Polygonal Current Model: An Effective Quantifier of Aromaticity on the

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Polygonal Current Model: An Effective Quantifier of Aromaticity on the Magnetic Criterion Stefano Pelloni, and Paolo Lazzeretti J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp406348j • Publication Date (Web): 19 Aug 2013 Downloaded from http://pubs.acs.org on August 24, 2013

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The Journal of Physical Chemistry

Polygonal current model: an effective quantifier of aromaticity on the magnetic criterion Stefano Pelloni and Paolo Lazzeretti∗ Dipartimento di Chimica dell’Università degli Studi di Modena via Campi 183, 41100 Modena, Italy. August 19, 2013

∗

E-mail: [email protected]

ACS Paragon Plus Environment


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Abstract To explain peculiar effects of electron delocalization on the magnetic response of planar cyclic molecules, a basic model which accounts for their actual geometrical structure has been developed by integrating the differential Biot-Savart law. Such a model, based on a single polygonal circuit with ideal features, is shown to be applicable to electrically neutral or charged monocyclic compounds, as well as linear polycyclic condensed hydrocarbons. Two theoretical quantities, easily computed via quantum chemistry codes - the out-of-plane components of the magnetizability, ξ∥ , and the magnetic shielding σ∥ (h) of points P on the symmetry axis orthogonal to the molecular plane, at distance h from the center of mass -, are shown to be linearly connected, e.g., for monocyclic structures, via the relationship σ∥ (h) = ±(µ0 /2π)ξ∥ D(h), where D(h) is a simple function of geometrical parameters. Equations of this type are useful to rationalize scan profiles of magnetic shielding - and nucleus independent chemical shift - along the highest symmetry axis. For a regular polygon, D(h) depends approximately on the third inverse power of the distance d of the vertices from the center, ξ∥ is proportional to the area of the polygon, i.e., ≈ d2 , hence the shielding σ∥ (0) and the related nucleus independent chemical shift NICS∥ (0) are unsafe quantifiers of magnetotropicity: they are biased by a spurious geometrical dependence on d−1 , incorrectly exhalting them in cyclic systems with smaller size. A more reliable magnetotropicity measure for a cyclic compound, in the presence of a magnetic field Bext applied at right angles to the molecular plane, is defined within the polygonal current model by the current susceptibility, or current strength, ∂I/∂Bext = −ξ∥ /Aef f , expressed in nano ampère per tesla, where Aef f is a properly defined area enclosed with the polygonal circuit. An extended numerical test on a wide series of mono- and polycyclic compounds and a comparison with corresponding ab initio current susceptibilities prove the superior quality of this indicator over other commonly employed aromaticity/antiaromaticity benchmarks on the magnetic criterion.

2


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1

Introduction

It is generally assumed that delocalized π electrons are quite free to move. According to a widespread consensus, they sustain “ring currents” in conjugated planar cyclic molecules, usually referred to as diatropic, 1,2,3,4 e.g., the archetypal aromatic system benzene, 5,6,7 in the presence of an external magnetic field with flux density B ext ≡ B ∥ at right angles to the molecular plane. On the same ground it is assumed that the π electrons of anti-aromatic systems, e.g., cyclobutadiene and the flattened cyclo-octatetraene (COT) model molecule, support paratropic currents delocalized all over the carbon ring. 8,9 Borazine and boroxine are instead regarded as non-aromatic systems, on account of electron flow localized about the nuclei of the hexagonal skeleton, displayed in maps of current density 8,10 and quantified by calculated current strengths. 11,12 The induced π ring currents in benzene are characterized by a leap-frog effect in the vicinity of the carbon nuclei, that is, they behave as stationary waves. 13 The in-plane components JxB and JyB of the magnetic-field induced current density in xy planes orthogonal to B ∥ ≡ B z exhalt the out-of-plane component ξ∥ ≡ ξzz of the magnetizability, and cause a paramagnetic (diamagnetic) shift of proton shielding in benzene (COT) by depressing (enH H ≡ σ∥H of the σαβ tensor. The same hancing) the out-of-plane component σzz

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statements would apply for cyclic permutations of x, y, z for a different choice of coordinate system. The effect is reversed for virtual probes at points inside the ring current circuit, e.g., the center of mass (CM). To avoid diffuse misunderstanding, it should be borne in mind that only the out-of-plane components of magnetizability and shielding tensors are biased by the ring currents. Therefore, only these components should be considered for an assessment of electron current delocalization and magnetotropicity, that is, aromaticity on the magnetic criterion. 14 For that reason, downfield proton chemical shifts measured via nuclear magnetic resonance spectroscopy, H H H H H H δav = (σREF − σav )/(1 − σREF ) ≈ σREF − σav ,

are not reliable aromaticity indicators, since they are average quantities, measured with respect to a reference compound. The isotropic value H H H H H σav ≡ (1/3)σαα = (σxx + σyy + σzz )/3,

accounts for only one third of the contributions provided by π-ring currents, H H , and σyy and contains spurious information from the in-plane components σxx

which are determined by the mixing of π and σ electrons in the presence of magnetic fields with Bx and By components. A more detailed discussion can be found in previous papers. 15,16,17,18 4


