1
Calculations of PAH anions: When are diffuse functions necessary?

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa1

2
The effect of including vs. excluding diffuse functions while calculating numerous parameters of PAH anions by various calculation methods is discussed.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

3
The omission of diffuse functions appears to have a negligible effect while calculating geometry parameters or total energy; thus, acceptable results may be obtained without them.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

4
The conclusions for charge density appear to be the same; however, limited results make an unambiguous claim unachievable.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

5
Calculating 1H- and 13C-NMR shifts undoubtedly requires the use of these functions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

Introduction

6
For some time now, computational chemistry has been an important tool for probing, observing, explaining and verifying what had been discovered experimentally.2

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met1

7
Calculations may verify or refute results still in question, or explain chemical, physical and other phenomena, whilst discovering previously unknown trends.

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met1

8
Combining experiment and theory results in a powerful and efficient instrument, enabling study of any kind of chemistry.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met2

9
Theoretical chemistry, therefore, is an important tool, having a critical position as a companion to experimental results, playing a significant role in the advancement and development of today's chemistry in general, and the research in polycyclic aromatic hydrocarbon (PAH) anions, in particular.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

10
In this paper we are concerned with the necessity of diffuse functions in the calculation of large conjugated anions, especially PAH anions.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

11
PAH anions have also been the focus of extensive research, as they exhibit extraordinary properties, pertaining to key questions of much interest in organic chemistry.3–6

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot2

12
The study of PAH anions is widespread, particularly in organic chemistry as they are models for charged graphite, fullerenes and other carbon-based compounds.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

13
Additionally, interest in PAHs and their anions ranges from synthetic,7 medicinal and biological8 chemistry, to environmental9 and interstellar10 chemistry.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

14
Theoretical characterization of PAHs and their anions is therefore critical to understanding their reactivity.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot3

15
The heart of straightforward calculations is the proper choice of basis sets.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

16
Improvement of a calculation, in order to obtain more reliable and accurate results, closer to experimental values, is possible via a change in the calculation protocol.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

17
For instance, one may use an extended basis set, utilizing three or more lobes to describe an orbital; or employ polarization functions,11 describing the change imposed on an orbital during the transition from free to bonded state.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met3

18
Diffuse functions constitute a further effort to improve theory.

Type: Method | Advantage: None | Novelty: Old | ConceptID: Met4

19
Just as the insertion of extra electrons into orbitals brings about their natural growth in order to reduce the Coulomb repulsion, so must a theoretical calculation allow for expression of such a change.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

20
Development of diffuse functions allowed orbital flexibility during calculation, allowing a more accurate description of the true, diminished electron repulsion, mainly for anions, and lowering the estimate of total energy, in accordance with the variation principle.12

Type: Method | Advantage: Yes | Novelty: Old | ConceptID: Met4

21
The use of diffuse functions originated from the calculation of atomic anions, such as for H−, F−, Li− and O−.13

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

22
The investigation of such anions demonstrated that without diffuse functions, even stable anions, such as O−, Cl− and F− are calculated to be unstable.14

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

23
Calculating anions without diffuse functions generally overestimates mutual electron repulsion,13 and even gives false interactions of each electron with itself.15

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

24
Various small anions have been shown to commonly need diffuse functions for useful calculations.13

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

25
Moreover, it has been shown that standard basis sets using only one set of diffuse functions for calculating properties of small anions may give inadequate results, indicating the necessity for double-diffuse functions.16

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

26
It is important to note that in all these cases, assumptions regarding the necessity of diffuse functions were based on calculations of anions with point charges.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

27
In such cases, the charge is almost entirely located on a single atom.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

28
These and other17 difficulties lead to the supposition that standard calculations are not appropriate for negative atomic and molecular anions, and eventually resulted in the inclusion of diffuse functions as a general prerequisite for reliable anionic calculations.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

29
This pertained to small molecular anions, as well as to atomic anions.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

30
Duke's work18 on the methyl anion showed that the inclusion of diffuse functions proved to be critical for adequate representation of the anion.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

