Article pubs.acs.org/JPCA

Delocalized Currents without a Ring of Bonded Atoms: Strong Delocalized Electron Currents Induced by Magnetic Fields in Noncyclic Molecules Stefano Pelloni,*,† Guglielmo Monaco,*,‡ Paolo Della Porta,‡ Riccardo Zanasi,‡ and Paolo Lazzeretti† †

Dipartimento di Scienze Chimiche e Geologiche, Universitá degli Studi di Modena, via G. Campi 183, 41100 Modena, Italy Dipartimento di Chimica e Biologia, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, Fisciano 84084 SA, Italy

‡

ABSTRACT: Some noncyclic small molecules, electrically neutral or charged, sustain interatomic electronic currents in the presence of a stationary, spatially uniform magnetic field. The existence of fairly large delocalized electron flow is demonstrated in H2O, BH3, NH3, CH4, CH3−CH3, H3O+, CH3+, and NH4+, by plots of quantum mechanical current density. Convincing quantitative evidence is arrived at by current strengths, defined via a flux integral of the ab initio current density. Application of a simple ring current model shows that the delocalized current strengths account for the out-of-plane component of the magnetic shielding tensor along the symmetry axis. A definition of delocalized electron current as a current flowing along a closed loop containing three or more atoms is discussed. three juxtaposed hydrogen molecules with D6h symmetry,26−28 first studied by London.13 The calculated current strength evaluated by accurate quantum mechanical methods is as big as 12.6 nA/T (nanoampere per tesla) for B∥ perpendicular to the plane of the hydrogen nuclei.26−28 This value is very close to 12.8 nA/T, the total current strength computed for benzene at the same level of theory.29 The cyclopropane molecule constitutes another remarkable example29−31 with a ring current strength of 10.2 nA/T.29,32 The above systems showing π- and σ-aromaticity share a cyclic nature; according to standard interpretation of interatomic distances they have a ring of bonded atoms. Therefore, one can imagine that intense delocalized currents should have a ring of chemically bonded atoms to sustain them. This guess is challenged by recent reports. Starting from cyclopropane, it has been reported that for a magnetic field B⊥ applied in the direction of a C2 symmetry axis, the current strength is 15.7 nA/T, approximately 1.5 times larger than 10.2 nA/T calculated29,32 for B∥ (incidentally, such a result implies that the large nucleus independent chemical shift (NICS),33 computed as −σCM aν = −44.9 ppm,30,31 is an unsafe measure of magnetic aromaticity of cyclopropane). Even more challenging are the reports of delocalization of electron flow taking place on the plane of the hydrogen nuclei of the CH3− group in methane and in D3d or D3h conformations of ethane in the presence of a magnetic field parallel to the C3 symmetry axis.34 Interatomic streams were observed even for water on a plane through the CM normal to a uniform external magnetic field B parallel to the binary symmetry axis.34 To what extent are the electrons of noncyclic compounds capable of sustaining delocalized flow in the presence of a

1. INTRODUCTION The electron density of atoms is set in rotation by an external magnetic field, and thus, by continuity of the flow, peripheral delocalized currents are expected for any molecule. The continued interest in delocalized currents1−11 originated from benzene and other unsaturated planar molecules. These systems were soon hypothesized 12,13 to sustain “ring currents”,14 although the very existence of such currents has been object of much debate, starting from the forceful dispute opposing Musher15−17 to Gaidis and West.18 Nowadays, it is generally assumed that delocalized π-electrons are quite free to move, and they sustain diatropic19−22 “ring currents” in “aromatic” planar cyclic molecules, for example, benzene,1,2,14 in the presence of B∥ orthogonal to the molecular plane. On the same ground it is assumed that the π-electrons of “antiaromatic” systems, for example, cyclobutadiene and the flattened cyclo-octatetraene (COT) model molecule, support paratropic currents delocalized all over the carbon ring.23,24 These delocalized currents have direct influence on magnetic properties. The π-ring currents, induced above and below the molecular plane by B∥ ≡ Bz applied at right angles, increase (decrease) the out-of-plane component ξzz of the magnetizability tensor of antiaromatic (aromatic) compounds. They determine a paramagnetic (diamagnetic) shift of proton shielding in benzene (COT) by diminishing (exhalting) the out-of-plane component σHzz. The effect is just the opposite for virtual probes lying in the inside of the carbon ring, for example, the center of mass (CM). Similar findings are at the root of the widespread interest in magnetic aromaticity:1,2,25 the definition of the elusive concept of aromaticity through the magnetic properties, determined by the onset of delocalized currents. The study of magnetic properties has been also expedient for the assessment of σ-aromaticity;5 delocalized currents occur in many planar saturated cylic systems. An archetypal example is provided by H6, the cyclic arrangement of © 2014 American Chemical Society

