, M. C.; Thaddeus, P. Astrophys. J. 2000, 529, L61–L64. (2) Fischer, I.; Chen, P. J. Phys. Chem. A 2002, 106, 4291–4300. (3) Yamaguchi, M. J. Mol. Struct. (THEOCHEM) 1996, 365, 143– 149. (4) Fortenberry, R. C.; King, R. A.; Stanton, J. F.; Crawford, T. D. J. Chem. Phys. 2010, 132, 144303–1-10. (5) Krylov, A. I. J. Chem. Phys. 2000, 113, 6052–6062. (6) Smith, C. E.; King, R. A.; Crawford, T. D. J. Chem. Phys. 2005, 122, 054110-1-8. (7) Szalay, P.; Gauss, J. J. Chem. Phys. 2000, 112, 4027–4036. (8) Hayes, C. J.; Merle, J. K.; Hadad, C. M. AdV. Phys. Org. Chem. 2009, 43, 79–134. (9) Webster, A. Mon. Not. R. Astron. Soc. 1993, 265, 421–430. (10) Herbst, E. Annu. ReV. Phys. Chem. 1995, 46, 27–53. (11) Dewar, M. J. S.; Longuet-Higgins, H. C. Proc. Phys. Soc. A 1954, 67, 795–804. (12) Longuet-Higgins, H. C.; Pople, J. A. Proc. Phys. Soc. A 1955, 68, 591–600. (13) Pariser, R.; Parr, R. G. J. Chem. Phys. 1953, 21, 466–471. (14) Pariser, R.; Parr, R. G. J. Chem. Phys. 1953, 21, 767–776. (15) Pople, J. A. Trans. Faraday Soc. 1953, 49, 1375–1385. (16) Pople, J. A. Proc. Phys. Soc. A 1955, 68, 81–89. (17) Currie, C. L.; Ramsay, D. A. J. Chem. Phys. 1966, 45, 488–491. (18) Tonokura, K.; Koshi, M. J. Phys. Chem. A 2000, 104, 8456–8461. (19) Callear, A. B.; Lee, H. K. Trans. Faraday Soc. 1968, 64, 308– 316. (20) Hudgens, J. W.; Dulcey, C. S. J. Phys. Chem. 1985, 89, 1505– 1509. (21) Reisler, H.; Krylov, A. I. Int. ReV. Phys. Chem. 2009, 28, 267– 308. (22) Blush, J. A.; Minsek, D. W.; Chen, P. J. Phys. Chem. 1992, 96, 10150–10154. (23) Minsek, D. W.; Blush, J. A.; Chen, P. J. Phys. Chem. 1992, 96, 2025–2027. (24) Minsek, D. W.; Chen, P. J. Phys. Chem. 1993, 97, 13375–13379. (25) Gilbert, T.; Fischer, I.; Chen, P. J. Chem. Phys. 2000, 113, 561566; Erratum. J. Chem. Phys. 2009, 131, 019903. (26) Gasser, M.; Schulenburg, A. M.; Dietiker, P. M.; Bach, A.; Merkt, F.; Chen, P. J. Chem. Phys. 2009, 131, 014304. (27) Ha, T.-K.; Baumann, H.; Oth, J. F. M. J. Chem. Phys. 1986, 85, 1438–1442. (28) Oliva, J. M.; Gerratt, J.; Cooper, D. L.; Karadakov, P. B.; Raimondi, M. J. Chem. Phys. 1997, 106, 3663–3672.