31
Even relatively small basis sets including diffuse functions, such as 3-21+G, were found to describe the geometries and proton affinities of small molecular anions, such as BeH−, BH2−, NH2− and OH−, fairly accurately.13

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

32
While calculating equilibrium geometries, the inversion barrier and other properties of CH3− and CH3 using ab initio methods, Driessler showed19 that SCF values obtained using diffuse functions were substantially more accurate than previous results calculated without them.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

33
Driessler's results reinforced Duke's,18 also showing that, lacking diffuse functions, one obtains positive orbital energy for the highest occupied orbitals for negative anions.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

34
Davidson and Schaefer both independently confirmed this later on.20

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

35
The necessity of diffuse functions in calculations and their contribution to various properties have been widely studied.21–26a–c The importance of diffuse functions has been investigated with respect to polarizibility and dipole and quadruple moments,21a–c relative,22a binding22b,c and reaction21c,22d energies, as well as electronic interactions.23

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

36
The effect of diffuse functions on vibrational frequencies,22b electron transfer,24 hyperfine splitting constants25 and conformational energy22d has also been studied.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

37
The necessity of diffuse functions for calculating electron affinity has especially been examined.21a,22b,24–26 The relation between geometry and diffuse functions has also been analyzed,21c,22b,25,26a although mostly for small and specific families of compounds.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

38
Generally speaking, in almost all cases, the introduction of diffuse functions into the calculation improved calculation results substantially.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

39
However, while the use of larger and/or more complex basis sets usually yields more accurate results, this is almost always accompanied by a significant increase in calculation effort and time.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

40
This is particularly evident for large molecules, such as PAHs.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

41
A calculation with diffuse functions for PAHs is difficult and often quite impractical, since these large molecules may contain a sizeable number of atoms.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

42
Such calculations require augmented computational resources, calculation times can be larger by a full order of magnitude, and failures in convergence are common.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

43
At the same time, the actual necessity of diffuse functions for these types of molecules can be questioned.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot4

44
The distribution of charge over a large spatial area reduces inter-electron repulsion.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

45
Additionally, charged PAHs may exhibit enhanced delocalization of π-electrons, for through this method, they attain aromatic stability.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

46
Thus, an examination of the actual contribution and necessity of diffuse functions for several specific cases has been undertaken.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

47
Is it possible that through size and delocalization PAH anions reduce inter-electron repulsion to such a degree that the use of diffuse functions is not necessary to accurately describe the system?

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

48
At the present time, the few DFT calculations performed on PAH anions have not extensively examined all parameters in this work.26,27

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot5

49
In this paper, we present calculated results regarding the above question.

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa2

Computational details

50
All computations were carried out using the GAUSSIAN '98 program package,28 with the exception of NBO analyses,29 which employed JAGUAR 4.0 software.30

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp1

51
All calculations of molecule parameters were carried out at a DFT level employing Becke's three-parameter hybrid density functional with the non-local correlation functional of Lee, Yang and Parr (B3LYP),31,32 and with Hartree–Fock methods.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met5

52
Four different basis sets were used: 6-31G*, 6-31+G*, 6-311G** and 6-311+G**;33,34 thus, each molecule was calculated with a total of eight different methods.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met6

53
All structures calculated were geometrically optimized within their highest symmetry point groups, and harmonic vibrational frequencies were computed to confirm that the stationary points correspond to minima.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met7

54
Only minimum energy structures were considered for NMR and NBO calculations.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj4

55
NMR chemical shifts were derived from additional single point calculations employing the GIAO35 method as implemented in GAUSSIAN '98.

Type: Experiment | Advantage: None | Novelty: None | ConceptID: Exp2

56
The isotropy values obtained in this way were subtracted from the respective isotropy values obtained for optimized TMS36 (see ESI) in order to obtain NMR chemical shifts.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met8

Methodology

57
Primary parameters evaluated in this work are geometry (bond lengths and intramolecular angles), total energy, 1H- and 13C-NMR chemical shifts and charge density (on carbons and protons, though the density on carbons are of main interest).