Received: March 12, 2014 Revised: April 16, 2014 Published: April 18, 2014 3367

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magnetic field? Are ring-shaped molecules carriers of stronger currents compared with noncyclic ones? The present investigation sets out to answer these questions by developing qualitative notions, criteria, and quantitative indicators of electron-current delocalization, which may be used to investigate whether noncyclic molecules support sizable interatomic electron flow in the presence of a stationary magnetic field. According to the conclusions expounded in Section 6 one can reasonably claim that strong delocalized currents can be observed also in noncyclic molecules.

3. PHASE PORTRAITS OF THE CURRENT DENSITY Although largely used in literature, the very definition of delocalized current has not been clearly given before. Such a definition is mandatory for our scope and we will borrow it as much as possible from that of electron delocalization. Electron delocalization is typical of systems whose structure is characterized by resonance hybrids according to the valencebond theory. Delocalized electrons do not have a specific location; they cannot be drawn in a simple Lewis structure.47 Allowing for a visualization reported in the IUPAC Gold Book,48 they are spread across a moiety including three or more atoms of a molecule. Examples48 illustrate the peculiar delocalization of charge in ionic conjugated systems, for example, R-CO2−, CH(CH2)2+. It is noteworthy that although the many different analyses of unperturbed densities proposed for the determination of electron delocalization49,50 can be applied for diatomic molecules as well as for polyatomic ones the IUPAC definition excludes them, as a clear reminescence of the many additive rules for molecular properties in terms of atomic and/or bond contributions developed along the history of chemistry. Additive rules are also known for magnetizabilities and chemical shifts, which are determined from perturbed densities, and it can be expected that much of the additive contributions stem from components of the perturbed density localized on atoms and/or bonds. Indeed, the component of magnetic properties stemming from circulations localized on atoms or bonds can be fairly big. In the case of ethylene for |B⊥| = 1 au orthogonal to the molecular plane, an intense current arises with maximum modulus 0.075 au51 (to be compared with the corresponding 0.08 au value for benzene), and current strength equal to 70% of the corresponding value for benzene.52 Therefore, coming to the delocalized current, we can tentatively define it as a current flowing along a closed loop containing three or more atoms. This definition, similar to that of electron delocalization, excludes the currents flowing around atoms and bonds. The theoretical approach needed to apply the above definition for a quantitative assessment of the delocalized currents, and thus exclude the circulations localized on atoms and bonds, is the topological analysis of the current density field through the phase portraits. Phase portraits of the first-order, magnetically induced current density vector field, that is, JB, are reported in Figures 1 and 2. These figures convey the entire information needed to understand the essential features of JB. They correspond to a section of the stagnation graph of JB on a suitable chosen plane and are interpreted via theoretical tools described in previous papers.21,22,35 Briefly, a stagnation graph collects and classifies the stagnation points, that is, the points where JB vanishes. Stagnation points are widely classified according to some features of the unsymmetrical 3 × 3 Jacobian matrix (∇J)r0, which, in the linear approximation, determines the nature of the flow close to a stagnation point r0,

