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Gauge-Origin Independent Calculations of the Anisotropy of the Magnetically Induced Current Densities Heike Fliegl, Jonas Juselius, and Dage Sundholm J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b03950 • Publication Date (Web): 20 Jun 2016 Downloaded from http://pubs.acs.org on June 23, 2016

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The Journal of Physical Chemistry

Gauge-origin Independent Calculations of the Anisotropy of the Magnetically Induced Current Densities Heike Fliegl,∗,† Jonas Jusélius,∗,‡ and Dage Sundholm∗,¶ Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, P.O.Box 1033 Blindern, 0315 Oslo, Norway, University of Tromsø, High Performance Computing group, Department of IT, N-9037 Tromsø, Norway., and University of Helsinki, Department of Chemistry, P.O. Box 55 (A.I. Virtanens plats 1), FIN-00014 University of Helsinki, Finland. E-mail: [email protected]; [email protected]; [email protected] Phone: +47 22845923; +47 47419869; +358 294150176. Fax: +47 22855441

∗

To whom correspondence should be addressed Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, P.O.Box 1033 Blindern, 0315 Oslo, Norway ‡ University of Tromsø, High Performance Computing group, Department of IT, N-9037 Tromsø, Norway. ¶ University of Helsinki, Department of Chemistry, P.O. Box 55 (A.I. Virtanens plats 1), FIN-00014 University of Helsinki, Finland. †

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Abstract Gauge-origin independent current density susceptibility tensors have been computed using the gauge-including magnetically induced current (GIMIC) method. The anisotropy of the magnetically induced current density (ACID) functions constructed from the current density susceptibility tensors are therefore gauge-origin independent. The ability of the gauge-origin independent ACID function to provide quantitative information about the current flow along chemical bonds has been assessed by integrating the cross-section area of the ACID function in the middle of chemical bonds. Analogously, the current strength susceptibility passing a given plane through the molecule is obtained by numerical integration of the current flow parallel to the normal vector of the integration plane. The cross-section area of the ACID function is found to be strongly dependent on the exact location of the integration plane, which is in sheer contrast to the calculated ring-current strength susceptibilities that are practically independent of the chosen position of the integration plane. The gauge-origin independent ACID functions plotted for different isosurface values show that a visual assessment of the current flow and degree of aromaticity depends on the chosen isosurface. The present study shows that ACID functions are not an unambiguous means to estimate the degree of molecular aromaticity according to the magnetic criterion and to determine the current pathway of complex molecular rings.

1

Introduction

Molecular systems sustain currents when they are exposed to external magnetic fields. 1–4 The current density flows largely perpendicularly to the direction of the external magnetic field around atoms, along chemical bonds, and around molecular rings. 5 Calculations of magnetically induced current densities provide information about the electron mobility in molecules and can be used for determining the current flow, the degree of aromaticity according to the magnetic criterion, and other properties that depend on the electron mobility. 6 Electron delocalization and molecular aromaticity are very important and useful chemical concepts 2


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that have been developed and evolved during the entire history of chemistry, even though they do not have any unambiguous physical explanation. The gauge including magnetically induced current method (GIMIC) is a reliable means to quantitatively determine the degree of molecular aromaticity. 5,7–9 Other approaches to provide insights about the degree of molecular aromaticity have also been reported. 4,10–12 In quantum mechanical systems, the direction of the magnetically induced current flow can either be in the classical diatropic (clockwise) direction or in the opposite paratropic (anti-clockwise) one. 13 Calculations of magnetically induced current densities show that the diatropic electron flow occurs at the outer surface of the electron density. 5,13–17 For aromatic molecular rings, the diatropic current at the outer periphery of the ring is stronger than the paratropic current inside it, 13 whereas for antiaromatic molecules the paratropic current inside the ring dominates. 13,18–20 For nonaromatic rings, the strength of the diatropic and paratropic currents are equal leading to a vanishing net ring-current strength. 7,13 In studies of magnetically induced current densities, the direction of the external magnetic field is usually applied perpendicularly to the molecular plane. However, for nonplanar molecules there is no obvious direction of the magnetic field. In our studies of the aromatic character and current pathways of Möbius twisted molecules, several possible directions of the magnetic field were employed. 21–23 The degree of aromaticity was finally assessed using the magnetic field direction for which the maximum ring-current strength susceptibility was obtained. Molecules in the gas or liquid phase tumble freely around implying that there is no fixed direction of the external magnetic field relatively to the molecular frame. Thus, when investigating the aromatic character of complicated ruffled molecular structures, it is desirable to employ a computational approach that provides information about the magnetic response without assuming any preselected direction of the applied magnetic field. In the well known anisotropy of the magnetically induced current density (ACID) approach, the function representing the magnetic response is independent of the magnitude and the direction of the magnetic field, because the ACID function is obtained by adding products of the tensor