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Delocalized σ-electron currents were also found for planar saturated molecules, e.g., H6 , the cyclic arrangement of three hydrogen molecules with D6h symmetry, 5,12,19 first studied by London in a seminal paper. 20 The calculated current strength is as big as 12.6 nA/T (nano ampère per tesla) for B ∥ perpendicular to the plane of the hydrogen nuclei. 12,19 Also cyclopropane sustains σ-electron currents, 21,22,23 for B ∥ perpendicular and B ⊥ parallel to the σh symmetry plane. Quite surprisingly, the total current strength, for a magnetic field B ⊥ applied in the direction of a C2 symmetry axis, is 15.7 nA/T, approximately 1.5 times larger than 10.2 nA/T calculated 24,23 for B ∥ . In fact, current density maps show an intense B ⊥ -induced flow, delocalized about the non-cyclic CH2 -CH2 moiety, which possibly provides a major contribution to the in-plane shielding component calculated at CM, CM σ⊥ = 50.9 ppm. This value is ≈ 18 ppm bigger than σ∥CM , the out-

of-plane component, and yields the dominant contribution to the average CM CM σav = (1/3)(2σ⊥ + σ∥CM ). 25,26 Such a result implies that the average nuCM cleus independent chemical shift (NICS), 27,28 defined as −σav = −44.9 ppm,

is an unsafe measure of magnetic aromaticity of cyclopropane. 21,22 These findings imply that a sound approach, useful to rationalize the phenomenology, and possibly aimed at defining aromaticity via magnetic criteria, should be based on reliable models of electron flow in molecules perturbed by a stationary magnetic field. 5



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A heuristic ring current model (RCM), based on the differential BiotSavart law (DBSL), assuming a perfectly circular, infinitely thin loop, 29,30 had been developed for cyclic conjugated sytems. 11,12,19 Previous papers report definitions of current susceptibility providing a benchmark of magnetotropicity and a relationship useful to rationalize σ∥ ≡ σzz along the loop axis of cyclic conjugated molecules (and scan profiles of NICS∥ ≡ −σ∥ investigated in the recent literature 31,32,33,34,35,36,37,38,39,40 ), in connection with ξ∥ , the “perpendicular” magnetizability of the loop. 11,12,19 The present paper sets up to enlarge on the earlier RCM proposal, 11,12,19 by taking into account the actual structure of planar conjugated molecules and ions, which are not characterized by a circular shape, see Section 2. A model for current flow in monocyclic systems, based on DBSL, 29,30 is presented in Section 2.1. It is extended in Section 2.2 to linear polycyclic aromatic hydrocarbons (PAH), which are intractable within the RCM. 11,12,19 The strength of delocalized π- and σ-electrons currents, providing a reliable measure of magnetotropicity, is estimated via a simple computational recipe in Section 2.3. Extended basis set calculations at the Hartree-Fock level of accuracy were carried out for a large series of neutral and charged molecules to estimate σ∥ and ξ∥ values needed to build up the current model. The details are outlined in Section 3 and concluding remarks are given in Section 4. Inexact assumptions of an earlier ARCS model 41,42,43 are discussed in the 6


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Appendix reported as Supplementary Information Available (SIA), in which correct procedures for testing quality of the model via logarithmic plots, are described.

2

A polygonal model for delocalized currents in planar conjugated molecules

The RCM described in previous papers 11,12,19 has been modified to account for the actual geometry of the closed path around which a circulating electric current flows, that is, for the form of an n-sided polygon typical of planar cyclic molecules. A general and simple representation, referred to as polygonal current model (PCM), is outlined hereafter.

2.1

PCM for monocyclic molecules

We consider one infinitely thin loop with Dnh symmetry, having the shape of a regular polygon with n sides, carrying a current of intensity I, induced by a spatially uniform external magnetic field B ext at right angles to it. Such a model is well-suited to describe delocalized σ-electron flow 20 in H6 , the cyclic system with D6h symmetry formed by three juxtaposed hydrogen molecules. It is assumed to have a correct asymptotic behavior for planar conjugated compounds, characterized by π-electron ring currents with the shape of two Waugh-Fessenden tori, 44 above and below the nodal σh symmetry plane. The current loop is supposed to be superconducting, 20 that is, the induced 7



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current I persists until the applied field is switched off. In Fig. 1, the isosceles triangle OQQ′ has base QQ′ and height OO′ , respectively of length 2L and s, coinciding with side and apothem of the polygon. Let us consider a point P on the Cn symmetry axis, at distance h from the center O, and R from the midpoint O′ of QQ′ . The direction of the current at points T in QO′ , at a distance l from O′ , with 0 ≤ l ≤ L in modulus, is specified by the local line element, i.e., the vector dl. The distance PT of P from these points is denoted by r, with R ≤ r ≤

√ L2 + R2 .

′

In Fig. 1, QO and QP correspond to the maximum l and r values. According to the DBSL, 29,30 an element of magnetic field induced at P is given by dB =

µ0 Idl × r . 4π r3

(1)

Its direction is specified by the right-hand curled fingers rule. The rectangular triangles PQO′ and PQ′ O′ have the same base and height, so that only the circuit segment QO′ = O′ Q′ is taken into account. Therefore, the contribution of the polygon side QQ′ to the magnetic field perpendicular to the plane of the PQQ′ triangle at point P, in modulus, is evaluated from µ0 IR B= 2π

∫

L

0

(l2

dl , + R2 )3/2

(2)

for a current flowing from Q to Q′ , using R = r sin ϕ and r2 = l2 + R2 . The integral is easily calculated by substituting l = R tan ζ, ζ = π/2 − ϕ, 8


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obtaining B(h) =

µ0 IL(s2 + h2 )−1/2 (L2 + s2 + h2 )−1/2 2π

(3)

for the magnetic field as a function of h. From Fig. 1 one has s = R sin θ = d cos α, L = d sin α and s2 + L2 = d2 , where d is the distance of a vertex from the polygon center and α = π/n, then the component B∥ = Bs/R along the Cn axis, induced by the current flowing in one polygon side, is

B∥ (h) =

µ0 IsL D(h), 2π

(4)

by defining D(h) =

[(

d2 + h2

)1/2 (

d2 cos2

)]−1 π + h2 . n

(5)

Since sL is the area of the triangle OQQ′ and A = nsL that of the polygon, the total magnetic field induced by the current flowing along the perimeter of a polygon with n sides is given by n contributions of type (4), that is B∥ (h) = ±

µ0 IA D(h), 2π

(6)

positive (negative) for a current flowing anticlockwise (clockwise). The component of induced magnetic field perpendicular to Cn at P is not taken into account, as the sum of the contributions from the polygon perimeter vanishes by symmetry.