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8857 (29) Aquilante, F.; Jensen, K. P.; Roos, B. O. Chem. Phys. Lett. 2003, 380, 689–698. (30) Koch, H.; Christiansen, O.; Jørgensen, P.; Olsen, J. Chem. Phys. Lett. 1995, 244, 75–82. (31) Stanton, J. F.; Gauss, J.; Ishikawa, N.; Head-Gordon, M. J. Chem. Phys. 1995, 103, 4160–4174. (32) Krylov, A. I. Annu. ReV. Phys. Chem. 2007, 59, 433–463. (33) Li, X.; Paldus, J. J. Chem. Phys. 2008, 129, 174101-1–15. (34) Fortenberry, R. C.; King, R. A.; Stanton, J. F.; Crawford, T. D. A benchmark study of the vertical electronic spectra of the linear chain radicals C2H and C4H. J. Chem. Phys., in press. (35) Fischer, I. Chem. Soc. ReV. 2003, 32, 59–69. (36) Houle, F. A.; Beauchamp, J. L. J. Am. Chem. Soc. 1978, 100, 3290– 3294. (37) Schussler, T.; Deyerl, H. J.; Dummler, S.; Fischer, I.; Alcaraz, C.; Elhanine, M. J. Chem. Phys. 2003, 118, 9077–9080. (38) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. Chem. Phys. Lett. 1989, 157, 479–483. (39) Gauss, J.; Stanton, J. F.; Bartlett, R. J. J. Chem. Phys. 1991, 95, 2623–2638. (40) Watts, J. D.; Gauss, J.; Bartlett, R. J. Chem. Phys. Lett. 1992, 200, 1–7. (41) Binkley, J. S.; Pople, J. A.; Dobosh, P. S. Mol. Phys. 1974, 28, 1423–1429. (42) Gauss, J.; Lauderdale, W. J.; Stanton, J. F.; Watts, J. D.; Bartlett, R. J. Chem. Phys. Lett. 1991, 182, 207–215. (43) Lauderdale, W. J.; Stanton, J. F.; Gauss, J.; Watts, J. D.; Bartlett, R. J. Chem. Phys. Lett. 1991, 187, 21–28. (44) Watts, J. D.; Gauss, J.; Bartlett, R. J. J. Chem. Phys. 1993, 98, 8718–8733. (45) Rittby, M.; Bartlett, R. J. J. Phys. Chem. 1988, 92, 3033–3036. (46) Peterson, K. A.; Dunning, T. H. J. Chem. Phys. 1995, 102, 2032– 2041. (47) Stanton, J. F.; Gauss, J. J. Chem. Phys. 1993, 99, 8840–8847. (48) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. J. Chem. Phys. 1992, 96, 6796–6806. (49) Stanton, J. F.; Bartlett, R. J. J. Chem. Phys. 1993, 98, 7029–7039. (50) Koch, H.; Christiansen, O.; Jørgensen, P.; de Meras, A. M. S.; Helgaker, T. J. Chem. Phys. 1997, 106, 1808–1818. (51) Christiansen, O.; Halkier, A.; Koch, H.; Jørgensen, P.; Helgaker, T. J. Chem. Phys. 1998, 108, 2801–2816. (52) Crawford, T. D.; Sherrill, C. D.; Valeev, E. F.; Fermann, J. T.; King, R. A.; Leininger, M. L.; Brown, S. T.; Janssen, C. L.; Kenny, J. P.; Seidl, E. T.; Allen, W. D. J. Comput. Chem. 2007, 28, 1610–1616. (53) CFOUR, a quantum chemical program package, written by J. F. Stanton, J. Gauss, M. E. Harding, P. G. Szalay with contributions from A. A. Auer, R. J. Bartlett, U. Benedikt, C. Berger, D. E. Bernholdt, Y. J. Bomble, O. Christiansen, M. Heckert, O. Heun, C. Huber, T.-C. Jagau, D. Jonsson, J. Juse´lius, K. Klein, W. J. Lauderdale, D. A. Matthews, T. Metzroth, D. P. O’Neill, D. R. Price, E. Prochnow, K. Ruud, F. Schiffmann, S. Stopkowicz, M. E. Varner, J. Va´zquez, F. Wang, J. D.Watts. The integral packages MOLECULE (J. Almlo¨f and P. R. Taylor), PROPS (P. R. Taylor), and ABACUS (T. Helgaker, H. J. Aa. Jensen, P. Jørgensen, and J. Olsen). ECP routines by A. V. Mitin and C. van Wu¨llen. (All 2009). (54) Hirota, E.; Yamada, C.; Okunishi, M. J. Chem. Phys. 1992, 97, 2963–2970. (55) By “spin-orbital formulation” we mean that the second-quantized operators used to construct the coupled cluster energy and amplitude equations do not yield eigenfunctions of the Sˆ2 operator. The programmed equations make use of spin-factored (and spin-integrated) expressions for efficiency, but the final ground- and excited-state wave functions are not spin adapted.
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