Assessment of Ring Current Models for Monocycles - The Journal of

Article pubs.acs.org/JPCA

Assessment of Ring Current Models for Monocycles Guglielmo Monaco* and Riccardo Zanasi Dipartimento di Chimica e Biologia, Universitá degli Studi di Salerno, Via Giovanni Paolo II, 132, Fisciano 84084 Salerno, Italy S Supporting Information *

ABSTRACT: Interpretation of both measured and computed values of chemical shift in (poly)cyclic molecules is widely based on ring current models (RCMs). Few improvements have been considered to date for the reference RCM, consisting of an infinitely thin circular loop of current (ICLOC). In this paper, six analytical RCMs (three of which are proposed for the first time) have been discussed, and they have been graded by comparing their ability to reproduce the π and σ contributions to the ab initio ring current strengths and to the scans of the parallel component of the magnetic shielding (σ∥ = −NICS∥) for a set of 33 organic and inorganic monocycles. For π currents, two vertically displaced ICLOCs (ICLOC2 model) are the preferred choice to have a very good reproduction of the scans, while the toroidal widening of the loops, proposed long ago by Farnum and Wilcox, is unable to give a significant improvement. For σ currents, the best model is the novel ICLOC2C, formed by two concentric ICLOCs. An off-plane extremum in the NICS∥ scan is not a general indication for the presence or absence of π-aromaticity. In agreement with the models proposed here, such an extremum is absent in the NICS∥,π scans of large rings; however, it characterizes most of the NICS∥,π scans of small rings and can also appear in NICS∥,σ scans.

■

with its many variants18,19 is still the most popular indicator of aromaticity, though often criticized.8,12,20−31 To overcome limitations of the NICS in the retrieval of the magnetotropicity, one can consider NICS scans, once again first proposed for fullerenes32 and then applied to a wide set of molecules23,33−35 (an azimuthal, rather than an out-of-plane scan of the magnetic shielding, was reported much earlier36). Either computed in a single point or along a line, the NICS is an indirect measure of current flow; a more direct indicator is the current susceptibility29,30,37−40 (also referred to as current strength),38,40,41 which can be computed in polycyclics42,43 to get quantitative measures of local aromaticity and is a modern version of previous well-established procedures based on tight-binding approaches.44 The interpretation of these novel computational data generally refers to ring current models (RCMs),45,46 which arose long ago in the analysis of the exaltation of magnetic susceptibilities of aromatic compounds47 and later were reestablished by the analysis of the peculiar values of proton chemical shifts in (poly)cyclic compounds.48−54 The analysis of complicated data in terms of simple models is commonplace in chemistry; indeed, many chemical concepts like bond, aromaticity, atomic charge, etc., have no immediate definition at the fundamental quantum-mechanical level but require the adoption of sound models.55 The ring current can be considered one such concept: the richness of detail contained in

INTRODUCTION The development of effective theoretical methods1−6 for the study of magnetic properties not only allowed a (semi)quantitative reproduction of experimental data (magnetizabilities and magnetic shieldings at nuclei) but also gave access to new qualitative and quantitative data. On one hand, the induced current density field J, which nowadays can be computed pointwise in three-dimensional space, has a rich topology,1 which has been used as a source of chemical information.7,8 On the other hand, computed magnetic properties have been increasingly used as benchmarks for delocalization. The use of computed properties in place of the experimental data offered two novel features, in addition to the obvious possibility of having information whenever the experiments cannot be easily performed. First, the availability of the three-dimensional J field allowed the definition of the overall magnetic properties in terms of the magnetizability density and the nuclear magnetic shielding density.9 Within the quantum theory of atoms in molecules, this feature has been used to obtain intrabasin and interbasin contributions to the magnetizability,2 and the latter are indicative of current delocalization;2,10−12 moreover, the nuclear shielding density also bears signatures of aromaticity.13 The second novel feature offered by computations is the possibility of obtaining the magnetic shielding at any point in space. The tight-binding14 and ab initio15,16 computations of the magnetic shielding at the atomfree center of fullerenes are likely among the first of such applications. Later, Schleyer computed the magnetic shielding at atom-free ring centers and changed its sign, introducing the nucleus-independent chemical shift (NICS),17 which, together © 2014 American Chemical Society

Received: November 14, 2013 Revised: February 6, 2014 Published: February 6, 2014 1673

dx.doi.org/10.1021/jp411194m | J. Phys. Chem. A 2014, 118, 1673−1683

The Journal of Physical Chemistry A

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the current density field cannot be condensed in an RCM defined in terms of a finite number of ring currents. Nevertheless, the loss of information caused by the adoption of an RCM can still be compatible with a good description of certain molecular properties, just as models with finite numbers of charges can well-reproduce molecular electric properties.56−59 The capability of different RCMs to account for a given set of data can be investigated to grade them and to select the simplest and most effective as a sound interpretative tool. Only a few RCMs have been considered in the literature, i.e., a single loop and two out-of-plane loops either infinitely thin31,60−63 or of a toroidal shape.52 After completion of our work, a fourth RCM, called the polygonal circuit model, has also been published.31 The purpose of this paper is the investigation of six RCMs, three of which are novel, and the assessment of their performance in describing the ab initio molecular magnetic response for a wide class of monocycles. Target properties for such an assessment will be axial scans of a component of the magnetic shielding tensor (corresponding to the NICSzz scans34,35) and the ring current strengths,5,38,38−41,64 two of the most used indicators of magnetic aromaticity. Definition of the RCMs. In the presence of an external magnetic field, the electrons of a molecule generate an induced current density J, which in turn generates an induced magnetic field, whose value is given, in both classical electrodynamics and quantum mechanics,65,66 by the Biot−Savart law B(r0) =