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The magnetic dipole induced in the loop by the external magnetic field with modulus Bext , orthogonal to it, can be cast in the form m∥ = ±IA = ξ∥ Bext ,

(7)

where ξ∥ is the out-of-plane component of the magnetizability tensor ξαβ of the polygonal loop. m∥ is positive (negative) for a current flowing anticlockwise (clockwise) along the polygon sides. The perpendicular component of the induced magnetic field is written B∥ (h) = −σ∥ (h)Bext ,

(8)

allowing for the Ramsey definition of magnetic shielding. 45 Therefore, from Eqs. (6)-(8), the out-of-plane component of the magnetic shielding tensor at point P is σ∥ (h) = −

µ0 ξ∥ D(h). 2π

(9)

This perceptive relationship connects the shielding at point P with the magnetizability ξ∥ . If the PCM based on Eqs. (4)-(9) is used to rationalize the π-electron contribution to the magnetic properties of a conjugated molecule, ξ∥ and σ∥ (h) are consistently interpreted as corresponding π-contributions to the experimental tensor components. In other words, Eq. (9) is useful to investigate π-electron contributions to NICS∥ (h) = −σ∥ (h) scan profiles in monocyclic planar conjugated systems, 31,32,33,34,35,36,37,38,39,40 10


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for a range of h values larger than a lower bound, hmin , 11,12,19 along the Cn symmetry axis. The conceptual importance of this result cannot be overemphasized: if a (possibly refined) PCM model were extended to explain the magnetic response of total π- and σ-electron currents, induced in a conjugated molecule by a magnetic field orthogonal to the molecular plane, Eq. (9) could be used to predict physically meaningful σ∥ (h) values. 46 Since the out-of-plane component ξ∥ of the magnetizability tensor is an experimentally accessible quantity, 47 values of NICS∥ (h) would, at least in principle, be potentially measurable in a certain range of h. Assuming that ξ∥ is known, Eq. (9) can be solved for h = 0, obtaining σ∥ (0) = −

ξ∥ µ0 . 3 2π d cos2 (π/n)

(10)

Relationships (7) and (10) should be carefully considered if either ξ∥ or NICS∥ (0) ≡ −σ∥ (0) are adopted as magnetotropicity quantifiers. Both are functions of geometrical parameters of the current loop, which strongly determine their magnitude. The former depends on the area A = nd2 sin α cos α, see Eq. (7), the latter on the inverse of the distance d−1 , according to Eqs. (7) and (10). Therefore, it is easily achieved that circuits of different size, supporting currents with the same intensity I, will be characterized by values of ξ∥ or NICS∥ (0) which may be largely different from one another, as 11



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documented in Section 3. In such a case, neither ξ∥ nor NICS∥ (h) can provide physically meaningful measures of absolute magnetotropicity of planar conjugated systems. Nor are they suitable for grading relative aromaticities on the magnetic criterion in a series of compounds. For that purpose, more sophisticated theoretical tools, e.g., current density maps displaying shape and size of a current loop, are needed.

2.2

PCM for polycyclic molecules

Eqs. (6) defines the contribution of one side to the out-of-plane component of the induced magnetic field in a regular polygon. In a rectangular circuit, with base 2a, height 2b, and diagonal 2d, see Fig. 2, integration of the DBSL relationship, Eq. (1), gives, for a diatropic circulation, µ0 Iab π (a2 + h2 )(d2 + h2 )1/2 Iab µ0 − π (b2 + h2 )(d2 + h2 )1/2 µ0 = ξ∥ F (h)Bext , 4π

B∥ (h) = −

(11)

where F (h) =

d2 + 2h2 . ) ( (d2 + h2 )1/2 d2 sin2 α + h2 (d2 cos2 α + h2 )

(12)

In Eqs. (11) and (12), h is the height of a point P on the C2 axis orthogonal to the plane of the rectangular loop. Therefore, allowing for a procedure similar to that used to arrive at Eq. (9), the out-of-plane component of the 12


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magnetic shielding tensor of a rectangular circuit with semi-diagonal d is σ∥ (h) = −

2.3

µ0 ξ∥ F (h). 4π

(13)

Current strengths in cyclic molecules from PCM

From Eqs. (6)-(8) one finds σ∥ (h) = −

∂B∥ µ0 ∂I = − A D(h) , ∂Bext 2π ∂Bext

(14)

therefore, equating this relationship with Eq. (9), one arrives at the following definition of current susceptibility IB =

ξ∥ ∂I =− , ∂Bext A

(15)

which can be adopted as a measure of strength of the induced current density. The relationships (6), (9), and (15) hold in the limit of the one-loop polygonal model based on classical BSL. 29,30 It is not expected to work with acceptable accuracy for a planar cyclic molecule sustaining delocalized currents flowing in many concentric loops, which have different areas and are placed at different heights from the plane of the loop. Therefore it is expedient to guess an effective area Aef f , greater than A, on account of the fact that larger loops provide a greater contribution to the magnetizability. 12 By solving Eq. (10) for d, one can define an effective value [ ]1/3 ξ∥ µ0 def f = − 2π σ∥ (0) cos2 (π/n) 13