2. CALCULATIONS Near Hartree−Fock calculations employing a computational scheme based on continuous translation of the origin of the current density−diamagnetic zero (CTOCD-DZ)3,35−38 and large basis sets of uncontracted Gaussian functions have been carried out, using the approach implemented in the SYSMO package,39 for a set of noncyclic, electrically neutral or charged molecules either planar (borane BH3, boron trifluoride BF3, methylcation CH3+, carbonate dianion CO32−, and nitrate anion NO3−, water), or nonplanar (ammonia, methane, ethane, hydronium H3O+, and ammonium NH4+ cations), assuming B parallel to the principal axis (that is, orthogonal to the molecular plane for planar systems). The water molecule was studied assuming B (i) parallel to the C2 symmetry axis (ii) orthogonal to the molecular plane. Molecular geometries were optimized at the Hartree−Fock level of theory by the Gaussian 03 code40 using the same basis sets:34 • for H, (8s) from ref 41, plus three sets of 2p functions from ref 42 and one set of 2p function with exponent 6.269, plus one set of 3d functions from ref 42. • for C, (13s8p) from ref 41, plus two sets of 2p functions with exponents 1512.9 and 355.1, three sets of 3d functions from ref 42 and two sets of 3d functions with exponents 5.262 and 0.08, and two sets of 4f functions from ref 42. • for O, (13s8p) from ref 41 plus one s function with exponent 0.0741898, three sets of 2p functions with exponents 3694.545, 859.598, 0.04821, three sets of 3d functions from ref 42 and three sets of 3d functions with exponents 10.962, 0.152. 0.05204, two sets of 4f functions from ref 42 and one set of 4f functions with exponent 0.27677 • for B, (13s8p) from ref 41 plus one s function with exponent 0.022845, three sets of 2p functions with exponents 880.0, 220.0, 0.01588267, three sets of 3d functions from ref 42, four sets of 3d functions with exponents 3.330, 9.990, 29.970, 0.048333, two sets of 4f functions from ref 42 and three sets of 4f functions with exponents 2.646, 0.103667, and 0.03455567 • for F, (13s8p) from ref 41 plus two sets of 2p functions with exponents 1057.4139, 0.05933, three sets of 3d functions from ref 42 and three sets of 3d functions with exponents 15.042, 0.1953, 0.0651, two sets of 4f functions from ref 42, and two sets of 4f functions with exponents 10.686, 0.38267 • for N, (13s8p) from ref 41 plus two sets of 2p functions with exponents 2282.782 and 537.7287, five sets of 3d functions, three from ref 42 and two with exponents 8.31463 and 0.11593, and two sets of 4f functions with exponents 2.666 and 0.685. Nuclear shieldings have been computed using Gaussian 0943 at the MP2 GIAO44−46 level with the same large near Hartree− Fock basis set, to test the relevance of electron correlation.

JB (r) = (r − r0) ·(∇J)r0

(1)

To be more specific, a stagnation point is classified as (rank, signature), where the rank is defined as the number of nonvanishing eigenvalues of (∇J)r0, and the signature is the excess of positive over negative eigenvalues if they are real or pure imaginary. The continuity equation for stationary flow implies that the Jacobian matrix is traceless all over the JB field, and thus only two eigenvalues are linearly independent. This places a limit on the possible values of rank and signature and 3368

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Figure 1. Phase portraits of the current density vector field. Color code for nuclei: light gray for hydrogen, yellow for boron, dark gray for carbon, blue for nitrogen, and red for oxygen. The phase portraits of the magnetically induced first-order current density display paratropic centers by small red dots, diatropic centers by small green dots, saddle points by blue crosses; light black lines connecting saddle points separate the different domains of circulation. The inducing magnetic field is perpendicular to the plotting planes and is pointing outward, that is, diatropic/paratropic current densities are clockwise/anticlockwise. In each plot, a bold black line represents the current trajectory for the maximum value (larger red dot) of the current density cross section over the plane bisecting the line that connects the atomic pair chosen to estimate the current strengths in Table 1.

Figure 2. Phase portraits of the current density vector field. The graphical conventions are the same as in Figure 1.