μ0

(r − r )

∫ |r − r 0|3 4π 0

Figure 1. Acronyms of the six nested RCMs considered in this paper. Models with 4, 3, and 2 parameters occupy areas of increasing darkness of gray to indicate that models with fewer parameters can be obtained as limiting cases of one or more models with more parameters.

suggested by the reports of contra-rotating currents reported many times for monocycles using σ-electron39 and allelectron2,37 calculations. The six RCMs share a circular symmetry, have a number of parameters not exceeding four, and are nested: their simpler versions are obtained by giving appropriate values to some parameters of the more general ones (Figure 1), a useful property in assessing the usefulness of refinements of the simpler RCMs. In the following subsections we will give the expressions for the RCMs. ICLOC Model. A magnetic field of modulus B∥, perpendicular to an infinitely thin circular loop of current of radius s, induces an electronic circulation, whose signed intensity I ̅ is, within the linear approximation, I ̅ = I BB∥. For this system, the integration of eq 1 gives the magnetic shielding at height h, i.e., the negative of the induced magnetic field for a unitary inducing field B∥, as

× J d3r (1)

where r0 is the point where the induced magnetic field is computed, r the position of the volume element d3r with respect to the origin, and the vacuum permeability μ0 = 4π × 10−7 N A−2 = 4π × 10−6 Å (nA)−1T. Knowledge of the induced magnetic field allows the computation of magnetizability and magnetic shielding.66 Magnetic properties, and their parent current densities, can be computed from first principles according to different basic models (e.g., neglect of intermolecular interactions, vibrational effects, correlation effects, finite basis set sizes). When applied to cyclic systems, such models have been previously called RCMs.13,67 Here we use the term RCM in a much more limited meaning. Considering a surface element whose normal da lies in the anticlockwise direction (as seen form the positive end of the applied field) of a circular loop perpendicular to the ring axis, dI ̅ = J · da is an infinitesimal signed current flowing tangentially around the loop. The signed current strength conveniently keeps trace of the handedness of the circulation: a positive (negative) signed current strength I ̅ indicates an anticlockwise (clockwise) current, i.e., a paratropic (diatropic) current producing a magnetic field parallel (antiparallel) to the inducing magnetic field. An RCM will be determined by a small number of loops, each with a well-defined shape, and sustaining a finite current. The most widely studied RCM consists of a single infinitely thin circular loop of current (ICLOC) in the molecular plane; few papers have considered improved RCMs.31,49,52,63 To improve ICLOC, one can conceive more general RCMs, differing either in the number and the location of loops or in their shapes. The six RCMs to be discussed here are sketched in Figure 1; the RCMs already considered in the literature, either analytically or numerically, are ICLOC, ICLOC2, and TCLOC2. TCLOC and ETCLOC are readily conceived as coarse-grained pictures of TCLOC2, whereas ICLOC2C is

σ (h) = −

μ0 2

IB

s2 (s 2 + h2)3/2

(2)

Two ICLOCs: ICLOC2 and ICLOC2C Models. The shape of π orbitals readily suggests the consideration of two identical ICLOCs displaced by ± z from the molecular plane.49,63 Each of the ICLOCs will contribute half of the total current strength I B and magnetizability; thus σ (h) = −

μ0 4

2

IB ∑ p=1

s2 {s 2 + [h + ( −1) p z]2 }3/2

(3)

This RCM will be referred to as ICLOC2. Rather than displaced along the vertical axis, the two loops can be concentric. In this case the two loops are not symmetrical and one can have as many as four parameters: σ (h) = −

⎤ μ0 ⎡ B s2 s2 ⎢ I1 2 1 2 3/2 + I2B 2 2 2 3/2 ⎥ 2 ⎣ (s1 + h ) (s 2 + h ) ⎦ B

I1B

(4)

I2B.

The total current strength will be I = + This model will be referred to as ICLOC2C. Finite Width: ETCLOC, TCLOC, and TCLOC2Models. We will now consider a bundle of closely spaced ICLOCs, as initially proposed by Farnum and Wilcox.52 Defining the function F (h , s , z ) = 1674

s2 (s 2 + (h − z)2 )3/2

(5)



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Table 1. Estimates of Ab Initio π Signed Current Strengths IπB According to the Six RCMs species

ICLOC

ICLOC2

ICLOC2C

TCLOC

ETCLOC

TCLOC2

IσB (nA T−1)

H6 C3H3+ C3H6 C4H42− C4H42− C4H4 C5H5− C6H6 C6H3+ C6I6 C6I62+ C7H7+ C8H82+ C8H C8H8 C9H9− C10H10 furan pyrrole thiophene N42− N6 O3 O42+ borazine boroxine Al3− Al42− Si6H6 P6 S6 Ga3− Ga42−

0.0 −4.0 −0.2 −4.5 −10.6 17.6 −12.3 −12.6 −7.2 −10.8 −10.8 −12.8 −12.9 −20.4 19.5 −20.9 −21.0 −8.0 −10.2 −9.5 −9.9 −12.0 −0.6 −6.9 −1.9 −0.6 −4.1 −5.0 −13.2 −12.7 0.9 −4.2 −4.7