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj5

58
While other properties, some more sensitive to diffuse functions, such as electron affinities, dipole moments and polarizabilities could also be examined, we nevertheless chose to focus on parameters central to PAH anion research generally, and particularly to systems previously studied in our laboratory.37

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj5

59
Various aspects, such as size, conjugation ability, charge states and systems with or without five- and six-membered rings were sampled.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj6

60
For these goals, calculations were made on a large series of different anions (mostly PAHs), namely, methyl (1), ethyl (2), n-propyl (3), iso-propyl (4), n-butyl (5), tert-butyl (6), formate (7), acetate (8), indenyl (9), fluorenyl (10) and cyclopentadienyl (11) anions (Scheme 1); and the acenaphthalene (12), pentalene (13), pyracylene (14), pyrene (15), phenanthrene (16), anthracene (17), tetracene (18), fluoranthene (19), indenofluoranthene (20), and cyclooctatetraene (21) dianions (Scheme 2).

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj7

61
Various comparisons were made in order to determine the effect of the use of diffuse functions on the different parameters calculated.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met9

Results and discussion

62
Taking indenyl anion (9 and Scheme 3) as a general example, the results arising from calculations with diffuse functions are practically identical to those arising from calculations without these functions.38

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

63
The comparisons typically performed weigh parameters for the four calculations performed with diffuse functions (HF/6-31+G*, HF/6-311+G**, B3LYP/6-31+G*, B3LYP/6-311+G**) against the analogous ones without them.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met10

64
The largest difference relating to the molecular bond lengths, between results arising from the HF/6-31+G* calculation and those acquired from using the HF/6-31G* method is only 0.004 Å.38

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

65
This corresponds to only 0.28% of the respective bond length.39

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

66
For all other comparisons (i.e. between other basis sets and calculation methods), the difference is at most the same value.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res3

67
The largest difference in intramolecular angles (between the same two calculation methods and basis sets) was 0.194°, 0.15% of the respective angle.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

68
Only the difference between B3LYP/6-31+G* calculated values and those obtained from B3LYP/6-31G* is larger; 0.202°, also 0.15% of its respective angle.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

69
Examining the calculated total energy, one finds similar results: ΔE is 0.032 au between B3LYP/6-31+G* and B3LYP/6-31G*, and smaller still for all other three comparisons.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

70
In comparison, the differences relating to charge densities and particularly to NMR shifts were somewhat larger.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

71
The largest difference relating to charge density on carbons was 0.02 charge units (HF/6-31+G* vs. HF/6-31G*), equaling a 7.63% difference of charge density on the corresponding carbon atom.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

72
The largest differences for NMR shifts were 7.60 ppm for carbons (B3LYP/6-31+G* vs. B3LYP/6-31G*), and 0.70 ppm for protons (HF/6-311+G** vs. HF/6-311G**).40

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

73
On the whole, most differences appear to be quite small, sometimes exceptionally so, to a point where they are insignificant.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

74
As we shall see, this appears to be the case in most instances for calculated PAH anions in the present work.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con5

Geometry

75
By comparing the basis sets 6-31G* with 6-31+G* and 6-311G** with 6-311+G** for HF and B3LYP separately, and by plotting the maximum differences for each parameter against the number of atoms in the skeleton,41 one can obtain four different figures for each calculated parameter.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met11

76
Fig. 1 shows that the comparison between the B3LYP/6-311G** and 6-311+G** calculations,42 methods that differ only by a single diffuse function, show highly different behavior for four different types of anions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

77
Anions derived from regular saturated alkanes, such as the methyl and ethyl anions, have a fairly large difference for Δrmax.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

78
Thus, without the use of diffuse functions, calculated geometries of small anionic molecules with significant charge density show considerable errors, reinforcing Duke18 and Driessler's19 results. n-Propyl (3) and n-butyl (5) behave similarly.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con6

79
However, the alteration of these anions to iso-propyl (4) and tert-butyl (6) give rise to a considerable reduction in Δrmax.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

80
The average figures are approximately three-fold smaller than those for their n-alkyl analogues.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