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elements of the current density susceptibility function. 24,25 The ACID function, which is independent of the magnetic field by construction, is a scalar function that depends only on the internal coordinates of the molecule rendering visual inspection uncomplicated. The ACID approach was initially proposed more than ten years ago by Herges and Geuenich who suggested that calculations of the anisotropy of the magnetically induced current-density tensor can be used as a computational tool in studies of the electron delocalization in molecules. 24,25 The main motivation for developing the ACID method was that they considered visualization of the vector functions such as the current density awkward as compared to scalar functions. The anisotropic part of the current density susceptibility tensor investigated in ACID studies is a scalar function similar to the electron density. 24,25 However, the previously employed ACID approach suffers from problems that significantly affect the reliability of the computational method. In Herges and Geuenich’s implementation of the ACID method, the current density susceptibility and the ACID function are calculated using ordinary basis functions instead of gauge including atomic orbitals (GIAOs), leading a very slow basis set convergence. Thus, one can expected that an improvement of the ACID method is achieved by using GIAOs, which have proven to be highly useful in current density calculations. 3,5,7,8,26–33 The use of GIAOs renders current density susceptibilities gauge-origin independent and they consider first-order corrections to the orbitals due to the external magnetic field, which results in a significantly improved basis-set convergence as compared to calculations using ordinary basis functions. In the calculations of current densities, the GIAOs remove the redundant division of the current density in paratropic and diatropic contributions, that is, the calculations yield directly the total current density. 5,7,8 However, GIMIC calculations yield well-defined diatropic or paratropic characters of the current density, which can be identified from the direction of the current relatively to the applied field. The tropicity of the current density provides additional information about the electron flow in molecules. 20 We have shown in a number of studies that quantitative information about the current strength susceptibilities and current pathways can indeed be

4


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obtained in GIMIC calculations by integrating appropriate linear combinations of the vector components of the current density. 5,7 Here, we present for the first time calculations of gauge-origin independent ACID functions that are obtained using gauge-including atomic orbitals. The present implementation to calculate the ACID function is an extension of the gauge-including magnetically induced current (GIMIC) method and includes a similar integration scheme for the ACID function as used for the current density flow. We show that the cross-section area of the ACID function is strongly dependent on the exact location of the integration plane resulting in different interpretations depending on the chosen plane. The paper is organized as follows. The employed methods of the electronic structure calculations are given in Section 2. In Section 3, the GIMIC method is briefly presented and the ACID method is discussed in Section 4. The results of the GIMIC and ACID calculations are compared and discussed in Section 5. The main conclusions are drawn in Section 6.

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Computational methods

The molecular structures of the studied molecules were optimized at the density functional theory (DFT) level using the Karlsruhe triple-ζ (def2-TZVP) basis sets. The DFT calculations were performed using Becke’s three-parameter functional in combination with the Lee-Yang-Parr exchange-correlation functional (B3LYP). 34,35 The Karlsruhe split-valence (def2-SVP), triple-ζ (def2-TZVP), and quadruple-ζ (def2-QZVP) basis sets augmented with polarization functions were employed in the nuclear magnetic resonance (NMR) shielding calculations, 36,37 which yield input data for the subsequent current density calculations. The shielding calculations were performed at the B3LYP and restricted Hartree-Fock (RHF) levels, whereas current densities were obtained using the GIMIC program. 7,8 In the text, the def2 prefix is omitted for clarity. The electronic structure calculations were performed with Turbomole. 38,39 The visualization of the ACID function was done with Paraview version 5.0. 40 Note, the ACID function as implemented in GIMIC is written out in atomic units.

