Density-functional calculations of the nuclear magnetic shielding and indirect nuclear spin–spin coupling constants of three isomers of C

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Vol. 106, No. 20, 20 October 2008, 2357–2365

RESEARCH ARTICLE

Density-functional calculations of the nuclear magnetic shielding and indirect nuclear spin–spin coupling constants of three isomers of C

20

Ola B. Lutnæs^a, Trygve Helgaker^a* and MichalJaszun´ski^b

aCentre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, Oslo, Norway;

bInstitute of Organic Chemistry, Polish Academy of Sciences, Warsaw, Poland

(Received 20 August 2008; final version received 9 September 2008)

The parameters of the nuclear magnetic resonance (NMR) spectrum – shielding constants and indirect spin–spin coupling constants – of three isomers of C₂₀are studied using density-functional theory. The performance of different exchange–correlation functionals is analysed by optimising the geometry for the ring, bowl and cage isomers, followed by a computation of the NMR constants at the optimised structure. The results are analysed and rationalised by performing comparisons of the three isomers with one another and with related systems such as polyynes (for the ring), o-benzyne (for the bowl) and C₆₀ (for the cage). The shielding and spin–spin parameters calculated using the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional are sufficiently reliable to assist in future experimental NMR studies of C20and, in particular, the identification of its isomers.

Keywords: C₂₀isomers; NMR; shielding constants; spin–spin coupling constants; DFT

1. Introduction

During the last decade or so, density-functional theory (DFT) has become the method of choice for the quantum-chemical calculation of NMR spectroscopic constants. There are many studies that present comparisons of the performance of different DFT functionals for nuclear shielding and indirect nuclear spin–spin coupling constants – see, for instance, [1–7]

and the reviews [8–10]. For most systems of chemical interest, DFT yields satisfactory results, considering its low cost. Nevertheless, certain systems still present a challenge to DFT, even with the best modern exchange–correlation functionals. Thus, in a recent o-benzyne study, the difficult calculation of NMR parameters of strained systems was considered [11].

In such systems, different choices of basis sets and exchange–correlation functionals can lead to widely different results – in particular, for the spin–

spin coupling constants, due to nearby triplet instabilities.

The most important conclusion from theo-benzyne study is that the most reliable spin–spin coupling constants are obtained when the same combination of exchange–correlation functional and basis set is used for the geometry optimization and for the subse- quent calculation of NMR constants [11]. In the present paper, we consider these problems further by undertaking a study of the NMR parameters of

the three important isomers of C20: the ring, the bowl and the cage. The C₂₀system is interesting since it allows both for a comparison of different isomers of varying degree of strain and for a comparison with related systems, such as o-benzyne and polyyne chains.

There is a large number of theoretical studies of C20, primarily devoted to the stability and energetics of its three isomers – the cage, bowl and ring structures.

Most of these studies have been carried out with density-functional theory (DFT), with the general conclusion that variations in the exchange–correlation functional and in the one-electron basis set lead to widely different results – see, for instance, [12–14]

and the reviews [15,16] for comparison and discussion of DFT results. A number of wave-function- based electronic-structure methods have also been applied to C20, including second-order Møller–

Plesset (MP2) theory [17], multi-reference MP2 theory [18], coupled-cluster singles-and-doubles-per- turbative-triples CCSD(T) theory [19,20], and quantum Monte Carlo theory [13]. However, these studies also are inconclusive. First, accurate ab initio methods were typically applied only at selected geometries, optimised at a cruder level of theory.

For instance, in a recent such study, a single-point CCSD(T) calculation was performed at the B3LYP cage geometry [21]. Next, distorted isomer structures

*Corresponding author. Email: [email protected]

ISSN 0026–8976 print/ISSN 1362–3028 online

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have not always been considered and the basis sets have often been small. Finally, the comparison with experiment may be difficult in that the energy differences between the C20 isomers are small and strongly affected by zero-point vibration and temperature effects – in particular, at high tempera- tures. For a discussion of these matters – see, for example, [20,22].

With the recent development of powerful electronic-structure methods for the calculation of NMR parameters, a new successful approach to the elucidation of molecular structure is to compare the parameters of the experimental NMR spectrum with those computed for a series of hypothetical structures.

This approach is particularly useful for distinguish- ing between isomers of nearly the same energy such as those of C20 . With this in mind, the NMR chemical shifts of six C₂₀ isomers were calculated by Romero et al. [23]. However, their study was carried out at a rather crude level of theory, using a plane- wave basis and effective core potentials. We also note that confirmation of the synthesis of C₂₀ and elucidation of its structure have primarily been based on an analysis of its ions by, for example, photoelectron spectroscopy [24,25]; so far, no experimental NMR spectra of C20 has in fact been recorded.

