Explicitly Correlated Coupled Cluster Calculations for the Benzenium

ARTICLE pubs.acs.org/JPCA

Explicitly Correlated Coupled Cluster Calculations for the Benzenium Ion (C6H+7 ) and Its Complexes with Ne and Ar Peter Botschwina* and Rainer Oswald Institut f€ur Physikalische Chemie, Universit€at G€ottingen, Tammannstraße 6, 37077 G€ottingen, Germany

bS Supporting Information ABSTRACT: Explicitly correlated coupled cluster theory at the CCSD(T)-F12x (x = a, b) level (Adler, T. B.; Knizia, G.; Werner, H.-J. J. Chem. Phys. 2007, 127, 221106) has been employed in a study of the benzenium ion (C6H7+) and its complexes with a neon or an argon atom. The ground-state rotational constants of C6H7+ are predicted to be A0 = 5445 MHz, B0 = 5313 MHz, and C0 = 2731 MHz. Anharmonic vibrational wavenumbers of this cation were obtained by combination of harmonic CCSD(T*)-F12a values with anharmonic contributions calculated by double-hybrid density functional theory at the B2PLYP-D level. For the complexes of C6H7+ with Ne or Ar, the lowest energy minimum is of π-bonded structure. The corresponding dissociation energies D0 are estimated to be 160 and 550 cm 1, respectively. There is no indication of H-bonds to the aromatic or aliphatic hydrogen atoms. Instead, three nonequivalent local energy minima were found for nuclear configurations where the rare-gas atom lies in the ring-plane and approximatly points to the center of one of the six CC bonds.

1. INTRODUCTION The benzenium ion (C6H+7 ) is clearly one of the fundamental cations of organic chemistry (see e.g., refs 1 and 2). Spectroscopic studies of the free cation in the gas-phase are still very scarce and are essentially limited to investigations by infrared multiphoton dissociation (IRMPD) spectroscopy, carried out with an infrared free electron laser light source. Two bands observed at 1228 and 1433 cm 1 were assigned to C6H+7 , two others at 1439 and 1572 cm 1 to C6D6H+.3 Further IRMPD results were subsequently published by Dopfer and co-workers.4 6 Additional gasphase information derives from IR photodissociation (IRPD) spectroscopy of weakly bonded complexes of C6H+7 with ligands such as rare-gas (Rg) atoms or simple molecules (N2, H2O, and CH4).7 9 The work of Solca and Dopfer7,8 was restricted to the range of the CH stretching vibrations above 2700 cm 1, whereas the more recent study of Douberly et al.9 made use of an optical parametric oscillator (OPO) laser system operating in the wide range between 700 and 4400 cm 1. In the case of C6H+7 3 Ar, as many as 15 different bands could be observed (see Figure 2 of ref 9). In the characteristic region of the CH2 stretching vibrations around 2800 cm 1, both IRPD spectroscopy groups report three band positions which agree within 2 cm 1 . 7 9 However, there are significant differences in the assignments. Solca and Dopfer7,8 assigned the two bands with lowest wavenumbers (2795 and 2810 cm 1) to the symmetric and asymmetric CH2 stretching vibrations, respectively, and attributed the band at 2819 cm 1 to a “C C stretch overtone”. However, Douberly et al.9 left the band at 2809 cm 1 unassigned, whereas the peaks at 2793 and 2820 cm 1 were interpreted as asymmetric and symmetric CH2 stretching vibrations, respectively. The IRPD spectroscopic studies of Solca and Dopfer7,8 and of Douberly et al.9 were accompanied by quantum-chemical r 2011 American Chemical Society

calculations. The former authors used second-order Møller Plesset perturbation theory (MP2) in conjunction with the moderately large 6-311G (2df, 2dp) basis set. Intermolecular interaction energies were corrected for the basis set superposition error by the counterpoise (CP) procedure of Boys and Bernardi.10 Three minima on the potential energy surface (PES) of C6H+7 3 Ar (see Figure 1 of ref 8) were reported. These were described as a π-bonded structure, where the ligand binds to the π-system of the (partially) aromatic ring, and as two H-bonded structures, with a weak hydrogen-bond either to the aliphatic CH2 group (CH2 bonded) or to the aromatic CH group opposite to the CH2 group (CH bonded). The largest equilibrium dissociation energy (De = 434 cm 1) was calculated for the π-bonded structure. Low De values of 293 and 220 cm 1 were reported for the CH2 bonded structure and the minimum of C2v symmetry, respectively. The more recent study of C6H+7 3 Ar by Douberly et al.9 made use of hybrid density functional theory at the B3LYP/6-311+ G (d,p) level and found two isomers for the complex. CP corrected De values of 211 and 179 cm 1 were reported. Since dispersion was not considered in these calculations and CP corrected values without basis set extrapolation are usually underestimates, the quoted De values appear to be much too low. Following our recent theoretical work on complexes of type C3H+3 3 L (L = Ne, Ar, N2, CO2 and O2),11 13 we will present here a comparable study for the complexes C6H+7 3 L with L = Ne and Ar. Like in the previous work, explicitly correlated coupled cluster theory at the CCSD(T)-F12x (x = a, b) level14,15 is employed. Special emphasis will be devoted to the energetics of Received: August 17, 2011 Revised: October 7, 2011 Published: October 08, 2011 13664

dx.doi.org/10.1021/jp207905t | J. Phys. Chem. A 2011, 115, 13664–13672

The Journal of Physical Chemistry A

ARTICLE

Table 1. Calculated and Recommended Equilibrium Structures for the Benzenium Cationa method basis