0.0 −4.0 −0.6 −4.1 −10.2 16.0 −11.7 −12.0 −6.8 −10.3 −10.3 −12.2 −12.4 −19.7 18.6 −20.3 −20.5 −7.6 −9.7 −8.9 −9.3 −11.4 −0.9 −6.6 −1.9 −0.8 −4.2 −4.6 −12.3 −12.0 0.8 −4.4 −4.5

0.0 −4.2 −0.4 −4.3 −10.5 17.0 −12.0 −12.3 −7.0 −10.5 −10.5 −12.5 −12.6 −20.0 19.0 −20.5 −20.7 −7.8 −10.0 −9.2 −9.6 −11.7 −0.8 −6.7 −2.0 −0.9 −4.5 −4.9 −12.7 −12.3 0.6 −4.7 −4.8

0.0 −4.2 −0.5 −4.3 −10.4 16.8 −11.9 −12.3 −7.0 −10.5 −10.5 −12.4 −12.6 −19.9 18.9 −20.4 −20.6 −7.7 −9.9 −9.1 −9.5 −11.6 −0.9 −6.7 −2.0 −0.9 −4.5 −4.9 −12.6 −12.2 0.8 −4.7 −4.8

0.0 −4.1 −0.7 −4.2 −10.3 16.5 −11.8 −12.1 −6.9 −10.4 −10.4 −12.3 −12.5 −19.8 18.8 −20.3 −20.5 −7.7 −9.8 −9.0 −9.4 −11.5 −1.0 −6.6 −1.9 −0.8 −4.4 −4.8 −12.4 −12.1 0.8 −4.6 −4.8

0.0 −4.1 −0.6 −4.1 −10.4 16.0 −11.8 −12.1 −6.8 −10.4 −10.4 −12.2 −12.5 −19.9 18.6 −20.4 −20.5 −7.6 −9.8 −9.0 −9.5 −11.4 −0.9 −6.6 −1.8 −0.8 −4.4 −4.8 −12.5 −12.1 0.8 −4.6 −4.7

0.0 −4.0 −0.5 −4.0 −10.2 15.6 −11.6 −11.9 −8.5 −9.9 −9.8 −12.1 −12.3 −19.6 18.4 −19.9 −20.1 −7.4 −9.5 −8.7 −9.4 −11.4 −0.9 −6.6 −1.9 −0.8 −4.4 −4.6 −12.1 −11.8 0.8 −4.5 −4.6

SD av % error

0.73 9.5

0.37 3.1

0.50 6.2

0.47 4.7

μ0 2

F (h , s , z )d I B

(6)

σ (h) = −

where d I B is the infinitesimal signed current strength passing through an infinitesimal section dsdz. For the sake of simplicity, we assume that the current density profile varies independently on the s and z coordinates, and we can write d I B = d I Bι(s)dsκ(z) dz, where ι(s) and κ(z) are probability density functions, and thus ∞ everywhere positive and normalized, ∫ ∞ 0 ι(s) ds = ∫ −∞κ(z) dz = 1, allowing compact expressions for average functions, e.g. ⟨f (s)⟩ =

∫0

⎛

⎜

⎛ ∂ 2F ⎞ +⎜ ⎟ ⎝ ∂s∂z ⎠⟨s⟩, ⟨z⟩ +

σ (h) = −

2

I

B

∞

∫−∞ ∫0

(7)

∫ ∞

F(h , s , z)ι(s)dsκ(z)dz

⎞

⎟

∫ (z − ⟨z⟩)κ(z)dz

⟨s⟩, ⟨z⟩

By integration of eq 6 μ0

⎡ μ0 I B ⎢ ⎛ ∂F ⎞ F(h , ⟨s⟩, ⟨z⟩) + ⎜ ⎟ ⎝ ∂s ⎠⟨s⟩, ⟨z⟩ 2 ⎢⎣

∫ (s − ⟨s⟩)ι(s)ds + ⎝ ∂∂Fz ⎠

∞

f (s)ι(s)ds

0.39 3.7

the variables s and z. When the expansion is limited to the second order, we have

the infinitesimal contribution to the chemical shift from each of the ICLOCs is compactly written as dσ (h) = −

0.41 4.3

(8)

∫ (s − ⟨s⟩)ι(s)ds ∫ (z − ⟨z⟩)κ(z)dz

1 ⎛ ∂ 2F ⎞ ⎜ ⎟ 2 ⎝ ∂s 2 ⎠⟨s⟩, ⟨z⟩

⎛

2

⎞

∫ (s − ⟨s⟩)2 ι(s)ds + 12 ⎜⎝ ∂∂zF2 ⎟⎠

⟨s⟩, ⟨z⟩

⎤ 2 (z − ⟨z⟩) κ(z)dz ⎥ ⎥ ⎦

Perusal of eq 7 in the above expression shows vanishing of linear and bilinear terms. Calculation of derivatives for the remaining terms leads to

To estimate the integral in eq 8, we can write a Taylor expansion for F(h, s, z) around the average values ⟨s⟩ and ⟨z⟩ of 1675



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Figure 2. Top-left (top-right): π (σ) contributions to the magnetic shielding component σ∥, perpendicular to the molecular plane of benzene, computed ab initio (small dots) and fitted according to 1-loop (top) and 2-loops (middle) RCMs. Bottom: π (left panel) and σ (right panel) current density maps superposed on the planes used to integrate the current strengths. Contour lines in the integration planes limit domains D2, D16, D128, and D1024, where domain Dn contains data with modulus higher than jmax/n, where jmax is the maximum absolute value of the current in the plane. For the sake of clarity, only the larger arrows, falling in the D2 domain, have been plotted.