2. Computational details

Following theo-benzyne study [11], we have here used exchange–correlation functionals that are particularly well suited to the study of nuclear shielding and indirect spin–spin coupling constants – namely, the KT1, KT2, and PBE functionals [26,27] together with the correlation-consistent valence and core–

valence triple-zeta basis sets cc-pVTZ and cc-pCVTZ [28,29] (denoted VTZ and CVTZ in our tables, respectively). We have restricted ourselves to the commonly studied planar-ring, bowl and cage isomers of C20 depicted in Figures 1 to 3. In all cases, the same combination of exchange–

correlation functional and one-electron basis sets was used in the geometry optimization and in the sub- sequent NMR calculation. Moreover, each optimised structure was verified to represent a stable minimum by performing a harmonic-frequency analysis, thereby confirming that all frequencies are real.

Shielding constants were calculated with gauge-including atomic orbitals (GIAOs) [30,31]. Electronic excitation energies were calculated using linear response theory as described in [32,33]. All calculations were carried out using Dalton [34].

3. Molecular geometries and energies

The geometry parameters of the optimised structures of C₂₀ are listed in Table 1, while Tables 2 and 3 contain the corresponding total energies and excitation energies, respectively. The optimised structures are depicted in Figures 1–3.

3.1. Ring

For all exchange–correlation functionals, the optimised ring structure in Figure 1 has alternating long

B B

A A

A

B B

B

A

B

A

B

Figure 1. Ring C₂₀ isomer structure. The optmised geometries at the KT1/cc-pVTZ, KT2/cc-pVTZ, KT2/cc-pCVTZ, and PBE/cc-pVTZ levels have equal bond angles along the ring (D_10hsymmetry), whereas at the KT1/cc-pCVTZ and PBE/cc-pCVTZ levels, there are two bond angles alternating along the ring (C10hsymmetry).

A

A A

A

B B

C

B

C

C C

B C

C

A

Figure 2. Bowl C₂₀isomer structure.

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(132.5–134.0 pm) and short (123.9–125.4 pm) bonds, see Table 1. At the PBE/cc-pCVTZ level of theory, the difference between the long and short bond lengths – that is, the bond-length alternation (BLA) – is 9.1 pm. This result is in good agreement with the BLA value of 8.9 pm obtained for long polyyne chains at the B3LYP/TZVP level in [35]

and with the BLA value of about 9 pm obtained for infinite polyyne chains at the B3LYP/6-31G* level in [36], although we note that CAM-B3LYP and BHHLYP calculations gave a significantly larger BLA value of about 13 pm [36]. In any case, the C₂₀ ring structure clearly resembles that of polyyne rather than that of cumulene chains.

The optimised equilibrium structure of the ring has a D_10h symmetry of bond angle 162^!, except in two cases of lower C_10h symmetry, where the angle alternates between 166.1^! and 157.9^! (KT1/cc- pCVTZ) and between 162.3^! and 161.7^! (PBE/cc- pCVTZ). However, in these two cases, the C_10h and D_10henergy differences are very small (10^"3eV or less).

We therefore expect the vibrational averaged structure of the ring to be of D_10h symmetry also at the PBE and KT1 levels of theory, the structure fluctuating between two adjacent C_10h wells on the potential energy surface.

3.2. Bowl

The bowl symmetry is reported as C_5vin the literature, which is also the symmetry found in our study – see Figure 2. Bond lengths and two distances selected to quantify the depth of the bowl are listed in Table 1.

The best agreement between the MP2 and DFT results is observed for the PBE functional.

The bowl structure in Figure 2 can be viewed as consisting of five fusedo-benzyne rings, with the triple bonds on the perimeter. Indeed, comparing bond lengths, we find almost identical triple-bond lengths of 125.3 and 125.4 pm in C₂₀ and o-benzyne, respectively, at the PBE/cc-pCVTZ level of theory.

The remaining bonds are 2–3 pm longer in C20than in o-benzyne, but otherwise follow the same pattern:

141.5, 143.8 and 142.7 pm in C20(moving along the six- atom ring, away from the triple bond), compared with 138.3, 141.6 and 140.4 pm, respectively, ino-benzyne.