81
The difference in connectivity of these sets of molecules is apparently significant to their stability.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con7

82
The ability for hyperconjugation in these anions apparently disperses the charge in a much more effective way than in the n-alkyl anions; thus, the effect of diffuse functions in the calculation becomes less noticeable.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con8

83
It is important to note that comparisons for all other three pairs of basis sets show similar results.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

84
The two models for resonance-stablized anions in this study, formate and acetate, indeed both show a much smaller Δrmax than for n-alkyls, even ones with the same number of atoms in their skeleton.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

85
It can be rationalized that the negative charge is much less localized in these resonance-stabled anions compared with those of the n-alkyls, where the charge resembles more of a point charge.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con9

86
Bond lengths in resonance stabilized molecules are therefore less susceptible to changes due to extra charge compared with n-alkyls, and parameters calculated with functions taking this charge into consideration are therefore less likely to be significantly different from those calculated without these functions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con9

87
For cyclopentadienyl anion (11) and cyclooctatetraene dianions43 (21), Δrmax drops off even further.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

88
For the PAHs, Δrmax is of the same order.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

89
This is understandable, as most PAHs discussed here contain both aromatic stabilization (similar to the resonance stability of formate (7) and acetate (8) anions) and a large number of atoms in their skeleton.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

90
The order of magnitude of maximum bond length difference obtained for the indenyl anion, the smallest PAH studied, is already extremely small, and repeats itself for almost all other PAHs whose geometrical values were calculated in this work.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

91
The only anomaly appears to be for the calculated differences of tetracene dianion (18), between 6-311G** and 6-311+G** in the Hartree–Fock method.38

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

92
This may be due to the known fact that bond lengths in neutral tetracene deviate largely from each other: the longest bond being 1.452 Å, the shortest, 1.367 Å.44

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con10

93
Bond lengths at the central ring of phenanthrene show similar magnitudes of Δr.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

94
Structural criteria are important in defining aromatic molecules.45

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

95
Nevertheless, neither the aromaticity of tetracene nor that of phenanthrene46 is questioned, and in any case, this aberration does not recur for the other parameters calculated for tetracene.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

96
Intramolecular angles show the same trend as that seen for bond lengths.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res20

97
Fig. 2 shows that the comparison between B3LYP/6-311G** and 6-311+G** methods.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

98
While methyl anion has an average Δαmax of about eight degrees, and the rest of the alkyls an average Δαmax of ∼4–5, there is a decline in these figures when moving to the iso-alkyl anions (ca.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

99
0.7–1.4°) and furthermore when proceeding to the resonance-stabilized anions of formate (7) and acetate (8) (ca.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

100
0.57 a degree and less).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

101
This interesting decline continues for the PAH anions, as no difference is greater than 0.43°.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res22

102
The acetate anion, containing only 4 atoms in its skeleton, shows a difference only slightly larger than that of pentalene dianion (13), the PAH with the largest Δαmax.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res23

103
After reaching a minimal aromatic41b hydrocarbon size (ca.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res24

104
5 carbons) the introduction of additional carbons into the skeleton or even additional rings, does not appear to have much effect on the difference, as long as conjugation remains possible.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res24

105
Similar conclusions to the fact that calculated geometry related parameters of PAH anions are not affected greatly by the addition of diffuse functions have been achieved before, most recently by Schaefer and Schleyer,26a although these conclusions have been made for smaller and more specific families of PAH anions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con11

106
Comparisons for all other pairs of basis sets showed similar results.38

Type: Result | Advantage: None | Novelty: None | ConceptID: Res25

107
No such correlations were found (for either bond lengths or intramolecular angles or any of the other parameters discussed in this work) when attempting to associate the differences with other factors, such as the total number of rings, the number of five-membered rings, or the ratio between the number of carbons to the number of rings.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res26

Energy

108
Energy is a clear case in which one cannot compare the absolute differences, as total energy varies considerably from molecule to molecule, especially when moving from small anions to PAHs, which have a very large number of wavefunctions.47