(16)


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and an effective area Aef f = nd2ef f sin α cos α

(17)

of the polygon, by a criterion analogous to that discussed for a circular loop, 12 so that Eq. (15) is replaced by B Ief f =

ξ∥ ∂I =− ∂Bext Aef f

(18)

as a more accurate measure of current strength. Using α = arctan(b/a) in Fig. 2, effective values [

def f

ξ∥ µ0 = − 4π σ∥ (0) sin2 α cos2 α

]1/3 ,

Aef f = 4d2ef f sin α cos α

(19)

are achieved for a rectangular circuit by the same reasoning behind Eq. (16). Therefore, Eq. (18) is used also for interpreting the magnetic response of linear PAH’s.

3

Calculations

Extended computational tests were made to value the PCM model described above. In particular, we checked whether it is suitable for an assessment of diatropicity/paratropicity, that is, aromaticity/antiaromaticity on the magnetic criterion, in a wide series of monocyclic, and a few linear polyciclic, systems. The magnetic properties needed to determine the parameters of the model were calculated at the coupled Hartree-Fock (CHF) level, within 14


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the procedure allowing for continuous translation of the origin of the current density-diamagnetic zero (CTOCD-DZ). 48,49,50,51,52 Large basis sets of Gaussian functions for H, B, C, N and O atoms developed ad hoc for magnetic properties, employed in previous studies, 11,12,22 were used. The aug-cc-pVTZ basis sets for Al, P, S and Si nuclei are from Woon and Dunning 53 and the basis set for Ga is taken from Wilson et al. 54 The same basis sets were also used to optimize molecular geometries (using Gaussian03 55 ) and to evaluate, via the SYSMO package, 56 σ- and π-electron contributions to the out-of-plane magnetizability, ξ∥ ≡ ξzz

1 ∂ = − ϵzβγ 2 ∂Bext

∫ JβB rγ d3 r,

(20)

and to the out-of-plane component of the magnetic shielding for points at distance R from the origin, µ0 ∂ σ∥ (R) ≡ σzz (R) = − ϵzβγ 4π ∂Bext

∫ JβB

Rγ − rγ 3 d r. |R − r|3

(21)

Tensor notation is employed in these definitions, e.g., ϵαβγ is the Levi-Civita third-rank pseudotensor, and the Einstein convention of summing over repeated Greek indices is in force. The Cn Hn aromatic systems with Dnh symmetry studied here are energetically stable. Jahn-Teller distortion cause symmetry breaking in the antiaromatics, 25 whose optimized geometries 55 are characterized by lower Abelian symmetries, which, however, does not limit applicability of PCM. 15



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All the CTOCD-DZ predictions of magnetizability presented here for center-symmetric cyclic molecules are invariant in a translation of the gauge origin. 48,49,57 CTOCD-DZ magnetic shielding tensors are origin independent for all molecular point group symmetries. At any rate, it should be recalled that the three components of the shielding (and NICS) tensor, at the center of mass of cyclic sistems with Dnh symmetry, are origin independent even if they are calculated by gaugeless basis sets within the common origin approach. 57 Thanks to the use of very large basis sets, calculated properties are of near Hartree-Fock quality. A further reliable benchmark of Hartree-Fock accuracy is given by sum rules for charge conservation and translational invariance. 58,59 Calculated values, in SI units, are reported in Table 1. Other details are given as SIA. Three parameters were analyzed for an assessment of the theoretical approach: • Approximate current strengths from Eqs. (16)-(18) were considered acceptable if they match corresponding ab initio estimates, 24,23,60,61,62 evaluated via a flux-integral B Iab initio

∫ =

J B · ds

(22)

of the quantum mechanical current density J B over a suitably defined 16


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“bond basin”. 22,23,62 • Values of ξ∥graph , i.e., the slope of the straight line, Eqs. (9) and (13), expressing σ∥ (h) as a function of the independent variables D(h), Eq. (5) and F (h), Eq. (12), were compared with the ab initio estimates computed from Eq. (20). The PCM is considered not applicable unless these values are sufficiently close to one another. • The functions D(h), Eq. (5), F (h), Eq. (12), and consequently σ∥ (h), Eqs. (9) and (13), tend to 0 for h → ∞, see Figs. 3 and 4. Therefore, on fitting computed ξ∥ and σ∥ (h) values by Eqs. (9) and (13), a Bravais correlation coefficient close to 1 and a vanishingly small intercept were assumed to probe the practicality of a PCM for a given compound. 63 The conclusions arrived at allowing for the theoretical predictions reported in Table 1 are the following: B • For monocyclic systems, the I B (Ief f ) current susceptibilities from Eq.