CCSD(T*)-F12a VDZ-F12

CCSD(T*)-F12a VTZ-F12

MP2

B3LYP

VQZ

B2PLYP-D

VTZb

VQZ

VTZ

VQZ

recom.c

lengths (Å) C1C2

1.4675

1.4663

1.4556

1.4648

1.4644

1.4648

1.4637

1.4628

C2C3

1.3703

1.3692

1.3698

1.3649

1.3645

1.3674

1.3664

1.3657

C3C4

1.4105

1.4095

1.4025

1.4064

1.4062

1.4073

1.4062

1.4060

C1H1

1.1050

1.1049

1.1045

1.1057

1.1051

1.1039

1.1034

1.1029

C2H3

1.0838

1.0836

1.0817

1.0825

1.0819

1.0813

1.0808

1.0816

C3H4

1.0814

1.0811

1.0791

1.0805

1.0799

1.0791

1.0786

1.0791

C4H5

1.0850

1.0848

1.0829

1.0835

1.0830

1.0825

1.0819

1.0828

angles (°) H5C4C3

118.42

118.42

118.48

118.52

118.51

118.48

118.49

118.42

C4C3C2

118.91

118.91

119.05

119.10

119.09

119.05

119.06

118.91

C3C2C1

120.93

120.94

120.53

120.85

120.86

120.83

120.82

120.94

C2C3H4

121.27

121.26

121.01

121.10

121.11

121.13

121.13

121.26

C3C2H3

120.90

120.91

120.91

120.99

120.98

120.98

120.97

120.91

H1C1H2

100.93

100.88

99.78

99.82

99.82

100.52

100.43

100.88

a See Figure 1 for numbering of atoms. Valence electrons are correlated in the CCSD(T*)-F12a and MP2 calculations. b Slightly different values have been published earlier.39 c Recommended equilibrium structure (see the text).

different isomeric structures of the Rg complexes of C6H+7 . The partially aromatic cation is more complex than the experimentally known C3H+3 isomers (cyclopropenyl and propargyl cation) and may thus offer a larger number of bonding sites. In particular, we are interested in finding out whether weak hydrogen bonds may be formed between a Rg atom and any of the seven hydrogen atoms of the benzenium ion.

2. DETAILS OF CALCULATIONS The electronic structure calculations of the present work made use of explicitly correlated coupled cluster (CC) theory, which allows for a much more rapid convergence of the correlation energy with respect to the basis set than obtainable with standard CC theory (see, e.g., refs 16 and 17 for recent reviews). To be more specific, the CCSD(T)-F12x (x = a, b) methods of Werner and co-workers14,15 were employed. These are very good approximations to standard CCSD(T)18 close to the basis set limit, but require a much lower computational effort. For details, we refer to the literature, here we restrict ourselves to a few details of the computations. For the benzenium ion, the systematically convergent basis sets for explicitly correlated wave functions as developed by Peterson et al.19 were used as the atomic orbital (AO) basis sets. They are denoted as cc-pVnZ-F12 (n = D, T, and Q) or briefly VnZ-F12. An accurate description of the static dipole polarizabilities of the rare-gas atoms requires more flexible basis sets and we therefore use the Dunning-type aug-cc-pVnZ basis sets20 23 for this purpose. The basis sets chosen for the argon atom actually include additional tight d functions as described by Dunning et al.23 For brevity, the ligand AO basis sets are denoted by AVnZ. For the complexes C6H+7 3 L we use the (cation, ligand) basis set combinations (VDZ-F12, AVTZ), (VTZ-F12, AVQZ), and (VQZ-F12, AV5Z). In the following, these will be further abbreviated as (D, T), (T, Q), and (Q, 5), respectively. Within the complementary auxiliary basis set (CABS) approach,24 the optimized auxiliary basis sets of Yousaf and Peterson25,26 for the

VnZ-F12 and AVnZ basis sets were employed. JKFIT and MP2FIT basis sets were taken from refs 27 29. The CABS singles correction15 was used to improve the Hartree Fock reference energies. Most of the explicitly correlated coupled cluster calculations made use of variant x = a and the contributions of connected triple substitutions were scaled (CCSD(T*)-F12a).15,16 Like in our previous work,11 13 the geminal exponent was chosen to be 1.2 a0 1 for the largest basis set (Q, 5) and 1.0 a0 1 for basis sets (T, Q) and (D, T). Valence electrons were correlated in the explicitly correlated and standard coupled cluster calculations of the present work, which were carried out with the MOLPRO package of ab initio programs.30 Since analytical energy gradients and second derivatives are not yet available with explicitly correlated coupled cluster theory, we have made use of (double) hybrid density functional theory to study anharmonicity effects for C6H+7 and harmonic vibrational wavenumber shifts for its complexes with Ne and Ar. These calculations were carried out with the Gaussian 09 suite of programs31 using the well-known hybrid density functional B3LYP and Grimme’s double-hybrid functional B2PLYP-D,32,33 which includes a dispersion correction. Second-order perturbation theory in normal coordinate space (VPT2) was used to calculate vibration rotation coupling constants and anharmonic contributions to vibrational levels.34 13 Fermi resonances and 3 Darling Dennison resonances were approximately taken into account as described in ref 34.

3. RESULTS AND DISCUSSION 3.1. Benzenium Ion (C6H+7 ). Calculated equilibrium structures

for the benzenium ion in its energetically most favorable nuclear configuration of C2v symmetry are listed in Table 1. Besides CCSD(T*)-F12a results as obtained with two VnZ-F12 (n = D, T) basis sets, data obtained by MP2 with the cc-pVQZ (briefly VQZ) basis set are included as well. Furthermore, Table 1 quotes results obtained by the (double) hybrid density functional methods B3LYP and B2PLYP-D. Dunning’s cc-pVnZ (VnZ) basis 13665

dx.doi.org/10.1021/jp207905t |J. Phys. Chem. A 2011, 115, 13664–13672


ARTICLE

sets20 with n = T and Q were used in those calculations. According to our recent experience with H2CCC35 and H2C3H+,12 Table 2. Calculated and Recommended Equilibrium Rotational Constants of C6H+7 method

basis

Ae (MHz)

Be (MHz)

Ce (MHz)

CCSD(T*)-F12a

VDZ-F12

5446.1

5315.7

2732.7

CCSD(T*)-F12a

VTZ-F12

5453.8

5323.2

2736.5

MP2

VQZ

5487.3

5369.9

2756.5

B3LYP

VTZ

5472.0

5345.0

2746.2

B3LYP

VQZ

5475.2

5347.4

2747.6

B2PLYP-D

VTZ

5466.0

5338.9

2743.4

B2PLYP-D

VQZ

5473.1

5346.5

2747.3

5480.4

5348.8

2749.8

recommendeda a

Calculated from the recommended equilibrium structure given in the last column of Table 1.