3.3. Cage

For the Ihsymmetry of the C20cage isomer, the highest occupied molecular orbital (HOMO) is four-fold degenerate but contains only two electrons, leading to a Jahn–Teller distortion [17]. Indeed, at all levels of theory considered here, the cage structure of lowest energy has D_2hsymmetry. In this reduced symmetry, each of the three distinct symmetry planes contains two carbon–carbon bonds, denoted LL (long), MM (medium) and SS (short), see Figure 3; the atoms in these bonds are likewise denoted L, M, and S, respectively. The remaining eight carbon atoms in the cage, labelled A in Figure 3, are symmetry equivalent, forming a rectangular parallelepiped.

The optimised cage bond lengths and parallelepiped edge lengths are listed in Table 1. The LL bond length of 153.5 pm (PBE/cc-pCVTZ) is close to that of a carbon–

carbon single bond, whereas the remaining bonds are shorter, between 140.4 and 148.6 pm. Broadly speaking, these shorter bond lengths are similar to the two distinct bond lengths in C60: 139.1 and 145.5 pm, calculated at the B3LYP level of theory in [37].

The D_2hcage symmetry obtained with DFT agrees with the results of the MP2 model [18]. Quantitatively, the best agreement with the MP2 values is obtained with the PBE functional, see Table 1. Except for a difference of 2.5 pm in the LL bond distance, all PBE and MP2 distances agree to within 0.7 pm.

We have also studied some of the lower-symmetry cage structures, which have occasionally been found to be the most stable at other levels of theory.

The optimised PBE structure of D_3d symmetry has an energy only slightly above the D_2h structure.

A A

A L L

L L

M

M M

M

S S S S

A

A A

Figure 3. Cage C20 isomer structure. The stapled lines display the parallelepiped which is rectangular for a D_2h symmetry structure.

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The same is true for the optimised C2hstructure at all levels, whereas a C2optimization returned to the D2h

minimum. In general, the different cage minima are essentially isoenergetic, with small variations in bond distances. All NMR calculations were therefore carried out only at the D2hgeometry.

3.4. Energetics

As noted in Section 1, there is little consensus on the relative stability of the C20 isomers in the literature.

In Table 2, we compare the relative stabilities of the

isomers, at the levels of theory considered here. While stability increases in the order cage–ring–bowl for the PBE functional, it increases in the order ring–

bowl–cage for the KT1 and KT2 functionals.

In Table 3, we have listed the lowest singlet and triplet excitation energies for the three isomers of C20, calculated using the PBE, KT1 and KT2 functionals in the cc-pCVTZ basis. For the ring and cage isomers, the excitation energies are particularly small. In such cases, it is mandatory to calculate triplet properties (such as the dominant Fermi-contact contribution to spin–spin coupling constants) at the optimised minimum of the potential-energy surface, as discussed foro-benzyne [11]. In this manner, problems related to the presence of nearby triplet instabilities are reduced. In this study, Table 1. Bond distances R(pm) and bond angles ! (^!) of the ring (D_10hand C_10h), bowl (C_5v) and cage (D_2h) isomers of C₂₀in the notation of Figures 1–3.

PBE PBE KT1 KT1 KT2 KT2 MP2^a

CVTZ VTZ CVTZ VTZ CVTZ VTZ

ring R_short 124.5 124.5 125.3 125.1 124.0 123.9 125.1

R_long 133.6 133.5 133.9 133.8 132.7 132.5 133.7

!^b 161.7 162.0 157.9 162.0 162.0 162.0 162.0

!^b 162.3 162.0 166.1 162.0 162.0 162.0 162.0

bowl RAA 142.7 142.7 143.6 143.4 142.1 141.7 142.3

RAB 143.8 143.8 144.6 144.3 143.1 142.7 143.4

RBC 141.5 141.4 142.2 142.1 140.7 140.5 141.1

RCC 125.3 125.3 126.2 126.1 124.9 124.8 126.9

R_5A-5B^c 46.6 46.6 47.4 46.6 46.4 45.3

R_5A-5C^c 71.8 71.9 72.8 71.5 71.4 69.6

cage R_LL 153.5 153.5 155.1 154.9 153.2 153.0 151.0

R_MM 144.5 144.4 145.6 145.4 144.0 143.8 144.5

R_SS 140.4 140.4 141.4 141.2 139.9 139.7 141.1

R_LA 142.6 142.6 143.7 143.4 142.0 141.8 142.9

R_MA 144.6 144.6 145.6 145.4 144.0 143.8 144.6

R_SA 148.6 148.6 149.5 149.3 147.8 147.6 148.4

R_AAl^d 236.3 236.3 237.6 237.3 235.1 234.7

R_AAm^d 236.9 236.9 238.3 238.1 235.7 235.3

R_AAs^d 224.7 224.7 225.8 225.5 223.4 223.4

aMP2 values from [18], calculated in a polarised valence triple-zeta basis.

bThe D_10hstructures have bond angles of 162.0^!, the other structures are of C_10hsymmetry.

cR_5X-5Yis the distance between the planes defined by five X atoms and five Y atoms, respectively.

dR_AAparallel to the LL, MM, and SS bonds is labelledR_AAl,R_AAm, andR_AAs, respectively.