A common feature of all plots with the exception of water with B parallel to the C2 axis is given by saddle points connected by asymptotic streamlines that provide the boundary among different circulation domains. For the systems here considered, three connected saddle points, two for water with B perpendicular to the molecular plane, dissect the molecular space in two main regions, one containing the local circulations and another characterized by a delocalized diatropic flow of current. Local circulations can be easily associated with atoms and bonds: circulations localized on the central atom are present in all systems, circulations localized on the bonds are visible in all systems but BF3 and H2O with the magnetic field along the C2v axis, circulations localized on the peripheral atoms are seen in BF3 and the two planar anions (for these two systems one could also tentatively assign additional localized flows to electron pairs on the oxygen atoms). A similar association is not possible for the delocalized currents flowing outside the saddle points. A continuous, ring-shaped, peripheral bold black line, representing the trajectory of the delocalized current for the maximum modulus |JB|, has been superimposed to the phase portraits of Figures 1 and 2. Details on the realization of phase portraits are as follows. A cross section of JB on a plane bisecting the line which connects a pair of peripheral nonbonded atoms has been computed, assuming a unitary inducing magnetic field parallel to the main symmetry axis of each molecule. For CH4, B is parallel to a C3

the stagnation point can only be (3,±1), (2,0), or (0,0). The (2,0) points in turn can either be saddles (real eigenvalues) or vortices (imaginary eigenvalues), and the latter are either diatropic or paratropic as can be determined by the vorticity, the local curl ∇ × J. Even in three dimensions, a flow can be two-dimensional if it does not depend on one of the coordinates; in this case, the (2,0) points form straight lines that characterize the flow completely. Upon departure from this idealized case, isolated singularities (3,±1) can occur, and they are classified as saddle-nodes (all real eigenvalues of the Jacobian) or foci (two imaginary eigenvalues). Figure 1 collects the phase portraits of planar molecules (BH3, BF3, CH3+, CO32−, NO3−, H2O perp), those for nonplanar molecules (CH4, CH3CH3, NH3, NH4+, H3O+, H2O), are displayed in Figure 2. The inducing magnetic field is always perpendicular to the plotting planes and points outward. 3369

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Table 1. Through Space Current Strengths IB = |IB| in nA/Ta,b molecule

APc

Dd

|JBmax|e

D2f

D16f

D128f

D1024f

BH3 BF3 CH3+ CO32− NO3− H2O⊥g CH4h CH3−CH3 NH3 NH4+ H3O+ H2O

H−H F−F H−H O−O O−O H−H H−H H−H H−H H−H H−H H−H

2.056 2.240 1.869 2.216 2.112 1.504 1.765 1.743 1.615 1.649 1.607 1.504

0.0443 0.0144 0.0934 0.0453 0.0926 0.1871 0.0816 0.0814 0.1216 0.1388 0.1912 0.2261

1.986 0.522 2.802 1.210 2.006 5.632 3.474 3.521 5.182 4.844 5.790 6.072

4.151 1.142 5.890 2.206 3.684 12.051 6.659 6.674 10.464 11.181 10.568i 13.139

4.535 1.266 6.431 2.297 3.840 13.246 7.152 7.142

4.598 1.283 6.519 2.303 3.849 13.456 7.231 7.214

12.039

12.169

14.253

14.768

All signed current strengths are negative, consistently with the diatropic flows shown in Figures 1−4. bFor all the molecules, the magnetic field is parallel to the principal symmetry axis with exceptions otherwise indicated. cAtom pair defining the current density cross-section plane, that is, the plane perpendicular to and cutting half way the line connecting the two atoms. dDistance between atoms in Å. eMaximum absolute value of the current density cross section in au. fIntegration domains D2, D16, D128, and so forth are enclosed within contour lines of decreasing value given by | JBmax(Red))|/2, |JBmax(Red))|/16, |JBmax(Red))|/128..., respectively (see red contours in figures). gThe magnetic field is perpendicular to the molecular plane. hThe magnetic field is parallel to a C3 symmetry axis. iD8 integration domain. a