σ (h) = −

μ0 I B ⎡ ⟨s⟩2 + ρ2 ⎢ 2 ⎣ (⟨s⟩2 + (h − ⟨z⟩)2 )3/2 2

2

⟨s 2⟩ = ⟨s⟩2 + ρ2 =

⟨s⟩ (ζ + 5ρ ) 3 2 (⟨s⟩2 + (h − ⟨z⟩)2 )5/2

+

15 (h − ⟨z⟩)2 ⟨s⟩2 ζ 2 + ⟨s⟩4 ρ2 ⎤ ⎥ 2 (⟨s⟩2 + (h − ⟨z⟩)2 )7/2 ⎦

(9)

where the horizontal and vertical variances are ρ2 = ⟨s⟩2 − ⟨s⟩2

σ (h) = − (10)

and ζ = ⟨z ⟩ − ⟨z⟩ , respectively. For future convenience we introduce the positive ratio 2

α=

2

+

2

ρ2 ≤1 ⟨s 2⟩

(12)

where the last expression is valid for small α. Equation 9 requires as many as 5 parameters ( I B, ⟨s⟩, ⟨z⟩, ρ, ζ). Simpler forms can be obtained fixing one or two of them. If ⟨z⟩ = 0,

2

−

⟨s⟩2 ≃ ⟨s⟩2 (1 + α) (1 − α)

μ0 I B ⎡ ⟨s⟩2 + ρ2 3 ⟨s⟩2 (ζ 2 + 5ρ2 ) ⎢ − 2 2 3/2 2 ⎣ (⟨s⟩ + h ) 2 (⟨s⟩2 + h2)5/2

15 h2⟨s⟩2 ζ 2 + ⟨s⟩4 ρ2 ⎤ ⎥ 2 (⟨s⟩2 + h2)7/2 ⎦

(13)

Although the assumption of ⟨z⟩ = 0 should be most appropriate for σ currents, it can also be considered in a coarse-grained description of π currents. Equation 13 will be referred to as elliptic toroidal circular loop of current

(11)

which allows writing the average square radius as 1676



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(ETCLOC). Its further simplified version with ζ2 = ρ2 = α⟨s2⟩, to be called toroidal circular loop of current (TCLOC), is σ (h) = − +

=−

magnetizabilities, and current strengths.5,39,64 The latter, in particular, are obtained by integrating the component of the current density which is perpendicular to a plane bisecting a bond of the monocycle. Integration is carried out around each maximum of the component J⊥ perpendicular to the bisecting plane. Typical π (σ) currents, as found in benzene, showing two vertically (horizontally) displaced homotropic (heterotropic) maxima, are displaced at the bottom left (bottom right) of Figure 2.39 Similar plots for all systems studied here are reported in the Supporting Information. Nonlinear optimizations have been performed in MATLAB7 (lsqnonlin code of the Optimization Toolbox). Five or more guesses of the parameters have been used in the optimizations, possibly taking into account the results of the optimizations for the nested models.

μ0 I B ⎡ ⟨s⟩2 + ρ2 9⟨s⟩2 ρ2 ⎢ − 2 ⎣ (⟨s⟩2 + h2)3/2 (⟨s⟩2 + h2)5/2

15 h2⟨s⟩2 ρ2 + ⟨s⟩4 ρ2 ⎤ ⎥ 2 (⟨s⟩2 + h2)7/2 ⎦ ⎛ ⟨s⟩2 ⎞ 1 3 α − 1 ⎜ ⎟ 2π (⟨s⟩2 + h2)3/2 ⎝ 2 ⟨s⟩2 + h2 ⎠

(14)

μ0 ξ

(15)

where we have considered that, according to the independence of s and z variables and to eq 7 ξ = I Bπ

∫0

∞

s 2ι(s)ds = I Bπ ⟨s 2⟩

■

RESULTS AND DISCUSSION The wide spectrum of systems considered in this paper has been chosen to thoroughly test the RCMs, rather than for discussing the individual values of current strength, which are partly known from previous works29,39 and are broadly in line with general trends reported elsewhere.29,37,39,40 Detailed discussion of the ab initio current strength certainly provides useful information (see refs 29, 30, 40), but it is beyond the scope of the present paper. General Trends of the Dissected NICS Plots. The scan of magnetic shielding σ = −NICS along the axis perpendicular to a ring has been used several times to discuss aromaticity.23,33,35,61−63 It has been noted that for (anti)aromatic systems the out-of-plane component (the parallel component σ∥ in our notation) generally determines the shape of the isotropic chemical shift profile,23 and there is consensus that the most appropriate information regarding aromaticity would be obtained taking only the π contribution to this parallel component.19,35,79 We have analyzed the π and σ contributions to the scans of σ∥ = σ∥,π + σ∥,σ; this should be the best choice to ascertain π and σ aromaticity on the magnetic criterion. Before discussing the results, we include some remarks on the general shapes of the plots. Studies on NICS scans first suggested curves decreasing monotonously to zero for σ-aromatic systems and curves with an off-plane maximum for π-aromatic systems.61,62 Later, those results were confirmed for the NICS∥ = −σ∥ component,23 and a monotonous decrease was reported for antiaromatic systems. A subsequent effort to extract the contribution σ∥,π led to the conclusion that the extremum found in the all-electron plots was due to σ electrons.35 These expectations can be reconsidered in light of the simplest RCMs described above. As all RCMs considered here have zero slope at h = 0 and tend to zero for large h, an off-center extremum can be anticipated if the curvature at h = 0 has the same sign as σ(0). This is not the case for ICLOC, for which