Table 2. Energies at the optimised geometries of the three C₂₀isomers at various levels of theory. For the cage isomer the total energies in E_hare given, for the bowl and the ring, the differences (in eV) with respect to the corresponding cage energies are listed.

ring bowl cage

PBE VTZ "0.35 "0.44 "760.837

CVTZ "0.28 "0.39 "760.867

KT1 VTZ 2.73 0.71 "772.684

CVTZ 2.88 0.73 "772.733

KT2 VTZ 3.06 0.91 "776.086

CVTZ 3.23 0.92 "776.142

Table 3. Lowest C₂₀ excitation energies in the cc-pCVTZ basis (eV).

singlet triplet

PBE KT1 KT2 PBE KT1 KT2

ring 0.76 0.70 0.71 0.39 0.38 0.58

bowl 2.25 2.13 2.21 1.97 1.94 1.71

cage 0.87 0.91 0.92 0.43 0.46 0.28

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therefore, all NMR constants have been computed with the same combination of exchange–correlation functional and one-electron basis sets as in the optimisation of the geometry – for a similar approach, see [38].

4. Nuclear shielding constants

The calculated shielding constants of the three C20

isomers are listed in Table 4 both on an absolute scale

" and on a scale # relative to the carbon shielding

constant in benzene.

The KT1 and KT2 functionals are known to perform well for shielding constants [2]. From Table 4, we note that the constants obtained with these two functionals are in good agreement with each other but that they differ somewhat from the PBE constants. Moreover, because of error cancellation, the agreement is better among the chemical shifts than among the absolute shieldings. The inclusion of core functions (going from the cc-pVTZ basis to the cc-pCVTZ basis) changes the shielding constants significantly, in all cases reducing the constants (thereby deshielding the nuclei).

4.1. Ring

There are either one or two distinct shielding constants in the ring, depending on whether the optimised symmetry is D_10h or C_10h, respectively. The distorted C_10hsymmetry occurs only in the PBE/cc-pCVTZ and KT1/cc-pCVTZ calculations, splitting the shieldings by 2.4 and 33 ppm, respectively. However, because of zero- point vibrations, this splitting would not be observed in an experimental NMR spectrum. Comparing with PBE/cc-pCVTZ calculations on HC20H, we find that the shielding constants in the linear polyyne chain are 10–20 ppm larger than those in the ring, which in turn are more than a factor of two larger than those in the bowl and in the cage (vide infra).

4.2. Bowl

The A and C shieldings in the bowl are similar to each other (about 35 ppm) and only about one half of the B shielding (about 60 ppm), compared to approxi- mately 110 ppm in the ring. Our shifts agree with those of Romero et al. [23], who do not report absolute shieldings.

4.3. Cage

The four distinct shielding constants in the cage are for the most part smaller than those in the ring and bowl.

Moreover, this isomer contains the only deshielded carbon atom encountered in our C20 study – namely, the cage M nucleus ("11.4 ppm at the KT2/cc-pCVTZ level of theory). Our results agree qualitatively with the D_2h values of Romero et al.[23] – see their Figure 3, where the intensity indicates that their C atom corresponds to our A atom. In the analysis of cage structures of various symmetries [14], it has been concluded that these are highly fluctional, thus – similarly to the ring – the NMR spectrum of the cage isomer would not correspond to a single minimum structure.

5. Indirect nuclear spin–spin coupling constants The calculated spin–spin coupling constants of C₂₀are listed in Tables 5–7. As a general observation, there is a reasonably good agreement between the PBE and KT1 results – in particular, in the larger cc-pCVTZ basis. The KT2 functional performs more erratically, as also observed in [1,11]. For instance, we note the unphysical KT2/cc-pCVTZ values for ⁴JSS (110.5 Hz) and⁵JSS("117.4 Hz) in the cage. The larger cc-pCVTZ basis gives better agreement between the KT1 and PBE results than does the smaller cc-pVTZ basis – as Table 4. C₂₀ nuclear shielding constants " and shifts