the CCSD(T*)-F12a/VTZ-F12 equilibrium bond lengths should be well within 0.001 Å of the basis set limit of this method (within the frozen-core approximation). For benzene, analogous calculations yield Re(CC) = 1.3947 Å and re(CH) = 1.0828 Å, to be compared with the recommended equilibrium bond lengths of 1.3915 ( 0.0010 Å and 1.0800 ( 0.0020 Å as published by Gauss and Stanton.36 The deviation in the CH equilibrium distance of 0.0028 Å is somewhat larger than expected. For comparison, CCSD(T*)-F12a/VTZ-F12 calculations for HCCH and HC4H yield 1.0633 Å and 1.0632 Å, respectively. These are larger than accurate literature values37,38 by only 0.0017 Å in both cases. The CC equilibrium distance is overestimated by 0.0029 Å for HCCH, while the errors for HC4H amount to 0.0029 Å (shorter CC bond) and 0.0034 Å (longer CC bond). The latter is probably better suited for a comparison with C6H+7 . On the basis of this discussion we apply simple corrections of Δre (CH) = 0.0020 Å and ΔRe (CC) = 0.0035 Å to all CH or CC bonds of C6H+7 in order to arrive at a recommended equilibrium

Table 3. Calculated Harmonic (ω) and Anharmonic (ν) Vibrational Wavenumbers (in cm 1) for C6H+7 MP2/VQZ Symmetry

ω (A)a

1a1 2a1

B3LYP/VTZ

CCSD(T*)-F12ab

B2PLYP-D/VTZ

ω (A)a

ν ω

ω (A)a

ν ω

ω

νc

3257 (3)

3213 (2)

128

3235 (2)

142

3223

3081

ν1

3230 (8)

3193 (5)

107

3212 (6)

116

3199

3083

ν2

3a1

3218 (1)

3182 (0)

107

3202 (0)

125

3187

3062

ν3

4a1

2981 (91)

2951 (62)

145

2969 (68)

151

2964

2813

ν4

5a1

1652 (50)

1642 (76)

38

1642 (70)

49

1639

1590

ν5

6a1

1486 (21)

1485 (27)

28

1484 (26)

32

1478

1446

ν6

7a1

1244 (107)

1277 (112)

39

1275 (112)

41

1280

1239

ν7

8a1 9a1

1209 (36) 1032 (0)

1216 (25) 1017 (0)

15 16

1209 (23) 1016 (0)

17 16

1205 1012

1188 996

ν8 ν9

10a1

994 (2)

1008 (1)

19

1003 (1)

20

998

979

ν10

11a1

916 (26)

901 (14)

9

900 (15)

10

898

888

ν11

12a1

588 (0)

597 (0)

5

593 (0)

5

585

580

ν12

1a2

1135 (0)

1152 (0)

38

1149 (0)

44

1148

1104

ν13

2a2

1019 (0)

1019 (0)

20

1018 (0)

18

1000

982

ν14

3a2

798 (0)

808 (0)

5

805 (0)

7

796

789

ν15

4a2 1b1

325 (0) 2994 (36)

333 (0) 2943 (24)

5 159

331 (0) 2974 (26)

4 166

320 2974

316 2808

ν16 ν17

2b1

1073 (1)

1080 (3)

19

1077 (3)

12

1057

1045

ν18

3b1

1045 (0)

1055 (0)

21

1055 (0)

21

1036

1016

ν19

assoc.d

4b1

837 (20)

852 (16)

20

851 (18)

21

841

820

ν20

5b1

659 (49)

663 (58)

6

660 (57)

8

647

640

ν21

6b1

385 (1)

419 (2)

1

408 (1)

3

393

390

ν22

7b1

137 (7)

216 (13)

+16

181

197

ν23

1b2 2b2

3256 (12) 3229 (5)

116 131

3221 3198

3105 3067

ν24 ν25

3b2 4b2

+18

193 (14)

3211 (6) 3192 (2)

106 128

3233 (8) 3211 (3)

1604 (38)

1575 (2)

+3

1579 (0)

9

1578

1568

ν26

1554 (162)

1487 (180)

34

1500 (194)

37

1488

1452

ν27

5b2

1440 (7)

1422 (4)

35

1429 (0)

40

1432

1392

ν28

6b2

1361 (13)

1368 (15)

53

1366 (15)

23

1361

1338

ν29

7b2

1212 (10)

1209 (18)

14

1202 (16)

14

1199

1185

ν30

8b2

1145 (3)

1152 (0)

20

1145 (1)

21

1143

1122

ν31

26 4

986 577

960 573

ν32 ν33

9b2 10b2

994 (23) 582 (6)

991 (17) 591 (5)

21 4

988 (17) 587 (5)

a Absolute IR intensities (in km mol 1) are given in parentheses. b Basis: VDZ-F12. c Including anharmonicity contributions, calculated by B2PLYP-D/ VTZ. d Conventional numbering of anharmonic vibrations: ν1-ν12(a1), ν13-ν16(a2), ν17-ν23(b1), ν24-ν33(b2).