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Therefore our isovalues are about an order of magnitude smaller as those reported by Herges and Geuenich. 24,25

3

Magnetically induced current densities

In the limit of zero magnetic field, the magnetically induced current density (Jγ (r)) is linearly proportional to the magnitude of the applied magnetic field (Bτ ). The current density can be expressed as a Taylor series expansion with respect to the strength of the external magnetic field

where

∂Jγ (r) ∂Bτ

X ∂Jγ (r) Bτ + O(Bτ2 ) Jγ (r) = ∂B τ Bτ =0 τ ∈x,y,z

Bτ =0

(1)

is the current density susceptibility tensor. An expression for calculating

gauge-origin independent current density susceptibilities can be derived when the magneticfield dependent GIAOs in Eq. (2) are employed as basis functions. 7 i χµ (r) = exp − (B × [Rµ − RO ] · r) χ(0) µ (r) 2

(2)

(0)

In Eq. (2), χµ (r) denotes a standard Gaussian-type basis function with Rµ as center and RO is the chosen gauge origin. The gauge-origin independent GIMIC expression for calculating the tensor elements of the current density susceptibility is then given by 7,8 X X ∂χ∗µ (r) ∂h(r) ∂Jγ (r) ∂h(r) ∂χν (r) + χ (r) + Dµν χ∗µ (r) = D ν µν K ∂Bτ Bτ =0 ∂Bτ ∂mγ ∂mK ∂Bτ γ µν µν X ∂Dµν µν

∂Bτ

In Eq. (3), Dµν and

χ∗µ (r)

∂Dµν ∂Bτ

∂h(r) χν (r) − ǫγτ δ ∂mK γ

"

X

2

Dµν χ∗µ (r)

µν

#

∂ h(r) χν (r) . ∂mK γ ∂Bδ

(3)

are the density matrix and the magnetically perturbed density

matrices in atomic orbital basis, respectively, which are obtained when performing NMR shielding calculations. 29,41,42 The derivatives

6

∂ 2 h(r) ∂mK γ ∂Bτ

and

∂h(r) ∂mK γ


are the numerator of the

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electronic interactions between the Cartesian components of the external magnetic field and the nuclear magnetic moments (mK γ ). ǫγτ δ is the Levi-Civita tensor. An analogous expression has been derived for the orbital contributions to the spin currents of open-shell molecules. 8,43 The current density susceptibility is a tensor function with nine components, corresponding to the electron flow in the three Cartesian directions when the external magnetic field is applied in the Cartesian directions. A vector function describing the current flow for a given magnetic field is obtained by contracting the current density susceptibility tensor with the Cartesian components of an applied magnetic field as in Eq. (1).

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Anisotropy of the current induced density

The calculation of the anisotropy of the magnetically induced current density (ACID) func ∂Jγ (r) τ tions also involves evaluation of the current density susceptibility tensor Jγ (r) = ∂Bτ , Bτ =0

which is defined in Eq. (3). The use of GIAOs in the calculation of the current density susceptibility renders the ACID function gauge independent. A faster basis-set convergence is obtained for the ACID function when the perturbation-dependent GIAOs are employed instead of ordinary basis functions. Following Herges and Geuenich, 24,44 the anisotropy of the induced current density is given by i 2 2 1h x 2 y y z z x Jx (r) − Jy (r) + Jy (r) − Jz (r) + (Jz (r) − Jx (r)) + ∆J (r) = 3 2 i 2 1h y 2 x z x z y Jx (r) + Jy (r) + (Jx (r) + Jz (r)) + Jy (r) + Jz (r) 2 2

(4)

The ACID function (∆J 2 (r)) is a scalar function that does not depend on the magnitude nor the direction of the applied magnetic field. ∆J 2 (r) can be used for visualization and it provides information about the electron mobility and the aromatic pathways in molecules. Quantitative information about the strength of the current flow can be estimated by integrating the cross-section area of the ACID function in planes perpendicularly to chemical

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bonds. However, as shown in the next section, the ACID cross-section area depends on where the cut plane intersects chemical bonds, implying that the calculated strength of the current flow depends on that choice.