(15) (Eq. (18)) are usually larger than (nearly the same as) those from the circular RCM, 11,12 which implies that a polygonal model is sound, but it does not improve the quality of the theoretical estimates. Nonetheless, only a PCM is applicable to linear PAH’s and large graphene fragments with a polygonal shape. 64

17



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• For the series of Cn Hn monocyclic compounds, charged or neutral, usually considered aromatic on the (4n + 2) π-electron H¨ uckel rule, the trend of PCM current strengths is exactly the same as that observed for B B B the ab initio Iab initio estimates. Theoretical predictions of I ’s (Ief f ’s)

from Eq. (15) (Eq. (18)) are systematically larger (smaller) than correB sponding Iab initio ’s, which indicates that (i) a PCM is valid for these hyB B drocarbons, (ii) the current susceptibility Ief f is closer to Iab initio , and,

for that reason, more accurate than I B , (iii) the effective loop area, Eq. (17), is larger than the geometric A = nd2 sin α cos α, (iv) a value of the effective distance def f more accurate than that obtained from Eqs. (10) and (16) should be determined in future investigations aimed to B predict current strengths closer to Iab initio ’s, (v) rigorous criteria are

necessary to estimate physically meaningful def f values, 46 however Eq. (18) can be used for an educated guess. Allowing for the ab initio current susceptibilities from Eq. (22), one obtains ( d = − B

ξ∥ B I n sin α cos α

) 21 ,

(23)

which gives values bracketed by d and def f , e.g., 1.74, 1.67, 1.72, 1.83, − and 1.97 ˚ A, respectively for the π-isoelectronic subset C4 H−− 4 , C5 H5 , ++ ˚ C6 H6 , C7 H+ 7 , and C8 H8 , and 2.20, 2.31, and 2.46 A, respectively for − B B C8 H−− 8 , C9 H9 , C10 H10 . Calculated Iab initio and Ief f values steadily in-

18


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crease with n from 3 to 10, in the same order as calculated ξ∥ ’s. The latter could in fact be adopted to rank relative diatropicities at a qualitative level. On the other hand, ξ∥ ’s become larger and larger on increasing the loop area, as shown by Eq. (7), which makes them unsuitable to grade absolute diatropicities. Computed values of NICS∥ (0) = −σ∥ (0) ++ decrease in the 6-electron subset C6 H6 → C7 H+ 7 → C8 H8 , and in

the 10-electron subset C8 H−− → C9 H− 8 9 → C10 H10 on account of the d−1 dependence predicted by Eq. (10). These trends are just the opposite of those established via current susceptibilities, which implies that NICS∥ (0) = −σ∥ (0) is an unreliable measure of either relative or absolute diatropicities for Cn Hn monocyclic hydrocarbons. • Three six-membered compounds, C6 H6 , Si6 H6 and P6 are characterized B B by nearly the same π-electron current strengths, either Ief f or Iab initio , B and should be considered diatropic to the same extent. Both Ief f and B Iab initio calculated for N6 are smaller. For these compounds neither ξ∥

nor NICS∥ are acceptable quantifiers of aromaticity on the magnetic criterion: calculated values are strongly biased by geometrical parameters, d2 and d−1 respectively. The huge NICS∥ (0) = −41.4 ppm of N6 , the less diatropic ring within this set, gives striking evidence of intrinsic inadequacy. Earlier conclusions 65 should be revised in the light of the

19



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present findings. • Borazine B3 H6 N3 and boroxol B3 H3 O3 are non-aromatic on the curB rent susceptibility criterion, on account of the smallness of their Ief f,

whereas triazine C3 H3 N3 is somewhat less diatropic than benzene at variance with a previous claim. 65 • Strong discrepancies with the ARCS estimates 41,66 were found for cyclopentadienide anion C5 H− 5 , pyrrole C4 H5 N, furan C4 H4 O and thiophen C4 H4 S. The current strengths reported by Jusélius and Sundholm 41 are clearly in error (e.g., the diatropicity computed 41,66 for C5 H− 5 exceeds that of C6 H6 by a factor greater than 2, that of pyrrole is 4/3 that of benzene), due to incorrect assumptions and methodology criticized in the Appendix to the present paper. According to the current susceptibilities reported in Table 1 of the present work, the five-membered heterocyclic compounds are less diatropic than benzene, with diatropicity increasing in the order C4 H4 O → C4 H5 N → C4 H4 S. • Among the systems studied, the current loop of H6 , represented as a set of six 1s orbitals with D6h symmetry, is probably the one best suited to prove practicality of the PCM, since it fulfills the ideal singlecircuit hypothesis quite nicely. The hydrogen ring, in the presence of a magnetic field at right angles to the molecular plane, sustains a strong 20


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delocalized diamagnetic σ-electron current 5,20 exhibiting a small leapB B frog effect. 13 The current susceptibilities, Iab initio = 12.6 and Ief f = 13.1

nA/T, quite close to each other and larger than benzene’s corresponding values, 11.7 and 9.99 nA/T, provide credible magnetic aromaticity measures. Information from other magnetic criteria is contradictory: σ∥CM is 68.2 ppm, much larger than the π-contribution to σ∥CM in benzene (37.2 ppm), while ξ∥ is −59.6 × 10−29 JT−2 , approximately one third smaller than benzene’s. These conflicting indications are to be imputed essentially to the much shorter geometrical d (effective def f ) radius of the hydrogen ring: 0.98 (1.33) ˚ A in H6 vs 1.40 (1.86) ˚ A in benzene. In particular, NICS∥ = −σ∥CM overemphasizes the diatropic character of the former due to its spurious dependence on the distance of the hydrogen nuclei from the polygon center. Accordingly, calculated results provide an additional piece of evidence in favour of the PCM, and the current susceptibility is confirmed to be a preferable yardstick of magnetotropicity. • The peculiar character of cylopropane’s magnetic response has been recalled in the introductory Section 1. The unrealistic predictions of a PCM for delocalized σ-electron flow in this molecule (one order of magnitude difference between d and def f , large discrepancy between