13666


The Journal of Physical Chemistry A structure for this cation. It is given in the last column of Table 1. We expect the recommended equilibrium bond lengths to be accurate to better than 0.001 Å. Interestingly, the B2PLYP-D/VQZ results show excellent agreement with the present recommended values, the largest deviation in equilibrium bond lengths amounting to only 0.0009 Å. Likewise, the differences in equilibrium bond angles are always smaller than 0.5°. Calculated and recommended equilibrium rotational constants (Ae, Be, and Ce) are listed in Table 2. The recommended values should be accurate to ca. 0.1%. Ground-state rotational constants (A0, B0, and C0) are obtained by combining the recommended equilibrium rotational constants with the vibrational contributions as calculated from the vibration rotation coupling constants αAr , αBr , and αCr . The latter, calculated at the B2PLYP-D/VTZ level, are supplied as Supporting Information (Table S1). Therewith, the ground-state contributions are calculated to be ΔA0 = 35.78 MHz, ΔB0 = 36.19 MHz, and ΔC0 = 18.40 MHz. Adding these data to the recommended equilibrium values, we arrive at A0 = 5444.6 MHz, B0 = 5312.6 MHz, and C0 = 2731.4 MHz. These predictions may be of help to forthcoming studies of C6H+7 by laboratory rotational spectroscopy and eventually radio astronomy. Calculated harmonic and anharmonic wavenumbers of the fundamental vibrations of C6H+7 are listed in Table 3. Among the harmonic vibrational wavenumbers, the CCSD(T*)-F12a/VDZF12 values are expected to be most accurate. Analogous calculations for H2C3H+ and H2C312,35 showed close agreement (mean absolute deviation: 5 cm 1) with the results of standard CCSD(T) calculations with the large V6Z basis set.40 In comparison with the results of the present CCSD(T*)-F12a/VDZ-F12 calculations, both B3LYP/VTZ and B2LYP-D/VTZ perform mostly very well. The former method has a tendency to underestimate the harmonic wavenumbers of the seven CH stretching vibrations by ca. 10 cm 1 while overestimates by a similar amount are observed for B2PLYP-D/VTZ. The latter method agrees with CCSD(T*)-F12a/VDZ-F12 in predicting the harmonic wavenumber of the asymmetric CH2 stretching vibration (ω17) to be larger than that of the symmetric one (ω4), while the opposite holds for B3LYP/VTZ. The present MP2/VQZ calculations and the earlier MP2/6-31G* calculations of Solca and Dopfer7 also predict ω17 > ω4. Table 3 also lists calculated anharmonic contributions (ν-ω) to the fundamental vibrations, obtained by VPT2 (including an approximate treatment of anharmonic resonances) and quartic force fields computed by both B3LYP/VTZ and B2PLYP-D/ VTZ. The latter values are then added to the harmonic wavenumbers obtained by CCSD(T*)-F12a/VDZ-F12 in order to arrive at improved predictions for the anharmonic vibrational wavenumbers. For the most intense vibration ν27 (4b2), we predict a wavenumber of 1452 cm 1, to be compared with the rather broad band at ca. 1433 cm 1 observed by IRMPD spectroscopy.3 Due to the multiphoton nature of this experiment, a red-shift of ca. 20 cm 1 is not uncommon. The band with the second-highest intensity is predicted to be the CH2 scissoring vibration ν7 (7a1) at 1239 cm 1, whereas the value derived from a second very broad IRMPD band is 1228 cm 1. For comparison, empirically scaled harmonic B3LYP/6-311++G** values are ν27 = 1434 cm 1 and ν7 = 1237 cm 1.3 More recent IRMPD values (see Table 2 of ref 4) are ν7 = 1225 cm 1 and ν27 = 1438 cm 1. In addition, two more bands at 1190 and 1585 cm 1 were observed which

ARTICLE

Table 4. Calculated Equilibrium Dipole Moments μe (in D) of C6H+a 7 method

VDZ-F12

VTZ-F12

VQZ-F12

SCF

0.8106

0.8042

0.7989

SCF+CABS-Sb DF-MP2-F12c

0.8007 0.7265

0.7990 0.7277

0.7987 0.7293

DF-SCS-MP2-F12d

0.6483

0.6499

0.6513

CCSD(T)-F12a

0.7434

0.7506

0.7531

CCSD(T)-F12b

0.7447

0.7510

0.7533

CCSD(T*)-F12a

0.7481

0.7525

0.7541

CCSD(T*)-F12b

0.7493

0.7529

0.7542

a

All calculations are carried out at the recommended equilibrium structure of Table 1. b Including CABS singles correction.15 c Explicitly correlated MP2 using density fitting. Ansatz 3C(FIX) is employed.42 d Including spin-component scaling.43

compare well with our predictions of ν8/ν30 = 1188/1185 cm 1 and ν5 = 1590 cm 1. The anharmonic contributions (ν-ω) to the symmetric and asymmetric CH2 stretching vibrations are calculated to be 151 and 166 cm 1, respectively, so that a small positive value of 5 cm 1 results for the difference ν4 ν17. The present calculations are possibly still not accurate enough to unambiguously answer the question which of the two vibrations has the higher wavenumber. The situation may well be complicated by the presence of combination tones or overtones in the considered region of the spectrum. In the range of the aromatic CH stretching vibrations, the highest wavenumber is predicted for the vibrational mode 1b2 with ν24 = 3105 cm 1. Within the double harmonic approximation, this is calculated to be the most intense aromatic CH stretch, an absolute IR intensity of 8 km mol 1 being obtained by B2PLYP-D/VTZ. The two totally symmetric vibrations ν1 and ν2 appear at almost the same wavenumber and their summed intensity about equals the intensity of the ν24 band. The asymmetric vibration ν25 (2b2) is predicted at 3067 cm 1 with a calculated intensity of 3 km mol 1. The ν3 band at 3062 cm 1 is calculated to have almost zero intensity. The corresponding normal vibration was not characterized in the earlier work of Douberly et al.9 and is graphically displayed in figure F1 of the Supporting Information. All five aromatic CH stretching fundamentals thus fall into a relatively small range of 43 cm 1. This may be compared with the benzene molecule where the separation between experimental values for ν1(1a1 g) and ν5(1b1u) is 59 cm 1.41 Combined CCSD(T*)-F12a/ B2PLYP-D calculations analogous to the present ones for C6H+7 yield a difference of 57 cm 1, very close to the experimental value. It is conceivable that higher-order anharmonic resonances may be present in the range of the five aromatic CH stretches of the benzenium ion (similar to benzene), but—owing to the small coupling elements typical for semirigid molecules—they will hardly shift vibrational wavenumbers by more than a few cm 1. The electric dipole moment of C6H+7 , which is an important quantity for pure rotational spectroscopy and radio astronomy, has been calculated at the recommended equilibrium structure (cf. Table 1). The dependence on method and basis set is reported in Table 4. The largest basis employed is VQZ-F12,19 which comprises 760 contracted Gaussian-type orbitals (cGTOs). With this basis, the four variants of explicitly correlated coupled cluster theory employed yield almost identical μe values of 0.7531 up to 0.7542 D. 13667



ARTICLE

Figure 1. Structure and atomic numbering for C6H+7 and definition of intermolecular coordinates R, θ1, and θ2 for C6H+7 3 Rg.

Figure 3. CCSD(T*)-F12a radial energy profiles for migration of an argon atom around C6H+7 . Full lines: basis (T, Q); dashed-dotted lines: basis (D, T).

Figure 2. CCSD(T*)-F12a radial energy profiles for migration of a neon atom around C6H+7 . Full lines: basis (T, Q); dashed-dotted lines: basis (D, T).