(16)

For π currents one can consider two identical TCLOCs symmetrically displaced by ± ⟨z⟩ from the plane. Each of the TCLOCs will contribute half to the current strength and the magnetizability. For this model, which will be called TCLOC2, we have in analogy with ICLOC2 (eqs 3 and 16) σ (h) = −

■

μ0 ξ 4π

2

∑ p=1

1 {⟨s⟩ + [h + ( −1) p ⟨z⟩]2 }3/2 2

⎧ ⎫ ⟨s⟩2 3 ⎬ × ⎨1 − α 2 p 2 2 ⟨s⟩ + [h + ( −1) ⟨z⟩] ⎭ ⎩

(17)

COMPUTATIONAL METHODS The RCMs discussed above have been applied to a set of 33 organic and inorganic monocycles as listed in Table 1. The chosen set is formed by monocycles previously considered in many investigations on aromaticity (e.g., refs 29, 30, 34, 37, 39−41, 63, 68−71); it spans a wide range of chemical elements, and, as will be shown below, it covers many different patterns of ring current maps. Large basis sets of Gaussian functions for H, B, C, N, and O atoms developed ad hoc for magnetic properties29,40 were used. The aug-cc-pVTZ basis sets for Al, P, S and Si nuclei are from Woon and Dunning,72 and the basis set for Ga is taken from Wilson et al.73 For the calculations of the two rings containing iodine, we adopted the 6-311G* basis set,74 previously used in ref 70, as recovered from the Basis Set Exchange Database.75 The same basis sets were used to optimize planar geometries, using Gaussian03,76 and to compute the magnetic properties via integration of the current density at the coupled Hartree−Fock (CHF) level,3,4 using the SYSMO package77 (in a few cases, such as C8H8 and C6I62+,70 the planar form is a saddle point at the HF level). Gauge-independence has been achieved by the CSGT method originally proposed by Keith and Bader,1 which has been extended and renamed “continuous transformation of the origin of the current densitydiamagnetic zero” (CTOCD-DZ).4,66 The same method is also widely addressed as “ipsocentric”,78 an adjective which immediately recalls that for this method the origin of the current density is always put in the same point where the calculation is performed. As all systems have a σh symmetry plane, σ and π orbitals are identified unambiguously as those symmetric and antisymmetric with respect to the mirror reflection, and they are used to compute the σ and π contribution to the current density and therefore, via suitable integrations,66 to magnetic shieldings,

⎛ d2σ ⎞ 3σ (0) ⎜⎜ 2 ⎟⎟ = − 2 s ⎝ dh ⎠ 0

(18)

and the induced magnetic field has an extremum only at the ring center, in agreement with what was reported for σ-aromatic systems.61,62 For ICLOC2 ⎛ d2σ ⎞ 3σ (0)(s 2 − 4z 2) ⎜⎜ 2 ⎟⎟ = − (s 2 + z 2)2 ⎝ dh ⎠ 0

1677

(19)



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Figure 3. Difference between TCLOC2 and ICLOC2 with parameters as close as possible to those used in ref 52 (continuous line) and with the best-fit parameters obtained here (dashed line). The asterisks are the values published by Farnum and Wilcox.52

ascribed to peculiar patterns of current; indeed, aside form traditional ring currents, additional in-plane localized vortices have been reported for all of the four systems: cyclopropane,39 cyclic ozone, borazine,29,80 and boroxine.29,80 A special case is S6, whose current density pattern, reported in the Supporting Information, shows a paratropic CLOC inside a diatropic CLOC, as well as six diatropic atomic loops. These five examples show that local diatropic vortices can occur in cyclic molecules of maingroup elements and are not specific to transition-metal compounds, as recently supposed.12 Coming now to the performance of the RCMs, for benzenelike π scans, occurring for most large π (anti)aromatic rings, all models but ICLOC have enough flexibility to allow reproducing the σ∥ (h) profile to a good graphical accuracy: both the widening of the loop in TCLOC and the addition of a smaller heterotropic loop carrying an amount of current lower than the main (external) loop in ICLOC2C can reasonably reproduce the pattern given by ICLOC2 (left panels of Figure 2). From the plots reported in the Supporting Information, it can be appreciated that for the four systems with additional extrema, only TCLOC2 and ETCLOC give a good reproduction of the profile, while the change of sign of σ∥(h) observed for S6 is reproduced only by ICLOC2C. Leaving aside these few special cases with additional extrema and/or change of sign, we turn our attention to the standard situation of a single out-of-plane extremum at most with no change of sign. For these cases the computed profiles are very close, and quantitative consideration of the residuals is needed for model selection. The residual sum of squares (RSS) reported in the Supporting Information can be used to select models with the same number of parameters. Using this criterion, we can select ICLOC2 over TCLOC 26 times over 32, TCLOC2 over

which indicates that at the ring center, the magnetic response increases with height, rather then decreasing, if (s2 − 4z2) < 0, i.e., s/z < 2, a condition which should be best satisfied by small cycles. Thus, we expect that the lack of an off-plane extremum, considered typical for π scans of aromatic and antiaromatic systems,35 could be challenged for small rings. Incidentally, for TCLOC ⎛ d2σ ⎞ 3σ (0)(2 − 5α) ⎜⎜ 2 ⎟⎟ = − ⟨s⟩2 (3α − 2) ⎝ dh ⎠ 0