21



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ξ∥calc and ξ∥graph ) are manifest signs of its impracticability. In fact, the quantum mechanical current density vector field looks like that of a long solenoid extending well beyond the hydrogen atoms, 21,22,23 which makes the PCM hypothesis of extra-thin conducting wire totally untenable. Therefore, an apparent lack of success of the model may suggest a rather different underlying physics, to be investigated by more sophisticated theories. B • The magnitude of the Ief f current susceptibilities calculated via the

PCM for antiaromatic hydrocarbons reported in Table 1 is systematically and grossly underestimated. This failure, far from constituting a breach of the model, yields a remarkable proof of its physical value and predictive ability: it implies that more refined theoretical tools are needed. In fact, I B ’s of antiaromatic systems in Table 1 are much B B smaller than the ab initio Iab initio ’s since the radius d of the param-

agnetic π-electron vortex about the Cn axis is shorter than the average distance d of the C nuclei from the center of mass, as demonstrated via the ab initio quantum mechanical current density maps of Fig. 5. • The rectangular PCM, Eqs. (11)-(13), was found quite useful to interpret magnetic properties of linear PAH’s. For naphthalene, the satisfactory agreement between the π-contributions to the out-of-plane 22


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magnetizability, ab initio ξ∥calc = −166 × 10−29 JT−2 and ξ∥graph = −191 × 10−29 JT−2 from the fitting displayed in Fig. 4, corroborates the hypothesis of electronic currents flowing with nearly constant intensity above and below the molecular C-skeleton, confirmed by the maps of Fig. 6. On the other hand, the striking discordance between ξ∥calc and ξ∥graph found for anthracene and tetracene does not mean that the rectangular current model is flawed with wrong assumptions. Just the other way round, it enabled us to discover the underlying phenomenology. This strong disagreement implies either that the intensity I of the delocalized π currents is not uniform along the peripheral Cloop, or that more than one current loop exists. In fact, the current density maps of Fig. 6 clearly demonstrate the presence of stronger currents about the central ring of C14 H10 and inner naphthalene moiety of C18 H12 . The modulus |J B | gets weaker about the carbon rings at opposite ends of these PAH’s, and a single loop PCM is not applicable. The existence of two current loops makes that the ab initio σ∥ (h)’s on the r.h.s. of Eq. (9) used in the fitting are larger than those predictable via the PCM. Consequently, the slope of the straight line is higher, i.e., ξ∥graph >> ξ∥calc , and the intercepts at the origin do not vanish. − − B • Ief f ’s calculated for three-membered inorganic cyclic ions Al3 and Ga3

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suggest that their diatropicity is comparable with that of C3 H+ 3 , whose central shielding σ∥CM is twice as big, since d and def f are much smaller for the cyclopropenyl cation. Also in this case, NICS∥ is an unsuitable standard of aromaticity on the magnetic criterion. • A similar judgment is reached by comparing the results obtained for some four-membered compounds. Calculated isotropic NICS(0)=4.0 ppm, NICS(1)=−1.5 ppm, and NICSzz (1) = −8.6 ppm of the tetra33 azadianaion N2− No sensible assessment of di4 have been reported.

atropicity is possible from these data. The π-contribution to σ∥CM calculated in this study is 35.1 ppm, ≈2 ppm smaller than that of benzene. The ab initio π-ring current susceptibility is fairly large, I B =9.4 nA/T, but smaller than benzene’s. These results would indicate that N2− 4 is strongly π-diatropic. The present calculations confirm the weak π-diatropism of Al2− 4 dianion, an “all-metal aromatic” comB pound. 33,61,67,65,68,69,70,71,72,73,74 The Iab initio value of 4.6 nA/T is lower

than 9.4 of N2− 4 .

24


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4

Concluding remarks and outlook

The polygonal current model outlined in Section 2, tested via near HartreeFock calculations on a large set of monocyclic and linear polycyclic compounds characterized by planar structure and delocalized π- and/or σ-electrons, was proved successful to interpret important aspects of their magnetic response. Two simple relationships were arrived at, (i) connecting the out-of-plane component of the magnetizability ξ∥ to the out-of-plane component of the magnetic shielding σ∥ (h) of points at distance h from the σh symmetry plane, along the Cn symmetry axis of a regular polygon with n sides, (ii) defining an B easy-to-calculate current susceptibility Ief f = −ξ∥ /Aef f , which depends on

some effective loop area Aef f , as a quantifier of aromaticity/antiaromaticity on the magnetic criterion. The former rationalizes NICS∥ (h) scan profiles and can be used to gauge practicality of the model by comparing the value of ξ∥ from a regression between σ∥ (h) and functions D(h), F (h) with the corresponding quantity computed ab initio. The latter was used to grade absolute and relative magnetotropicity of 35 aromatic, nonaromatic and antiaromatic systems. Theoretical results obtained unequivocally demonstrate that neither ξ∥ nor NICS∥ (0) are safe measures of magnetotropicity, since they are heavily

25



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biased by geometrical parameters: the former depends on the area enclosed within the current loop, the latter from the inverse of the effective distance of the nuclei at the vertices of a given polygon from its center. Wide eviB dence of higher reliability of Ief f was obtained for 4-, 5-, and 6-membered

compounds, having the same current strength but different geometry, and consequently quite different ξ∥ and NICS∥ (0). Crucial computer experiments based on the polygonal current model for C6 H6 , Si6 H6 , N6 , and P6 , and cyclic hydrocarbons Cn Hn leave no doubt on the inadequacy of NICS∥ (0) as a magnetotropicity measure. The heuristic power of PCM was demonstrated also for antiaromatic cyclic hydrocarbons, by proving that the paratropic π-electron vortex is restricted within an area smaller than that bound by the C perimeter. This feature is exactly the opposite of that observed for aromatics, in which the ring currents extend well beyond this perimeter. Whenever the predictions of the ultra-thin one-loop polygonal model appear unrealistic, for a system with delocalized electrons perturbed by a magnetic field, it means that the basic assumptions on which it relies, outlined in Section 2, are not applicable. This is the case of anthracene and tetracene molecules, characterized by the existence of two loops, and of cyclopropane, in which the delocalized σ-electron currents has the structure of a long solenoid extending beyond the plane of the hydrogen atoms. 26


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The polygonal current model can easily be applied to larger polycyclic condensed hydrocarbons. Extended applications to planar graphene sheets with polygonal shape are presently underway.