3.2. C6H+7 3 Ne and C6H+7 3 Ar. Like in our previous work on

C3H+3 3 L complexes (L = Ne, Ar, N2, CO2, and O2),11 13 we have first calculated energy profiles for rare-gas (Rg) migration around the rigid benzenium ion (cf. Table 1 for equilibrium structures). Two different paths are considered, path I proceeding in a plane which is perpendicular to the ring-plane of the benzenium ion and which contains its C2 axis, whereas path II proceeds within the ring-plane. Points lying on path I or II are described by the variables (Ropt, θ1) for path I or (Ropt, θ2) for path II, Ropt being the optimized value of the distance between the Rg nucleus and the center-of-mass (CM) of the benzenium ion. The angles θ1 and θ2 are described in Figure 1. CCSD(T*)-F12a radial energy profiles for C6H+7 3 Ne and C6H+7 3 Ar are displayed in Figures 2 and 3, graphs obtained with basis sets (D, T) and (T, Q) being quoted in both cases. The corresponding points on the paths and their relative energies with

respect to the fragments (Erel) are supplied as Supporting Information (Tables S2 S5). Along path I, both complexes exhibit a pronounced energy minimum at an angle θ1 of 96 97°. This minimum will be termed Perp M1 in the following. The energy profiles for migration of the Rg atom within the ring-plane have a more complex shape, with three minima of not strongly differing well depth being separated by an angle of Δθ2 ≈ 60°. The deepest minimum occurs for an angle θ2 of 152°; it will be termed Ipl M1. The maxima of the energy profiles are calculated at angles close to 0°, 60°, 120°, and 180°. Accordingly, there is no signature of weak hydrogen bonds between the hydrogen atoms of the partially aromatic ring and the Rg atoms. Our findings are in contrast to the work of Solca and Dopfer,7,8 who reported weak hydrogen bonds to the aromatic hydrogen atoms. Figure 4 attempts to shed some light on that issue. It plots potential curves for C6H+7 3 Ar as a function of R for three different values of the angle θ1 (0°, 30°, and 60°). At long-range, the orientation with θ1 = 0° is energetically most favorable and might appear to lead to the formation of a weak H-bond with atom H5 (see Figures 1 and 5). However, exchange repulsion sets in rather early such that the minimum of the corresponding potential curve occurs at Ropt (0°) = 5.2398 Å (basis set: (T, Q)). Upon approach of the argon atom toward the center of an aromatic CC bond (θ1 = 30°), repulsion sets in much later. The two potential curves cross at R ≈ 5.4 Å, and Ropt (30°) is as small as 4.7740 Å, with the corresponding relative energy being lower by 98 cm 1 as compared to θ1 = 0°. Optimized structures and relative energies for four nonequivalent minima as well as the stationary points with θ1 = θ2 = 0° and θ1 = θ2 = 180° are given in Table 5. Throughout, the small (D, T) basis set performs quite well both for structures and energetics. For the more strongly bonded Ar complexes, differences in the separations Re and Ropt between results obtained with the two basis sets do not exceed 0.007 Å and the angles agree 13668



ARTICLE

Table 5. CCSD(T*)-F12a Optimized Structures and Relative Energies for C6H+7 3 L (L = Ne and Ar)a L = Ne structure Perp M1

(D, T) Re (Å) θ1e (°)

Ipl M1

Ipl M2

Erel (cm 1) Re (Å)

259.3 4.4702

238.5 4.4857

649.1 4.7643

624.9 4.7663

θ2e (°)

151.67

151.88

151.60

151.61

Erel (cm 1)

212.3

195.9

524.5

507.9

Re (Å)

Re (Å)

Ropt(Å)

C2v (180°) Ropt (Å) Erel (cm 1)

Figure 5. Energetically most favorable structures for C6H+7 3 L (L = Ne and Ar) as obtained by CCSD(T*)-F12a/(T, Q). De values in cm 1.

within a few hundredths of a degree. Deeper well depths are obtained with the smaller basis in all cases, probably as a result of a larger basis set superposition error. The relative differences in Erel for the minima amount to 8 10% for the neon complexes and to 3 4% for the argon complex.

3.4570 96.13

Erel (cm 1)

Figure 4. Potential curves for C6H+7 3 Ar as a function of R (distance between Ar nucleus and CM of C6H+7 ) for different in-plane angles θ2.

3.4500

(T, Q)

96.10

θ2e (°) Erel (cm 1)

a

3.1861

(D, T)

96.80

Erel (cm 1)

C2v (0°)

(T, Q)

96.77

θ2e (°) Ipl M3

3.1624

L = Ar

4.5827 28.60 189.4 4.5859 89.49 188.8 5.0216 136.0 4.8320 153.6

4.6039 28.56 172.9 4.6074 89.56 171.9 5.0401 122.0 4.8487 140.6

4.8675 28.26 469.4 4.8730 89.81 464.0 5.2302 366.6 5.0911 397.8

4.8704 28.27 453.3 4.8764 89.84 447.2 5.2398 352.7 5.0942 385.3

Monomers are kept fixed in their equilibrium structures (see Table 1).

At the minima Perp M1 and Ipl M1-M3 as obtained by CCSD(T*)-F12a with the (T, Q) basis set, single-point calculations with the large (Q, 5) basis set have been carried out. The results are given in Table 6. The table quotes rigid-monomer equilibrium dissociation energies De for eight different methods accounting for electron correlation. Consideration of correlation effects is crucial; note that the Hartree Fock approximation provides repulsive contributions to the relative energies at the optimized structures of the complexes. Apparently, dispersion interaction plays a major role for all the four different structures of each complex considered. The differences in De values between the four explicitly correlated coupled cluster variants including perturbative triples are very small. Differences between versions a and b vary between 1.2 and 2.9 cm 1. Scaling of the triples contributions increases the dissociation energies by less than 0.4% in all cases. The influence of perturbative triples is quite significant. Their inclusion increases the De values of the inplane structures Ipl M1-M3 by 17 18% and the energetically most favorable structures Perp M1 by even 22%. Besides the results of explicitly correlated coupled cluster calculations, Table 6 also contains data obtained by DF-MP2-F12 (explicitly correlated MP2 using density fitting)42 and its spin-componentscaled43 variant (DF-SCS-MP2-F12). The performance of DFMP2-F12 shows no clear trends. DF-SCS-MP2-F12 significantly underestimates the dissociation energies in all eight cases considered. Results obtained by the double-hybrid density functional method B2PLYP-D and basis set combination (VTZ, AVTZ) are shown in Table 7. Compared with the accurate CCSD(T)F12x results from Table 6, the B2PLYP-D dissociation energies for the neon complexes are significantly too high (by ca. 40%). For the argon complexes, the situation is much better. For Perp M1, Re = 3.4459 Å and θle = 97.35° are calculated, to be compared with (3.4570 Å, 96.13°) as obtained by CCSD(T*)F12a with the basis set combination (T, Q). The corresponding dissociation energies are 612.8 cm 1 (B2PLYP-D) and 620.9 cm 1 (CCSD(T*)-F12a, large (Q, 5) basis set). An harmonic 13669