magnetic field parallel (antiparallel) to the inducing magnetic field. Then, for each molecule, as discussed in Section 3, the cross section of JB has been computed on a plane bisecting the line which connects a pair of peripheral atoms and the location and value of its maximum modulus have been determined for an external magnetic field perpendicular to the current loop, that is, parallel to the z axis. Table 1 reports the atomic pairs and maximum modulus values in columns 2 and 4, under the headings AP and |JBmax|, respectively. Integration domains for eq 2 have been defined as the areas inside the contour lines of the current cross section detected for the values |JBmax|/2, |JBmax|/16, | JBmax|/128, and |JBmax|/1024, which provide a convenient series of decreasing values to test the convergence of the computed integrals. Calculated current strengths are reported in Table 1, under the headings D2, D16, D128, and D1024. For NH3 and NH4+, D128 and D1024 integration domains were not determined due to some complexity of the cross section scalar field. As can be observed, current strengths converge nicely to rather large values. Domain D16 provides the 90% of the current strength on average, indicating that much of the delocalized current flows in a “tube” of reduced size. This can be best appreciated looking at plots of Figures 3 and 4, where D2, D16, and D128 integration domains are superimposed to the computed current density vector field represented by wide arrows of area proportional to the current magnitude. In particular, cutoffs have been applied to show only the field portion crossing the D2 domain. The delocalized currents represented by phase portraits (Figure 2) and vector plots (Figure 4) for H2O and NH3 molecules and H3O+ and NH4+ cations can be classified as strong on account of the current susceptibilities evaluated ab initio via the flux-integral, eq 2, reported in Table 1. Quite noticeably, the ab initio current strenght for NH3, H3O+, and NH4+ are close to the benzene yardstick cited above (12.8 nA/ T29). In fact, values computed for water in two different orientations with respect to B are ∼1−2 nA/T larger. The current strength calculated for CH4 and methyl group in CH3CH3 is smaller, ∼7.2 nA/T, but large enough to evidence sizable magnetic-field induced electron currents, delocalized

symmetry axis; for water, the case of B perpendicular to the molecular plane has been considered too. Then, cross sections have been inspected to find out their maximum magnitude within the delocalized circulation domain. These maximum values are reported in the fourth column of Table 1 under the heading |JBmax| and marked with a larger red dot in Figures 1 and 2. As can be observed, |JBmax| values are in the majority of cases larger or comparable to the corresponding value of 0.08 au calculated for the benzene π-electron ring current.29 Current density cross sections within the delocalized circulation domains are found to reach their maximum magnitude on the molecular plane of planar molecules and somewhere in between the central atom and the plane containing the hydrogen atoms in the case of non planar molecules. In either case, these positions have been adopted to compute the phase portraits displayed in Figures 1 and 2, by cutting the stagnation graphs of JB with planes containing these points and perpendicular to B.

4. AB INITIO CURRENT STRENGTHS An accurate method to quantify the magnetotropicity of a molecule relies on the concept of electronic current strength, defined as the flux of the B -induced quantum mechanical current density vector field JB from a planar molecular domain crossing a bond region,4,6,26,27,29,31,32,52−54 evaluated ab initio via the integral I B = B−1

∫ JB ·da

(2)

where J is the current density induced at first order by the external magnetic field B, and da is the normal of a surface element of a plane parallel to the applied field B of modulus B. Current intensity as well as its derivative with respect to the external magnetic field are positive scalar quantities. However, the choice of an orientation for da implies that the above integral is signed, it gives a signed current strength.54 Choosing the orientation of da in the anticlockwise direction as seen from the positive end of the applied field conveniently keeps trace of the handedness of the circulation: a positive (negative) signed current strength IB indicates an anticlockwise (clockwise) current, that is, a paratropic (diatropic) current producing a B

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Figure 3. Integration domains of the current density cross section. Perspective view of the integration domains (red contour lines) superimposed to the through-space current density represented by arrows with size proportional to |JB|. For the sake of comparison, the same scale factor has been used for all the figures except for BF3 and CO32−, which have been reduced by a further factor of 0.80 and 0.85, respectively. Integration of the current density cross section within such domains provides the current strengths reported in Table 1. Smaller/intermediate/larger domains correspond to D2/D16/D128, see caption to Table 1

Figure 4. Integration domains of the current density cross section. The graphical conventions are the same as in Figure 3. For NH3, only D2 and D16 were evaluated. For H3O+, only D2 and D8 were determined.