(20)

which still implies a nonmonotonous decrease of the magnetic response for sufficiently wide loops, i.e., α > 2/5 = 0.4. π Currents. The left part of Figure 2 shows values of the σ∥,π function computed along the symmetry axis of benzene (small dots). Superposed on the ab initio values we also report the profiles computed according to the three 1-loop models (top panels) and the three 2-loops models (middle panels). Similar figures for the other systems are reported in the Supporting Information. In 15 cases, the ab initio shape of the σ∥,π(h) = −NICS∥,π(h) function is as in Figure 2, showing a monotonous drop toward zero and thus in agreement with ref 35. However, in other 12 cases, the scan shows a maximum (minimum) for aromatic (antiaromatic) systems. In agreement with eq 19, these cases occur for the smaller cycles: all three- and four-membered cycles (with the exception of O42+), and a few five-membered cycles (tiophene, C5H−5 ). Still in qualitative agreement with eq 19, for the larger homocycles with six or more ring atoms, the π contribution to the magnetic response, |σ∥|, decreases monotonously without exception. More complicated shapes occur in five cases. In four of them, two well-defined off-plane extrema are found. This feature can be 1678



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Table 2. Estimates of Ab Initio σ Signed Current Strengths IσB According to the Six RCMs species

ICLOC

ICLOC2

ICLOC2C

TCLOC

ETCLOC

TCLOC2

IσB (nA T−1)

H6 C3H3+ C3H6 C4H42+ C4H42− C4H4 C5H5− C6H6 C6H3+ C6I6 C6I62+ C7H7+ C8H82+ C8H82− C8H8 C9H9− C10H10 furan pyrrole thiophene N42− N6 O3 O42+ borazine boroxine Al3− Al42− Si6H6 P6 S6 Ga3− Ga42−

−13.2 −5.3 −8.7 3.9 3.4 6.1 1.0 1.5 −9.2 2.2 −11.9 1.5 1.2 0.7 2.4 0.7 0.5 1.0 1.1 −1.6 8.0 3.2 −11.4 8.0 1.7 1.8 −8.3 −23.8 0.8 1.1 −3.8 −6.6 −24.9

−13.2 −6.3 −9.4 3.8 3.3 5.9 1.1 1.5 −9.2 2.1 −14.8 1.5 1.2 0.7 2.3 0.7 0.6 1.1 1.1 −2.5 7.8 3.1 −11.4 7.8 1.7 1.8 −8.3 −23.8 0.9 1.1 −3.6 −6.8 −24.9

−13.2 −6.5 −9.6 1.7 1.2 4.6 −2.2 −1.1 −9.4 1.0 −15.9 −0.8 −1.2 −1.4 0.9 −1.3 −1.4 −2.2 −2.2 −3.2 7.3 1.2 −11.4 7.4 −0.4 −0.1 −8.3 −23.8 −1.1 −2.0 −4.4 −6.8 −24.9

−13.2 −6.5 −9.5 3.8 3.4 6.0 −2.1 1.5 −9.2 2.2 −16.0 1.5 1.2 0.7 2.4 0.7 −1.3 −2.0 −2.0 −3.1 8.0 3.2 −11.4 7.9 1.7 1.8 −8.3 −23.8 0.8 −1.8 −4.3 −6.8 −24.9

−13.2 −6.5 −9.5 3.8 3.3 6.0 −2.1 0.6 −9.4 2.2 −16.0 0.7 0.6 −1.3 1.9 −1.2 −1.3 −2.0 −2.0 −3.1 7.8 2.5 −11.5 7.8 0.7 0.9 −8.4 −23.9 −0.9 −1.8 −4.3 −6.8 −25.0

−13.2 −6.5 −9.5 3.7 3.3 5.8 −2.1 1.5 −9.2 2.1 −16.0 1.5 1.2 0.7 2.3 0.7 −1.3 −2.0 −2.0 −3.1 7.7 3.1 −11.4 7.7 1.7 1.8 −8.3 −23.8 0.9 −1.8 −4.3 −6.8 −24.9

−13.3 −6.7 −9.7 1.7 1.1 5.0 −2.3 −1.2 −8.0 0.3 −17.3 −1.0 −1.4 −1.9 1.0 −1.8 −1.6 −2.1 −2.2 −2.5 7.3 1.3 −11.6 7.6 −0.4 −0.1 −8.5 −23.8 −1.4 −1.7 −4.3 −6.6 −25.1

1.3 1.7

1.2 1.5

0.4 0.3

1.1 1.1

0.8 0.7

1.1 1.0

SD av error

Supporting Information. Considering that the maximum of the π current strength in benzene is at roughly 1 atomic unit above the plane, the height of the loop is likely overestimated by ICLOC2 and underestimated by TCLOC2. The standard deviations are generally rather large; they are comparable to the ring radii in most cases, indicating highly asymmetric distributions. It is interesting to note that when the radial and vertical standard deviations are allowed to change independently, i.e. in the ETCLOC model, the fitting of the σ∥,π function most often leads to negligible or null values for the radial spread ρ. The worst retrieval of the current strength upon inclusion of the finite width is well-defined, albeit very small. To acquire more confidence in the smallness of the effect, we went back to the original proposition of the double-toroidal model of Farnum and Wilcox,52 and we recomputed the difference arising from the drop of the infinitely thin approximation of the loop as the difference of model TCLOC2 and ICLOC2 with parameters taken as close as possible to those of ref 52 (i.e., I B = 13.4 nA T−1,49 s = 1.39 Å, ρ = ⟨z⟩ = s/2). The result is shown as a continuous curve in Figure 3, together with asterisks indicating the original values computed numerically by Farnum and Wilcox.52 Also shown on the plot is the same difference