ARTICLE

Table 6. Rigid-Monomer Equilibrium Dissociation Energies (in cm 1) for C6H+7 3 L (L = Ne and Ar) from Calculations with the Large (Q, 5) Basis Seta L = Ne method

L = Ar

Perp M1

Ipl M1

Ipl M2

Ipl M3

Perp M1

Ipl M1

Ipl M2

Ipl M3 68.2

SCF

116.3

39.7

35.9

41.8

231.1

51.3

51.2

SCF+CABS-Sb

118.0

41.1

36.9

42.9

230.6

51.0

50.8

67.8

DF-MP2-F12c DF-SCS-MP2-F12d

239.3 162.9

170.9 123.2

150.4 109.1

148.5 105.9

770.6 561.0

539.6 409.8

482.1 368.4

475.9 358.3

CCSD-F12a

183.7

159.4

141.0

138.2

483.7

417.5

373.6

363.4

CCSD-F12b

185.4

161.1

142.2

139.5

486.6

420.2

375.8

365.6

CCSD(T)-F12a

234.9

192.9

170.0

168.7

618.6

503.6

449.7

443.3

CCSD(T)-F12b

236.6

194.5

171.3

170.0

621.5

506.2

451.9

445.5

CCSD(T*)-F12a

235.6

193.2

170.4

169.1

620.9

505.0

450.9

444.7

CCSD(T*)-F12b

237.3

194.9

171.6

170.4

623.8

507.6

453.1

446.9

a

Optimized CCSD(T*)-F12a/(T, Q) structures from Table 5 are used in all the underlying calculations. b Including CABS singles correction.15 c Explicitly correlated MP2 using density fitting. Ansatz 3C(FIX) is employed.42 d Including spin-component scaling.43

Table 7. B2PLYP-D Optimized Structures and Equilibrium Dissociation Energies for C6H+7 3 L (L = Ne and Ar)a structure Perp M1

L = Ne Re (Å) θ1e (°)

Ipl M1

Ipl M2

97.35

De (cm 1) Re (Å)

320.8 4.3259

612.8 4.7263

θ2e (°)

151.74

151.46

De (cm 1)

272.2

511.8

Re (Å) De (cm 1) Re (Å) θ2e (°) De (cm 1)

a

3.4459

97.56

θ2e (°) Ipl M3

3.0506

L = Ar

4.4409 28.88 244.0 4.4476 89.51 242.3

4.8289 29.10 453.5 4.8350 89.41 451.0

Basis set: (VTZ, AVTZ).

wavenumber analysis has been carried out for the C2v structure with θ1 = θ2 = 0°. In agreement with the results of the explicitly correlated coupled cluster calculations (cf. Figure 3), a secondorder saddle points with two imaginary wavenumbers (17.0 i and 23.6 i cm 1) was found. B2PLYP-D/VTZ harmonic wavenumber shifts have been calculated for all four nonequivalent minima of both C6H+7 3 Ne and C6H+7 3 Ar. Those for the neon complexes are very small and thus of no concern. Results for the different minima of C6H+7 3 Ar are given in Table S6 (see Supporting Information). For the great majority of the vibrations, the shifts are still quite small. Compared with free C6H+7 , the harmonic CH2 stretching vibrations experience a significant degree of localization, associated with a change in the intensity ratio (symm./asymm.) from 2.63 to 1.54 (B2PLYP-D results). Since anharmonicity has changed the order of asymmetric and symmetric CH2 stretching vibrations in the case of free C6H+7 (cf. Table 3), we may expect that the band with higher wavenumber will be the more intense vibration. Owing to the subtle effects of vibrational anharmonicity in the 2800 cm 1 region of the spectrum, the calculated harmonic wavenumber shifts for the CH2 stretching vibrations (cf. Table S6 of the Supporting Information) are possibly not very reliable estimates for

the anharmonic shifts. The calculated shifts are quite small, however, and so the theoretical anharmonic values for free C6H+7 of ν4 = 2813 cm 1 and ν27 = 2808 cm 1 are expected to be close to those for the argon complex. We are therefore tempted to associate them with the bands observed at 2820 and 2809 cm 1. The weakest band in the IRPD spectrum at 2793 cm 1 might be a combination tone, a possible candidate being (ν27 + ν29). On the basis of deuterium experiments, Solca and Dopfer7 assigned the band observed at 2819 cm 1, which is the most intense vibration in the range 2790 2820 cm 1, to a “C C stretch overtone (2σC C)”. However, Douberly et al.9 presented strong experimental evidence against this assignment. According to the present calculations for free C6H+7 (cf. Table 3), only 2ν6, 2ν27, and 2ν28 appear in a reasonably acceptable wavenumber region for an assignment to the band at 2819 cm 1. VPT2 calculations at the B2PLYP-D/VTZ level yield wavenumbers of 2904, 2916, and 2782 cm 1 for these overtones. Only the last value is fairly close to the observed band. The major problem with the assignment of Solca and Dopfer is the high intensity of the band at 2819 cm 1. From where should a “C C-stretch overtone” derive its intensity? The fundamentals ν6 and ν28 of free C6H+7 have small calculated intensities of 26 and 0 km mol 1, respectively (see Table 3). Only ν27 has a relatively high intensity of almost 200 km mol 1, but overtone intensities of asymmetric CC ring stretching vibrations will hardly have more than 10% of the intensity of the corresponding fundamental. According to the present B2PLYP-D calculations for all isomers of C6H+7 3 Ar, the harmonic shifts of the five aromatic CH stretching vibrations are all smaller than 5 cm 1 (see Table S6 of the Supporting Information), in most cases less than 0.5 cm 1. We may thus compare the anharmonic values from Table 3 with the IRPD data.7 9 The observed peak with highest wavenumber at 3110 cm 1 or 3107 cm 1 (see refs 7 9) is in very close agreement with the most intense band ν24 calculated at 3105 cm 1. The ν1 and ν2 peaks predicted at 3081 and 3083 cm 1 may overlap in an IRPD spectrum and are probably associated with the peak observed at 3078 cm 1. The band at 3006 cm 1 in the spectrum of Douberly et al.9 is too low to be assignable to a fundamental vibration of C6H+7 3 Ar. It differs in wavenumber from the strongest aromatic CH stretch by as much as 101 cm 1, much larger than the range of calculated values for the aromatic 13670