the most popular indirect computational approach is certainly based on the NICS,33 whose positive (negative) value in ring centers has been considered an indicator of diatropic (paratropic) ring currents (although the acronym NICS first appears in ref 33, the idea that delocalized currents could be detected by any magnetic nucleus inside them has been advanced several times before59−61). The assumption that the effects of ring currents would be detected by an isotropic NICS33 has been confuted by the crucial computer experiment on cyclopropane29−31 recalled in the Introduction. More correctly, one should only consider NICS∥ the contribution of the out-of-plane component to the average value.25,62−64 On the other hand, the intrinsic inadequacies of NICS∥ have been documented in recent papers:26−28 its spurious dependence on some inverse power of the ring radius leads to NICS∥ changes that are not consistent with the changes in the ring current strengths. Other reports about an inefficient assessment of magnetic aromaticity via NICS calculations can be easily found in literature.10,65−67 Rather than using a single value of σ∥, it can be argued that a scan of σ∥ can allow a better quantification of magnetic aromaticity.68 Recently, the development of ring current models based on the Biot-Savart law of classical electromagnetism69 have successfully been applied to retrieve the current strengths of cyclic systems from NICS ∥

about a circuit encompassing the hydrogen nuclei. Strong currents, induced by B parallel to a bond direction, are present also in the H3O+ and NH4+ cations, see Table 1.

5. MAGNETIC SHIELDINGS The presence of delocalized currents in cyclic systems was generally inferred indirectly from experimental values of magnetizabilities and nuclear shieldings. Though somewhat paradoxical, this approach is even more common after the development of ab initio methods. Indeed, these methods could be used to compute the parent current density, but they are more often adopted to compute the derived magnetic properties; computations are now routinely performed before the experimental measures and even for systems which have not been synthesized. In the case of magnetizability, its partition in terms of atomic basins defined within Bader’s QTAIM theory55 allows the retrieval of intrabasin and interbasin contributions.56,57 The values of the latter in particular favorably correlate with the ab initio current strengths.10,58 Besides that, 3371

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Table 2. Magnetic Shielding Tensor Components, Isotropic Shieldings σav, and Anisotropies Δσ in ppma CTOCD-DZ

molecule BH3 BF3 CH3+ CO32− NO3− H2O⊥ CH4 CH3−CH3 NH3 NH4+ H3O+ H2O

MP2 GIAO

σX⊥

ΔσX∥

ΔσXav

ΔσX

σX⊥

ΔσX∥

ΔσXav

ΔσX

σLC ⊥

ΔσLC ∥

ΔσLC ∥

ΔσLC

σLC ⊥

ΔσLC ∥

ΔσLC av

ΔσLC

−27.23 110.40 −368.12 −19.82 −315.18 81.64 195.49 34.79 182.88 29.41 280.04 64.20 243.52 53.13 332.11 85.26 343.04 45.86

145.92 90.14 182.74 72.06 56.05 159.21 195.49 43.94 200.15 46.50 238.12 63.97 243.52 67.11 287.22 89.47 323.29 79.46

30.49 103.65 −184.50 10.80 −191.41 107.49 195.49 37.85 188.63 35.13 266.07 64.12 243.52 57.79 317.16 86.67 336.45 57.06

173.15 −20.26 550.86 91.89 371.23 77.57 0.00 9.15 17.27 17.09 −41.92 −0.23 0.00 13.98 −44.89 4.21 −19.75 33.60

−44.45 103.51 −398.81 −9.40 −137.49 82.39 201.33 37.05 186.25 30.69 295.09 65.11 251.33 54.29 342.47 86.25 350.25 45.83

147.08 83.45 185.52 57.50 24.57 175.42 201.33 47.92 206.21 50.24 247.61 67.94 251.33 70.33 296.46 93.67 345.57 80.59

19.39 96.82 −204.03 12.90 −83.47 113.40 201.33 40.68 192.91 37.21 279.26 66.05 251.33 59.64 327.13 88.72 348.69 57.42

191.53 −20.06 584.33 66.90 162.06 93.03 0.00 10.87 19.96 19.55 −47.48 2.83 0.00 16.04 −46.01 7.42 −4.68 34.76