Acknowledgements The authors wish to thank Dr. G. Monaco and Prof. R. Zanasi for making their results available prior to publication. Financial support from the Italian MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) via PRIN 2009 scheme is gratefully acknowledged.

Supporting Information Available This section contains the Appendix ”Logarithmic plots and assessment of RCM and PCM - Graphs of Eqs. (9) and (13) used in the fittings”. Misconceptions flawing the ARCS approach 41,42,43 are pointed out. A large set of regression lines used to obtain information reported in Table 1 is reported. This information is available free of charge via the Internet at http://pubs.acs.org.

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Page 28 of 43

Table 1: Results of the calculation in SI units.†

Molecule

d

def f

σ∥CM

IB

B Iab initio

B Ief f

ξ∥calc

ξ∥graph

C3 H+ 3

0.78 1.73 −9.44

−12.7

14.6

11.9

3.99

2.43

C4 H++ 4

1.02 1.57 −13.7

−18.2

14.1

6.58

4.04

2.77

C4 H−− 4

1.02 1.99 −62.9

−61.0

31.8

30.2

10.4

7.92

C5 H− 5

1.20 1.86 −77.2

−79.8

36.9

22.6

11.6

9.43

C6 H6

1.40 1.86 −89.8

−98.0

37.2

17.6

11.7

9.99

C7 H+ 7

1.61 1.97

−111

−118

36.0

15.6

12.1

10.5

C8 H++ 8

1.81 2.09

−135

−142

34.6

14.6

12.3

10.9

C8 H−− 8

1.85 2.31

−267

−245

50.8

27.6

19.5

17.7

C9 H− 9

2.03 2.39

−306

−307

50.5

25.7

19.9

18.5

C10 H10

2.24 2.55

−358

−348

47.7

24.3

20.1

18.7

C3 H3 N3

1.31 1.72 −62.0

−60.3

32.4

13.9

−

8.05

B3 H 6 N 3

1.42 2.56 −28.2

−23.4

4.50

5.38

1.90

1.66

B3 H 3 O 3

1.36 3.17 −16.8

−13.9

1.40

3.50

1.60

0.64

C4 H4 O

1.15 1.75 −46.0

−43.8

26.3

14.6

−

6.33

C4 H5 N

1.16 1.75 −58.0

−59.3

32.9

18.2

−

7.93

C4 H4 S

1.16 1.82 −64.6

−61.2

32.9

20.3

−

8.23

Continued on the next page

28


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Table 1 – Continued from previous page Molecule

d

σ∥CM

IB

B Iab initio

B Ief f

def f

ξ∥calc

ξ∥graph

P6

2.08 2.57

−171

−179

26.7

15.2

11.7

9.91

N6

1.28 1.60 −63.0

−68.2

41.4

14.8

10.6

9.53

Si6 H6

2.21 2.83

−209

−207

24.5

16.5

11.8

10.0

Al− 3

1.50 3.63 −41.4

−54.9

6.92

14.2

−

2.42

Ga− 3

1.51 3.85 −49.0

−58.4

6.86

16.5

−

2.54

N−− 4

0.95 1.67 −41.2

−42.8

35.1

22.8

9.40

7.35

Al−− 4

1.82 3.07 −55.3

−73.3

7.65

8.35

4.60

2.94

Ga−− 4

1.84 3.66 −70.6

−71.2

5.78

10.5

−

2.64

O++ 4

0.90 1.41 −21.0

−21.1

29.9

13.1

−

5.27

C3 H6

0.86 2.73 −82.3

−65.3

32.4

85.0

10.2

8.50

H6

0.98 1.33 −59.6

−53.4

68.2

23.9

12.6

13.1

C4 H4

1.02 1.35

33.3

54.6

−55.8 −16.0

−15.6

−9.32

C5 H+ 5

1.21 1.40

98.1

147

−109 −28.3

−

−21.0

C7 H− 7

1.62 1.90

492

546

−177 −68.7

−

−49.93

C8 H8

1.83 2.41

163

205

−27.4 −17.2

−18.3

−9.95

C9 H+ 9

2.04 2.11

342

357

−82.6 −28.4

−

−26.6

C10 H8

2.83 2.91

−166

−191

−

11.4

36.7

12.1

Continued on the next page

29



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Page 30 of 43

Table 1 – Continued from previous page Molecule

†

d

σ∥CM

IB

B Iab initio

B Ief f

def f

ξ∥calc

ξ∥graph

C14 H10

3.88 3.64

−240

−323

44.3

11.9

−

13.5

C18 H12

5.04 4.46

−245

−388

38.7

9.04

−

11.5

In homonuclear systems, d denotes the actual distance of a nucleus from the polygon