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac14

109
Using the B3LYP/6-311G** and B3LYP/6-311+G** methods, for instance, one calculates −39.829 au and −39.856 au for the methyl anion (1), respectively.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

110
The much larger hyperconjugated tert-butyl anion (6) has energies of −157.809 au and −157.836 au, calculated by the above two methods, respectively.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

111
Lastly, the even larger PAH fluoranthene dianion (19) has energies of −615.775 au and −615.802 au, respectively.38

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

112
The difference between the two calculation methods for all three molecules comes to approximately 0.027 au.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res27

113
However, one realizes immediately that this difference of 0.027 au does not carry the same significance for each anion.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res28

114
One cannot compare a difference of 0.027 au out of approximately −40 au with the same difference out of roughly −615 au.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res28

115
Therefore, in this case, the figures shown are those for relative ΔEvs. the number of atoms in the skeleton.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con12

116
Fig. 3 depicts the differences arising from the comparison between B3LYP/6-311G** and 6-311+G** methods.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res29

117
As for geometry, comparable trends are found for all comparisons38 pertaining to ΔE as well.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res29

118
It is important to note that unlike in the gas phase, where electrons of highly charged anions are considered to be unbound, in all calculations in this work, the electrons are restricted to be attached to the skeleton of the molecule.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

119
Effectively, this means that values obtained here (regardless of basis set used and the existence of diffuse functions), only result in an estimate for the ground-state energy.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con13

120
In other words, the values should most probably only be considered qualitatively.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con13

121
In this regard, and with respect to the historical difficulties reported,18–20 it is highly important to note that all total energies for PAHs turn out to be negative, as expected from experiment.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con14

122
While most studies in the past concluded that diffuse functions are critical to the improvement of calculated energies,22b–d these were based on knowledge mainly from atomic anions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con15

123
For PAHs, we conclude that the use of diffuse functions for calculating total energy improves the energy by only ca.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res30

124
0.01% or less.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res30

125
These trends are in good agreement with conclusions reached by Schleyer and Schaefer for the more specific family of cyclopentadiene-annulated PAHs.26a

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con16

NMR shifts

126
One of the main objectives of this work was to deduce conclusions regarding NMR shifts, as this is of utter importance and a main tool in examining aromaticity and aromatic-related properties in charged PAHs.48

Type: Goal | Advantage: None | Novelty: None | ConceptID: Goa3

127
Specifically, 13C-NMR shifts are of more interest to our study, as these shifts show an empirical correlation between the additional charge each carbon obtains upon reduction and the shift change of each carbon introduced due to charging of the molecule.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj8

128
NMR shifts, therefore, show clear relationships to various important parameters, such as charge distribution, anisotropy, aromaticity,49a,50 anti-aromaticity,49b–e,50 the HOMO–LUMO energy gap and paramagnetism,49b–e and are of major importance to organic and theoretical chemists.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj8

129
Fig. 4 shows the maximum differences, ΔδC max, between B3LYP/6-311G** and 6-311+G** methods vs. the number of atoms in the skeleton.41b,51 Here, the difference must be viewed as the absolute difference, as NMR shifts (relative to TMS) may differ considerably from solvent to solvent, and the choice of solvent is arbitrary.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

130
Furthermore, the choice of TMS as a standard is arbitrary as well.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met12

131
The above is especially true for 13C-NMR, where the typical field is much larger; aromatic carbons typically resonate at 110–150 ppm, while aliphatic ones show resonances mainly in the 10–30 ppm region.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

132
While PAH anions generally produce a difference of 12 ppm or smaller, the differences for the small alkyl anions are less, and PAH anions give differences often larger than those calculated for the n-alkyl or iso-alkyl carbanions.38

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

133
ΔδC max values typically in the order of 10 ppm surely are not negligible.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res31

134
Calculations utilizing B3LYP methods accentuate this fact.38

Type: Result | Advantage: None | Novelty: None | ConceptID: Res32

135
These findings are as expected, and agree with those of Jaszuński et al52. that the use of diffuse functions for NMR shieldings are less important; larger basis sets with more flexible core functions should be employed.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con17