R1

0.047 0.042 0.039 0.028 0.025 0.041

The shielding at the central nucleus X (either B, C, O, or N) is denoted by σ , that of a virtual probe at the center of the loop of delocalized flow, on the plane of the nonbonded hydrogen atoms in nonplanar molecules, is denoted by σLC. For both CTOCD and MP2 GIAO calculations, the same large basis set described in Section 2 has been used. For nonplanar systems, the two methods have been also used to compute 101 σ∥ = −NICS∥ points at steps of 0.2 au from the CM up to 20 au in the direction of the delocalized loop. The disagreement factor R1 of the two σ∥ scans factor is CTOCD (hi)|/∑|σCTOCD (hi)|. computed as R1 = ∑|σMP2 ∥ (hi) − σ∥ ∥ a

X

scans.10,26−28,54,64,70 Detailed investigation has shown that, although the retrieval of the total signed current strength in monocycles is almost quantitative, in the presence of multiple concentric loops the indirect determination of the parameters of the individual loops from NICS∥ scans faces the problem of a strong correlation of the parameters themselves.54 This complication characterizes also the systems investigated here, where the current density maps show delocalized currents together with currents localized on the central atom(s), and therefore we will not undertake any fitting of NICS scans. Trying to apply some of the above knowledge to our noncyclic systems, we have calculated the principal components of the Cartesian magnetic shielding tensor, and we have used them to compute the isotropic shielding σav = 1/3(σxx + σyy + σzz) and the anisotropy Δσ = σ∥ − σ⊥. For the systems studied, the parallel component σ∥ is always equal to the σzz, and σ⊥ = (σxx + σyy)/2. The shielding tensor components have been computed both at the CTOCD-DZ and the GIAO-MP2 level using the same large basis set discussed in Section 2. The results are reported in Table 2. Comparison of the CTOCD-DZ and the GIAO-MP2 magnetic shielding at nuclei indicates the occurrence of large quantitative variations, even exceeding the 50%. The larger variations occur for the planar systems which have degenerate resonance hybrids, that is, NO3− and CO32−. Variations are smaller for other planar systems and definitely smaller (typically a few percentual points) for the nonplanar system, which can be written with a single Lewis structure, and are typical examples of nonconjugated molecules. In all cases, the sign of the individual shielding components is preserved: the only negative shielding components are the perpendicular ones for CO32−, CH3+, and NO3−. These negative values, in the case of the latter systems are large enough that even the isotropic values are negative. In addition to the individual components,

also the sign of the anisotropies is preserved upon inclusion of correlation with the single exception of the shielding calculation at the center of the delocalized loop in NH3. For nonplanar systems, those for which we have performed axial scans the overall disagreement factors along the scans are always below 5%, definitely smaller than the relative disagreements observed at nuclei. The superposition of the CTOCD-DZ1 and GIAOMP2 scans shown in Figure 5 graphically documents the overall good capability of the lower level ab initio computations to follow the main changes in the shielding scan, especially far from the nuclei. In order to appreciate how the current strengths determined above influence the NICS scans, we have used the simplest ring current model, the Infinitely thin Circular Loop Of Current (ICLOC), for which σ (h) = −

μ0 I B

s2 2π (s 2 + h2)3/2

(3)

where μ0 is the vacuum permeability, s is the loop radius, and h is the distance from the loop plane. As can be seen in Figure 5, taking s equal to the distance from the axis of the atoms involved in the delocalized current and IB as the ab initio signed current strength (eq 2), this single ICLOC model, with no adjustable parameter, is able to describe the shape of the NICS∥ scan at sufficiently large distances: the delocalized current discussed above are the main source of the shielding at large distances. The worse agreement at large distances is observed for H2O in a field B parallel to the C2 axis. It is noteworthy that this system has another peculiar feature: it is the only one with a phase portrait devoid of saddle points separating the inner and the outer current flow (bottom-right panel in Figure 2). The fact that the outer current is isotropic with the current of the central oxygen atom casts a shadow on any simple 3372