5

Results and discussion

5.1

Basis-set studies

For benzene, the ring-current strengths and the cross-section areas of the ACID function were calculated in a plane perpendicularly to the C–C bond intersecting in the middle of the bond. To assess the basis-set dependence, three different basis sets were employed. The obtained values show that practically converged current strengths and cross-section areas are obtained with the SVP basis sets. The use of GIAOs results in a very fast basis-set convergence also for the ACID function. The conventional ring-current strength for benzene, which is obtained with the magnetic field oriented perpendicularly to the molecular ring, is 11.9 nA/T at the B3LYP/QZVP level. The ACID cross-section areas might also be used as a quantitative measure of the strength of the current flow. For benzene, the cross-section area at the center of the C–C bond is 3.4 nA/T at the B3LYP/QZVP level as compared to 3.3 nA/T at the B3LYP/SVP level. The basis-set studies of the ring-current strength and the ACID cross-section area of benzene are summarized in Table 1. For molecules without rings, the mobility of the electrons along chemical bonds can be estimated by calculating the strength of the current passing in one direction along chemical bonds. Such an approach was recently employed to estimate the strength of internal hydrogen bonds. 6 Calibration calculations of the strength of the hydrogen bond showed that the hydrogen-bond strength correlates linearly with the current strength passing the hydrogen bond in one direction. The mobility of the electrons along the single and double bonds of 1,3-butadiene can be estimated from the current strengths. In Table 2, we compare the 8


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Table 1: Ring-current strengths for benzene are compared to the cross-section area of the ACID function. The ring-current strengths J (in nA/T) are ob1 tained with the magnetic field applied perpendicularly to the molecule. |∆J 2 | 2 (in nA/T) is the ACID cross-section area. The numerical integrations were performed at the center of one of the C–C bonds. The current density was calculated at the B3LYP level using different basis sets. Grid-point spacings of 0.02 were used in the numerical integrations. Basis SVP TZVP QZVP

J 11.9 12.1 11.9

1

|∆J 2 | 2 3.3 3.4 3.4

Figure 1: Comparison of the ring current strength (J) and the ACID cross-section area 1 (A = |∆J 2 | 2 ) for benzene. Current strength profiles are plotted as a function of the extent of the cross-section. The cut plane is perpendicular to the benzene ring and starts in the middle of the ring, passing the center of the C–C bond as indicated by the red arrow.

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Table 2: The diatropic (Jdia in nA/T) and paratropic (Jpara in nA/T) ring-current strengths, i.e., the current strengths passing in the two opposite directions along the chemical bonds of 1,3-butadiene are compared to the ACID cross-section 1 area (|∆J 2 | 2 in nA/T). The numerical integrations were performed at the center of the chemical bonds. The current densities were calculated at the RHF and B3LYP levels using different basis sets. Grid-point spacings of 0.02 were used in the numerical integrations.

Basis SVP TZVP QZVP SVP TZVP QZVP

Bond Single Single Single Double Double Double

Jdia 8.0 7.8 7.9 11.1 10.3 10.7

RHF Jpara -8.3 -7.8 -7.9 -11.2 -10.3 -10.8

2

|∆J | 2.4 2.4 2.5 2.6 2.6 2.5

1 2

Jdia 8.0 7.9 7.8 10.7 10.7 10.3

B3LYP Jpara -8.4 -8.0 -7.7 -10.8 -10.9 -10.2

1

|∆J 2 | 2 2.4 2.5 2.4 2.6 2.5 2.5

current strengths passing in one direction along the chemical bonds of 1,3-butadiene with its ACID cross-section area. Calculations using SVP basis sets yield practically the same results as obtained with the QZVP ones. The strengths of the diatropic and paratropic currents are somewhat larger for the double bond than for the single bond, whereas the sum of the two contributions is almost zero as it should be when the charge is conserved. For 1,3-butadiene, the ACID cross-section area is almost the same for the single and double bond. Comparison of the current strengths for benzene calculated at the RHF and B3LYP levels shows that the correlation effects are small.