136
Anthracene dianion (17) shows the largest difference, over 30 ppm in the 6-311G** vs. 6-311+G** comparison in the B3LYP method.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res33

137
This relatively large difference is an irregularity and is discussed later.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res33

138
In any case, this difference for the anthracene dianion (17) is far smaller in the other comparisons made.38

Type: Result | Advantage: None | Novelty: None | ConceptID: Res34

139
With respect to ΔδC max, there is no apparent decline in value in advancing from small aliphatic anions to π-conjugated systems.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res35

140
Since the only physical parameter easily obtainable for comparison with calculated data is NMR shifts, evaluation of these experimental values with respect to those calculated is obviously called for and necessary.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con18

141
Fig. 5 shows examples of correlations between experimental5313C-NMR values, and those calculated in this work with and without diffuse functions, for various PAH dianions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res36

142
While fairly small dissimilarities exist between the three graphs in each figure, the overall picture is clear: while some significant differences between calculated data with, and without, diffuse functions do occur, the pattern is replicated fairly well.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con19

143
The square of the correlation factor between these two sets of data54 is 0.965 for the acenaphthalene dianion (12), and even higher values for R2, of 0.999, 0.999 and 0.992 are obtained for the same comparison for the anions of fluorene (10), indene (9) and pyracylene (14),38,54 respectively.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res37

144
R2 is largely of the same order of magnitude for comparisons between calculated data (either with diffuse functions or without) and experimental values.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res38

145
R2 is often even better for comparisons between experimental values and calculations made without diffuse functions than with calculations employing these functions.38

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac16

146
This information clearly shows that while diffuse functions are essential in calculating 13C-NMR shifts, calculated NMR shifts without diffuse functions may provide for assistance in experimental assigning.48c

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con20

147
1H-NMR shifts show similar behavior with relation to the effect of diffuse functions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res39

148
Fig. 6 shows the ΔδH max difference between B3LYP/6-311G** and 6-311+G** calculated methods.41b,51 ΔδH max is frequently on the order of 1 ppm or more.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res39

149
This is even a more critical error than for ΔδC max.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res40

150
Here, too, ΔδH max for PAH anions and other, non-aromatic anions seem to be on the same order of magnitude, again agreeing with the results of Jaszuński et al.52

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con21

151
This trend, which appears to be a major distinction from the other figures in general and from ΔδC max in particular, presumably reflects the fact that peripheral protons' NMR resonances are affected in a crucial way by anisotropic effects.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con22

152
This is in sharp contrast to peripheral carbons, which are insensitive to anisotropic effects.55

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con23

153
These anisotropic effects may possibly explain the anomaly in anthracene, which, as well as many linear acenes, is customarily regarded as less stable than its angular counterpart.56

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con24

154
Randić suggested57 that this is due to a smaller number of (4n + 2) π-electron conjugated circuits possible in such species.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

155
The linear systems cause significant dilution of the aromatic sextet.56a Schleyer proposed58a that anthracene has only one resonant sextet located in the central ring.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

156
The NICS value for this ring is significantly larger than that of benzene; conversely, NICS values for the outer two rings are smaller than that of benzene.58

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

157
The works of Ligabue et al59. concerning density maps of anthracene, phenanthrene and triphenylene; and later on, of Ruiz-Morales60 on HOMO–LUMO gaps of PAHs, further supported Schleyer's conclusions.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

158
All of the above bring support to the fact that anthracene and anthracene dianion (17) may contain significant anisotropic effects.49b–e Thus, 1H-NMR shifts are most affected and, as mentioned above, slightly less for 13C-NMR.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con25

159
Non-magnetic related parameters, such as geometry (and energy to some extent), are hardly affected.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

160
Comparison to experimental38,54,611H-NMR values (Fig. 7) gives rise to a similar image.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res41

161
While R2 between a specific calculated data (either with diffuse functions or without) and experimental values varies from poor (indenyl, 9) through fair (acenaphthalenyl, 12) to quite good (fluorenyl, 10),62 correlation factors between calculated data obtained with, and without, diffuse functions are good throughout.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res41