center. In borazine and boroxine, d is the average distance of the skeleton nuclei from the center of mass. d and effective def f values, computed via Eqs. (16) and (19), are in m×10−10 . For all the compounds, except H6 and C3 H6 sustaining σ-electron currents, ξ∥calc ≡ ξ∥ from Eq. (20), in JT−2 × 10−29 , is the contribution of the π electrons to the out-of-plane component of the magnetizability calculated by Eq. (20) at the CTOCD-DZ2 CHF level of accuracy. 48,49,75 ξ∥graph is the out-of-plane component of the magnetizability tensor from the intercept of straight lines, Eqs. (9) and (13), Figs. 3 and 4. σ∥CM , in ppm, is the π-electron (σ-electron for H6 and cyclopropane C3 H6 ) contribution to the out-of-plane component of the magnetic shielding at the center of mass, calculated via B Eq. (21). The last three columns report estimates of current susceptibilities I B , Iab initio , 19 B B and (18), in nA/T. The Iab and Ief initio value for f , respectively from Eqs. (15), (22),

cyclopropane is from Ref. 22 .

30


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dB P

θ h R r

Q’

O’

α s

φ l

O dl

d

T Q

Figure 1: The element of magnetic field induced by a current I flowing in the direction of dl along the QQ′ polygon side, from the differential Biot-Savart law, Eq. (1).



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α b d

a

Figure 2: The rectangular circuit model for diatropic π-electron current density J B in anthracene, Eq. (11). The applied magnetic field is orthogonal to the plot plane, directed upward.


σ|| (h ) [ppm]

C 5H 5 11 10

f(x ) = 159.63 x + 0.13

9

2

R = 1.00

8 7 6 5 4 3 2 1 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

D (h ) [A−3]

C 6H 6 σ|| (h ) [ppm]

8 7

f ( x ) = 196.00 x− 0.03

6

2

R = 1.00

5 4 3 2 1 0 −1

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

D (h ) [A−3] +

σ|| (h ) [ppm]

C 7H 7 16 14

f (x) = 235.79 x + 0.04 2

12

R = 1.00

10 8 6 4 2 0

0

0.01

0.02

0.03

0.04

0.05

0.06

D (h ) [A−3] _

C 9H 9 σ|| (h ) [ppm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


|

Page 33 of 43

16 14

f (x)= 614.71 x+ 0.06 2

12

R = 1.00

10 8 6 4 2 0

0

0.005

0.01

0.015

0.02

0.025

D (h )[A−3]

Figure 3: The asymptotic behavior of the π-electron contribution to the outof-plane component of σ∥ (h) magnetic shielding for points on the Cn axis, as a function, Eq. (9), of D(h), Eq. (5), with h the distance from the center of aromatic Cn Hn compounds, for n=5, 6, 7 and 9. Points used in the fitting of Eq. (9) are represented by crosses. Non-linear behavior occurs in the proximity of the polygon center, on the right of the graphs.



Page 34 of 43

40 35

f (x ) = 190.61 x− 0.30 R2 = 1.00

30 25 20 15 10 5 0 −5

0

0.05

0.1

0.15

0.2

F (h ) [A−3] σ || (h ) [ppm]

C 14H 10 14 12

f (x )= 323.02 x− 0.78 R2 = 1.00

10 8 6 4 2 0 −2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

F (h )[A ] −3

C 18 H σ || (h ) [ppm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

σ|| (h ) [ppm]

C 10H8

12

6

f (x ) = 387.74 x− 0.18 R = 1.00

5 4 3 2 1 0 −1

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

F (h )[A−3]

Figure 4: The asymptotic behavior of the π-electron contribution to the out-of-plane component of σ∥ (h) magnetic shielding for points on the C2 axis perpendicular to the molecular plane in naphthalene, anthracene and tetracene, as a function, Eq. (13), of F (h), Eq. (12), with h the distance h from the center of mass. Points used in the fitting of Eq. (13) are represented by crosses. Non-linear behavior occurs in the proximity of the polygon center, on the right of the graphs.


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Figure 5: Quantum mechanical J B current density maps for planar aromatic and flattened antiaromatic conjugated systems. The size of the bulging arrows is proportional to the modulus |J B |. On the left (right) the diamagnetic (paramagnetic) π-electron vortex of the current density, induced by a magnetic field applied at right angles to the σh symmetry plane of C4 H2− 4 (C4 H4 ), − + + − 2− − + C5 H5 (C5 H5 ), C7 H7 (C7 H7 ), C8 H8 (C8 H8 ), C9 H9 (C9 H9 ), directed upward. Nuclei are identified by circular symbols: red (C), grey (H). Only the Waugh-Fessenden 44 torus below σh is displayed, the same pattern would be observed above the molecular plane. The effective Aef f , Eq. (17), is smaller (bigger) than geometric area for the antiaromatic (aromatic) compound, as implied by the theoretical current strengths displayed in Table 1. ACS Paragon Plus Environment


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Figure 6: The Waugh-Fessenden current densities 44 below the molecular plane of benzene, naphthalene, anthracene and tetracene induced by a magnetic field orthogonal to the σh symmetry plane and directed upward. For all the molecules the size of the bulging arrows is proportional to the modulus |J B |, slightly higher in the proximity of the C nuclei, but nearly constant all over the carbon skeleton of C6 H6 and C10 H8 . For anthracene and tetracene, the maps show two circuits of delocalized π-electron flow (i) about the molecular C perimeter (ii) about the central ring in C14 H10 and the central naphthalene fragment in C18 H12 , which explains why a single-loop current model predicts |ξ∥graph | >> |ξ∥calc |, see Table 1.


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[6] von Ragué Schleyer, P. Chem. Rev. 2001, 101, 1115–1117, and articles therein.

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TOC graphic

α b d

a

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