ETCLOC 16 times over 32, and TCLOC2 over ICLOC2C always, except for S6, the system with contrarotating π currents. Thus, ICLOC2 is performing better than TCLOC, while ETCLOC and TCLOC2 are almost comparable and better than TCLOC. The same conclusion is reached considering the signed current strengths (Table 1). Comparison of models with a different number of parameters cannot be accomplished with RRS. For nested models one can use a standard 0.05 F-test, whose application to our data shows that for any couple of nested models the general model is always preferred, except in the case of TCLOC2, where in seven cases the fit gave parameters the same as those of ICLOC2 (i.e., ρ ≃ 0) and the reduced model was preferred. The other RCM parameters, aside from signed current strengths, give further information on RCM performances. Loop radii (Table 1 of Supporting Information) are generally larger than the geometrical value sP = P/2π, where P is the ring perimeter (for a regular polygon, sP gives a value intermediate between the inradius and the circumradius). The only model which is qualitatively poor in reproducing the σ∥,π function, i.e., the simple ICLOC, gives also the most significant ovestimation of the radii. The radii of the other models overestimate sP by a rather similar extent, but TCLOC2 gives closer values. Still other parameters can be found in Table 3 of the 1679



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Figure 4. Comparison of the ab initio scan of σ∥,σ obtained for C4H42+ with the ICLOC2C model computed for two different set of parameters I1, s1, I2, and s2. The current strengths and the radii displayed in the box are in nanoamperes per tesla and in angstoms, respectively.

that in nine cases TCLOC2 was able to reproduce the pattern with a change of sign. This is because very large values of the ρ2 variance can give values of α close to 1 and then, according to eq 17, σ∥ can change sign for small h and ⟨z⟩. This change of sign is a spurious effect due to the finite order of the Taylor expansion used in the definition of the model, which in its intentional form, two vertically displaced homotropic tori, cannot lead to a change of sign for σ∥(h). The average error (Table 2) for ICLOC2C is significantly smaller than for any other model and is also close to the accuracy of integration, which, according to our experience, is close to 0.1 nA T−1. Considering the well-defined preference of the ICLOC2C model, we did not investigate any further statistics for σ currents. The pattern with a single off-plane extremum observed for the σ electrons of cyclopropane and C3H3+ can be rationalized because the inner heterotropic current is much smaller than the outer one and is able to decrease the central shielding, but not enough to let it change sign. Thus, once again the assessment of π aromaticity from the occurrence of a maximum in the σ∥(h) profile23,61,62 is infeasible.35 A final word of caution should be included about the best fit parameters of ICLOC2C. Let us first consider the case of two CLOCs of equal radius. In this case the general model ICLOC2C

evaluated with the best-fit parameters obtained in this paper. Besides the observation of a nice agreement between the 1967 numerical computation and ours, it can be noted that both the original parameters and the best-fitted parameters give a difference between the models which is rather low compared with σ∥ (Figure 2). Despite its smallness, the worsening in the description of the target data caused by the introduction of the finite width according to TCLOC2 is well-defined; thus, this model, which can be considered an analytical clone of the model proposed by Farnum and Wilcox,52 is not supported by the ab initio calculations. σ Currents. The right part of Figure 2 shows values of the σ∥,σ function computed along the symmetry axis of benzene (small dots). Superposed on the ab initio values we also report the profiles computed according to the three 1-loop models (top panel) and the three 2-loops models (middle panel). Similar Figures for the other systems are reported in the Supporting Information. Out of 32 systems, in only 8 cases (H6, C6H3+, O3, O4+, Al3−, Al42−, Ga3−, and Ga42−) the σ∥,σ(h) function decreases monotonically with increasing h, as previously reported.60−62 Instead, in the most common situation, both the profile and its derivative change sign before tending to zero. Notably, in 3 cases (C3H3+, C3H6, and C6I62+) the scan is similar to that of small π rings: an extremum at finite height with no change of sign. The monotonous decrease should be well-interpreted by ICLOC, but this is not the case for the less trivial shapes. The shape with a single off-plane extremum should be well-described by ICLOC2, but the profile with a change of sign should be interpreted only by ICLOC2C, where the central loop has a smaller intensity and contributes only for moderate heights, while the outer heterotropic loop has a higher intensity and prevails at larger heights. These considerations are effectively born out by the fits. However, quite unexpectedly, we observed

becomes an ICLOC model with I1B = I1B + I2B, and the two parameters I1B and I2B are not meaningful if considered individually. This situation has been found in a few cases (O3, Al42−, Ge42−). Considering to separate gently the two concentric loops, it can be expected to find a region in parameter space where the intensities of the two current loops are strongly correlated. Indeed, at second order in the differences of the radii from a common value s0, a sum of concentric CLOCs is 1680



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⎡ μ0 I B ⎢ ⎛ ∂F(h , s , 0) ⎞ F(h , s0 , 0) + ⎜ σ (h) ≃ − ⎟ (⟨s⟩I − s0) ⎢ ⎝ ⎠s 2 ⎣ ∂s 0 +