162
R2 between these two sets of data is 0.985 for the acenaphthalene dianion (12) and 0.995 and 0.981 for the anions of fluorene (10) and indene (9), respectively.63

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

163
This further supports the supposition that although diffuse functions are critical in calculating 1H-NMR shifts, if calculation is meant only to assist in assignment, then one may possibly do so from calculated data excluding diffuse functions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con26

Charge density

164
For charge density, the trends exhibited for geometry and energy are reiterated.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res42

165
Fig. 8 shows Δρmax for the B3LYP/6-311G** and 6-311+G** comparison.41b,51,64,65 As in the previous cases mentioned, also here one can see the distinction between the π-conjugated system/PAH region and the alkyl anion region.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res43

166
Even the smaller PAHs have relatively smaller Δρmax than the alkyl anions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res44

167
Similar trends are found for all other comparisons.38

Type: Result | Advantage: None | Novelty: None | ConceptID: Res45

168
However, since far fewer points appear on these graphs than on the corresponding ones for geometry and energy,51,64 one should be cautious when drawing conclusions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res46

169
At any rate, NBO calculations apparently confirm the underlying working assumption; PAHs have a much smaller density of charge within their carbon-skeleton framework.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con27

170
Scheme 4 graphically shows that PAHs have a far less density of charge in their carbon skeleton due to their far larger size and π-conjugation ability.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res47

171
These findings clearly indicate that the contribution of diffuse functions to the calculation of charge density in PAH anions would be quite small.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res48

172
This contrasts the conclusions concerning small anions, as in the investigation of the hydride ion by Shore et al., who showed66 that without diffuse functions, the local exchange approximation of the ground state failed, giving an incorrect ground state, with partial delocalization of the electrons.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con28

173
Therefore, one can, albeit vigilantly as mentioned above, conclude that the inclusion of diffuse functions in calculations of charge density does not cause the results obtained without them to change significantly.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con29

General

174
From looking at all figures and data,38 one can see some general trends common to all cases:

Type: Result | Advantage: None | Novelty: None | ConceptID: Res49

175
(i) For almost all parameters calculated, the difference between larger basis sets (6-311G** and 6-311+G**) is less significant than the difference between smaller ones (6-31G* and 6-31+G*).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res50

176
This is understandable, as larger basis sets allow more flexibility in the calculation; consequently, allowing further flexibility using diffuse functions is of less importance in these cases.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con30

177
When charge delocalization is present, inter-electronic repulsion is already being dealt with by larger basis sets and polarization functions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con30

178
The exception to this case is NMR differences: in these instances the differences follow no specific trend.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res51

179
(ii) Generally, n-alkyl and iso-alkyl anions exhibit the property that differences between two basis sets are larger for a B3LYP calculation than for the analogous calculations using HF theory.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res52

180
This is in harmony with Truhlar's results22d on alkyls and other relatively small molecules.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con31

181
Diffuse functions appear to be of much more significance for DFT than for Hartree–Fock in these cases.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res53

182
For PAHs, however, these differences are, on the whole, much smaller.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res54

183
In fact, the differences between two B3LYP calculations are almost identical to those obtained from the same basis sets in HF.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res55

184
For a sizeable number of occurrences, the difference is actually smaller for B3LYP calculations than for those calculated in the HF method.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res56

185
(iii) Since the differences reported are absolute values, it is noteworthy to examine trends within the differences (not only the maximum differences).