# relative to the carbon shielding constant in benzene calculated at the corresponding level of theory (ppm)^a.

PBE PBE KT1 KT1 KT2 KT2

CVTZ VTZ CVTZ VTZ CVTZ VTZ

ring "_A 96.1 101.4 92.3 117.2 112.3 117.6

"_B 98.5 101.4 125.3 117.2 112.3 117.6

#_A 49.5 48.8 32.0 49.8 50.0 47.9

#_B 51.9 48.8 65.0 49.8 50.0 47.9

bowl "_A 19.7 26.5 35.9 44.0 37.2 45.7

"_B 45.4 51.3 56.7 63.7 59.1 66.5

"_C 17.1 23.5 33.5 41.0 34.9 42.7

#_A "26.9 "26.1 "24.4 "23.4 "25.1 "24.0

#_B "1.2 "1.4 "3.6 "3.7 "3.2 "3.2

#_C "29.5 "29.1 "26.8 "26.4 "27.4 "27.0

cage "_L 36.8 43.5 50.9 58.5 53.7 61.7

"_M "26.2 "18.4 "15.5 "5.9 "11.4 "1.4

"_S 14.0 21.3 18.0 26.7 23.6 32.6

"_A "9.4 "1.7 "2.7 6.6 2.1 11.7

#_L "9.7 "9.1 "9.4 "8.9 "8.6 "8.0

#_M "72.8 "71.0 "75.8 "73.2 "73.7 "71.1

#_S "32.6 "31.3 "42.3 "40.7 "38.7 "37.2

#_A "56.0 "54.3 "62.9 "60.8 "60.2 "58.1

aThe calculated carbon shielding constants in benzene are (in ppm): PBE/VTZ: 52.6, PBE/CVTZ: 46.6, KT1/VTZ: 67.4, KT1/CVTZ: 60.3, KT2/VTZ: 69.7 and KT2/CVTZ: 62.3.

The benzene chemical shift (129.26 ppm [39]) may be used to convert to the TMS scale.

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expected, tight core functions are important for the reliable calculation of the coupling constants. In general, an even better core description is preferable for the calculation of spin–spin couplings. Our choice here was dictated by our recent experience with the similaro-benzyne molecule, where the cc-pCVTZ basis gives coupling constants within a few Hz of the basis- set limit for the difficult triple-bond coupling [11]. In the following, all references are to the PBE/cc-pCVTZ results, unless otherwise stated.

5.1. Ring

The spin–spin coupling constants in the ring resemble closely those in linear polyynes. Thus, the two distinct one-bond coupling constants in the ring, corresponding to the short and long carbon–carbon bonds, are 236.4 and 185.1 Hz, respectively – slightly larger than the corresponding average values 219 and 175 Hz in HC20H. On the other hand, the two-bond constants in the ring (about 15 Hz) are smaller than those in the linear polyynes (about 20 Hz). However, with an

increasing number of intervening bonds, the coupling constants in the linear chain decay more rapidly than those in the ring, which remain surprisingly large – for instance, the ten-bond coupling constants are"7.6 Hz in the ring, compared with about"1.5 Hz in the linear chain. The sign alternations are identical in the two systems.

In Figure 4, we have plotted the total spin–spin coupling constants and their Fermi-contact Table 6. Spin–spin coupling constants in the bowl isomer, in Hz.

Coupling^a CVTZ VTZ CVTZ VTZ CVTZ VTZ

1J_AA^b 63.5 73.4 63.8 49.0 71.3 51.3

1J_AB^b 51.8 58.9 53.8 35.4 63.1 38.6

1JBCb 95.5 101.9 95.1 67.3 115.1 75.1

1JCCb 204.1 244.1 202.5 159.4 274.5 178.1

2JAA 9.9 2.3 10.0 "1.1 9.7 "2.4

2JABb 2.5 0.7 2.5 0.8 "0.6 "1.0

2JACb

"1.2 "4.3 "1.1 "0.3 "14.3 "6.4

2JBCb 6.8 5.1 5.6 4.5 "4.9 0.2

2J_CC 2.2 1.5 5.1 3.5 0.2 "0.1

3J_AB 2.6 2.8 2.5 1.9 2.6 1.7

3J_cAC^bc 10.1 7.1 11.9 "0.2 22.9 3.2

3J_tAC^c 0.8 1.2 0.5 0.4 0.5 0.4

3J_BB^b 1.5 "1.0 1.4 "5.4 4.2 "5.2

3J_CC 3.9 3.4 3.8 1.1 3.0 1.1

4J_cAC^d 1.3 1.1 1.3 0.4 2.2 1.0

4J_tAC^d "0.1 "0.2 "0.2 "0.3 "0.1 "0.2

4J_BB 0.4 0.4 0.4 0.5 0.2 0.6

4J_BC "1.8 "1.9 "1.6 "1.4 "2.9 "1.9

4J_CC 4.7 4.2 4.7 1.2 10.3 3.5

5J_pBC^e "0.6 "0.7 "0.5 "0.8 "0.7 "1.4

5JrBCe 0.0 0.0 0.0 "0.1 "0.1 "0.2

5JCC 3.9 4.0 4.4 4.0 5.9 4.2

6JpCCf 0.9 0.9 0.8 1.1 "1.3 0.8

6JrCCf 2.2 1.9 2.6 1.2 5.7 2.0

aThe atoms are labelled according to Figure 2.