5.2

Comparison of current densities with ACID functions 1

The profiles of the ring currents and the ACID cross-section area (|∆J 2 | 2 ) have been calculated for benzene in a cut plane starting from the middle of the molecular ring and being stepwise extended by 0.1 bohr until it passes the center of the C–C bond. The ring-current profiles in Figure 1 show that the ring currents consist of the paratropic part inside the molecular ring and a dominating diatropic contribution outside it. The square root of the ACID function is positively definite by construction.

10


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The ring-current strengths and the ACID cross-section areas have been calculated for a number of cut planes. The cut planes are perpendicular to the ring, beginning at the ring center. The planes extend radially from the ring center and cut the C–C bond in different locations. Figure 2 shows that the ring-current strengths are practically independent of the direction of the cut plane, whereas the ACID cross-section area strongly depends on the angle of the integration plane cutting the C–C bond. The ACID function has much larger amplitudes at the nuclei than at the bond center. Thus, the electron mobility cannot be reliably quantified using ACID cross-section areas nor by plotting the ACID function.

1

Figure 2: The integrated current strength (J) and the ACID cross-section area (A = |∆J 2 | 2 ) have been calculated for benzene as a function of the direction of the integration plane, which is oriented perpendicularly to the benzene ring. The origin corresponds to the center of the C–C bond. At 30 degrees, the integration plane passes through the C and H atoms. The strengths of the magnetically induced currents obtained with a fixed direction of the magnetic field and the ACID cross-section area are reported for a number of saturated, unsaturated, anti-aromatic, and aromatic hydrocarbons in Table 3. The aromatic character and relative degree of aromaticity are obtained in the calculations of the ring-current strengths. The ACID function for naphthalene suggests that an aromatic ring current also passes the C–C bond in the middle of the molecule. However, due to symmetry the net cur-

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rent strength must vanish. Current-density calculations show that mainly local paratropic currents inside the two benzoic rings flow along the central C–C bond. 16 Table 3: The ring-current strengths are compared to the cross-section area of the 1 ACID function. The current strength is denoted J (in nA/T) and |∆J 2 | 2 (in nA/T) is the ACID cross-section area. The numerical integrations were performed at the center of the chemical bonds. The current densities were calculated at the B3LYP/TZVP level. Grid-point spacings of 0.02 were used in the numerical integrations. Molecule Benzene (C6 H6 ) Naphthalene (C10 H8 ) Cyclobutadiene (C4 H4 ) Cyclobutadiene (C4 H4 ) Cyclohexane (C6 H12 ) Cyclohexene (C6 H10 ) Cyclohexene (C6 H10 ) Cyclohexadiene (1,2,4,5-C6 H8 ) Cyclohexadiene (1,2,4,5-C6 H8 ) Cyclohexadiene (1,2,3,4-C6 H8 ) Cyclohexadiene (1,2,3,4-C6 H8 ) Cyclohexadiene (1,2,3,4-C6 H8 ) Cyclohexadiene (1,2,3,4-C6 H8 )

Bond

Single Double Single Double Single Double Single 1 Single 2 Single 3 Double

J 12.1 13.0 -20.0 -19.9 0.3 0.6 0.3 -0.6 -0.5 -0.3 -0.2 -0.3 -0.3

1

|∆J 2 | 2 3.4 3.5 5.5 5.5 2.7 2.8 2.4 2.6 2.4 2.8 2.8 2.4 2.3

For nonaromatic molecules, the local bond currents lead to different values for the currents passing in one direction along the C–C bonds. However, the net ring-current strengths of the nonaromatic molecules are close to zero. For the studied hydrocarbons, the ACID cross1

section areas are between 2.4 nA/T and 5.5 nA/T. The largest |∆J 2 | 2 value of 5.5 nA/T was 1

obtained for cyclobutadiene, whereas the aromatic and nonaromatic molecules have |∆J 2 | 2

values in the interval of [2.4, 3.5] nA/T, implying that the ACID cross-section area in the middle of the C–C bonds is not a very sensitive measure of the electron mobility. The ACID isosurfaces in Figures 3a-3f show that the ACID functions have larger amplitudes at the conjugated and double bonds than for the single bonds. Current strengths and ACID cross-section areas have also been calculated for free-base trans-porphyrin (t-PH2 ). The obtained current strengths and ACID cross-section areas for