Type: Method | Advantage: None | Novelty: New | ConceptID: Met13

186
It should be pointed out at the start that all trends reported now are essentially the same regardless of the difference in basis sets or calculation methods.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res57

187
The addition of diffuse functions does not appear to result in a trend concerning the sign of the difference, i.e. calculations including diffuse functions give larger values than calculations excluding them roughly the amount of instances than vice versa.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con32

188
In the specific case of Δr, PAH anions show a slight leaning towards negative differences, i.e. larger values for results obtained without diffuse functions than those obtained with them.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res58

189
In energy calculations, more than 95% of the cases show a negative difference for ΔE.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs7

190
Since the total energy is a negative value, this means that in almost all cases, calculations made with basis sets containing diffuse functions give a better estimate67 of the energy compared to calculations made with basis sets omitting them.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res59

191
This supports previous claims21a,22b,d,26a that diffuse functions are necessary for improving various energies calculated for anions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con33

192
Most molecules show an equal amount of negative and positive differences for ΔδC.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res60

193
However, PAHs show a slight bias towards positive differences, i.e. the 13C-NMR shifts obtained using diffuse functions are larger than those obtained without them.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res61

194
For 1H-NMR shifts, however, the picture is considerably different: the shifts are almostalways larger for calculations made using basis sets including diffuse functions compared with those obtained using basis sets excluding them.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res62

195
In other words, ΔδH is almost always positive.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res63

196
The picture vis-à-vis Δρ resembles that of ΔE.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res64

197
The use of basis sets containing diffuse functions almost always (>95% of values calculated) results in more negative values than using the same method and basis sets excluding them.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con34

Conclusions

198
Rightfully so, diffuse functions have widely been recognized for some time as a tool necessary for efficiently calculating various properties of anions.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot6

199
High charge density affects numerous properties, such as bond length, energy, and electron affinity.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac18

200
However, PAHs may use their size and, apparently more importantly, their π-system, for dispersion of charge over the carbon skeleton, reducing Coulomb repulsion and stabilizing themselves.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac19

201
Differences between using and not using diffuse functions while calculating parameters of geometry are considerably affected by the size of the molecule, but more so by the conjugational availability.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res65

202
We conclude that the use of diffuse functions for calculating geometrical parameters for PAH anions in general is unnecessary and does not improve the calculated results significantly.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con35

203
Energy calculations are affected in much the same way.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res66

204
PAH anions show little difference whether their energies were calculated including or excluding diffuse functions.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res67

205
Diffuse functions almost always improve the calculated energy for PAH anions, although trivially.67

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con36

206
It is therefore of reduced importance using these functions when calculating total energy or geometrical parameters for PAH anions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con37

207
The effect on NMR shifts is quite different.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res68

208
For the most part, the absolute differences, Δδ, are not insignificant, clearly demonstrating that diffuse functions are crucial in calculating NMR shifts, regardless of the nature of anion or nucleus in question.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con38

209
If actual absolute values are desired (for instance, if no experimental data is available), then accuracy is available only to a certain degree.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con39

210
If, however, the calculation is only meant to assist assignment, then, in most cases, regardless of whether 13C- or 1H-NMR shifts are being calculated, full assignment may be realized quite reliably through correlation between experimental and calculated values, without use of diffuse functions.

Type: Method | Advantage: None | Novelty: New | ConceptID: Met14

211
On the whole, charge density calculations show that Δρ is always smaller for PAH anions than for smaller, aliphatic ones.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res69

212
NBO calculations do appear to confirm the much lower charge density for PAH anions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con40

213
However, lack of results due to technical difficulties64 and for reasons stated in the computational details section51 prevent us from making unequivocal conclusions in this regard, and caution should be taken in attempting to extrapolate from the results presented.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con41

214
Some PAH anions exhibit irregularity regarding differences calculated; however, closer examination of these anomalies may provide insight as to why they are such, and chemical reasoning may elucidate aberrations.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con42

215
Irregularity of various points on such figures may thus possibly be a tool for ascertaining specific and distinct chemical properties of particular anions.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con43

216
Such comparisons may therefore assist in the realization of which properties are similar for PAHs and which are dissimilar.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con44

Supporting information available

217
Isotropy values obtained for optimized TMS using the various methods (1 table), tables of data and statistics for indenyl anion exemplifying methodology (2 tables), tables of full data for all anions calculated with all comparisons made (21 tables), schemes for all parameters set for anions excluding indenyl anion (20 schemes), and all three analogous figures not presented in paper for all properties (18 total) are available.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8