bThe corresponding PBE/cc-pCVTZ values foro-benzyne are

1J_AA¼71.3 Hz, ¹J_AB¼48.9 Hz, ¹J_BC¼74.7 Hz,

1J_CC¼208.7 Hz, ²J_AB¼ "1.5 Hz, ²J_AC¼ "13.0 Hz,

2J_BC¼4.1 Hz,³J_cAC¼11.9 Hz and³J_BB¼2.2 Hz, see [11].

cThe notation cAC and tAC refers to cis and trans arrangements of A and C.

dFor the four-bond AC couplings, going through the atoms A-A-A-B-C, cAC and tAC refer to cis and trans arrangements about the AB bond.

eFor the two five-bond BC couplings, one of the atom pairs can be reached by moving along the periphery of the molecule (denoted⁵J_pBC); for the other coupling, one has to move through the centre ring of the molecule (denoted

5J_rBC).

fFor the two six-bond CC couplings, one of the atom pairs can be reached by moving along the periphery of the molecule (denoted⁶J_pCC); for the other coupling, one has to move through the centre ring of the molecule (denoted

6J_rCC).

Table 5. Spin–spin coupling constants in the ring isomer, in Hz.

PBE PBE KT1 KT1 KT2^b

Coupling^a CVTZ VTZ CVTZ VTZ VTZ

1J_s 236.4 254.3 235.9 208.2 197.2

1J_l 185.1 180.5 187.4 150.4 141.4

2JAA 15.2 18.4 14.1 24.3 30.7

2JBB 15.4 16.8

3Js 18.3 15.1 19.7 9.3 6.1

3Jl 13.4 11.2 15.0 4.5 2.8

4JAA "8.0 "6.0 "10.0 "1.6 "4.2

4JBB "7.9 "8.7

5J_s 6.0 2.9 7.6 "3.0 0.2

5J_l 7.5 5.1 8.8 "0.8 3.7

6J_AA "6.1 "3.9 "8.1 0.2 "2.0

6J_BB "6.0 "6.8

7J_s 4.7 2.2 6.0 "2.3 "5.2

7J_l 6.8 4.5 7.8 "0.6 "2.8

8J_AA "6.9 "4.7 "8.2 "0.8 7.5

8J_BB "6.8 "7.1

9J_s 6.3 3.8 6.9 "0.8 "13.8

9J_l 7.1 4.7 7.6 "0.1 "12.9

10J_AA "7.6 "5.4 "8.5 "1.5 13.3

10J_BB "7.6 "7.6

aAtoms are labelled according to Figure 1. For the coupling constants between nuclei separated by an odd number of bonds, s denotes coupling with more short bonds, and l denotes coupling with more long bonds between the coupled nuclei.

bSpin–spin coupling constants at the KT2/CVTZ level of theory are not reported due to problems indicating an instability in the calculation.

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contributions against the number of intervening bonds in the ring. Comparing the plots, we see that the coupling is dominated by the Fermi-contact mechan- ism, which is also responsible for the large long-range constants. The constants first decay rapidly with increasing number of intervening bonds but then remain constant beyond three bonds, alternating between equally large positive and negative values, in agreement with the Dirac vector model.

The strong long-range coupling in the ring can be partly explained by the existence of two distinct coupling paths in the system – for example, we can interpret a ⁵J constant as consisting of a five-bond clockwise contribution and a 15-bond counterclock- wise contribution of the same sign. Adding contributions in this manner, using the averages of all relevant ⁿJ_CC from the polyyne chain, we obtain

(with calculated ring values in parentheses): ⁴J¼

"5.1 Hz ("8.0 Hz), ⁵Js¼10.0 Hz (6.0 Hz), ⁵Jl¼5.8 Hz (7.5 Hz),⁶J¼ "3.9 Hz ("6.0 Hz),⁷J_s¼7.8 Hz (4.7 Hz),

8J¼ "3.3 Hz ("6.9 Hz), ⁹Js¼6.5 Hz (6.3 Hz),

9Jl¼5.8 Hz (7.1 Hz) and ¹⁰J¼ "3.1 Hz ("7.6 Hz).