⎤ 1 ⎛ ∂ 2F(h , s , 0) ⎞ 2 2 ⎥ ⟨ s ⟩ − ⟨ s ⟩ s + s ( 2 ) ⎟ ⎜ I I 0 0 ⎥ 2⎝ ∂s 2 ⎠s ⎦ 0

Farnum and Wilcox double-toroidal model,52 of which TCLOC2 can be considered an analytical clone. It is likely that the setup of a model more accurate than ICLOC249,63 for π currents should take into account the marked asymmetry which is clearly appreciated from the cross sections of the current density.39 However, the effects on magnetic properties expected from similar refinements are quite small. As a matter of fact, ICLOC2 allows retrieving the ab initio π signed current strength with an average error as low as 3.1%. Similar good agreements for σ currents are obtained with the ICLOC2C model, whose parameters are however strongly correlated, when retrieved from unconstrained optimizations of σ∥,σ scans as has been done here. This problem does not affect the ICLOC2C model, for which, instead, we have recently shown that the nonlinear optimization needed for retrieving the parameters can be avoided by a computationally friendlier linearized ICLOC2 model, with a moderate increase of the error (7%) in estimating the signed current strengths.82 Finally, the large set of systems considered and the interpretation of the profiles according to the analytical equations have allowed us to show that the assessment of aromaticity from the shape of dissected π and σ components of the |σ∥(h)| = |NICS∥(h)| function requires more care than previously supposed.35,61,62 Indeed, an off-plane extremum is absent in the NICS∥,π scans of large rings, but it characterizes most of the NICS∥,π scans of small rings; it also appears in some NICS∥,σ scans (cyclopropane, C3H3+, and C6I62+). On the other side, NICS∥,σ scans do not generally show a monotonous drop of the magnetic response: in most cases, as a consequence of the two heterotropic concentric circulations present in most of them, they show an off-plane extremum together with a change of sign before dropping to zero.

(21)

where F is defined by eq 5, I B = ∑i IiB, and ⟨s n⟩I = (∑ IiBsin)/ I B i

(22)

Therefore, for a set of loops with radii close to s0, it should be possible to change intensities and radii of the loops while keeping I B, ⟨sI⟩ and ⟨s2⟩I constant, without affecting the overall scan. Incidentally, it can be observed that if one takes s0 = ⟨s⟩I, eq 21 degenerates to a three-parameter model, which is just an ETCLOC model (eq 13) with ζ = 0 and ρ2 = ⟨s2⟩I − ⟨s⟩2I . This simplification can be invoked for most π scans, but not for σ scans, whose ⟨s⟩I values generally differ significantly from the individual radii and can even be negative. In any case, the four ICLOC2C parameters can be anticipated to be strongly correlated, inasmuch as they are derived from axial scans of the magnetic shielding. Figure 4 illustrates this point, showing that the σ∥,σ scan of C4H42+ is well-reproduced by two very different sets of parameters, although the value of I B = I1B + I2B are close (1.80 and 1.68 nA T−1). Similar correlations affect most of the ICLOC2C models, whose parameters are not to be trusted individually, although the sum of the intensities is meaningful as also indicated by the good agreement with the ab initio values. A more precise definition of all four ICLOC2C parameters should require information beyond the axial scans considered here.

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CONCLUSIONS The development of good ring current models (RCMs) has been an object of many investigations in past47−52 and recent29−31,40,41,53,54,60−63,81 literature. Besides the different approaches for parametrization, those reports are based on either one or two infinitely thin circular loops of current (models ICLOC and ICLOC2 in our notation). The introduction of a finite width of the loops, as proposed by Farnum and Wilcox,52 sounds more as a line-of-principle proposal than as a settled improvement of the basic RCM, as, after its appearance, it has never been considered for any quantitative description. Here, the numerical results of Farnum and Wilcox have been reproduced by an analytical model (TCLOC2), and two other models with loops of finite width (models TCLOC, ETCLOC) have been introduced. A further model with two concentric inifinitely thin circular loops (ICLOC2C) has also been introduced for the description of σ currents. The six RCMs have been tested for their ability to reproduce shielding scans and signed current strengths in a set of 33 monocycles. Rather than the finite width, the most important feature needed to achieve quantitative agreement between the ab initio values and the RCM values is the introduction of a second loop, either vertically displaced for π electrons (ICLOC2 model) or concentric with the first one for σ electrons (ICLOC2C model, proposed here for the first time). The finite width of the loops, as described by TCLOC2, leads to a slightly worse retrieval of the ring current strength; moreover, the improvement in the reproduction of the σ∥ scan caused by the additional parameter is not statistically significant. Therefore, the ab initio calculations give no support to the

ASSOCIATED CONTENT

* Supporting Information S

Scans of the π and σ contributions to the parallel component of the magnetic shielding computed ab initio and fitted with the 6 RCMs, three-dimensional current density maps, and contour levels of the Cartesian component of the current density used to obtain the signed current strengths for all the 33 monocycles; tables of best-fit parameters (average and standard deviations of loop radii, distances from the molecular plane, etc.) and values of the resulting residual sum of squares. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +39 89 969570. Fax: +39 89 969603. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are thankful to S. Pelloni and P. Lazzeretti for discussions and for providing the geometries and wavefunctions of most of the molecules reported here. Financial support from the MIUR is gratefully acknowledged. This paper is dedicated to Prof. A. Zambelli. Zambelli, who spurred to combine conformational and chemical shielding models to effectively tackle the complicated NMR spectra of tactic polymers in the pre-ab initio period.83,84 1681



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