ARTICLE

Table 8. Comparison of Experimental and Theoretical Vibrational Wavenumbers (in cm 1) for C6H+7 3 Ara band

IRPDb

ν21

p-H2c

theor.d

641

639

ν27 + ν29e

2793

band

IRPDb

theor.d

ν20

831 (829)

819

819

ν17

2809 (2879)

2814

ν11

903 (878)

894

889

ν4

2820 (2887)

2812

ν32

964 (964)

988

961

ν5 + ν28e

3006

ν18

1058 (1039)

1043

ν26 + ν27e

3035

ν8

1198 (1177, 1183)

1185

1188

ν25

ν7

1239 (1256)

1226

1239

ν3

ν29 ν27

1334 (1319) 1456 (1450)

ν2 ν1

3078 (3133)

1452

1338 1452

3083 3081

ν5

1607 (1600)

1603

1590

ν24

3107 (3135)

3105

3067 3062

a

The same numbering of vibrations as for free C6H+7 is employed (see Table 3). b ref 9. Scaled harmonic wavenumbers obtained by B3LYP/6-311+G(d, p) are given inparentheses. c From IR spectra in solid p-H2 (ref 44.). d Results for C6H+7 from the second-last column of Table 3 are combined with harmonic argon shifts for the energetically most favorable isomer Perp M1 from Table S6 of the Supporting Information. e Tentative assignments.

CH stretching vibrational wavenumbers of free C6H+7 (43 cm 1). A possible candidate is provided by the combination band (ν5 + ν28). We are also sceptical that the band observed at 3035 cm 1 has to be assigned to an aromatic CH stretch since the difference from the band at 3107 cm 1 also appears to be too large. A more reasonable assignment is provided by the combination tone (ν26 + ν27). The sum of the fundamentals ν26 and ν27 is calculated to be 3035 cm 1 by using the ν data of Table 3, while B2PLYP-D yields a small value of 5.5 cm 1 for the anharmonicity constant X27, 26. Within VPT2, this constant corresponds to the difference (ν26 + ν27) ν26 ν27. Finally, Table 8 makes a comparison of experimental and theoretical vibrational wavenumbers for C6H+7 3 Ar. The table also includes data from IR spectra of C6H+7 isolated in solid p-H2, about which the authors of the present paper were informed after completion of their work.44 Agreement between theory and experiment for the 10 bands observed in the mid-IR region of the spectrum (600 2000 cm 1) is very good. In particular, the differences in wavenumbers for the two most intense vibrations ν27 and ν7, and ν5 do not exceed 5 cm 1. According to the harmonic B2PLYP-D calculations, the intensity of the strongest ν27 band is reduced by complex formation from 194 to 183 km mol 1, while the ν7 and ν5 bands experience changes of +3 and 4 km mol 1, respectively. The right-hand half of Table 8 compares the vibrations in the range above 2800 cm1. These have been discussed in detail above and so no further comment is necessary.

4. CONCLUSIONS Explicitly correlated coupled cluster theory and double-hybrid density functional theory has been employed in a theoretical study of the benzenium ion and its complexes with the Rg atoms Ne or Ar. For the free cation, the ground-state rotational constants (A0 = 5445 MHz, B0 = 5313 MHz, and C0 = 2731 MHz) have been predicted with an accuracy of ca. 0.1%, thereby providing important information for forthcoming studies by pure rotational spectroscopy and radio astronomy. For the complexes C6H+7 3 Rg, a π-bonded structure lies significantly lower in energy than three nonequivalent local minima where the rare-gas atom is in the ring-plane. Calculated Ar shifts do not exceed 6 cm 1. For the two most intense fundamentals of C6H+7 3 Ar (ν27 and ν7) agreement between theory (Perp M1) and experiment is excellent, differences amounting to less than 5 cm 1. The five aromatic CH stretching vibrations are predicted

to occur in a small wavenumber range spanning only 43 cm 1. For intensity reasons and due to overlapping bands, it is quite reasonable that only two bands are detectable by lowresolution spectroscopy. On the basis of our most extensive explicitly correlated coupled cluster calculations, the De values of the π-bonded structures of C6H+7 3 Ne and C6H+7 3 Ar are estimated to be 235 and 620 cm 1, respectively. Nonrigidity effects on De have been studied by B2PLYP-D and were found to be negligibly small (below 2 cm 1). Within the harmonic approximation, the zeropoint vibrational contributions to the dissociation energies were computed as 75 and 70 cm 1 such that we arrive at D0 values of 160 and 550 cm 1. Since the harmonic approximation may have significant errors for the intermolecular vibrations, the uncertainties in the D0 estimates may well be of the order of 30 cm 1. Interestingly, no hydrogen-bonded structures could be found to be local minima of the potential energy surfaces for C6H+7 3 Ne and C6H+7 3 Ar. This situation is thus different from that for the complexes of type c-C3H+3 3 Rg studied earlier.11 13 In that work, three equivalent H-bonded local minima were found. Work in progress is devoted to the more strongly bonded complexes of C6H+7 with L = N2 and CO2 and will be published separately.

’ ASSOCIATED CONTENT

bS

Supporting Information. Vibration rotation coupling constants for C6H+7 . CCSD(T*)-F12a optimum distances and relative energies for Rg (Rg = Ne, Ar) migration around C6H+7 along two different paths (perpendicular and in-plane). B2PLYPD harmonic wavenumbers for C6H+7 and shifts for C6H+7 3 Ar. Normal vibration of aromatic CH stretch with lowest wavenumber. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors acknowledge financial support by the Fonds der Chemischen Industrie. Thanks are due to the Gesellschaft f€ur 13671


The Journal of Physical Chemistry A wissenschaftliche Datenverarbeitung (GWDG) for providing computation time.