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Electron motion all over an interatomic circuit, induced by a static magnetic field, has been documented in a series of small noncyclic molecules, electrically neutral or charged, via the quantum mechanical current density. The strength of delocalized electron flow taking place in small planar (borane BH3, boron trifluoride BF3, methylcation CH3+, carbonate dianion CO32−, and nitrate anion NO3−, water) and nonplanar molecules (ammonia, methane, ethane, hydronium H3O+, and ammonium NH4− cations) was determined via current susceptibilities calculated by rigorous ab initio methods as flux integral over a carefully defined planar molecular domain (Figures 3 and 4). Assuming the calculated value IBbenzene = −12.8 nA/T for total electron stream in benzene as a benchmark, the current susceptibilities are 116 and 105% of it for water in two different orientations and 95% for the ammonium cation. Fairly intense currents flow in a loop encircling the three hydrogen atoms of the methyl group in methane and ethane, with strength 56% of IBbenzene. Slightly weaker current susceptibilities are found for the isoelectronic planar system CH3+ and BH3, respectively 51 and 36% of the benzene standard. The delocalized currents in BF3, carbonate dianion CO32−, and nitric anion NO3− are less intense due to electron withdrawing exerted by the electronegative peripheral atoms. In the light of these findings, one could tentatively propose a definition of delocalized current consistent with the IUPAC acceptation of delocalized charge,48 as a current flowing along a closed loop containing three or more atoms. Such a definition would seem appropriate for CH3+ and BH3, as well as NH3, H3O+, and NH4−. It is important to consider that in the presence of a central atom the above definition should be complemented by the occurrence of saddle points in the phase portraits. Indeed, these saddles enable a clear-cut separation of local and extended domains in the current density maps. Such saddles are present in five of the six nonplanar systems considered here, but not for H2O in a field B parallel to the C2 axis, as commented above. The relevance of the delocalized currents discussed in this paper is well exemplified by their ability to justify the NICS∥ scans (Figure 5), just as happens with traditional ring currents. The fact that the nonplanar molecules investigated here are considered archetypal localized structures as for the electron delocalization, yet show strong delocalized currents, can be seen as an intriguing challenge for the magnetic aromaticity criterion.

Figure 5. Scans of the parallel component of the magnetic shielding σ∥ = −NICS∥ for the six nonplanar molecules considered in the paper, computed at the CTOCD-DZ level (continuous line) and at the MP2 GIAO level (asterisks) with the same large basis set described in Section 2. The dashed lines have been computed by an ICLOC model (eq 3) with radius s is equal to the distance of the peripheral hydrogen atoms from the z axis and IB taken from the integral 2 of the ab initio current density. The height h is null at the level of the nonbonded hydrogen atoms involved in the delocalized currents.

evaluation of the peripheral delocalized current and is likely the source of the major disagreement found in the simple calculation of the NICS scan. It is here appropriate to add a further comment on the shielding components reported in Table 2. The out-of-plane components σX∥ of the magnetic shielding tensor computed on the central nuclei Xs are big. It can be noticed that the magnitude of the parallel component σ∥ is not the largest among the diagonal components for H2O, CH3+ and NO3− (just as found for cyclopropane); moreover, as a consequence, for the latter two molecules the sign itself of the isotropic shielding σav is reversed with respect to σzz. Of course, such changes should not be added to the long list of problems in the application of isotropic NICS, because the shieldings in discussion fall outside the basic philosophy of NICS calculations, as they refer to points in space which are not devoid of nuclei. This is not the case, however for the σLC values obtained for nonplanar molecules at loop centers, that is, along the symmetry axis at the level of the hydrogen atoms interested in the delocalized current. If NICS∥ were a reliable quantifier of “absolute” aromaticity on the magnetic criterion,62 one should arrive at the absurd conclusion that all six nonplanar molecules are more aromatic than benzene itself, by comparing values of σ∥(0) in Table 2 with the π-contribution to σ∥(0) at the center of the benzene ring (36.3 ppm62). This paradox, even without considering the perturbation caused by the central atom, is immediately dismantled on account of eq 3: H2O, NH3, and CH4 sustain a delocalized current, but the huge central shielding is determined by the much smaller radius of the loop about the hydrogen atoms.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: +39 89 969570. Fax: +39 89 969603. *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 S.P. and P.L. gratefully acknowledge financial support from the Italian MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca) via PRIN 2009 scheme. G.M. and R.Z. gratefully acknowledge financial support from the MIUR (FARB 2012).

6. CONCLUDING REMARKS The ability to sustain delocalized currents is not the exclusive prerogative of molecules characterized by a ring structure. 3373

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