This comparison indicates that additional enhance- ment effects, possibly related to delocalization, are present for the longest-range couplings due to the closure of the ring.

5.2. Bowl

In the bowl isomer, there is a large one-bond coupling of 204 Hz associated with the triple bond, the other one-bond couplings being smaller, between 50 and 100 Hz. The triple-bond coupling of 204 Hz agrees with the corresponding triple-bond coupling of 209 Hz in o-benzyne, as expected from the similar bond distances. In fact, the observed pattern of the spin–spin coupling constants of each benzyne fragment in the bowl bears a quite striking resemblance with the pattern of an individual o-benzyne molecule, keeping in mind that the carbon atoms outside the triple bond are bonded to hydrogens ino-benzyne and to carbons in C20. Thus, moving away from the triple bond, the spin–spin coupling constants of a benzyne fragment in the bowl are 95, 52, and 64 Hz, compared with 75, 49 and 71 Hz, respectively, ino-benzyne – see Table 6. In passing, we note that the spin–spin coupling constants appear to be more transferable than the shielding constants – for example, whereas the triple-bonded carbon atoms are deshielded in o-benzyne ("20.5), they are shielded in the bowl (17.1 ppm).

Comparing the bowl and ring isomers, we note that the triple-bond coupling constant of 204 Hz in the bowl is significantly smaller than the corresponding constant of 236 Hz in the ring. The remaining one-bond coupling constants are much smaller (by a factor of two or more) in the bowl than in the ring. This difference arises since the non-triple bond distances in the bowl (between 141 and 144 pm) are longer than the non-triple carbon–carbon bond distance (134 pm) in the ring.

With respect to long-range spin–spin coupling constants, we note that those of the bowl are much smaller than the unusually large long-range constants in the ring and the cage (discussed below). The slow decay of the long-range coupling constants in the ring and cage may be connected with the very small triplet excitation energies of these systems – see Table 3. In the bowl, the lowest excitation energies are about five times larger than those in the ring and cage.

Table 7. Spin–spin coupling constants in the cage isomer, in Hz.

Coupling^a CVTZ VTZ CVTZ VTZ CVTZ VTZ

1J_LL 61.2 64.8 57.4 36.2 85.2 42.5

1J_SS 96.9 102.4 98.3 52.4 204.1 68.3

1JMM 53.5 63.4 53.7 35.7 63.4 37.7

1JLA 68.0 77.2 68.6 41.9 89.9 46.2

1JMA 54.7 64.7 54.9 35.7 64.8 37.5

1JSA 62.5 68.5 63.7 38.8 97.7 47.4

2JLS 8.2 3.9 8.5 1.1 8.0 0.0

2JLM 6.5 2.3 5.9 1.7 2.0 0.2

2J_SM 8.6 4.5 8.5 2.8 8.1 2.4

2J_LA 7.4 3.3 6.8 2.3 1.6 1.5

2J_SA 3.1 "0.4 2.1 1.7 "23.5 "4.0

2J_MA 5.8 1.8 5.5 0.9 3.0 0.0

2J_ALA 9.6 4.4 10.0 4.0 12.5 6.8

2J_AMA 7.9 3.5 7.9 1.5 14.8 4.4

2J_ASA 9.0 4.0 8.9 0.9 4.3 "3.2

3J_LS 3.6 3.6 4.2 2.4 5.0 2.7

3J_MS 3.6 3.9 3.8 3.1 4.7 3.5

3J_LM 4.2 4.2 4.4 2.9 6.2 3.0

3J_LA "0.8 0.3 "1.6 1.4 "8.7 1.0

3J_MA 5.7 6.0 6.1 4.8 7.7 5.1

3JSA "2.5 "1.3 "3.2 1.4 "26.4 "2.9

3JALLA 1.8 2.1 1.8 1.3 "3.6 "2.1

3JAMMA 4.9 4.6 5.8 3.2 12.8 6.7

3JASSA 8.4 8.3 9.0 6.4 19.7 10.8

4JLL 11.8 8.9 10.8 1.9 33.6 4.8

4J_SS 22.3 16.8 22.5 4.9 110.5 17.0

4J_MM 1.8 1.7 2.6 3.5 "0.3 2.9

4J_LA 4.4 3.6 4.5 2.5 9.9 3.5

4J_MA 1.8 1.7 2.4 3.2 0.0 2.6

4J_SA 8.5 7.1 9.8 4.7 33.4 9.9

5J_LL "7.8 "4.6 "11.0 2.3 "36.7 "4.1

5J_MM 13.6 12.5 15.8 12.0 19.2 12.6

5J_SS "17.6 "11.9 "12.8 3.8 "117.4 "22.3

5J_AA 7.5 7.4 9.4 8.8 1.4 4.6

aThe atoms are labelled according to Figure 3.