12


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(a)

(b)

(d)

(c)

(e)

(f)

Figure 3: The ACID isosurfaces of a) cyclohexane, b) cyclohexene, c) 1,2,4,5-cyclohexadiene, d) 1,2,3,4-cyclohexadiene, e) benzene, and f) naphthalene. For the visualization, an isosurface cutoff value of 0.003 was used.

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t-PH2 are summarized in Table 4. Calculations of the current strengths along the inner and outer pathways of t-PH2 shows that 68% of the current takes the outer route at the pyrrole with an inner hydrogen and 45% passes the outer C=C bond of the pyrrole without the inner hydrogen. Calculations of the ACID cross-section areas suggest that 54% of the current passes the C=C bond of the pyrrolic ring with an inner hydrogen and 40% of the ring current flows via the outer route of the pyrrole without an inner hydrogen. Both the current-strength and the ACID-function calculations show that neither the 18π [18]annulene picture, 45–49 where the inner NH groups act as inert bridges, nor the 18π [16]annulene inner cross picture 50,51 can be considered to be the correct description of the aromatic pathway of t-PH2 . 17 The ACID isosurface is shown for two cutoff values in Figure 4a and 4b. The plots of the ACID functions indicate incorrectly the traditional 18π [18]annulene aromatic pathway of t-PH2 , even though the integration of the ACID cross-section areas and the calculated ring current strengths clearly show that all pathways are involved in the current flow around the porphyrin ring. 17,52–55 The present study on t-PH2 shows that one has to be careful when determining current pathways for porphyrinoids using approaches based on nucleus independent chemical shifts and ACID functions, 12,24,25,50,51 also when the calculated ACID functions are gauge-origin independent as obtained when using GIAOs. Table 4: Ring-current strengths (in nA/T) and the ACID cross-section area (in nA/T) for free-base trans-porphyrin (t-PH2 ). The current strength is denoted J 1 and |∆J 2 | 2 is the ACID cross-section area. Pyrrolic rings with inner hydrogens are denoted by the superscript H . The numerical integrations were performed at the center of the chemical bonds. The current density is calculated at the B3LYP/TZVP level. Grid-point spacings of 0.02 were used in the numerical integrations.

Molecule t-PH2 t-PH2

Ring AH B

C=C 1 J |∆J 2 | 2 18.5 4.0 12.3 2.9

C–N–C 1 J |∆J 2 | 2 8.7 3.4 15.2 4.4

Total 1 J |∆J 2 | 2 27.2 7.4 27.5 7.3

Finally, we have investigated the ACID function for gaudiene (C72 ), which is an allcarbon molecule forming a hollow structure of Oh symmetry. 56,57 Gaudiene consists of six 14


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(a)

(b)

Figure 4: The ACID isosurface of free-base trans-porphyrin obtained with isosurface cutoff values of a) 0.003 and b) 0.005. four-sided rings having alternating triple and single bonds. The molecular cage is closed by the eight hexadehydro[12]annulene rings surrounding the four-sided rings on each side. The hexadehydro[12]-annulene rings are conjugated consisting of alternating single, double, single, and triple bonds. The hexadehydro[12]annulene rings considered alone is antiaromatic, 18,58,59 whereas the four-sided rings would be nonaromatic as separate molecules, because the carbons in each corner of the ring are formally four-coordinated with single bonds in the direction of the ring. The molecular structure of gaudiene consists of 24 triple, 12 double and 48 single bonds. Recent GIMIC calculations on the hollow C72 showed that it is aromatic sustaining a ring current of 44.3 nA/T around the molecular cage. 56 Figure 5a-5d shows the ACID function for three isosurfaces of C72 , where half of the surface is filtered out for clarity. Plotting the ACID function shows that it has a somewhat larger amplitude at the triple and double bonds than for the single bonds as also observed for the other molecules studied. However, it is not easy to judge from the calculated ACID functions whether C72 is aromatic, nonaromatic or antiaromatic, whereas integation of the current strengths reveals that a strong diatropic current circles around the aromatic C72 cage.