’ REFERENCES (1) Olah, G. A. Acc. Chem. Res. 1997, 4, 240. (2) Olah, G. A., Schleyer, P.v.R., Eds.; Carbonium Ions; John Wiley and Sons: New York, 1970; Vol. II. (3) Jones, W.; Boissel, P.; Chiavarino, B.; Crestoni, M. E.; Fornarini, S.; Lemaire, J.; Maitre, P. Angew. Chem., Int. Ed. 2003, 42, 2057. (4) Dopfer, O.; Solca, N.; Lemaire, J.; Maitre, P.; Crestoni, M.-E.; Fornarini, S. J. Phys. Chem. A 2005, 109, 7881. (5) Dopfer, O. J. Phys. Org. Chem. 2006, 19, 540. (6) Lorenz, U. J.; Solca, N.; Lemaire, J.; Maître, P.; Dopfer, O. Angew. Chem., Int. Ed. 2007, 46, 6714. (7) Solca, N.; Dopfer, O. Angew. Chem., Int. Ed. 2002, 41, 3628. (8) Solca, N.; Dopfer, O. Chem.—Eur. J. 2003, 9, 3154. (9) Douberly, G. E.; Ricks, A. M.; v. R. Schleyer, P.; Duncan, M. A. J. Phys. Chem. A 2008, 112, 4869. (10) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (11) Botschwina, P.; Oswald, R. J. Chem. Phys. 2011, 134, 044305. (12) Botschwina, P.; Oswald, R.; Rauhut, R. Phys. Chem. Chem. Phys. 2011, 13, 7921. (13) Botschwina, P.; Oswald, R.; Dopfer, O. Phys. Chem. Chem. Phys. 2011, 13, 14163. (14) Adler, T. B.; Knizia, G.; Werner, H.-J. J. Chem. Phys. 2007, 127, 221106. (15) Knizia, G.; Adler, T. B.; Werner, H-.J. J. Chem. Phys. 2009, 130, 054104. (16) Werner, H.-J.; Knizia, G.; Adler, T. B.; Marchetti, O. Z. Phys. Chem. 2010, 224, 493. (17) Werner, H.-J., Knizia, G., Adler, T. B., Manby, F. R. in Recent Progress in Coupled Cluster Methods: Theory and Applications; Carsky, P., Ritter, J., Paldus, J., Eds.; Springer: New York, 2010. (18) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. Chem. Phys. Lett. 1989, 157, 479. (19) Peterson, K. A.; Adler, T. B.; Werner, H.-J. J. Chem. Phys. 2008, 128, 084102. (20) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007. (21) Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. J. Chem. Phys. 1992, 96, 6796. (22) Woon, D. E.; Dunning, T. H., Jr. J. Chem. Phys. 1993, 98, 1358. (23) Dunning, T. H., Jr.; Peterson, K. A.; Wilson, A. K. J. Chem. Phys. 2001, 114, 9244and references therein. (24) Valeev, E. F. Chem. Phys. Lett. 2004, 395, 190. (25) Yousaf, K. E.; Peterson, K. A. J. Chem. Phys. 2008, 129, 184108. (26) Yousaf, K. E.; Peterson, K. A. Chem. Phys. Lett. 2009, 476, 303. (27) Weigend, F. Phys. Chem. Chem. Phys. 2002, 4, 4285. (28) Weigend, F; K€ohn, A.; H€attig, C. J. Chem. Phys. 2002, 116, 3175. (29) H€attig, C. Phys. Chem. Chem. Phys. 2005, 7, 59. (30) Werner, H.-J., Knowles, P. J., Lindh, R., Manby, F. R., Sch€utz M. et al., MOLPRO version 2009.1, a package of ab initio programs, 2009, see http://www.molpro.net. (31) Frisch, M. J., Trucks, G. W., Schlegel, H. B., Scuseria, G. E., Robb, M. A., Cheeseman, J. R., Scalmani, G., Barone, V., Mennucci, B., Petersson, G. A., Nakatsuji, H., Caricato, M., Li, X., Hratchian, H. P., Izmaylov, A. F., Bloino, J., Zheng, G., Sonnenberg, J. L., Hada, M., Ehara, M., Toyota, K., Fukuda, R., Hasegawa, J., Ishida, M., Nakajima, T., Honda, Y., Kitao, O., Nakai, H., Vreven, T., Montgomery, Jr., J. A., Peralta, J. E., Ogliaro, F., Bearpark, M., Heyd, J. J., Brothers, E., Kudin, K. N., Staroverov, V. N., Kobayashi, R., Norm, J., Raghavachari, K., Rendell, A., Burant, J. C., Iyengar, S. S., Tomasi, J., Cossi, M., Rega, N., Millam, N. J., Klene, M., Knox, J. E., Cross, J. B., Bakken, V., Adamo, C., Jaramillo, J., Gomperts, R., Stratmann, R. E., Yazyev, O., Austin, A. J., Cammi, R., Pomelli, C., Ochterski, J. W., Martin, R. L., Morokuma, K., Zakrzewski, V. G., Voth, G. A., Salvador, P., Dannenberg, J. J., Dapprich, € Foresman, J. B., Ortiz, J. V., Cioslowski, J., S., Daniels, A. D., Farkas, O.,

ARTICLE

Fox, D. J. Gaussian 09, Revision A.02, Gaussian, Inc.: Wallingford CT, 2009. (32) Grimme, S. J. Chem. Phys. 2006, 124, 034108. (33) Schwabe, T.; Grimme, S. Phys. Chem. Chem. Phys. 2007, 9, 3397. (34) Barone, V. J. Chem. Phys. 2005, 122, 014108 and references therein. (35) Botschwina, P.; Oswald, R. J. Phys. Chem. A 2010, 114, 9782. (36) Gauss, J.; Stanton, J. F. J. Phys. Chem. A 2000, 104, 2865. (37) Lievin, J.; Demaison, J.; Herman, M.; Fayt, A.; Puzzarini, C. J. Chem. Phys. 2011, 134, 064119. (38) Thorwirth, S.; Harding, M.; Muders, D.; Gauss, J. J. Mol. Spectrosc. 2008, 251, 220. (39) Garkusha, I.; Fulara, J.; Nagy, A.; Maier, J. P. J. Am. Chem. Soc. 2010, 132, 14979. (40) Wilson, A. K.; van Mourik, T.; Dunning, T. H., Jr. Mol. Struct. (THEOCHEM) 1996, 388, 339. (41) Erlekam, U.; Frankowski, M.; Meijer, G.; von Helden, G. J. Chem. Phys. 2006, 124, 171101. (42) Werner, H.-J.; Adler, T. B.; Manby, F. R. J. Chem. Phys. 2007, 126, 164102 and references therein. (43) Grimme, S. J. Chem. Phys. 2003, 118, 9095. (44) Lee, Y.-P., Bahon, M., Book of Abstracts, 31th International Symposium on Free Radicals, Port Douglas, Australia, 24 29 July 2011.

13672


Explicitly Correlated Coupled Cluster Calculations for the Benzenium

Recommend Documents