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5.3. Cage

In general, the cage isomer has longer bond lengths and smaller one-bond coupling constants than do the ring and bowl isomers. The largest coupling constant in the cage is ¹JSS¼97 Hz, less than one half of the largest coupling constants in the ring and bowl; the remaining constants are smaller than 70 Hz. Except for the fact that the shortest bond has the largest one-bond spin–

spin coupling constant, there is little correlation between bond lengths and coupling constants in the cage. For instance, whereas¹J_LL¼61 Hz for the longest LL bond (153.5 pm) in the cage (and in all isomers of C20), the much shorter MM bond (144.5 pm) has a smaller coupling constant of¹JMM¼54 Hz.

As in the ring isomer, the long-range coupling constants of the cage are unusually large – for instance,

4J_SS¼22 Hz and ⁵J_SS¼ "18 Hz. Again, this behavior may be understood from the small triplet excitation energies of this isomer, noting that the coupling constants are dominated by the Fermi-contact and also spin-dipole contributions. Moreover, the long- range coupling constants may be larger than in other isomers since there are many different pathways connecting the coupled nuclei.

6. Conclusions

The differences in the NMR parameters between the three isomers of C₂₀ are obviously much more pronounced than the differences in energy. Therefore, to distinguish among the different C20 isomers, a comparison of approximate values of the shielding and spin–spin coupling constants appears to be more useful than a comparison of the small differences in the total energies, even when (as in the literature) the energies have been computed at a significantly higher level of approximation.

The shielding constants are large in the ring (about 100 ppm) and smaller (by a factor of two or more) in the bowl and cage. The cage is the only C₂₀isomer that

contains deshielded nuclei. The calculated shielding constants and, in particular, the chemical shifts do not depend strongly on the choice of exchange–correlation functional and basis set. We note, however, the large splitting of the shielding constants in the ring, which occurs only at the KT1/cc-pCVTZ level of theory, reflecting a distortion of the ring at this level of theory.

The one-bond spin–spin coupling constants in the ring and bowl are similar to the corresponding coupling constants in the linear polyynes and o-benzyne, respectively. The largest one-bond coupling constants in C₂₀are those of the triple bonds in the ring and in the bowl – 236 and 204 Hz, respectively, at the PBE/cc-pCVTZ level of theory; in the cage, the largest constant is only 97 Hz. The coupling constants over multiple bonds in the ring and in the bowl resemble less closely those of the polyynes and o-benzyne – in particular, the long-range couplings in the ring are much larger than those in the linear chain, although the same alternating pattern is observed. The long- range spin–spin coupling constants in the cage and ring are much larger than those in the bowl, which is probably related to the smaller triplet excitation energies for the cage and ring.

For the spin–spin coupling constants, the choice of exchange–correlation functional and basis set plays an important role. For all three isomers of C₂₀, the PBE and KT1 spin–spin coupling constants are in reason- able agreement with each other, whereas the KT2 results are either very different or completely unphysical, confirming that this functional is not suitable for spin–spin coupling calculations. The basis-set depen- dence of one-bond spin–spin coupling constants is much larger with the KT1 functional than with the PBE functional. In part, this dependency is a geometrical effect – the cc-pVTZ and cc-pCVTZ bond lengths are similar with the PBE functional but noticeably different with the KT1 (and KT2) functionals. We note, however, that the KT1 and PBE coupling constants are similar in the cc-pCVTZ basis even though the geometries differ.

2 3 5 7 9

100

200 total spin–spin coupling constants

2 3 5 7 9

100

200 FC contributions

Figure 4. The PBE/cc-pCVTZ indirect spin–spin coupling constants in the C₂₀ring isomer, plotted as a function of the number of intervening bonds (Hz). Each plotted value is the average over the two distinct coupling constants for the given number of intervening bonds.

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As in our previous work ono-benzyne [11], the PBE functional seems to offer the most consistent performance throughout all steps of the computational study – the optimization of the geometry, the calculation of the shielding constants, and the calculation of spin–spin coupling constants.

Acknowledgements

This research was supported by the Norwegian Research Council through the CoE Centre for Theoretical and Computational Chemistry (Grant No. 179568/V30). MJ acknowledges many helpful discussions with G. Dolgonos and H. Dodziuk. We would like to thank D.J. Tozer for helpful discussions.

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