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(a)

(b)

(c)

(d)

Figure 5: The ACID isosurface for gaudiene with isosurface cutoff values of a) 0.008, b) 0.007, c) 0.007 and d) 0.004, respectively. Half the surface has been filtered out for clarity. When decreasing the isosurface cutoff value, the ACID surface displays chemical bonds by first connecting the triple bonds, which can be seen in (a). For a slightly smaller isosurface cutoff value, the electron mobility of the double bonds becomes visible in (b) and (c), which is a rotated view of (b). When further decreasing the isosurface cutoff value, the plotted ACID function becomes continuous along all chemical bonds as shown in (d). Note that the differences in the isosurface cutoff values are very small.

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6

Summary and conclusions

We have implemented a gauge-origin independent method for calculating the anisotropy of the magnetically induced current density (ACID) functions using gauge-origin independent current density susceptibilities that are obtained by employing gauge-including atomic orbitals (GIAOs or London orbitals) as implemented in the gauge-including magnetically induced current (GIMIC) method. 5,7,8 Current density calculations show that current density susceptibilities are very insensitive to the size of the basis set thanks to the use of perturbation dependent GIAO basis sets. Standard basis set sizes can be employed rendering calculations on large molecules feasible. In this work, properties of the ACID functions have been investigated by calculating them for a number of simple aromatic and nonaromatic hydrocarbons with single and double bonds. The feasibility of the approach to study larger molecules has been demonstrated by calculations on free-base trans-porphyrin (t-PH2 ) and gaudiene (C72 ). The extent of the electron mobility along chemical bonds was estimated by computing ACID cross-section areas, which however were found to depend strongly on the exact location of the intersecting plane. This is in sharp contrast to current strengths that are obtained by integrating vector component of the current density susceptibility along the normal of the integration plane. The present calculations show that the ACID function is not a reliable means to assess the degree of molecular aromaticity, because the ACID cross-section area at the center of the chemical bonds depends only slightly on the extent of electron mobility. The calculations also show that different aromatic properties can be deduced from visual inspections of the ACID functions and from numerical integration of the cross-section area of the ACID function. Integration of the cross-section area of the ACID function suggests that aromaticity of t-PH2 is not corrcetly described by the traditional 18π [18]annulene aromatic pathway, which is supported by current density calculations, whereas the visualization of the ACID function for t-PH2 may lead to incorrect conclusions concerning the aromatic pathway. Calculation and visualization of the ACID function for gaudiene show where the double 17



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and triple bonds are, whereas the ACID plots do not provide much information about the aromatic properties of C72 . We conclude that GIMIC calculations are physically more sound than ACID studies providing much more reliable and detailed information about the magnetically induced current pathways, the current strengths, and molecular aromaticity. Since the complexity and computational costs of ACID and GIMIC calculations are practically the same, there is no reason for performing ACID calculations. Thus, current-strength calculations instead of ACID calculations are strongly recommended in computational studies of molecular aromaticity and the extent of electron mobility.

Acknowledgement This work has been supported by the Academy of Finland through projects (266227 and 275845). DS thanks the Magnus Ehrnrooth Foundation, the Swedish Cultural Foundation in Finland, the Alexander von Humboldt Stiftung, and the Fulbright Foundation for financial support. We thank the Norwegian Research Council for financial support through the CoE Centre for Theoretical and Computational Chemistry (Grant No. 179568/V30 and 231571/F20). The computational resources have been provided by the Norwegian Supercomputing Program NOTUR (Grant No. NN4654K) and by CSC – the Finnish IT Center for Science.

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Graphical TOC Entry

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The anisotropy of the induced current density function (ACID) using gauge including atomic orbitals has been implemented in the gauge including magnetically induced current (GIMIC) program. A comparison between the ACID function and gauge including magnetically induced currents is presented illustrating some limitations of the ACID approach.


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Gauge-Origin Independent Calculations of the ... - ACS Publications

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